ARENBERG DOCTORAL SCHOOLFaculty of Engineering Science

Multi-scale modelling of spatialvariability in textile compositesUncertainty quantification based on experimental

data of internal geometry

Andy Vanaerschot

Dissertation presented in partialfulfillment of the requirements for the

degree of Doctor in Engineering Science

August 2014

Multi-scale modelling of spatial variability in textile

composites

Uncertainty quantification based on experimental data of internal

geometry

Andy VANAERSCHOT

Examination committee:Prof. dr. ir. H. Neuckermans, chairProf. dr. ir. D. Vandepitte, supervisorProf. dr. ir. S.V. Lomov, co-supervisorProf. dr. ir. D. MoensProf. dr. ir. S. VandewalleProf. dr. ir. J. Van DyckProf. dr. ir. M. Arnst

(Université de Liège)Prof. dr. B.N. Cox

(Teledyne Scientific Co. LCC)

Dissertation presented in partialfulfillment of the requirements forthe degree of Doctorin Engineering Science

August 2014

© 2014 KU Leuven – Faculty of Engineering ScienceCelestijnenlaan 300B box 2420, B-3001 Heverlee (Belgium)

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Preface

. . .

i

Abstract

The advantages of using composites for design, manufacturing and during operationare well known. Besides a lower weight for the same or an enhanced performance, thisnew type of material further reduces maintenance costs and enables a full integrationof individual parts for aeronautical components. In automotive, composite materialsare even crucial to realise the upcoming regulations in further reducing theCO2

exhaust gases. Though, the introduction of composites is hampered by the relativelyhigh cost of raw material and the uncertain quality of high-performance compositestructures. In order to assure the design requirements, high safety factors and strictmanufacturing tolerances are enforced that hinder composites to be a competitivematerial for design.

An improved assessment of the quality of any composite part is achieved byidentifying the irregularity in the tow reinforcement. Thevariability in macroscopicperformance is dominated by the randomness in the geometrical characteristics at thelower scale, especially for textile products. However, no evident step in this directionhas been made over the past decades. Sources of variability remain poorly understoodand computational methods are lacking for building representative numerical models.The majority of the state-of-the-art restricts to local features, without regarding thespatial dependency of tow path parameters at different locations. In addition, due tothe lack of measured data, researchers content themselves with assumptions on theinput distribution and correlation structures, which leads to incorrect estimates of theactual limits of material properties. The next step in this development should consistin a modelling approach that introduce scatter at the different levels and is calibratedwith experimental work.

This dissertation provides a multi-scale framework for generating realistic virtualtextile specimens. A roadmap is provided to characterise the spatial scatter in theinternal structure of any textile composite and simulate random models possessingthe measured statistical information on average. Therefore, it is a first step towardsa systematic modelling approach for textile composites where powerful simulationprocedures are applied in combination with experimental data. This contribution

iii

iv ABSTRACT

complies with the industrial interest of virtual testing towardsfirst time right. High-fidelity simulations are gaining ever more importance and decisions are increasinglypursued based on simulation results, demonstrating the need for instruments such asrealistic models.

Virtual textile specimens with random reinforcement are acquired in three mainsteps. First, an experimental methodology is presented to characterise the geometricalvariability in terms of the centroid coordinates and cross-sectional parameters onthe short-range (meso-scale) and long-range (macro-scale). Non-destructive state-of-the-art inspection techniques such as X-ray micro-computed tomography, opticalimaging or strain field analysis are applied to measure the fabric architecture in areliable and efficient way across the composite volume. The inherent scatterofeach tow path parameter in each tow direction is quantified interms of an averagetrend, standard deviation and correlation length by applying the reference periodcollation method. Secondly, a stochastic multi-scale modelling approach is developedto reproduce the measured uncertainty in the tow reinforcement within the unit celland between neighbouring unit cells. Random instances of tow paths are acquiredby combining the deduced average trends with generated zero-mean fluctuationspossessing the experimental standard deviation and correlation lengths on average.Zero-mean deviations which are only correlated along the tow path are produced bythe Monte Carlo Markov Chain for textile structures, while uncertain quantities thatare dependent along and between tow paths are generated using the cross-correlatedSeries Expansion method. In the last step, virtual composite specimens with randomfibre architecture are created in the WiseTex format by an intrusive approach. Nominaltow path descriptions are overwritten with realistic tow representations obtained fromthe previous step, while preserving the original fibre mechanics and matrix properties.

This framework is demonstrated for a carbon-epoxy 2/2 twill woven compositeproduced by resin transfer moulding. Substantial differences in tow path informationare observed for warp and weft direction attributed to the manufacturing process of theweave. For all tow properties, a good correspondence is obtained for the experimentaland simulated deviations trends in terms of wavelengths of the centrelines and extremevalues.

Beknopte samenvatting

De gunstige voordelen van composietmaterialen in ontwerp,productie en gedurendegebruik zijn goed gekend. Naast een lager gewicht voor een zelfde of verhoogdeperformantie, laat dit materiaaltype ook een verdere besparing in onderhoudskostentoe en maakt een volledige integratie van afzonderlijke onderdelen mogelijk voorluchtvaartcomponenten. Composietmaterialen zijn zelfs cruciaal om de aanstaanderegelgeving in een verdere reductie vanCO2 uitlaatgassen voor de auto-industrie terealiseren. Ondanks deze voordelen, verloopt de introductie van composietmaterialennog moeizaam. Eén van de redenen is de relatief hoge kostprijs van grondstoffen,en een andere belangrijke hinderpaal is de onzekere kwaliteit van hoog-performantecomposieten structuren. Dit resulteert in hoge veiligheidsfactoren en strikte productie-toleranties die op hun beurt de concurrentie met conventionele materialen belemmert.

Een verbeterde inschatting van de kwaliteit van een composieten stuk wordt verkregendoor het identificeren van de imperfecties in de vezelversterking. De variatie inmacroscopische performantie wordt namelijk gedomineerd door de spreiding in degeometrische karakteristieken aanwezig op de lagere schaal, en dit vooral voortextieltopologieën. Nochtans is er de laatste decennia geen duidelijke stap gezetin deze richting. Bronnen van spreiding zijn ondermaats gekend en numerieketechnieken slagen er niet in om representatieve modellen opte stellen. De meerderheidvan de state-of-the-art beperkt zich tot lokale kenmerken,zonder de ruimtelijkeafhankelijkheid van een vezelpad parameter tussen de verschillende posities inrekening te nemen. Bovendien stellen onderzoekers zich vaak tevreden met aannamesover de invoergegevens van statistische verdelingen en de correlatiestructuren wanneerexperimentele data ontbreekt, wat leidt tot foutieve grenzen van betrouwbaar gebruikvan materiaaleigenschappen. De volgende stap in de ontwikkeling moet nu bestaanuit het samenbrengen van modelleringsbenaderingen met experimenteel werk.

Dit proefschrift stelt de ontwikkeling voor van een meerschalen strategie tot hetbekomen van realistische virtuele textielmodellen. Een stappenplan is uitgewerktom de ruimtelijke spreiding in de inwendige structuur van een textielcomposiette karakteriseren en vervolgens te reproduceren in modellen die deze statistische

v

vi BEKNOPTE SAMENVATTING

informatie gemiddeld bezitten. Het betreft een eerste werkdat een systematischeaanpak voor het modelleren van textielcomposiet voorsteltwaar geavanceerdesimulatietechnieken worden toegepast in combinatie met experimentele gegevens.Deze modelleeraanpak ondersteunt de industriële verschuiving naarfirst time rightdoor middel van virtuele testen. Een verhoogde betrouwbaarheid van simulatieswordt onontbeerlijk vermits steeds meer beslissingen worden nagestreefd op basisvan numerieke testen, wat de noodzaak aantoont voor instrumenten zoals realistischenumerieke modellen.

Virtuele textielmonsters met random vezelversterking zijn verkregen in drie stappen.Een eerste stap betreft het doorlopen van een voorgesteld experimenteel kaderom realistische gegevens over de posities van de hartlijn ende doorsnede vanvezelbundelpaden op korte afstand (meso-schaal) en lange afstand (macro-schaal) teverzamelen. Niet-destructieve state-of-the-art inspectietechnieken zoals X-ray micro-computertomografie, optische beeldvorming of rekveld analyse worden toegepast omde vezelversterking op een nauwkeurige en doeltreffende wijze over de volledigeuitgestrektheid van het composiet te onderzoeken. De inherente ruimtelijke spreidingvan elke vezelbundel parameter in elke weefrichting wordt gekwantificeerd metbetrekking tot een gemiddelde trend, standaardafwijking en correlatielengte doortoepassing van de referentie periode methode. In een volgende stap wordt een sto-chastisch meerschalen aanpak ontwikkeld om de onzekerheidin de vezelversterkingte reproduceren binnen eenzelfde eenheidscel en tussen naburige eenheidscellen. Wil-lekeurige realisaties van vezelbundel paden worden verworven door het combinerenvan de experimenteel verkregen gemiddelde tendens met gegenereerde deviaties diede gemeten standaardafwijking en correlatie lengtes gemiddeld bereiken. Afwijkingendie enkel gecorreleerd zijn langsheen het vezelbundel pad worden geproduceerd metde Monte Carlo Markov Keten voor textielstructuren, terwijl deviaties die zowelgecorreleerd zijn langsheen en tussen vezelbundel paden worden gegenereerd meteen kruis-gecorreleerde Series Expansie techniek. In de laatste stap worden virtuelemodellen met willekeurige vezelarchitectuur aangemaakt in de WiseTex softwaredoor gebruik te maken van een intrusieve aanpak. Nominale vezelbundel padbeschrijvingen worden overschreven met realisaties uit devorige stap, met behoudvan de oorspronkelijke vezelmechanica en matrixeigenschappen.

De gehele methodologie is aangetoond voor koolstof-epoxy 2/2 keper composietgeproduceerd met resin transfer moulding. Aanzienlijke verschillen in de vezelbundelpaden zijn waargenomen voor schering en inslag richting, toegeschreven aan hetproductieproces van het weefsel. Een goede vergelijking isverkregen tussen deexperimentele en gesimuleerde afwijkingen met betrekkingtot de golflengtes van dehartlijnen en extremen waarden voor alle vezelbundel parameters.

List of Abbreviations

1-D One-dimensional2-D Two-dimensional3-D Three-dimensionalCOV Coefficient Of VariationCT Computed TomographyDPI Dots Per InchFE Finite ElementK-L Karhunen-LoèveLHS Latin Hypercube SamplingPC Polynomial ChaosPDF Probability Density FunctionRMSD Root Mean Square DeviationRTM Resin Transfer MouldingRVE Representative Volume ElementSSFEM Spectral Stochastic Finite Element Method

vii

ix

x LIST OF SYMBOLS

List of Symbols

Latin Symbols

a Grid spacing in the Markov Chaina1 Minor elliptical axisa2 Major elliptical axisa1 Directional vector of the minor axis of the ellipsea2 Directional vector of the major axis of the ellipseA Tow cross-sectional areaAgap Gap areaAtrans,0 Initial tri-diagonal probability transition matrixAtrans Calibrated probability transition matrixAR Tow cross-sectional aspect ratiob Bi-normal vector to the centreline of the towC Correlation functionCauto Auto-correlation functionCcross Cross-correlation functionCdata,i Experimental correlation dataCexp Exponential correlation functionC f it,i Fitted correlation dataCsq,exp Squared exponential correlation functionC(ǫǫ′)

c Correlation between deviations of the same genus at the samelocationC(ǫ,t)

c Correlation between neighbouring tows of the same genusC(ǫwaǫwe)

c Correlation between a warp and weft tow at cross-over locationCo Covariance functioncov Covariance matrixcorr Correlation matrixD Block cross-correlation matrix~e Unit vectorEres Sum of squares of the residuals betweencdata,i andcf it,i

f (x) Probability Density Functiong(x) General set of deterministic spatial functions

LIST OF SYMBOLS xi

hgap Gap heightH Random fieldH Approximated random fieldk Integer valuel Integer value different fromkm Number of intervals in the Markov ChainM Number of locations for the correlation computationMpq Moments of the domain of a tow in cross-sectionn Number of pairs in correlation computationn Normal vector to the centreline of the towN Size of the data set for a particular genusNdata Number of data pointsNF Number of fields in the Series ExpansionNf ,r Number of eigenvalues and -vectors of the cross-correlation matrix after truncationNG Number of grid locations to represent a single random fieldNi Number of locations along the gridN j Number of tows belonging to a particular genusNr Multiplication of Nf ,r with Nvar

Nred Number of terms used to compute the truncation errorNsim Number of realisationsNvar Number of K-L termsP Probability distribution vectorp Probability of a single element of the probability distribution vectorR Rotation matrixS Sign factor for the smoothing operationsptow Tow spacingt Warp of weft tow genust Tangent vector to the centreline of the towtp Ply thicknessts,b Ply location bottomts,e Ply location topT Transpose operationwgap Gap widthwtow Tow widthx Spatial coordinate along the x-axis, used to indicate the centroidX Vector of scalarsy Spatial coordinate along the y-axis, used to indicate the centroidY Vector of scalars different fromXz Spatial coordinate along the z-axis, used to indicate the centroid

xii LIST OF SYMBOLS

Greek Symbols

α One of the three parameters to calibrate the probability transition matrixαs Misalignment angle of a ply around z-axisβ One of the three parameters to calibrate the probability transition matrixβs Misalignment angle of a ply around x-axisγ One of the three parameters to calibrate the probability transition matrixγs Misalignment angle of a ply around y-axisδ Equidistant spacing between the set of points used to compute the correlation∆ Normalised difference between experimental and simulated parameterǫi Zero-mean deviation from average value at locationiζ Parameter in the probability transition matrix depending on choice ofαη Vector of independent standard Gaussian random variablesθ Tow orientation in cross-sectionι Depth of a dip in the tow pathκ Slope of a dip in the tow pathλA Eigenvalues of the auto-correlation matrixλC Eigenvalues of the cross-correlation matrixλD Eigenvalues of the block cross-correlation matrixλx Periodic length of the unit cell along the x-axisλy Periodic length of the unit cell along the y-axisΛ Triangular matrix representing the Cholesky rootµ Mean valueν grid spacing of the lattice on which the tow paths are definedξ Correlation lengthξauto Auto-correlation lengthξcross Cross-correlation lengthξexp Correlation length obtained from exponential fitξsq,exp Correlation length obtained from squared exponential fitΞ Systematic variationπ Random permutation operatorρ In-plane centroid location for the warp or weft genusσ Standard deviationτ Absolute difference between two spatial coordinatesυ Parameter value of which the∆ is computedΥ General set of random variablesφA Eigenvectors of the auto-correlation matrixφC Eigenvectors of the cross-correlation matrixφD Eigenvectors of the block cross-correlation matrixχD Cross-correlated random matrixψ Shear angle between warp and weft directionω Function of the domain of a tow cross-sectionεi Uncertain parameter at locationi< εi > Average value at locationiϑ Used to indicate the randomness ofH Ply orientation Normalised sum of eigenvalues or truncation errorς Difference of ply systematic curves with overall systematic

LIST OF SYMBOLS xiii

Miscellaneous symbols

E Expectation< • > Average operationF−1 Inverse of the cumulative normal distribution function

•i Grid Location• j Tow index for a particular tow genus•s Ply or sample index•sl Slice index

• Generated property•lr Data acquired from the long-range characterisation•sr Data acquired from the short-range characterisation•comb Combined data set•spec Specimen data set•target Target value

Contents

Abstract iii

List of Abbreviations vii

List of Symbols ix

Contents xv

List of Figures xix

List of Tables xxvii

I Executive Summary 1

1 Introduction 2

1.1 Need for realistic representations of textile reinforcements . . . . . . 2

1.2 Challenges in modelling geometrical variability . . . . .. . . . . . . 4

1.3 Research objectives and contributions . . . . . . . . . . . . . .. . . 5

1.4 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . .7

1.4.1 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

xv

xvi CONTENTS

2 State-of-the-art 11

2.1 Composite Materials and representations . . . . . . . . . . . .. . . . 12

2.2 Sources of variability and types of geometrical irregularities . . . . . 15

2.3 Uncertainty quantification: definitions and methods . . .. . . . . . . 18

2.3.1 Definitions of uncertainty . . . . . . . . . . . . . . . . . . . 18

2.3.2 Probabilistic uncertainty quantification for spatial fields . . . . 19

2.3.3 Characterisation of a spatial field by the correlationlength . . 21

2.3.4 Target statistical data to describe the spatial uncertainty . . . . 23

2.4 Experimental techniques to characterise variability .. . . . . . . . . 23

2.4.1 Techniques to collect experimental data . . . . . . . . . . .. 24

2.4.2 Statistical analysis of experimental data . . . . . . . . .. . . 26

2.4.3 Selection of characterisation techniques . . . . . . . . .. . . 27

2.4.4 Assumptions and errors in the characterisation procedure . . . 27

2.5 Uncertainty modelling in textile composites . . . . . . . . .. . . . . 28

2.5.1 Overview of modelling approaches . . . . . . . . . . . . . . 28

2.5.2 Non-intrusive simulation techniques . . . . . . . . . . . . .. 30

2.5.3 Intrusive simulation techniques . . . . . . . . . . . . . . . . 30

2.5.4 Selection of modelling strategies . . . . . . . . . . . . . . . .33

2.5.5 Assumptions and errors in the modelling strategy . . . .. . . 34

2.6 Link of mechanical performance with geometrical randomness . . . . 35

2.7 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . 36

3 Development of a stochastic multi-scale modelling framework 39

3.1 Multi-scale framework . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Collection of experimental data and statistical analysis (step 1) . . . . 42

3.2.1 Experimental framework . . . . . . . . . . . . . . . . . . . . 42

3.2.2 Application to a woven textile composite . . . . . . . . . . .45

CONTENTS xvii

3.3 Stochastic multi-scale modelling of the reinforcement(step 2) . . . . 51

3.3.1 Overview of the modelling strategy . . . . . . . . . . . . . . 51

3.3.2 Average trends . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.3 Simulation of auto-correlated deviations by the Monte CarloMarkov Chain method . . . . . . . . . . . . . . . . . . . . . 54

3.3.4 Simulation of auto- and cross-correlated deviationsby thecross-correlated Karhunen-Loève Series Expansion . . . . . .59

3.4 Construction of virtual specimens in the WiseTex software (step 3) . . 67

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Concluding remarks and future research topics 71

4.1 Main conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.1 General framework . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.2 Application to a woven textile composite . . . . . . . . . . .74

4.2 Recommendations for future research . . . . . . . . . . . . . . . .. 75

4.2.1 Model improvement and extension . . . . . . . . . . . . . . . 76

4.2.2 Propagation of geometrical uncertainty to the macroscopicperformance . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.3 Application of the framework to simulate uncertaintyin othercomposite types and properties . . . . . . . . . . . . . . . . . 77

II Key publications 79

Paper I:Stochastic framework for quantifying the geometrical variability oflaminated textile composites using micro-computed tomography 81

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2 Stochastic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3 Experimental procedure (step 1) . . . . . . . . . . . . . . . . . . . . 84

4 Data processing and analysis of cross-sections (step 2) . .. . . . . . 85

xviii CONTENTS

4.1 Image segmentation and ellipse fitting . . . . . . . . . . . . . 85

4.2 Definition coordinate system and sample alignment correction 86

5 Statistical characterisation of the tow paths (step 3) . . .. . . . . . . 88

5.1 Definition of reference period . . . . . . . . . . . . . . . . . 89

5.2 Determination of the systematic trend for each tow parameter 90

5.3 Determination of the statistical properties of the tow parameters 91

6 Results and comparison of statistical analysis between plies . . . . . . 92

6.1 The choice of genus . . . . . . . . . . . . . . . . . . . . . . 92

6.2 Results for systematic trends . . . . . . . . . . . . . . . . . . 92

6.3 Results for stochastic deviations . . . . . . . . . . . . . . . . 96

6.4 Correlation lengths for deviations . . . . . . . . . . . . . . . 97

7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Paper II:Stochastic characterisation of the in-plane tow centroid in textilecomposites to quantify the multi-scale variation in geometry 104

Paper III:Stochastic multi-scale modelling of textile composites based oninternal geometry variability 122

Paper IV:Simulation of the cross-correlated positions of in-plane towcentroids in textile composites based on experimental data 146

Bibliography 167

Curriculum Vitae 179

List of publications 181

List of Figures

1.1 Use of fibre-reinforced polymer composite materials in the A380 [66]. 3

2.1 Different 2-D textile fabric architectures: (a) plain weave , (b) braidwithout inlay and (c) knitted. . . . . . . . . . . . . . . . . . . . . . . 13

2.2 The normalised elastic constants in function of the fibreorientation [66]. 14

2.3 Variations of the tow centroid positions occurring at the short-range,i.e. within unit cell periodic lengthsλ, and long-range, i.e. lengthsexceeding the unit cell dimensions. Tow paths are extractedfromexperimental samples with a 2/2 twill woven topology. . . . . . . . . 15

2.4 Scatter in the textile geometry is present in the tow positions, toworientations and tow dimensions: (a) shows the internal randomnessin the tow positions inside a C-SiC composite, (b) represents thetow waviness in a plain weave glass fabric and (c) demonstrates thevariability in the tow cross-sectional dimensions of a 2/2 twill composite. 17

2.5 Correlation graph of the aspect ratio of weft tows in a 2/2 twillwoven composite laminate of seven plies: (a) correlation data perply and (b) correlation data for the combined data set. The overallcorrelation graph is fitted with an appropriate correlationfunction andapproximated by a linear fit. . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Different experimental techniques to characterise the textilegeometry:(a) optical imaging of a 2/2 twill woven carbon-epoxy composite, (b)X-ray micro-CT slice of 3-D glass woven fabrics and (c) renderedsurface of an ceramic interlock weave from strain field analysis. . . . 26

xix

xx LIST OF FIGURES

2.7 RVE of the considered 2/2 twill woven carbon-epoxy composite by aschematic representation and as a simulated WiseTex model,using themanufacturer’s data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Multi-scale framework. . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Definition of spatial dependencies of deviations demonstrated for thein-plane centroid of two weft tows: auto-correlation (along the tow)and cross-correlation (between neighbouring tows). . . . . .. . . . . 44

3.3 WiseTex model of a 2/2 twill woven reinforcement. The x-axis andy-axis of the coordinate system are respectively parallel to the warpand weft direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Digital image of a cross-section in weft direction obtained frommicro-CT. Tows in cross-section are first identified using imagesegmentation and afterwards fitted with an ellipse shape to deducethe tow information. This procedure is demonstrated for ply7. . . . . 46

3.5 Optical scan of a one-ply 2/2 twill woven carbon fibre fabricimpregnated with epoxy resin. Warp tows are oriented horizontally,while weft tows are positioned in vertical direction. The redsquare indicates the region of interest where the in-plane position ischaracterised. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.6 Periodic and handling trends of a 2/2 twill woven carbon-epoxycomposite. Periodic trends are represented for one unit cell distance,while the handling effect is shown over a distance of ten unit cells. . . 50

3.7 Stochastic multi-scale modelling approach of the reinforcement. . . . 52

3.8 Procedure of generating a discretised tow representation, demon-strated for the out-of-plane centroid position. . . . . . . . . .. . . . 53

3.9 Different views of the average reinforcement description combiningthe systematic and handling trends of all tow path parameters. . . . . 54

3.10 Warp out-of-plane centroid deviations trend for 28 warp tows: (a)experimental vs. (b) smoothed deviations obtained from simulations.The simulated path is presented for a grid of 16 points to compare withthe experimentally characterised values. . . . . . . . . . . . . . .. . 57

3.11 The random out-of-plane warp tow path for a length of fiveunit cellsgenerated with the Markov Chain algorithm. The smoothing operationreduces the spikes present in the path. . . . . . . . . . . . . . . . . . 57

LIST OF FIGURES xxi

3.12 Correlation graph showing the experimental and simulated data of thewarp z-centroid coordinate. A linear approximation of the first lagdata is performed to deduce the correlation length. . . . . . . .. . . . 58

3.13 The unit cell statistics of the generated out-of-planewarp centroidpositions (a) without and (b) with smoothing. Simulated dataachieve the target statistics on average. When smoothing is applied,the simulated standard deviations are slightly affected, while allcorrelation lengths are increased. . . . . . . . . . . . . . . . . . . . .60

3.14 Warp in-plane centroid deviations trend for 80 warp tows: (a)experimental vs. (b) simulated deviations. . . . . . . . . . . . . .. . 64

3.15 Comparison of the warp input and simulated (a) auto-correlation and(b) cross-correlation structure. A perfect fit is obtained with minorfluctuations for the highest point spacings. . . . . . . . . . . . . .. . 65

3.16 Simulated (a) auto- and (b) cross-correlation lengthsof the warp in-plane centroids. The experimental value is simulated on average. . . . 66

3.17 WiseTex representation of a virtual specimen. The in-plane dimensionof the virtual composite shows the random in-plane positions, whilethe unit cell description indicates the variations in tow cross-sectionand out-of-plane centroid. . . . . . . . . . . . . . . . . . . . . . . . . 68

3.18 Map of gaps distributed over the virtual specimen presented infigure 3.17. The area of the significant gaps [mm2] are indicated andcategorised in five intervals. . . . . . . . . . . . . . . . . . . . . . . . 70

I1 WiseTex model of a 2/2 twill woven fabric. The coordinate axissystem is chosen to have the x-axis and y-axis respectively parallelto the warp tows and weft tows. . . . . . . . . . . . . . . . . . . . . . 84

I2 Digital image of a cross-section obtained by micro-CT. (a) Fullyprocessed 2-D cross-section in warp direction with areas indicatedwhere the boundary between warp and weft tows is blur. Ellipseshapes are fitted to the warp cross-sections in ply 3. (b) Enlargedimage of a part of the cross-section with fitted binarised cross-sectional shapes obtained from the image segmentation. Thefittingshows perfect comparison with the actual shapes. . . . . . . . . .. . 86

I3 Extraction of nineteen equally spaced slices from the reconstructed 3-D volume. The definition of a reference period is presented for theperiodic length of one unit cell. . . . . . . . . . . . . . . . . . . . . . 89

xxii LIST OF FIGURES

I4 Systematic curves for the warp out-of-plane and in-planecentroidpaths of ply 1. The systematic value at each location is determinedas the mean value of all data points at that location. . . . . . . .. . . 90

I5 Systematic out-of-plane singe-ply warp genuses for all plies. Asimilar path is observed between plies which makes the assumptionof one all-ply warp genus feasible. . . . . . . . . . . . . . . . . . . . 93

I6 The frequency of the dip orientation for the single-ply warp genuses(left) and single-ply weft genuses (right). . . . . . . . . . . . . .. . . 95

I7 Mean systematic curves of the tow shape parameters for theall-plyweft genus. The aspect ratio and area are correlated with thecross-over locations. The orientation exhibits a particular trend for thislimited size of data set. . . . . . . . . . . . . . . . . . . . . . . . . . 96

I8 (a) Cumulative distribution function of the ply normalised deviationsof the AR parameter for the warp tows. No differences between pliesare observed. (b) Normal probability plot of the normaliseddeviationsfor all components using data for the all-ply warp genus. . . .. . . . 97

I9 (a) Autocorrelation graph of the warp genus z-centroid coordinate. Afast decay is present. (b) Autocorrelation graph of the warpgenus y-centroid coordinate. The in-plane centroid exhibits largecorrelationfor distances exceeding the unit cell dimensions. . . . . . . . .. . . . 98

II1 WiseTex model of a 2/2 twill woven reinforcement. The x-axis andy-axis of the coordinate system are respectively parallel to the warpand weft direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

II2 Optical scan of a one-ply 2/2 twill woven carbon fibre fabricimpregnated with epoxy resin. Warp tows are oriented horizontally,while weft tows are positioned in the vertical direction. . .. . . . . . 108

II3 Map of gaps distributed over sample 1. The area of the significantgaps (mm2) are indicated and categorised in five intervals. . . . . . . 111

II4 Procedure to define the in-plane centroid deviations applied on sample1. (a) Detail image of best-fit grid to the experimental cross-overlocations. (b) Deviations pattern of all warp tows after subtractionof the experimental cross-over data from the grid values. . .. . . . . 112

II5 Deviations trend of warp tows decomposed in (a) handlingeffect and(b) stochastic deviations. . . . . . . . . . . . . . . . . . . . . . . . . 113

II6 Normal probability plot for (a) warp and (b) weft tow deviationsshowing approximately normal behaviour of the in-plane deviations. . 114

LIST OF FIGURES xxiii

II7 Definition of spatial dependencies of deviations demonstrated fortwo weft tows: auto-correlation (along the tow) and cross-correlation(between neighbouring tows). . . . . . . . . . . . . . . . . . . . . . . 114

II8 Correlation graphs of the warp tows for the (a) auto-correlation and (b)cross-correlation. The data points in lighter colour are not consideredfor the fitting procedure. . . . . . . . . . . . . . . . . . . . . . . . . 116

II9 Correlation graphs of the weft tows for the (a) auto-correlation and (b)cross-correlation. The data points in lighter colour are not consideredfor the fitting procedure. . . . . . . . . . . . . . . . . . . . . . . . . 116

III1 WiseTex model of a 2/2 twill woven reinforcement. The x-axis andy-axis of the coordinate system are respectively parallel to the warpand weft direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

III2 Digital image of a cross-section in weft direction obtained by micro-CT. Ellipses are fitted to the warp cross-sections of the unitcell of ply3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

III3 The translated out-of-plane centroid systematic curves for the differentwarp ply genuses. All ply systematic curves have a similar path whichpermits the derivation of an overall mean systematic curve.. . . . . . 128

III4 Procedure of the warp tow discretisation on an equally spaced grid of32 points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

III5 Overview of the stochastic multi-scale modelling approach. . . . . . . 133

III6 Comparison of the systematic curves resulting from thestochasticcharacterisation and the nominal WiseTex model. (a) No significantdifferences in systematic curves are observed for the out-of-planecoordinate except for the presentdips. (b) The in-plane coordinatecurve from the experimental procedure is fluctuating aroundthe zero-axis without a clear trend. . . . . . . . . . . . . . . . . . . . . . . . . 134

III7 (a) Correlogram of in-plane warp centroid for the experimentallymeasured deviations and simulated deviations with the Markov Chain.The Markov Chain reproduces correlation values that coincide withthe linear fit of experimental data used to deriveζy. (b) Generation ofan in-plane warp tow path with and without applying the smoothing.The smoothing operation removes small amplitude spikes. . .. . . . 137

xxiv LIST OF FIGURES

III8 The unit cell statistics of the out-of-plane warp centroid (a) withoutand (b) with smoothing. The simulated standard deviation isslightlyshifted after smoothing. Altough, a good comparison with theexperimental statistics is obtained on average. . . . . . . . . .. . . . 138

III9 The unit cell statistics of the in-plane warp centroid without (a) andwith smoothing (b). No significant effect of smoothing is observedfor the standard deviation. A good comparison with the experimentalstatistics is obtained on average. . . . . . . . . . . . . . . . . . . . . 140

III10 Orientation vectors of a tow cross-section used in WiseTex [57]. . . . 141

III11 Different views of the created random virtual unit cell in the WiseTexrepresentation: (a) general overview, (b) top view and (c) tilted sideview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

IV1 WiseTex model of a 2/2 twill woven reinforcement. The x-axis andy-axis of the coordinate system are respectively parallel to the warpand weft direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

IV2 Optical scan of a one-ply 2/2 twill woven carbon fibre fabricimpregnated with epoxy resin. Warp tows are oriented horizontally,while weft tows are positioned in the vertical direction. The red squareindicates the region where the in-plane position of the centroid ischaracterised. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

IV3 Experimental deviation trendsǫ( j,t) of warp (left) and weft (right) tows. 151

IV4 Definition of spatial dependencies of deviations demonstrated fortwo weft tows: auto-correlation (along the tow) and cross-correlation(between neighbouring tows). . . . . . . . . . . . . . . . . . . . . . . 152

IV5 Grid representation of the simulated in-plane deviations. . . . . . . . 154

IV6 Handling trend of warp (left) and weft (right) tows. . . . .. . . . . . 154

IV7 Normalised sum of the eigenvalues of the auto-correlation (left) andcross-correlation (right). The red line indicates when thenormalisedsum is equal to 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

IV8 Simulated warp (left) and weft (right) deviations trendfor 80 warp andweft tows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

IV9 Mean tow orientations of weft in-plane deviations: experimental trendof sample 2 (left) vs. simulated trend of an arbitrary specimen (right). 160

IV10 Comparison of the warp input and simulated auto-correlation (left)and cross-correlation (right) structure. . . . . . . . . . . . . . .. . . 161

LIST OF FIGURES xxv

IV11 Simulated auto- (left) and cross-correlation (right)lengths of warp in-plane centroids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

List of Tables

2.1 Comparison of stiffness components of a nominal model and randommodels (1000 generated random unit cells) of a 2/2 twill wovencarbon-epoxy composite [103]. . . . . . . . . . . . . . . . . . . . . . 36

3.1 Mean values forAR, A andθ computed over all plies in the unit cellsample of [104]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Standard deviation of the tow path parameters from the short-range[104] and long-range characterisation [107], respectively indicated bysr andlr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Correlation lengths of the tow path parameters from the short-range[104] and long-range characterisation [107], respectively indicated bysr and lr . Only for the in-plane position a cross-correlation lengthisdefined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Standard deviation and auto-correlation length for thecombined dataset of warp deviations produced with the Markov Chain algorithm.Smoothed results are indicated bysm. . . . . . . . . . . . . . . . . . 59

3.5 Mean of the standard deviation and auto-correlation length of the warpdeviations belonging to single unit cells, produced with the MarkovChain algorithm. Smoothed results are indicated bysm. . . . . . . . . 61

3.6 Input correlation functions and applied truncation forsimulating thein-plane fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.7 Standard deviation and correlation lengths for the (i) combined dataset of in-plane positions and (ii) mean of the individual specimens,generated with the cross-correlated Series Expansion technique. . . . 65

xxvii

xxviii LIST OF TABLES

3.8 Standard deviation and correlation lengths for individual 1-D randomfields, representing the in-plane centroid, produced with the cross-correlated Series Expansion technique. . . . . . . . . . . . . . . . .. 66

3.9 Sample mean and coefficient of variation of the gaps from theexperimental samples (region of thirteen by thirteen unit cells) andmean of ten simulated virtual specimens (region of ten by tenunit cells). 69

I1 Geometrical characteristics of the plies. . . . . . . . . . . . .. . . . 87

I2 Differences between the systematic curves for single-ply genuses andits corresponding all-ply genus. . . . . . . . . . . . . . . . . . . . . . 92

I3 Mean values forAR, A andθ computed over all plies. . . . . . . . . . 93

I4 Dip quantification by depth and slope for the single-ply genuses. . . . 95

I5 Standard deviations of each component using data of the all-ply warpand weft genuses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

I6 Correlation length values for all warp and weft components using dataof the all-ply genuses and single-ply genuses. . . . . . . . . . . .. . 99

II1 Unit cell periods obtained from long range data, short-range data andmanufacturer’s data . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

II2 Tow width and spacing of the one-ply 2/2 twill woven fabric. . . . . . 109

II3 Sample mean and coefficient of variation of the gaps in sample 1(N=1762) and sample 2 (N=1862) . . . . . . . . . . . . . . . . . . . 110

II4 Standard deviation of warp and weft tows for the combineddata set. . 115

II5 Auto- and cross-correlation lengths obtained from exponential andsquared exponential function fitting for the warp and weft tows usingthe combined data set. . . . . . . . . . . . . . . . . . . . . . . . . . . 117

III1 Experimental standard deviations and correlation lengths of thecentroid coordinates using the all-ply statistics data set. . . . . . . . . 128

III2 Experimental correlation length values for all warp and weft centroidcoordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

III3 Inter-tow correlations of (i) different centroid components at one gridlocation (data set size of 520) and (ii) the same centroid componentsbetween different tows of the same genus (data set size of 390). . . . . 130

LIST OF TABLES xxix

III4 Cross-correlations of the centroids at cross-over locations for a dataset size of 112. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

III5 Results of the statistics of the generated warp centroid deviations: withand without smoothing. . . . . . . . . . . . . . . . . . . . . . . . . . 139

IV1 Skewness and kurtosis values of stochastic in-plane deviations. . . . . 152

IV2 Standard deviation, auto- and cross-correlation lengths of the in-planecentroid using the combined data set. . . . . . . . . . . . . . . . . . . 153

IV3 Input correlation functions for simulating the in-plane fluctuations. . . 158

IV4 Truncation parameters of the series expansion method. .. . . . . . . 159

IV5 Standard deviation and correlation lengths for the combined data setand average values for the individual specimens. . . . . . . . . .. . . 160

IV6 Standard deviation and correlation lengths for individual one-dimensionalrandom fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Part I

Executive Summary

1

Chapter 1

Introduction

1.1 Need for realistic representations of textilereinforcements

Composite materials have excellent mechanical properties. The combination of a highstrength and stiffness with a low weight offers advantages in energy efficiency for air,ground and water transport. Other benefits to conventional materials are the durabilityand favourable fatigue properties. Moreover, composite components are efficientlydesigned to have a particular mechanical performance whileclaiming a minimalmaterial usage. Several applications do already use composites in their structure, withthe aerospace industry leading this tendency. Figure 1.1 shows the usage of polymercomposites in the Airbus A380. Though, the introduction of composites proceedswith difficulty, particularly for safety critical components. One reason is the costwhile another major obstacle remains the uncertain reliability and quality of compositestructures, especially the way a user can understand the reliable performance of a loadcarrying component in a structural application.

A good quality demands that all mechanical properties meet the requirements ateach location for each realisation of the product. Sufficient instruments are availablefor the characterisation of the mechanical properties by experiments and numericalsimulation. Although, there is a lack of understanding on how the mechanicalproperties vary across the composite product. Due to the specific nature of eachcomposite with its specific manufacturing process, scattercan be very pronouncedwhich impedes a correct estimate of the quality of the composite component. Thisresults in large safety factors and strict manufacturing tolerances which in turn lead tocostly and inefficient products.

2

NEED FOR REALISTIC REPRESENTATIONS OF TEXTILE REINFORCEMENTS 3

Outer wing

Ailerons

Flap track fairings

Outer flap

Radome

Fixed leading edge

upper and lower panels

Main landing

gear leg fairing

door

Main and

center landing

gear doors

Nose landing

gear doors

Central torsion box

Pylon

fairings,

nacelles,

and

cowlings

Pressure

bulkhead

Keel beam

Tail cone

Vertical

stabilizer

Horizontal

stabilizer

Outer boxes

Over-wing panel

Belly fairing skins

Trailing edge upper and lower

panels and shroud box

Spoilers

Wing box

Figure 1.1: Use of fibre-reinforced polymer composite materials in the A380 [66].

Variability in the macroscopic performance is directly linked with scatter in theinternal structure and constituents at the lower scales. The mesoscopic variation is aresult of the various design parameters for manufacturing the fibre architecture, whichare inevitably subjected to fluctuations. On top of this scatter, the production processcan also significantly influence the end product. To analyse the variability, probabilitydensity functions (PDFs) can be established for all uncertain design parameters, takinginto account the correlation between these parameters. A correct interpretation of suchstatistical analysis is indispensable, knowing that probability of every output quantitydepends on the input probabilities and their correlations.To obtain an accurateresponse of a random structure, all input data must be validated.

The research community recently pays more attention to the characterisation andsimulation of the stochastic properties of composites thatare due to imperfections inthe fibre reinforcement. Though, many steps still need to be performed since sourcesof variability remain poorly understood and computationalmethods are lacking in theaccurate prediction of the effects of variation on the mechanical performance. Themajority of the publications is restricted to local features, without considering thespatial dependency across the structural composite which is essential for adequatelysimulating the internal reinforcement. Analysts are forced to make assumptionsregarding the distribution parameters the uncertain parameter, leading to incorrectestimates of the actual limits of material properties. Further, advanced techniquesfor the analysis of spatial variability are well developed in the field of stochastic

4 INTRODUCTION

mechanics. However, there is no systematic approach to apply them to compositematerials in combination with realistic experimental data.

Charmpis et al. [10] describe these needs for experimental data and a suitable multi-scale approach. Experimental data on the spatially correlated random fluctuationsof uncertain properties should be first collected and subsequently used to derive theprobabilistic information for macroscopic properties from the lower scale mechanicalcharacteristics. The development of realistic models of the material structure at thedifferent scales is an essential tool to make such accurate predictions. By replicatingthe real internal geometry of textiles on average, more reliable results about themechanical response are acquired. Therefore, it supports the material design towardshigh-performance composite solutions in the aerospace industry. When the materialis fully identified, material safety factors can be reconsidered [63, 84] and less rawmaterial is employed. In addition, the certification time ofa design and correspondingcost can be strongly reduced [14, 84]. This methodology can also be exploited tooptimise a single production step that has a strong effect on the desired compositeresponse, determined from a sensitivity analysis. The robustness of the productionprocess is improved, while the design cost decreases by lessscrap and waste disposalfor the same functionality.

