Fibre-orientation measurements in short-glass-fibre composites—II: a quantitative error estimate...

Fibre-orientation measurements in short-glass-fibre composites—II:a quantitative error estimate of the 2D image analysis technique

Colin Eberhardta, Ashley Clarkea,*, Michel Vincentb, Thomas Giroudb, Sylvain Flouretb

aMolecular Physics and Instrumentation Group, Department of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UKbEcoles des Mines de Paris, F-06904 Sophia Antipolis, Cedex, France

Received 19 October 2000; received in revised form 3 July 2001; accepted 17 July 2001

Abstract

Fibre-orientation measurement by two-dimensional (2D) image analysis of polished cross-sections is a rapid and highly efficientmethod for determining the fibre orientation distribution over large sample areas. In a recent paper, a new technique was presented

for measuring fibre orientation to a high level of accuracy by the use of a confocal microscope. In this paper, the confocal techniqueis used to evaluate independently the performance of the 2D image analysis technique and the errors are presented in full. Theresults reveal a significant systematic error in orientation measurements and the effect of image resolution is considered. A methodis proposed for correcting this systematic error and its validity is verified experimentally. A technique for determining the sampling

error of a fibre-orientation measurement is presented, enabling the calculation of confidence limits about the derived orientationtensor components. It is shown how confidence limits aid the comparison of two fibre-orientation measurements of similar samples.Furthermore, the sampling error should enable a more meaningful comparison of numerical simulation of injection moulded

composites and their experimentally manufactured counterparts. # 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Short fibre composites; Glass fibres; Optical microscopy; Fibre orientation

1. Introduction

Over the past 40 years short-fibre-reinforced thermo-plastics (SFRT) have started to replace traditionalmaterials in many high-performance applications. Whilethe use of long-fibre composites is restricted because ofthe difficult fabrication process, the injection-mouldingtechnique enables the fabrication of components withcomplex morphologies. The fibre-orientation distribu-tion (FOD) within the finished component is of greatimportance because a number of the properties of thebulk material are highly dependant on this aspect of thematerials microstructure.

Recent advances in fibre suspension rheology andcomputer simulation techniques have made it possibleto perform accurate simulations of the injection mould-ing process. These simulations can provide quantitativedata on the FOD of the finished component. The mould-filling parameters may then be modified within the com-puter simulation and their effect on the component FOD

observed. This method of optimisation of the compo-nent microstructure is clearly much quicker and cheaperthan a purely experimental method.

In order to validate the results from computer simu-lations they must be compared to FOD data acquiredfrom real specimens. A number of different methods formeasurement of FOD exist, although most techniqueshave the similarity that they involve the inspection of thefibre cross-sections on a carefully prepared specimen.Some of the first FOD measurements were performed bymanual measurement of the elliptical parameters by hand[1], although this method is impractical when largequantities of data are required. Later measurements wereperformed interactively by the use of a computer and adigitiser tablet, significantly increasing the rate at whichfibre parameters could be measured [2]. More recently,the increase in affordable computer power has fuelledresearch into image analysis to automatically extract theorientation data from these images. Image processingproblems which have now been largely overcome are theremoval of broken fibre fragments and splitting oftouching fibre cross-sections which frequently occur inimages of high packing fraction specimens [3]. Together

0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.

PI I : S0266-3538(01 )00106-3

Composites Science and Technology 61 (2001) 1961–1974

www.elsevier.com/locate/compscitech

* Corresponding author.

E-mail address: [email protected] (A. Clarke).

with an XY translation stage and pattern matchingsoftware, this technique can be used to determine fibreorientations at a rate of thousands of fibres per minute[4].

Any comparison between experimental data andnumerical simulation requires that the experimentalmethods be scrutinised and any errors or uncertainties bequantified. When injection moulded parts are comparedto their numerical simulations the accuracy of the FODmeasurements need verification. Furthermore, when anumber of components are manufactured with exactly thesame processing parameters, small variations in FODbetween specimens would be expected. A comparison offibre orientation predicted by numerical simulation andtheir respective injection moulded components shouldalso take this into account, although the quality of dataagain hinges on the accuracy of the 2D image-analysistechnique.

