DEPARTMENT OF MECHANICAL ENGINEERING

Residual Strength Prediction of Composite Laminates

Containing Impact Damage

Master Thesis in Solid Mechanics Linköping University

February 2005

Erik Nilsson

LITH-IKP-EX—05/2219--SE

Institute of Technology, Dept of Mech Eng, SE-581 83 Linköping, Sweden

Datum Date 2005-02-10

Avdelning, institution Division, Department Div of Solid Mechanics Dept of Mechanical Engineering SE-581 83 LINKÖPING

URL för elektronisk version

Titel Residual Strength Prediction of Composite Laminates Containing Impact Damage Title

Författare Erik Nilsson Author

Sammanfattning Abstract

This report concerns rapid semi-analytical methods for calculating residual strength of damaged composite structures. The study is divided into three parts. The first part deals with modification of a semi-analytical method which concerns buckling and growth of delaminations in damaged composite structures. The work consists of correction and modification of preliminary computer code and verification of the method. In the first part, the verifications are performed by comparing results from the semi-analytical method with results from FE-simulations. The second part deals with characterization of impact damaged regions in composites as regions with reduced stiffness. In an attempt to estimate the relative inclusion stiffness, i.e. the stiffness reduction coefficient of damaged composite, a comparison between the results from a semi-analytical method used at Saab, and an alternative method is made. The semi-analytical method calculates stresses for an infinite plate with an elliptical opening or inclusion. The third part deals with verification of the methods mentioned above. In the comparison, experimental test results and an alternative method based on experimental testing is used.

Nyckelord: Impact Damage, Composite, Residual Strength, Composite Damage, Delamination Keyword

ISBN ISRN _________________________________________________________________ Serietitel och serienummer Title of series, numbering

LiTH-IKP-EX—05/2219--SE

Språk Language Svenska/Swedish x Engelska/English ________________

Rapporttyp Report category Licentiatavhandling x Examensarbete C-uppsats D-uppsats Övrig rapport _____________

Preface This thesis is the final assignment for the examination as Master of Science in Mechanical Engineering at Linköping Institute of Technology. The work has been performed at Saab Aerostructures in Linköping, Sweden. I would like to thank my supervisor, Dr. Tonny Nyman, and his colleague, Anders Bredberg, for their encouragement and support throughout this study. I also want to thank the manager of the department, Anders Rydbom, and all the staff at the department of Structural Strength for their support. Finally I want to thank my examiner, Kjell Simonsson, for his review of this report. Linköping 2005-02-10 Erik Nilsson

Abstract This report concerns rapid semi-analytical methods for calculating residual strength of damaged composite structures. The study is divided into three parts. The first part deals with modification of a semi-analytical method which concerns buckling and growth of delaminations in damaged composite structures. The work consists of correction and modification of preliminary computer code and verification of the method. In the first part, the verifications are performed by comparing results from the semi-analytical method with results from FE-simulations. The second part deals with characterization of impact damaged regions in composites as regions with reduced stiffness. In an attempt to estimate the relative inclusion stiffness, i.e. the stiffness reduction coefficient of damaged composite, a comparison between the results from a semi-analytical method used at Saab, and an alternative method is made. The semi-analytical method calculates stresses for an infinite plate with an elliptical opening or inclusion. The third part deals with verification of the methods mentioned above. In the comparison, experimental test results and an alternative method based on experimental testing is used. Concerning the semi-analytical method which deals with buckling and growth of delaminations in damaged composite structures: The different comparisons for artificial circular delaminations show that the calculated buckling and critical load show good correspondence to the results provided by the FE-based method. Generally, the differences in calculated buckling loads by the two methods are small and the semi-analytical method is more conservative. Furthermore, the semi-analytical method shows good agreement with the results provided by an alternative method and available experimental test results. Due to the small amount of test results available, more experimental tests need to be done for different load conditions to confirm the accuracy and applicability of the semi-analytical method. Concerning the semi-analytical method which deals with rapid assessment of stresses and strains of notched composite structures where the damaged region is treated as a soft inclusion or an open hole: Comparison with alternative methods show that the method will be too conservative if the damaged region is treated as an open hole when the damage region is small, while it provides good results for large damaged regions. Soft inclusion, and methods to approximately determine its relative inclusion stiffness, is needed, especially in the case of tensile loading and for small damaged regions. To be able to establish a simple estimation of the relative stiffness reduction, a close interaction between experiments and modelling is necessary. For tensile dominated load cases a simplified approach of soft inclusion is proposed, since residual strength after impact is usually not the sizing criteria for a structural part. An empirical equation for the stiffness reduction is derived in this work.

Table of Contents 1 Introduction .................................................................................................................. 2 1.1 Impact Damages - Background............................................................................................................ 3 1.2 Damage Characterization..................................................................................................................... 4 1.2.1 Matrix Cracking .............................................................................................................................. 4 1.2.2 Delamination ................................................................................................................................... 4 1.2.3 Fibre Breakage ................................................................................................................................ 5 1.2.4 Damage Initiation and Growth ........................................................................................................ 5 1.3 Objectives ............................................................................................................................................ 5 1.4 Material Properties............................................................................................................................... 6 2 Buckling and Growth of Delaminations..................................................................... 7 2.1 Introduction.......................................................................................................................................... 7 2.2 Semi-analytical Method ....................................................................................................................... 8 2.2.1 Delamination Buckling Theory ....................................................................................................... 9 2.2.2 Approximate Calculation of Strain Energy Release Rate.............................................................. 10 2.2.3 Characterization of Damage.......................................................................................................... 13 2.2.4 Numerical Implementation............................................................................................................ 13 2.3 FE-based Method ............................................................................................................................... 15 2.3.1 General .......................................................................................................................................... 15 2.3.2 Introduction ................................................................................................................................... 15 2.3.3 Strain Energy Release Rate Calculation Using the VCC-Technique ............................................ 16 2.3.4 Models........................................................................................................................................... 16 2.3.5 Boundary Conditions..................................................................................................................... 17 2.3.6 FE-results ...................................................................................................................................... 19 2.4 Comparison of Results ....................................................................................................................... 20 2.4.1 Buckling Strain.............................................................................................................................. 20 2.4.2 Critical Strain ................................................................................................................................ 24 2.4.3 Discussion ..................................................................................................................................... 29 2.5 Effects of Reduced Stiffness on Delamination Buckling ................................................................... 30 2.5.1 Effects of Different Failure Modes on Delamination Buckling .................................................... 30 2.5.2 Effects of Equally Reduced Stiffness on Delamination Buckling................................................. 31 2.5.3 Discussion ..................................................................................................................................... 32 2.6 Conclusions........................................................................................................................................ 32 3 Soft Inclusion Theory................................................................................................. 33 3.1 Introduction........................................................................................................................................ 33 3.2 Soft Inclusion Theory (SIT) ............................................................................................................... 33 3.3 Failure Criterion................................................................................................................................. 35 3.4 Approximate Determination of Relative Inclusion Stiffness ............................................................. 36 3.5 Conclusions........................................................................................................................................ 38 4 Verification of Methods ............................................................................................. 39 4.1 Compression ...................................................................................................................................... 39 4.2 Tension............................................................................................................................................... 40 4.3 Compressive-Tensile Loading ........................................................................................................... 41 4.4 Discussion.......................................................................................................................................... 42 5 Conclusions ................................................................................................................. 43

References ................................................................................................................... 45

Appendices A Delamination Buckling Theory E Determinate Buckle Size, Non-allowable B Approximate Calculation of Strain Energy Release Rate Solutions

F Virtual Crack Closure Technique C Approximate Calculation of Strain Energy Release Rate – A Formerly Used Approach at Saab G Laminate Theory, Laminate Classification

D Energy Formulation of Anisotropic Plates

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1 Introduction Laminated polymer composites are increasingly used in many advanced structural applications. The major advantage of composite materials is that the orientations of the plies can be ideally tailored to the loading situation so that the laminate has the desired elastic properties and failure behaviour at minimum weight. By choosing an appropriate combination of reinforcement and matrix material, manufacturers can produce properties that exactly fit the requirements for a particular structure for a particular purpose. Modern aviation, both military and civil, is a prime example. In JAS 39 Gripen combat aircraft e.g. the wings, the radome, the canard and the fin are made of composite, see Figure 1. Carbon fiber reinforced plastics (CFRP) are increasingly used in aircraft structures due to superior specific strength and stiffness. The conventional materials such as aluminium are more and more replaced by composite materials. Composite materials have lower density, greater strength and better stiffness than aluminium. Having a smaller and lighter structure carry the same load has significant effect on performance, weight, design and cost.

Figure 1 Parts on JAS Gripen aircraft made of composites Composite materials are relatively complex in comparison with metal materials and require specific skills in areas like design, production and quality-control. In aircraft design there is a strong need to be able to predict the durability and damage tolerance of a structural component. One of the main obstacles to efficient use of composite materials is their susceptibility to impact damage. Current design philosophy is based on no damage growth criteria and is realised by conservative strain limitations and verification testing on components and built up components. Consequently, expensive testing is frequently performed by the industry. The problem is usually represented by a test pyramid where each stage refers to an investigation level in terms of specimen category, see Figure 2.

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Figure 2 Schematic of testing requirements, Rouchon´s test pyramid

1.1 Impact Damages - Background Impact damage may cause significant reductions in stability and strength of composite structures. Low velocity impact damage from e.g. bird strike, runway debris, dropped tools during fabrication or maintenance operations may cause damages below the “barely visible impact damage” (BVID) limit. This type of damage may not be visible to the naked eye and has to be observed by e.g. ultrasonic C-scan. These types of damages could lead to catastrophic failure if proper design precautions are not taken, and are therefore important to take into consideration when designing a composite structure. Since such damage is difficult to detect, especially in-service, structures must be able to function safely with BVID present. The complex problem of determining the effects of impact damage may be divided into two sub problems - Impact damage resistance, which deals with the response and damage caused by a certain impact, and - Impact damage tolerance, which deals with the reduced strength and stability of the structure due to damage [1]. The aim of the work by Saab concerning impact on composites is to develop reliable methods to evaluate the effects of damage and the residual strength properties after impact, in other words to determine Impact tolerance, see Figure 3. The study of this report mainly concerns impact damage tolerance and residual strength prediction of composite structures containing impact damage of a given size.

Experimental testing is an efficient way to determine the effects of impact damage. Due to the fact that testing is expensive and time-consuming, there is a great need to develop calculation methods that are rapid and reasonably accurate that provides the opportunity to perform parametric studies from an engineering point of view.

Figure 3 Impact tolerance divided into two sub problems [1].

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In studies of impact damage on CFRP:s three major damage types are of concern; fiber breakage, matrix cracks and multiple interlaminar delaminations. The largest effects of impact damage are observed in compression [1]. The reduction of compressive strength due to impact is more significant than the reduction in tensile strength and other strengths. Therefore, the work on residual strength is focused on delamination buckling which reduce the flexural properties of the damaged laminate and may cause significant reduction in compressive strength.

1.2 Damage Characterization Impact damage characterization is of importance both for impact damage resistance and for impact damage tolerance. In the latter case damage characterization is of major importance due to the fact that the geometry of the damage will determine which damage mechanisms that will control the failure strength. Further, the complex geometry of the damage precludes exact modelling of the damage and simplifications have to be done. To accomplish correct simplifications, elaborate damage characterizations are vital to obtain accurate residual strength analyses of impacted composite structures. In this report, a few important observations are illustrated. Extensive work in the area of impact damage characterization and development of methods concerning determination of the residual strength after impact in composite materials has been performed at The Aeronautical Research Institute of Sweden (FFA) [1],[2]. According to studies at FFA, impact damage may affect structural failure in several ways; delamination growth due to buckling of sub laminates, reduced panel buckling load due to the presence of impact damage, and in-plane failure due to stress concentrations at the damage.

1.2.1 Matrix Cracking In low-velocity impact on composite structures, damage growth generally occurs through matrix cracking, followed by delamination growth and finally fibre fracture [3]. The matrix cracks are initiated by high transverse shear stresses, membrane stresses and/or flexural stresses. Matrix cracks and delaminations generally interact. For example, matrix cracks divert into delaminations if the cracks reach adjacent plies with a different fibre direction. Results produced by FFA show that the region with matrix cracks is larger than the region with fibre breakage, but smaller than the region with delaminations.

1.2.2 Delamination Delaminations are initiated by matrix cracks and typically occur between plies of different fibre orientations. Delaminations generally increase in size with thickness and mismatch angle of the plies [4]. Delaminations are more or less peanut shaped with the major axis parallel to the fibres of the lower ply. Delamination growth is considered to be the most energy consuming damage mechanism and therefore represents the dominating damage process during impact [5]. The majority of the energy absorbed in the laminate during impact dissipates into delamination growth.