1.2 Challenges in modelling geometrical variability

The hierarchical structure of textile composites allows anaccurate description of thegeometry and its mechanical behaviour. Still, a complete characterisation of thegeometrical variation is missing for many types of textile composites, precludingthe reproduction of the materials internal geometry. Most uncertainty approachesonly consider variation in the global reinforcement parameters and are limited to themeso-level; the multi-scale characteristics are not fullyexploited. Furthermore, thestatistical analysis of collected experimental data is very limited without consideringany correlation in most cases. However, only when dependencies for a single propertywithin the composite and between properties are adequatelyrepresented, replicas ofthe real structure can be simulated. The next step in this development should consistin a modelling approach that introduce scatter at the different levels and is calibratedwith experimental work. This dissertation follows such a generic modelling approachfor textile composites, that is supported by experiments asproposed by Charmpis etal. [10]. Two main bottlenecks with associated challenges can be distinguished.

Lack of experimental data Although the need for experimental data to calibratethe simulations was already identified two decades ago [88, 110], it appearsthat no evident step in this direction has been made. This canbe attributedto various reasons such as [10]: (i) the cost of measurements, (ii) the

RESEARCH OBJECTIVES AND CONTRIBUTIONS 5

amount of measurements to have good statistical resemblance of the population,and (iii) the shortage of experimental frameworks to deducegeometricalparameters. Due to the lack of measured data, researchers content themselveswith assumptions on the input distribution and correlationstructures basedon engineering experience and judgement. The definition of statisticalparameters is than subjective, and so is the result. Freudenthal [30] statesthat the “ignorance of the cause of variation does not make such variationrandom.” Experimental data are essential to validate the input informationof the stochastic modelling technique to attain a reliable representation of theheterogeneous structure and corresponding performance.

Limited introduction of randomness in multi-scale modelling procedures A newmodelling framework is requested since variation in the reinforcement structureis frequently omitted or only partially introduced in simulations [34]. In mostcases, composite components are modelled as identical repetitive unit cellswith regular tow path descriptions based on deterministic inputs. However,researchers have already demonstrated that real physical samples consist ofspatially distributed unit cells that differ from neighbour to neighbour [67]. Acorrect representation of the spatial uncertainty requires a modified approachwith an appropriate scaling technique calibrated with experimental data [10].

1.3 Research objectives and contributions

To ensure the quality of fibre-reinforced composites, it is of utmost importanceto accurately describe the inherent spatial scatter in the internal geometry. Oncethe composite microstructure is identified, a reliable computational model can beestablished that forms a direct link between the irregularities in the reinforcementstructure and the stochastic material properties. This dissertation addresses this needby following the approach proposed by Charmpis et al. [10], now applied to textilecomposites. The research objectives can be distinguished in two complementary tasks:

Collection of experimental data with statistical analysisThe collection and pro-cessing of objective data is pursued in this dissertation. To study the geometricalscatter at the short- (meso-scale) and long-range (macro-scale), state-of-the-art characterisation techniques need to be selected that can be appliedfor all types of textile composites. The desired reinforcement parameters(centroid coordinates and cross-sectional parameters) must be detected withsignificant accuracy, while a statistical analysis describes the correct type ofdistribution, distribution parameters and possible correlation information. Foreach geometrical parameter, the stochastic behaviour is defined in terms of anaverage trend, standard deviation and correlation data.

6 INTRODUCTION

Development of a stochastic multi-scale modelling approach Advanced numericalprocedures need to applied in a multi-scale modelling strategy to adequatelysimulate the spatial variation of any uncertain reinforcement parameter. Forthis purpose, techniques from stochastic mechanics can be considered whichare not yet used to simulate the geometry of composite materials. A set ofmeasured fluctuations is reproduced by calibrating the modelling procedureswith the experimental statistical information. In a next step, the random towpath descriptions have to be introduced in a nominal WiseTexmodel to buildvirtual specimens that are actual replicas of the physical samples.

The proposed methodology is devoted to typify the spatial geometrical randomness atthe meso- and macro-scale of high-performance composite, mainly used in aerospaceapplications. Fluctuations at the micro-level, such as fibre distribution and resincontent inside a single tow, are discarded. The material characterisation and modellingchain is demonstrated for a typical two-dimensional (2-D) polymer woven composite,but the concepts and frameworks can be extended to all types of textiles consideringonly minor modifications. This entire approach delivers large textile models, i.e.consisting of multiple unit cells, with a reinforcement structure that possesses thesame statistical information as quantified from the experimental samples.

Four main steps were completed throughout the PhD trajectory that describe theoverall strategy, with a carbon-epoxy 2/2 twill woven composite produced by ResinTransfer Moulding (RTM) acting as demonstrator material. Adetailed discussion ofall steps is elaborated in each of the key publications that form this dissertation:

• First, an experimental framework is developed to derive statistical informationof the tow reinforcement on the short-range, i.e. smaller orequal to theunit cell size. Samples of unit cell size are inspected usinglaboratory X-ray micro-computed tomography (micro-CT) to collect geometrical data. Thestatistical processing of these data are performed with thereference periodcollation method [3], previously applied to a ceramic interlock weave andnow extended to laminated polymer composites. The methodology is testedon a seven-ply carbon-epoxy unit cell sample with 2/2 twill woven topology.The internal geometry of this woven composite exhibits significant variability.Average systematic trends of the out-of-plane centroid coordinate and tow shapeparameters are found to be periodic and depending on the cross-over locations.The in-plane centroid coordinate is subject to the largest variability from itssystematic curve and is correlated along the tow with distances exceeding theunit cell dimensions. Both the systematic trends and deviations are found to besimilar between the different plies.

• In most cases, the short-range characterisation needs to becomplemented withlong-range data to quantify variations that persist beyonda single unit cell.

OUTLINE OF THE DISSERTATION 7

For the considered 2/2 twill woven composite, additional data are collectedof the in-plane centroid on larger one-ply samples. An optical scan of the in-plane dimension permits to identify the in-plane centroid positional behaviourover a region of ten unit cells by ten unit cells. In contrast to the other towpath parameters, the mean in-plane centreline varies without a periodic pattern,representing the handling effect of the fabric before production, while deviationsare also correlated between neighbouring tows of the same type. Significantdifferences are observed for tows in warp and weft direction. Weft tows aremore variable, affecting neighbouring tows over a distance exceeding the unitcell size. Warp tows have a large correlation length along the tow, with a limitedcross-correlation.

• Thirdly, random virtual unit cells are generated in a geometrical format,compatible with virtual textiles software, for example WiseTex [113]. Anefficient Monte Carlo Markov Chain method for textile structures [6] is appliedwhich is calibrated with the database of deviation values, standard deviationand auto-correlation length of each tow component. Fluctuations belonging toa single tow are generated in one step and subsequently addedto the averagepattern of the tow parameter to represent a random tow path. This approachis illustrated to simulate the random centroid coordinatesof the 2/2 twillwoven composite. Produced specimens possess the experimental statisticalinformation of the tow path properties on average.

• When a tow parameter is correlated along the tow and between neighbouringtows, a more complex simulation approach is required than the Markov Chainprinciple [6]. To accurately reproduce the dependency structures, tow pathdeviations of all tows of the same type must be generated simultaneously. Thisstrategy is followed in a cross-correlated Series Expansion technique developedby Vorechovský [114]. All cross-correlated fields, within a single specimen, areexpanded using the Karhunen-Loève (K-L) expansion technique with the samespectrum of eigenvalues and -vectors, but the sets of randomvariables usedfor the expansion of each field are cross-correlated with neighbouring fields.For the subject 2/2 twill woven composite, only the in-plane centroid positionsare simulated with this technique. Each tow is represented by an individualGaussian random fields which is cross-correlated with neighbouring tows. Theauto- and cross-dependency are accurately reproduced for the entire correlationstructure.

1.4 Outline of the dissertation

This dissertation consists of two main parts. The first part describes the mainobjectives, reviews the current state-of-the-art and specifies a roadmap how to

8 INTRODUCTION

develop realistic virtual specimens of any textile composite using the approaches andtechniques developed throughout the four years of research. The second part bundlesthe key publications where each of the different concepts in the modelling strategy arefurther clarified and details can be found. All steps of the developed framework aredemonstrated for a carbon-epoxy 2/2 twill woven composite.

Throughout the manuscript, symbols of the key publicationsare matched between thedifferent papers to preserve consistency in the nomenclature. This allows a reader torapidly search for more details without any confusion in symbols.

1.4.1 Part I

This introductory chapter briefly states the need for virtual modelling of textilecomposites, presents the main research objectives and achievements. The contributionto the current state-of-the-art is discussed in the following chapters:

• Chapter 2 provides an overview of the current state-of-the-art in uncertaintyquantification and modelling of composite materials. The distinct sourcesof variability in the composite structure are highlighted and related to theproduction process. Different characterisation methods are studied that areable to quantify the geometrical scatter with enough detail, and probabilisticstatistical concepts are defined to analyse these uncertainties. Further, existingmodelling strategies and recent developments in simulating tow paths arereviewed and compared. The chapter ends with a short discussion of the effectof tow path irregularity on the mechanical performance.

• Chapter 3 is written with a double intention. It is meant as anextensive summaryof the performed research towards realistic representations of compositestructures, demonstrated for a typical woven composite used in liquid mouldingprocesses. The main scientific contributions are highlighted in this chapter.Alternatively, it serves as a manual how to (i) collect statistical informationof the reinforcement structure and (ii) simulate virtual textile composites withcorrect spatial dependencies, supported by experimental data. The performedstatistical analysis and simulation tools are genericallydescribed, such that thedeveloped procedures are applicable to model any textile topology consideringonly minor adjustments.

• The last chapter of Part I summarizes the main conclusions and suggests futureresearch topics for the uncertainty assessment of textile composites.

OUTLINE OF THE DISSERTATION 9

1.4.2 Part II

The second part of the manuscript encompasses the key publications as outcomes ofthe performed research during the PhD trajectory. The followed approaches for eachof the different steps towards virtual specimens are described in fourseparate peer-reviewed publications. It is an indispensable overview of each characterisation andsimulation technique, in case the reader would be interested in additional details andreference material. Three publications are published by different international peer-reviewed journals. A fourth publication is published as a book chapter which is part ofa dedicated IUTAM symposia on Multiscale Modeling and Uncertainty Quantificationof Materials and Structures.

• Paper I is published in Composites Part A (2013) and discusses the experimentalframework to derive statistical information about the tow reinforcement at theunit cell level using micro-CT. The methodology is demonstrated for a carbon-epoxy 2/2 twill woven composite [104].

• Paper II is published as a chapter in the book collecting the Proceedings of theIUTAM Symposium on Multiscale Modeling and Uncertainty Quantification ofMaterials and Structures (2014). The in-plane centreline position of the 2/2 twillwoven composite is investigated on larger experimental samples using an opticalimaging technique to characterise its long-range statistical behaviour [107].

• Paper III is published in Computers & Structures (2013), as special issue ofthe Computational Fluid and Solid Mechanics conference. The Monte CarloMarkov Chain method for textile structures is used to simulate virtual textileunit cells with random reinforcements, with application for the subject woventextile. Only the short-range statistical information is used to calibrate themethod; no correlation between neighbouring tows is considered [105].

• Paper IV is published in the journal of Composite Structures(2014). A cross-correlated Series Expansion technique, originally used tomodel correlatedconcrete material properties, is adopted to simulate geometrical textile towparameters which are correlated along the tow and between tows. Thisprocedure is applied to produce the in-plane centroid coordinates of the 2/2twill woven composite [106].

Besides these publications, which discuss the developmentof realistic virtual speci-mens, a (co)-authored preceding journal article is published in Polymer Composites(2012) [76]. This work presents data on the internal geometry of two wovencomposites, obtained by optical imaging, which is afterwards introduced in anumerical multi-scale modelling approach to evaluate the macroscopic stiffnessstatistics of a unit cell sample. A set of tow parameters are randomly sampled from

10 INTRODUCTION

the experimental distributions using a Monte Carlo framework without consideringany correlation. The numerical modelling approach is able to appoint the maincontributors to the stiffness dispersion due to geometrical variability. When theexperimental Young’s modulus is compared with the obtainednumerical Young’smodulus, a significantly lower scatter is observed for the latter value. The overallvariability of the stiffness value is not only linked to the internal geometry, butalso to other sources. The uncertainty in each stage of the production process andexperimental determination of the stiffness value must be considered.

Chapter 2

State-of-the-art

Composite materials are considered as substitution material for metals in manyweight-critical components in the transportation industry. For aerospace applications,this weight-saving lowers fuel consumption and offers an increase in payload.Composites also reduce the number of components and fasteners, have a higherfatigue and corrosion resistance, and offer wide opportunities for tailored solutionsaccording to the design requirements [66]. Although, the quality of compositecomponents is often insufficiently known, a lower impact damage is acquired andsome mechanisms of damage development and growth still needto be explored. ThisPhD dissertation contributes to the field of uncertainty quantification and simulationof the reinforcement of textile composites by providing a stochastic multi-scaleframework to assess the quality of the final composite, such that it can be improved ina next step.

The chapter starts with a brief description of composite materials and the differenttypes of textile composites. Next, the sources of variability throughout compositeproduction are discussed and resulting geometrical irregularities are identified. Spatialgeometrical variation is described with a probabilistic uncertainty analysis of whichdefinitions and characteristics are given in section 2.3. The general approach of thisuncertainty analysis consists of two steps: (i) collectionof experimental data witha complete statistical characterisation and (ii) proposition of a stochastic multi-scalemodelling scheme to simulate virtual specimens that possess the measured statisticalinformation on average. An overview of different experimental methodologies thatare able to assess this variability in geometry is summarised in section 2.4, while thedifferent stochastic modelling approaches suggested in the literature are evaluated insection 2.5. Further, a short discussion of the effect of variation in the geometry at themeso- and macro-scale on the mechanical properties of a composite is given. At the

11

12 STATE-OF-THE-ART

end of the chapter, a summary is provided.

2.1 Composite Materials and representations

Composite materials consist of two or more constituents. For the general case oftwo-material elements, one phase act as reinforcement while another phase fixes thisreinforced structure. New properties are created or an averaged out combination ofthe properties of both constituents is obtained. Structural applications for transportneeds to consider a fibre architecture of high strength and modulus, designed to carrythe load, that is bind with a polymeric or ceramic matrix. Thematrix serves asload transfer medium and barrier against environmental influences (temperature andhumidity).

Fibre architectures appear in a wide variety of topologies with different kinds of fibresand resins. Fibres are grouped in bundles, ortows, and arranged in a composite toobtain favourable properties and/or to facilitate the processing step. Of the mostcommonly used fibre types (glass, carbon, natural, ceramic), carbon is frequentlyconsidered for high-performance applications due to its exceptionally high tensilestrength-weight ratio as tensile modulus-weight ratio. Depending on the application, apolymeric, metallic or ceramic composite is used to form a rigid structure. Continuousfibre reinforcements can be one-dimensional (1-D), two-dimensional (2-D) or three-dimensional (3-D). All tows in 1-D composites are oriented in the same directionresulting in the highest strength and modulus in the fibre direction, but a much lowertransverse strength and modulus. Such composites are oftenused to buildlaminates(multi-layer composites) where each layer, orply, can have the same or a different toworientation to reduce the strong anisotropy in the mechanical properties. Yet, manyinterlaminar problems exist with such composites [66]. 2-DComposites have fibretows in two or more directions to acquire the same or predefined mechanical propertiesfor different material orientations. Expect for 2-D composites with randomly orientedfibres or laminates with different unidirectional plies, these composites are calledtextiles. Textile fabrics are manufactured by weaving, knitting, orbraiding, eachhaving unique characteristics as shown in figure 2.1. These processing techniquesprovide different opportunities to tailor the structural performance and produce a widerange of structural shapes and forms. Generally, 1-D and 2-Dlaminated compositescannot transport load in the thickness direction and are therefore easily exposed todelamination. 3-D Composites have an additional Z-tow or stitching threads suchthat a stack of e.g. woven fabrics have an improved through-thickness behaviour thatreduces delamination risks. For each combination of reinforcement architecture, shapeand size of the final component and loading requirements, a particular productionprocess is preferred. The most common production methods for mass production arecompression moulding, pultrusion and filament winding [66]. Also liquid composite

COMPOSITE MATERIALS AND REPRESENTATIONS 13

moulding with RTM is often used where a heated thermoset resin is impregnated intoa dry preform with a closed mould. Other alternatives, specific for lower productionrates, are bag-moulding processes with autoclave.

(a) Woven composite (b) Braided composite (c) Knitted composite

Figure 2.1: Different 2-D textile fabric architectures: (a) plain weave , (b) braidwithout inlay and (c) knitted.

The mechanical behaviour of composites is strongly dependent on the direction andcannot be considered as isotropic. Composites are classified as orthotropic materialswith three principal material directions according to the three orthogonal planes ofsymmetry. This orthotropic nature results in a variation ofproperties as indicated infigure 2.2 that shows the variation of the elastic constants of a continuous glass fibrelaminate. The Young’s modulusExx is shown in function of the fibre orientation,demonstrating that a significant decrease in modulus takes place when variationsin fibre directions are present, withE22 < Exx < E11. Besides this deterministicvariation, significant stochastic scatter is observed in the mechanical properties ofdifferent realisations of the same composite, designed with equal specifications andproduction method [5]. A non-uniform quality is obtained caused by the designflexibility of a composite material which inevitably results in a wide range of uncertainmaterial data. Of all design choices, such as raw material and number of plies, thearchitecture of the fibre reinforcement, orientation of single layers and total fibrevolume fraction are most crucial to control in order to limitthe randomness in thephysical and mechanical performance of the final product. Asindicated by Li et al.[52], a high or low variability in the reinforcement reflects, respectively, poor or highquality of the processed component.

Specific for textile composites, this variation in mechanical properties is mainlyattributed to the variation in the reinforcement structurethat is characterised atmultiple scales. Themicro-scalerepresents local features with reference lengthsof 10−6 − 10−4m. This is the scale of individual fibres that are bundled within asingle tow. For larger reference lengths of 10−4 − 10−2m, determined as themeso-scale, individual fibres cannot be distinguished anymore but appear in groups that

14 STATE-OF-THE-ART

Gxy /G12

Exx /E11

1

0

08 458 908

Fiber orientation angle, q

xy / 12

Figure 2.2: The normalised elastic constants in function ofthe fibre orientation [66].

correspond to the tows. Tows are arranged following a topology that defines themacroscopic mechanical performance. Themacro-scaleaverages out the small effectsof the heterogeneous structure at tow level and permits to quantify any long-rangevariations with lengths of 10−2 − 10−0m. Most types of composites are built using aperiodic reinforcement structure that can be identified at the meso-scale. This periodiccell is namedrepresentative volume element(RVE) or unit cell. Such a descriptionis exploited in several software packages to model the textile geometry [62, 113].The textile pre-processor WiseTex, further considered in this work, constructs amechanical model based on the hierarchical approach of the textile and a limitednumber of input data on the fibre tow properties, specific topology and fibre bundlespacing. The tow path is subsequently constructed via the minimum energy principleof the fibre bundle deformation. The obtained numerical model is an idealisedrepresentation of the textile composite since real physical textile composites do showvariations at the short-range, i.e. within unit cell periodic lengthsλ, and long-range, i.e.lengths exceeding the unit cell dimensions. This multi-scale scatter is demonstratedfor the centroid positions in figure 2.3 representing real experimental data. Regulartow paths and unit cell periodicity are never realised in practice.

SOURCES OF VARIABILITY AND TYPES OF GEOMETRICAL IRREGULARITIES 15

10

Long-range variations

spanning multiple unit cells

Short-range variations at

the unit cell level

Figure 2.3: Variations of the tow centroid positions occurring at the short-range, i.e.within unit cell periodic lengthsλ, and long-range, i.e. lengths exceeding the unitcell dimensions. Tow paths are extracted from experimentalsamples with a 2/2 twillwoven topology.

2.2 Sources of variability and types of geometricalirregularities

Any reliability study of a structure requires the identification of the uncertainty in thematerial properties, geometry, and applied loading. When all these uncertainties areadequately described, high-quality simulations can be carried out and the mechanicalperformance is evaluated with high confidence. While for moststructural materials,typically metals and unreinforced polymers, the material properties are relatively wellknown, this is not the case for reinforced composites. In composite materials, scatterin the material properties is dependent on the prevailing variation in the internalreinforcement. A thorough investigation and proper simulation of the geometricalvariation permit to quantify the uncertainty in the performance.

Randomness in composite structures is introduced from fabric manufacturing tillcomposite production. The main sources of uncertainty during composite processingare [94]:

16 STATE-OF-THE-ART

• Variations in the fibre architecture, which are initiated bythe manufacturingparameters, handling or storage of the fabric

• Variations in the matrix material, which is caused by batch variation in the resincomposition

• Variations in the production parameters and environmentalconditions

Scatter in the tow reinforcement may affect the outcome of a forming step [94] andgenerates uncertainty in the textile permeability [27, 80,116], determining the amountof impregnation of resin in the fabric structure, and thermal behaviour [33]. It directlyinfluences the structural performance of the composite due to local misalignment andorientations of tows [22, 29, 76]. Especially for damage initiation and progression,variation in the local geometry can have a detrimental effect and thus requires realisticmodels [15, 26, 54, 65, 120, 122]. Discrepancies in resin viscosity influence the resininjection and curing step, and can lead to the forming of entrapped voids, whichare zones without resin or fibres inside the composite [27, 28, 68]. Productioneffects are related to uncertainty in process conditions (compression effects, curingconditions), tooling properties (cutting and stacking of fabric layers in mould),boundary conditions (temperature and pressure) and environmental situation [52, 85].The magnitude of uncertainty for each of the different steps in composite processingdefines the quality of the final composite. In addition to these uncertainties, an extratype of variation must be considered when the composite is subjected to a physical orchemical mechanism for some time. Such scatter in time is notthe scope of this workand will not be considered.

These sources of variation translate into a spatial geometrical scatter of the finalcomposite product at the meso- and macro-scale, as shown in figure 2.4 for differenttypes of textiles. Spatial randomness means that the geometrical parameters are notconstant over the area or the volume of one and the same component in one specificmaterial. The magnitude of this type of uncertainty is strongly dependent on the textiletopology and production process. Topology defines the positions of the fibre bundlesand causes deterministic variation in the dimension of tow cross-sections, while theproduction method can influence e.g. the fibre bundle orientation due to curing or thetow thickness variation due to a fixed laminate thickness. For later stages, variation ingeometry is also caused by the onset of fatigue damage. Spatial geometrical variabilityis characterised by inspecting the scatter of individual tow path parameters across theentire reinforcement. Fluctuations at the micro-level, such as fibre distribution andresin content inside a single tow, are not considered in thisdissertation. Meso- andmacro-scale variations in the textile geometry are associated with differences in:

• Tow positions which are described by the centroid coordinates (x, y, z) thatrepresent the centrelines of single tows [1, 3, 118]. Depending on the manu-facturing procedure, tow movement in one direction can be more significant

SOURCES OF VARIABILITY AND TYPES OF GEOMETRICAL IRREGULARITIES 17

(a) Tow positions [3]

(b) Tow orientations [1] (c) Tow dimensions [76]

Figure 2.4: Scatter in the textile geometry is present in thetow positions, toworientations and tow dimensions: (a) shows the internal randomness in the towpositions inside a C-SiC composite, (b) represents the tow waviness in a plain weaveglass fabric and (c) demonstrates the variability in the towcross-sectional dimensionsof a 2/2 twill composite.

than other directions. These fluctuations are also related to the variation in towspacing with a Coefficient of Variation (COV), which is the ratio of standarddeviation on the mean value, ranging from 1 to 9% [22, 27, 47, 76].

• Tow orientations that can be regarded as the mean orientation of a ply orbundles of tows within a composite, usually vary with standard deviationsaround 1 degree, but maxima till 5 degrees also occur depending on the textilestructure [1, 76, 94]. These tow oscillation is also identified as the wavinessof the tows along their path which is related to the fluctuation of the in-planecentroid coordinates [29, 47, 102].

• Tow dimensions, more specific the variations in thickness, width and shape ofa tow in cross-section and corresponding aspect ratio and area. Despite the typeof textile, the largest variation is observed in the tow thickness with COV’s inthe range of 6-17%, while this is 2-10% for the tow width. Tow area and aspectratio have scatter in the range of 5-12% [3, 22, 31, 47, 76, 92,119].

18 STATE-OF-THE-ART

The magnitude of variation depends largely on the topology and production processas already indicated. High-performance composites are usually susceptible to lowervariations, but still these textiles have non-negligible scatter in the tow reinforcement.This list is indicative but not exhaustive. A more thorough discussion about each ofthese irregularities in textile geometry can be found in [19].

Depending on the typical fibre architecture of any type of textile, additional randomparameters that define the reinforcement structure can be characterised, such as e.g.the location and distortion of the stitching threads in non-crimp fabrics [47] andthe braiding angle in knitted composites [37]. The magnitude of scatter in the towpositions, waviness and dimensions in a laminate is also expected to be related to thenesting of the plies [61]. This means that the statistics obtained for a zero-orientationlaminate and a single-ply composite made from the same topology and material andproduced with equal production parameters (pressure and temperature), can exhibit adifferent stochastic behaviour.

2.3 Uncertainty quantification: definitions andmethods

An extensive record of the history of the sample and a thorough statistical analysis arerequired to quantify the uncertainty in the geometry of composites. Used definitionsof uncertainty and concepts for performing a statistical analysis are defined andelaborated.

2.3.1 Definitions of uncertainty

Advanced simulation of composites requires anon-deterministicanalysis. Non-determinism refers to the response of a system that is not precisely predictable becauseof the existence of uncertainty of the system, environment or human interaction [75].In such analysis, it is important to differentiate the terminologyvariability, uncertaintyanderror [74, 75]:

Variability This refers to the inherent variation or intrinsic randomness associatedto the modelled physical system. The considered variable israndom and has adifferent value each time it is observed. This variability is also calledaleatory orobjective uncertainty. It is assumed to be irreducible since more measurementswill not reduce the uncertainty in the value of the variable.

Uncertainty The term covers the potential inaccuracy due to a lack of knowledge.The fundamental cause is incomplete information; withpotentialstressing that

UNCERTAINTY QUANTIFICATION: DEFINITIONS AND METHODS 19

the inaccuracy may or may not exist. It is also assigned asepistemic orsubjective uncertainty. Taking more or improved measurements provides abetter knowledge about the value of the variable and reducesthe uncertainty.

Error An error is anticipated to categorise the recognizable shortcomings of themodelling approach which are not due to inherent variation or a lack ofknowledge.

Given this distinction in terminology, different researchers [48, 108] argue that it isdifficult to assign a type of uncertainty as either aleatory or epistemic, since they arenot mutually exclusive but rather determined by the modelling choice.

This work deals with randomness in composites to identify the variability ofgeometrical parameters and properties. The associated inherent uncertainty of asingle parameter is quantified using a probabilistic uncertainty approach. When thevariation in the constituents and mechanical properties need to be characterised, anon-probabilistic uncertainty approach could be favourable. The specific uncertaintyintroduced by human interaction and environmental conditions, when quantifyingmaterial properties from e.g. tensile tests, are subjectedto epistemicuncertainty.These are not known and difficult to predict, but can be reduced when manyexperiments are performed. This requires a non-probabilistic analysis using intervalor fuzzy numbers [69, 111]. Such type of variation and analysis technique is out ofthe scope of this work and is not considered.

2.3.2 Probabilistic uncertainty quantification for spatial fields

In a probabilistic approach, each uncertain parameter is defined over a range of valuesas a distributed quantity that is determined by performing asignificant amount ofmeasurements on experimental samples. Probability theoryquantifies the probabilityof occurrence by probability density functionsf (x) (PDFs), which for a certaininterval [a,b] of an uncorrelated parameterX is expressed as:

P (a < X ≤ b) =∫ b

afX(x) dx (2.1)

An experimental distribution is completely characterisedin terms of the type ofdistribution, the expected or mean valueE[X] = µX and the varianceσ2

X.

This representation however assumes that the uncertain property has a constant valueacross the structural domain. When spatial information needs to be determined,these statistics must be complemented with dependency information by defining acovariance matrix covX that specifies the correlation between numerical values of thesame parameter at two different locations{Xk,Xl}, calledauto-correlation, or between

20 STATE-OF-THE-ART

two different parameters at a certain location{Xk,Yk}, namedcross-correlation. Bydemonstration, the covariance matrixcovX for different point distances is obtained bycomputing the covarianceCo(Xk,Xk+l):

Co(Xk,Xk+l) = E[(Xk − µXk)(Xk+l − µXk+l )] = E(XkXk+l) − µXkµXk+l (2.2)

with k = 1..M, reflecting theM locations of that property, andl = 1..M − 1) a fixedvalue for each covariance calculation. SinceCo(Xk,Xk) = σ2

Xk, the main diagonal of

the covariance matrix is equal to the variance of the parameter X at each location.

Throughout the dissertation, a standardised covariance orcorrelation C(Xk,Xk+l) isconsidered defined as:

C(Xk,Xk+l) =Co(Xk,Xk+l)σXkσXk+l

(2.3)

Following the Cauchy-Schwarz inequality, this results in the following property−1 ≤C(Xk,Xk+l) ≤ 1. When constructing thecorrelation matrix corrX, this means that themain diagonal elements are unity.

The correlation matrix based on the covariance is the most suitable statistical spatialdescriptor to characterise the microstructural anisotropy on both the short- and long-range. As alternative, a pair-correlation function can be considered to quantify thespatial arrangement for a peculiar orientation. However, most other existing spatialfunctions, such as nearest neighbour distance distribution and radial distributionfunctions are insensitive to the direction of analysis and therefore inappropriate toquantify the variation in space across textile composites [99, 100].

The availability of a correlation matrix permits to expressthe uncertain quantity asa random field[109]. This mathematical concept describes the spatial variation ofa model parameter on the basis of the spatial evolution of thefirst two statisticalmoments of the random parameter and the corresponding correlation functionCX. Thestochastic field is represented byH(x, ϑ) with x the spatial coordinate andϑ indicatingthe randomness. As in most cases, the assumption of anisotropicrandom field is validfor the textile problem since the correlation function onlydepends on the absolutedistanceτ between two pointsτ =| x− x′ | and is thus invariant to all rigid translations[79]. For numerical applications, a discretisation of the random field is performed. Acommon technique is the K-L series expansion [32, 86, 97], where the auto-correlationstructure of a single property is reproduced by expanding the random field into a setof independent random variables and deterministic spatialfunctions. Specific for K-Ltechnique, these deterministic functions are obtained by spectral decomposition of theauto-correlation functionCauto(x, x′):

Cauto(x, x′) =

∞∑

i=1

λAi φ

Ai (x)φA

i (x′) (2.4)

with λA and φA respectively the eigenvalues and -vectors of the auto-correlationstructure, acquired by solving the eigenvalue problem. A single realisation of a

UNCERTAINTY QUANTIFICATION: DEFINITIONS AND METHODS 21

random fieldH(x, θ) is represented as [32, 97]:

H(x, ϑ) =∞∑

i=1

√

λAi ηi(ϑ)φA

i (x) (2.5)

with {ηi , i = 1..∞} a set of independent orthonormal random variables. In practice,a truncation of the random fieldH(x, θ) is performed by using a finite set of randomvariables{ηi , i = 1..Nvar}. The number of K-L termsNvar in the series is defined byordering the eigenvalues in a descending series and considering only theNvar largereigenvalues that capture most of the randomness. The eigenvector basisφA

i (x) in theK-L expansion is optimal in the sense that the mean-square error resulting from a finiterepresentation ofH(x, ϑ) is minimised. For sufficiently largeNvar, the second-momentproperties ofH(x, ϑ) can thus be approximated by the second-moment properties ofH(x, ϑ):

H(x, ϑ) ≈ H(x, ϑ) =Nvar∑

i=1

√

λAi ηi(ϑ)φA

i (x) (2.6)

2.3.3 Characterisation of a spatial field by the correlationlength

Spatial information can also be quantified by acorrelation lengthξ, which “measuresthe distance of two different stochastic field locations, over which the correlationbetween the respective random variables approaches zero ora practically very smallvalue” [10]. Generally, this length is estimated by a linearapproximation orby evaluating appropriate correlation functions that approximate the experimentalcorrelation structure. A linear approximation is desired when only little data isavailable, with the correlation length being inversely proportional to the slope ofthis curve. When data sets are large, a correlation function is a more suitable wayto define the dependency information. Appropriate functions need to be definedthat are dependent on the correlation length, represent thedependency informationand are physically reasonable. Frequently, conventional correlation functions areconsidered to represent the spatial dependency, with the exponential correlationfunction Cexp = exp

(

− τξ

)

[79] often used as correlation kernel. However, otherfunctions should be regarded when a clearly different correlation structure is present.A typical experimental correlation graph orcorrelogram, expressing the correlationdata in function of different point spacings, is given in figure 2.5. The left-hand sideshows the correlation information deduced for individual plies in a laminate, while theright-hand side demonstrates the process of identifying the correlation behaviour bylinear approximation or function fitting.

It is the author’s opinion that any computed correlation length is subjected to alarge uncertainty. On the one hand, the fitting procedure of aproposed correlation

22 STATE-OF-THE-ART

0 5 10 15−0.5

0

0.5

1

Distance [mm]

Correlation [−]

(a) Correlation data per ply

15 0 5 10 15−0.5

0

0.5

1

distance [mm]

correlation [−]

combined data set

function fit

linear fit (k<6)

(b) Correlation data combined data set

Figure 2.5: Correlation graph of the aspect ratio of weft tows in a 2/2 twill wovencomposite laminate of seven plies: (a) correlation data perply and (b) correlation datafor the combined data set. The overall correlation graph is fitted with an appropriatecorrelation function and approximated by a linear fit.

function to the experimental correlation data is extremelysensitive to small deviations,which is present for small data set sizes as indicated from the ply correlation datain figure 2.5(a). For example, the correlation length in an exponential correlationfunction appears in the denominator, meaning that it controls the curvature of thepresent correlation trend. This reflects the inherent variation of the correlation lengthof a single parameter, which is reduced when more data are collected. On the otherhand, several choices in the derivation of the correlation length are not fixed andneed to be defined by the researcher. First, the choice of the correlation functionis ambiguous since no general tools are defined to numerically evaluate if a functionis a good approximation of the experimental correlation trend or not. In this work,the decision is made based on a least-square error between the fitted and experimentalcorrelation data. Secondly, the analyst can consider all correlation data or only apart of it for matching a correlation function to estimate the correlation length. Forlimited data set sizes, especially the ones belonging to thelargest point spacings, thecorrelation trend is affected by outliers and can therefore be inappropriate for fitting ageneral tendency. This is demonstrated by the more fluctuating correlation data for thehigher lags in figure 2.5. All these required choices in deriving a correlation length,make it an operator-dependent process. It is necessary to always provide a completedescription of how this parameter is determined such that a reader can reproduce theresults. However, correlation lengths will always be defined over a large range. Anestimate of the uncertainty of±25% is assumed for the computed correlation lengthsfor a C/SiC composite in [6].

EXPERIMENTAL TECHNIQUES TO CHARACTERISE VARIABILITY 23

2.3.4 Target statistical data to describe the spatial uncer-tainty

In summary, a complete spatial description of an uncertain parameter requires thefollowing actions: (i) determination of the type of PDF and distribution parametersof the field variable (sample average and standard deviation), (ii) computation ofthe correlation structure across the experimental sample,and (iii) estimation of thecorrelation length using representative correlation functions.

The group of variabilities must be further divided intointra-plate variability, which isproperty of one particular realisation due to spatial scatter, or asinter-plate variabilitythat is the property of a series of nominally identical realisations of a particular productand can be a direct consequence of intra-plate variation. When different compositesamples are considered, which are typically cut from a larger plate, a record mustbe kept to remain a clear distinction from which plate the sample originates. Inter-plate variability in composites is discussed by Berube et al. [5] where five differentmanufacturers had the assignment to fabricate identical laminates based on equalspecifications and production method. The results show large differences in tensileas compressive modulus and strength. For both properties, COVs are acquired from2% to 10% which is a similar interval found in the literature [22, 67, 73, 76]. This workonly quantifies intra-plate variability, where the spatialgeometrical scatter within onecomposite specimen is investigated.

2.4 Experimental techniques to characterise vari-ability

The irregularities in the reinforcement of composites are quantified by (i) measuringthe geometrical parameter across experimental samples, and (ii) performing astatistical analysis. Geometrical quantities can be measured by a wide range ofexperimental procedures that are classified in either destructive or non-destructiveinspection techniques. Depending on the type of experiment, a direct, indirect orcombined characterisation approach is regarded to quantify one or more tow pathparameters in a single experimental set-up. In a direct method, the desired geometricalquantity is measured from images made from the internal geometry of a composite.An indirect technique derives geometrical data from measuring an entity or propertythat can be related to a geometrical feature.

24 STATE-OF-THE-ART

2.4.1 Techniques to collect experimental data

The geometrical parameters of interest are the tow path position, orientation anddimensional information, as described in section 2.2. Except for the in-plane positionsand orientations, these characteristics require internalinformation of a compositestructure. While commonly destructive inspection techniques are considered bytaking cross-sections at discrete locations in the composite, non-destructive techniquesare recently emerging as a more reliable and efficient way to measure the fabricarchitecture over the extent of the composite. Although, the majority of the existingnon-destructive methods (ultrasonic, acoustic, thermography, ect.) are only applicableto detect near-surface flaws (<0.1-1cm depth) with resolutions ranging from a fewhundred microns till a few microns [25, 43, 46, 91]. Only advanced techniques suchas X-ray micro-CT provide high-resolution images that are recommended for thecharacterisation step to quantify geometrical imperfections which are situated in therange of 10-1000 microns. All relevant destructive and non-destructive inspectiontechniques used in the literature to characterise the internal structure of a textilecomposite can be summarised as:

Optical imaging This technique can be employed to inspect the internal geometry orinformation from the surface of a composite. Internal data are acquired frommaterial cross-sections which are cut at a predefined location and polished.The process to prepare a cross-section is labour intensive and therefore oftencountered when spatial information, requiring numerous cross-sections, need tobe analysed. From the individual cross-sections, data on the tow dimensions(tow width, tow height), tow shape and tow spacing can be extracted. Whencombining subsequent cross-sections along the tow path, additional informationabout the out-of-plane and in-plane tow path positions are quantified. Opticalimaging is also applied to inspect the in-plane dimension ofa composite. Nosectioning is made since the aim is to gather a digital image of the outercomposite surface to quantify in-plane tow positions, tow widths and towspacing. Depending on the kind of information to be collected, two opticalimaging techniques are frequently used:

• Optical microscopyfor analysing the internal geometry. A digital image istaken from an illuminated magnified representation using image software.Resolutions can be achieved in the order of ten micron per pixel [22, 47,76, 92]. An example of a digital image using optical microscopy is givenin figure 2.6(a).

• Scanning technique or Photographfor analysing the composite surface.In-plane data are determined by scanning the surface with resolutionsfrom twenty till five micron per pixel [22, 47] or taking a digital imageusing a camera [1, 27, 94]. Gan et al. [31] collected photographs of

EXPERIMENTAL TECHNIQUES TO CHARACTERISE VARIABILITY 25

light transmitted through a translucent composite sample to extract the keyspatial information. In all cases, the material is accompanied by a ruler oranother measurement system to determine the pixel resolution.

Optical imaging is characterised as time intensive and difficult to automatize.The digital images can be further processed to obtain the geometrical charac-teristics in a direct approach using Matlab algorithms or anindirect way usingother techniques, such as the Fourier transform [94], to identify the variation inunit cell positions or tow waviness.

X-ray micro-CT Structural parameters defining the entire tow reinforcement canbe directly derived from high-resolution micro-CT images.A 3-D volumerepresentation of the sample is acquired based on differences in absorptionsof X-rays through the material [50]. From this volume, virtually 2-D slicesin all directions can be extracted without any need of physical sectioning.Geometrical information across the composite is obtained in a single experi-ment, which means an enormous gain in experimental time compared to opticalmicroscopy. This technique has as disadvantages that the equipment cost is highplus a high degree of skill and experience are required to define the correct X-ray acquisition parameters (voltage and current of the beam) that are uniquefor each type of material. It is also limited to small-scale samples that fitinside the device. The density of the constituents of the considered compositemust significantly differ to allow a clear difference in contrast. However,image analysis and filtering options are addressed to improve the contrast inconstituents when needed. In general, this experiment allows large flexibility inimage processing and can be more automatized. Resolutions under one micronper volume pixel can be achieved, such that boundaries between tows can bebetter distinguished than with optical imaging [3, 22, 35, 119]. This makesmicro-CT an indispensable tool to also inspect porosity [64, 72] and damage intextile composites [4, 7, 87, 89, 90, 93, 117]. Figure 2.6(b)shows a processed2-D slice of a C/SiC composite.