The performance of the 2D image-analysis techniquehas previously been considered by a number ofresearchers, and their findings will be discussed in thefollowing sections. The recent development of a con-focal technique for FOD measurement has finally pro-vided a means of independently measuring the errorsand uncertainties in the 2D technique.

2. Fundamentals

2.1. Tensor description of fibre orientation

A brief description follows of how the second-orderorientation tensor is calculated. More detailed discus-sions on the formation and application of orientationtensors can be found elsewhere [5].

The orientation of a single fibre is described in polarcoordinates by two angles, � and �. Alternatively, theorientation of individual fibres can be characterised bythe components of a vector p which lies parallel to thefibre as illustrated in Fig. 1. The components of thesecond-order orientation tensor, a, which provides aconcise description of the orientation state of n fibresare given below,

aij ¼1

n

Xnk¼1

akij ¼1

n

Xnk¼1

pki pkj

!ð1Þ

where pki pkj is a dyadic product of the kth fibre’s vector

components. Throughout this paper the following con-vention will be adopted: tensor components with super-scripts, akij, denote the orientation tensor for singlefibres, whereas tensor components without any super-script, aij denote the orientation average of a group offibres.

The probability that a fibre will be intersected by arandomly placed section plane is related to its length, L,diameter, D, and orientation as follows [6],

P �ð Þ ¼ Lcos� þDsin� ð2Þ

In order to derive an unbiased estimate of the orien-tation tensor, the tensor components of each fibre mustbe weighted by the inverse of its probability of intersec-tion, Fk. This results in the following equation for theweighted orientation tensor,

aij ¼

Pnk¼1

Fkakij

Pnk¼1

Fkð3Þ

3. 2D Image analysis techniques

3.1. Overview

The measurement of FOD by 2D image analysisrequires the preparation of a cross-section at a pre-scribed location in the finished component. The methodof preparation depends on the technique used to imagethe specimen, e.g. scanning electron microscopy (SEM)or reflected light microscopy. The cost of a light micro-scope is considerably less than that of a SEM, thereforeif similar images can be achieved with the former, thiswill inevitably be the preferred choice. The preparationof a specimen for inspection by reflected light micro-scopy requires high contrast between the fibre andmatrix, which may not be achieved simply by polishing

Fig. 1. The orientation of a single fibre can be expressed in polar

coordinates by the two angles (�, �) or in Cartesian coordinates by the

components of a vector p, (p1, p2, p3).

1962 C. Eberhardt et al. / Composites Science and Technology 61 (2001) 1961–1974

the specimen. A major increase in contrast is observed ifthe specimen is etched with oxygen ions. The matrix isroughened resulting in greater light scattering and imageswith contrast comparable to those obtained by SEM [7].

The orientation of a fibre and the location of itsintersection with the section plane are described by theparameters (�k, �k) and (xk, yk), respectively. Theseparameters are straightforward to derive from an ellip-tical fibre cross-section by image analysis. The para-meters that describe an ellipse are its position (xc, yc),axis lengths (a, b) and orientation, �, as illustrated inFig. 2. The orientation of a fibre is derived from theparameters of its elliptical cross-section as follows:

�k ¼ arccos b=að Þ

�k ¼ � or �þ 180 ð4Þ

The two possible values for �k are the result of anambiguity in determining the orientation of a fibre, dueto fibres with orientations � and �+180 having identicalcross-sections. The diagonal components of the orien-tation tensor are unaffected by this ambiguity.

A number of different automated techniques formeasurement of elliptical parameters by binary imageanalysis have been developed, including the Houghtransform, least squares and second moments techni-ques. The Hough transform is relatively computation-ally intensive, involving the plotting of each point fromthe image plane as a curve in the parameter space, withan ellipse requiring a 5D parameter space [8]. The leastsquares technique (LST) and second moments technique(SMT) are both much more simple in terms of compu-tation. Both have been used as methods of measuringthe FOD of composite materials.