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1.2.3 Fibre Breakage In tension loading, fibre breakage is the dominating failure mode and the residual strength is primarily controlled by the extent of fibre breakage during impact. Generally, the distribution of fibre breakage through-the-thickness is more or less uniform for all laminates and the extension in the width direction is quite narrow. Fractographic characterizations of impact damages performed at FFA indicates that the fibre breakage is centred under the point of impact and extends to a radius of one third to one half of the maximum delamination width [2].

1.2.4 Damage Initiation and Growth Damage growth generally occurs through matrix cracking, followed by delamination growth and finally fibre fracture [3]. In thick laminates with a span-to-thickness ratio of 10 to 20 the delaminations initiate close to the impacted side and in thin laminates close to the midplane of the laminate. Further, the span-to-thickness ratio is observed to have a significant influence on the damage extension during impact. For thick laminates, with small span-to-thickness ratio, the transverse shear effect is more pronounced, causing a “barrel” shaped delamination distribution. For thin laminates, with larger span-to-thickness ratio, the largest delaminations are located close to the back surface of the laminate and the distribution tends to be more conical [6]. Laminates with a large span-to-thickness ratio experience less delamination growth and more extensive fibre fracture as a result of membrane effects.

Figure 4 Delamination growth sequence in thick and thin laminates [7].

1.3 Objectives The study is divided into three parts. The first part deals with modification of a semi-analytical method which concerns buckling and growth of delaminations in damaged composite structures. The work consists of correction and modification of a preliminary computer code and verification of the method. In the first part, the verifications are performed by comparing results from the semi-analytical method with results from FE-simulations.

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The second part deals with characterization of impact damaged regions in composites as regions with reduced stiffness. In an attempt to estimate the relative inclusion stiffness, i.e. the stiffness reduction coefficient of damaged composite, a comparison between the results from a semi-analytical method used at Saab, and an alternative method is made. The semi-analytical method calculates stresses for an infinite plate with an elliptical opening or inclusion. The theory of the alternative method is confidential and is therefore not presented in this report. The third part deals with verification of the methods mentioned above. In the comparison, experimental test results and an alternative method based on experimental testing is used. The alternative method is the same as mentioned above.

1.4 Material Properties In this study, two different carbon fiber reinforced plastics are dealt with; HTA/6376C, a commonly used material at Saab, and CYTEC/977-2. The mechanical properties of the HTA/6376C material are E11= 140 GPa, E22= 9 GPa, E33= 9 GPa, G12= 4 GPa, G13= 4 GPa, G23= 3 GPa, ν12= 0.3, ν13= 0.5, ν23= 0.4 and the CYTEC/977-2 material E11= 130GPa, E22= 5GPa, G12= 5GPa, ν12= 0.35. The constitutive properties of the two materials are fairly similar and in the case of delamination buckling a rough comparison of results based on the two materials can be done. The nominal ply thickness is 0.13 mm for both the materials. Three different lay-ups are studied. The quasi-isotropic lay-up is one of the most commonly utilized lay-up at Saab and is therefore mainly used in this study. In each case the laminate consists of 48 plies, which gives a total thickness of 6.24 mm.

Quasi-isotropic lay-up 0°-dominated lay-up 90°-dominated lay-up

[0/90/45/-45]6S [02/90/02/-45/02/45/02/90]2S [902/0/902/-45/902/45/902/0]2S

Figure 5 Stacking sequence

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2 Buckling and Growth of Delaminations

2.1 Introduction One of the most common failure modes for composite structures is interlaminar delamination. Delamination as a result of impact, high stress concentrations from geometrical discontinuity or a manufacturing defect can cause a significant reduction in the compressive load-carrying capacity of a structure. When a delamination is subjected to in-plane compressive dominated load, local buckling of the delaminated region may occur before global buckling of the laminate. In some cases a mixed mode buckling may occur, which is a combination of local and global buckling as shown is Figure 6. Normally, the global buckling load represents the failure load of the delaminated composite panel at global buckling, and no post buckling behaviour of the delamination needs to be dealt with [8]. (This failure mode is not treated in this report). For local and mixed buckling modes, growth of delamination is generally the failure mode of the delaminated composite and post buckling analysis of the delamination is therefore necessary.

Figure 6 Buckling modes The laminate’s lowered ability to resist compressive loads greatly depends on the location of the delamination in the through-the-thickness direction but also on the area and shape of the delamination. An increase of the delamination area leads to a decrease in buckling load; an effect which is more pronounced when the delamination is located closer to the surface [8]. The delamination location has a significant influence, especially on the buckling mode. Delaminations closer to the surface of the laminate generally buckle at a lower load. However, immediate failure will not occur if the delamination does not grow, which is often the case for delaminations close to the surface, since the energy level in the buckled plies is low. Experiments indicate and confirm that delaminated composite plates may undergo increased load after buckling until delaminations grow [8]. Therefore, it is necessary to both understand the effects of delamination buckling and initial growth of delamination to determine the residual strength after impact, commonly named Compression After Impact (CAI). Current design philosophy at Saab is based on a no damage growth criteria. For solid laminates a ply failure load is taken as the laminate failure load.

8

A Rayleigh-Ritz method is used to formulate an eigenvalue buckling problem and to predict buckling loads and deformation mode shapes. The method is used to determine the strain at which delamination buckling will occur for a composite plate containing a single delamination. The two-dimensional analysis assumes a delamination elliptical or circular in shape. Most analyses of delamination growth are based on the fracture mechanics approach and evaluation of the energy release rate. Post-buckling behaviour and delamination growth can not be predicted by the eigenvalue buckling problem as it is beyond the capability of this eigenvalue analysis [9]. In this report, different methods using the total energy release rate to determine the initial growth of delamination are used and presented. In Appendix C, a method to calculate the initial growth of delamination, which was formerly used at Saab, is presented. A new theory of delamination growth is presented in Chapter 2.2.2 and Appendix B. The delamination buckling theory and the theory of delamination growth will together determine the residual strength in delaminated composite plates. The theories of delamination buckling and determination of critical strains due to crack growth that are described in the following chapters are implemented in the Fortran code CODEIN.

2.2 Semi-analytical Method A plate is assumed to contain a single delamination and the plate is assumed to be thin relative to its span, such that buckling instead of compressive failure occurs. The delamination is assumed to be elliptical or circular in shape, with local axes of symmetry which coincide or may be at an angle relative to the global axes. The origin of the delamination coincides with the origin of the global axes. The delamination is oriented by an angle θ between the local coordinate system (x´, y´) and the global coordinate system (x, y), and its size is described by the lengths 2a and 2b along the x´- and y´-axes, see Figure 7.

Figure 7 Characterisation of the damage The damaged region of the plate is divided into two sub laminates. The sub laminates located above and beneath the delamination are referred to as the delaminated region and the base laminate (or base region), respectively (see Figure 8). To avoid misunderstanding, the term “sub laminate” will also refer to the delaminated region. The base laminate is assumed to be rigid and only the delaminated region will be subjected to local buckling. In other words, thin plate linear buckling theory is assumed to be valid.

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Figure 8 Loadings and definitions

When analysing elliptical delaminations the present buckling analysis is strictly speaking valid only for those load cases where the shear component is relatively small [10]. In the analysis the load is transformed to a load which coincides with the principle axes of the sub laminate in the x´y´-system. For example, in the case of pure shear the loading is rotated 45 degrees resulting in a compressive-tensile loading with no applied shear. For circular delaminations no restriction on load cases exists. For engineering purposes, when designing an aircraft structure, it is most practical to consider circular delaminations with diameter equal to the largest of 2a and 2b as measured on impact specimens. This is a conservative approach. A survey of the theories of delamination buckling and of delamination growth is presented in the following chapter. More detailed presentations are given in Appendix A and B.

2.2.1 Delamination Buckling Theory The theory is based on the Rayleigh-Ritz method, which is used for calculating buckling strains of elliptic and circular delaminations in orthotropic plates [10], [11]. The procedure is divided into the following main steps: (1) Selection of an admissible transverse displacement function. (2) Calculation of the total potential energy. (3) Creation of eigenvalue equation. Assumption on transverse displacement field for the delaminated region is made:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ′

+⎟⎠⎞

⎜⎝⎛ ′

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ′

−⎟⎠⎞

⎜⎝⎛ ′

−=2

2

2

10

222

1byC

axCC

by

axw (1)

where C0, C1 and C2 are the so called generalized displacements. The total potential energy, ∏, of the sub laminate is the sum of the strain energy, U, and the potential energy of applied loads, V. VU +=Π (2)

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The assumed transverse displacement field, w, is substituted into the expressions for the strain energy and potential energy of external loads. After necessary differentiation and integration the strain energy is given by { } [ ]{ }00 CKCU T= (3) and the potential energy of external loads is given by { } [ ]{ } λ00 CKCV g

T= (4) where { } ( )2100 ,, CCCC T = , [ ]K and [ ]gK are two different stiffness matrices andλ is a scalar

factor of the reference load. The subscript “0” in { }T0C indicates that the vector is normalized.

An eigenvalue problem is created by taking the first invariant of the total potential energy: [ ]{ } [ ]{ } 000 =+ λCKCK g (5) The solution of equation (5) results in three eigenvalues corresponding to three different buckling modes. The lowest absolute eigenvalue corresponds to the first buckling mode, hence the only one of interest. The critical strain, εbuck, is given by the lowest eigenvalue multiplied with the applied strain.

2.2.2 Approximate Calculation of Strain Energy Release Rate The method is based on work done by H. Chai and C. D. Babcock [12] and L.M. Kachanov [14]. The analysis of initial growth of delaminations is based on a fracture mechanic approach and Griffith’s linear elastic theory for crack propagation. Griffith based his considerations on a global balance of energy in an entire structure. The energy balance equation deals with the energy needed to create a new crack, energy stored in the structure as elastic strain energy, and the work done by external loads. When a crack grows it obtains an increment in length while the potential energy of the body decreases and there is an energy release. According to Griffith’s theory of brittle fractures the energy release is used to create new crack surfaces. The total potential energy, ∏, of the sub laminate is the sum of the strain energy, U, and the work done by applied loads, V. VU +=Π (6) Crack growth is possible when the energy release rate, G, reaches a critical value, i.e. when critGG = (7) where the energy release rate is defined as

dAdG Π

−= (8)

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The calculations of the total potential energy is divided into two stages; pre buckling (ε < εbuck) and in buckling (ε > εbuck). The strain energy and the potential energy of external loads are calculated differently in the two stages and the terms (Π0, U0, V0) and (Π, U, V) refer to pre buckling and in buckling, respectively. In brief, pre buckling, no transverse deflection is assumed to take place and membrane energy is dominating. In buckling, the potential energy consists of membrane energy and bending energy but the bending energy is dominating if the deflection is larger than the thickness of the sub laminate [14]. The expressions are presented in Appendix B. Pre buckling (ε < εbuck), the elastic potential energy, Π0, is given by 000 VU +=Π (9) and in buckling (ε > εbuck) 0Π++=Π VU (10) No interlaminar stresses, and consequently no energy release rate, develop at the delamination front until the delaminated region buckles. Therefore, in buckling, the total energy release rate of the ellipse becomes:

( )⎟⎠⎞

⎜⎝⎛

∂Π∂

−=ab

Ga π1

( )⎟⎠⎞

⎜⎝⎛

∂Π∂

−=ba

Gb π1 (11)

where Ga and Gb are the energy release rate along the “a” axis (“b” fixed) and along “b” axis (“a” fixed), respectively (see Figure 9). The potential energy, ∏0, is calculated using the strains and the load intensities referring to the initial buckling state.

Figure 9 Growth of elliptic delamination

In buckling, the derivatives of the bending energy with respect to a and b may be expressed by

( ) { } [ ]{ } { } [ ]{ }( )λCKCCKCaa

VUg

TT +∂∂

=∂+∂

( ) { } [ ]{ } { } [ ]{ }( )λCKCCKCbb

VUg

TT +∂∂

=∂+∂ (12)

12

where λ is a scalar factor of the reference load. Thus, λ is equal to the eigenvalue when buckling occurs. The normalized eigenvector{ } ( )2100 ,, CCCC T = is obtained when solving the eigenvalue problem, equation (5). The vector represents the shape of the buckle, not the size of it. To be able to calculate G, the scalar factor (D0/C0) needs to be determined in{ } ( )( )21000 ,, CCCCDC T = . The constant D0 is simply referring to the height at the centre of the buckle. Ref [14] presents an approach using a strip model to determine this constant, see Figure 10.