Strain fields analysis Thin composite panels are subjected in a static test to, ona macro-scale, nominally homogeneous uniaxial and biaxialstress states inthe plane of the plate and to bending loads. The identification of the towcharacteristics is determined indirectly by 3-D digital image correlation toimage the surface topography in terms of in-plane and out-of-plane variations.Analysis of deviations can be further performed by computing the spectralcontent of the spatial derivatives of the deviation components using Fourieranalysis. Resolution in the order of twenty micron can be obtained [83], withan example of a reconstructed geometry presented in figure 2.6(c).

26 STATE-OF-THE-ART

(a) Optical imaging [76] (b) X-ray micro-CT [22]

(c) Strain field analysis [83]

Figure 2.6: Different experimental techniques to characterise the textilegeometry: (a)optical imaging of a 2/2 twill woven carbon-epoxy composite, (b) X-ray micro-CTslice of 3-D glass woven fabrics and (c) rendered surface of an ceramic interlockweave from strain field analysis.

2.4.2 Statistical analysis of experimental data

In a next step, statistical information is derived from the inspected tow parametersthat define the entire reinforcement. To obtain a representative distribution of eachcharacteristic, experiments are repeated for a number of nominal identical componentsfrom the same material and production process. The inherentvariability of eachparameter is quantified using a probabilistic uncertainty approach, of which theprocedure and definitions are given in section 2.3. Depending on the tow propertiesthat can differ in different directions and manufacturing of the textile, a distinction instatistical characterisation must be made even for equal tow parameters but originatingfrom different tow directions. A complete spatial description of each uncertaintow parameter requires the derivation of an average trend, standard deviation andcorrelation length. However, almost all published statistical analyses are limited toaverage values and standard deviation, ignoring the scatter in space. The authoronly knows about correlation analysis of textile geometrical parameters performedby Skordos and Sutcliffe [94] and Bale et al. [3], demonstrating once more theneed of textile data to generate reliable material models. The statistical data obtainedfrom several materials must be stored under consideration of geometrical parameters,material topology and material production.

EXPERIMENTAL TECHNIQUES TO CHARACTERISE VARIABILITY 27

2.4.3 Selection of characterisation techniques

In this dissertation, spatial information is collected on the short- and long-rangegeometrical characteristics using the above state-of-the-art methods. As alreadyindicated, X-ray micro-CT is more efficient than optical imaging when internalgeometry need to be quantified across the different tow paths and provides high-resolution images. This technique is selected to quantify the tow oscillations on thesmall scale (meso-level). Long-range data (macro-level) requires samples that are atleast one magnitude larger than the unit cell dimensions which do not fit inside themicro-CT equipment. One possibility is to cut the large samples in smaller pieces thatare suitable for micro-CT analysis, but this is very time consuming and difficulties canarise due to the large data sets that need to be combined. A better alternative wouldbe optical imaging or strain field analysis, which can also capture all details sincelong-range fluctuations rather appear with standard deviations in the order of 100-1000 micron. Which technique is most advisable depends on which path parametersexhibit a long-range fluctuation. A proper specification fordata collection is requiredto have a good repeatability of the experiments. An experimental framework is definedwhere each characteristic gets a definition of the measurement procedure, with thedetermination of the density of the measuring grid and the choice of data storage. Asmentioned in section 2.3.4, only intra-plate variability is studied.

2.4.4 Assumptions and errors in the characterisation proce-dure

It is important to assess the errors that are introduced (i) when quantifying the internalstructure of a composite by experiments and (ii) by making assumptions during thestatistical analysis. First, the characterisation of a geometrical parameter is done byimage processing where inaccuracies are introduced e.g. inthe material segmentationof an image and which geometrical shape is considered to represent a tow in cross-section. It is also important to verify if the resolution is sufficient, since the order ofmagnitude of errors made throughout the characterisation procedure (mostly manuallyinterference) can be similar to the pixel resolution of the digital image. In the statisticalanalysis, a distribution type is assumed from the histogramrepresentation of thedeviations. This approximates the real physical PDF and itscorresponding distributionparameters. Further, a large uncertainty is present for theestimation of the correlationlength. A representative correlation function can only be derived from correlationdata that are based on sufficiently large data sets for each of the point distances inthe correlation graph. When only a few data are available, thecorrelation resultis susceptible to outliers. This uncertainty can be reducedif the size of the dataset is increased, but fitted correlation functions still remain to approximate the realdependency structure and matching correlation length. It is important to minimise

28 STATE-OF-THE-ART

these errors to have an as accurate as possible statistical description of the material.The aim of the modelling procedure is to reproduce the input standard deviation andcorrelation information. A thorough characterisation of the tow path of experimentalsamples is thus mandatory.

2.5 Uncertainty modelling in textile composites

2.5.1 Overview of modelling approaches

The reinforcement of any textile component is adequately modelled by exploitingthe hierarchical principle. Predictive models are constructed following a sequencefrom fibre, tow, textile, preform, to the final composite [56]. To representits internal geometry, a periodic meso-level unit cell model is considered wheretow path characteristics are computed based on deterministic inputs such as fibremechanics, topology, tow dimensions (shape, width, height) and tow spacing. Sucha representation is shown in figure 2.7 for the considered 2/2 twill woven composite.While the tow cross-sectional dimensions are kept constant and equal to the inputvalues, the tow path itself is determined using e.g. the minimum energy principle ofthe fibre bundle deformation [113]. These unit cell descriptions are considered to berepetitive along the entire structure without any variation in the tow position, shapeand dimension, while representing the mechanical behaviour of the whole system.However, physical samples do show randomness in the deterministic geometricalparameters within a single unit cell and between neighbouring unit cells; the tow pathproperties are spatially distributed across the composite. Realistic modelling of itsinternal geometry should permit to introduce local variations along each individualtow path and at the different scales (micro-meso-macro), depending on the responseof interest.

Figure 2.7: RVE of the considered 2/2 twill woven carbon-epoxy composite by aschematic representation and as a simulated WiseTex model,using the manufacturer’sdata.

UNCERTAINTY MODELLING IN TEXTILE COMPOSITES 29

Among the different strategies to accurately simulate the randomness in compositesusing appropriate scaling techniques [8, 45, 77, 95], the approach of Charmpiset al. [10] emphasizes that realistic modelling of composite performance canonly be achieved when the numerical multi-scale modelling scheme is supportedby experimental data. The procedure applied to textile composites consists oftwo main steps: (i) collection of material data about the uncertain tow properties(uncertainty quantification and characterisation) and (ii) proposition of a stochasticmulti-scale modelling scheme where the macroscopic material properties are derivedfrom realistic geometrical characteristics at the lower scale (definition of a stochasticmodel). The aim is to generate specimens with a reinforcement that possess thesame statistical information on the short- and long-range distance as measured fromexperimental samples. Advanced composite modelling requires the implementation offlexible equations that produce local tow parameters along and between the tow paths.The deterministic inputs and predictive model construction based on the minimumenergy principle must be replaced by a technique that only has as purpose to producedeviations that match the statistical information of a characterised composite specimenin its final form. Especially correlation data is essential to simulate the spatialdistribution of each parameter along the reinforcement, asit will strongly affect theperformance of the component [95]. Further, each compositeproduct requires a fullmodel of the entire structure rather than a unit cell description.

Simulation of random composite models is already performedwith different method-ologies, but not yet following the approach of Charmpis et al[10]. Either statisticaldata are incomplete without any correlation analysis, or the applied techniques do notsimulate spatial variation within a single unit cell or between unit cells. The lack of athorough statistical analysis of material data and techniques to adequately model thegeometrical irregularities, results in composite models which are not actual replicas ofthe real structure. Possible randomness which is not reflected in computational modelscould result in an important difference in estimated properties and apparent propertiesof the textile composite.

Depending on the required response, geometrical descriptions are sometimes trans-formed to simplified micromechanical models that are more computational advan-tageous [45, 77]. In other cases, more realistic structuresare acquired withoutcollecting experimental data [96]. The given overview of current realistic compositemodelling only discusses procedures that develop a random representation of thefull internal geometry inside a RVE or larger composite model, and is precededby a geometrical characterisation or experimental test to define the mechanicalperformance. These simulation methods can be generally classified into non-intrusiveand intrusive approaches. Intrusive techniques can be further decomposed in threegroups depending if spatial dependency is discarded, considered only along the towpath or taken into account along and between neighbouring tow paths.

30 STATE-OF-THE-ART

2.5.2 Non-intrusive simulation techniques

Non-intrusive simulation techniques introduce variationin the geometry withoutchanging the existing modelling schemes. The model input parameters, typicallyrelated to the tow spacing and cross-sectional dimensions,are given small deviationsfrom their nominal value. These deviations are randomly sampled from theexperimental statistical distribution of each parameter.In combination with a MonteCarlo framework, many specimens can be generated and the scatter in correspondingmechanical properties can be computed [22, 76]. However, such procedures frequentlyacquire unrealistic fibre tow volume fraction due to the calibration of the overall unitcell fibre volume fraction to the target value. This is a result of the inability ofnon-intrusive approaches to introduce local geometrical scatter along the tow path.To accurately represent local variations, intrusive methods are requisite with spatialdescriptions that replace the single parameter description of one property.

2.5.3 Intrusive simulation techniques

2.5.3.1 Methods without considering spatial dependency

The first group of intrusive methods extends the nominal equations of existingmodelling techniques in their functionality. Additional path parameters are definedand/or statistical descriptors are used to adjust the microstructure such that theresulting macroscopic mechanical behaviour of the model compares with performedexperimental results [73, 92, 119]. In other cases, the model is constructed from idealshapes which are subjected to mechanical loading. Controlling parameters such asthe modulus, friction and yield strength of tows lead to a realistic representation ofthe internal geometry [35]. All these techniques are based on the assumption that asimilar overall mechanical response of two systems can onlybe obtained when thegeometrical structure is similar.

2.5.3.2 Methods that consider spatial dependency along their path

In the second group of intrusive methods, the description ofgeometrical parametersis considered asrandom fieldswhich are simulated by a stochastic process. Thefirst two moments, mean and correlation data, are sufficient to fully characterise astationary Gaussian process, which is applicable to model the randomness in geometry.The spatial behaviour of each tow parameter is accurately represented in each of thedifferent realisations of the stochastic fields.

UNCERTAINTY MODELLING IN TEXTILE COMPOSITES 31

In cases where correlation data are lacking, assumptions need to be made on theprocess properties. Examples of such modelling in textilesis found in [28] and[1], where trigonometric functions are used to discretise the random field where theamplitude, wavelength and/or phase are varied randomly between a fixed interval.Although these intervals can be determined from the range ofobserved variations,trigonometric functions assume an idealised shape of the tow position or dimensionalparameter. The real trend of the geometrical irregularities and its correspondingdependency structure remains approximated.

Once experimental correlation information is available, the stochastic fields of thegeometrical uncertain parameters are able to reproduce thecorrect dependencystructures. The correlation matrix is constructed from a fitted function whichapproximately describes the experimental correlation trend for all point spacings.Two options of correlation functions are possible: choose aknown Gaussianprocesses to express the spatial distribution of the randomfield, or propose othercorrelation functions from empirical measurement that satisfy the positive semi-definite requirement to be a correlation matrix [32, 79]. Subsequently, zero-meanrealisations of random fields are obtained by decompositionof the correlation matrixin combination with a vector of independent standard Gaussian random variables:

Cholesky factorisation Random realisationsX ∼ N(µ,CX) obtained using theCholesky matrix decomposition are computed using:

X = µ + Λη andCX = ΛΛT (2.7)

with the vectorη ∼ N(0,1) representing the independent standardised randomvariables andΛ a triangular matrix representing the Cholesky root.T denotesthe transpose operation.

EigendecompositionRandom instances acquired using the eigendecomposition ofthe correlation matrix is incorporated in the K-L Series Expansion technique,described in section 2.3.2.

Both procedures appear in the literature. Skordos and Sutcliffe [94] modelled theanisotropic spatial correlation structure of tow orientations as an Ornstein-Uhlenbeckprocess of which the parameters are estimated by the likelihood function. Next,realisations of woven sheets are obtained using the Cholesky decomposition ofthe covariance structure of warp and weft direction in combination with normallydistributed independent variables. Yushanov and Bogdanovich [118] generated arandom reinforcement, characterised by its mean value and covariance matrix, usingthe Series Expansion technique. Of both approaches, the eigendecomposition ismore efficient for highly correlated vectors [78], and allows a reduction of thedimensionality of the process. Simulation of random fields based on Series Expansionis thus more appropriate, especially when only a few terms ofthe K-L expansion,

32 STATE-OF-THE-ART

corresponding to the larger eigenvalues, need to be considered to capture most of therandom field fluctuations.

Correlated deviations can also be produced without decomposing the correlationmatrix by applying a Monte Carlo Markov Chain formulation for textile structuresdeveloped by Blacklock et al. [6]. Tow fluctuations are not simulated as realisationsof a random field, but as a set of discretised points by marching systematically alongthe tow path using a probability transition matrix. This matrix, calibrated with themeasured standard deviation and nearest neighbour correlation data, is implementedin a Monte Carlo scheme to produce realisations of the tow deviations by first choosinga uniform random number which is afterwards rescaled by a mapping operationonto the probability distribution of the subjected parameter. The latter distributionis determined from the transition matrix given the deviation value at the previous gridlocation. All parameter deviations of a single tow are generated in a single step andindependently from other tow parameters and neighbouring tows. An additional post-processing smoothing operation is however required to remove unphysical spikes inthe generated tow path. Not only positional deviations [6],but also 3-D solid towrepresentations [81] are precisely simulated using this method.

2.5.3.3 Methods that consider spatial dependency along their path andbetween neighbouring fields

Both the continuous stochastic functions as the Markov Chain methodology are able toadequately reproduce the correlation in one direction. Whena geometrical deviationis not only determined by its location along the tow path, butalso depends on theneighbouring deviation value, adjustments to the techniques need to be made toaccurately simulate the dependency relations in all directions: along the tow andbetween neighbouring tows. Particularly the modelling of cross-correlation is not orpartially considered since not every property is cross-correlated or because it requiresspecified complex simulation effort. Only in the work of Skordos and Sutcliffe [94]both auto- and cross-correlation is considered to simulatetow orientations of a carbon-epoxy satin weave.

The simultaneous reproduction of the auto- and cross-correlation structure of a singleuncertain parameter demands more advanced generation techniques developed in thefield of stochastic mechanics. Methods developed for probabilistic simulation ofvarious mechanical problems, only or mainly tested with virtual data, can now beapplied to simulate the complete dependency structure measured in textile composites.An overview of extensions to previous procedures is:

• To acquire realisations of cross-correlated stochastic functions using theCholesky decomposition, an anisotropic correlation matrix must be defined that

UNCERTAINTY MODELLING IN TEXTILE COMPOSITES 33

represent both the correlation information along and between the tow directions.This is determined by defining a multi-dimensional normal distribution whichcan be described by e.g. a two-dimensional extension of the Ornstein-Uhlenbeck process which possess the required covariance matrix [94].

• A methodology developed by Vorechovský [114] permits to generate realisa-tions of cross-correlated random fields based on K-L Series Expansion. Theprocedure is generally applicable for the modelling of a setof stochastic fieldswhich share an identical auto-correlation structure and ofwhich the cross-correlation can be defined by a cross-correlation coefficient. When applied totextile composites, each stochastic field corresponds to the parameter deviationsof a single tow. The framework of Vorechovský uses the K-L Series Expansionequation 2.6 as basis. The key intention of the procedure is that all cross-correlated fields, within a single specimen, are expanded using the samespectrum of eigenvaluesλA

i and -vectorsφAi (x), but the sets of random variables

χDj,i used for the expansion of each field are cross-correlated with neighbouring

fields:

H j(x, ϑ) =Nvar∑

i=1

√

λAi χ

Dj,iφ

Ai (x) (2.8)

This cross-dependency is introduced by modal decomposition of the experimen-tally deduced cross-correlation matrix of each parameter.The random fieldrepresentation is also approximated by performing truncation of the series.

• The Monte Carlo Markov Chain method for textile structures [6] is originallypresented to simulate parameter deviations of which the dominant correlationsmust be those along a tow, with correlations between tows being relatively weakto satisfy the Markovian procedure. The extension towards cross-correlateddeviations is done by combining Markov processes with an inverse FourierTransform. At each grid point along the tow direction, tow packing densitydeviations are generated using random values of amplitude and phase of eachFourier component taken from experimental distributions.Realisations of theamplitude are produced by applying the Markov Chain algorithm, while thephase of the Fourier component is obtained via a random-walkalgorithm. Aninverse Fourier transform provides the corresponding deviation values at all gridpoints [15, 82].

2.5.4 Selection of modelling strategies

In this dissertation, a realistic representation of the internal geometry of a textilestructure is modelled following the strategy of Charmpis etal. [10]. Simulationsof uncertain tow parameters are performed at the different material scales supported

34 STATE-OF-THE-ART

by experimental data. Only intrusive methods are employed that reproduce theexperimental correlation structure without assuming a fixed trigonometric trend towhich the deviations need to comply. Single auto-correlated properties are producedby the Monte Carlo Markov Chain for textile structures [6] which has already provenits accuracy for a ceramic textile, without much complexity. Tow parameters whichare both auto- and cross-correlated, are generated using the cross-correlated SeriesExpansion method of Vorechovský [114]. It are these intrusive methods, calibratedwith experimental correlation data, which give the most adequate description of theinternal geometry for any position along the tow path. The obtained virtual textilecomposites can be used in a next step to acquire a more reliable estimate of the materialproperties.

While the discussed Markov Chain method is a recent reconstruction algorithmfor composites, K-L series expansion is already consideredto simulate commoncomposite material properties such as stiffness and shear moduli [11, 13, 67, 71].Specific for Mehrez et al [67], an adapted technique is used toconstruct such aspatial representation based on limited measured data. First the spatially stochasticstiffness properties of Twintex composite beams are characterised using dynamicmeasurements. In a next step, a stochastic model of the material properties isconstructed from the empirical covariance matrix using theK-L expansion, in whichthe random variables are expanded into the Hermite Polynomial Chaos (PC) basis.Besides material properties, K-L expansion can also applied to model other compositeproperties, e.g. the mesoscopic volume fraction of long fibre thermoplastic composites[36].

2.5.5 Assumptions and errors in the modelling strategy

Throughout the modelling procedure, unavoidable assumptions are introduced toacquire virtual reinforcements. Among these, (i) a normal distribution for alldeviations and (ii) proposition of an idealised correlation function to represent thedependency information, are the most important hypotheses. It is further assumedthat auto-correlated and both auto- as cross-correlated deviations can be generatedindependently from each other, which is only valid when there is no significant cross-correlation between the corresponding geometrical parameters. When a significantcross-correlation is present, adjustments to the simulation procedure must be made.

In case the generated virtual tow paths are not representative for the characterisedexperimental tow trends, these assumptions should be revised. This occurs whenthe experimental distribution of a geometrical parameter shows large deflections fromnormality or if the fitted correlation function does not comply with the experimentalcorrelation function.

LINK OF MECHANICAL PERFORMANCE WITH GEOMETRICAL RANDOMNESS 35

2.6 Link of mechanical performance with geomet-

rical randomness

Variability in the mechanical performance of textile composites is dominated bythe intrinsic scatter in the geometrical structure. Modelling based on periodicityand idealised representations will result in less accuratepredictions of the effectiveproperties, and underestimate failure and damage effects in composites [51, 101]. Anyvariation in the tow path characteristics at the mesoscopicstructural level propagatesto the macro-scale where it influences the distribution of each mechanical property.Scatter has a measurable effect on elastic properties, and a dominant effect on strength,fatigue life, and all properties related to damage mechanisms and work of fracture[19]. In general, performance simulation must apply realistic textile models in orderto obtain an adequate characterisation of the variation in the mechanical properties.This is essential towards reliable designing of composites.

When variation in the reinforcement is introduced in the simulations, more reliableestimates of the mechanical response are obtained that correspond to the averagevalues acquired from composite testing. However, simulated statistics (standarddeviation, correlation) can show small differences from experiments since additionalvariation is introduced by the experimental campaign itself. Among the experimentaluncertainties, the most significant errors are present in the (i) cutting precision of thesamples [18, 53], (ii) experimental preparation, set-up, and/or testing conditions, and(iii) post-processing computations on the data set [76]. These effects are howeverassumed to be averaged out for the mean value.

The magnitude of variability in performance, resulting from measured or assumedimperfections in the textile, has been partially reported with contributions consideringthe elastic mechanical properties [9, 22, 38, 73, 76], thermo-mechanical properties[33], formability [94], permeability [27, 28, 116] and damage initiation and prop-agation [12, 26, 120, 122]. Almost all published work considers micromechanicalmodels which are constructed without any type of correlation. Yet, spatial dependencymust be included when constructing composite models in order to achieve realisticdistributions of the material properties that contribute to the quality improvement ofnumerical analyses of composites.

The effect of randomness in the textile reinforcement on the mechanical properties isnot described in this dissertation, but is an obvious next step in the continuation ofthis research. Since scatter is adequately simulated over the extent of the compositestructure, a quantitative measure of the spatial variationin the properties can beobtained that varies from one to another location in the composite. A preliminaryinvestigation of the influence of geometrical imperfections on the main stiffnesscomponents is already performed by the author in [103]. Table 2.1 presents theseresults obtained from stochastic simulation of 2/2 twill woven unit cells. The stiffness

36 STATE-OF-THE-ART

Table 2.1: Comparison of stiffness components of a nominal model and randommodels (1000 generated random unit cells) of a 2/2 twill woven carbon-epoxycomposite [103].

E11 [GPa] E22 [GPa] G12 [GPa]

Nominal model - systematic trends 67.57 65.48 3.20Random model -mean 65.83 64.19 3.27Random model -COV 0.85% 0.87% 0.88%

values for theE11 and E22 moduli are dropped with 2 to 2.5 % compared to thestiffness values of the average model. This compares with the results of Olave etal. [76] where the same topology is considered using a similar approach, but with3K and 12K of carbon fibres per tow. The mean value ofG12 is increased with 2.2%, demonstrating that the misalignment of the tow paths contributes in an increase ofin-plane resistance when a shear load is applied. The effect of long-range variabilityon these conclusions should still be investigated. Besidesdetermining the effect onthe mechanical properties, representative textile modelsalso enables to establish arelationship between the different mechanical properties at the macro-level, whichare depending on the same geometrical information, but cannot be deduced fromexperimental tests. At last, these virtual specimens servean important purpose inanalysing more precisely the damage initiation and progression in textile compositesas discussed in [14, 15].

2.7 Summary and conclusions

This chapter discusses the inherent variability present inthe reinforcement structureof composite materials, with focus on textile composites. The sources of geometricalscatter are described and the most important probabilisticstatistical concepts aretouched upon. A complete statistical description of any uncertain parameter isdefined in terms of an average trend, standard deviation and correlation length alongthe tow path and between neighbouring tow paths. An overviewof the currentstate-of-the-art for uncertainty quantification and simulation of the reinforcement oftextiles demonstrates that no systematic approach is yet available to assess the spatialuncertainty over the extent of the structure. There is a lackof mature experimentalcharacterisation schemes with a complete statistical analysis to efficiently andaccurately determine the geometrical randomness at the short-range (meso-scale)and long-range (macro-scale). Modelling techniques frequently omit the spatialdependency in the generation method, due to the shortage on data or since it requires

SUMMARY AND CONCLUSIONS 37

specified complex simulation effort. Although advanced numerical procedures exist inthe field of stochastic mechanics, the introduction of thesemethods did not yet occurfor precisely simulating composites.

In the next chapter, a general framework is elaborated to develop random virtualtextile specimens that possess the same statistical information as measured from theexperimental samples. A combination of state-of-the-art inspection techniques suchas X-ray micro-CT, optical imaging or strain field analysis are applied to measure thefabric architecture in a reliable way across the composite volume. Random instancesof tow paths are acquired by using two of the discussed generation techniques thatare calibrated with the measured experimental data. Auto-correlated deviations areproduced by the Monte Carlo Markov Chain algorithm for textile structures, whilecross-correlated uncertain parameters are generated using the cross-correlated SeriesExpansion method.

Chapter 3

Development of a stochasticmulti-scale modellingframework

This chapter presents a stochastic multi-scale framework to develop virtual specimenswith random reinforcement. The needs in material data and appropriate scalingmethodologies, thoroughly described in chapter 2, are addressed to adequatelysimulate textile composites. It is an extensive summary of the performed researchthroughout the PhD, highlighting the main scientific contributions and sketchingthe usage of each procedure in a larger frame. All characterisation procedures andmodelling techniques are demonstrated on a carbon-epoxy 2/2 twill woven composite.Interested readers can use this chapter as a roadmap how to develop realistic virtualspecimens of any textile composite. The current state-of-the-art of the discussed stepsin the framework are indicated in each of the four key publications collected in part IIof this dissertation.

First, the general framework is defined to construct virtualspecimens consideringexperiments and simulations. Next, section 3.2 discusses the collection of statisticalinformation of uncertain geometrical parameters from experimental samples on theshort-range (paper I) and long-range (paper II). This information is used in section 3.3to calibrate advanced numerical procedures that reproducethe statistical informationfor auto-correlated parameters (paper III) and both auto- as cross-correlated towproperties (paper IV). Finally, large random virtual specimens, spanning multiple unitcells, are constructed using the WiseTex software. The chapter ends with a discussionof the obtained results.

39

40 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

3.1 Multi-scale framework

Realistic woven specimens are acquired that are replicas ofexperimental samplesfollowing the approach of Charmpis et al. [10]. Randomness is introduced inthe numerical models at the meso- and macro-level; scatter in the matrix and fibreproperties are not considered. The variability of each tow path is defined for thecentroid coordinates (x, y, z), tow aspect ratioAR, tow areaA and orientationθ incross-section which fully describe a woven reinforcement.Figure 3.1 presents anoverview of the multi-scale framework, where three main steps can be distinguishedto obtain such random representations:

1. Collection of experimental data and statistical analysis

(a) Characterisation of the short-range scatter (meso-scale) with samplesclose to the unit cell size.

(b) Characterisation of the long-range variation (macro-scale) with samplesspanning several unit cells.

(c) Statistical analysis of the tow path parameters in termsof average trends,standard deviation and correlation lengths.

2. Stochastic multi-scale modelling of the reinforcement

(a) Definition of systematic and handling trends from the experimental data.

(b) Generation of zero-mean deviations correlated along the tow path usingthe Monte Carlo Markov Chain method.

(c) Generation of zero-mean deviations correlated along and between neigh-bouring tow paths using the Cross-correlated Series Expansion technique.

3. Construction of virtual specimens in the WiseTex software

(a) Simulation of the nominal model with matrix and fibre properties from themanufacturer.

(b) Redefinition of the reinforcement information with the produced towpaths.

(c) Recalculations of the path orientation vectors and length, in addition to theupdated general unit cell properties.

The statistical analysis, generation algorithms and modelcomputations are performedusing Matlab as indicated in figure 3.1. XML-scripting is applied to generalise anominal WiseTex model to a stochastic representation. When other topologies thanwoven are considered, additional parameters should be quantified to allow a fulldescription, e.g. the braid angle for braids and the distortion of the Z-yarn in caseof non-crimp fabrics.

MULTI-SCALE FRAMEWORK 41

Figure 3.1: Multi-scale framework.

42 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

3.2 Collection of experimental data and statistical

analysis (step 1)

3.2.1 Experimental framework

As starting point of the methodology, statistical information of the reinforcementparameters is derived from physical samples as shown in figure 3.1. Differentexperimental techniques are selected to investigate the geometrical variability presentwithin the unit cell (short-range) and propagating over several unit cells (long-range),demonstrated for the tow centroid positions in figure 2.3. The outcome of this processis statistical data of the reinforcement parameters that will calibrate the simulationprocedures.

In a first step, it is recommended to acquire 3-D images of the internal geometryof the sample via laboratory X-ray micro-CT. This techniqueis efficient to quantifythe short-range spatial information of the entire reinforcement in a single experiment.From the 3-D volume representation, equally spaced 2-D slices are extracted in eachweave direction to trace the tow path data along its length. Tows in cross-section arefitted with a geometrical shape using the freeware ImageJ from which the centroidcoordinates and cross-sectional information can be deduced. A statistical descriptionof the tow parameters is afterwards obtained by applying thereference period method[3]. This approach groups tows that should be identical given the nominal periodicityof the textile, with such a representative tow calledgenus. Each characteristicis partitioned for every genus into periodic systematic variations and non-periodicstochastic deviations, over a grid withNi locations according to the periodic lengths.The decomposition can be represented withε representing one of the four parameters{ρ, z,AR,A} andρ = y or x respectively for each genus:

ε( j,t,s)i =< ε

( j,t,s)i > +ǫ

( j,t,s)i (3.1)

with ǫ( j,t,s)i the zero-mean deviation from the systematic value< ε

( j,t,s)i > at locationi

(i = 1..Ni) along the towj ( j = 1..N j) of tow genust in ply (for laminates) or sample(for one-ply samples)s. While the systematic variations describe the average trendsof each genus represented by the mean value of any parameter at each grid point in thereference period, random deviationsǫ( j,t,s)

i are further quantified in terms of standarddeviationσ and correlation information.

The standard deviation is computed for each parameter combining the data for all gridlocationsi and all towsj belonging to the genust:

σ(t,s)ǫ =

√

∑

i, j(ǫ( j,t,s)i )2

N − 1(3.2)

COLLECTION OF EXPERIMENTAL DATA AND STATISTICAL ANALYSIS (STEP 1) 43

with N =∑

i Ni . The standard deviation is assumed to be independent of the grid pointlocation.

Correlation is computed using the Pearson’s moment correlation parameter for pairsof data taken at distinct locations on the same tow spaced bykν (auto-correlationCauto), and pairs of data taken at the same grid location but on other neighbouring tows,spaced bykν′ (cross-correlationCcross). The deduced dependency can be presentedin a correlation graph that shows the correlation data in terms of distances betweenthe two data points, orlags, from which the correlation is computed. The directionof these dependencies are indicated in figure 3.2. As example, the auto-correlationparameter is computed by:

C( j,t,s)auto (k) =

∑n−ki=1 ǫ

( j,t,s)i ǫ

( j,t,s)i+k

√

∑n−ki=1 (ǫ( j,t,s)

i )2√

∑n−ki=1 (ǫ( j,t,s)

i+k )2

(3.3)

with n the number of pairs andk an integer value (k = 1..Ni − 1). In addition to thecross-dependencies between tows belonging to the same genus, correlation betweentwo different genuses also needs to be quantified at the cross-over locations wherewarp and weft positions overlap. Depending on the considered topology, correlationscan be negligible or prominent.

The Pearson’s moment coefficient is further used to estimate the correlation lengthξ,which is discussed in section 2.3.3. This particular lengthcan be derived from fittingfunctions to the apparent trend in the correlation graphs. This is however only possiblewhen sufficient data can be collected for several lag locations. Higher point spacingsare namely populated by much smaller data sets and can therefore be inappropriatefor fitting a general tendency, especially in the case of limited data set sizes. Thefunction fitting is performed in a least-square sense where the sum of squares of theresidualsEres at each lag between the experimental correlation dataCdata,i and thefitted correlation dataC f it,i are minimised:

Eres =

Ni∑

i=1

(Cdata,i −C f it,i)2 (3.4)

In this methodology, two different fitting approaches are proposed to define thecorrelation length: (i) linear approximation of the first lag locations for limiteddata sets and (ii) correlation function fitting using at least half of the lag locationsfor moderate data sets. When more data are acquired, additional lag locationsof higher point spacings can be included for the fitting procedure. Appropriatecorrelation functions need to be defined that represent the correlation information andare physically reasonable. The exponential correlation function is widely used [79],and also considered here as a reference:

Cexp= exp

(

−τ

ξ

)

(3.5)

44 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

For small data sets, a linear approximation of the standard exponential function isobtained by considering only the first two terms of its Taylorexpansion series. Thecorrelation length is defined as the slope of the straight line fitted to the correlationvalues:

C(k) = 1− kδ/ξ (3.6)

The linear fit is performed on only the first five data points (k ≤ 5) or until thecorrelation crosses the zero-correlation boundary. For larger sizes of data sets, thecorrelation function of equation 3.5 can be fitted directly to the correlation information.When correlation data crosses the zero-correlation before the desired number of lags,further data points are not considered since negative correlation is unphysical but canbe present in data due to a larger variation.

66

6.2

6.4

6.6

6.8

67

7.2

Tow j+1

Tow j

Auto-correlation

Cross-

correlation

Figure 3.2: Definition of spatial dependencies of deviations demonstrated for the in-plane centroid of two weft tows: auto-correlation (along the tow) and cross-correlation(between neighbouring tows).

Based on the derived auto- and cross-dependencies of the towparameters, additionallong-range characterisation could be required. The magnitude of the computedcorrelation length of an individual parameter is used as indicator to determine if thevariation occurs within the unit cell size or exceeds this dimension. However, enoughunit cell samples from different locations and distinct realisations are needed to verifythis statement. Apart from the correlation length, periodicity in correlation graph canbe present with again higher dependencies for point spacings exceeding the unit cellsize. Depending on the tow path parameter(s) that exhibit(s) such a long-range trend,additional micro-CT scans, optical imaging techniques or strain field analysis need tobe performed as discussed in section 2.4.1. Spatial quantification of the out-of-planecentroid and cross-sectional variations always need internal information, while the in-plane centroid can already be characterised using the faster and easier optical scanningtechnique.

COLLECTION OF EXPERIMENTAL DATA AND STATISTICAL ANALYSIS (STEP 1) 45

3.2.2 Application to a woven textile composite

3.2.2.1 Material

As demonstration of the methodology, a 2/2 twill woven carbon fibre Hexcel fabric(G0986 injectex) [40] is considered. Each dry unit cell reinforcement consists of fourequally spaced warp and weft tows with a nominal areal density of 285 g/m2. The unitcell topology is given in figure 3.3 withλx=11.43 mm andλy=11.43 mm, respectivelythe periodic lengths of the warp (x-axis) and weft (y-axis) tows. Laminated and single-ply samples are produced in a RTM process to quantify its geometrical variability onthe short- and long-range. Regarding the weaving process ofthe 2/2 twill wovenfabric, warp tows are represented by a singlegenus, and similar for the weft tows.

Figure 3.3: WiseTex model of a 2/2 twill woven reinforcement. The x-axis and y-axisof the coordinate system are respectively parallel to the warp and weft direction.

3.2.2.2 Short-range characterisation

Short-range variations are identified in paper I [104] from one seven-ply unit cellsample using laboratory micro-CT. A 3-D reconstructed volume is obtained and 2-D slices are extracted in warp and weft direction with a voxelsize of (6.75µm)3.Although the image quality of the cross-sections is optimised using different filteringtechniques, fully-automated material segmentation is notpossible. The similarmaterial density of the carbon fibres and epoxy resin resultsin a low contrast, as canbe seen from figure 3.4. The required manual input for image segmentation limits theamount of slices to nineteen to analyse the structure of the sample in a reasonable timebut capturing all the essential fluctuations in the tow path (set to every 0.75 mm alongthe tow path). The segmented tow shapes are fitted with an elliptical shape yieldinginformation about the tow path centroids (ρ, z) with ρ=y for the warp tows andρ=x forthe weft tows, tow aspect ratioAR, tow areaA and tow orientationθ in cross-section.

46 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

Figure 3.4: Digital image of a cross-section in weft direction obtained from micro-CT.Tows in cross-section are first identified using image segmentation and afterwardsfitted with an ellipse shape to deduce the tow information. This procedure isdemonstrated for ply 7.

The tow path parameters of each genus, based on data from fourtows (N j = 4), aredescribed on an equidistant grid with sixteen points (Ni = 16) and a total grid lengthset to the unit cell period of the particular genus (λx or λy). These periodic lengthsare derived from the experimental data using a minimisationalgorithm described inpaper I [104]. The reference period collation approach [3] is applied, where eachtow parameter is decomposed in non-stochastic, periodic systematic trends and non-periodic stochastic fluctuations according to equation 3.1with s= 1..7. The presenceof several plies maximises the data that can be derived from asingle sample, butalso permits to compare the statistics of each ply. Comparison of the ply systematictrends shows that no discernible differences are present for each single tow parameter.One warp and one weft systematic trend are enough to represent the mean tow pathof a laminate. The mean behaviour of the out-of-plane centroid coordinate, area,aspect ratio and orientation are correlated along the cross-over locations as outlinedin figure 3.6, with the overall mean values for the cross-sectional properties given intable 3.1. The in-plane centreline does not possess such a periodic trend. Similarconclusions are made for the deviations from the systematicvalues. The statistics ofthe fluctuations for different plies are indistinguishable, permitting the derivation ofthe standard deviation and correlation length of each parameter using the data set ofdeviations combining data from all plies. This increase of data set size is favoured,certainly for the correlation information which is sensitive to the amount of data.

Table 3.1: Mean values forAR, A andθ computed over all plies in the unit cell sampleof [104].

< AR> [-] < A > [mm2] < θ > [◦]

Warp genus 12.946 0.366 -0.539Weft genus 12.222 0.360 -1.266

Statistical information (σ, ξauto) quantified in paper I [104] is given in tables 3.2

COLLECTION OF EXPERIMENTAL DATA AND STATISTICAL ANALYSIS (STEP 1) 47

and 3.3. Correlation lengths are derived using the linear approximation of the firstfive lag points (equation 3.6) due to the small data set size. The deviations are fairlyrepresented by a normal distribution with the in-plane centroid subjected to the largestvariability. This reflects the difference in tow tension for warp and weft tows duringproduction. All tow characteristics, with exception of thein-plane centroid, varywithin the unit cell dimensions. This is indicated by the correlation length whichexceeds the unit cell size for the in-plane position as shownin table 3.3, while the out-of-plane centroid possesses a correlation length in the range of 2-4 mm. The in-planecomponent is less controlled than the out-of-plane centroid during RTM productionwhere flat platens fix the thickness of the laminate. The area has a correlation lengthsimilar to the cross-over spacings, while the aspect ratio and orientation vary withinthe unit cell size. Generally, large uncertainty is presentwhen deriving the correlationlengths using only data of the single plies. The limited sizeof the data set for thelargest point separations causes fitted lines to be affected by outliers, which is a majorsource of variability in the correlation length. Additional data must be collected tobring down this variation. Cross-correlations between tows of the same or differentgenuses are investigated in paper II [105] for neighbouringlocations. However, onlyweak dependencies are found thus no cross-correlation lengths are computed. Limitedcorrelation (0.4 - 0.6) between the in-plane centroid of neighbouring tows is present,both for the warp and weft genus. The correlation of the warp and weft tow at thecross-over locations is found to be weak for the z-centroid,which is expected due tothe RTM production process that fixes the thickness of the composite. For furtherdetails and discussion of the results the reader is referredto paper I and III in thesecond part of this dissertation.

3.2.2.3 Long-range characterisation

The short-range characterisation of the tow path concludesthat only the in-planecomponent has a long-range behaviour. Additional data of this centroid is collectedin paper II [107] by optical imaging of one-ply samples. Two one-ply reinforcements,spanning a region of thirteen unit cells by thirteen unit cells, are produced in a RTMprocess with equal production parameters as for the laminated sample. The productionof one-ply samples is more challenging than multiple-ply samples but offers theadvantage to obtain a high contrast between tow and resin regions for the imageprocessing step. The in-plane dimension of both samples (sample 1 & 2, s = 1..2)is scanned optically with a resolution of 1200 dots per inch (DPI), with figure 3.5showing the digital image of sample 1. On this image a square region of ten unit cellsis indicated, away from the edges to minimise possible edge effects and large enoughto be at least one order of magnitude larger than the sample used to collect the shortrange data. Next, the freeware GIMP is used to extract the in-plane centrelines of theforty warp and forty weft tows (N j = 40) within the region of interest. Boundaries of

48 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

the tows are marked based on visual recognition for prescribed grid spacings accordingto the cross-over distance, in x- (warp) and y- (weft) direction. Centroid locations aresubsequently defined as half the tow width at each grid location. These coordinatesare given as input to Matlab where they are transformed to a global coordinate systemand compensated for possible misalignment during scanningof the sample.

Figure 3.5: Optical scan of a one-ply 2/2 twill woven carbon fibre fabric impregnatedwith epoxy resin. Warp tows are oriented horizontally, while weft tows are positionedin vertical direction. The red square indicates the region of interest where the in-planeposition is characterised.

Figure 3.5 demonstrates that the in-plane centroid of a single tow does not follow astraight path over the extent of the sample. The dotted line in the image, representingthe in-plane tow path, clearly deflects from the straight dashed line. These in-planeundulations are quantified by computing the difference between the experimental towpaths and an ideal lattice description. Tows of this latticeare represented as straightlines with nominal spacing in x- and y-direction derived from the experimentallyobtained periodic lengthsλx andλy. In total, forty data points are extracted (Ni = 40).A best fit of this lattice with the experimental cross-over locations of the tows issought by a minimisation algorithm reducing the overall standard deviation of thefluctuations from the grid. The in-plane warp and weft deviations are considered inrespectively the y- and x-direction. These obtained differences are further decomposed

COLLECTION OF EXPERIMENTAL DATA AND STATISTICAL ANALYSIS (STEP 1) 49

in a mean trend and zero-mean deviations using equation 3.1 with s=1 or 2 referringto the sample. However, the particular average trend shouldnot be interpreted as asystematic periodic trend but as a handling trend. Due to thehandling of the materialduring storage, cutting and placement in the RTM mould, shearing of the fabric occursthat affects the in-plane movement of the tows. This average patternfor the warp andweft genus is shown in figure 3.6. The stochastic variations of sample 1 and 2 arecombined in one larger data set per tow genus, since no expected statistical differencesare present between the samples.

The random behaviour of the in-plane deviations is further described in terms ofstandard deviation and correlation length along the tow andbetween tows (tables 3.2and 3.3). In contrast to the short-range variations in paperI [104], in-plane deviationsshow larger deflections from normality. This is especially true for the weft deviations.However, from the limited amount of samples, it is not possible to verify if anothertype of distribution could be more relevant thus a normal distribution is furtherconsidered. The standard deviation of in the in-plane centroid is five times higherfor the weft direction. This is again attributed to the production process where warptows are tensioned while weft tows are inserted. The moderate size of the data set(high Ni andN j) permits to derive a representative correlation function from whichthe correlation length can be estimated. ExponentialCexp = exp

(

− τξ

)

and squared

exponentialCsq,exp = exp(

− τ2

ξ2

)

correlation functions show to be good fits to theexperimental graphs with a least-square errorEres less than or equal to 1%. Thisfitting procedure only considers the first twenty lags which correspond to the length offive unit cells or half the correlation data. The auto- and cross-correlation behaviouris well represented by an exponential correlation functionfor the warp in-planedeviations, while the squared exponential function is a better fit for both correlationdirections of the weft tows. The in-plane warp correlation lengthξauto along the tow isapproximately twice of the weft tows, reflecting the straightness of the warp tows andthe significant in-plane movement of the weft tows. Distortions in the weft in-planecentroid positions affect near- and further-neighbouring tows, appearing as bands inthe composite tow paths as shown in figure 3.5. This translates in a significantly highcross-correlation lengthξcross for the weft tows, exceeding the unit cell dimension,while the cross-dependency of neighbouring warp tows are limited within the unit cell.A thorough discussion of the procedure and results is given in paper II [107].

3.2.2.4 Summary of the statistical information of the carbon-epoxy 2/2twill woven composite

Figure 3.6 shows the warp and weft systematic trends of the out-of-plane centroidcoordinatez, aspect ratioARand areaA and the warp handling effect of the in-plane

50 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

0 2 4 6 8 10−0.1

0

0.1

x [mm]

z [m

m]

2 4 6 8 10−0.1

0

0.1

y [mm]

z [m

m]

0 20 40 60 80 100 120

−0.2

0

0.2

x [mm]

y [m

m]

0 20 40 60 80 100 120

−0.2

0

0.2

y [mm]

x [m

m]

0 2 4 6 8 10

12

14

x [mm]

AR

[−]

2 4 6 8 10

12

14

y [mm]

AR

[−]

0 2 4 6 8 100.3

0.35

0.4

x [mm]

A [m

m²]

2 4 6 8 100.3

0.35

0.4

y [mm]

A [m

m²]

Figure 3.6: Periodic and handling trends of a 2/2 twill woven carbon-epoxy composite.Periodic trends are represented for one unit cell distance,while the handling effect isshown over a distance of ten unit cells.

centroid positiony. The periodic trend is defined over the unit cell dimension, whilethe handling effect is typified for a length of ten unit cells.

The short- and long-range statistical information are described in tables 3.2 and 3.3.No cross-correlation between different tow genuses in the z-direction is contemplatedsince more data should be first collected to identify its significance. Except for the in-plane centroid component, information of the tow path parameters is taken from theshort-range characterisation. Significant other behaviour is observed for the in-planeposition of the short- and long-range characterisation. This demonstrates the need tocomplement tow path information with long-range data when the derived correlationlength exceeds the inspected sample size.

Both the average behaviour as the statistical data in terms of standard deviation andcorrelation lengths are used as to calibrate the simulationof virtual textile compositesin the next step.

STOCHASTIC MULTI-SCALE MODELLING OF THE REINFORCEMENT (STEP 2) 51

Table 3.2: Standard deviation of the tow path parameters from the short-range [104]and long-range characterisation [107], respectively indicated bysr andlr .

σsrx σsr

y σsrz σsr

AR σsrA σsr

θ σlrx σlr

y

[mm] [mm] [mm] [-] [mm2] [mm] [ ◦] [mm]

σwarp - 0.113 0.014 1.774 0.023 0.797 - 0.106σwe f t 0.063 - 0.015 1.440 0.024 0.833 0.615 -

Table 3.3: Correlation lengths of the tow path parameters from the short-range [104]and long-range characterisation [107], respectively indicated bysr and lr . Only forthe in-plane position a cross-correlation length is defined.

ξsrx ξsr

y ξsrz ξsr

AR ξsrA ξsr

θ ξlrx ξlr

y

[mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm]

ξwarpauto - 22.89 1.78 7.26 2.53 4.56 - 114.89ξ

warpcross - - - - - - - 4.49ξ

we f tauto 9.42 - 1.62 5.48 1.01 3.49 52.89 -ξ

we f tcross - - - - - - 13.16 -

3.3 Stochastic multi-scale modelling of the rein-forcement (step 2)

3.3.1 Overview of the modelling strategy

This section describes the methodology to generate random reinforcements thatpossess the same statistical information as quantified by the experiments. Thestatistical information is given as input to simulate tow path variations in both centroidcoordinates and cross-sectional parameters; no rotation of the tow in cross-section isconsidered. Stochastic instances of the tow reinforcementare acquired that are usedto develop random virtual specimens in the third step of the framework (figure 3.1).

3.3.1.1 Methodology

Figure 3.7 presents this modelling approach in detail. As pre-processing step, a 2-D lattice with grid spacingν is constructed with rows representing warp tows and

52 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

columns representing weft tows. The grid length in x- and y-direction is equal tothe experimentally derived unit cell periods in paper I [104]. Next, a stochasticdescription of the tow path is obtained for many specimens bysubsequently adding (i)the systematic or handling trend of each parameter to the appropriate grid locations,(ii) the fluctuations which are not cross-correlated, produced by a Monte CarloMarkov Chain algorithm, and (iii) the cross-correlated deviations generated by a cross-correlated Series Expansion.

Figure 3.7: Stochastic multi-scale modelling approach of the reinforcement.

3.3.1.2 Application to a woven textile composite

Virtual models are generated for the 2/2 twill woven textile spanning a region of ten byten unit cells, or forty warp and weft tows. The model is representative for a ply withina laminate. Each tow is discretised in 320 equidistant points such that the informationof one unit cell is defined over a grid of thirty-two points. This procedure is shownin figure 3.8 for the out-of-plane centroid coordinate. Variations that are not cross-correlated (out-of-plane centroid coordinate, area and aspect ratio) are produced withthe Monte Carlo Markov Chain method, while the cross-correlated in-plane centroidpositions are generated with a cross-correlated K-L SeriesExpansion technique. Atotal of 40000 warp and weft tows, with lengths equal to ten times the unit cell periods,are simulated to create thousand virtual specimens. Comparison with the experimentaltarget values is performed by analysing the histogram representations of the statisticsand evaluating a normalised difference∆ from the target values, defined as

∆ = |υexp− υsim

υexp| · 100% (3.7)

with υ equal to the standard deviation, auto-correlation length or cross-correlationlength.

The modelling procedure is based on several assumptions:

STOCHASTIC MULTI-SCALE MODELLING OF THE REINFORCEMENT (STEP 2) 53

Figure 3.8: Procedure of generating a discretised tow representation, demonstrated forthe out-of-plane centroid position.

• Deviations are assumed to be normally distributed

• Cross-section of a tow is approximated by an ellipse shape

• Short- and long-range deviations can be generated independently from eachother

• In-plane long-range deviations, characterised from a single-ply sample, are alsorepresentative for a multi-ply sample

• Short-range deviations do not have any repetitive long-range effect exceedingthe unit cell size

The first and second assumption are validated by the limited experimental data,however additional experiments should provide a decisive answer. A more realisticgeometrical shape of a tow in cross-section can be acquired using moment analysis(see appendix A of paper I) or using splines [44]. The lack of cross-correlationbetween the short-range parameters and the long-range in-plane centroid permits theindependent generation of short- and long-range deviations. The fourth hypothesissupposes that inter-ply effects have a limited effect on the in-plane centroid path. Thelast assumption refers to the out-of-plane centroid, tow area and tow aspect ratiothat are quantified from a unit cell sample. Although it is notexpected, additionalquantification over longer-range samples should be executed.

3.3.2 Average trends

The periodic and handling trends define the average tow path description of thegenerated random specimens. This tendency of each genus is derived from the

54 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

experimental data as the mean value per grid location of all different tow paths whichare translated to the reference period (see also paper I [104]).

Tow properties of the 2/2 twill woven textile that vary on the short-range (out-of-planecentroid, aspect ratio and area) show a systematic trend which is periodic for eachunit cell, as already shown in figure 3.6. A strong correlation is observed at the cross-over locations, with a similar behaviour for the warp and weft direction. The in-planecentroid fluctuates with non-periodic wavelengths exceeding the unit cell size, againillustrated in figure 3.6. No short-range pattern is considered since no clear trend wasobserved for this parameter in the unit cell sample of paper I[104].

To develop an average reinforcement description of the textile, individual trends firstneed to be interpolated to the equidistant grid locations over which the specimen isdefined and afterwards combined. Periodicity is exploited to construct the repetitivesystematic trend of the short-range parameters along the entire lattice. Figure 3.9displays the average reinforcement of the 2/2 twill woven composite.

0

50

100

0

50

100

−0.20

0.2

x [mm]y [mm]

z [mm]

0 20 40 60 80 100 120

0

20

40

60

80

100

120

x [mm]

y [mm]

Figure 3.9: Different views of the average reinforcement description combining thesystematic and handling trends of all tow path parameters.

3.3.3 Simulation of auto-correlated deviations by the MonteCarlo Markov Chain method

Tow path deviations which are not cross-correlated, neither with each other onneighbouring tows nor with other properties on the same tow,are generated withthe Monte Carlo Markov Chain method for textile structures [6]. The algorithmis implemented in Matlab and uses as input the database of thedeviation values,standard deviation and auto-correlation length of each towcomponent. Correlationinformation is calibrated by only regarding the nearest-neighbour correlation. All

STOCHASTIC MULTI-SCALE MODELLING OF THE REINFORCEMENT (STEP 2) 55

parameter deviations of a single tow are generated in a single step and independentlyfrom other tow parameters and neighbouring tows.

3.3.3.1 Methodology

First, the experimental deviations of the considered parameter ǫ are discretised onan interval with grid spacinga (a is chosen independently from lattice spacingν)and number of intervals 2m+1: {−ma,−(m− 1)a, ...,0, ..., (m− 1)a,ma} that satisfiesthe relationma = 3σǫ . The parameterm must be chosen not too low to avoid highamplitude of spikes in the produced deviations and unrealistic jumps in the tow pathfor subsequent deviation values. Although, a higher parameter valuem will increasethe computational time for a single run. The probabilities according to this discretisedinterval are collected in the distribution vectorPǫ

i for locationi:

Pǫi =

[

p(i)m p(i)

m−1 ... p(i)0 ... p(i)

−m+1 p(i)−m

]T(3.8)

with T denoting the transpose operation. The Markov process generates thedistribution vectorPǫ

i+1 of the particular parameterǫ at the next grid locationi + 1using the probability transition matrixAtrans:

Pǫi+1 = AtransP

ǫi (3.9)

In its initial form, the probability transition matrix is tri-diagonal consisting of threeparametersα, β, γ:

Atrans,0 =

α γ 0 . . 0β′ α γ 0 . .

0 β α . . .

. 0 . . γ 0 .

. . β α ζ 0 .

. 0 β α β 0 .

. 0 ζ α β . .

. 0 γ . . 0 .

. . . α β 0. 0 γ α β′

0 . . 0 γ α

(3.10)

The parameterα is given an arbitrary fixed value of 0.9, while the relative magnitudesof the parametersβ and γ are varied such that the produced standard deviation ofthe fluctuations is set equally to the experimental standarddeviation. Next, thecorrelation information of the nearest neighbour (k=1), deduced from the generatedauto-correlation length, is reproduced by iterative application of the tri-diagonal

56 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

transition matrixAtrans,0. The Markovian procedure is the core computation within theMonte Carlo based scheme which is repeated for all parameters [6], with a different(2m+1) by (2m+1) probability transition matrixAtrans for each tow parameter. Arandom fluctuation is acquired by first choosing an uniform random number whichis afterwards rescaled by a mapping operation onto the probability distribution of thesubjected parameter. This distribution is determined fromthe probability transitionmatrix, based on the deviation value at the previous grid location.

The produced deviations possess high-amplitude long-range wavelength fluctuations,similar to the auto-correlation length of the considered parameter, combined withlow-amplitude short-range wavelength variations. The latter effect can appear asunphysical sharp spikes which are not present in the experimental sample. A post-processing smoothing operation is required to reduce the amount of spikes using anaveraging procedure without affecting the statistics of the deviations too much. Anadapted version of the moving box average is regarded that conserves the standarddeviation [6]. However, when deviations with different signs are present in theaveraging interval, the conventional moving averaging rule must be applied whichdoes not conserve the standard deviation. Deviation valuesare smoothed usinginformation of e.g.±2 neighbouring grid points.

3.3.3.2 Application to a woven textile composite

The Monte Carlo Markov Chain approach is already employed for the generationof the zero-mean centroid deviations of a random unit cell structure in paper III[105] using the short-range data of tables 3.2 and 3.3, including the in-plane centroid.The sample standard deviation and auto-correlation lengthof each tow parameter arereproduced on average.

To simulate deviations for multiple unit cell structures, this approach can be easilyextended with slightly different acquisition parameters for the Markov Chain. Zero-mean fluctuations of the 2/2 twill woven composite are produced for the out-of-planecentroid positions, tow aspect ratio and tow area.

Each tow parameter is produced over a grid with a length of tenunit cells as discussedin section 3.3.1. Given the statistical information of the tow path parameters, itis sufficient to discretise the experimental deviations in twenty pieces (m = 10)with corresponding distribution vector. Smoothing is applied as post-processing stepusing information of±2 neighbouring grid points. The experimental wavelengthof fluctuations are reproduced by the smoothed deviations from simulations, asdemonstrated in figure 3.10 for the warp out-of-plane centroid coordinate. Theexperimental systematic trend is afterwards added with these generated deviations toobtain a stochastic instance of the tow parameter. Figure 3.11 illustrates this process

STOCHASTIC MULTI-SCALE MODELLING OF THE REINFORCEMENT (STEP 2) 57

0 5 10

−0.05

0

0.05

x [mm]

z [mm]

(a) Experiment

0 5 10

−0.05

0

0.05

x [mm]

z [mm]

(b) Simulation

Figure 3.10: Warp out-of-plane centroid deviations trend for 28 warp tows: (a)experimental vs. (b) smoothed deviations obtained from simulations. The simulatedpath is presented for a grid of 16 points to compare with the experimentallycharacterised values.

for the out-of-plane centroid data of of five subsequent unitcells, before and after thesmoothing step.

0 10 20 30 40 50 60

−0.1

−0.05

0

0.05

0.1

0.15

x [mm]

z [mm]

Systematic curve Generated deviations Smoothed deviations

Figure 3.11: The random out-of-plane warp tow path for a length of five unit cellsgenerated with the Markov Chain algorithm. The smoothing operation reduces thespikes present in the path.

A total of 40000 warp and weft tows are simulated by the MarkovChain algorithmto construct thousand long-range specimens. Resemblance with the experimentalstatistical information is demonstrated for the warp tows using the (i) combined dataset, collecting the deviations of all 40000 tows, and the (ii) data sets that representtow data of single unit cells. No additional comparison of the single tow statistics isperformed due to the lack of experimental information of individual tows with a lengthspanning ten unit cells.

58 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

0 2 4 6 8 10 12−0.5

0

0.5

1

distance [mm]

correlation [−]

Target correlation Linear fit data Markov Chain Linear fit simulation

Figure 3.12: Correlation graph showing the experimental and simulated data of thewarp z-centroid coordinate. A linear approximation of the first lag data is performedto deduce the correlation length.

Figure 3.12 presents the auto-correlation graph of the experimental and simulatedwarp z-centroid for the combined data set. The correlation length is approximatedby a linear fit to the nearest point spacings (k ≤ 5) using equation 3.6. Thecalibration ofAtrans with only the nearest neighbour (k = 1) results in an exponentialcorrelation course with minor fluctuations that crosses thezero-correlation boundaryfor large point spacings (k > 5). This is in contrast with the experimental correlationinformation that is typified by data that fluctuates around a trend, which is mainlyattributed to the limited experimental data set size. The simulated standard deviationand correlation length of each produced tow parameter is shown in table 3.4. Standarddeviations of all parameters are simulated with high accuracy (∆ < 0.12%), whilethe correlation length of the tow path parameters of the z-centroid and area showsignificant normalised errors∆ from 50 to 73% from the target data. Especially forthe tow parameters with a correlation length smaller than 5ν, higher correlation lengthsfor the produced deviations are retrieved. Application of smoothing results a generalincrease of normalised error∆, which is relatively larger for the standard deviations.In cases when one or more points in the averaging interval have a different sign, theconventional smoothing operation must be applied. Correlation lengths are slightlyincreased after the smoothing step, which is expected sinceneighbouring values aremade more dependent.

In addition, the produced and experimentally obtained average unit cell standarddeviation and auto-correlation length are compared. Experimental average valuesare determined from the laminated unit cell sample in paper I[104] by takingthe mean of the different standard deviations and auto-correlation lengths per ply.Produced unit cell statistics are computed by identification of tows belonging to a

STOCHASTIC MULTI-SCALE MODELLING OF THE REINFORCEMENT (STEP 2) 59

Table 3.4: Standard deviation and auto-correlation lengthfor the combined data setof warp deviations produced with the Markov Chain algorithm. Smoothed results areindicated bysm.

σz σAR σA ξz ξAR ξA

[mm] [-] [mm2] [mm] [mm] [mm]

Warp genus 0.014 1.772 0.023 3.08 8.77 3.79∆warp 0.11% 0.12% 0.02% 73.16% 20.90% 50.01%Warp genus -sm 0.013 1.755 0.023 3.65 10.86 4.56∆warp− sm 5.84% 1.09% 3.83% 104.79% 49.61% 80.45%

single unit cell in the virtual specimen. A good comparison in average values isobtained for all short-range parameters as presented in table 3.5. The generated unitcell standard deviations and auto-correlation lengths arecentred around the targetvalues, for both warp and weft tows. This is demonstrated forthe warp z-centroidcoordinate in figure 3.13. Similar to the results in paper III[105], smoothing haslimited effect on the unit cell standard deviation, while a high variance is observedfor the non-smoothed and smoothed correlation lengths. Thesensitive calculation ofthe correlation length as linear approximation of the first lags in the auto-correlationgraph (equation 3.6), and the relatively small number of data points (128 per unit cell),results in significant variability. As indicated by table 3.5, the normalised difference∆ increases substantially for the smoothed statistics to around 10% for the standarddeviation and from 50% till 170% for the correlation lengths.

Results from the combined data sets and from the unit cell inspection illustrate thatthe necessity of the smoothing operation affects the results, but the magnitude of thestandard deviations and smoothed correlations remains thesame. The relatively largedifference for the auto-correlation length of the non-smoothedarea deviations canbe explained by the small value and the limited number of datapoints to derive thisstatistical parameter.

3.3.4 Simulation of auto- and cross-correlated deviations bythe cross-correlated Karhunen-Loève Series Expansion

Tow path fluctuations which are correlated along the tow and between neighbouringtows require a simultaneous generation of all its deviations belonging to the samegenus within a single specimen. A methodology developed by Vorechovský [114]follows this strategy where each property is represented bya Gaussian randomfield which is cross-correlated with neighbouring fields. The procedure is generally

60 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

0 0.005 0.01 0.015 0.02 0.0250

2000

4000

6000

8000

10000

σz

Frequency

0 5 10 150

0.5

1

1.5

2

2.5x 10

4

ζz

Fre

qu

en

cy

simulated experiment

(a) Statistics of non-smoothed deviations

0 0.005 0.01 0.015 0.02 0.0250

2000

4000

6000

8000

10000

σz

Fre

qu

en

cy

0 5 10 150

0.5

1

1.5

2

2.5x 10

4

ζz

Fre

qu

en

cy

simulated experiment

(b) Statistics of smoothed deviations

Figure 3.13: The unit cell statistics of the generated out-of-plane warp centroidpositions (a) without and (b) with smoothing. Simulated data achieve the targetstatistics on average. When smoothing is applied, the simulated standard deviationsare slightly affected, while all correlation lengths are increased.

applicable for the modelling of a set of stochastic fields which share an identicalauto-correlation structure and of which the cross-correlation can be defined by across-correlation coefficient. When applied to textile composites, each stochasticfield corresponds to a single tow and cross-correlation can only be reproduced in theorthogonal direction. The algorithm is implemented in Matlab and calibrated with theexperimental standard deviation, the auto- and cross-correlation information.

STOCHASTIC MULTI-SCALE MODELLING OF THE REINFORCEMENT (STEP 2) 61

Table 3.5: Mean of the standard deviation and auto-correlation length of the warpdeviations belonging to single unit cells, produced with the Markov Chain algorithm.Smoothed results are indicated bysm.

< σz > < σAR > < σA > < ξz > < ξAR > < ξA >

[mm] [-] [mm2] [mm] [mm] [mm]

Target [104] 0.014 1.774 0.023 2.27 6.84 1.70Warp tows 0.013 1.569 0.022 2.15 7.00 2.80∆warp 4.29% 11.56% 3.91% 5.29% 2.32% 64.59%Warp tows -sm 0.013 1.540 0.021 3.45 11.82 4.54∆warp− sm 10.71% 13.22% 8.70% 51.78% 72.82% 166.95%

3.3.4.1 Methodology

Deviations are generated that belong to all random fields of one property withina single specimen using a cross-correlated Series Expansion technique. Themethod is a generalisation of the standard K-L expansion [32, 97], which simulatesindependent 1-D orunivariate random fields, to cross-correlated 1-D fields. SeriesExpansion techniques reproduce the auto-correlation structure of a single propertyby expanding the univariate random field into a set of independent random variablesand deterministic spatial functions. Specific for K-L technique, these deterministicfunctions are obtained by spectral decomposition of the auto-correlation functionCauto(x, x′):

Cauto(x, x′) =

∞∑

i=1

λAi φ

Ai (x)φA

i (x′) (3.11)

with λA andφA respectively the eigenvalues and eigenvectors of the auto-correlationstructure, acquired by solving the eigenvalue problem. A single realisation of arandom fieldH(x, ϑ), with ϑ demonstrating the randomness, is represented as [32, 97]:

H(x, ϑ) =∞∑

i=1

√

λAi ηi(ϑ)φA

i (x) (3.12)

with {ηi , i = 1..∞} a set of independent orthonormal random variables.

The framework of Vorechovský [114] simulates cross-correlated random fields usingequation 3.12 as basis. The key intention of the procedure isthat all cross-correlatedfields, within a single specimen, are expanded using the samespectrum of eigenvaluesλA

i and -vectorsφAi (x), but the sets of random variables used for the expansion of

each field are cross-correlated with neighbouring fields. This cross-dependency isintroduced by modal decomposition of the experimentally deduced cross-correlation

62 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

matrix of each parameter. Specific for the generation of tow representations of textilecomposites, subsequent steps need to be followed to acquireNsim realisations ofGaussian cross-correlated fields (thousand specimens) [106, 114], withNG equal tothe number of grid locations to represent a single random field andNF the number offields which are cross-correlated:

1. Perform modal decomposition of the auto-correlation structure and applytruncation:λA

i , φAi (x) with i = 1..Nvar (Nvar ≤ NG)

2. Perform modal decomposition of the cross-correlation structure and applytruncation:λC

i , φCi (x) with i = 1..Nf ,r (Nf ,r ≤ NF)

3. GenerateNr × Nsim (Nr = Nvar · Nf ,r ) Gaussian uncorrelated random variablesηr using Latin Hypercube Sampling (LHS)

4. Construct the block cross-correlation matrixχD = φDλDηr of which theelements are uncorrelated for a single tow and cross-correlated between tows

5. Simulate all individual random fields for a single realisation by

H j(x, ϑ) =Nvar∑

i=1

√

λAi χ

Dj,iφ

Ai (x) (3.13)

Step 1 and 2 are carried out using equation 3.11. In step 3, random variablesηr aregenerated using LHS. This sampling technique is preferred to gain an acceptable levelof accuracy of the statistical description for small numberof simulations. Step 4 iscrucial in the procedure and reproduces the cross-correlation structure between thedifferentNF random fields using the matrix decomposition method. The eigenvaluesλD and -vectorsφD are determined from a squared block cross-correlation matrix D oforderNvarNf ,r . This matrix possesses squared blocks (Nvar·Nvar) of unit matrices on itsdiagonal, while off-diagonal blocks represent the cross-correlation betweeneach twosets of random variables using the eigenvaluesλC and eigenvectorsφC from the cross-correlation matrix. The obtained random matrixχD consists ofj blocks (j = 1..NF),where each block delivers theNvar standard Gaussian uncorrelated random variablesused inH j(x, ϑ) expression to represent a single tow in the virtual specimen, while thevectorsχD

i andχDj are cross-correlated. Discretised representations of random fields

are obtained in the last step using equation 3.13 withχDj,i as random variables. The

random field representationH(x, ϑ) of equation 3.12 is approximated by performingtruncation of the series. The number of K-L termsNvar is determined by ordering theeigenvaluesλA

i in a descending series and considering only theNvar larger eigenvaluesλA

i and corresponding eigenvectorsφAi (x) that capture most of the randomness. All

produced fieldsH j(x, ϑ) will possess the experimental auto- and cross-correlationstructure between neighbouring fields. A more detailed discussion of the methodology,applied to textile composites, is elaborated in paper IV [106].

STOCHASTIC MULTI-SCALE MODELLING OF THE REINFORCEMENT (STEP 2) 63

In addition to this framework, Vorechovský proposes an improvement of theaccuracy of the simulated auto-correlation structure by (i) applying correlation controltechniques [115] and (ii) the anticipation of additional grid locations on the sides foreach field. This first operation encounters the problem of spurious correlation whichis sometimes introduced along the random variablesχD

j,i of a single field. The effectof such techniques needs to be assessed for each new topologyto determine if thiscomputational expensive procedure is required or not. Furthermore, extra side pointscan be considered in the case disturbances are present in thegenerated values at theedges of the field.

3.3.4.2 Application to a woven textile composite

Only the in-plane centroid is cross-correlated for the considered 2/2 twill wovencomposite. In-plane centroid positions within a virtual specimen are modelled asGaussian random fields where tows belonging to the same genusshare an identicalauto-correlation structure and are cross-correlated at the same time with neighbouringtows of the same type. The methodology is repeated for both the warp and weftgenus with forty individual tows per genus in one specimen (NF = 40). Each singlefield, representing one tow, is described over an equidistant grid of forty-one points(NG = 41) that span a total length that is a bit larger than ten timesthe periodiclength. This is required to exclude extrapolation when the data is interpolated overthe lattice of 320 points on which the systematic trend, handling effect and short-range deviations are defined (section 3.3.1). The experimental correlation matricesare isotropic (τ = |x2 − x1|) and constructed by projecting the fitted correlationfunctions onto this grid (see table 3.6). In this approach, it is assumed that forty-onelocations along one tow are sufficient to represent the long-range trend of the in-planecentroid, since no short-range variation of this parameteris observed as discussed insection 3.2.2.2.

Table 3.6: Input correlation functions and applied truncation for simulating the in-plane fluctuations.

Warp tows Weft tows

Auto-correlationCauto σ2wa exp

(

− τξ

warpauto

)

σ2weexp

(

− τ2

ξwe f tauto

2

)

Cross-correlationCcross exp(

− τξ

warpcross

)

exp(

− τ2

ξwe f tcross

2

)

Nvar 33 4Nf ,r 40 13

64 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

Random fieldsH j(x, ϑ) of the in-plane centroids of warp and weft tows are computedusing the truncated series of equation 3.13. After sorting the eigenvalues, only theNvar or Nf ,r largest eigenvalues and corresponding eigenvectors are considered in theprocedure instead of respectivelyNG andNF . An appropriate measure of the capturedvariability is given by the normalised sum (or truncation error) which is fixed tominimum 0.9975:

=

∑Nred

i=1 λi∑N

i=1 λi

≥ 0.9975 (3.14)

with Nred equal toNvar or Nf ,r . The applied truncation for warp and weft deviationsis given in table 3.6. Throughout the procedure, no correlation control techniquesare considered to reduce possible spurious correlation between the random variablesχD

j,i . A sensitivity analysis concluded that this additional operation does not add insignificant accuracy for the resulting statistics. Also no grid locations are foreseen atthe sides since no effect is observed.

In-plane centroid fluctuations are generated for 40000 towsof each genus. The shortwavelength of the experimental warp deviations and long wavelength of the measuredweft variations are reproduced in the simulated data. The correspondence for thewarp tows in-plane deviation trend is demonstrated in figure3.14. As opposed to thedeviations produced with the Markov Chain algorithm in section 3.3.3.2, no additionalsmoothing operation is required. All fluctuations describea tow path which is lessspiked than observed in the experiments. This is not only attributed to the SeriesExpansion technique, but also to the normality assumption of the in-plane deviationswhich diminish the presence of larger spikes in the simulated path.

0 20 40 60 80 100 120−0.4

−0.2

0

0.2

0.4

x [mm]

y [mm]

(a) Experiment

0 20 40 60 80 100 120−0.4

−0.2

0

0.2

0.4

x [mm]

y [mm]

(b) Simulation

Figure 3.14: Warp in-plane centroid deviations trend for 80warp tows: (a)experimental vs. (b) simulated deviations.

The conformity between the experimental and simulated statistics are validated forthe warp tows using data of thousand generated virtual specimens. This information

STOCHASTIC MULTI-SCALE MODELLING OF THE REINFORCEMENT (STEP 2) 65

is investigated for the (i) combined data set, (ii) individual specimens consisting ofhundred unit cells and (iii) individual 1-D fields, each representing one tow with alength of ten unit cells.

Statistics of the combined data set are precisely reproduced with normalised differ-ences∆ for standard deviationσ and correlation lengthsξauto, ξcross which are lessthan 1% (table 3.7). Target correlation functions (table 3.6) and simulated correlationstructures perfectly overlap with small differences for the largest lag spacings, shownin figure 3.15 for the warp auto- and cross-correlation graph.

0 20 40 60 80 100 120−0.5

0

0.5

1

distance [mm]

correlation [−]

Target correlation

Generated correlation

(a) Auto-correlation

0 20 40 60 80 100 120−0.5

0

0.5

1

distance [mm]

correlation [−]

Target correlation

Generated correlation

(b) Cross-correlation

Figure 3.15: Comparison of the warp input and simulated (a) auto-correlation and (b)cross-correlation structure. A perfect fit is obtained withminor fluctuations for thehighest point spacings.

When the statistical data per specimen (<σ>, <ξauto>, <ξcross>) are considered, againgood correspondence is obtained as indicated in table 3.7. The produced auto- andcross-correlation lengths for the warp tows of all thousandreinforcement descriptionsare shown in figure 3.16. All generated correlation lengths have normalised errors∆ which are less than 3.48% for the standard deviation and maximum 1.72% for thecorrelation lengths.

Table 3.7: Standard deviation and correlation lengths for the (i) combined data setof in-plane positions and (ii) mean of the individual specimens, generated with thecross-correlated Series Expansion technique.

σcomb < σspec> ξcombauto < ξ

specauto > ξcomb

cross < ξspeccross>

Warp tows [mm] 0.106 0.103 115.81 114.00 4.54 4.42∆warp 0.09% 3.48% 0.80% 0.78% 1.03% 1.72%

66 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

0 50 100 150 200 2500

50

100

150

200

ξcomb

auto [mm]

Fre

quency

simulated

experiment

(a) Auto-correlation

0 5 10 15 200

50

100

150

200

ξcomb

cross [mm]

Fre

quency

simulated

experiment

(b) Cross-correlation

Figure 3.16: Simulated (a) auto- and (b) cross-correlationlengths of the warp in-planecentroids. The experimental value is simulated on average.

An overview of the individual warp field statistics, in termsof standard deviation andauto-correlation length, are presented in table 3.8. The experimental information ofthe individual tow standard deviation and correlation information is computed fromarranging the data in paper IV [106]. While the generated standard deviation isobtained within 3% error, the produced auto-correlation length only approximatesthe target value with similar order of magnitude. LHS sampling of the independentdeviationsηr ensures a good similarity with the target mean and standard deviation.The difference in target and simulated correlation length is due to the normalityassumption of the deviations distribution and the ideal fitted input correlationfunctions. These provoke the duplication of the statistical information on theindividual tow level.

Table 3.8: Standard deviation and correlation lengths for individual 1-D randomfields, representing the in-plane centroid, produced with the cross-correlated SeriesExpansion technique.

σtarget σ1D ξtargetauto ξ1D

auto

Warp tows -mean[mm] 0.051 0.053 20.69 32.06∆warp,mean - 2.93% - 54.93%

CONSTRUCTION OF VIRTUAL SPECIMENS IN THE WISETEX SOFTWARE (STEP 3) 67

3.4 Construction of virtual specimens in the WiseTex

software (step 3)

The last step of the multi-scale framework (figure 3.1) creates textile specimens withrandom geometry. A virtual representation of the textile composite is obtained inthe WiseTex format [113] by overwriting the original tow path data of a nominalWiseTex model with the stochastic tow path realisations. The latter realisations of thetow reinforcement are acquired by combination of the systematic and handling trendsof section 3.3.2 with the zero-mean deviations generated insections 3.3.3 and 3.3.4.The WiseTex XML-structure is used which permits scripting of local reinforcementinformation without the need of understanding the internalcomputational procedure[60]. Except for the tow path description, new path lengths are computed andorientation vectors are defined that fix each cross-section in space [56, 57, 105]. Inaddition, total specimen dimensions and properties are redefined.

An arbitrary virtual specimen of the 2/2 twill woven composite is presented infigure 3.17 with the warp and weft tows respectively orientedhorizontally andvertically. The typical diagonal appearing in the twill structure also fluctuates fromits straight line, as was observed in the experimental sample of figure 3.5.

3.5 Discussion

The developed virtual representations of the textile geometry possess the samestatistical information as quantified from experimental samples. In this procedure,geometrical scatter is quantified and simulated at the meso-and macro-level. Variationin the fibre properties, in fibre distribution within a singletow or in resin materialare not considered. These properties are expected to have a negligible effect onthe randomness of the effective properties. Although, additional characterisation atthe micro-scale improves the understanding of damage initiation when high-fidelitydamage simulations are performed. This investigation requires additional micro-CTscans to inspect the fibre level, with a voxel size which is at least half of the used voxelsize (6.75µm)3 to characterise the tow level of the 2/2 twill composite.

Deviations generated with the Monte Carlo Markov Chain or cross-correlated SeriesExpansion approximate the experimental statistical information with a differentaccuracy. Generally, the target statistics are better achieved by using the more complexSeries Expansion technique. This method calibrates the generation procedure withfitted correlation functions such that the entire dependency structure is reproduced,instead of only the nearest neighbour information used in the Markov Chain algorithm.The discretisation of the deviations distribution in the Markov Chain also affects the

68 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

Figure 3.17: WiseTex representation of a virtual specimen.The in-plane dimensionof the virtual composite shows the random in-plane positions, while the unit celldescription indicates the variations in tow cross-sectionand out-of-plane centroid.

accuracy due to the mandatory smoothing procedure to reduceunphysical spikes inthe tow path. Application of smoothing significantly increases the error∆. However,the simple concept of a transition matrix approximates the target information with anacceptable error∆ and is thus considered when parameters are not cross-correlated.One must also be careful when statistics are evaluated from data belonging toindividual tows or a single unit cell. These typically smalldata sets exhibit largerdifferences from the target values induced bybatch variation, which is present forlimited data sizes, in addition to thesampling variationwithin a single data set.

The simulated in-plane wavelengths of warp and weft tows arealso represented inthe open gap distribution. These are regions within the one-ply specimen which arefully occupied with resin. Using the same procedures as in paper II [107], gaps arecharacterised from the virtual description using rectangular shapes in terms of gapwidth wgap, heighthgap and areaAgap. Each of these dimensions is Weibull distributedwith the statistical information of the experimental samples and the virtual sampleof figure 3.17 presented in table 3.9. The measured gap widthwgap in the virtualtextile matches these of the experimental samples, while the mean gap heighthgap isroughly doubled. This effect is pursued to the gap areaAgap with a resulting porosityof the entire sample which is more than doubled. The discrepancy is caused by using

DISCUSSION 69

Table 3.9: Sample mean and coefficient of variation of the gaps from the experimentalsamples (region of thirteen by thirteen unit cells) and meanof ten simulated virtualspecimens (region of ten by ten unit cells).

Experimental Experimental Specimensample 1 sample 2 figure 3.18

Ndata 1762 1862 1499wgap - mean [mm] 0.418 0.451 0.448wgap - COV 61.46% 53.49% 43.80%hgap - mean [mm] 0.184 0.177 0.363%hgap - COV 56.85% 59.18% 42.93%Agap - mean [mm2] 0.075 0.080 0.166Agap - COV 84.93% 83.35% 65.74%Porosity full sample 0.56% 0.65% 1.86%

the short-range data instead of the long-range informationto calibrate the simulationof the cross-sectional variations. For the experimental characterisation, differenttechniques are applied to quantify the tow dimensions (sections 3.2.2.2 and 3.2.2.3),resulting in mean tow widths for the short-range sample thatare respectively 0.20 mmand 0.13 mm lower in absolute values for the warp and weft tows. Especially for thesimulated warp tows, oriented horizontally in figure 3.17, this results in more openspace between neighbouring tows and thus an overall larger gap height. The limiteddifference in the mean tow width of the weft tows is cancelled by the high variabilityof this tow centreline, providing a good resemblance of the sample gap width. Alllocations are further used to construct maps that representthe spatial distributionof the gaps. Figure 3.18 presents this map for the virtual textile in figure 3.17,showing the location and the range of area for each individual entity. When tows aretouching, no gap is present and the location remains empty. The gap characterisationis performed from the virtual tow path descriptions in Matlab, which is more accuratethan quantifying these regions in the binarised image of theWiseTex representationsin figure 3.17 due to its limited resolution. Such maps demonstrate that single towfluctuations can span several unit cells but do not persist over the entire length of thetow, as quantified by the correlation information in section3.2.2.4. This spatial gapinformation, dominating the through-thickness permeability, can be applied in flowsimulations to improve the understanding of local flow behaviour.

Besides the description of the heterogeneous structure, virtual specimens can befurther employed to (i) predict its mechanical performance, such as stiffness, in orderto have a quantitative measure of the spatial variation overthe structure, (ii) analysemore precisely the damage progression in textile composites, or (iii) accurately

70 DEVELOPMENT OF A STOCHASTIC MULTI-SCALE MODELLING FRAMEWORK

0 20 40 60 80 100 120

0

20

40

60

80

100

120

x [mm]

y [m

m]

0.005 <Agap

< 0.1 0.1 <Agap

< 0.2 0.2 <Agap

< 0.3 0.3 <Agap

< 0.4 0.4 <Agap

< 0.5 0.5 <Agap

< 0.632

Figure 3.18: Map of gaps distributed over the virtual specimen presented in figure 3.17.The area of the significant gaps [mm2] are indicated and categorised in five intervals.

simulate the resin impregnation of component-size fabrics. The WiseTex formatis directly compatible with tools for micromechanical analysis [42, 56, 76] andpermeability simulations [112]. However, when transforming the WiseTex modelinto a finite element (FE) model, small adaptations are required of the tow pathdescription. As observed in Figure 3.17, limited interpenetration appears in the virtualspecimens. This is a result of the independent generation ofall tow properties, exceptthe in-plane positions. Interpenetration between different genuses is present at thecross-over locations for each specimen. Warp and weft tows already intersect whencombining the systematic and the in-plane handling trends.Interpenetration betweenneighbouring tows of the same genus also occurs, but less frequently, which is mainlycaused by the independent generation of the tow width from the in-plane position.However, a correlation between the two parameters is not found in the experimentaldata. Within these models, tows need to be translated until the interpenetrationis removed, but such that topological rules stay satisfied [57, 81] and statisticalinformation is not altered.

Chapter 4

Concluding remarks andfuture research topics

A major bottleneck in composite design remains the requiredhigh safety marginsto cope with the variability in the mechanical response. Theworst-case scenariois pursued to ensure the robustness of a component, hindering the advantages ofcomposite materials to be fully exploited. Additionally, the raw material andproduction cost are still higher than conventional materials, while the overall qualityis often insufficiently known. A wide industrial application of compositesrequirefaster, cheaper and more accurate production techniques incombination with a betterunderstanding of the materials behaviour.