Figure 10 Strip model

The contraction ∆1 of the strip due to bending (in buckling) is

xdxwa

a

′⎟⎠⎞

⎜⎝⎛

′∂∂

=∆ ∫−

2

1 21 (13)

The contraction ∆2 due to compression after initial buckling is a202 ε=∆ (14) where the strain ε0 refers to the added strain after buckling, i.e. ε0 = ε - εbuck where ε is the applied strain. Equating 21 ∆=∆ leads to an expression for D0

( )⎟⎟⎠⎞

⎜⎜⎝

⎛+

= 20

21

20

20

0 11321155

CCCa

Dε

(15)

When initial buckling occurs, the energy release rate is calculated by setting the applied strain equal to the buckling strain, ε = εbuck. If the critical energy release rate is reached when buckling occurs, the buckling strain is set to be the critical strain. Otherwise the critical far field strain, εcrit, related to the critical strain energy release rate, Gcrit, is found in an iterative manner by changing the applied loads, which in its turn will determine the height D0. The critical strain thus depends on the C-vector, through the increase of the buckle (D0), and the external loads, which contributes to the work done.

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2.2.3 Characterization of Damage In ref. [15], three techniques to characterise impact damage are presented. All three techniques are implemented in CODEIN. In the first technique (T1), it is assumed that an ellipse covers the projected damage area and each individual delamination is assumed to have the same size as the ellipse, see Figure 11. In the second (T2) and third technique (T3), each delamination is assumed to have a peanut shape in the direction of the neighbouring lower ply. In this report, only the first technique is dealt with.

Figure 11 Technique 1

2.2.4 Numerical Implementation The theories described in previous chapters are implemented in the Fortran code CODEIN. The flow chart of the program is shown in Figure 12. The input data are material properties, ply thickness, dimensions of the elliptical or circular damage (the major axis a and the minor axis b of the ellipse), in-plane loads, stacking sequence, the critical strain release rate value, Gcrit, and choice of technique (T1, T2 or T3). Depending on the characterization of the damage three techniques T1, T2 and T3 can be used in the program. The program works in such a way that the single delamination advances through the stack of the laminate and the buckling strains and the critical strains are determined for each sub laminate. The thickness of the sub laminate increases with one ply per calculation. In accordance with the flow chart below the strains and stresses are calculated for the undamaged laminate. Further on, the stiffness and the stresses of the sub laminate are determined. The stresses are transformed to the local coordinate system of the ellipse, i.e. the local x-axis coincides with the major axis of the ellipse. The eigenvalue problem is created in the local system. The solution of eigenvalue problem results in three eigenvalues corresponding to three different buckling modes. The lowest absolute eigenvalue corresponds to the first buckling mode, hence the only one of interest. The eigenvalue multiplied with the applied strain determines the initial buckling strain. Finally, the critical strain is determined.

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Figure 12 Flow chart of the program CODEIN

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2.3 FE-based Method

2.3.1 General Using a FE-based method is another way to determine the residual strength of a damaged composite structure. The advantages of using FE-based methods are many; structures and damaged regions with complicated geometry can be easily modelled; different material properties (especially reduced stiffness due to different types of damages), complicated boundary conditions and load cases can be easily used as well. A semi-analytical method, focusing on local effects such as local buckling, may need to take global effects into consideration, using correction factors. The FE-based method takes both local and global effects into account, depending on the model. In short, the FE-based method should be used when a more accurate prediction of reality is needed. The disadvantages of the FE-based method are the time-consuming modelling and analysis. This is one of the main reasons to why a rapid and efficient (semi-)analytical method is needed to calculate residual strength of damaged composite structures.

2.3.2 Introduction An FE-based method is used to evaluate the accuracy of the results produced by the semi-analytical method described in previous chapters. Two different models, dealing with different load cases, have been created using solid linear brick elements. The damaged region is simulated by an artificial circular delamination located at different interfaces close to the surface of the specimen. The loaded specimen is a quadratic flat plate with the side length equal four times the diameter of the delamination. These dimensions are needed to neglect effects of the boundaries. There are two kinds of buckling analyses; linear buckling analysis and non-linear analysis, which can be used to predict the buckling load and buckling mode shape of the delaminated region. Linear buckling analysis determines a bifurcation point of a perfect structure, where two or more load-deflection curves intersects. In reality, a structure that contains imperfections and geometrical non-linearities generally makes the results from the linear buckling analysis unconservative. In brief, a non-linear buckling analysis is a non-linear static analysis with a gradually increasing load to seek a load level at which the solution starts to diverge and the structure becomes unstable [8]. Due to the fact that post buckling involves geometrical non-linearities and large-deflections, non-linear analyses are used in this work. The non-linear analyses are performed using ABAQUS, version 6.4, ref. [26]. I-DEAS version 9, ref. [27], is used in the pre-and post processing, to create models and print results. A commonly used method called Virtual Crack Closure Technique (VCCT) is used to evaluate the strain energy release rate at the delamination front.

16

2.3.3 Strain Energy Release Rate Calculation Using the VCC-Technique When a sub laminate begins to buckle, three different crack opening modes are generated, Mode I, II and III, with the associated strain energy release rates GI, GII and GIII, respectively. The differences between the modes are described in short by Figure 13. If the energy release rate at the delamination front exceeds the critical strain release rate of the material Gcrit, the delamination will begin to grow.

Figure 13 Crack opening modes

A commonly used method called Virtual Crack Closure Technique (VCCT) is utilized to evaluate the energy release rate at the delamination front. The technique is based on the assumption that when a crack extends by a small amount, the energy released is equal to the work required to close the crack [16]. The different mode components of the energy release rate, GI, GII and GIII, are calculated from the displacements and nodal forces obtained from the solution of the FE-analysis. A more detailed description of the method is found in Appendix F.

2.3.4 Models A large number of FE-models are created with an artificial circular delamination in a quadratic plate. The sides of the plate are chosen to be four times the diameter of the circular delamination. These dimensions are needed to neglect the local effects of the boundary conditions on the sides of the plate. Mostly, a diameter of 60 mm is used and a plate side length of 240 mm. The FE-model is made of 15 300 solid linear brick elements (ABAQUS notation: C3D8) with a finer mesh in the front of the delamination. The number of nodes are 17 931 and the mesh of the model is shown in Figure 14. Solid linear brick elements are used instead of solid parabolic brick elements. The reason is that non-linear analysis with higher order elements is more time-consuming, especially when contact elements are utilized. Tests made show that the results do not differ notably due to the large amount of elements close to the delamination front. The element width at the delamination front is 0.5 mm and it is considered sufficiently small. Contact elements are used between the delaminated region and the base laminate to avoid non-allowed solutions as penetration of one surface into another. The delamination area is simulated by double nodes separated 0.1 µm from each other. The surfaces in the opening form a contact pair with contact elements in between. The stiffness matrix for different laminates and lay-ups are obtained using the program COST, which is developed at Saab. The matrices are converted to the ABAQUS input file. The program COST generates the same stiffness matrix as the pre- and post processing

17

program Patran. The program COST calculates the combined stiffness for a three-dimensional orthotropic material model of a laminate containing n plies. The plies in the laminate are treated as one unit, i.e. the stacking sequence is not taken into account. The latter may affect the load-displacement coupling that normally is found in laminated materials and one or a few couplings are not accounted for depending on the lay-up, see Appendix G. To lower the negative effects of these disadvantages of the technique used in COST, only delaminations located a few plies below the surface of the plate are examined. In this work, the delaminations are located at interface 6,7,8,9 and 10. Three different lay-ups are used in the FE-analyses; Quasi-isotropic lay-up [(0/90/45/-45)6]S , 0°-dominated lay-up [02/90/02/-45/02/45/02/90]2S , 90°-dominated lay-up [902/0/902/-45/902/45/902/0]2S. The material is HTA/6376C (carbon/epoxy), with the mechanical properties given in Chapter 1.4.

2.3.5 Boundary Conditions Different load cases are analysed needing different boundary conditions. The load cases are pure compression and different combinations of compressive and tensile loads in the x- and y-direction, see Figure 15. In all analyses, the back surface of the plate is clamped in the z-direction to avoid effects of global buckling. This simulates the thin film approach, used in the semi-analytical method, where no global buckling is assumed to take place. An imperfection, in form of a small point force applied on the centre node of the delaminated region, is added to initiate a buckle and to force the sub laminate to buckle in the “correct” direction. This force has an insignificant effect on the results due to its small magnitude.

18

Top side

Left

side

Right side

Bottom side

Figure 14 Model mesh and notations

Pure Compression In the case of pure compression, the left side of the plate is clamped in all directions, with the force applied on the right side in negative x-direction. On the right side, no translation in the y-direction is permitted and displacement control in the x-direction keeps the edge straight when loading. Compressive-Tensile Loading The left side and the bottom side are only permitted to translate in the y- and x-direction, respectively. The forces are applied on the top side and the right side, in the y- and x-direction, respectively. The right side is loaded in compression, while the top side is loaded either in compression or in tension.

19

(a) Pure compression (b) Compressive-Tensile loading

Figure 15 Loadings

2.3.6 FE-results In the case of a combination of compressive-tensile loading in x-and y-direction, the buckling mode is similar to the case of pure compression. In general, the greatest crack opening displacement, in the case of no applied shear load, occurs at a point located at the delamination front perpendicular to the compressive load direction, in accordance point “b” in Figure 16. The maximum strain energy release rates are reached in these points. At an angle of 90 degrees to these points, generally the delaminated region is in contact with the base laminate (see point “a” in Figure 16). The strain energy release rate is neglected at these locations, on the assumption that Mode I is the dominating crack opening mode. The FE-analyses show that the dominating crack opening mode is Mode I, followed by Mode II. In general, Mode III can be neglected.

Figure 16 Pure compression

20

2.4 Comparison of Results A selected amount of FE-analyses are performed to evaluate the accuracy of the results reached by the semi-analytical method described in previous chapters, concerning local buckling of delaminated regions. The FE-analyses are carried out using varying laminate lay-ups, load cases, and locations and sizes of the delaminations. The results are presented in this chapter. The chosen critical strain energy release rate, Gcrit, is 300N/m which is a reasonable value according to ref [23]. In FE-analysis the critical strain is equal to the applied strain when the total strain energy release rate, Gtot, exceeds the critical value, i.e. Gtot = GI + GII + GIII = Gcrit. The comparison is divided into two chapters due to the fact that the theory behind the semi-analytical method is divided into two parts; calculation of buckling strain and calculation of critical strain. The buckling strain and critical strain used in the comparisons are directly related to the load applied in the x-direction, the compressive load direction.

2.4.1 Buckling Strain A comparison showing the differences in calculated buckling strain at different load cases is performed. The plate has a quasi-isotropic lay-up and the circular delamination is located at interface 6 and 8, with a diameter of 60 mm. The combination of compressive-compressive loads is described as:

direction - xin the load eCompressivdirection -y in the load eCompressivRatio =

and for compressive-tensile loads:

direction - xin the load eCompressivdirection -y in the load TensileRatio =

21

0

0,05

0,1

0,15

0,2

0,25

0,3

0 0,25 0,5 0,75 1

Ratio [Compression in y-direction / Compression in x-direction]

Buc

king

stra

in [%

]

FEM -Interface 8Codein -Interface 8FEM - Interface 6Codein - Interface 6

Figure 17 Buckling strain versus different load conditions (Compression – Compression)

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 0,25 0,5 0,75 1 1,25 1,5

Ratio [Tension in y-direction / Compression in x-direction]

Buc

klin

g st

rain

[%]

FEM - Interface 8Codein - Interface 8

FEM - Interface 6Codein - Interface 6

Figure 18 Buckling strain versus different load conditions (Tension – Compression)

22

A comparison showing the differences in calculated buckling strain at different damage sizes is performed. The comparison is carried out using four different diameters of a circular delamination, 30, 45, 60 and 80 mm, located at interface 8 in a plate with a quasi-isotropic lay-up. The plate is loaded in compression.

0

0,2

0,4

0,6

0,8

1

1,2

30 45 60 75

Diameter of delamination [mm]

Buc

klin

g st

rain

[%]

FEMCodein

Figure 19 Buckling strain versus damage size – at interface 8

The following three graphs show the differences in calculated buckling strain for a delamination located at different interfaces in plates with three different lay-ups. The plates are loaded in compression and the lay-ups are the same as mentioned in Chapter 1.4. The delamination is 60 mm in diameter. The semi-analytical method uses the Reduced Bending Stiffness approximation (see Appendix A). The effect of not using the approximation is shown in the following graphs.

0

0,1

0,2

0,3

0,4

0,5

6 7 8 9 10

Interface

Buc

klin

g st

rain

[%]

FEMCodeinCodein (without RBS)

Figure 20 Buckling strain at different interfaces (quasi-isotropic lay-up)

23

0

0,05

0,1

0,15

0,2

0,25

6 7 8 9

Interface

Buc

klin

g st

rain

[%]

FEM

Codein

Codein (without RBS)

Figure 21 Buckling strain at different interfaces (0°-dominated lay-up)

0

0,1

0,2

0,3

0,4

0,5

0,6

6 7 8 9

Interface

Buc

klin

g st

rain

[%]

FEMCodeinCodein (without RBS)

Figure 22 Buckling strain at different interfaces (90°-dominated lay-up)

24

2.4.2 Critical Strain A comparison showing the differences in calculated critical strain at different load cases is performed. The plate has a quasi-isotropic lay-up and the circular delamination is located at interface 6 and 8, with a diameter of 60 mm. The load ratio used in the graphs is described in Chapter 2.4.1.