This dissertation is a contribution towards an improved comprehension of the materialperformance by identifying the inherent uncertainty in thereinforcement geometry. Aroadmap is provided to characterise the scatter in the internal structure of any textilecomposite component and simulate realistic models possessing the same statisticalinformation on average. It serves as an essential tool to relate the variation in themechanical properties with the geometrical characteristics at the lower scale. Themechanical response can be adequately estimated and safetyfactors can be redefinedleading to a more optimal solution in terms of a lower weight and less scrap.

This chapter lists the main conclusions and contributions together with suggestions forfuture research subjects.

71

72 CONCLUDING REMARKS AND FUTURE RESEARCH TOPICS

4.1 Main conclusions

4.1.1 General framework

A generic framework is developed to generate realistic virtual textile specimens.In contrast to the majority of the current state-of-the-art, validated input data areconsidered to reproduce the geometrical variations which are actually present in ahigh-performance composite material. This dissertation presents a first step towardsa systematic modelling approach for textile composites where powerful simulationprocedures are applied in combination with realistic experimental data. Such astrategy is essential to obtain a reliable estimate of the uncertain material properties ofany type of material in a next step [10]. Throughout the methodology, the multi-scalecharacteristics are fully exploited by identifying the uncertain geometrical parametersat both the meso- and macro-level. Furthermore, correlation information is collectedand used to calibrate the advanced numerical procedures. Virtual reinforcements areacquired that vary within the unit cell and between neighbouring unit cells.

The modelling chain is typified by three main steps to characterise and simulatethe scatter at the different levels using a minimum of hypotheses. Concepts andprocedures of this framework are developed for woven composites, but only minormodifications are required for other textile topologies than woven structures. Thevariability of the tow reinforcement is defined using probability theory in terms of thecentroid coordinates and cross-sectional parameters.

In a first step, experimental data are collected on the short-range, i.e. within theunit cell size, and long-range, spanning multiple unit cells, using the suggestedmethodologies in the manuscript. Non-destructive inspection techniques are appliedto measure the fabric architecture in a reliable and efficient way across the compositevolume. Only advanced state-of-the art techniques are considered to quantifygeometrical imperfections in the range of 10-1000 microns with enough detail.To inspect the internal geometry of the reinforcement at themeso-scale, it isrecommended to perform a X-ray micro-CT scan of a unit cell sample using thedeveloped experimental framework in this dissertation. Long-range data requiressamples that are at least one magnitude larger than the unit cell dimensions, which canbe characterised by multiple micro-CT scans, optical imaging or strain field analysis.Which technique is most suitable depends on which path parameters exhibit a long-range fluctuation. Next, the inherent scatter of each tow path parameter in each towdirection is quantified using a probabilistic uncertainty approach. Geometrical scatteris computed in terms of an average trend, standard deviationand correlation lengthby applying the reference period collation method [3]. The data set is maximised bygrouping nominally identical tows in single genus, of whichthe statistical behaviour isanalysed. Mean trends along the tow path can be periodic, according to the systematic

MAIN CONCLUSIONS 73

unit cell description, or non-periodic, caused by the handling of the fabric beforeproduction. Further, depending on the tow parameter, correlation is present alongthe tow path and between neighbouring tows. This statistical information forms acomplete description of the spatial variation inside a textile composite and is used asinput for the numerical modelling procedure.

The second step covers the stochastic multi-scale modelling of the reinforcement.Advanced procedures from stochastic mechanics are appliedand calibrated with theexperimental statistical data to reproduce the measured spatial variability. Randominstances of the tow paths are acquired by combining the deduced systematic andhandling trends with zero-mean fluctuations possessing theexperimental standarddeviation and correlation lengths on average. Two different statistical techniques areregarded to provide an adequate description of the internalgeometry for any positionalong the tow path. Auto-correlated deviations are produced by the Monte CarloMarkov Chain for textile structures, which has already proven its accuracy for aceramic textile [6]. This Markov approach represents each uncertain parameter bya set of discretised points by marching systematically along the tow path using aprobability transition matrix. All parameter deviations of a single tow are generatedin a single step and independently from other tow parametersand neighbouringtows. An additional post-processing smoothing operation is however required toremove unphysical spikes in the generated tow path. Both auto- and cross-correlateduncertain quantities are generated using the cross-correlated Series Expansion methodof Vorechovský [114]. This procedure is generally applicable for the modelling of aset of random fields which share an identical auto-correlation structure and of whichthe cross-correlation can be defined by a cross-correlationcoefficient. When applied totextile composites, each uncertain tow path quantity alonga single tow is representedby a 1-D random field. An adapted Karhunen-Loève Series Expansion method withcross-correlated random variables reproduces the dependency structure along the towand between neighbouring tows. All fluctuations for all towswithin a specimen aresimultaneously produced.

Virtual composite specimens with a random fibre architecture are created in thethird step by an intrusive approach. Stochastic models are developed in theWiseTex software [113] using the XML-structure. This format permits scriptingof local reinforcement information without the need of understanding the internalcomputational procedure. First, a nominal WiseTex model ofthe same material isconstructed based on the manufacturer’s data. Next, original tow path descriptions ofthis idealised model are overwritten with realistic tow representations obtained fromthe previous step, together with a redefinition of the path length, orientation vectorsand total specimen dimensions and properties. Nominal dataabout the fibre mechanicsand matrix properties are not altered in this virtual model.

The result of this work, in particular the availability of realistic representations of thematerial data with spatial scatter, can be used for studies on the reliability and quality

74 CONCLUDING REMARKS AND FUTURE RESEARCH TOPICS

of composites.

4.1.2 Application to a woven textile composite

The entire methodology is demonstrated for a carbon-epoxy 2/2 twill wovencomposite produced by RTM. The geometrical variability of the subject high-performance textile is found to be significant with substantial differences for warpand weft direction attributed to the manufacturing processof the weave. Duringthe weaving process, warp tows are kept in tension, while weft tows are inserted byhandles. Short-range data acquired from a seven-ply unit cell sample by a micro-CT scan demonstrate a different statistical behaviour of both tow directions. Thein-plane centroid positions show the largest standard deviation of all tow propertieswith a correlation length exceeding the unit cell dimensions. Therefore, additionaldata are gathered by optical imaging of the in-plane dimension of multiple unit cellone-ply composites to inspect the in-plane positions over alarger distance. The mainobservations from the experimental campaign are summarised as follows:

• All tow characteristics, with exception of the in-plane centroid position, exhibitsystematic periodic trends that are correlated with the cross-over locationsinside the fabric. The mean trend of the in-plane centroid position does notpossess any periodic pattern but reflects the handling effect of the fabric beforeproduction.

• The out-of-plane centroid and tow cross-sectional properties vary within theunit cell dimensions with similar statistics for warp and weft direction. Thecorrelation lengths of the out-of-plane centroid coordinate and area are smallerthan the cross-over spacing in the weave. This is expected toarise from the intra-ply cross-over effects and inter-ply nesting effects inside the laminate. Aspectratio and orientation of the tow have correlation lengths within the unit cell size.Especially for the aspect ratio, the slightly higher correlation value is possiblyattributed to the original stacking of the tows on the pulleyduring the unwindingstep of the weaving process.

• The in-plane centreline is subject to the largest variationwith an auto- and cross-correlation length exceeding the unit cell dimensions. Significant differences areobserved for the warp and weft tows, with the weft fluctuations persisting withsignificantly longer wavelengths compared to the warp deviations. Warp towspossess fewer in-plane movement with a small standard deviation, indicated bythe large auto-correlation length that endures for more than ten unit cells. Across-correlation effect is observed that is limited to a few tows. Weft towsare less tensioned during manufacturing, translating to a much larger standarddeviation and a much smaller auto-correlation length that is roughly half of

RECOMMENDATIONS FOR FUTURE RESEARCH 75

the warp tows. This tow direction exhibits an appearing bundling behaviour ofneighbouring tows that is quantified by the cross-correlation length which spansaround five tows.

In a next step, virtual reinforcements are simulated that span a region of ten by tenunit cells and are representative for a ply within a laminate. Deviations of the out-of-plane centroid coordinate, aspect ratio and area are produced using the MarkovChain method, while the cross-correlated in-plane centroid position is generated bythe Series Expansion procedure. A good comparison in terms of wavelengths andextreme values is obtained between the experimental and simulated deviations trendsfor all properties. Further, all simulated tow deviations achieve the target statisticson average. The conformity of the statistics with the experimental data is performedat the level of full specimens, unit cells and individual tows. Standard deviations arereproduced with a maximum normalised error from the target values of less than 15%,while the difference from the target values for the correlation lengths ofseveral towparameters exceed the 100% normalised error. However, the same order of magnitudeis obtained for these characteristic lengths. Generally, ahigher accuracy is observedfor the Series Expansion technique. In the latter method theentire dependencystructure is reproduced, instead of only the nearest neighbour information used inthe Markov Chain algorithm. Deviations generated with the Markov Chain will alsohave much higher normalised errors for the correlation length when smoothing isapplied. However, the simple concept of a transition matrixapproximates the targetinformation with an acceptable error.

By overwriting the nominal tow path description of an idealised WiseTex model,virtual specimens are acquired that possess the target statistical information of thephysical samples on average. These realistic textile representations can be furtherexploited to evaluate the mechanical performance, performfailure predictions basedon local stress-strain behaviour or simulate the impregnation of dry fabrics.

4.2 Recommendations for future research

The proposed framework delivers virtual textile specimensthat are statisticallyidentical to real textile composites, measured in a first step. It is a crucial instrumentto improve the reliability of current analyses of compositecomponents and to assessthe quality of textile composites. Advanced techniques forthe analysis of spatialvariability are systematically applied to simulate the reinforcement of compositematerials in combination with realistic data. New experimental methodologies aredeveloped and some generation techniques have not yet been applied in the fieldof composites, revealing the potential of the chosen procedures in future textilemodelling. Still, this conducted research in realistic geometry generation is, of course,

76 CONCLUDING REMARKS AND FUTURE RESEARCH TOPICS

only a small contribution in the general progression towards high-fidelity simulationsof composite structures. Many challenges still remain and can be tackled in futureresearch tracks. The author distinguished three main research topics for futureconsideration: (i) model improvement and extension, (ii) propagation of geometricaluncertainty to the macroscopic performance and (iii) application of the framework tosimulate uncertainty in other composite types and properties.

4.2.1 Model improvement and extension

In this dissertation, textile reinforcements are adequately simulated with high accuracyand introduced in existing modelling schemes to develop representative randomvirtual specimens. However, someadjustments to the modelling strategycan beconsidered to improve the applicability of the model. Currently, a small amount ofinterpenetration appears between tows of the same type and different type which mustbe removed when transforming the WiseTex model into a FE model. Methods basedon contact algorithms [98] or topological rules [57, 81] canbe assessed to eliminatethis undesired feature in order to evaluate the performancein a FE analysis.

Numerical random representations of textile composites can further extended by alsoconsideringvariation at the micro-scale: scatter in the constituents and the fibredistribution within a tow. Uncertainty in the constituentsshould be investigated forthe resin viscosity and the fibre mechanics. To inspect the fibre distribution, additionalmicro-CT scans need to be performed with a suggested voxel size of at least half thefibre diameter. These small scale variations are expected tobe almost negligible forthe effective properties, but essential to simulate damage initiation and progression[14, 15].

4.2.2 Propagation of geometrical uncertainty to the macro-scopic performance

Random specimens form a directly link between the variationin the macroscopicmechanical properties and the geometrical variability at the lower scale. These virtualtextiles can be applied to investigate theeffect of the intrinsic scatter in the textilestructure on the macroscopic mechanical behaviour. The accurate description ofgeometrical scatter over the extent of the composite allowsto (i) predict its mechanicalperformance, such as stiffness, to define a quantitative measure of the spatial variationover the structure, (ii) perform damage simulations with a higher fidelity, or (iii)precisely simulate the resin impregnation of component-size fabrics. Either the entirerandom model can be considered, or only a few tow properties can be made variable toperform a sensitivity analysis. The random specimens in theWiseTex format, without

RECOMMENDATIONS FOR FUTURE RESEARCH 77

any adjustments, are directly compatible with tools for micromechanical analysissuch as stiffness evaluation [42, 56, 76] and permeability simulations [112]. Othersimulations, especially damage investigation, require the FE representation of therandom specimens.

Once the variation in the mechanical behaviour is determined, random field expres-sions of the uncertain properties can be established using the K-L expansion. Thesecan be subsequently introduced in reliability analyses of load carrying structuressuch as the spectral stochastic finite element method (SSFEM) to propagate theuncertainty of material properties to the overall composite responseof interest.Examples of this procedure are already completed for the case laminated compositeplates [11], fibre metal laminates [13], UC composite panelsin [71]. Besides thevariation in the material properties, uncertainty in overall geometry and loading canalso be regarded.

The representative textile models also enable toestablish a proper relationshipbetween the different mechanical properties, since material parameters at themacro-level depend on the fundamental geometrical and material information atthe meso-level. Different properties at the macro-level thus rely on the samebasic information, suggesting that a correlation should exist between them. Unlikeexperimental characterisation, this multi-scale approach is capable of correctlyrepresenting this relation.

4.2.3 Application of the framework to simulate uncertainty inother composite types and properties

The proposed framework is developed for textile compositeswith woven topology, butthe examined procedures are generally applicable tocharacterise and simulate anyother fabric structure considering only minor adjustments. Fibre architectures of e.g.UD, braids and non-crimp fabrics can be statistically simulated without any restrictionin fibre or matrix material as long as the assumptions for the modelling step are valid.

Especially fornatural fibre composites there is still a substantial lack of treatinguncertainty in the geometrical architecture. Initiativesare already undertaken toemploy the developed procedures to have a more realistic description of the internalstructure. In this type of composites, the cross-sectionalshape and size, orientation ofthe fibre yarns and the fibre length are subjected to the largest scatter.

Besides adequately modelling the tow reinforcement of a variety of composites, theadvanced characterisation and generation techniques can also be used in awidefield of engineering developments and applications to quantify and represent thespatial variation. Any feature that varies over the extent of a component can benefit

78 CONCLUDING REMARKS AND FUTURE RESEARCH TOPICS

from the conducted work, such as e.g. the thickness over a composite plate or thesubstantial variation in mechanical properties in additive manufacturing.

Part II

Key publications

79

Paper I - Stochasticframework for quantifyingthe geometrical variability oflaminated textile compositesusing micro-computedtomography

80

81

Published in Composites Part A 44 (2013) 122-1311

Stochastic framework for quantifying the geometrical variability of laminatedtextile composites using micro-computed tomography

Andy Vanaerschota, Brian N. Coxb, Stepan V. Lomovc and Dirk Vandepittea

aKatholieke Universiteit Leuven, Department of MechanicalEngineering, Leuven, BelgiumbTeledyne Scientific Co. LLC, Thousand Oaks, CA, USA

cKatholieke Universiteit Leuven, Department of Metallurgyand Materials Engineering, Leuven, Belgium

Abstract

Reference period collation, a method recently proposed foranalysing the stochasticnature of a nominally periodic textile reinforcement, is extended to allow applicationto a laminate of stacked, nested plies. The method decomposes the characteristics ofthe fibre reinforcement into non-stochastic periodic (or systematic) trends and non-periodic stochastic fluctuations. The stochastic character of every tow is analysedin terms of the centroid position, aspect ratio, area, and orientation of its cross-section. The collation method is tested using X-ray micro-computed tomography datafor a seven-ply 2/2 twill woven carbon-epoxy composite produced by resin transfermoulding.

All tow characteristics, with exception of the in-plane centroid position, exhibitsystematic trends that show only mild differences between plies. They correlate moststrongly with cross-over points within a single ply. Of the various parameters, the in-plane centroid position is subject to the largest tow-to-tow variability, with deviationscorrelated over distances exceeding the unit cell size.

1Symbols can differ with original publication to preserve consistency in theentire manuscript.

82 Paper I

1 Introduction

The variability of reinforced polymer composites can be substantial, especiallywhen the reinforcement is a textile product. With the sources of variability poorlyunderstood, methods are lacking for its control or the accurate prediction of the effectsof variability on the quality and reliability of composite structures. This is particularlyimportant for safety critical components used in aerospaceand for lifetime estimationof various applications, e.g. wind turbine blades.

Uncertainty quantification for composite materials is getting more attention lately[10, 47, 95]. The definitions of uncertainty and variabilityare taken from Oberkampfet al.[74] and Moens et al.[70]. The magnitude of variability in performance,resulting from imperfections generated at various stages of production, has beenpartially reported with contributions considering elastic mechanical properties [22],formability [94], permeability [27, 28] and damage propagation [26, 120]. However,the inadequacy of experimental data [10] and realistic numerical models of compositesfor establishing Virtual Tests [14] are still obstacles. Almost all published workdeals with randomness of local properties, without considering the correlation ofproperties between different positions in a component. Yet, experimental work [67]has already proven that spatial variation must be taken intoaccount to achieve acorrect representation of the material. Significant advances in realistic materialmodelling can only be achieved by [10] (i) collecting enoughexperimental data onthe spatially correlated random fluctuations of uncertain material parameters for shortand long range deviations, and (ii) forming a probabilisticlink between macroscopicmechanical properties and the lower scale mechanical characteristics. This paperreports and analyses data for short range deviations, i.e. deviations correlated overdistances less than or compared to the size of the unit cell, while the collection oflong range deviations and the relation of microstructural deviations to macroscopicproperties will be addressed in future publications.

The spatial geometrical variability over scales of 1-15 mm is characterised usingthree-dimensional (3-D) images acquired via X-ray microfocus computed tomography(micro-CT). This non-destructive technique is nowadays used in composite materialsto characterise the geometry for modelling purposes [2, 89]or to analyse damageinitiation [87, 93, 117]. The use of micro-CT for quantifying geometrical variability[3, 22] offers advantages over the optical imaging processes when spatial informationneeds to be analysed over the extent of a structure. Unlike optical imaging, whichyields data on discrete, separated cross-sections [55, 76], micro-CT images everyvoxel throughout a scanned volume.

Variability quantification of geometry using micro-CT has already been performedwith different approaches but with the same objective: quantifying the scatterin geometry introduced during the textile manufacturing. Desplentere et al.[22]

STOCHASTIC FRAMEWORK 83

quantified variability in the structure of four different 3-D glass warp-interlacedfabrics. Local tow parameters such as tow dimension and spacing are statisticallyquantified in terms of the mean value and standard deviation.The measuresare performed from cross-sectional images taken at an arbitrary location in thetextile. Bale et al.[3] statistically quantified the entiretow path by introducing thereference period collationprocedure which maximises the information derived froma small specimen and exploits the nominal periodicity of thetextile. This methodwas introduced to distinguish non-stochastic periodic variations from non-periodicstochastic deviations. Two ceramic textile samples are investigated using synchrotronmicro-CT to quantify the tow centroid locations stochasticity with variations in theaspect ratio, area and orientation of tow cross-sections interms of its standarddeviation and correlation length. This correlation lengthis essential to determine thespatial dependencies between parameters on the same tow andbetween tows.

The current paper extends the characterisation procedure of Bale et al.[3] to polymertextiles with different topology and arranged in a laminate. The objectives are to(i) present a stochastic framework to derive a spatially statistical description of anarbitrary textile composite using micro-CT at the laboratory scale, (ii) apply thismethodology to a seven ply carbon-epoxy 2/2 twill woven textile sample producedby resin transfer moulding (RTM) and (iii) discuss the variability in geometry withinone ply and between plies.

2 Stochastic Framework

A statistical description of the internal geometry of a textile composite is experimen-tally acquired following three general steps:

1. Perform micro-CT scan of a composite specimen

(a) Preparation of samples consisting of several unit cells(at least one).

(b) Definition of the radiation source parameters.

(c) Acquisition of reconstructed 3-D volume of the composite specimen.

2. Data processing and analysis of cross-sections

(a) Image segmentation (identification of distinct material domains) for anumber of equally spaced cross-sections.

(b) Ellipse fitting to each tow in transverse section.

(c) Transformation of data into a new coordinate system withcorrection forsample alignment during micro-CT scan.

84 Paper I

3. Statistical characterisation of the tow paths within each ply

(a) Definition of a reference period.

(b) Determination of the systematic trend for each tow component.

(c) Determination of stochastic deviations from this systematic trend withdefinition and quantification of standard deviation and correlation length.

The sample material does not require any special treatment for the CT analysis, whichis an important advantage of this framework. Samples cut from different plates areso recorded, because variability can arise both within one plate,intra-plate variabilityand also between plates,inter-plate variability[108].

3 Experimental procedure (step 1)

The dimension of each sample is chosen as a trade-off, given the fixed total numberof voxels, between the desired degree of detail in the images(smaller voxel sizepreferred) and the number of tows that can be analysed from one scan (larger voxelsize preferred with specimen volume at least one unit cell).The sample material in thiswork is a seven-ply polymer textile composite. Each ply consists of a twill 2/2 wovencarbon fabric from Hexcel (G0986 injectex) [40], with arealdensity 285 g/m2 andnominal unit cell dimensions of 11.4 by 11.4 mm. The seven plydry reinforcementis impregnated with epoxy resin in a RTM process, achieving afibre volume fractionof 55.3%. Figure I1 shows the WiseTex virtual model of this fabric. One unit cellconsists of 4 warp tows which are all equally spaced byλy/4 over the period in they-directionλy, and 4 weft tows which are equally spaced with periodλx/4 in the x-directionλx.

Figure I1: WiseTex model of a 2/2 twill woven fabric. The coordinate axis system ischosen to have the x-axis and y-axis respectively parallel to the warp tows and wefttows.

DATA PROCESSING AND ANALYSIS OF CROSS-SECTIONS (STEP 2) 85

The X-ray source parameters (voltage and current of the beam) and acquisitionparameters of the micro-CT inspection technique are chosenbased on differences inabsorption of X-rays through the material [50]. One sample of 12.5 by 12.5 mm withabove material properties is mounted in a GE Nanotom. The equipment has a 180kV/15W nanofocus X-ray tube and can obtain minimum voxel sizes of 500 nm. Forthe current type of material with the given sample size, X-ray source parameters areset to 33 kV and 295µA to acquire high quality images with voxel size of (6.75µm)3.

4 Data processing and analysis of cross-sections

(step 2)

4.1 Image segmentation and ellipse fitting

The micro-CT scan provides a 3-D volume representation of a composite sample.Equally spaced 2-D slices are extracted from this volume in each weave direction.The warp tow cross-sections are characterised from slices normal to the weft directionand weft tow cross-sections from slices normal to the warp direction.

The steps used in image analysis are dictated partly by the imperfect quality of images.The carbon fibres and epoxy matrix material have a similar material density and tendto give poor contrast between the tows of different fibre orientation if the image voxelsize is not small compared to the fibre diameter. Contrast enhancement techniques areavailable for characterising carbon-epoxy materials in micro-CT equipment [23, 24],but require additives used to coat tows during production. The image quality of thecurrent sample is optimised by subsequent post-processingsteps in the image analysissoftware VG Studio MAX 2.1 to reduce the noise in the images with median filters,enhance contrast with filters, and accentuate edges with edge-preserving smoothingalgorithms. A processed cross-sectional image is presented in figure I2(a), where themarked areas indicate locations where the boundary betweenwarp and weft tows arestill hard to distinguish. Manual input for image segmentation is required, whichlimits the amount of slices used to analyse the internal structure of the sample in areasonable time (set to around 6 hours). Nineteen slices were analysed for both warpand weft directions, yielding information at 0.75 mm intervals. Given the tow widths,each tow cross-over contact area is represented in each direction by three data slices.Image segmentation is performed separately for warp and weft slices with the Avizov6.1 software2. Figure I2(b) shows perfect comparison between the binarised cross-sections obtained from the image segmentation process and the actual cross-sectionalshape as present on the processed digital image.

2software licence Lawrence Berkeley National Laboratory, Berkeley, CA, USA

86 Paper I

(a) 2-D digital cross-section in warp direction from the 3-Dvirtual volume

(b) Detail image of right-hand side

Figure I2: Digital image of a cross-section obtained by micro-CT. (a) Fully processed2-D cross-section in warp direction with areas indicated where the boundary betweenwarp and weft tows is blur. Ellipse shapes are fitted to the warp cross-sections in ply 3.(b) Enlarged image of a part of the cross-section with fitted binarised cross-sectionalshapes obtained from the image segmentation. The fitting shows perfect comparisonwith the actual shapes.

The geometrical characteristics of the tows are determinedby fitting ellipses to theimaged tow cross-sections using the freeware ImageJ. The shapes of tow cross-sections can differ from the most often assumed elliptical contour, as discussed forexample in [41, 47]. Nevertheless is the assumption of an ellipse shape acceptable toacquire the geometrical data for the current topology as demonstrated in [76], wherelaminates with the same reinforcement were studied. A more general approach toidentify the tow cross-sections for all kinds of topologiesappear in Appendix A. Theellipse fitting yields the tow path centroid coordinates (x, y, z), tow aspect ratioAR,tow areaA and tow orientationθ of a tow’s cross-section on each image slice (seefigure I2). These data collected for each image slice in a series with appropriatespacing adequately characterise the locus and shape of the tow over its entire length.

4.2 Definition coordinate system and sample alignment cor-rection

The tow data obtained from ellipse fitting are converted to the global materialcoordinate system of figure I1, with the x-axis as warp direction and y-axis as weftdirection. The data set shows that the average warp and weft orientations are not

DATA PROCESSING AND ANALYSIS OF CROSS-SECTIONS (STEP 2) 87

orthogonal. This is caused by the presence of limited shear,which is expected to besignificant for this topology due to the deformability of thedry fabric.

An optimisation technique is used to determine the misalignment anglesαs,βs,γs

of the sample when mounted in the GE Nanotom for each plys and at the sametime derive the periods of the ply unit cellsλx,λy (see figure I1). First an initialguess of misalignment angle (1◦) and periods, equal to the manufacturer’s data, istaken. Second, all tow data will be transformed into the candidate axis systemusing the appropriate rotation matrix and afterwards translated: warp tows are tobe superimposed on the first warp tow and similar for weft. Thelast step consistsin defining the root mean square deviation (RMSD) of the superimposed tows fromtheir mean tow axis coordinate. This RMSD is minimised in thealgorithm. For themisalignment angle in the xy-plane, the procedure is repeated separately for warp andweft direction to define the shear angle asψs = abs(αs

x − αsy).

Table I1: Geometrical characteristics of the plies.

λx [mm] λy [mm] ψ [◦] [◦] tp [mm] ts,b [mm] ts,e [mm]

Ply 1 11.32 12.02 87.50 0 0.40 1.76 2.16Ply 2 11.42 10.91 89.71 -0.79 0.43 1.48 1.91Ply 3 11.65 11.56 91.60 -1.45 0.44 1.12 1.56Ply 4 11.38 11.74 88.12 -0.04 0.45 0.87 1.32Ply 5 11.54 11.39 90.66 -1.61 0.49 0.53 1.02Ply 6 11.54 11.22 92.14 -3.55 0.41 0.28 0.69Ply 7 11.98 11.52 92.86 -2.23 0.42 0.00 0.42

Mean 11.55 11.48 90.37 -1.38 0.43 - -σ 0.22 0.36 2.026 1.260 0.028 - -COV [%] 1.92 3.11 2.242 - 6.44 - -

The tow data set is afterwards transformed for each individual ply and tow typeusing the rotation matrixR(s,wa) = Rx(γs)Ry(βs)Rz(αs

x) for warp tows andR(s,we) =

Rx(γs)Ry(βs)Rz(αsy) for weft tows. From this information, the ply orientations

is derived by comparingαsx, with ply 1 as a reference, and the shear angleψs by

comparingαsx with αs

y. The results are summarised in table I1. The periodic lengthof the unit cell is similar for warp and weft direction as expected for this balancedfabric, with the weft exhibiting a slightly higher coefficient of variation (COV). Theshear angle and ply angle, defined as the average orientationof the warp tows in eachply, exhibit low variations between the plies. The thickness of each ply is definedas the difference between the lowest and highest z-coordinates of all tow boundarieswithin each ply; these are defined as respectively the startts,b and end positionts,e.

88 Paper I

The data show that tows of neighbouring plies can possess overlapping z-coordinatesat different locations in the xy-plane, due to the nesting of the plies in the laminateas already observed in figure I2. A quantification of this nesting can be obtainedby summing the individual ply thicknesses, as computed above and dividing this bythe laminate thickness. The nesting value is 1.40, which is higher than previouslyobserved for a nine-ply composite with 12K carbon fibre in thesame topologyproduced by autoclave [76]. However, for both materials, a different definition ofthe individual ply thickness is used. In [76] the nominal plythickness is used, whichis equal to twice the tow thickness. In the current work, a different definition is appliedbased on actual ply images which is more credible. Another cause of this discrepancyin nesting value could also be attributed to a different compaction pressure in RTMand autoclave production. Since no information about the compaction pressure isavailable for RTM, this can not be investigated. Nesting will have an effect on thestatistical behaviour of the tow path.

5 Statistical characterisation of the tow paths (step3)

The statistical analysis uses the method ofreference period collationto collect data fortows assigned to the samegenus, i.e., tows that should be identical given the nominalperiodicity of the textile [3]. Considering the weaving process for the particular 2/2twill woven fabric, the warp tows can be represented by one tow genus and the wefttows by another tow genus. Distinct genuses are introduced for warp or weft towsin different plies, because there is no expected spatial relationship between tows indifferent plies.

The tow data set is presented by{ρ( j,t,s)sl , z( j,t,s)

sl ,AR( j,t,s)sl ,A( j,t,s)

sl , θ( j,t,s)sl } with ρ = y for

warp tows andx for weft tows. Data are taken from each slicesl=1...19, tow labelj=1...4 within the tow genus sett= warp or weft and plys=1...7. A statisticalrepresentation of the tow paths is obtained by partitioningeach of the characteristicsinto periodic systematic variations and non-periodic stochastic deviations. Stochasticconcepts of standard deviation and correlation parameterswill quantify the randomdeviations from the systematic part to fully characterise the geometry. This procedureis based on the methodology used by Bale et al.[3] for a 3-D interlock ceramiccomposite.

STATISTICAL CHARACTERISATION OF THE TOW PATHS (STEP 3) 89

5.1 Definition of reference period

A reference periodλi for each tow genus (warp or weft) is selected that coincideswith the centroid positions of the first warp or weft tow for the calculated periodicwavelength. A uniform grid is defined over this reference period with sixteen gridpoints labelledi=1...16. The grid spacingδ, equal toλi/16, is chosen to be similar tothe spacing of the image slices described above, but need notequal it. The procedure isshown in figure I3. It is assumed that the tow at grid point 17 isstatistically equivalentto the tow at grid point 1. All tows are discretised with an equal number of grid pointsand the grid spacing of their tow genus. The decomposition ofeach tow genus can berepresented, forε representing one of the five parameters{ζ, z,AR,A, θ}, as:

ε( j,t,s)i =< ε

( j,t,s)i > +ǫ

( j,t,s)i (I1)

with ǫ( j,t,s)i the zero-mean deviation from the systematic value< ε

( j,t,s)i > at locationi

along the towj of tow genust in ply s.

Figure I3: Extraction of nineteen equally spaced slices from the reconstructed 3-Dvolume. The definition of a reference period is presented forthe periodic length ofone unit cell.

The process of reference period collation requires an interpolation scheme to calculatethe parameter values at each grid point, because their locations do not coincidewith those of the nineteen image slices (see figure I3). A piecewise cubic Hermiteinterpolating polynomial is chosen for this purpose. The interpolating polynomialsare also used to determine the cross-over locations where warp and weft tows haveequal centroid coordinates in the xy-plane.

90 Paper I

5.2 Determination of the systematic trend for each towparameter

The systematic variations of parameters describe the average trends of each genusrepresented by the mean value of any parameter at each grid point in the referenceperiod. The systematic variation is calculated using all available data. Tows with equalgenus classification are translated to the reference periodusing the vector (kλx/4~ex +

lλy/4~ey) with k andl integers. Although each tow length in the data images is largerthan the reference period (see figure I3), all data are used, by translating the grid pointslying outside one period to the corresponding reference point within the referenceperiod. The systematic variationΞ(t,z)

i of each tow genus is obtained by:

Ξ(t,s)i =

1Ni

∑

j

[ε( j,t,s)i − (k

λx

4~ex + l

λy

4~ey)] (I2)

with ~ex and ~ey the unit vectors andNi the number of data points belonging to thereference pointi. The systematic variation of each towj at the location occupied bythat tow in the textile can be generated subsequently by applying a similar translationoperation to the systematic trendΞ(t,s)

i for the reference period:

< ε( j,t,s)i >= Ξ

(t,s)i + (k

λx

4~ex + l

λy

4~ey) (I3)

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

x [mm]

z [m

m]

tow 1 tow 2 tow 3 tow 4 systematic curve

0 2 4 6 8 10−0.2

−0.1

0

0.1

0.2

x [mm]

y [m

m]

Figure I4: Systematic curves for the warp out-of-plane and in-plane centroid paths ofply 1. The systematic value at each location is determined asthe mean value of alldata points at that location.

STATISTICAL CHARACTERISATION OF THE TOW PATHS (STEP 3) 91

5.3 Determination of the statistical properties of the towparameters

With the systematic trends predicted everywhere in the sample, the deviations ofparameters from the systematic values evaluated at all points can be used to (i)determine the standard deviation of the deviations for any parameter and tow genusand (ii) evaluate correlations between the deviations of parameter values within onetow.

The standard deviation is calculated for each parameter combining the data for all gridlocationsi and for each towj belonging to the genust

σ(t,s)ǫ =

√

∑

i, j(ǫ( j,t,s)i )2

N − 1(I4)

with N =∑

i Ni whereNi is equal to the number of data points for grid pointi. Thestandard deviation is assumed to be independent of the grid point location

Correlations along a tow are summarised by evaluating Pearson’s correlation parame-ter for pairs of data{ǫ( j,t,s)

i ǫ( j,t,s)i+k } taken from different locations on the same tow spaced

by kδ. The autocorrelation coefficient of a parameterǫ( j,t,s) is determined by

C( j,t,s)(k) =

∑n−ki=1 ǫ

( j,t,s)i ǫ

( j,t,s)i+k

√

∑n−ki=1 (ǫ( j,t,s)

i )2√

∑n−ki=1 (ǫ( j,t,s)

i+k )2

(I5)

with n the number of pairs andk an integer value. The autocorrelation parameteris used to define the correlation length, which is a measure ofthe range over whichfluctuations at a certain position still have an influence on the fluctuations at anotherlocation. For each tow genus, this parameter is determined by fitting a straight line tothe variation of the correlation parameter, using data for small k:

C( j,t,s)(k) = 1− kδ/ξ( j,t,s) (I6)

with ξ( j,t,s) the correlation length. This linear fit uses only the first fivedata points(k ≤ 5). Higher spacingsk > 5 are populated by much smaller data sets and also tendto lie beyond the zone of significant correlation, and are therefore inappropriate forevaluating the correlation length.

92 Paper I

6 Results and comparison of statistical analysis

between plies

6.1 The choice of genus

In collating data from different plies, two definitions of genus are useful. The first,single-ply genus, defines one distinct genus for tows of type warp and one for tows oftype weft for each ply in the laminate. The second definition,all-ply genus, combinesall the tows of type warp from all plies into a single genus andall the tows of typeweft into a second genus. Both definitions of genus are used indata analysis, thefirst allowing inter-ply variations to be studied in detail,the second maximising thedata set available for analysing any tow characteristic. But the choice of definitionis ultimately not arbitrary: the validity of combining towsfrom different plies intosingle genuses depends on whether differences are found between plies when data areanalysed.

6.2 Results for systematic trends

Systematic curves were first determined using-ply genuses.Figure I4 showsexemplary systematic curves for the centroid coordinates of the warp genus in ply 1.The four warp tows are translated to the reference period interval. The data belongingto grid points that would lie outside of the period are translated to the correspondingreference point within the reference period, as discussed in section 5.2.

Table I2: Differences between the systematic curves for single-ply genuses and itscorresponding all-ply genus.

ςsx [mm] ςs

y [mm] ςsz [mm] ςs

AR [-] ςsA [mm2] ςs

θ [◦]

Warpςp - 0.027 0.012 0.817 0.014 0.680Weft ςp 0.024 - 0.014 1.110 0.017 0.996

Systematic curves obtained using single-ply genuses can becompared to checkwhether significant differences are present between plies. To do this, the systematiccurves computed for a single plys,< ε(t,s) >, are translated to a single reference periodand their mean,< ε(t,µ) >, is computed for each position along the tow path. Thedifference between the systematic curve for any single plys and the mean systematic

RESULTS AND COMPARISON OF STATISTICAL ANALYSIS BETWEEN PLIES 93

curve is than characterised by the norm

ςs =

√

∫

i(< ε(t,s)

i > − < ε(t,µ)i >)2dx (I7)

Figure I5 shows the systematic curves for the warp out-of-plane coordinate for allplies, together with their mean, while table I2 presents thedifference measureςs foreach ply’s systematic curve relative to the mean systematiccurve. The mean values ofthe tow shape parameters are given as reference in table I3. The difference measureςs for the systematic curve determined for any parameter for any ply s is less than thestandard deviation of the stochastic variations of that parameter within a single ply.Therefore, the plies are approximately indistinguishable. Yet one mild feature doesdistinguish different plies, which is described as follows.

Table I3: Mean values forAR, A andθ computed over all plies.

< AR> [-] < A > [mm2] < θ > [◦]

Warp genus 12.946 0.366 -0.539Weft genus 12.222 0.360 -1.266

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

x [mm]

z [m

m]

ply 1 ply 2 ply 3 ply 4 ply 5 ply 6 ply 7

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

x [mm]

z [m

m]

systematic

Figure I5: Systematic out-of-plane singe-ply warp genusesfor all plies. A similarpath is observed between plies which makes the assumption ofone all-ply warp genusfeasible.

For the out-of-plane centroid coordinate path,dipsare seen in the middle of the twill

94 Paper I

float3 (see figure I5). This effect is a plausible result of the compaction and mutualinteractions of tows during production. When a preform is compressed through itsthickness, a tow can deflect more easily at the centre of a floatthan near the ends ofthe float, where down-trending segments can act as column-like supports. Thus thecharacteristic dip observed in mid-float can be formed. The tows lying on top of thesurface plies do not exhibit dips as long as they are the closest tows to the mouldsurfaces, which might be attributed to the presence of flat platens in the mouldingprocess to which the surface plies conform. The difference observed in the dips insurface and non-surface plies is the only difference observed between the systematiccurves for different plies. It is also important to remark that the float region for innerplies (not on the surface) does not necessary have one or two dips along the tow lengthsince the entire path itself can be distorted. Due to nestingof the different plies andthe scatter of the in-plane tow path, these dips are highly fluctuating and no trends arefound for these inner plies. Table I4 describes the dips by quantifying the depthι andslopeκ for all the single-ply genuses. A dip is set apart from a smallfluctuation of thez-centroid if the depth is larger than 10µm, accordingly chosen that it must be largerthan the nominal diameter of a single carbon fibre (7µm). The depth is defined as thedifference in maximum and minimum value of the dip; while the slope is calculatedby fitting a line between the locations of these extreme values. The depth is typicallytwice the standard deviation of the z-coordinate of the tow centroid. The averageslope is around 1.5◦, but individual tow floats show dip slopes up to 5◦ as shownin table I4 and figure I6 showing the dip orientation frequency. Plies can have one,two or even no dips (e.g. the single-ply warp genus of ply seven). In fact, the dipsvary in form substantially from instance to instance (figureI5), rarely exhibiting thesymmetry implied by their representation in the averaged, systematic curve. The dipswill cause stress concentrations when applying a load that is aligned with a tow, whichwill tend to either straighten or buckle the tow. A misalignment of 5◦ is comparable totypical misalignments in other textile composites [16, 17,20] and sufficient to act asa preferred site for fatigue damage initiation, for example[21].

The systematic variations of aspect ratio, area and orientation computed for a weftgenus that combines tows from all plies, the all-ply weft genus, are presented infigure I7. The aspect ratio and the area of the tow are correlated with the locationsof the cross-overs. The aspect ratio is maximum at the cross-over locations, wherethe area is minimum. This effect is also observed for ceramic composites [3]. Thecross-over locations within each ply are shifted from an equally spaced grid due to theinteraction between plies that can locally shift tows. Thiscauses the maxima in themean systematic curves for all the plies to be unequally spaced over the tow length,as observed in figure I7. The orientation of the tow cross-section is slightly rotatedclockwise when the cross-section is viewed along the positive direction of the tow axis.

3A float, in weaving jargon, is a long, relatively straight towsegment where a tow is passing over oneor more tows of a different type, which ends in curved segments where the tow bends down to pass underother tows.

RESULTS AND COMPARISON OF STATISTICAL ANALYSIS BETWEEN PLIES 95

Table I4:Dip quantification by depth and slope for the single-ply genuses.