0

0,2

0,4

0,6

0,8

0 0,25 0,5 0,75 1

Ratio [Compression in y-direction / Compression in x-direction]

Com

pres

sive

stra

in in

x-d

irect

ion

[%]

FEM Interface 8 - Buck.strainFEM Interface 8 - Crit.strainCodein Interface 8 - Buck.strainCodein Interface 8 - Crit.strainFEM Interface 6 - Buck.strainFEM Interface 6 - Crit.strainCodein Interface 6 - Buck.strainCodein Interface 6 - Crit.strain

Figure 23 Buckling strain and critical strain versus different load conditions (Compression – Compression)

25

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

0 0,25 0,5 0,75 1 1,25 1,5Ratio [Tension in y-direction / Compression in x-direction]

Com

pres

sive

stra

in in

x-d

irect

ion

[%FEM Interface 8 - Buck.strain

FEM Interface 8 - Crit.strain

Codein Interface 8 - Buck.strain

Codein Interface 8 - Crit.strain

FEM Interface 6 - Buck.strain

FEM Interface 6 - Crit.strain

Codein Interface 6 - Buck.strain

Codein Interface 6 - Crit.strain

Figure 24 Buckling strain and critical strain versus different load conditions (Tension – Compression)

A comparison showing the differences in calculated critical strain at different sizes of damage is performed. The comparison is carried out using four different diameters of a circular delamination, 30, 45, 60 and 80 mm, located at interface 8 in a plate with a quasi-isotropic lay-up. The plate is loaded in compression.

0

0,2

0,4

0,6

0,8

1

1,2

30 45 60 75

Diameter of delamination [mm]

Com

pres

sive

stra

in [%

]

FEM - Buck.strain

FEM - Crit.strain

Codein - Buck.strain

Codein - Crit.strain

Figure 25 Critical strain versus damage size – at interface 8

26

The following three figures show the differences in calculated critical strain for a delamination located at different interfaces in plates with three different lay-ups. The plate is loaded in compression and the lay-ups are the same as mentioned in previous chapters. The delamination is 60 mm in diameter.

0

0,1

0,2

0,3

0,4

0,5

0,6

6 7 8 9 10

Interface

Com

pres

sive

stra

in [%

]

FEM - Buck.strain

FEM - Crit.strain

Codein - Buck.strain

Codein - Crit.strain

Figure 26 Critical strain at different interfaces (quasi-isotropic lay-up)

0

0,1

0,2

0,3

0,4

0,5

0,6

6 7 8 9

Interface

Com

pres

sive

stra

in [%

]

FEM - Buck.strainFEM - Crit.strainCodein - Buck.strainCodein - Crit.strain

Figure 27 Critical strain at different interfaces (0°-dominated lay-up)

27

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

6 7 8 9

Interface

Com

pres

sive

stra

in [%

]

FEM - Buck.strainFEM - Crit.strainCodein - Buck.strainCodein - Crit.strain

Figure 28 Critical strain at different interfaces (90°-dominated lay-up)

The FE-analyses indicate that the magnitude of the strain energy release rate of Mode I, GI, is more or less half the magnitude of the total strain release rate, when Gtotal = Gcrit. That is GI / (GI + GII + GIII) ≈ 50% and it is independent of lay-up. The strain release rate of Mode III can be neglected.

28

Figure 29 shows the predicted buckling strain and critical strain versus the damage radius for a circular delamination located at different interfaces in a plate with quasi-isotropic lay-ups and material HTA/6376C. The plate is loaded in compression and the lay-up is the same as mentioned in Chapter 1.4. The continuous and dashed lines refer to the, by Codein, predicted buckling strain and critical strain, respectively. The dots refer to the results produced by non-linear FE-analysis.

0

0,1

0,2

0,3

0,4

0,5

0,6

10 20 30 40 50 60 70

Radius [mm]

Com

pres

sive

stra

in [%

]

Buckling strain - Interface 6 Critical strain - Interface 6FEM: Buckling strain - Interface 6 FEM: Critical strain - Interface 6Buckling strain - Interface 8 Critical strain - Interface 8FEM: Buckling strain - Interface 8 FEM: Critical strain - Interface 8Buckling strain - Interface 9 "Critical strain - Interface 9FEM: Buckling strain - Interface 9 FEM: Critical strain - Interface 9Buckling strain - Interface 10 Critical strain - Interface 10FEM: Buckling strain - Interface 10 FEM: Critical strain - Interface 10

Figure 29 Buckling and critical strain versus damage radius (quasi-isotropic lay-up)

29

2.4.3 Discussion The elastic stiffness of a laminate are defined differently in the FE-analysis as compared to the semi-analytical method. In the FE-analysis, the stiffness of the solid brick elements is defined by COST. The program COST uses a technique by which it is possible to represent the three-dimensional elastic stress-strain behaviour of an entire laminate, consisting of n orthotropic plies, by one orthotropic Hooke´s matrix. The technique treats the plies in the laminate as one unit without taking the original stacking sequence into consideration. In the semi-analytical method the classic laminate theory is used to determine the constitutive equation of the laminate, i.e. the ABD-matrix. The laminate theory is described in Appendix G. In a typical laminate there can exist several undesired load-deformation couplings and these are related to certain components in the ABD-matrix, see Appendix G. Some of these effects are not accounted for when utilizing the technique of COST, where for example the bending-stretching coupling is not accounted for. To lower the negative effects of the mentioned disadvantages in the technique of COST, only delaminations located a few plies below the surface of the plate are examined. In the Figures 20-22, the effect of not using the Reduced Bending Stiffness approximation (see Appendix A) in the semi-analytical method is shown. The results indicate that the bending-stretching coupling will decrease the buckling load. In general, the FE-results predict a slightly higher buckling strain than the semi-analytical method, which could be explained by the discussion above. The anisotropy of the laminate is more pronounced when a 0°- and 90°-dominated lay-up is used. Consequently, a higher buckling strain will be predicted by the FE-method which is shown Figure 21 and 22. The author of this report recommends a further use of the Reduced Bending Stiffness approximation, which will provide conservative predictions of the buckling load. The FE-analysis takes material contact between the delaminated region and the base laminate into account, the semi-analytical does not. To avoid miscalculations contact regions are treated as non-allowable solutions and are disregarded. More information is found in Appendix E. The fracture problem encompasses all three components of G as well as their critical values. A mixed mode type fracture approach does not seem feasible with the present theory of the semi-analytical method due to the difficulties in analytically resolving G into its three components. According to [13] in the early stages of buckling the difference between the far-field strain and the membrane strain of the buckled sub laminate is small. Consequently, the shearing modes (i.e. GII and GIII) are expected to be relatively small so that G ≈ GI and Gcrit ≈ GIC. GIC is shown to be a material property, independent of test specimen geometry and of orientations of plies from both sides of the delaminating interface [17]. The FE-results show that the magnitude of the strain energy release rate of Mode II is approximately 0-50% of the total strain energy release rate for different load cases. This emphasizes the problem of determining the critical strain energy release rate. In the FE-analysis, a critical strain energy release rate of 300N/m is used which will generate slightly underestimated values of the critical strain due to the discussion above and the fact that a mixture of modes requires higher critical fracture energy than Mode I, [18].

30

2.5 Effects of Reduced Stiffness on Delamination Buckling The studies presented in the following chapters are based on non-linear FE-analysis. The analyses are performed in ABAQUS, version 6.4. I-DEAS is used in the pre-and post processing, to create models and print results. The FE-models, materials, lay-up, stacking sequence etc. are the same as presented in previous chapters. The FE-models are created with an artificial circular delamination, located at interface 8, in a quadratic plate with a total thickness of 6.24mm (48 plies). The laminate of the plate has quasi-isotropic lay-up and is made of the material HTA/6376C. The diameter of the delamination is 60 mm and the plate side length is 240 mm. The plate is loaded in compression. The damaged region, treated as a soft inclusion with reduced elastic properties, is assumed to be cylindrical with a diameter equal to the diameter of the delamination and a height equal to the total thickness of the laminate. That is the damage is assumed to exist through the entire thickness of the laminate. Notice that this is an extreme way of modelling the damage.

2.5.1 Effects of Different Failure Modes on Delamination Buckling As failure occurs in a ply of a laminate, material properties of that failed ply are changed by the material property degradation rule which deals with matrix cracks, fibre breakage, fibre-matrix shear out and delaminations. For each failure mode, there exists a proposed sudden material property degradation rule [17], see Table 1. These rules determine the elastic properties of the soft inclusion, i.e. the damaged region. The value one for the elastic constants means that it remains its original value. Zero means that it is reduced to a value slightly above zero.

Table 1 Sudden material property degradation rule Failure mode Exx Eyy Ezz Gxy Gyz Gxz νxy νyz νxz Matrix Tension 1 0 1 1 1 1 0 1 1 Matrix Compression 1 0 1 1 1 1 0 1 1

Fibre Tension 0 0 0 0 0 0 0 0 0 Fibre Compression 0 0 0 0 0 0 0 0 0

Fibre-Matrix Shear out 1 1 1 0 1 1 0 1 1

Delamination Tension 1 1 0 1 0 0 1 0 0 Delamination Compression 1 1 0 1 0 0 1 0 0

The sudden material property degradation rule is used in non-linear FE-analysis, to study the effects of each failure mode on the delamination buckling behaviour. The results are shown in Figure 30. The buckling strain and the critical strain predicted by Codein at interface 8 are also shown, where the critical energy release rate is assumed to be 300N/m. The result presented under the name “Undamaged” are derived from FE-analysis where no reduction of the elastic properties is made. These material properties are comparable to the ones used in the program Codein.

31

0

0,5

1

1,5

2

0 0,1 0,2 0,3 0,4 0,5Strain [%]

z-di

spla

cem

ent [

mm

]

0

300

600

900

1200

1500

1800

Stra

in e

nerg

y re

leas

e ra

te [N

/m]

Undamaged: z-dispMatrix failure: z-dispShearout failure: z-dispDelamination: z-dispCODEIN: Buckling strainUndamaged: GtotMatrix failure: GtotShearout failure: GtotDelamination: GtotCODEIN: Gtot

Figure 30 Effects of different failure modes on delamination buckling

2.5.2 Effects of Equally Reduced Stiffness on Delamination Buckling Non-linear FE-analysis is used in an attempt to estimate the effects on the delamination buckling behaviour when the damaged region is treated as a soft inclusion with equally reduced elastic properties. The Poisson’s ratio is assumed to be unaffected by the damage and the elastic moduli of the material is equally reduced by a reduction coefficient. The FE-analyses and models are identical to the ones in the previous chapter. The plate is loaded in compression.

0

0,5

1

1,5

2

0 0,1 0,2 0,3 0,4 0,5Strain [%]

z-di

spla

cem

ent [

mm

]

0

300

600

900

1200

1500

1800S

train

ene

rgy

rele

ase

rate

[N/m

]SRC 0%: z-dispSRC -25%: z-dispSRC -50%: z-dispSRC -75%: z-dispCODEIN: Buckling strainSRC 0%: GtotSRC -25%: GtotSRC -50%: GtotSRC -75%: GtotCODEIN: Gtot

Figure 31 Effect of reduced stiffness on delamination buckling (SRC = Stiffness Reduction

Coefficient)

32

2.5.3 Discussion The first comparison, presented in Figure 30, shows an extreme way of presenting the effects of each failure mode on the delamination buckling behaviour. The results should only be seen as a comparison of the tendencies of the different failure modes separately. In reality, the extension of the damage region is lesser and the damage consists of a mixture of the failure modes. Further, the distribution of the damage has its maximum centred under the impact point and it is decreasing in the radial direction. The case of fibre failure mode is not analysed due to the fact that it is most unlikely that fibre breakage can occur in the entire damage region, and would thus make the results incomparable to each other. The fibre failure mode in a ply is a catastrophic mode of failure and when it occurs, the material in that region cannot sustain any load. Figure 30 shows that a large extent of delaminations, with consequently reduced transverse stiffness Ezz, implies significantly reduced buckling load. The other failure modes have minor influence on the buckling load. The post-buckling behaviour shows the same tendencies. In this comparison, the critical strain calculated by Codein predicts an underestimated value even if failures are initiated. The second comparison, presented in Figure 31, shows the effects of characterizing the damaged region as a soft inclusion with equally reduced elastic properties. The results indicate that the buckling strain is significantly reduced by increased stiffness reduction. It is interesting to note that the critical strain calculated by Codein predicts an underestimated value even if the stiffness reduction coefficient is high.