Warp genus ιwarp [mm] κwarp [◦] Weft genus ιwe f t [mm] κwe f t [◦]

Ply 1 0.024 1.075 Ply 1 0.011 0.460Ply 2 0.031 1.347 Ply 1 0.046 1.780Ply 3 0.021 1.689 Ply 2 0.030 1.376Ply 4 0.043 2.006 Ply 2 0.033 1.536Ply 4 0.019 0.851 Ply 3 0.033 1.519Ply 5 0.018 0.839 Ply 3 0.034 1.482Ply 5 0.021 0.921 Ply 4 0.029 1.277Ply 6 0.051 2.290 Ply 5 0.045 2.078Ply 6 0.015 0.631 Ply 6 0.035 1.538Ply 7 - - Ply 7 0.057 4.637

Mean 0.027 1.295 Mean 0.035 1.768σ 0.012 0.580 σ 0.012 1.090

The largest rotations occur at locations where the tow z-coordinate increases and thesmallest where the z-coordinate decreases.

0 1 2 3 4 50

1

2

3

4

5

6

7

Warp genus dip orientation [°]

Fre

quen

cy

0 1 2 3 4 50

1

2

3

4

5

6

7

Weft genus dip orientation [°]

Fre

quen

cy

Figure I6: The frequency of the dip orientation for the single-ply warp genuses (left)and single-ply weft genuses (right).

96 Paper I

2 4 6 8 1010.5

11

11.5

12

12.5

13

13.5

y [mm]

AR

[−]

2 4 6 8 100.3

0.32

0.34

0.36

0.38

0.4

y [mm]A

[mm

²]

2 4 6 8 10−3

−2

−1

0

y [mm]

θ [°

]

Figure I7: Mean systematic curves of the tow shape parameters for the all-ply weftgenus. The aspect ratio and area are correlated with the cross-over locations. Theorientation exhibits a particular trend for this limited size of data set.

6.3 Results for stochastic deviations

Deviations from the systematic trend were first analysed using a distinct genus fortows from each ply. As an example, figure I8(a) shows the cumulative probabilitydensity function for normalised deviations of the aspect ratio for warp tows in all plies(seven single-ply genuses). The difference between ply one to three and the remainingplies is not considered to be significant since no a-priori difference in the deviations ofthe three top plies to the other plies is expected. The RTM production process uses thesame flat platens on both sides of the mould. The small deviations can be attributedto local nesting of the first three plies for this particular specimen. No significantdifferences are thus observed between all the deviations, whatever ply they belongto. Similar observations are found for each parameter, and also for the weft tows.Therefore, further analyses of deviations were carried outusing two distinct all-plygenuses.

The deviations for each parameter are approximately normally distributed, as is shownby figure I8(b). The normal probability plot of each parameter (normalised values)is calculated, where perfectly normally distributed data should fall on the straightlines. The distributions adhere to normal behavior except possibly for extreme values,defined here as deviations which are close to or larger than 3σ from the mean value.These extreme values are more frequent in the data for area, orientation, and z-coordinate than expected for a normal distribution. Curiously, in the angle interlockweave studied by Bale et al.[3], extreme values tended to be less frequent. However,the sample size for extreme values in both studies was small;larger data sets wouldbe required to test whether the difference is real. It is potentially important, becauseof the importance of extremes in local geometry to material failure.

The standard deviations computed for all parameters, usingall-ply genuses, arepresented in table I5. The in-plane centroid coordinates for both the warp and weft

RESULTS AND COMPARISON OF STATISTICAL ANALYSIS BETWEEN PLIES 97

Table I5: Standard deviations of each component using data of the all-ply warp andweft genuses.

σx [mm] σy [mm] σz [mm] σAR [-] σA [mm2] σθ [◦]

Warp genus - 0.113 0.014 1.774 0.023 0.797Weft genus 0.063 - 0.015 1.44 0.024 0.833

genuses have the largest deviations. This is expected sincethe in-plane path is not asrestricted as the out-of-plane path during RTM production.

6.4 Correlation lengths for deviations

The autocorrelation of each component is determined, first by single-ply genuses andthen for the all-ply genuses. Figure I9 shows the correlograms (plot of the Pearson’scorrelation parameter in function of the separation of two points on a tow) for thewarp genus z-coordinate and y-coordinate. The data for the all-ply warp genusare represented by circles and the data for single-ply genuses by lines. Correlationlengths are inversely proportional to the slope of the curves at zero separation. Thecorrelograms for z-coordinate deviations (figure I9(a)) indicate correlation lengths inthe range 2 - 4 mm. For separations exceeding these values, the value of Pearson’scorrelation parameter is small and occasionally changes sign, indicating the absence

Figure I8: (a) Cumulative distribution function of the ply normalised deviations ofthe AR parameter for the warp tows. No differences between plies are observed. (b)Normal probability plot of the normalised deviations for all components using data forthe all-ply warp genus.

98 Paper I

of significant correlation. The correlograms for y-coordinate deviations (figure I9(b))indicate much longer correlation lengths, sometimes exceeding the dimensions of theunit cell and therefore the specimen dimensions in the present study.

0 5 10 15−1

−0.5

0

0.5

1

Distance [mm]

Corr

ela

tion [−

]

all plies

0 5 10 15−1

−0.5

0

0.5

1

Distance [mm]

Corr

ela

tion [−

]

all plies

(a) (b)

Figure I9: (a) Autocorrelation graph of the warp genus z-centroid coordinate. A fastdecay is present. (b) Autocorrelation graph of the warp genus y-centroid coordinate.The in-plane centroid exhibits large correlation for distances exceeding the unit celldimensions.

Figure I9 shows significant variance in the correlograms fordifferent plies andtherefore in the deduced correlation lengths. The limited size of the data set for thelargest point separations causes fitted lines to be affected by outliers, which is a majorsource of large fluctuations in the correlation length. A second source of uncertaintyfor the y-coordinate deviations is that the correlation length is the reciprocal of a smallnumber (the negative slope of the curve): small errors in theslope translate to largeerrors in the correlation length.

Correlation lengths obtained for the all-ply genuses are listed in table I6, along withthe mean< ξt,s > and coefficient of variationCOVt,s

ξ of the correlation lengthscalculated using single-ply genuses. The correlation lengths of the out-of-planecentroid coordinates are smaller than the cross-over spacing in the weave, as wasobserved in [3]. This is consistent with out-of-plane deviations being influencedby a combination of intra-ply cross-over effects and inter-ply nesting effects. Thesame conclusion is made for the correlation length of the area. The in-plane centroidcoordinate has much higher correlation lengths, which exceed the unit cell dimensionsfor the warp genus. This is consistent with the in-plane tow displacements beingrelatively unconstrained by the fabric architecture, within-plane tow deflectionsable to span a number of unit cells. Thus a long-range deviation is present forthis displacement component, which must be investigated onlarger samples. Thecorrelation lengths for the tow parameters of orientation and aspect ratio are within theunit cell size. The aspect ratio has slightly higher correlation values and correlation

DISCUSSION 99

length, which could possibly be attributed to the original stacking of the tows on thepulley during the unwinding step for the weaving.

All correlation lengths exhibit significant uncertainty (table I6). Additional data mustbe collected to achieve a lower variation.

Table I6: Correlation length values for all warp and weft components using data of theall-ply genuses and single-ply genuses.

ξx [mm] ξy [mm] ξz [mm] ξAR [mm] ξA [mm] ξθ [mm]

ξwarp - 22.89 1.78 7.26 2.53 4.56< ξwarp,s > - 18.19 2.27 6.84 1.70 4.43COVwarp,s

ξ - 71.43 % 38.53 % 60.38 % 78.72 % 45.38 %ξwe f t 9.42 - 1.62 5.48 1.01 3.49< ξwe f t,s > 10.90 - 1.95 5.40 1.54 3.09COVwe f t,s

ξ 66.16 % - 33.15 % 33.95 % 68.38 % 57.27 %

7 Discussion

The statistical description of the tow path enables an improved multi-scale modellingstrategy taking into account variability in the reinforcement structure at meso-scale.On the one hand, it can calibrate a geometry generator principle based on a MarkovChain for generating the stochastic part [6], while on the other hand it can also beused to extend deterministic numerical modelling softwaresuch as WiseTex [113] tostochastic modelling. The objective is to reconstruct a virtual composite specimenthat has the same statistical properties as the considered sample. Such realisticdescriptions of internal geometry can subsequently be usedto evaluate the effects ofstochastic microstructure on the mechanical behaviour of the polymer composite. Theidealisation of the tow cross-section by an ellipse hardly influences the acquisitionof the data. When modelling the structure, it is also consistent to represent thereinforcement as overlapping ellipsoids when evaluating the stiffness properties usingthe Mori-Tanaka scheme [42]. However, minor modifications to the elliptical cross-section for tows in the reinforcement need to be considered for damage modelling andstress-strain computations, where the actual shape of the tow is of higher importance.

The detection of dips in floats in the systematic curves for the z-coordinatedemonstrates how sensitively the process of reference period collation can identifyweak but structurally rational features. The lack of symmetry of individual instancesof these dips must be considered when developing virtual specimens. It would be

100 Paper I

an error to represent all dips by the characteristics seen inthe systematic curves: thevirtual specimens should be generated with variable dips, by superimposing randomdeviations of appropriate magnitude on the systematic curve. Since the statistics of thedips are predominantly related to the statistics of the nesting and the variation of the in-plane centroid, a post-processing step could be required toalter the dip configurationin the particular case if there is a high probability that a dip occurrence does not takeplace in a reality. Yet apart from the presence of dips, a relatively mild discrepancy,both the systematic curves and the deviations from the systematic curves that arecomputed for all parameters are very similar for different plies. It is an unexpectedbut important outcome that the statistics of tow positioning and shape are not stronglyaffected by the fact that plies in this laminate are offset from one another by randomdistances comparable to the unit cell size. In model representations, the absence ofdependence on ply permits the assignment of tows of type warpor weft from all pliesto a single genus. Only where the minor difference in dip magnitude is expected tosignificantly affect composite properties is the use of a distinct genus for each plywarranted.

The deviations of the in-plane centroid position are correlated over relatively longdistances, exceeding the dimensions of a typical CT specimen. Since the surfaceplies are statistically representative of all plies, the required large-specimen datacould possibly be extracted from ordinary optical images ofthe specimen surface.The combination of the current short-range data with this long-range geometricalinformation will calibrate numerical models at the macro-scale consisting of manyunit cells, as required to model component-scale structures.

At last, it is important to notice that errors are introducedat several steps throughoutthe procedure: (1) image segmentation and ellipse fitting, (2) optimisation techniqueto define the reference periods, (3) assumption of normal distribution for the deviationsand (4) derivation of the correlation length from a linear fitto the correlogram. Thefirst and fourth error are assumed to be the dominant errors introducing uncertaintyin the deduced statistical information. Although the errorin image segmentationis limited, the ellipse fitting can introduce small errors when the tow cross-sectiondeviates largely from an ellipse. No large deviations of thecross-section from anellipse shape are expected for this kind of topology as discussed in section 4.1. A largeuncertainty is however present for the correlation length as discussed in section 6.4.This is caused by the least-square linear fit to the autocorrelation values which is verysusceptible to outliers. This uncertainty can be reduced ifthe size of the data setis increased. An estimate of the uncertainty of±25% is assumed for the deducedcorrelation lengths, which is similar as the for the C/SiC composite in Blacklock etal.[6].

CONCLUSIONS 101

8 Conclusions

The present work illustrates that the reference period collation method, previouslyapplied to a ceramic angle interlock weave [3], can be extended by modifying themanner in which tows are grouped into genuses, to laminated polymer composites.

Further, the geometrical variability of the subject high-performance carbon-epoxytextile laminate is found to be significant. The out-of-plane centroid coordinate, aswell as the tow shape parameters (aspect ratio, area and orientation) have systematiccurves that depend on the cross-over locations. However, nomajor differences in thismean behaviour are observed between plies, indicating minimal effect of the randomnesting of plies or their position in the laminate sequence.

The in-plane centroid coordinate is subject to the largest variability from its systematiccurve and is correlated along the tow with distances exceeding the unit cell dimensions.The deviations from systematic patterns are also similar for all plies, with nodiscernible inter-ply effects. The deduction of correlation lengths exceeding the unitcell size points to the need to complement micro-CT data withdata from much largerspecimens, exceeding the unit cell size by at least an order of magnitude.

The statistics reported here are the calibrating data required for generating virtualsamples possessing the same stochastic properties as thosecomputed for theexperimentally characterised samples.

Acknowledgements

This study is supported by the Flemish Government through FWO-Vlaanderen(project G.0354.10) and by the US Air Force Office of Scientific Research (Dr. AliSayir) and NASA (Dr. Anthony Calomino) under the National Hypersonics ScienceCenter for Materials and Structures (AFOSR Contract No. FA9550-09-1-0477).Prof. M. Wevers, Dr. G. Kerckhofs and Dr. G. Pyka (KU Leuven) are gratefullyacknowledged for the help with the micro-CT acquisition. Dr. H. Bale (UC Berkeley),Dr. M. Blacklock and Dr. R. Rinaldi (UC Santa Barbara) are also acknowledged forthe support in image segmentation and analysis.

102 Paper I

Appendix A. Alternative procedure to identify the

tow boundary and cross-section

Statistics for a tow cross-section can alternatively be gathered by analysing themoments of the distribution of matter in the cross-section.Taking a warp tow asan example, let the functionω(x′, y′) define the domain occupied by the tow cross-section in an image slice, withω=1 inside the tow andω=0 elsewhere and the axes(x′, y′) related to the global x- and y-axes by a rotation. Define the moments ofω by

Mpq =

∫ ∫

ω(x′, y′)xpyqdx′dy′ (I8)

The area of the cross-section is given byM00 and the coordinates of its centroid by(M10/M00,M01/M00). A useful definition of semi-axes,a1 anda2, is obtained by firstfinding the value ofθ that maximisesM20/M02, which angle becomes the rotation of

the tow cross-section, and then settinga1 =

√

M20 − M210 and a2 =

√

M02 − M201.

These quantities are related to the semi-axes of an ellipse.

The moment analysis yields six independent parameters (2 centroid coordinates, area,orientation, semi-axisa1 and semi-axisa2), whereas the ellipse-fitting procedureyields only five (2 centroid coordinates, area, orientationand aspect ratio). Assumingan elliptical shape imposes a constraint between the area, aspect ratio, and semi-axes.Therefore, while ellipse-fitting is conveniently providedin standard image analysispackages, other methods of deducing tow statistics may be preferable in cases wherethe shape is significantly different from elliptical.

Paper II - Stochasticcharacterisation of thein-plane tow centroid intextile composites to quantifythe multi-scale variation ingeometry

103

104 Paper II

Published as a book chapter in the Proceedings of the IUTAM Symposium on Multiscale Modeling and

Uncertainty Quantification of Materials and Structures, Springer (2014) 187-2024

Stochastic characterisation of the in-plane tow centroid in textile composites toquantify the multi-scale variation in geometry

Andy Vanaerschota, Brian N. Coxb, Stepan V. Lomovc and Dirk Vandepittea

aKatholieke Universiteit Leuven, Department of MechanicalEngineering, Leuven, BelgiumbTeledyne Scientific Co. LLC, Thousand Oaks, CA, USA

cKatholieke Universiteit Leuven, Department of Metallurgyand Materials Engineering, Leuven, Belgium

Abstract

Optical imaging is performed to quantify the long-range behaviour of the in-plane towcentroid of a 2/2 twill woven textile composite produced by resin transfer moulding.The position of the carbon fibre tow paths is inspected over a square region of tenunit cells and characterised by decomposing the centroid data into a non-periodic non-stochastic handling effect and non-periodic stochastic fluctuations. A significantlydifferent stochastic behaviour is observed for warp and weft direction. Variabilityof the in-plane coordinate, identified by the standard deviation, is found to be sixtimes higher in weft direction. The spatial dependency of deviations along the towdemonstrates a correlation length of ten unit cells for warptows, which is twice thelength computed for weft tows. The observed bundling behaviour of neighbouringtows of the same type is quantified by a cross-correlation length. Warp tow deviationsaffect neighbouring centroid values within the unit cell dimension, while this effectexceeds the unit cell size for weft tows. The stochastic information reflects thedifference in tow tensions during the weaving of the fabric.

4Symbols can differ with original publication to preserve consistency in theentire manuscript.

INTRODUCTION 105

1 Introduction

Fibre reinforced composite materials are subjected to a significant amount of scatterin the geometrical structure, leading to a remarkable variability in performance. Thenominal periodicity in the tow reinforcement of a textile composite, prescribed bythe manufacturer, is only approximated in real samples. Different work alreadydemonstrated that the tow paths in textile composites should not be represented asdeterministic, but as stochastic entities where deviations are fluctuating around amean trend [22, 28, 31]. Mapping the variation in geometry and material propertieswill support material design and certification of structural composites [84, 121]. Itincreases the reliability of numerical analyses of composite structures.

Almost all published research deals with randomness of local properties withoutconsidering the correlation of a property at different positions along a tow, orcorrelation between different properties at the same position on a tow. Although,experimental work [67] already has demonstrated that spatial variability must beconsidered to achieve an accurate description of the material. Also the sourcesof variability remain poorly understood and the inadequacyof experimental data[10, 108] result in assumptions for the input probability density functions of numericalmodelling techniques. Further, only a few tools are available to partially model thegeometrical variation of textile reinforcements [14]. Significant advances in realisticmaterial modelling can be achieved by [10]: (i) collecting sufficient experimentaldata on the spatially correlated random fluctuations of the uncertain tow pathparameters and (ii) deriving probabilistic information for the macroscopic propertiesfrom the lower scale mechanical characteristics. This workis part of a seriesof papers following the approach of Charmpis [10]. The objective is to createvirtual specimen of polymer reinforced composites possessing the same statisticalinformation as observed in experimental samples. Such random composite structuresare subsequently used to define the spatial variability in the mechanical propertiescaused by geometrical variation in the tow path.

Variation in the geometry is a multi-scale phenomenon in textiles: geometrical scattershould be investigated on the short range, i.e. deviations correlated over distancesless than or compared to the size of the unit cell, complemented with long rangeinformation, i.e. spanning several unit cells. The methodology is tested on a carbon-epoxy 2/2 twill woven composite produced with resin transfer moulding (RTM). Shortrange tow path data are already quantified in [104], while this paper reports andanalyses the collection of the long range deviations of the in-plane position.

While data about the out-of-plane centroid and cross-sectional variations demand theinvestigation of the internal geometry, in-plane centroidinformation can be deducedfrom scans performed of the top view of the composite. It doesnot require sectioningor need a full three-dimensional representation. Optical imaging of the surface of

106 Paper II

textile composites has already been applied to characterise the in-plane geometricalvariations for several woven and stitched composites. Endruweit et al. [28] linksfabric irregularities with permeability variations. Optical images are taken of severalwoven fabrics to inspect the scatter in tow width, tow spacing and inter-tow angle.Depending on the structure of the fabric, higher or limited fibre tow mobility isallowed. Skordos and Sutcliffe [94] investigated the influence of fibre architecturalparameters on the forming of woven composites. Variabilityin tow directions andunit cell size are quantified for a pre-impregnated carbon-epoxy satin weave textileusing the Fourier transform of a grey-scale image. A two-dimensional spectrum withdirectional structure is obtained which corresponds to thephysical tow directions. Thismethodology is found to be effective, but does not permit to analyse the local centroidcoordinate along the tow. Gan et al. [31] used an optical technique to quantify thevariability of three different glass reinforcement structures. The translucent propertyof glass fibres is exploited to set up an automated characterisation procedure usingMatlab. Samples spanning several unit cells are quantified in areal weight variations,with additional local tow orientations, tow spacings and widths for the periodicreinforcements. This automated procedure can however not be applied for carbonfibres due to its opacity.

A full characterisation of the in-plane centroid of the 2/2 twill woven textile is obtainedby scanning the top surface of the impregnated composite. Local and correlatedinformation of the tow centroids are investigated over a region spanning multiple unitcells. The objectives of the paper are to (i) perform opticalimaging with derivationof the in-plane tow centroid of a one-ply 2/2 twill woven carbon epoxy composite,(ii) develop a procedure to define the in-plane centroid deviations (iii) compute themean trend and statistical information of the deviations interms of standard deviationand correlation length. The statistical information is prepared to be used as input in astochastic multiple unit cell modelling technique.

2 Material

The inspected tow reinforcement is a 2/2 twill woven Hexcel fabric (G0986 injectex)[40]. The unit cell topology is given in figure II1 withλx andλy, respectively theperiodic length in warp (x-axis) and weft (y-axis) direction. Nominal areal densitymeasures 285 g/m2 with an ends/picks count of 3.5 resulting in unit cell dimensionsof 11.4 by 11.4 mm.

Two one-ply reinforcements of this fabric, spanning a region of thirteen unit cellsby thirteen unit cells, are impregnated with epoxy resin in aRTM process. Theproduction of one-ply samples is more challenging than multiple-ply samples butoffers the advantage to obtain a high contrast between tow and resin regions for the

IMAGE PROCESSING & ANALYSIS 107

Figure II1: WiseTex model of a 2/2 twill woven reinforcement. The x-axis and y-axisof the coordinate system are respectively parallel to the warp and weft direction.

image processing step. Characterisation of the in-plane centroid on these multipleunit cell samples provides new information of the geometrical scatter on the longrange. Deviations correlated over distances less than or compared to the size ofthe unit cell are already quantified using laboratory micro-computed tomographyin [104]. The tow path is statistically characterised for the centroid location (in-and out-of-plane), area, aspect ratio and orientation in cross-section. The referenceperiod collation method [3] is applied, where each tow parameter is decomposed innon-stochastic, periodic systematic trends and non-periodic stochastic fluctuations.Average behaviour of the tow parameter is represented by thesystematic trend,while the stochastic characteristics are given in terms of the standard deviation andcorrelation length. The procedure and statistical information is described in [104].

The investigation of short range variations pointed out that only the in-plane centroidcomponent of the tow path possesses a long range effect, indicated by the correlationlength along the tow which exceeds the unit cell dimensions.The out-of-planecentroid and tow cross-sectional properties vary within the unit cell dimensions.

3 Image processing & analysis

An optical scan of the in-plane dimension of both the samples(sample 1 & sample2) is performed with a resolution of 1200 dots per inch (DPI).The obtained imageof the first sample is shown in figure II2. A region of 10 unit cells by 10 unit cellsis indicated where the in-plane tow data are analysed. This area of interest is chosenaway from the edges to minimise possible edge effects and large enough, roughly onemagnitude larger than the short range data.

The freeware image processing tool GIMP is used to extract the centroid line of thetow reinforcement in order to quantify the in-plane position of warp and weft tows.In a first step, boundaries of the tows are marked based on visual recognition for

108 Paper II

Figure II2: Optical scan of a one-ply 2/2 twill woven carbon fibre fabric impregnatedwith epoxy resin. Warp tows are oriented horizontally, while weft tows are positionedin the vertical direction.

prescribed grid spacings of 125 pixels in x (warp) and y (weft) direction. Thesedistances correspond to the nominal tow spacing of a 2/2 twill weave. In a secondstep, the centroid locations are computed as half the tow width at each grid location.Typical patterns in the centroid positions are further investigated to quantify the globaldeformation of the fabric.

The discrete representations of the tows are given as input to Matlab. Before furtherinspection, data are translated to a global axis system and rotated. Average warpand weft angles are computed to verify if a rotation of the entire data set is requiredto compensate a possible shift in the sample data due to manual placement in thescanning device. A continuous representation of the discrete tow data is afterwardsobtained by cubic spline interpolation.

Using the same approach as for the unit cell sample in [104], the unit cell periodscan be defined by a minimisation algorithm. Table II1 compares the periodic lengths(i) obtained from the considered long range samples, (ii) derived from the unit cellsample and (iii) given by the manufacturer. The experimental data for the short andlong range demonstrate that the unit cell period of the warp tows is slightly longer

IMAGE PROCESSING & ANALYSIS 109

Table II1: Unit cell periods obtained from long range data, short-range data andmanufacturer’s data

λ [mm] λ [mm] λ [mm] λ [mm] λ [mm]sample 1 sample 2 combined short-range manufacturer

[104] [40]

Warp direction 11.43 11.53 11.48 11.55 11.40Weft direction 11.38 11.37 11.38 11.48 11.40

Table II2: Tow width and spacing of the one-ply 2/2 twill woven fabric.

Warp tows Weft tows

wtow - mean [mm] 2.64 2.49wtow - COV 4.96% 7.17%sptow - mean [mm] 2.86 2.90sptow - COV 3.62% 8.73%

than the weft tows.

Geometrical characteristics of the single ply 2/2 twill woven fabric, such as towspacingsptow and widthwtow, can be defined from the in-plane dimensional image.These parameters are derived from the boundary points and centroid locations of thetows and presented in table II2. Weft tows have on average a smaller width, while thetow spacing is similar for warp and weft tows. The variation is higher for the weftdirection.

The digital image also allows to quantify the open gaps between neighbouringtows. Pattern of these gaps originates from the fluctuating in-plane centroid ofthe tow path over the experimental sample and represent regions fully occupied byresin. After thresholding of the digital image, characterisation of the gaps is furtherperformed automatically in Matlab to define the location andshape. The approximatedrectangular shape of the gaps is represented by the width andheight at each gaplocation. To eliminate disturbances due to e.g. dust particles in the image, a gapis considered to be significant if it has at least an area of 16 pixels2 with a widthand/or height of at least 4 pixels. Maps of the significant gaps located over the sampledimension can be constructed such as presented in figure II3 for sample 1. Each gapsize is categorised in one of the five considered intervals ofthe gaps area as indicatedby the legend. Such mapping of gaps demonstrates bundling behaviour of the tows,

110 Paper II

which is more noticeable for the weft tows. Local shifts in a tow affects neighbouringtows over a certain distance. The fluctuations along a singletow spans several unitcells, but do not persist over the entire length of the tow. This is reflected by the opengaps which occur in bands over the entire sample.

Statistics of the gaps (widthwgap, heighthgap and areaAgap) are described in table II3in terms of sample mean value and coefficient of variation (COV). Individual gapdimensions are Weibull distributed with a scale parameter approximating the zerovalue and shape parameter between 1.2 and 1.9. The width of the gapswgap (x- orwarp direction) is 2 to 2.5 times the height of the gapshgap (y- or weft direction). Thisreflects the larger variability in the weft in-plane tow pathcompared to the warp tows,as will be discussed in section 4. A higher width leads to significant gaps betweenneighbouring weft tows in the warp direction. The amount of gaps in the one-plysample is less than 1% of the entire area, which dominates thethrough-thicknesspermeability. It can be used to optimise flow simulation in accurately describing thelocal flow and minimising the number of voids.

4 Statistical characterisation of the in-plane cen-

troid

Figures II2 and II3 demonstrate that the in-plane tow centroid does not follow straightpaths with equal tow spacing. These in-plane undulations and shifts in tow spacing arequantified by comparing the experimental tow paths with an ideal lattice description.Tows of this lattice are represented as straight lines, withnominal spacing equal toδ =λy/4 andδ′ = λx/4 respectively for the warp and weft spacing (λx, λy taken from theexperimental unit cell periods in table II1). A best-fit of this grid with the experimental

Table II3: Sample mean and coefficient of variation of the gaps in sample 1 (N=1762)and sample 2 (N=1862)

Sample 1 Sample 2

wgap - mean [mm] 0.418 0.451wgap - COV 61.46% 53.49%hgap - mean [mm] 0.184 0.177hgap - COV 56.85% 59.18%Agap - mean [mm2] 0.075 0.080Agap - COV 84.93% 83.35%Porosity full sample 0.56% 0.65%

STATISTICAL CHARACTERISATION OF THE IN-PLANE CENTROID 111

0 20 40 60 80 100 120 140 160

0

20

40

60

80

100

120

140

160

x [mm]

y [m

m]

0.007 <Agap

< 0.1 0.1 <Agap

< 0.2 0.2 <Agap

< 0.3 0.3 <Agap

< 0.4 0.4 <Agap

< 0.51

Figure II3: Map of gaps distributed over sample 1. The area ofthe significant gaps(mm2) are indicated and categorised in five intervals.

cross-over locations of the tows is searched by a minimisation algorithm reducing theoverall standard deviation of the fluctuations from the grid. This procedure is shownin detail in figure II4(a), where the lattice is fitted to the centroid data of the left bottomside.

4.1 Analysis of the in-plane deviations

Deviations from the nominal architecture are determined bycomputing the differencebetween the experimental tow path and the lattice at each grid location. The in-planewarp and weft fluctuations are considered respectively in y-and x-direction. Thisprocedure results in a deviations pattern for the warp tows,combination of sample 1and 2, as given in figure II4(b).

The obtained deviations are further represented asε( j,t,s)i , with i the grid location

(i = 1..Ni andNi = 40), j the tow index (j = 1..N j andN j = 40) in each direction,t= warp or weft tows ands=1 or 2 referring to the sample. The in-plane centroiddeviations along the different tows have a particular non-periodic trend. This patternis shown for the warp tows in figure II5(a), by considering themean value per grid

112 Paper II

0 5 10 15 20 25 30 35 40

0

5

10

15

20

25

30

35

40

x [mm]

y [m

m]

(a) Best-fit grid to cross-over locations

0 20 40 60 80 100 120−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x [mm]

deviations [mm]

(b) Deviations pattern

Figure II4: Procedure to define the in-plane centroid deviations applied on sample1. (a) Detail image of best-fit grid to the experimental cross-over locations. (b)Deviations pattern of all warp tows after subtraction of theexperimental cross-overdata from the grid values.

point < ε( j,t,s)i >. The lack of periodicity signifies that this tendency shouldnot

be interpreted as a systematic trend, representing the repetitive mean behaviour ofthe tow path, but as an effect due to handling. Variability already originates in thein-plane centroid before production due to storage and handling of dry fabrics, e.g.unwinding of the fabric from the pulley and preparing the stacking sequence in theRTM mould. This kind of variation should not be considered asstochastic, but as anadded deterministic effect. Subtracting the handling pattern per sample< ε

( j,t,s)i >

in figure II4(a) from the deviation valuesε( j,t,s)i in figure II4(b), results in stochastic

variationsǫ( j,t,s)i which are attributed only to the loom itself. Decompositionof in-

plane deviations in a deterministic and stochastic part is summarised as

ε( j,t,s)i =< ε

( j,t,s)i > +ǫ

( j,t,s)i (II1)

The deviationsǫ( j,t,s)i are presented in figure 6.5(b) for the warp tows.

Equal results can also be obtained following the reference period method [3, 104].The proposed procedure uses a similar approach by defining the nominal tow spacing,expressed asδ in the ideal lattice, equal toλ/4. The periodic lengthsλx andλy areexperimentally obtained using the same minimisation algorithm. The similarity isindicated in figure II5(a) where the cross marks indicate thesystematic trend obtainedusing the reference period technique. However, this systematic curve should beinterpreted as a handling effect.

The transformed deviationsǫ( j,t,s)i are further analysed to obtain stochastic information

about the geometrical scatter of the in-plane tow centroid.Both sample deviations aregrouped in one data set, which is permitted since no special relationship is expected

STATISTICAL CHARACTERISATION OF THE IN-PLANE CENTROID 113

0 20 40 60 80 100 120

−0.2

−0.1

0

0.1

0.2

x [mm]

devi

atio

ns [m

m]

Sample 1Sample 2CombinedRef. period

(a) Handling effect

0 20 40 60 80 100 120−0.4

−0.2

0

0.2

0.4

x [mm]

deviations [mm]

(b) Stochastic deviations

Figure II5: Deviations trend of warp tows decomposed in (a) handling effect and (b)stochastic deviations.

between the different samples after subtraction of the mean trend. The warp and weftdeviations approximately follow a normal distribution. The normal probability plot ofthe in-plane deviations of the warp tows (figure II6(a)) showgood agreement, exceptfor the tails where a lower frequency of deviations is present. The weft deviations onthe other hand do show significant differences from normality (figure II6(b)), with ahigher frequency of values in the left tail of the distribution and a higher frequencyof deviations values around the zero mean. The weft fluctuations behaviour couldbe depicted as a combination of two normal distributions: one with low standarddeviation (peaked curve) and one with high standard deviation (wide curve). However,there is no physical reason why these fluctuations would follow such a distribution.Therefore, the experimental data are considered for now as if they are normallydistributed.

The random behaviour of the in-plane position is described in terms of standarddeviation and correlation length. These statistics are required to generate virtualrandom specimen possessing the same statistical information as the experimentalsamples. Correlation of the centroid is considered along a single tow, further calledauto-correlation, but also between neighbouring tows of the same type, named cross-correlation (figure II7). The latter correlation type is notobserved for the other towproperties in the short range data [104], but is significantly present for the in-planecentroid as indicated by the bundling of tow positions in figures II2 and II3.

4.2 Standard deviation

The standard deviationσ for the combined data set of warp and weft tows arepresented in table II4. Weft in-plane centroids are subjected to a much larger variation

114 Paper II

−0.4 −0.2 0 0.2 0.4

0.00010.001

0.01

0.1

0.5

0.9

0.99

0.9990.9999

Data

Pro

babi

lity

Normal fit to deviations

(a) Warp tows

−2 −1 0 1 2

0.00010.001

0.01

0.1

0.5

0.9

0.99

0.9990.9999

Data

Pro

babi

lity

Normal fit to deviations

(b) Weft tows

Figure II6: Normal probability plot for (a) warp and (b) wefttow deviations showingapproximately normal behaviour of the in-plane deviations.

66

6.2

6.4

6.6

6.8

67

7.2

Tow j+1

Tow j

Auto-correlation

Cross-

correlation

Figure II7: Definition of spatial dependencies of deviations demonstrated for two wefttows: auto-correlation (along the tow) and cross-correlation (between neighbouringtows).

which can be explained by the production process. Warp tows are put under tensionduring fabrication of the fabric, while weft tows are inserted. Weft tows are thereforeless restricted in their in-plane movement. Similar results are obtained by Skordosand Sutcliffe [94] for a carbon epoxy five harness satin weave where the variability inlocal tow orientations, which can be related to the in-planedeviations, is higher forthe weft direction.

4.3 Correlation information

Correlation information is summarised by evaluation of thePearson’s momentcorrelation parameter for pairs of data taken at distinct locations on a single tow,spaced bykδ (auto-correlationCauto), and pairs of data on neighbouring tows but

STATISTICAL CHARACTERISATION OF THE IN-PLANE CENTROID 115

Table II4: Standard deviation of warp and weft tows for the combined data set.

Sample 1 Sample 2 Combined

σwarp [mm] 0.108 0.105 0.106σwe f t [mm] 0.701 0.515 0.615

fixed at the same grid location on a tow, spaced bykδ′ (cross-correlationCcross). Thecorrelation parameter for computing the auto-correlationis given by:

C( j,t,s)auto (k) =

∑n−ki=1 ǫ

( j,t,s)i ǫ

( j,t,s)i+k

√

∑n−ki=1 (ǫ( j,t,s)

i )2√

∑n−ki=1 (ǫ( j,t,s)

i+k )2

(II2)

with k the lag index (k = 1...Ni − 1 in warp and weft direction), andδ, δ′ the gridspacings in warp and weft direction.

Next, exponential functions are fitted to represent the computed correlation infor-mation and to estimate the correlation lengthξ. The objective is not the find theoptimal function perfectly representing the experimentalcorrelation information, butto consider conventional functions which are physically reasonable and give a goodestimate of the centroid behaviour. However, when more dataare collected an optimalcorrelation function can be searched. In this work, only twotypes of exponentialfunctions are considered which approximately represent the correlation information.These are functions ofτ = |x2−x1| = kδ (or replace bykδ′ in case of cross-correlation):

Cexp(τ) = e−|τ|ξ = e−

kδξ (II3)

Csq,exp(τ) = e−|τ|2

ξ2 = e−

(kδ)2

ξ2 (II4)

Both functions are fitted in a least-square sense to the correlation graphs, also calledcorrelograms, which represent the correlation values in function of the lagkδ. Forthis procedure, a maximum of 20 data points are considered, corresponding to thefirst 20 lags or a length of five unit cells. Correlation information of larger pointspacings are not used for fitting since these correlation data are based on a smallerdata set size leading to a larger variability. In the case that the correlation data crossthe zero-correlation before 20 lags are reached, no furtherdata points are consideredsince negative correlation is not expected but can be present in the data due to a largervariation. To obtain the optimal fit, the sum of squares of theresidualsEres at each lagbetween the experimental correlation dataCdata,i and the fitted correlation dataC f it,i

should be minimised:

Eres =

n∑

i=1

(Cdata,i −C f it,i)2 (II5)

116 Paper II

0 20 40 60 80 100 120−1

−0.5

0

0.5

1

distance [mm]

correlation [−]

Data

Cexp

Csq,exp

(a) Auto-correlation

0 20 40 60 80 100 120−1

−0.5

0

0.5

1

distance [mm]

correlation [−]

Data

Cexp

Csq,exp

(b) Cross-correlation

Figure II8: Correlation graphs of the warp tows for the (a) auto-correlation and (b)cross-correlation. The data points in lighter colour are not considered for the fittingprocedure.

0 20 40 60 80 100 120−1

−0.5

0

0.5

1

distance [mm]

correlation [−]

Data

Cexp

Csq,exp

(a) Auto-correlation

0 20 40 60 80 100 120−1

−0.5

0

0.5

1

distance [mm]

correlation [−]

Data

Cexp

Csq,exp

(b) Cross-correlation

Figure II9: Correlation graphs of the weft tows for the (a) auto-correlation and (b)cross-correlation. The data points in lighter colour are not considered for the fittingprocedure.

Figures II8 and II9 respectively present the correlation information of the warp andweft tows. The most appropriate function is evaluated by considering the sum ofsquares error estimateEres and observation of the correlation graphs. The deducedcorrelation lengths are described in table II5. It is demonstrated that either theexponential or square exponential function is preferred with an errorEres less thanor equal to 1%.

The warp auto- and cross-correlation information are well represented by anexponential correlation functionCexp. The correlation length along the warp tow isfound to be very high and of similar size of the area of interest: around ten unitcells. This value reflects the straightness of the warp tows,which corresponds to theproduction process of a 2/2 twill fabric where the warp tows are kept straightened. Theobservation of the cross-correlation data shows that shifts of the in-plane warp centroid

TOWARDS VIRTUAL MODELLING OF REALISTIC MULTIPLE UNIT CELL STRUCTURES 117

Table II5: Auto- and cross-correlation lengths obtained from exponential and squaredexponential function fitting for the warp and weft tows usingthe combined data set.

ξexp Eres ξsq,exp Eres

[mm] [mm]

Warp tows -Cauto 114.89 0.5% 70.03 8.9%Warp tows -Ccross 4.49 0.3% 4.55 6.8%Weft tows -Cauto 75.72 18.6% 52.89 1.0%Weft tows -Ccross 13.89 6.7% 13.16 0.6%

only affect near-neighbouring tows within one unit cell distance. The auto- and cross-correlation of the weft tows have a squared exponential correlation behaviourCsq,exp.In-plane deviations along the weft direction seem to persist between four and fiveunit cells. This corresponds to only half the warp auto-correlation length, causedby the larger variability in the weft tow path. This is already reflected in the highstandard deviation of this tow type (table II4) and the higher COV in tow width andspacing (table II2). The cross-correlation length of the weft tows shows an influenceexceeding the unit cell size. Positions of the weft tows are less restricted due to thelack of tensioning during production. This affects near- and further-neighbouring tows,causing the band behaviour to appear in the composite tow paths as mentioned insection 3 and shown by the gaps distribution in figure II3.

A different dependency structure is present than for the tow orientations of a fiveharness satin weave [94]. There, a higher auto-correlationis observed for the wefttows, while auto- and cross-correlation of the warp tow orientations are negligible.These dissimilarities can be attributed to the differences in the manufacturing processof the weave.

5 Towards virtual modelling of realistic multipleunit cell structures

A full characterisation of the short and long range deviations of the different towpath parameters enables to construct realistic descriptions of the tow geometry. Allstatistical information is given as input to a stochastic multi-scale modelling strategywhich has the objective to generate random reinforcements that possess the samestatistical information as quantified by the experiments.

118 Paper II

Virtual specimens are built by combining the systematic andhandling trends with zero-mean deviations. The systematic trends at the short-range and handling effect at thelong-range can be taken directly from the experimental data. The zero-mean stochasticdeviations need to be generated such that they represent thesample standard deviationand correlation lengths of each tow parameter.

Different generator techniques are used to represent the full randomness of the towreinforcement at the meso- and macro-scale. Tow path parameters which vary withinthe unit cell size (out-of-plane centroid, tow area and tow aspect ratio) are generatedusing a Monte Carlo Markov Chain algorithm for textile structures, recently proposedby Blacklock et al. [6] and already successfully applied in the generation of randomunit cell structures of the 2/2 twill woven composite [105]. Generation of the long-range in-plane centroid deviations requires a different approach due to the occurenceof cross-correlation. For this purpose, a methodology described by Vorechovský [114]is applied where series expansion methods based on Karhunen-Loève decompositionproduce cross-correlated Gaussian random fields. The in-plane position of each towis represented by a single random field, sharing an identicalauto-correlation structurefor all tows, which is cross-correlated at the same time withneighbouring tows of thesame type.