2.6 Conclusions The different comparisons for artificial circular delaminations show that the calculated buckling and critical load show good correspondence to the results provided by the FE-analysis. Generally, the differences in calculated buckling loads by the two methods are small and the semi-analytical method is more conservative. The approximate calculation of strain energy release rate generally generates underestimated critical strains in comparison with FE-analysis. This is seen as a major advantage due to the fact that it is difficult to estimate the effects of reduced stiffness of the damaged region on the buckling behaviour.

33

3 Soft Inclusion Theory

3.1 Introduction Impact damage in composite structures usually contains fiber breakage, matrix cracks and interlaminar delaminations. The largest effects of impact damage are observed in compression, due to delamination buckling. Delaminations reduce the flexural properties of the damaged laminate and may cause significant reduction in compressive strength and stability. In tensile dominated load cases, at which delamination buckling will not occur, the loss of stiffness due to matrix cracks and fiber breakage could be of major importance. However, for a laminate that is also designed to sustain stress concentrations, for example open holes or fastener installations, the strength reduction in tensile dominated load cases due to impact damage is not very significant. Therefore a simplified conservative approach may be used for establishing this strength reduction. The characterization of impact damage in composite structures is complex and an abundance of different parameters will have to be considered when trying to characterize it; the impact energy level, the ductility of the materials, laminate thickness, fiber volume fraction, fiber lay-up, stacking sequence etc. In this chapter a semi-analytical method based on Lekhnitskii´s complex variable technique for rapid assessment of stresses and strains of notched composite structures is presented. The damaged region is treated as a soft inclusion with a reduced stiffness or an open hole. The theory is implemented into computer code and the program is used at Saab under the name “Conan” (COmposite Notch ANalysis). A concise description of the method is presented; see ref. [20], [21] for more details.

3.2 Soft Inclusion Theory (SIT) Consider an infinite plate subjected to general in-plane loading where p, q and t are the general in-plane load components, see Figure 32.

Figure 32 In-plane loading for a plate containing an inclusion

34

The stresses in and around the inclusion can be obtained by using the complex mapping technique by Lekhnitskii [20].

( ) ( )[ ]222211

212 zzpx Φ′+Φ′ℜ+= µµσ (16)

( ) ( )[ ]22112 zzqy Φ′+Φ′ℜ+=σ (17)

( ) ( )[ ]2221112 zztx Φ′+Φ′ℜ+= µµτ (18)

where

( ) ( ) ( ) ( ) ( )( )[ ]1

2221

111

21

ξµµ

µµabitCaqBbipAz −−+−−−

−=Φ (19)

( ) ( ) ( ) ( ) ( )( )[ ]2

1121

221

21

ξµµ

µµabitCaqBbipAz −−+−−−

−=Φ (20)

and a, b = major and minor axes of the ellipse

( ) =′′′ xyyxCBA τσσ ,,,, constant inclusion stresses ξ1 and ξ2 are defined as:

biabazz

k

kkkk µ

µξ

−−−+

=2222

, k = 1, 2 (21)

which originates from a conformal mapping operation and the complex variables zk are defined as

yxz kk ′+′= µ (22) where µ1 and µ2 are the roots of

( ) 0222 22262

66123

164

11 =+−++− aaaaaa µµµµ (23) and aij are the in-plane compliances of the plate. The theory deals with an inclusion in an infinite plate. A finite plate needs a finite width correction term to accurately determine the stresses around a soft inclusion. If necessary FE-analysis is used to determine the correction term. An inclusion with arbitrary lay-up and material can be used, as well as an open hole. When utilizing a soft inclusion, a characterization of the damage has to be done, i.e. determine

ijij aam ′= where ija′ is the in-plane compliance of the inclusion.

35

3.3 Failure Criterion The modes of failure of composite materials are complicated and there are many different failure criteria dealing with different failure modes. The Maximum Strain Criterion is commonly used at Saab and is therefore used in this study as the failure criterion in Conan. A brief description of the criterion is given in this chapter. An evaluation of the criterion and a discussion about its applicability are not given in this report. In the maximum strain criterion, the failure load is defined as the lowest load that makes one of the following inequalities satisfied: tensionxx ,εε ′≥ tensionyy ,εε ′≥ ncompressioxx ,εε ′≥ ncompressioyy ,εε ′≥ (24) shearshear γγ ′≥ where the superscript ´ refers to the allowable strains. The ply strength in tension and compression in the fibre direction and the transverse fibre direction, and the shear strength, is determined in laboratory tests. The maximum strain criterion does not take interactions among different failure modes into consideration and is therefore most suitable for failure prediction in fibre controlled laminates, i.e. laminates with at least 10% fibres in all four directions 0°, 90°, 45° and -45°. The Point Stress Criterion is used to determine the stresses which are connected to the strains in (24). The stresses are calculated in the fibre directions at certain distance from the edge of the elliptical opening or inclusion, see Figure 33. This distance is called the characteristic distance and is determined by experimental tests for different materials and lay-ups. To use the PSC, the stresses at a characteristic distance from the edge of the opening or inclusion have to be known. These are calculated by Conan in eight points according to Figure 33.

Figure 33 Evaluation points for the Point Stress Criterion

36

3.4 Approximate Determination of Relative Inclusion Stiffness In an attempt to estimate the relative inclusion stiffness, i.e. the stiffness reduction coefficient of damaged composite, a comparison between results from Conan and an alternative method is made. In an iterative manner, the stiffness of the inclusion is adjusted in Conan until the critical load is the same as for the alternative method. The lay-up and stacking sequence of the soft inclusion and the undamaged material are identical. The alternative method is a method based on experimental testing. It is a method that has been qualified for certifying aircraft structures to FAR/JAR 25 regulations, but can not be presented here in detail for reasons of confidentiality. The Poisson’s ratio is assumed to be unaffected by the damage and the elastic moduli of the material is reduced by a reduction coefficient. The stiffness of the soft inclusion in CONAN is iterated until the same critical strain is achieved as for the alternative method. The laminate used in the study consists of the material CYTEC/977-2 with a quasi-isotropic lay-up (according to Chapter 1.4). The study is focused on tensile dominated load cases and an equation is created using curve fitting from the results given in the case of pure tension, see Figure 35. The load cases used in the study are: pure tension, pure compression and different combinations of tensile-compressive loads. The load definitions used in the graphs below are explained in Table 2 and Figure 34.

Table 2 Load definitions Load case Definition

Tension Pure tension (uni-axial) Compression Pure compression (uni-axial)

Ten/Comp=2/1 12

direction -y in the load eCompressivdirection - xin the load Tensile

=

Ten/Ten=2/1 12

direction -y in the load Tensiledirection - xin the load Tensile

=

Comp/Ten=2/1 12

direction -y in the load Tensiledirection - xin the load eCompressiv

=

Figure 34 In-plane loading

37

0

0,2

0,4

0,6

0,8

1

0 2000 4000 6000 8000 10000 12000

Damage size [mm^2]

Red

uctio

n co

effic

ient

TensionCompressionComp/Ten=2/1Reduction coefficient = [ exp(-(Damage area)^(2/3)/100)+0.048 ]

Figure 35 Reduction coefficient versus damage size – three different load cases

Equation of stiffness reduction:

Reduction coefficient ( )

048.0e100

area Damage 32

+=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

(25) The reduced stiffness of the damaged material is equal to the stiffness of undamaged material multiplied by the reduction coefficient. A comparison between different tensile dominated load cases is carried out using the equation of stiffness reduction (25). Figure 36 shows the applicability of using the equation at different tensile dominated load cases.

0

0,2

0,4

0,6

0,8

1

0 2000 4000 6000 8000 10000 12000

Damage size [mm^2]

Crit

ical

tens

ile s

train

[%]

Conan [Tension]Alternative method [Tension]Conan [Ten/Ten=2/1]Alternative method [Ten/Ten=2/1]Conan [Ten/Comp=2/1]Alternative method [Ten/Comp=2/1]

Figure 36 By Conan predicted critical strain using the equation of stiffness reduction (25),

compared to the alternative method

38

3.5 Conclusions The results shown in Figure 36 show the difficulties of determining the relative stiffness reduction in damaged composites. A simple approach, like in this case, will not be conservative when compared to different load cases, even though the results do not differ greatly. Furthermore, different failure modes are associated with different load conditions. However, for tension dominated load cases a simple approach may be sufficient when the purpose is not to perform exact calculations but to handle different load conditions. To be able to establish a simple estimation of the relative stiffness reduction, a close interaction between experiments and modelling is necessary. The relative stiffness reduction needs to be established for separate load cases, which are associated to different failure modes and failure criteria. A major disadvantage of the soft inclusion theory presented above is that a constant relative stiffness reduction is required and that the theory does not deal with the variation of the stiffness reduction within the inclusion. The reduced stiffness significantly depends on the extent and distribution of the different failure modes depending on e.g. matrix cracks, fibre breakage, and a comparable constant relative stiffness reduction is difficult to determine. Therefore, simple approximations have to be made based on experimental testing. For example, experimental tests concerning three different load conditions could be studied; compression, tension and shear load. In Conan, an equation of stiffness reduction with respect to the damage area could be established for each load condition, based on the experimental test results. The differences in e.g. material, lay-up, stacking sequence needs to be taken into consideration. In the case of a mixture of different loadings, an eligible combination of the three equations could be used by simple applying the superposition principle.

39

4 Verification of Methods Three different methods for calculating residual strength of impact damaged composite structures are presented in this report; Conan (a computer program used at Saab, calculates stresses for an infinite plate with an elliptical opening or soft inclusion, intended to be used for tensile loading), Codein (predicts critical loads due to buckling and growth of delaminations) and an alternative method (based on experimental testing, valid for both tension and compression). Results from the alternative method are based on the material CYTEC/977-2. A number of comparisons between the methods are performed. The following graphs show the differences in predicted critical strain versus damage size. Two different CFRP-materials are studied; CYTEC/977-2 and HTA/6376C. The constitutive properties of the two materials are fairly similar and a rough comparison of results based on the two materials can be done. A quasi-isotropic lay-up is studied; [(0/90/45/-45)6]S. The laminates consist of 48 plies, which give a total thickness of 6.24 mm. The material and lay-ups are described in more detail in Chapter 1.4. The damaged region is assumed to be circular in shape. For reference purposes, results for an open hole are included as well (calculated in Conan). In Conan the damage region is simulated by a through-the-thickness cylindrical cut-out and no finite width correction is taken into consideration. The experimental test-results are from experimental studies at Saab, where some of the experiments have been performed in cooperation with FFA [3], [21]. These results are shown in Figure 37 under the name “FKH-results”. The laminates are made of HTA/6376C material with a quasi-isotropic lay-up and a total thickness of 6.2-6.24 mm.

4.1 Compression In Figure 37 the predicted critical strain is presented versus the damage size. The dashed lines refer to results based on the material HTA/6376C and the continuous lines CYTEC/977-2. Results from an old version of the program Codein are added. Its theory is described in Appendix A and C, while the theory for the current version (developed in this work) is described in Appendix A and B. The Conan results are presented for reference only, and are valid for an open hole.

40

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0 2000 4000 6000 8000 10000 12000

Damage area [mm^2]

Crit

ical

com

pres

sive

stra

in [%

]Codein (Cytec/977-2)Codein (HTA/6376C)Conan Cut-out (Cytec/977-2)Conan Cut-out (HTA/6376C)Alternative method (Cytec/977-2)FKH-resultsCodein Old version (HTA/6376C)

Figure 37 Critical compressive strain versus damage area (quasi-isotropic lay-up)

4.2 Tension In Figure 38 the critical tensile strain is presented versus the damage area. In Conan the damage region is simulated by a circular cut-out and a soft inclusion respectively. The material CYTEC/977-2 is used in both Conan and the alternative method.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0 2000 4000 6000 8000 10000 12000

Damage area [mm^2]

Crit

ical

tens

ile s

train

[%]

Alternative methodConan Cut-outConan Soft Inclusion (Equation 25)

Figure 38 Critical tensile strain versus damage area (quasi-isotropic lay-up)

41

4.3 Compressive-Tensile Loading The load definitions used in the graphs below are explained in Table 3 with x- and y-direction in accordance with Figure 34. The critical strain refers to the applied compressive strain in the x-direction. The material is CYTEC/977-2.