More details and results of this generation of multiple unitcell structures is ongoingwork and will be addressed in future publications. The realistic representationsof internal geometry can be applied to (i) improve the understanding of damageprogression in textile composites and (ii) obtain a quantitative measure of the spatialvariation of the mechanical properties over the extent of composite components causedby variation in the reinforcement structure.

6 Conclusions

The long range statistical behaviour of the in-plane centroid of a one ply 2/2 twillwoven carbon epoxy composite is investigated over a region of ten unit cells by tenunit cells. The in-plane position is decomposed in a mean trend and stochastic zero-mean deviations. The mean trend represents the handling effect of the fabric beforeit is impregnated with resin and is distinct for each individual sample. No periodicsystematic pattern is present for the in-plane centroid. The scatter around the meantrend shows significant differences in warp and weft direction. The weft tows aresubjected to a much larger variability, quantified by the standard deviation which issix times higher compared to the warp direction. Also the spatial dependency of thedeviations significantly differs. Correlation of the in-plane centroid along the towpath (auto-correlation) is found to be twice as high for the warp tows, spanning tenunit cells. The correlation of deviations between different tows (cross-correlation)

CONCLUSIONS 119

demonstrates that the neighbouring warp tows are only affected within the unit celldimension, while this effect exceeds the unit cell size for the weft tows. This resultsinbundling behaviour which is mainly present for the weft tows. A possible explanationfor the difference in scatter can be attributed to the manufacturing process of the weave,where the warp tows are kept in tension, while the weft tows are inserted by handles.The stochastic information is prepared such that it can be used as input for a stochasticmultiple unit cell modelling procedure.

Acknowledgements

This study is supported by the Flemish Government through the Agency for Innovationby Science and Technology in Flanders (IWT) and FWO-Vlaanderen.

Paper III - Stochasticmulti-scale modelling oftextile composites based oninternal geometry variability

121

122 Paper III

Published in Computers & Structures 122 (2013) 55-645

Stochastic multi-scale modelling of textile composites based on internalgeometry variability

Andy Vanaerschota, Brian N. Coxb, Stepan V. Lomovc and Dirk Vandepittea

aKatholieke Universiteit Leuven, Department of MechanicalEngineering, Leuven, BelgiumbTeledyne Scientific Co. LLC, Thousand Oaks, CA, USA

cKatholieke Universiteit Leuven, Department of Metallurgyand Materials Engineering, Leuven, Belgium

Abstract A stochastic model of an experimentally measured unit cell structure is com-puted using the multi-scale textile software WiseTex. The statistical characteristics ofa sample, derived in prior work, are used to calibrate the recently proposed MarkovChain algorithm for textile fabrics. The generated variable tow reinforcements aretransformed in the WiseTex format that is compatible with tools for micromechanicalanalysis and permeability simulation.

The application is a seven ply polymer textile composite, with each ply consisting ofa twill 2/2 woven carbon fabric in an epoxy matrix. The developed modelpossessesrandom tow centroid paths with nominal cross-sectional properties.

5Symbols can differ with original publication to preserve consistency in theentire manuscript.

INTRODUCTION 123

1 Introduction

The variability in the internal geometry of textile reinforced polymer composites canbe substantial. It is important to quantify the effect on the mechanical performanceto improve the quality and reliability of numerical analysis for composite structures.Mapping the variation in material properties will support material design andcertification of structural composites for e.g. aerospace applications [84, 121].

Within the field of uncertainty quantification, the overall uncertainty is differentiatedin aleatory uncertaintyandepistemic uncertainty[70, 74]. The former uncertainty isdue to inherent randomness of the structure, while epistemic uncertainty is due to alack of knowledge and has an unknown source. The objective inthis study is to reducethe epistemic uncertainty in the geometrical parameter distributions and the variationin the mechanical properties of textile composites. Uncertainty quantification incomposite materials is attaining more attention lately [10, 49]. Introducing variabilityat different scales in computational models is found to be essential or even critical.Zohdi et al. [122, 123] demonstrate that the macroscopic failure behaviour ofballistic fabrics changes from abrupt to gradual failure when variation in the filamentalignment within a tow is considered. Variation in the geometrical model of compositematerials must be included to obtain random media models [45, 77] that can be usedto investigate the effect of random design parameters on e.g. the effective elasticproperties.

When performing uncertainty analysis, it is important to verify the assumptions madethroughout the procedure. The sources of variability in composites remain poorlyunderstood and the inadequacy of experimental data [10, 108] results in assumptionsfor the input probability density functions of numerical modelling methods. Further,only a few tools are available to partially model the geometrical variation of textilereinforcements [14]. Almost all published work considers randomness of localproperties without correlation. However, recent experimental work [67] has proventhat spatial variation of the stiffness properties is present over the extent of a compositestructure. Significant advances in realistic material modelling can be achieved by[10] (i) collecting sufficient experimental data on the spatially correlated randomfluctuations of uncertain material properties for short andlong range deviations, and(ii) deriving probabilistic information for macroscopic properties from the lower scalemechanical characteristics of the material. In prior work [104], statistical informationis collected on the short range deviations of a woven textilecomposite, i.e. deviationscorrelated over distances less than or compared to the size of the unit cell. This datais used as input for a multi-scale modelling technique to generate virtual specimens.

The developed virtual specimens will differ from computational models where thegeometry is transformed to be more useful for numerical analysis. For these models,the reader is referred to the work of Kaminski [45] and Ostoja-Starzewksi [77]. In

124 Paper III

this study, virtual specimens are generated that are replicas of the actual samplesi.e. the random tow reinforcement is fully modelled. The availability of the fullstructure permits performing micromechanical analysis, permeability calculations oreven creating a finite element model for stress-strain analysis, using the same randommodel. Modelling of random textile reinforcements for evaluation of performancehas already been analysed with different approaches [1, 6, 92, 118]. In a recentwork of Abdiwi et al. [1], full-field variability of the tow directions across flatsheets is modelled. First, the variability of the inter-towangles is determined fora range of woven fabrics. Second, a geometry mesh is generated with a pin-jointed net kinematics code where variability is added by introducing (i) horizontalstretching or contraction of elements along the horizontalcentreline of the meshand (ii) additional perturbations of the nodes based on functions over the horizontaland vertical centreline. This code is combined into a genetic algorithm controllingsix parameters to ensure that the generated full-field geometric mesh reproducesthe variability of the measurements. Yushanov and Bogdanovich [118] presented ageneric theory to generate a random reinforcement path. Theinput information islimited to the mean and standard deviation of measured tow paths. The stochasticreinforcement is generated by defining a suitable position vector with a random vectorfunction which is expanded into a deterministic component and a random component.A stochastic local basis is afterwards introduced to uniquely define the directionalcosines of the arbitrary reinforcement path, specified by the position vector. Theelastic characteristics are evaluated using a stochastic generalisation of the orientationaveraging approach. Sejnoha and Zeman [92] use the concept of statistical equivalentperiodic unit cell by considering a two-layer idealised unit cell to be constructed withseven independent parameters, including the tow spacing and thickness. Numericalvalues are accordingly chosen such that the resulting macroscopic behaviour of theunit cell compares with that of a measured material. The correct values are found bya minimisation procedure of a set of equations of experimentally obtained two-pointprobability functions and lineal path functions. Blacklock et al. [6] define a MarkovChain algorithm to generate virtual specimens in which eachtow is represented bya one-dimensional (1-D) locus in the three-dimensional (3-D) space. Fluctuations inthe coordinates are generated by marching systematically along the tow’s length anddeviations depend only on the deviation of the previous point. The transition matrixfor moving from one point in a single tow to the next is calibrated with the standarddeviation and the correlation length measured for that typeof tow. The dominantcorrelations must be those along a tow, with correlations between tows being relativelyweak, to satisfy the Markovian procedure. If one of these conditions is not met, avariant of the proposed algorithm needs to be developed. A virtual specimen is builtby combining the systematic variations of each parameter along the tow length, whichis defined by experimental data, with the generated deviations by the Markov Chainprocedure.

The current paper develops a stochastic virtual specimen inthe deterministic WiseTex

STATISTICAL DATA OF THE MATERIAL 125

software [113] by applying the Markov Chain algorithm [6]. Virtual samples aregenerated that possess the same statistical information ofan experimentally measuredcarbon-epoxy 2/2 twill woven textile laminate produced by resin transfer moulding(RTM) [104]. The systematic, periodic (or mean) patterns are first constructed fromWiseTex according to a predefined grid along each tow length.Second, the MarkovChain algorithm, with specific input parameters for the material under consideration,generates deviations for the centroid coordinates for eachparticular tow type. Theaddition of these deviations to the systematic centroid values at each grid locationresults in a random structure that can be used for mechanicalproperty evaluation. Theobjectives of the paper are summarised as (i) the generationof the centroid deviationsof each tow type for the particular 2/2 twill woven material with verification of theMarkov Chain assumptions, and (ii) the construction of the stochastic virtual specimenin the WiseTex software.

2 Statistical data of the material

2.1 Stochastic characterisation of the woven material

The subject material is a seven ply polymer textile composite. Each ply has a 2/2twill woven carbon fabric from Hexcel (G0986 injectex) [40], with areal density 285g/m2 and ends/picks count of 3.5. The seven ply dry reinforcement is impregnatedwith epoxy resin using RTM as production process. The unit cell representation ofthe fabric is given in figure III1 withλx andλy respectively the period in the x- andy-direction. One unit cell includes four equally spaced warp tows and four equallyspaced weft tows with nominal unit cell dimensions of 11.43 mm by 11.43 mm.

Figure III1: WiseTex model of a 2/2 twill woven reinforcement. The x-axis and y-axisof the coordinate system are respectively parallel to the warp and weft direction.

In prior work [104], statistical information of this material is collected using 3-Dimages acquired via laboratory micro-CT. Use of micro-CT offers advantages over

126 Paper III

the optical imaging processes when spatial information needs to be analysed over theextent of a structure. A sample of unit cell dimensions is positioned in a GE Nanotomwith X-ray source parameters set to 33 kV and 295µA to acquire high resolutionimages with a voxel size of (6.75µm)3. From the 3-D volume representation, two-dimensional (2-D) slices are extracted in warp and weft directions. The warp cross-sections are characterised from slices normal to the weft direction and vice versa forthe weft tow cross-sections. Nineteen slices are analysed for each tow direction,yielding information at 0.75 mm intervals, so that each tow cross-over contact ispopulated by three data slices. The tow cross-sections along the path are afterwardsfitted with ellipses (see figure III2), which is a valid assumption for the tow shapeof the current topology as discussed in Olave et al. [76]. This procedure yieldsinformation about the tow path centroid coordinates (x, y, z), tow aspect ratioAR, towareaA and tow orientationθ in its cross-section.

Figure III2: Digital image of a cross-section in weft direction obtained by micro-CT.Ellipses are fitted to the warp cross-sections of the unit cell of ply 3.

Each ply is further analysed individually. First, geometrical information of the plies isderived with as most important outcome the mean of the unit cell periods: λx=11.55mm andλy=11.48 mm. Second, the tow paths in each ply are statisticallycharacterisedusing the method ofreference period collationwhere each tow parameter is projectedon a defined grid along the tow length [3]. This process exploits the nominalperiodicity of the textile to maximise the information derived from a small specimen.Data is collected for tows assigned to the samegenus, i.e. tows that should be identicalgiven the nominal periodicity of the textile. The particular 2/2 twill woven fabric canbe represented by two different genuses: one for the warp tows and one for the wefttows.

Information is collected about the tow parameters{ρ( j,t,s)sl , z( j,t,s)

sl ,AR( j,t,s)sl ,A( j,t,s)

sl , θ( j,t,s)sl }

with ρ = y for warp tows andx for weft tows. Data are taken from eachslice sl=1...19, tow label j=1...4 within the tow genus sett= warp or weft andply s=1...7. A statistical representation for the warp and weft genus is acquiredby partitioning each of the characteristics into periodic systematic variations andnon-periodic stochastic deviations. The decomposition ofeach tow genus can berepresented, byε representing one of the five parameters{ρ, z,AR,A, θ}, as:

ε( j,t,s)i =< ε

( j,t,s)i > +ǫ

( j,t,s)i (III1)

STATISTICAL DATA OF THE MATERIAL 127

with ǫ( j,t,s)i the zero-mean deviation from the systematic value< ε

( j,t,s)i > at locationi

along the towj of tow genust in ply s.

The systematic variations of parameters represent the average trends of each genus.The statistical properties of the tow parameters are definedin terms of the standarddeviation and correlation length of the deviations from thesystematic trend. Thecorrelation length is determined by linear approximation of the first autocorrelationvalues for small spacing of grid locations along one tow:

C( j,t,s)(k) = 1− kδ/ξ( j,t,s) (III2)

with C( j,t,s) the autocorrelation coefficient andξ( j,t,s) the correlation length. This linearfit is performed for the autocorrelation values of consecutive points up to five gridpoints (k ≤ 5) in order to exclude correlation values that are populatedby muchsmaller data sets and tend to lie beyond the zone of significant correlation6.

For the current data set, the systematic curves and deviations from the systematicpatterns are determined for each ply and each of the five parameters{ρ, z,AR,A, θ}.The results of this statistical analysis show that the ply systematic curves for eachparameter do not present significant differences between plies, except for the mildfeature seen asdips in the paths [104]. This effect is a result of compaction andmutual interaction of tows during the RTM production process. In figure III3, theply systematic out-of-plane warp genuses are presented foreach ply. All systematictrends are translated to have an equal starting point in order to derive an overall meansystematic curve. The ply systematic values for each parameter fluctuate aroundthis overall mean systematic curve with a standard deviation that is smaller thanthe standard deviation computed for the deviations of all measured tows to their plysystematic trend. This observation is also present for the weft tows and leads to theconclusion that the warp and weft tows from all the plies can be represented by oneoverall warp systematic curve and one overall weft systematic curve.

Similar conclusions are made for the fluctuations of each parameter around its plysystematic curve. The comparison of the cumulative probability density functionsof the parameter deviations for each ply shows no significantdifferences betweenthe plies for each single parameter [104]. Since the deviation statistics computed fordifferent plies are indistinguishable, the standard deviationand correlation length ofany parameter are derived using the data set of deviations combining data from allplies: the all-ply statistics data set. This increase in data set size is a necessity for thederivation of the correlation length. Significant variancein the correlograms (plot ofPearson’s correlation parameter in function of the separation of two points on a tow)for different plies is observed. All correlation lengths exhibit a significant amountof uncertainty, which can be reduced by increasing the data set size. For details and

6Two random variables are considered to be statistically independent if the correlation coefficient is lessthan±0.3 [39]. The zone of significant correlation is given by: [−1,−0.3] ∪ [0.3,1]

128 Paper III

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

x [mm]

z [m

m]

ply 1 ply 2 ply 3 ply 4 ply 5 ply 6 ply 7

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

x [mm]

z [m

m]

systematic

Figure III3: The translated out-of-plane centroid systematic curves for the differentwarp ply genuses. All ply systematic curves have a similar path which permits thederivation of an overall mean systematic curve.

discussion of the results, the reader is referred to [104].

The deviations generator in this paper is calibrated with the statistical informationof the centroid locations derived from the all-ply statistics data set. The standarddeviations and correlation lengths of these components areshown in table III1. Inaddition, the mean and the coefficient of variation (COV) of the derived correlationlengths for each individual ply are presented in table III2 to stress the large uncertaintypresent for this value. The generated virtual unit cells will also possess this highvariation in correlation length.

Table III1: Experimental standard deviations and correlation lengths of the centroidcoordinates using the all-ply statistics data set.

σx [mm] ξx [mm] σy [mm] ξy [mm] σz [mm] ξz [mm]

Warp genus - - 0.113 22.886 0.014 1.780Weft genus 0.063 9.419 - - 0.015 1.621

STATISTICAL DATA OF THE MATERIAL 129

Table III2: Experimental correlation length values for allwarp and weft centroidcoordinates.

ξwarp,sy [mm] ξ

warp,sz [mm] ξ

we f t,sx [mm] ξ

we f t,sz [mm]

< ξp > 18.19 2.27 10.90 1.95COVp

ξ 71.43% 38.53% 66.16% 33.15%

2.2 Verification of the Markov Chain assumptions

The generation of deviations with the Markov Chain algorithm is based on severalassumptions. In order to reproduce the same statistics of the measured sample,these conditions need to be verified or a modification needs tobe considered. Theassumptions are described in [6] with the generated deviations,ǫ, denoted with atildeto distinct from the experimental values (e.g. ˜ǫ=y or z for warp tows):

1. The distribution of any component of the simulated deviations ǫ, is the same forall data points on tows of the same genus.

2. The values taken by each component ˜ǫ at different points on the same tow arecorrelated to a degree that diminishes with the separation of the points.

3. The different components ˜ǫ at each grid point are statistically independent ofone another.

4. Each component ˜ǫ takes statistically independent values on different tows,regardless whether the tows are of the same or different genus.

Verification of these assumptions to the 2/2 twill woven textile data shows thatassumptions 1 and 2 are valid.

Assumptions 3 and 4 are checked by performing additional inter-tow correlationcalculations to the data set of the centroid locations. Three possible dependencies areanalysed: (i)C(ǫǫ′)

c : correlation between parameter deviationsǫ( j,t,s)i andǫ

′( j,t,s)i in one

tow and at one location, where the latter represents anothertow property at that samelocation i, (ii) C(ǫ,t)

c : correlation between neighbouring tows of the same genus for aparticular tow parameter:ǫ( j,t,s)

i andǫ( j′,t,s)i , and (iii) C(ǫwaǫwe)

c : correlation at typicalcross-over locations in the fabric, locations where warp and weft tows have equalcentroid coordinates in the xy-plane, between the warp componentǫ(wa,s)

cr,i and weft

componentǫ(we,p)cr,i . These dependencies are evaluated using the Pearson’s correlation

parameter for the particular data set. The first two are calculated for data of the same

130 Paper III

genus. As example, the correlation coefficient for evaluating the first dependencyC(ǫǫ′)c

is determined by

C(ǫǫ′)c =

∑ni=1 ǫ

( j,t,s)i ǫ

′( j,t,s)i

√

∑ni=1 (ǫ( j,t,s)

i )2√

∑ni=1 (ǫ

′( j,t,s)i )2

(III3)

with n the number of pairs{ǫ( j,t,s)i ǫ

′( j,t,s)i } extracted from all tows of the same genus and

over the seven plies. Since the cross-correlations at cross-over locations are calculatedbetween two different genuses, the correlation coefficientC(ǫwaǫwe)

c is computed as:

C(ǫwaǫwe)c =

∑ncr

i=1 ǫ(wa,s)cr,i ǫ

(we,s)cr,i

√

∑ncr

i=1 (ǫ(wa,s)cr,i )2

√

∑ncr

i=1 (ǫ(we,s)cr,i )2

(III4)

with ncr the total number of cross-overs in the sample. For each unit cell, sixteenlocations are present where warp and weft tows cross each other in the xy-plane (seefigure III1).

Table III3: Inter-tow correlations of (i) different centroid components at one gridlocation (data set size of 520) and (ii) the same centroid components between differenttows of the same genus (data set size of 390).

C(xy)c C(xz)

c C(yz)c C(x,t)

c C(y,t)c C(z,t)

c

Warp tows 0.009 0.007 -0.012 1 -0.615 0.155Weft tows 0.006 0.029 0.041 -0.438 1 0.233

Results of these analyses are presented in tables III3 and III4. No dependency ispresent between the different centroid parameters at one grid location sinceC(ǫǫ′)

c isvery low. The cross-correlationC(ǫ,t)

c has intermediate values for the in-plane centroidof the warp and weft genus. A weak correlation for this centroid is expected due to thefact that the average void length7 between consecutive warp or weft tows, equal to 0.26mm, is at least two times the warp or weft standard deviation for this coordinate (seetable III1). The dependency of the warp and weft tow components at the cross-overlocations is negligible except for the z-centroid. An intermediate correlation valuefor this centroid is expected due to the RTM production process that presses the dryfabric in the z-direction to the desired thickness before impregnating the resin. Thecross-correlations at cross-over locations follow the same trend as the interlock weavedescribed in [3], but with a significantly lowerC(zwazwe)

c correlation.

Although the size of the data set is large, the values of the correlations are biased dueto the fact that consecutive points are present on the same tow (a total of 28 tows are

7The void length is the free length between two neighbouring tows calculated as the nominal tow spacingsubtracted with the nominal tow width.

STOCHASTIC MODELLING OF THE 2/2 TWILL WOVEN TEXTILE COMPOSITE 131

present per genus in the sample). The limited data set results in cross-correlations thatare considered as weak for any of the three dependencies. Therefore, it is furtherassumed that the inter-tow correlations between parameters of the same genus orbetween different genuses are negligible and not taken into account in the generationof deviations.

Table III4: Cross-correlations of the centroids at cross-over locations for a data setsize of 112.

C(ywaxwe)c C(ywazwe)

c C(zwaxwe)c C(zwazwe)

c

-0.030 0.031 -0.012 0.603

3 Stochastic modelling of the 2/2 twill woven

textile composite

3.1 Modelling approach

The aim is to generate virtual random 2/2 twill woven unit cells that have the samestatistics for the centroid positions as in the measured sample. The cross-sectionalparameters along the tow path are fixed to their nominal value. This modellingapproach uses the WiseTex software [113] to create the stochastic model based onthe tow path information. WiseTex is able to construct the internal geometry of a widerange of textile reinforcements [56, 57, 59], including this 2/2 twill woven composite,for a limited set of input parameters: tow properties, particular topology and towspacing. These parameters include all the textile mechanics corresponding to the towpath. A numerical model is built using an energy minimisation algorithm of the towdeformation for a given crimp value. The nominal WiseTex centroid description ofthe subjected reinforcement path is generalised to stochastic representation.

The random tow instances in the unit cells are created with the all-ply statistics dataset in order to be representative for any ply. In these structures, the present dips in theout-of-plane centroid path (see figure III3) are not taken into account in the models.This feature varies in form substantially with no dip at the surface plies due to the flatplatens of the RTM moulding process [104]. When more data about these dips arecollected from different experimental specimens, the out-of-plane centroid path canbe updated with a variable dip size for unit cells representative for the inner laminateplies.

132 Paper III

Figure III4: Procedure of the warp tow discretisation on an equally spaced grid of 32points.

Each tow is represented by a discrete set of values originating from a grid along itstow direction as presented in figure III4. The grid length forwarp and weft tows isaccordingly chosen to the mean value of all measured ply periods for both directions(λx=11.55 mm,λy=11.48 mm). The reinforcement of an arbitrarily sized specimen isafterwards built by translating the generated tows using the vector (kλx/4~ex+ lλy/4~ey),with ~ex, ~ey the unit vectors andk, l taking an integer value{0,1,2,3}, depending onthe tow type and location within the unit cell. No additionalin-plane twist of thetows is considered. The procedure results in virtual unit cells with fixed dimensions inwarp and weft direction. These dimensions are only enlargedwhen the most outwardtows, close to the unit cell boundary, exceed the boundary location. In all other cases,the unit cell dimensions are set toλx=11.55 mm andλy=11.48 mm. This is a validassumption since, due to the variation, no relationship is found between the tow pathlength and its corresponding unit cell dimension. The thickness dimension of eachunit cell is variable, depending on the individual out-of-plane tow path variation. Atotal of 32 grid points are chosen along the grid length to obtain a grid spacing which isseveral times less than the correlation length of the z-centroid. This parameter has thesmallest correlation length for warp and weft tows (see table III1). The resulting gridspacing ensures that the final smoothing step for generatingthe tow deviations will notchange the statistics too much. Smoothing is required to obtain a tow shape withoutany non-physical irregularities since the Markov Chain procedure can generate spikesat short distances.

An overview of the stochastic multi-scale modelling approach is given in figure III5.The systematic value of each grid location for any parameteris constructed with theWiseTex representation of the woven fabric. The grid valuesare extracted by readingan XML-output from WiseTex that contains all information ofthe WiseTex model. Ina next step, the deviations of all centroids are generated inMatlab by implementationof the Markov Chain. The final grid values along the tow lengthare obtained byadding the deviations to the systematic values at each grid location. These values, in

STOCHASTIC MODELLING OF THE 2/2 TWILL WOVEN TEXTILE COMPOSITE 133

Figure III5: Overview of the stochastic multi-scale modelling approach.

combination with a new definition of the orientation vectorsdefining the tow path,are used to overwrite the original values of tow path description in the XML-file ofthe woven model. The XML-file is afterwards the input in WiseTex to create thestochastic model of the unit cell or to micromechanical analysis tools for evaluationof the mechanical properties.

3.2 Systematic patterns generated by WiseTex

From the stochastic characterisation of the woven materialin [104], the average towpath trends, or systematic variations, of each genus are available for each tow pathparameter. When comparing these trends with the tow path representation of thesame topology in WiseTex, only minor differences are observed. The out-of-planeand in-plane centroid systematic curves, computed from previous analysis and fromWiseTex, are presented in figure III6. The out-of-plane systematic trend is similar tothe z-centroid determined by WiseTex, except for the present dips in the middle ofthe twill float. Using WiseTex to extract the systematic curve removes this featurein the developed stochastic model. The in-plane systematiccurve obtained from thestochastic analysis is fluctuating around the zero-axis with a standard deviation that isone magnitude lower than the standard deviation of the in-plane deviations (table III1).Since no clear pattern is observed in this experimental path, the mean behaviour of thein-plane component is taken as a straight line in the construction of virtual specimen.

The systematic values of each grid location along the tow length (32 in total) aretaken from the original WiseTex representation. The WiseTex model has unit celldimensions and tow properties derived from the stochastic analysis. When larger

134 Paper III

0 5 10 15−0.1

−0.05

0

0.05

0.1

x [mm]

z [m

m]

data WiseTex

(a) Warp out-of-plane centroid systematic

0 5 10 15−0.02

−0.01

0

0.01

0.02

x [mm]

y [m

m]

data WiseTex

(b) Warp in-plane centroid systematic

Figure III6: Comparison of the systematic curves resultingfrom the stochasticcharacterisation and the nominal WiseTex model. (a) No significant differences insystematic curves are observed for the out-of-plane coordinate except for the presentdips. (b) The in-plane coordinate curve from the experimental procedure is fluctuatingaround the zero-axis without a clear trend.

differences in systematic trends are observed, which is the casefor the tow cross-sectional shape parameters (see [104]), the systematic values are also taken from thestochastic characterisation. This does not introduce any additional difficulties.

3.3 Generation of centroid deviations with the Markov Chainalgorithm

The centroid fluctuations around the mean centroid patternsare generated with theMarkov Chain algorithm proposed by Blacklock et al. [6]. As for the angle interlockweave of Bale et al. [3], no significant inter-tow correlations for the same genus arepresent. Also the considered 2/2 twill woven topology does not exhibit significantinter-tow correlations at the cross-overs, which was the case for the z-centroid ofthe angle interlock weave. This allows reproducing the deviations of warp and wefttows for the 2/2 twill woven fabric independently without any modificationwhencombining warp and weft tows into a unit cell structure.

The input parameters of the generation algorithm are accordingly chosen to thestatistical information of the material. The all-ply statistics data set of each parameteris discretised on an interval{−ma,−(m− 1)a, ...,0, ..., (m− 1)a,ma} with grid spacinga and number of intervalsm that satisfy the relationma = 3σǫ . The possible valuesof generated deviations are limited by the 3σ spread in the experimental data set. Dueto the high standard deviation of the in-plane coordinate for warp and weft tows,

STOCHASTIC MODELLING OF THE 2/2 TWILL WOVEN TEXTILE COMPOSITE 135

m is given the value of 20. A lowerm value, with corresponding higher intervalwidths a, would have several undesired effects: (i) high amplitude of spikes in thegenerated deviations along the tow length and (ii) unrealistic jumps in the centroidspath according to the data. A higherm value is not required and would lead to longercomputation times.

The probabilities corresponding to the discretised interval are collected in thedistribution vectorPǫ

i for locationi, with T denoting the transpose operation:

Pǫi =

[

p(i)m p(i)

m−1 ... p(i)0 ... p(i)

−m+1 p(i)−m

]T(III5)

The probability of the next grid locationi+1 for the same parameterPǫi+1 is determined

by the Markovian procedure:Pǫ

i+1 = AtransPǫi (III6)

with Atrans the probability transition matrix. In the initial form, this transition matrixis represented as:

Atrans,0 =

α γ 0 . . 0β′ α γ 0 . .

0 β α . . .

. 0 . . γ 0 .

. . β α ζ 0 .

. 0 β α β 0 .

. 0 ζ α β . .

. 0 γ . . 0 .

. . . α β 0. 0 γ α β′

0 . . 0 γ α

(III7)

All elements have values between [0,1] and each column adds to unity. The generateddeviations ˜ǫi are controlled by varying the relative magnitudes ofβ andγ. The value ofα is fixed and arbitrarily chosen to be 0.9 [6]. The transition matrix is calibrated withthe standard deviation and correlation length of the considered parameter. First, theproduced standard deviation of the fluctuations is set to be equal to the experimentalstandard deviation. This condition permits determining the element values ofβ andγ.Knowing these values, the generated correlation length is fitted to the experimentallyobtained value by iterative application of the tri-diagonal transition matrixAtrans,0 tillthe experimental correlation length is simulated. This ensures that the transition rule,with Atrans the transition matrix in its final form, will generate deviations that havethe target statistics of the all-ply statistics data set. The Markovian procedure isimplemented in an operating algorithm, described in [6], tocreate any fluctuationvalue at each grid point by marching along the specific genus.The procedure isrepeated for one parameter at a time since the tow centroids are not correlated at

136 Paper III

one grid point as discussed in section 2.2. Each parameter has its (2m+1) by (2m+1)tri-diagonal probability transition matrix.

The output is a set of deviations along each tow possessing high-amplitude long-range wavelength variations equal to the correlation length of the measured sample,with low-amplitude short-range wavelength fluctuations. The latter fluctuations canappear as sharp spikes which are not physical and also not observed in the measuredsample. A smoothing operation is proposed to reduce these spikes by applying a localaveraging method over an interval length that is significantly smaller than the shortestexperimentally derived correlation length. The procedureis an adapted version of themoving average that ensures that the standard deviation of the generated deviationsis preserved. The smoothed deviation values ˜ǫ

(s)i are obtained for the current material

and settings by:

ǫ(s)i = S

√

√

√

13

1∑

j=−1

ǫ2i+ j (III8)

with S = ±1 a sign factor having the sign of all the non-zero deviationsin the sum ofequation III8. The deviation value at one location is smoothed by using informationof ±2 neighbouring grid points (recursion depth= 2). If the subjected deviationsfor one smoothing step have different signs, the proposed smoothing operation is notconsidered. The conventional moving averaging rule is thanapplied (see appendix Bof [6] for more information). This procedure is applied for all generated deviations.

As validation of the algorithm, a total of 4000 warp and weft tow centroid deviationsare generated to construct 1000 unit cells with variable towreinforcement. Results ofthe procedure are demonstrated for the warp genus.

When the deviations of all warp tows are combined in one data set, the correlogramof the simulated deviations can be compared with the experimentally obtainedautocorrelation values for equal spacing between neighbouring points. Figure III7(a)presents the correlogram for the simulated and experimentally obtained deviations ofthe warp in-plane centroid. The autocorrelation graph shows good agreement betweenthe simulated and experimentally determined values. This proves that the MarkovChain generates in-plane warp centroid deviations that have the same statistics as theexperimental sample. Figure III7(b) shows a set of deviations for the warp in-planecentroid. The simulated deviation values along the tow length demonstrate the needfor the smoothing operation.

The conformity of the statistics for each particular randomunit cell to the experimentalunit cell is also verified. As example, figure III8 and III9 report the unit cell statisticsof the warp tow parameters in histograms, respectively for the out-of-plane centroidand in-plane centroid. Both the standard deviation and correlation length of each unitcell are presented (non-smoothed and smoothed) and compared with the results of theexperimental characterisation of the material. The experimental standard deviations

STOCHASTIC MODELLING OF THE 2/2 TWILL WOVEN TEXTILE COMPOSITE 137

0 2 4 6 8 10 120

0.5

1

Distance [mm]

Cor

rela

tion

[−]

DataLinear fitMarkov Chain

(a) Correlogram of the warp in-plane centroid

0 2 4 6 8 10 12−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

x [mm]

y [m

m]

Systematic curveGenerated deviationsDeviations after smoothing

(b) Series of simulated warp in-plane centroid deviations for one tow

Figure III7: (a) Correlogram of in-plane warp centroid for the experimentallymeasured deviations and simulated deviations with the Markov Chain. The MarkovChain reproduces correlation values that coincide with thelinear fit of experimentaldata used to deriveζy. (b) Generation of an in-plane warp tow path with and withoutapplying the smoothing. The smoothing operation removes small amplitude spikes.

are those computed from the all-ply statistics data set since no differences in thesevalues are present between the plies. Simulated correlation lengths are compared withthe mean values of the experimental data set consisting of the correlation length ofeach individual ply (see table III2).

The histograms of the standard deviations of both centroid parameters are centredaround the experimental standard deviation. When smoothingis applied to removethe sharp spikes in the generated tow paths, the standard deviation of the out-of-planecoordinate is slightly affected. The proposed averaging procedure, with conservationof the standard deviation, cannot be used when one or more points in the averaginginterval have a different sign. This is an effect of a small correlation length, such as forthe out-of-plane centroid, causing the deviations to oftenchange sign. The standard

138 Paper III

0 0.01 0.02 0.030

20

40

60

80

100

120

σz

Fre

quen

cy

0 5 100

50

100

150

200

250

ζz

Fre

quen

cy

simulated experiment

(a) Statistics of non-smoothed deviations

0 0.01 0.02 0.030

20

40

60

80

100

120

σz

Fre

quen

cy

0 5 100

50

100

150

200

250

ζz

Fre

quen

cy

simulated experiment

(b) Statistics of smoothed deviations

Figure III8: The unit cell statistics of the out-of-plane warp centroid (a) without and (b)with smoothing. The simulated standard deviation is slightly shifted after smoothing.Altough, a good comparison with the experimental statistics is obtained on average.

moving averaging method is than applied that does not conserve the standard deviation.The simulated in-plane component’s standard deviations show large variations in thehistogram caused by the high experimental standard deviation (see table III1). This isalso demonstrated in table III5. The y-coordinate deviations of the warp tows are notaffected by the smoothing.

The simulated correlation lengths histograms show alreadyhigh variation for the non-smoothed values. The cause of this high variation is (i) the high sensitivity of the

STOCHASTIC MODELLING OF THE 2/2 TWILL WOVEN TEXTILE COMPOSITE 139

correlation length to outliers and (ii) the limited data setfrom which the correlationlength is derived. This high variation is also present when considering the high COVvalues for correlation lengths of different plies shown in table III2. When smoothingis applied, the correlation length values are shifted to higher values. This is anobvious result when the value of a certain parameter is made more dependent on itsneighbouring points. However, the magnitude of the correlations is still acceptable asshown in table III5.

Table III5: Results of the statistics of the generated warp centroid deviations: with andwithout smoothing.

σy [mm] σz [mm] ξy [mm] ξz [mm]

Warp simulated -no smoothing 0.114 0.014 24.162 2.260Warp simulated -with smoothing 0.114 0.013 39.642 3.570

Replicas of each tow parameter are thus created by marching sequentially along thelength of the tow. The Markov Chain algorithm is able to reproduce the statisticalinformation correctly on average.

3.4 Stochastic representation in the WiseTex software

Stochastic instances of the tow reinforcement are obtainedby combining thesystematic values from WiseTex with the generated deviations from the Markov Chain.The coordinates are translated up and down from their systematic value to obtain thefinal centroid coordinates.

Each new tow path is constructed by defining the path length and orientation of thetow parameters. A cubic spline is fitted to the centroid positions to derive the length ofthe centroid line of all segments, defined by successive gridlocations. The directionof each tow segment is further determined by the set of vectors {t,a1,a2,n,b} [56, 57].The direction of the centroid line of a tow is indicated by thetangent vectort = dr/ds.The cross-section is represented by vectorsa1 anda2 according to the minor and majoraxis of the ellipse shape. Both latter vectors define one plane with the tangent vectort normal to it, as presented in figure III10. The remaining normal n and bi-normalbvectors are defined respectively asdt/dsandt × n.

The values of centroid coordinates, path length and orientation vectors in the nominalWiseTex XML-file are overwritten by the updated values for each tow to obtainstochastic realisations of the unit cell. Figure III11 shows different views of the virtual2/2 twill woven specimen created in the WiseTex software. Eachvirtual unit cell has

140 Paper III

0 0.1 0.2 0.30

20

40

60

80

100

120

σy

Fre

quen

cy

0 50 100 1500

50

100

150

200

250

ζy

Fre

quen

cy

simulated experiment

(a) Statistics of non-smoothed deviations

0 0.1 0.2 0.30

20

40

60

80

100

120

σy

Fre

quen

cy

0 50 100 1500

50

100

150

200

250

ζy

Fre

quen

cy

simulated experiment

(b) Statistics of smoothed deviations

Figure III9: The unit cell statistics of the in-plane warp centroid without (a) andwith smoothing (b). No significant effect of smoothing is observed for the standarddeviation. A good comparison with the experimental statistics is obtained on average.

fixed in-plane dimensions when no tows cross the unit cell boundary, as discussed insection 3.1, but a different thickness depending on the variable out-of-plane towpathbehaviour.

DISCUSSION 141

Figure III10: Orientation vectors of a tow cross-section used in WiseTex [57].

4 Discussion

The individual generation of random tows, without any correlation between neigh-bouring tows of the same type or different type, can lead to possible interpenetration.Although the procedure in this paper only shifts the centroid positions and keeps thecross-sectional parameters to their systematic values, this interpenetration is presentfor a few specimens when tow parameters are added with high deviation values (closeto 3σ) as could be generated by the Markov Chain. When variability of the ellipse axesin the tow cross-section is introduced, interpenetration has a much higher probabilityto occur. This demands for an adapted approach where tows canbe translatedaccording to topological rules [57, 81]. When performing these additional operations,it is important to check that the statistics of the model do not change, but also thatthe fibre volume fraction of the unit cell still matches the experimentally measuredvariation.

The current model is built with elliptical cross-sections.Although the tow cross-sections do not represent ideal elliptical shapes, fitting ellipses is still a reasonableapproach. It permits to capture all the tow path statistics approximately correct [104].When modelling the structure, it is also consistent to represent the reinforcementas overlapping ellipsoids when evaluating the stiffness properties using the Mori-Tanaka scheme [42]. However, minor modifications to the elliptical cross-section fortows in the reinforcement need to be made for damage modelling and stress-straincomputations where the actual shape of the tow should be considered.

The XML format of the virtual textile in the WiseTex softwarepermits scriptingof local information on reinforcement, without the need of access or knowledgeof the internal computational procedure. The output formatcan be further usedfor simulation of the composite unit cell reinforcement properties and behaviour,

142 Paper III

(a) (b)

(c)

Figure III11: Different views of the created random virtual unit cell in the WiseTexrepresentation: (a) general overview, (b) top view and (c) tilted side view.

with compatibility of micromechanical analysis [42, 56, 76, 113] and permeabilitysimulations [112]. The meso-level model can also be translated to a meso-level finiteelement model [57, 58]. The results from the meso-level mechanics can be afterwardstransferred to the macro-level for application of structural, forming or impregnationanalysis.

5 Conclusions

A unit cell with random positioning of the tow paths is constructed in WiseTex thatpossesses the same statistics as derived from an experimental sample. The tools tocreate a stochastic unit cell are implemented independently from the WiseTex softwareby using the XML-output format of a nominal WiseTex model. This permits using theefficient Markov Chain algorithm to generate the zero-mean deviations of the tow

CONCLUSIONS 143

centroid parameters that are afterwards added to the systematic patterns of the towpaths. The random reinforcement model is obtained by changing the values of thetow properties input, concerning centroid locations and tow path orientations, in theoriginal XML-file of the nominal model. The tow path fluctuations in the modelshave a standard deviation and correlation length that correspond to the statisticalcharacterisation of the same 2/2 twill woven composite in prior work. When manyvirtual samples are constructed, the statistics of the mechanical properties can beevaluated using meso-level models of micromechanics whichare compatible with theWiseTex format.

Acknowledgements

This study is supported by the Flemish Government through FWO-Vlaanderen(project G.0354.10), the Agency for Innovation by Science and Technology inFlanders (IWT) and by the Air Force Office of Scientific Research (Dr. Ali Sayir)and NASA (Dr. Anthony Calomino) under the National Hypersonic Science Centerfor Materials and Structures (AFOSR Contract No. FA9550-09-1-0477). Dr. M.Blacklock and Dr. R. Rinaldi from UC Santa Barbara are acknowledged for theiruseful discussions about the Markov Chain algorithm.