Table 3 Load definitions Definition Load case

Compression Pure compression

Comp/Comp=2/1 12

direction -y in the load eCompressivdirection - xin the load eCompressiv

=

Comp/Comp=1/1 11

direction -y in the load eCompressivdirection - xin the load eCompressiv

=

Comp/Ten=2/1 12

direction -y in the load Tensiledirection - xin the load eCompressiv

=

Comp/Ten=1/1 11

direction -y in the load Tensiledirection - xin the load eCompressiv

=

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0 2000 4000 6000 8000 10000 12000

Damage area [mm^2]

Crit

ical

com

pres

sive

stra

in in

x-d

irect

ion

[%]

Codein [Compression]

Codein [Comp/Comp=2/1]

Codein [Comp/Comp=1/1]

Alternative method [Compression]

Alternative method [Comp/Comp=2/1]

Alternative method [Comp/Comp=1/1]

Figure 39 Critical strain at Compressive-Compressive loading (quasi-isotropic lay-up)

42

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0 2000 4000 6000 8000 10000 12000

Damage area [mm^2]

Crit

ical

com

pres

sive

stra

in in

x-d

irect

ion

[%]

Codein [Compression]

Codein [Comp/Ten=2/1]

Codein [Comp/Ten=1/1]

Alternative method [Compression]

Alternative method [Comp/Ten2/1]

Alternative method [Comp/Ten1/1]

Figure 40 Critical strain at Compressive-Tensile loading (quasi-isotropic lay-up)

4.4 Discussion The different comparisons show that the critical strain, calculated by Codein, show good correspondence to the results provided by the alternative method. In compressive loading, all the methods predict lower critical strains than the experimental test results. In Figure 40, the comparison of result provided by Codein and the alternative method shows that the semi-analytical method is more conservative in the case of compressive-tensile loading. In the comparison between the semi-analytical method in Codein and FE-analyses it can be seen that the difference between the buckling strain and the critical strain decrease when the tensile load increase, see Figure 24. The comparisons show that it will be too conservative to treat the damaged region as an open hole when the damage region is small, referring to the Conan results in Figure 37 and 38. Soft inclusion, and methods to approximately determine its relative inclusion stiffness, is needed, as shown in Chapter 3.4. This is especially noticeably in the case of tensile loading, see Figure 38. Furthermore, in Conan the damage region is simulated by a circular cut-out and according to Figure 38, this will give results that correspond well to the results provided by the alternative method when the damage area are larger than 6000 mm2.

43

5 Conclusions This report concerns rapid semi-analytical methods for calculating residual strength of damaged composite structures. The study is divided into three parts. The first part deals with modification of a semi-analytical method which concerns buckling and growth of delaminations in damaged composite structures. The work consists of correction and modification of a preliminary computer code and verification of the method. In the first part, the verifications are performed by comparing results from the semi-analytical method with results from FE- simulations. The second part deals with characterization of impact damaged regions in composites as regions with reduced stiffness. In an attempt to estimate the relative inclusion stiffness, i.e. the stiffness reduction coefficient of a damaged composite, a comparison between the results from a semi-analytical method used at Saab, and an alternative method is made. The semi-analytical method calculates stresses for an infinite plate with an elliptical opening or inclusion. The third part deals with verification of the methods mentioned above. In the comparison, experimental test results and an alternative method based on experimental testing is used. The major conclusions of this work are listed below. Concerning the semi-analytical method which deals with buckling and growth of delaminations in a damaged composite structure: The different comparisons for artificial circular delaminations show that the calculated buckling and critical load show good correspondence to the results provided by a FE-based method. Generally, the differences in calculated buckling loads by the two methods are small and the semi-analytical method is more conservative. Because of the geometrical non-linearity of post critical deflection it is difficult to analyze the post critical behaviour of the buckled near surface laminate. Therefore, a simple model is preferable for the analysis of growth of delaminations. The results provided by the approximate method presented in this report concerning growth of delaminations show good correspondence to the results provided by alternative methods and experimental test results. Due to the small amount of test results available, more experimental tests need to be done for different load conditions to confirm the accuracy and applicability of the method. The author of this report recommends that the theory presented in Appendix A and B should be used for further development, in lieu of the formerly used theory at Saab presented in Appendix A and C. Concerning the semi-analytical method which deals with rapid assessment of stresses and strains of notched composite structures where the damaged region is treated as a soft inclusion or an open hole: Comparison with alternative methods show that the method will be too conservative if the damaged region is treated as an open hole when the damage region is small. Soft inclusion, and methods to approximately determine its relative inclusion stiffness, is needed, especially in the case of tensile loading. It is difficult to determine the relative stiffness reduction in damaged composites, due to the complex geometry with varying extent of matrix cracks, fibre breakage etc. A simple approach, like in this work, may not be conservative in comparison with different load cases, even though the results do not differ greatly. To be able to establish a simple estimation of the relative stiffness reduction, a close

44

interaction between experiments and modelling is necessary. The relative stiffness reduction needs to be applicable to different load cases, which are connected to different failure modes and failure criteria. However, for tensile dominated load cases a simplified approach of soft inclusion is proposed, since residual strength after impact is usually not the sizing criteria for a structural part. An empirical equation for the stiffness reduction is derived in this work.

45

References [1] Olsson R. (1999). Impact and damage tolerance of composites – status and future work

at FFA, FFA TN 1999-77, The Aeronautical Research Institute of Sweden, Bromma.

[2] Sjögren A. (1999). Fractographic characterization of impact damage in carbon fiber/epoxy laminates, FFA TN 1999-17, The Aeronautical Research Institute of Sweden, Bromma.

[3] Olsson R. (1999). A review of impact experiments at FFA during 1986 to 1998. FFA TN 1999-08, The Aeronautical Research Institute of Sweden, Bromma.

[4] Levin K. (1991). Characterization of delamination and fiber fractures in carbon reinforced plastics induced from impact, Mechanical Behavior of Materials – VI, Pergamon Press, Oxford, 1 pp.519-524.

[5] Hull D. and Shi Y. B. (1993). Damage mechanism characterization in composite damage tolerance investigations, Composite Structures, 23 pp.99-120.

[6] Gyunn E.G. and O’Brien T. K. (1985), The influence of lay-up and thickness on composite impact damage and compression strength, In proceedings of 26th AIAA Conference, Orlando FL, pp.187-196.

[7] Giugno D. (1998). Effect of geometry and boundary conditions on impact response and damage in composite materials, FFA TN 1998.17

[8] Hwang Shun-Fa, Mao Ching-Ping (2000). Failure of Delaminated Carbon/Epoxy Composite Plates under Compression, Department of Mechanical Engineering, National Yunlin University of Science & Technology, Touliu Taiwan.

[9] Hyonny K., Kedward K. T. (1999). A method for modelling the local and global buckling of delaminated composite plates. Department of Mechanical and Environmental Engineering, University of California, Santa Barbra, CA 93106, USA.

[10] Davidson B. D. (1990). Delamination Buckling: Theory and Experiment, Mechanical Systems Engineering and Research Division. Jet Propulsion Laboratory, California Institute of Technology, Pasadena, USA.

[11] Shivakumar K. N., Whitcomb J. D. (1984). Buckling of Sublaminate in a Quasi-Isotropic Composite Laminate, NASA Langley Research Center, Hampton, Virginia, USA.

[12] Chai H., Babcock C. D., Knauss W. G. (1981). One Dimensional Modelling of Failure in Laminated Plates by Delamination Buckling, Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, USA.

[13] Chai H., Babcock C. D. (1984). Two-Dimensional Modelling of Compressive Failure in Delaminated Laminates, California Institute of Technology, Pasadena, USA.

[14] Kachanov L. M. (1988). Delamination buckling of Composite Materials, Brookline, Massachusetts, USA: Lkuwer Academic Publishers, ISBN 90-247-3770-2.

46

[15] Nyman T., Bredberg A. and Schön J. (1998). Equivalent Damage and Residual Strength for Impact Damaged Composite Structures, In the proceedings of the American Society for Composites, 13th Annual technical Conference, September 21-23 1998, Baltimore, MD, USA, pp 1759-1775

[16] Shivakumar K. N., Whitcomb J. D. (1988). Strain-Energy Release Rate Analysis of Plates with Postbuckled Delaminations, Journal of Composite Materials, Vol. 23- July 1989, pp 714-734.

[17] Chai H. (1984). The Characterization of Mode I Delamination Failure in Non-Woven, Multidirectional Laminates, Composites (November 1984)

[18] Olsson R., Thesken J.C., Brandt F., Jönsson N.& Nilsson S. (1996). Investigations of delamination criticality and the transferability of growth criteria. FFA TN 1996-31, The Aeronautical Research Institute of Sweden, Bromma.

[19] Lessard B. L., Shokrieh M. M (1995). Fatigue Behaviour of Composite Pinned/Bolted Joints, Structures and Materials Laboratory, NRC, McGill University, Montreal, Canada.

[20] Lekhnitskii S. G. (1968). Anisotropic Plates 2 Ed. Ed. By S. W. Tsai and T. Cheron, Gordon and Breach Science Publishers.

[21] Bredberg A. (1999). Review of impact experiments at Saab AB since 1984, SWECOMP (NFFP 336+345), FKH-1999-0106, Linköping.

[22] Dym, C. L. and Shames, I. H. (1973). Solid mechanics – A Variational Approach, McGraw Hill, New York.

[23] Nyman T. (1996). Delamination buckling theory and associated strain energy release rate calculation. A simplified approach, Saab AB, Linköping.

[24] Dahlberg T., Ekberg A. (2002). Failure Fracture Fatigue – An introduction, Lund: Studentlitterature, ISBN 91-44-02096-1.

[25] Whitney J. M. (1987). Structural Analysis Of Laminated Anisotropic Plates, Lancaster, Pennsylvania: Technomic Publishing Company, INC., ISBN 87762-518-2.

[26] “ABAQUS/Standard User’s Manual, Version 6.4” (2003), Hibbit, Karlsson & Sorensen Inc.

[27] I-DEAS User’s Guide, Version 9 m2 (2002), Unigraphics Solutions Inc.

[28] Bredberg A. (1996). Summary of ABAQUS delamination buckling analyses, TUDL R-4037, Saab Military Aircraft, Linköping.

Appendix A Delamination Buckling Theory The theory is based on a Rayleigh-Ritz method, which is used for calculating buckling strains of elliptic and circular delaminations in orthotropic plates. The theory is taken from references [10] and [11]. The procedure is divided into the following main steps: (a) Selection of an admissible transverse displacement function. (b) Calculation of the potential energy. (c) Creation of eigenvalue equations. Assumption on transverse displacement field for the delaminated region is made that satisfies the clamped boundary conditions.

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ′

+⎟⎠⎞

⎜⎝⎛ ′

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ′

−⎟⎠⎞

⎜⎝⎛ ′

−=2

2

2

10

222

1byC

axCC

by

axw (1)

where C0, C1 and C2 are so called generalized displacements. The notations a and b are the major and minor axis of the ellipse according to Figure 7 in Chapter 2.2. The boundary conditions at the elliptical or circular boundary are

xwu′∂

∂= ,

ywv′∂

∂= , 0=== vuw (2)

i.e. clamped boundary conditions are assumed. Total Potential Energy The total potential energy, ∏, of the sub laminate is the sum of the strain energy, U, and the potential energy of applied loads, V.

VU +=Π (3) Strain Energy The strain energy, U, of the buckled sub laminate, which does not exhibit bending-stretching coupling, can be written as

∫∫⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛′∂′∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛′∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛′∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛′∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛′∂

∂=

ellipseA yxwD

yw

xwD

ywD

xwDU

22

662

2

2

2

12

2

2

2

22

2

2

2

11 4221

ydxdyx

wy

wDyx

wx

wD ′′⎪⎭

⎪⎬⎫⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛′∂′∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛′∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛′∂′∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛′∂

∂+

2

2

2

26

2

2

2

164 (4)

where Dij are the flexural moduli with respect to the local coordinate system (x´, y´). With the assumed transverse displacement field the in-plane compliance, Aij , and the coupling compliance will not contribute to the strain energy. The in-plane compliance matrix, Aij, is assumed to be neglected. To account for the bending-stretching coupling behaviour an approximation is made. In the Reduced Bending Stiffness (RBS) approximation the nonzero

bending-coupling matrix, Bij , is accounted for by modifying the classical laminated plate equations

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

κε

DBBA

MN

(5)

by

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

∗ κε

DA

MN

(6)

where

[ ] [ ] [ ][ ] [ ]BABDD 1−∗ −= (7) Substituting the assumed transverse displacement field into the strain energy and performing necessary differentiation and integration the expression for U becomes

{ } [ ]{ }00 CKCU T= (8) where { } ),,( 2100 CCCC T = and the stiffness matrix [ ]K is

[ ] [ ] [ ] [ ]( )66123222111 22 DDKKDKDK +++= (9) The coefficient of D16 and D26 vanish due to the symmetry of the deflection function, w, and the matrices [ ]1K , [ ]2K and [ ]3K are given by

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

15.005.05.005.035.15.05.05.00.4

31 abK π

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

35.005.05.005.015.15.05.05.00.4

32 baK π (10)

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

1.15.00.15.01.10.10.10.10.8

63 abK π

Potential Energy of External Loads The potential energy of external loads, V, is given by

ydxdyw

xwN

ywN

xwNV

ellipseA

dyx

dy

dx ′′

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛′∂

∂⎟⎠⎞

⎜⎝⎛

′∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛′∂

∂+⎟

⎠⎞

⎜⎝⎛

′∂∂

= ∫∫ ′′′′ 221

22

(11)

The potential energy of external loads is written in the local coordinate system and the superscript “d” indicates load intensities for the delaminated region.

Figure 1 Loadings and definitions There are no globally applied moments, [M] = 0. The load intensities, Nij, are given for the undamaged laminate in the global coordinate system. Calculating the strains in the same system, the strains for the sub laminate and the base laminate are determined. The strains of the sub laminate are converted to load intensities and transformed to the local coordinate system. The following equation is obtained after performing the necessary integration and differentiation

{ } [ ]{ } λ00 CKCV gT= (12)

whereλ is a scalar factor of the reference load Nx´ which lead up to a proportionality between the in-plane load components

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

′′

′

′

′′

LK

NNN

N

yx

y

x

ji

1λ (13)

The [Kg] matrix is symmetric, with elements given by

( )ab

aNbNK yx

g 3

22

11′′ +

=π

( )ab

aNbNK yx

g 1203 22

22′′ +

=π

( )ab

aNbNK yx

g 1203 22

33′′ +

=π

(14)

yg NbaK ′=

3012π

023 =gK

xg NabK ′=

3013π

Buckling Equation Applying the Trefftz criterion [23], i.e. 022 =∂Π∂ iC , i = 0,1,2 yields the eigenvalue

equation

[ ] [ ] 0KK g =+ λ (15) The solution of equation (15) results in three eigenvalues corresponding to three different buckling modes. The lowest absolute eigenvalue correspond to the first buckling mode, hence the only one of interest. The corresponding eigenvector are determined by solving

[ ]{ } [ ]{ } 000 =+ CKCK gλ (16) The elements in the eigenvector are the constants in the assumed displacement field, C0, C1, and C2. Note that the eigenvector is normalized and represent the shape of the buckle, not the size of it. The shape of the buckle is used when calculating the critical strain. If the out-of-plane deflection is relatively small, the post critical equilibrium shape is assumed analogous to the shape in the initial buckling state. The critical strain, εbuck, is given by the lowest eigenvalue multiplied with the applied strain.

Appendix B Approximate Calculation of Strain Energy Release Rate The method is based on work done by H. Chai and C. D. Babcook [12], [13] and L.M. Kachanov [14]. The analysis of delamination growth is based on a fracture mechanic approach and Griffith’s linear elastic theory for crack propagation. Griffith based his considerations on a global balance of energy in an entire structure. The energy balance equation deals with energy needed to create a new crack, energy stored in the structure as elastic strain energy, and the work done by external loads. When a crack grows it obtains an increment in length while the potential energy of the body decreases and there is an energy release. According to Griffith’s theory of brittle fractures the energy release is used to create new crack surface. The energy release rate G is defined as

dAdG Π

−= (17)

where G is rate of change in total potential energy with respect to crack area. The energy release rate G is a measure of the energy available to create the new crack and is usually called crack driving force. Crack growth is possible when the energy release rate G reaches a critical value, i.e. when

critGG = (18) Gcrit is a measure of the fracture toughness of the material. The calculations of the total potential energy is divided into two stages; pre buckling (ε < εbuck) and in buckling (ε > εbuck). The strain energy and the potential energy of external loads are calculated differently in the two stages and the terms (Π0, U0, V0) and (Π, U, V) refer to pre buckling and in buckling conditions, respectively. Pre Buckling (ε < εbuck) Pre buckling (ε < εbuck) the elastic potential energy, Π0, is given by

000 VU +=Π (19) where U0 is the membrane strain energy equal to the first part of expression (39) [Appendix D]:

( ) ( ) ( ){ } ydxdAAAAUellipseA

yxyyxx ′′+++= ∫∫ ′′′′′′2

662

22122

110 221 εεεεε (20)

and V0 is the potential energy of external loads given by expression (43) [Appendix D]:

{ } ydxdNNNVellipseA

yxd

yxydyx

dx ′′++−= ∫∫ ′′′′′′′′ εεε 20 (21)

The strains and load intensities are presented in the local coordinate system and the area integrals are taken over the elliptical boundary of the damage. Aij is the in-plane compliance matrix with respect to the local coordinate system (x´, y´). In Buckling (ε > εbuck) The potential energy of the buckled delaminated region consists of membrane energy and bending energy. According to ref. [14], in the post buckling equilibrium, it is possible to neglect the contribution of energy of compression as compared to the energy of bending, which is especially true for the case of snap buckling. The bending energy is dominating if the buckling deflection is larger than the thickness of the sub laminate [14]. In ref. [12], the magnitude of the membrane energy is shown to be much smaller than the bending energy, when buckling occurs. Therefore, the potential energy is assumed to be equal to the bending energy together with the contribution of the accumulated potential energy in the sub laminate before it buckles, Π0. The expressions for the bending energy are the same as used when determining the buckling strain, see previous chapter. Determination of G No interlaminar stresses, and consequently no energy release rate, develop at the delamination front until the delaminated region buckles. Therefore, in buckling, the total energy release rate of the ellipse becomes:

( )⎟⎠⎞

⎜⎝⎛

∂Π∂

−=ab

Ga π1

( )⎟⎠⎞

⎜⎝⎛

∂Π∂

−=ba

Gb π1 (22)

where 0Π++=Π VU and Ga and Gb are the energy release rates along the “a” axis (“b” fixed) and along “b” axis (“a” fixed), respectively. The potential energy, ∏0, is calculated by using the strains and load intensities referring to the initial buckling state, ε = εbuck. In buckling, the derivatives of the bending energy with respect to a and b may be expressed by

( ) { } [ ]{ } { } [ ]{ }( )λCKCCKC

aaVU

gTT +

∂∂

=∂+∂

( ) { } [ ]{ } { } [ ]{ }( )λCKCCKCbb

VUg

TT +∂∂

=∂+∂ (23)

where λ is a scalar factor of the reference load. Thus, λ is equal to the eigenvalue when buckling occurs. The normalized vector{ } ( )2100 ,, CCCC T = is obtained when solving the eigenvalue problem (16). The vector represents the shape of the buckle, not the size of it. To be able to calculate G, the scalar factor (D0/C0) needs to be determined in{ } ( )( )21000 ,, CCCCDC T = . The constant D0 is simply referring to the height at the centre of the buckle.

Due to the geometrical non-linearity of the post critical deflection it is difficult to analyze the post critical behaviour of the buckled near surface laminate. Therefore, simple models are preferable for the analysis of this problem. In post buckling, the shape of the buckle is assumed to remain the same, i.e. C0, C1 and C2 are constant. Ref [12] presents an approach using a strip model to determine the constant D0, see Figure 2.

Figure 2 Strip model

The contraction ∆1 of a strip of the delaminated region due to bending (in buckling) is

xdxwa

a

′⎟⎠⎞

⎜⎝⎛

′∂∂

=∆ ∫−

2

1 21 (24)

The contraction ∆2 of the base laminate due to compression after initial buckling is

a202 ε=∆ (25) where the strain ε0 refers to the added strain after initial buckling, i.e. ε0 = ε - εbuck where ε is the applied strain. Equating 21 ∆=∆ leads to an expression for D0

( )⎟⎟⎠⎞

⎜⎜⎝

⎛+

= 20

21

20

20

0 11321155

CCCa

Dε

(26)

When initial buckling occurs, the energy release rate is calculated by setting the applied strain equal to the buckling strain, ε = εbuck. If the critical energy release rate is reached when buckling occurs, the buckling strain is set to be the critical strain. Otherwise the critical far field strain, εcrit, related to the critical strain energy release rate, Gcrit, is found in an iterative manner by changing the applied loads, which in its turn will determine the height D0. The critical strain thus depends on the C-vector, through the increase of the buckle (D0), and the external loads, which contributes to the work done.

Appendix C Approximate Calculation of Strain Energy Release Rate – A Formerly Used Approach at Saab The method is based on work done by B. D. Davidson, ref. [10], K. N. Shivakumar and J. D. Whitcomb, ref. [11], H. Chai and C. D. Babcook, ref. [13] and L.M. Kachanov [14]. The theories are further developed at Saab AB [23]. The approach is using the result from the method of determination of buckling strain, presented in Appendix A. No interlaminar stresses, and consequently no energy release rate, develop at the delamination front until the delaminated region buckles. The energy release rate, or crack driving force, is defined as, ref. [24].

AG

∂Π∂

−= (27)

where the expression for the total potential energy, ∏, is the same as the expression used when determine the buckling strain, see Appendix A (equation (3), (4) and (11)). Only bending energy is considered. Crack growth is possible when the energy release rate G reaches a critical value, i.e. when

critGG = (28) Gcrit is a measure of the fracture toughness of the material and it may be considered a material property. The strain energy release rate, G, of the ellipse becomes, ref. [13]

abGa ∂

Π∂−=π1

baGb ∂

Π∂−=π1 (29)

where a and b are the major and minor axis of the ellipse according to Figure 3.

Figure 3 Self similar growth of elliptic delamination, from ref [13].

Determination of G The derivatives of the total potential energy with respect to a and b may be expressed in other terms

{ } [ ]{ } { } [ ]{ }( )λCKCCKCaa g

TT +∂∂

=∂Π∂

{ } [ ]{ } { } [ ]{ }( )λCKCCKCbb g

TT +∂∂

=∂Π∂ (30)

where λ is a scalar factor of the reference load. Thus, λ is equal to the eigenvalue when buckling occurs. The normalized vector{ } ( )2100 ,, CCCC T = is obtained when solving the eigenvalue problem (17). The vector represents the shape of the buckle, not the size of it. To be able to calculate G, the scalar factor (D0/C0) needs to be determined in{ } ( )( )21000 ,, CCCCDC T = . The constant D0 is simply referring to the height at the centre of the buckle. A similar approach using a strip model as described in Appendix B is used to determine the constant D0, see Figure 2. The contraction ∆1 of the strip due to bending (in buckling) is

xdxwa

a

′⎟⎠⎞

⎜⎝⎛

′∂∂

=∆ ∫−

2

1 21 (31)

where C1 and C2 in the transverse displacement, w, are disregarded:

( )0

222

1 Cby

axw

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ′

−⎟⎠⎞

⎜⎝⎛ ′

−= (32)

The contraction ∆2 due to compression is

a202 ε=∆ (33) where the strain ε0 refers to the reference load. Equating 21 ∆=∆ leads to an expression for D0

00 21016

εaD = (34)

When initial buckling occurs, the energy release rate is calculated by setting the applied strain equal to the buckling strain, ε0 = εbuck. Note that this is an approximate calculation of the energy release rate, due to the fact that no interlaminar stresses, and consequently no energy release rate, develop at the delamination front until the delaminated region buckles. If the critical energy release rate is reached when buckling occurs, the buckling strain is set to be the critical strain. Otherwise the critical far field strain, εcrit, related to the critical strain energy release rate, Gcrit, is found in an iterative manner by changing the height D0, and the

applied loads, corresponding to the buckling strain, remain constant during the iteration. The external load, contributing to the work done by external loads, V, is constant. The critical strain thus depends on the C-vector, implicit on ε, through the increase of the buckle (D0). This gives the relationship

2

buck

crit

buck

crit

GG

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

εε

(35)

Appendix D Energy Formulation of Anisotropic Plates The equations are taken from ref. [25]. Strain Energy – the Complete Expression The strain energy of an elastic body is given by the relationship

( )dxdydzUV

xzxzyzyzxyxyzzyyxx∫∫∫ +++++= εσεσεσεσεσεσ21 (36)

where the integration is preformed over the volume of the body. According to basic assumptions of laminated plate theory, 0=== yzxzz εεε . The constitutive relations of the plate are (see Appendix G):

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

xy

y

x

xy

y

x

xy

y

x

xy

y

x

DDDBBBDDDBBBDDDBBBBBBAAABBBAAABBBAAA

MMMNNN

κκκεεε

0

0

0

662616662616

262212262212

161211161211

662616262616

262212262212

161211161211

(37)

The strains can be expressed in terms of displacements:

2

20

xwz

xu

x ∂∂

−∂∂

=ε

2

20

ywz

yv

y ∂∂

−∂∂

=ε (38)

yxwz

xv

yu

xy ∂∂∂

−∂∂

+∂∂

=200

2ε

Reformulation of the expression of the strain energy, and integration with respect to z, gives the area integral for a plate of uniform thickness:

∫∫⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=A y

vAyv

xuA

xuAU

20

22

00

12

20

11 221

200

66

000

26

0

162 ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+xv

yuA

xv

yu

yvA

xuA

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

∂∂

−∂∂

∂∂

− 2

20

2

20

122

20

11 2yw

xu

xw

yvB

xw

xuB

⎥⎦

⎤⎢⎣

⎡

∂∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

−∂∂

∂∂

−yx

wx

uxv

yu

xwB

yw

yvB

2000

2

2

162

20

22 22

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂∂

−⎥⎦

⎤⎢⎣

⎡

∂∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

−xv

yu

yxwB

yxw

yv

xv

yu

ywB

002

66

2000

2

2

26 422

2

2

2

222

2

2

2

12

2

2

2

11 2 ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+ywD

yw

xwD

xwD

dxdyyx

wDyx

wywD

xwD

⎪⎭

⎪⎬⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂+

∂∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+22

66

2

2

2

262

2

16 44 (39)

There exist a coupling between bending and stretching due to the presence of the product of inplane displacements u0, v0 and the transverse displacement w. The coupling matrix, Bij , is zero in the case of symmetric lay-ups. Strain Energy - Pure Bending For pure bending problems the first expression can be considered as an arbitrary constant and the strain energy for transverse bending of a laminated plate can be written as follows

∫∫⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=A yx

wDyw

xwD

ywD

xwDU

22

662

2

2

2

12

2

2

2

22

2

2

2

11 4221

dxdyyx

wywD

yxw

xwD

⎪⎭

⎪⎬⎫⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+2

2

2

26

2

2

2

164 + C (40)

where Dij are the flexural moduli, w is the transverse displacement function and C is an arbitrary constant. The expression is identical to the expression of bending strain energy of a homogeneous anisotropic plate. In the case of specially orthotropic material, that means if D16 = D26 = 0, then the equation can be written as

dxdyyx

wDyw

xwD

ywD

xwDU

A∫∫

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=22

662

2

2

2

12

2

2

2

22

2

2

2

11 4221

+ C (41)

Potential Energy of External loads – Pure Bending In the case of constant plate thickness, the potential energy, V, is given by

dxdyyw

xwN

ywN

xwNV

Axyyx∫∫

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

= 221

22

(42)

where Nij are load intensities. There are no applied moments. Potential Energy of External Loads – No Out-of-Plane Deflection Before buckling, the potential energy, V, is given by

{ }dxdyNNNVA

xyxyyyxx∫∫ ++−= εεε 2 (43)

where Nij are load intensities. There are no applied moments.

Appendix E Determinate Buckle Size As described in Appendix B and C, a strip model is used to determine the constant D0 in the vector{ } ( )( )21000 ,, CCCCDC T = . The elements in the vector, C0, C1, and C2, are the constants in the assumed displacement field, see equation (1). The vector represents the shape and the size of the buckled delaminated region, where D0 is the height at the centre of the buckle. The approximate calculations of the constant D0 are different in the Appendix B and C. In the formerly used approach at Saab, presented in Appendix C, the constants C1 and C2 are disregarded and the same buckle shape is used in all calculations. The buckle shape is shown in Figure 4a and the height D0 is calculated by equation (34). In the new theory presented in Appendix B, the variation of the buckle shape is taken into consideration by utilizing the constants C1 and C2, which result in the expression for D0 in accordance with equation (26). The Figures 4a-c shows that the calculation of the height of the buckle is clearly dependent on the shape of it. This is especially true in the case of appearance of two peaks, see Figure 4c.

(a) { } ( )0,0,1=TC (b) { } ( )0,1,1 −=TC (c) { } ( )0,4,1=TC

Figure 4 Shape of the buckled delamination region

Non-allowable Solution – Contact Region The theory does not take contact between the delaminated region and the base laminate into consideration. By studying the magnitude of C1 and C2 in relation to the magnitude of C0 at a certain point, non-allowable solutions can be found, as shown in Figure 5. These points are treated as a point of contact where no crack growth is assumed to be possible, and therefore is disregarded.

{ } ( )0,2,1 −=TC

Figure 5 Non-allowable solution

Appendix F Virtual Crack Closure Technique The Virtual Crack Closure Technique [16] is used to calculate the strain energy release rate at the delamination front in plates with post buckled delaminations. The technique is three-dimensional and determines GI, GII and GIII over a short distance ∆a at the delamination front, see Figure 6. The VCC technique is implemented in a post-processing program to ABAQUS and used at Saab AB under the name g3d. A preliminary computer code of the program is corrected and adjusted by the author of this report.

Figure 6 Crack front region of delamination

The VCC technique is based on a theory where the forces needed to close a crack over an infinitesimal distance are used to calculate the strain energy release rate. The technique uses the relative displacement of nodes behind the delamination front and the stress distribution in terms of forces at nodes ahead of the delamination front. When using brick elements with four nodes on the surface of the delamination front, the strain energy release rate associate with the three different crack opening modes can be calculated by, [28].

( )( ) 22 21

222

wwauuF

G aábi +∆

−−= ′ (44)

where Fb2 is the nodal force, (ua´2 – ua2) is the relative displacement, ∆a is the element length, w is the distance between the corner nodes of the elements and the index i indicates the different crack opening modes I, II and III. The relative displacement and nodal forces in the n, r and t-direction (see Figure 7) refers to the modes I, II and III, respectively.

Figure 7 Crack front region of delamination

The Theory of The Virtual Crack Closure Technique The fundamental idea of the crack closure technique is that strain energy release rate is equal to the work per unit area required to close a crack over an infinitesimal distance. The crack front region of delamination is divided into three regions; the intact laminate (A), the sub laminate (B) and the base laminate (C) (see Figure 6). The normal to the midplane of the plate is assumed to remain normal in bending and the only gradient through the thickness is the linear variation of u and v due to rotation. The latter gives a simplified response in the thickness direction and there is an abrupt change in the response of the plate from the intact region to the cracked region. For example, the curvature κx of the intact region at the crack front is different from κx of the cracked region at the crack front. Closure of the crack front over an infinitesimal distance, ∆a, implies that over the distance ∆a, the two sub laminates A and B behave as a single intact laminate after closure. That is, u, v, εx, εy and εxy vary linearly and w is constant through the entire thickness of the combined laminate (region B + region C) over the distance ∆a. Since ∆a is infinitesimal, the strain distribution through the thickness of the sub laminates after closure is the same as in the intact laminate at the crack front. The work per unit area required to change the strains and curvatures to impose the closure is equal to the strain energy release rate. First, the strain distribution of the different regions is expressed as

( )( )C

Ci

Ci

Ci

BBi

Bi

Bi

Ai

Ai

Ai

zz

zz

z

−+=

−+=

+=

κεε

κεε

κεε

0

0

0

, 3,2,1=i (45)

where zB and zC are the z-coordinates of the midplanes of the sub laminates B and C, respectively. κ is the curvature; κ1 = κx, κ2 = κy and κ3 = κxy. A

i0ε , Bi0ε and C

i0ε are the midplane strains. ε1 = εx, ε2 = εy and ε3 = εxy. The strain distribution through-the-thickness of the sub laminates B and C needs to be the same as for A, which requires strain increments as follows

( ) ( )( )B

Bi

AiB

Ai

Bi

Ai

Bi

Ai

Bi zzz −−++−=−=∆ κκκεεεεε 00

( ) ( )( )CCi

AiC

Ai

Ci

Ai

Ci

Ai

C zzz −−++−=−=∆ κκκεεεεε 00 (46) 3,2,1=i

For each sub laminate the equations (46) can be expressed as

00

0

0

xyxyxyxy

yyy

xxx

z

zz

εκεε

κεεκεε

∆=′∆+∆=∆

′∆+∆=∆

′∆+∆=∆ (47)

where z´ is either z-zB or z-zC depending on which sub laminate being considered. The twist curvature xyκ∆ is zero [16]

To impose these changes of the midplane strains and curvature, the in-plane stress and moment resultants for each sub laminate needs to be determined. These are calculated by

[ ] [ ][ ] [ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

xy0

y0

x0

xy0

y0

x0

T

xy

y

x

xy

y

x

∆κ∆κ∆κ∆ε∆ε∆ε

DBBA

∆M∆M∆M∆N∆N∆N

(48)

The strain energy release rate is comparable to the work required to change the midplane strains and curvatures in the cracked region of the laminate at the crack front so that they are equal to the midplane strains and curvature in the uncracked region at the crack front. Finally, the strain energy release rate is expressed as

(

)C laminate subfor termsingcorrespond

21

000

000

+

+∆∆+∆∆+∆∆+

+∆∆+∆∆+∆∆=

Bxy

Bxy

By

By

Bx

Bx

Bxy

Bxy

By

By

Bx

Bx

MMM

NNNG

κκκ

εεε

(49)

Appendix G Laminate Theory A laminate is built from unidirectional plies with different fibre orientations. Each ply can be fabricated from various materials, having different mechanical properties. Classical laminate theory is used to determine the constitutive equation of the laminate. The theory is valid for linearly elastic materials. The generalized Hooke´s law for a ply subjected to a three dimensional state of stress is

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

6

5

4

3

2

1

665646362616

565545352515

464544342414

363534332313

262524232212

161514131211

6

5

4

3

2

1

εεεεεε

σσσσσσ

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

(50)

where ijC is the stiffness matrix where the stresses and strains in Voist notation are given by

126135234333222111

126135234333222111

,,,,,,,,,,εεεεεεεεεεεε

σσσσσσσσσσσσ======

====== (51)

Assuming an approximate state of plane stress, the transverse normal strain zε can be calculated in terms of plate stiffness through equation (50):

xyyxz CC

CC

CC

εεεε33

36

33

23

33

13 −−−= (52)

Using equation (52), the plane stress constitutive equation for the kth layer becomes

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

xy

y

x

kkk

kkk

kkk

kxy

ky

kx

QQQQQQQQQ

εεε

σσσ

662616

262212

161211

(53)

where the reduced stiffness terms ijQ are given by

33

33

CCC

CQ jiijij −= (54)

The laminate strains can be expressed in terms of the laminate midplane strains and curvatures:

xyxyxy

yyy

xxx

z

z

z

κεε

κεε

κεε

+=

+=

+=

0

0

0

(55)

where

x

ux ∂

∂=

00ε ,

yv

y ∂∂

=0

0ε , xv

yu

xy ∂∂

+∂∂

=00

0ε

2

2

xw

x ∂∂

−=κ , 2

2

yw

y ∂∂

−=κ , yx

wxy ∂∂

∂−=

2

2κ (56)

where 0u and 0v are tangential displacements of the middle-plane, w is the transverse displacement, 0

iε are the midplane strains and 0iκ are the curvatures. Classical Kirchhoff plate

theory for thin laminates states that a line originally straight and normal to the midplane of the laminate will remain straight and normal to the midplane when the laminate is extended and bent. This normal is assumed to have constant length. Further information could be found in [25]. Using (54) in conjunction with (55) and the stress and moment resultant definitions yield the following constitutive relations for the plate:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

xy

y

x

xy

y

x

xy

y

x

xy

y

x

DDDBBBDDDBBBDDDBBBBBBAAABBBAAABBBAAA

MMMNNN

κκκεεε

0

0

0

662616662616

262212262212

161211161211

662616262616

262212262212

161211161211

(57)

where Ni and Mi are line forces and line moments respectively, and

( ) ( )∫−

=2/

2/

2,,1,,h

h

kijijijij dzzzQCBA (58)

The integration of the stiffness matrix is done with respect to the z-axis. 0=z refers to the midplane of the laminate and the top and bottom of the laminate is 2/hz = and 2/hz −= , respectively.

Laminate Classification When using laminates in different applications, a number of features due to the anisotropy of the material need to be taken into consideration. In a typical laminate there can exist several undesired load-deformation couplings. The mechanical behaviour and couplings is related to certain components in the ABD-matrix, see Figure 8.

Symmetric Laminates A laminate is defined as symmetric if for each ply there is an identical ply located symmetrically with respect to the midplane of the laminate. In symmetric laminates, the coupling forces largely cancel out and the laminate as a whole will not distort, although there are still local stresses across the interlaminar boundaries. The coupling between stretching and bending vanish, i.e. 0=ijB . This means that no laminate bending is induced from the application of forces, only midplane strains result. Consequently, the applied moments can only cause curvatures but no midplane strains. Example: [0,90,90,0] = [0,90]S , [0,±30,60,60,±30,0] = [0,±30,60]S Balanced Laminate A laminate is defined as balanced if all laminate at angles other than 0° and 90° occur only in plus or minus pairs (not necessarily adjacent). A balanced lay-up will remove the coupling between stretching and shearing, i.e. 02616 == AA . The shear strain and shear force are uncoupled from normal strains and normal forces. Example: [0,30,90,-30] , [0,+45,-45,90,90,-45,+45,0] Quasi-Isotropic Laminates The quasi-isotropic laminate has the same in-plane stiffness independent of the coordinate system, i.e. [A] matrix is independent of orientation. 2211 AA = , 2616 AA = , 661211 2AAA =−

Figure 8 Coupling phenomena in laminated composite plates