Paper IV - Simulation of thecross-correlated positions ofin-plane tow centroids intextile composites based onexperimental data

145

146 Paper IV

Published in Composite Structures 116 (2014) 75-838

Simulation of the cross-correlated positions of in-plane tow centroids in textilecomposites based on experimental data

Andy Vanaerschota, Brian N. Coxb, Stepan V. Lomovc and Dirk Vandepittea

aKatholieke Universiteit Leuven, Department of MechanicalEngineering, Leuven, BelgiumbTeledyne Scientific Co. LLC, Thousand Oaks, CA, USA

cKatholieke Universiteit Leuven, Department of Metallurgyand Materials Engineering, Leuven, Belgium

Abstract In-plane centroids of textile composites are simulated as cross-correlatedrandom fields. Each tow position is defined as an average trendquantified fromexperimental data, added with zero-mean deviations produced as a stochastic field.Realisations of these fields are generated using a frameworkbased on the Karhunen-Loève series expansion that is calibrated with experimental information from priorwork. Positional deviations are obtained that are correlated along the tow and betweenneighbouring tows.

The application is a 2/2 twill woven carbon fibre reinforced epoxy consisting ofmultiple unit cells. Generated in-plane deviations of the warp and weft tows resemblethe experimental fluctuations with similar wavelengths. Simulation of thousandspecimens demonstrates that the virtual in-plane positions possess the experimentalstandard deviation and correlation lengths on average.

8Symbols can differ with original publication to preserve consistency in theentire manuscript.

INTRODUCTION 147

1 Introduction

The reinforcement of a textile composite is susceptible to asignificant amount ofvariability. Though, composites are often modelled without considering randomnessin the tow path descriptions. Such ideal representations donot appear in physicalsamples and lead to unreliable results because the performance of a textile is stronglylinked to its geometrical structure. Any variation in the tow path will influencethe properties of the final composite. The effect of geometrical scatter on themechanical performance is already investigated for the elastic mechanical properties[22, 76], formability [94], permeability [27, 28] and damage initiation and propagation[26, 120, 122]. A correct identification of the spatial geometrical fluctuations of atextile product improves the quality of numerical analysesand permits to quantify theeffect of variability on the mechanical performance.

This deficiency in realistic simulations is partially attributed to the lack of experi-mental data with a thorough statistical analysis. Researchers who are quantifying thevariation in geometry usually do not consider correlation.However, the description ofdependency of a single property along its tow length or between different parametersat one location, is required to reproduce the correct geometry of the material. Thisis already demonstrated for several types of composites [67]. Further, modellingtechniques are often inadequate to introduce local and long-range variations in thetow path and accurately simulate the desired correlation structures [14]. It is not theobjective to simulate more precise representations without considering experimentaldata, such as in [96], but to calibrate the numerical modelling procedure with themeasured variations. Realistic representations of textile composites are acquired by atwo-step procedure [10]: (i) collection of sufficient experimental data on the spatiallycorrelated short- and long-range geometrical variations,and (ii) derivation of themacroscopic mechanical properties from the lower scale geometrical characteristics.This paper describes the last step of the approach of Charmpis et al. [10] and ispreceded by several other publications [104, 105, 107]. A methodology is proposed togenerate the cross-correlated geometrical variability oftextiles typically present at thelong-range, and is demonstrated for a 2/2 twill woven carbon fibre reinforced epoxycomposite.

Long-range variations are frequently omitted when modelling component-size textilesamples. Generally, representative volume elements are assembled to generate anysize of composite, but this is not a full replica of the real reinforcement structure.Simulation of long-range variations in textile compositesis only sparsely tackledin literature. Skordos and Sutcliffe [94] investigated the in-plane behaviour of acarbon-epoxy satin weave for modelling the forming process. Tow orientations aresimulated as an Ornstein-Uhlenbeck process of which the parameters are estimatedby the likelihood function. Next, realisations of woven sheets are obtained usingthe Cholesky decomposition of the covariance structure of warp and weft direction

148 Paper IV

in combination with normally distributed independent variables. Values of warp andweft orientation and unit cell size are generated at discrete points that compromisethe warp and weft direction and size of the unit cell. Endruweit et al. [27]performed stochastic simulations of the resin injection inbi-directional non-crimpfabrics. Random fields of fibre distances are constructed using spectral representationswith trigonometric functions to implement spatial correlation along the tow. Thefrequency and phase of each function are determined randomly on given intervals,with an upper limit for frequencies determined by the tow mobility in the transversedirection observed in actual fabrics and the phases between0 andπ. Abdiwi et al.[1] reproduced full-field variability of the tow directionsacross flat sheets based onmeasured variability of inter-tow angles. The geometry mesh is generated with a pin-jointed net kinematics code where variability is added by stretching and additionalperturbations of the nodes. These perturbations are introduced using sinusoidalfunctions with particular wavelength and vertical amplitude to simulate the long-range correlation effects. No short-range variability is considered. This code isimplemented in a genetic algorithm with objective functions that minimises the meanand standard deviation of the measured inter-tow angle to reproduce similar statisticalvariations as measured. Series expansion techniques are also employed by Yushanovand Bogdanovich [118] to generate a random reinforcement path characterised byits mean value and covariance matrix. The stochastic reinforcement is generated bydefining a suitable position vector with a random vector function which is expandedinto a deterministic component and a random component. A stochastic local basisis afterwards introduced to uniquely define the directionalcosines of the arbitraryreinforcement path, specified by the position vector.

Except for the contribution of Skordos and Sutcliffe, all described publicationsconsider an approximated correlation function along the tow direction which isnot validated with experimental data. The presence of cross-correlation betweenneighbouring tows is also not tackled or is indirectly introduced. Experimentalvalidation of the input data and correct introduction of allcorrelation structuresare however mandatory to replicate the internal geometry. To precisely model thelong-range tow path parameters, a methodology developed byVorechovský [114] ischosen which is not yet applied in the field of composites. Vorechovský proposed aframework for generating cross-correlated random fields based on Karhunen-Loèveseries expansion. Each random field must share an identical auto-correlation, whilethe cross-correlation structure between each pair of fieldscan be described by a simplecoefficient.

This paper discusses the implementation of the procedure ofVorechovský forgenerating Gaussian random fields of the in-plane centreline of the tow path, calledcentroid. Each stochastic field corresponds to a single tow. The methodology iscalibrated with the experimental data of [107]. In summary,the paper discusses the (i)collection of statistical information from experimental samples and (ii) the application

EXPERIMENTAL DATA 149

of the method of Vorechovský [114] in the field of textile composites to simulatethe in-plane centroid positions. A comparison is made between the experimentaland simulated statistical information to demonstrate the accuracy of the generationprocedure.

2 Experimental data

The developed methodology is demonstrated for a 2/2 twill woven Hexcel fabric(G0986) [40] impregnated with epoxy resin. Each reinforcement unit cell is built withfour equally spaced warp and weft tows consisting of 6K carbon fibres, with a nominalareal density of 285 g/m2. The idealised unit cell topology is given in figure IV1 withλx=11.43 mm andλy=11.43 mm, respectively the periodic lengths of warp (x-axis)and weft (y-axis) tows as specified by the manufacturer.

Figure IV1: WiseTex model of a 2/2 twill woven reinforcement. The x-axis and y-axisof the coordinate system are respectively parallel to the warp and weft direction.

Data of the in-plane centroids are collected in [107] from two single-ply carbon fabricsimpregnated with epoxy in a resin transfer moulding (RTM) process. The in-planedimension of both samples (sample 1 & 2) is quantified using optical imaging over asquare region of thirteen by thirteen unit cells to investigate the long-range effect. In-plane coordinates of forty warp and forty weft tows are derived that represent a totalof hundred unit cells by manually marking the boundaries of tows for prescribed gridspacings. Centroid locations are subsequently defined as half the tow width at eachgrid location. A single tow in-plane centroid is afterwardsdefined by connecting theselocations along the entire grid. All coordinates are given as input to Matlab where theyare transformed to a global axis system and verified if a possible shift in the sampledata should be removed due to the manual placement of the composite in the scanningdevice [107].

The digital image of sample 1 in figure IV2 shows that the in-plane centroid of asingle tow does not follow a straight path over the entire sample. The dots in the

150 Paper IV

Figure IV2: Optical scan of a one-ply 2/2 twill woven carbon fibre fabric impregnatedwith epoxy resin. Warp tows are oriented horizontally, while weft tows are positionedin the vertical direction. The red square indicates the region where the in-planeposition of the centroid is characterised.

image, representing the in-plane tow path, clearly deflect from the straight dashed line.The in-plane movement in transverse direction of the path isquantified by computingthe difference between the experimental tow paths and a best-fitted ideal latticedescription. Tows of this lattice are represented as straight lines, with nominal spacingin x- and y-direction derived from the experimentally obtained periodic lengths. Thein-plane warp and weft fluctuations are considered respectively in y- and x-direction.

The obtained deviations are represented asε( j,t,s)i , with i the grid location (i = 1..Ni ,

Ni = 40), j the tow index (j = 1..N j , N j = 40) in each direction,t= warp or wefttows ands=1 or 2 referring to the sample. A non-periodic trend along thetowlength is observed which presumably originates from the handling of the materialduring storage, cutting and placement in the RTM mould. Thistrend< ε

( j,t,s)i >

(see figure IV6) is determined as the average value per grid location of all in-planedeviations in one sample. Next, in-plane deviationsε

( j,t,s)i are decomposed in:

ε( j,t,s)i =< ε

( j,t,s)i > +ǫ

( j,t,s)i (IV1)

EXPERIMENTAL DATA 151

with ǫ( j,t,s)i the stochastic variations attributed to the manufacturingprocess of the

fabric itself, further referred asdeviations. The stochastic variations of sample 1 and2 are combined in one larger data set per tow typeǫ

( j,t)i , assuming that no physical

differences are present between theǫ( j,t,s)i of the samples. Considering the weaving

process of the 2/2 twill woven fabric, the warp tows can be represented by onerepresentative tow, calledgenus, and similar for the weft tows. The experimentaldeviationsǫ( j,t)

i are presented in figure IV3 showing that the in-plane warp deviationshave a short wavelength, while the weft in-plane centroids fluctuate with a longerwavelength.

0 20 40 60 80 100 120−0.4

−0.2

0

0.2

0.4

x [mm]

y [mm]

0 20 40 60 80 100 120

−2

−1

0

1

y [mm]

x [mm]

Figure IV3: Experimental deviation trendsǫ( j,t) of warp (left) and weft (right) tows.

In contrast to the short-range variations in [104], which are fairly represented bynormal distributions, in-plane deviationsǫ( j,t)

i show large deflections from normality.Especially the weft deviations seem to follow a particular distribution. However, fromthe limited amount of samples it is not possible to verify if another type of distributioncould be more relevant. Any small shift in periodicity during fabric production alreadyresults in a very different weft tow spacing for such samples. The assumption ofnormality is still acceptable as is indicated by the kurtosis and skewness values9 intable IV1. Warp in-plane deviations are well approximated by a normal distribution,with the largest differences occurring in the tails where a lower frequency is present.

The statistical behaviour of the deviationsǫ( j,t)i are further described in terms of

standard deviationσ and correlation information. Correlation is investigatedusing thePearson’s moment correlation parameter for pairs of data taken at distinct locations ona single tow (auto-correlation Cauto), and pairs of data on neighbouring tows but fixedat the same grid location on a tow (cross-correlation Ccross). The deduced dependencycan be presented in a correlation graph that shows the correlation data in terms ofdistances, orlags, between the two data points from which the correlation is computed.

9The skewness describes the lack of symmetry in the distribution, while the kurtosis value refers to thepeakedness of the distribution

152 Paper IV

Table IV1: Skewness and kurtosis values of stochastic in-plane deviations.

normal distribution warp tows weft tows

Kurtosis 3 2.49 3.83Skewness 0 0.18 -0.52

Definitions of both correlation directions are given in figure IV4. The appearingtrend in the correlation graphs are fitted using exponentialand squared exponentialcorrelation functions to estimate the correlation lengthξ, which is a measure of therange over which fluctuations at a certain position still have an influence on thefluctuations at another location. This fitting procedure only considers the first twentylags which correspond to the length of five unit cells. Largerpoint spacings are notused to fit a correlation trend since these data are based on a smaller data set size andthus subjected to a larger variability. The auto- and cross-correlation behaviour is wellrepresented by an exponential correlation function for thewarp in-plane deviations,while the squared exponential function is a better fit for both correlation directions ofthe weft tows.

66

6.2

6.4

6.6

6.8

67

7.2

Tow j+1

Tow j

Auto-correlation

Cross-

correlation

Figure IV4: Definition of spatial dependencies of deviations demonstrated fortwo weft tows: auto-correlation (along the tow) and cross-correlation (betweenneighbouring tows).

Variation in the in-plane centroid, indicated by the standard deviation, is five timeshigher for the weft direction. The in-plane warp correlation lengthξauto along the towis approximately twice of the weft tows, reflecting the tensioning of the warp towsand the lack of tensioning of the weft tows during production. Individual weft towsare less controlled in their path, leading to bundling of neighbouring weft tows asobserved in figure IV2. This effect is translated in a higher cross-correlation lengthξcrossfor the weft tows, exceeding the unit cell dimension. A more detailed discussionof the procedure and results is given in [107].

SIMULATION OF THE IN-PLANE CENTROIDS 153

Table IV2: Standard deviation, auto- and cross-correlation lengths of the in-planecentroid using the combined data set.

Warp tows Weft tows

σ [mm] 0.106 0.615ξauto [mm] 114.89 52.89ξcross [mm] 4.49 13.16

3 Simulation of the in-plane centroids

3.1 Overview

The in-plane centroids of a thousand virtual textile modelsare generated that consistof forty warp and weft tows. Each tow centroid is representedby forty-one pointsalong an equidistant grid (figure IV5) such that the length between five consecutivegrid locations corresponds to the computed periodic lengthof the experimental warpor weft tow, respectively equal toλx = 11.48 andλy = 11.38 [107]. This limitedamount of grid locations is sufficient to represent the in-plane centroid since no short-range variations are observed in the unit cell sample of [104]. A stochastic descriptionof the tow path is obtained by combining the experimental handling trend with cross-correlated long-range deviations. The deviations of each tow are simulated as a singlerandom field using the statistical information of section 2.Warp and weft in-planedeviations are generated independently, assuming that thedeviations are normallydistributed.

3.2 Handling trend

Figure IV6 presents the average trend< ε( j,t,s)i > of the in-plane warp and weft genus.

As discussed in section 2, the lack of periodicity signifies that this tendency shouldnot be interpreted as a systematic trend, representing the repetitive mean behaviour ofthe tow path, but as an effect due to handling. Zero-mean deviations are added to thehandling trend to obtain random realisations of the in-plane coordinates.

3.3 Long-range in-plane deviations

Reproducing the cross-correlated long-range statistics requires a simultaneous gener-ation of all in-plane deviations belonging to the same genuswithin a single specimen.

154 Paper IV

1 41

Figure IV5: Grid representation of the simulated in-plane deviations.

0 20 40 60 80 100 120

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x [mm]

⟨ε(j,warp) ⟩ [mm]

Sample 1

Sample 2

Combined

0 20 40 60 80 100 120

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

y [mm]

⟨ε(j,weft) ⟩ [mm]

Sample 1

Sample 2

Combined

Figure IV6: Handling trend of warp (left) and weft (right) tows.

This strategy is assimilated in a methodology by Vorechovský [114] where cross-correlated material properties of a concrete structure aresimulated. The procedureis however generally applicable for the modelling of a set ofstochastic fields whichshare an identical auto-correlation structure and of whichthe cross-correlation canbe defined by a cross-correlation coefficient. The method is a generalisation of thesimulation of independent distinct one-dimensional (1-D)or univariaterandom fieldsto cross-correlated univariate fields, further referred asmultivariate fields. First abrief description of univariate fields is given to understand the principles used in theprocedure of Vorechovský.

SIMULATION OF THE IN-PLANE CENTROIDS 155

3.3.1 Description of the methodology

A stochastic property is described by a univariate random field H(x, ϑ) [109] thatrepresents its spatially correlated behaviour, withϑ demonstrating the randomness.Different realisations of this field can be simulated using series expansion techniques[32, 97]. These discretise the stochastic fieldH(x, ϑ) by expanding any realisationusing a set of random variablesΥi(ϑ) and deterministic spatial functionsgi(x) thatrepresent the auto-correlation structure:

H(x, ϑ) =∞∑

i=1

Υi(ϑ)gi(x) (IV2)

In the case of the Karhunen-Loève (K-L) series expansion, these deterministicfunctionsgi(x) are obtained by spectral decomposition of the auto-correlation functionCauto(x, x′):

Cauto(x, x′) =

∞∑

i=1

λAi φ

Ai (x)φA

i (x′) (IV3)

with λA andφA respectively the eigenvalues and eigenvectors of the auto-correlationstructure, acquired by solving the eigenvalue problem. Anyrealisation of the zero-mean random fieldH(x, ϑ) can now be presented as:

H(x, ϑ) =∞∑

i=1

√

λAi ηi(ϑ)φA

i (x) (IV4)

with {ηi , i = 1..∞} a set of independent orthonormal random variables. In practice, atruncation of the random fieldH(x, ϑ) is performed by using a finite set of randomvariables{ηi , i = 1..Nvar}. The number of K-L termsNvar in the series is defined byordering the eigenvalues in a descending series and considering only theNvar largereigenvalues that capture most of the randomness. The eigenvector basisφA

i (x) in theK-L expansion is optimal in the sense that the mean-square error resulting from a finiterepresentation ofH(x, ϑ) is minimised. For sufficiently largeNvar, the second-momentproperties ofH(x, ϑ) can thus be approximated by the second-moment properties ofH(x, ϑ):

H(x, ϑ) =Nvar∑

i=1

√

λAi ηi(ϑ)φA

i (x) (IV5)

For more information about the K-L series expansion and its properties, the reader isreferred to [32, 97]. This simulation technique is already applied to generate individualauto-correlated composite properties: stiffness properties of twintex composites [67],volume fraction of long fibre thermoplastic material [36] and material properties oflaminated composites in [11].

156 Paper IV

The framework of Vorechovský enables to simulate cross-correlated random fieldsusing the K-L series as basis. The key idea of the method is that all cross-correlated fields, within a single specimen, are expanded using the same spectrumof eigenvectors, but the sets of random variables used for the expansion of each fieldare cross-correlated with neighbouring fields. These latter sets must be correlatedwith respect to the cross-correlation function. The subsequent steps to produceNsim

realisations of Gaussian cross-correlated fields are:

1. Perform modal decomposition of the auto-correlation structure and applytruncation:λA

i , φAi (x) with i = 1..Nvar (Nvar ≤ NG)

2. Perform modal decomposition of the cross-correlation structure and applytruncation:λC

i , φCi (x) with i = 1..Nf ,r (Nf ,r ≤ NF)

3. GenerateNr × Nsim (Nr = Nvar · Nf ,r ) Gaussian uncorrelated random variablesηr using Latin Hypercube Sampling (LHS)

4. Construct the cross-correlated random matrixχD = φDλDηr of which theelements are uncorrelated for a single random field and cross-correlatedbetween neighbouring fields

5. Simulate all individual random fields for a single realisation by

H j(x, ϑ) =Nvar∑

i=1

√

λAi χ

Dj,iφ

Ai (x) (IV6)

Modal decomposition of the correlation functions is performed in the same wayas equation IV3, withNG equal to the number of grid locations to represent asingle random field andNF the number of fields which are cross-correlated. Theindependent random variablesηr are generated using LHS which is a more advancedsampling scheme compared to crude Monte Carlo sampling. In LHS, stratificationof the theoretical probability distribution function is performed such that an accuraterepresentation of the sampling distribution is already obtained for a limited number ofsimulations. Thek-th realisation (k = 1..Nsim) of the random variableη of locationi(i = 1..Nvar) is computed using:

ηi,k = F−1i

(

π j(k) − rand

Nsim

)

(IV7)

with F−1 the inverse of the cumulative normal distribution function, π a randompermutation of 1..Nsim, andrand a random uniform distributed number. The cross-correlated matrixχD is built using the eigenvalue decomposition method and suchthatit consists ofj blocks (j = 1..NF), where each block consists ofNvar standard Gaussianindependent random variables, while the vectorsχD

i andχDj are cross-correlated. This

SIMULATION OF THE IN-PLANE CENTROIDS 157

is the most important step in the procedure and reproduces the cross-correlationstructure between the differentNF random fields. The random field expression ofeach property or parameter is expanded using the same formula of equation IV5 butnow withχD

j,i as random variables. All the produced fields will possess thecorrect auto-and cross-correlation structure between neighbouring fields. A detailed description ofthe procedure is elaborated in [114].

Vorechovský also proposes additional operations that could enhance the accuracy: (i)application of correlation control techniques and (ii) generation of additional sidepoints. The first procedure encounters the problem of spurious correlation which issometimes introduced along the random variablesχD

j,i of a single field representing towj. This could alter the resulting correlation information ofthe generated deviations.To assess the effect, techniques to diminish the undesired correlation are proposedin [115]. However, no apparent difference is observed using correlation control forthe simulation of the 2/2 twill in-plane centroid position. Omitting such operationsignificantly downsizes the computational expense and correlation control will thusnot be considered. Additional side points are considered incase disturbances arepresent in the generated values at the edges. However, no effect is observed in theproduced in-plane deviations of section 3.3.2, so no additional grid locations areanticipated.

3.3.2 Application to the in-plane centroid deviations

The in-plane centroid deviations are generated using the framework of Vorechovský.A single tow is represented over a grid of forty-one points (NG = 41) by a zero-mean Gaussian random field that shares the same correlation structure along the towand of which the random variables are cross-correlated withneighbouring tows ofthe same genus. Warp and weft simulations are performed separately, with eachrealisation consisting of forty individual but cross-correlated fields (NF = 40). Thetarget statistics in table IV2 are reproduced by projectingthe correlation functions,given in table IV3, onto the equidistant grid of figure IV5. From this information,auto- and cross-correlation matrices are constructed. Since the mean and variance areconstant over the field and correlation is only dependent on the distance between twopoints (τ =| x2 − x1 |), the generated stochastic fields arehomogeneous.

Random fieldsH j of the in-plane centroids are computed using the truncated seriesin equation IV6. After sorting the eigenvalues in descending order, only theNvar

or Nf ,r largest eigenvalues and corresponding eigenvectors are considered. Anappropriate measure of the captured variability, as discussed in section 3.3.1, is givenby normalised sum (or truncation error) which is fixed to minimum 0.9975:

=

∑Nred

i=1 λi∑N

i=1 λi

≥ 0.9975 (IV8)

158 Paper IV

Table IV3: Input correlation functions for simulating the in-plane fluctuations.

Warp tows Weft tows

Auto-correlationCauto σ2wa exp

(

− τξwa,A

)

σ2weexp

(

− τ2

ξ2we,A

)

Cross-correlationCcross exp(

− τξwa,C

)

exp(

− τ2

ξ2we,C

)

with Nred equal toNvar or Nf ,r . The normalised sumover all eigenvalues is shown infigure IV7 for the warp and weft tows with the resulting truncation parameters givenin table IV4. Considering truncation is computationally advantageous for the weft in-plane centroids, while it is limited or even not possible forthe warp in-plane centroid,given the maximum chosen truncation error.

0 10 20 30 400

0.5

1

Eigenvalue index

Norm

alis

ed s

um

0 10 20 30 400

0.5

1

Eigenvalue index

Norm

alis

ed s

um

(a) Warp tows

0 10 20 30 400

0.5

1

Eigenvalue index

Norm

alis

ed s

um

0 10 20 30 400

0.5

1

Eigenvalue index

Norm

alis

ed s

um

(b) Weft tows

Figure IV7: Normalised sum of the eigenvalues of the auto-correlation (left) andcross-correlation (right). The red line indicates when thenormalised sum is equal to1.

In-plane deviations are produced for 40000 warp and weft tows as mentioned insection 3.1. The produced zero-mean deviations trends for two arbitrary specimens infigure IV8 demonstrates a good agreement with the experimental in-plane deviations

SIMULATION OF THE IN-PLANE CENTROIDS 159

Table IV4: Truncation parameters of the series expansion method.

Nvar Nf ,r

Warp tows 33 40Weft tows 4 13

0 20 40 60 80 100 120−0.4

−0.2

0

0.2

0.4

x [mm]

Hj (x,ϑ) [mm]

0 20 40 60 80 100 120

−2

−1

0

1

y [mm]

Hj (x,ϑ) [mm]

Figure IV8: Simulated warp (left) and weft (right) deviations trend for 80 warp andweft tows.

of both experimental samples shown in figure IV3; the short wavelength of theexperimental warp fluctuations and long wavelength of the measured weft deviationsare reproduced in the simulated data. The deviations possess fewer low-amplitudeshort-range fluctuations than observed in the experiments.This is not only attributedto the series expansion method, but also to the normality assumption of the in-planedeviations which diminish the presence of large short wavelength oscillations in thesimulations.

The deviations also have comparable patterns in neighbouring mean tow orientations,defined by the slope of a linear least-square fit to all deviation values along a single tow.Specifically for the weft in-plane centroid positions, neighbouring weft tows possesssimilar orientations due to the significant cross-correlation as presented in figure IV9.The simulated trend is more regular which is again caused by the assumption of anormal distribution for the in-plane centroid deviations.No such patterns are observedfor the warp in-plane deviations which have a lower standarddeviation and limitedcross-correlation.

The corresponding statistics to the thousand generated virtual specimens are verifiedin each tow direction for (i) the combined data set, (ii) individual specimens consistingof hundred unit cells and (iii) individual 1-D random fields representing a single towwith a length of ten unit cells. The normalised difference∆ from the target values in

160 Paper IV

0 10 20 30 40−0.02

−0.01

0

0.01

0.02

0.03

0.04

tow index

mean slope [−]

0 10 20 30 40−0.02

−0.01

0

0.01

0.02

0.03

0.04

tow index

mean slope [−]

Figure IV9: Mean tow orientations of weft in-plane deviations: experimental trend ofsample 2 (left) vs. simulated trend of an arbitrary specimen(right).

table IV2 is defined as∆ = | υexp−υsim

υexp | · 100%, withυ equal to the standard deviation,auto-correlation length or cross-correlation length. An overview of the statistics of thecombined data set and the individual specimens are given in table IV5, while thesefor the individual 1-D random fields are presented in table IV6. The latter table alsoshows the experimental information regarding the individual tow standard deviationand correlation information since this is not yet described.

Table IV5: Standard deviation and correlation lengths for the combined data set andaverage values for the individual specimens.

σcomb < σspec> ξcombauto < ξ

specauto > ξcomb

cross < ξspeccross>

Warp tows [mm] 0.106 0.103 115.81 114.00 4.54 4.42∆warp [-] 0.09% 3.48% 0.80% 0.78% 1.03% 1.72%Weft tows [mm] 0.613 0.570 52.95 52.76 13.10 13.05∆we f t [-] 0.28% 7.38% 0.12% 0.25% 0.45% 0.87%

The standard deviationσ and correlation lengthsξauto, ξcrossof the combined data setindicate that the series expansion method reproduces the experimental data within∆=1% difference of the experimental data. The input and simulated correlationstructures perfectly overlap, even for pairs of points withthe largest spacing asshown in figure IV10 for the warp auto- and cross-correlationstructure. When fewersimulations are performed small deflections for the largestlags are present. Theapplied truncation given in table IV4 does thus not prevent to obtain a good similaritywith the target statistics.

SIMULATION OF THE IN-PLANE CENTROIDS 161

The statistical information corresponding to the individual specimens also shows goodresemblance with the target statistics. In figure IV11, the histogram is shown of allsimulated auto- and cross-correlation lengths for the warptows. Similar results areobtained for the weft tows. The mean of the statistical information of all specimenshas a normalised error∆ which is less than 7.4% for the standard deviation andmaximum 1.72% for the correlation lengths. Although the weft deviations are onlyapproximated by a normal distribution (section 2), the error between the experimentalstandard deviation and the standard deviation obtained from the Gaussian process isstill acceptable. The large variation observed in the histograms demonstrates the largeuncertainty when deriving the correlation correlation lengths.

As third comparison, the individual tows are characterisedin terms of standarddeviation and auto-correlation. Table IV6 shows the results derived from theexperimental data with mean values and coefficient of variations (COV) for thesimulations. The experimental information of single tows,derived in the samemanner as described in section 2, considerably deviates from the specimen statisticsin table IV2. The high auto-correlation in combination witha significant cross-correlation, leads to individual tow information that can significantly differ from oneto another. This uncertainty is reflected in the high coefficient of variation. Especiallyfor the correlation length, data collected from only two one-ply samples is insufficientto obtain an accurate description. The simulated standard deviations for single towscorrespond very well with the experimental information as indicated by the normaliseddifference∆. The LHS sampling technique ensures that even at this level the targetmean and standard deviation are achieved. In contrast to these observations, simulatedcorrelation lengths of the 1-D random fields are only fairly reached with similar orderof magnitudes. The auto-correlation length of the warp towsis overestimated, whilethe weft auto-correlation length is underestimated. Besides this discrepancy, theCOV is much larger than for the experiments. Various reasons, besides the limited

0 20 40 60 80 100 120−0.5

0

0.5

1

distance [mm]

correlation [−]

Target correlation

Generated correlation

0 20 40 60 80 100 120−0.5

0

0.5

1

distance [mm]

correlation [−]

Target correlation

Generated correlation

Figure IV10: Comparison of the warp input and simulated auto-correlation (left) andcross-correlation (right) structure.

162 Paper IV

0 50 100 150 200 2500

50

100

150

200

ξcomb

auto [mm]

Fre

qu

en

cy

simulated

experiment

0 5 10 15 200

50

100

150

200

ξcomb

cross [mm]

Fre

qu

en

cy

simulated

experiment

Figure IV11: Simulated auto- (left) and cross-correlation(right) lengths of warp in-plane centroids.

data set size, can be raised: (i) the normality assumption ofthe distribution altersthe single tow correlation information and (ii) the input correlation functions do notaccurately represent the correlation at tow level. No effect of the fitting operation isexpected since similar least-square fitting errors are obtained for fitted functions to theexperimental and simulated data.

Table IV6: Standard deviation and correlation lengths for individual one-dimensionalrandom fields.

σtarget σ1D ξtargetA ξ1D

A

Warp tows -mean[mm] 0.051 0.053 20.69 32.06Warp tows - COV [-] 35.16% 36.62% 108.04% 132.00%∆warp,mean[-] - 2.93% - 54.93%Weft tows -mean[mm] 0.363 0.353 116.04 63.89Weft tows - COV [-] 62.79% 51.36% 86.77% 154.82%∆we f t,mean[-] - 2.67% - 44.94%

The similarities between experiments and simulations of the in-plane centroiddemonstrate that the methodology based on the K-L series expansion method is ableto correctly reproduce the statistical information on average.

DISCUSSION 163

4 Discussion

The observation of the experimental in-plane deviations trend in figure IV3 demon-strates a different behaviour of the in-plane centroid for both tow directions. Weftfluctuations seem to persist with significantly longer wavelengths compared to thewarp deviations. These tendencies reflect the distinct dependencies of the in-planeposition for the warp and weft tows given in table IV2. Correlation lengths stronglydiffer which is presumably induced by the weaving process of the fabric, where warptows are kept in tension while weft tows are inserted by handles. It has as a firsteffect that the warp tows will possess fewer in-plane movement with a small standarddeviation. This is indicated by the large auto-correlationlength that endures for morethan ten unit cells. In addition, neighbouring warp tows encounter limited effect fromeach other since the available open space between tows act asa buffer for these smallvariations. As a second effect, the in-plane position of the weft tows is more free due tothe lack of tensioning. This translates to a large standard deviation and a much smallerauto-correlation length that is roughly half of the warp tows. On the other hand, thelarge variations cause bundling behaviour of the weft tows which appears as bands inthe composite sample. The computed cross-correlation length now exceeds the unitcell size to around five tows. The discrepancy in warp and weftcorrelation lengthsis described here as fully attributed to the reinforcement manufacturing. Of course,the RTM production process will also affect the tow deviations, but it is expected tobe of a much lower magnitude compared to the fluctuations introduced by the fabricbuild up. During production, resin is injected from all sides so no preference in towdirection for the resin flow is present for the considered square fabric. When the in-plane centroid positions are simulated, warp and weft tow generation techniques needto be calibrated with its specific correlation information to accurately reproduce thedifferent in-plane tow behaviour.

Simulation results are in accordance to the statistical information of the experimentalsamples. The modelling procedure reproduces the input standard deviation andcorrelation information. A thorough characterisation of the tow path of experimentalsamples is thus mandatory to achieve virtual specimens thatare replicas of thephysical samples. Therefore, it is important to assess the errors that are introducedwhen quantifying the in-plane centroid variations from theone-ply samples in [107].The strongest assumptions are the approximated normal distribution for the in-planedeviations and the estimation of the correlation length from fitted correlation functions.In addition, uncertainty is also introduced during processing of the digital image witha resolution of 1200 DPI. The manual error in the determination of the tow boundariesis assumed to be±2 pixels, which corresponds to 0.042 mm. This is of similarmagnitude of the standard deviation of individual warp towsand is thus not negligible.It is important to minimise these errors to have an as accurate as possible statisticaldescription of the material.

164 Paper IV

In a next step, virtual specimens are created by a stochasticmulti-scale modellingapproach. The availability of these virtual textiles permits to evaluate the effectof geometrical variability on the mechanical response of the composite, as alreadyrevealed in the introduction. The short- and long-range geometrical variability aresimulated for the in-plane centroid, out-of-plane centroid, aspect ratio and area ofeach cross-section. Except for the in-plane deviations, all tow path parametersvary within the unit cell dimensions and are not cross-correlated with neighbouringtows. Each random reinforcement parameter is built by combination of an averagetrend with zero-mean deviations, using the statistical information collected in [104]and [107]. Average trends are determined from the experimental data and are thesame for all realisations. Deviations are simulated using the Monte Carlo MarkovChain algorithm for textile structures [6, 105] when not cross-correlated, while thecross-correlated in-plane centroid fluctuations are produced as Gaussian randomfields by applying the methodology described in section 3. A virtual model ofthe textile composite is obtained in the WiseTex format [113] by overwriting theoriginal tow path information of a nominal WiseTex representation. These realistictextile models can be subsequently used for evaluating the mechanical performance,damage modelling or impregnation analysis. The general procedure of this modellingapproach is addressed in a future publication.

5 Conclusions

A framework based on the Karhunen-Loève series expansion isadopted to simulatethe in-plane tow path variability of textile composites. The experimental auto- andcross-correlation matrices are represented in terms of itseigenvalues- and vectorsto introduce the spatial correlation information in each ofthe produced randomseries. Within a specimen, all tow centroids of equal genus are expanded usingthe same spectrum of eigenvectors that represent the auto-correlation structure.The sets of random variables, used for the expansion of each field, are cross-correlated with neighbouring fields to describe the positional dependency betweentows. Any realisation of the in-plane path position describes the long-rangegeometrical variability within the textile composite. In combination with the tow pathparameters varying on the short-range, realistic virtual specimens can be generatedthat span multiple unit cells.

The methodology is validated by simulating thousand specimens of a carbon-epoxy2/2 twill woven composite consisting of forty warp and weft tows. In-plane deviationspossess similar wavelengths as the experimental fluctuations quantified in prior work.The experimental statistical information of single specimens is reproduced with amaximum normalised error of 7.4% for the standard deviationand 1.72% for the

CONCLUSIONS 165

correlation lengths, while the auto-correlation length ofthe individual tows are onlyapproximated by the same order of magnitude.

Acknowledgements

This study is supported by the Flemish Government through the Agency for Innovationby Science and Technology in Flanders (IWT) and FWO-Vlaanderen.

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Curriculum Vitae

Personal data

Andy Vanaerschot, maleBorn 14 March 1987, Mortsel, Belgiumandy.vanaerschot@gmail.com

Professional Experience

Aug 2010 - today: PhD student - Research Engineer at KU LeuvenProject is a joint collaboration between the PMA division (MechanicalEngineering) and the CMG division (Materials Engineering).

Jan 2011 - Sep 2011: Conference Operations ManagerCoordinator of the ISMA2012 & USD2012 International Conferences andCourses in Leuven, 17-21 September 2011.

Aug 2011 - Sep 2011: Visiting PhD student at UC Santa Barbara (UCSB), USAResearch stay and collaboration with Brian N. Cox (TeledyneScientific) aboutstate-of-the-art techniques for generating virtual textile composite specimens.

Aug 2009 - Jul 2010: Master thesis at von Karman Institute for Fluid Dynamics (VKI)Master thesis topic:Development of a flat plate probe for off-stagnation testingof space re-entry vehicles.

179

180 CURRICULUM VITAE

Education

2010-2014:PhD Student with grant of IWT-VlaanderenDepartment of Mechanical Engineering, KU Leuven, Leuven (Belgium)

2008-2010:MSc Degree in Mechanical Engineering - option AerospaceGraduated Magna Cum LaudeKU Leuven, Leuven (Belgium)

2005-2008:BSc Degree in Mechanical (major) - Materials (minor) EngineeringGraduated Cum LaudeKU Leuven, Leuven (Belgium)

1999-2005:Secondary School (Science-Mathematics)Onze-Lieve-Vrouwecollege, Antwerpen (Belgium)

List of publications

Articles in international refereed journals

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D. Simulation of the cross-correlated positions of in-plane tow centroids in textile composites based onexperimental data.Composite Structures, 116:75-83, 2014.

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D. Stochastic multi-scale modelling of textile composites based on internal geometry variability.Computers& Structures, 122:55-64, 2013.

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D. Stochastic frameworkfor quantifying the geometrical variability of laminated textile composites usingmicro-computed tomography.Composites Part A, 4:122-131; 2013.

• Olave M., Vanaerschot A., Lomov S.V., Vandepitte D. Internal geometryvariability of two woven composites and related variability of the stiffness.Polymer Composites, 33(8):1335-1350, 2012.

Chapter in Academic Books

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D. Stochastic charac-terisation of the in-plane tow centroid in textile composites to quantify themulti-scale variation in geometry. InProceedings of the IUTAM Symposiumon Multiscale Modeling and Uncertainty Quantification of Materials andStructures, Santorini, Greece, September 8-11, 2013. Springer (2014):187-202.

181

182 LIST OF PUBLICATIONS

Articles in international conference proceedings

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D. Stochastic modellingof the geometrical variability in textile composites usingexperimental data.In Proceedings of the International Conference on Uncertainty in StructuralDynamics (USD 2014). Leuven, Belgium, September 15-17, 2014.

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D. Stochastic multi-scalemodelling of composite materials based on experimental data. InProceedings ofthe International Conference on Structural Dynamics (EURODYN2014). Porto,Portugal, June 30-July 2, 2014.

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D. Stochastic multi-scalemodelling of short- and long-range effects in textile composites based onexperimental data. InProceedings 16th European Conference on CompositeMaterials. Seville, Spain, June 22-26, 2014.

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D., Generation of stochasticmacroscopic structures using experimental data of random geometry. InProceedings of the 11th International Conference on Textile Composites(TexComp11), Leuven, Belgium, September 19-20, 2013.

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D. Stiffness evaluationof polymer textile composites subjected to internal geometry variability. InSAMPE 2013 Proceedings: Education& Green Sky - Materials Technologyfor a Better World. Long Beach, USA, May 6-9, 2013. p. 2630-2644

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D. Mechanical propertyevaluation of polymer textile composites by multi-scale modelling based oninternal geometry variability. InProceedings of the International Conference onUncertainty in Structural Dynamics (USD 2012). Leuven, Belgium, September17-19, 2012.

• Vanaerschot A., Cox B.N., Blacklock M., Kerckhofs G., Wevers M., LomovS.V., Vandepitte D. Statistical description of the internal geometry of a polymertextile composite using micro-computed tomography. InProceedings 15thEuropean Conference on Composite Materials. Venice, Italy, June 25-28, 2012.

• Vanaerschot A., Olave M., Lomov S.V., Vandepitte D. A stochastic multi-scaleframework for textile composites to evaluate the stiffness tensor. InProceedings14th AIAA Non-Deterministic Approaches Conference. Honolulu, USA, April22-26, 2012.

LIST OF PUBLICATIONS 183

Abstracts in international conference proceedings

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D. Stochastic multi-scalemodelling of macro-scale textile composites using experimental data of randomgeometry. InBook of Abstracts for 1st International Conference on Mechanicsof Composites (MECHCOMP2014), Stony Brook, USA, June 8-12, 2014.

• Vanaerschot A., Cox B.N., Lomov S.V., Vandepitte D. Stiffness evaluationof textile composites by multi-scale modelling based on internal geometryvariability. In Book of Abstracts for 10th World Congress on ComputationalMechanics. Sao Paulo, Brazil, July 8-13, 2012.

Other

• Olave M., Vanaerschot A., Lomov S.V., Vandepitte D. Effect of geometricvariability on the elastic properties of composites.Society of Plastics Engineers,September 2012.

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