3.2. Second moments technique

The parameters of an elliptical fibre cross-section maybe derived from the object’s second moments of inertia

[9]. Firstly, the image is converted into a binary imageby applying a simple threshold. Objects are locatedwithin the image by a process called pixel connectivity,where groups of pixels are ‘grown’ until all the pixelswithin the object have been added to the group. Thesums required for the formation of the second momentsare calculated during the formation of each pixelgrouping (or object) and they relate to the ellipse para-meters as follows:

xc ¼MX

yc ¼MY

a2 ¼ 2 MXX þMYYð Þ þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMXX �MYYð Þ

2þ4MXY

2

qb2 ¼ 2 MXX þMYYð Þ � 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMXX �MYYð Þ

2þ4MXY

2

q2� ¼ arctan

2MXY

MXX �MYY

� �ð5Þ

where MX and MY are the object’s first moments andMXX, MYY and MXY are the object’s second moments ofinertia.

The SMT considers all the pixels within the object inorder to calculate the ellipse parameters. Therefore, inorder to obtain accurate FOD measurements using thistechnique each fibre cross-section must exist in itsentirety. Any missing information, whether it be fromthe objects perimeter or within its body, will contributeto an error in fibre orientation measurement. A numberof fibre cross-sections and their elliptical fit by the SMTare illustrated in Fig. 3(i). Clearly this technique has apoor performance for partial fibre cross-sections asdemonstrated by the fibre marked ‘A’ in the figure.These partial cross-sections may occur when a fibre hasbeen cut close to one of its ends, or as an artefact of thespecimen preparation process.

In order to improve the performance of the SMT it isbeneficial to filter the data to remove some of the erro-neous elliptical fits. There are many parameters whichmay be defined in order to filter the data, although someprovide better results than others. Two parameters thathave been found to give good results are the minor axislength and elliptical fit factor. The minor axis length ofa fibre cross-section is equal to the diameter of the fibre.Therefore, if the fibres within the specimen have a welldefined diameter distribution, filtering can be applied tothe fitted ellipses to remove data with abnormally highor low minor axis lengths. The second filtering techni-que uses the elliptical fit factor [10],

s ¼A

�abð6Þ

which compares the area of the pixelated object, A, tothat of its fitted ellipse. If, s � 1, the object is highly

Fig. 2. The position and orientation of a fibre can be determined from

the parameters of its elliptical cross section (a, b, xc, yc, �). This

method results in a 180� ambiguity in determining �.

C. Eberhardt et al. / Composites Science and Technology 61 (2001) 1961–1974 1963

elliptical, whereas if s << 1, it is quite dissimilar inshape to an ellipse. Applying a threshold to the fit factorof say, s > 0:9 will ensure that all the recorded fibreshave resulted from objects which are themselves ellipses.

When specimens with a high fibre volume fraction areanalysed, fibre cross-sections may appear to be touch-ing. This effect is emphasised by the point spread func-tion of the microscope leading to a slight lateral blurringof the images, thus increasing the number of ‘touching’fibres. The SMT requires that these touching fibres areseparated before analysis which inevitably increases theimage processing time and may cause the truncation ofboth fibre’s cross-sections.

3.3. Least squares technique

An alternative method for determining the parametersof an ellipse is the least squares technique [11]. The LSTdiffers from the SMT in that only the perimeter pixels ofan object are considered. Pixels on an objects perimetermay be identified during pixel connectivity, i.e. any pixelwhich has a neighbouring pixel below the threshold isflagged as a perimeter pixel.

All curves of second order satisfy the following equa-tion:

Ax2 þ Bxþ CxyþDyþ Ey2 þ F ¼ 0 ð7Þ

which describes ellipses and other hyperbolae. Formulaethat relate the coefficients in the above equation to thoseof an ellipse, (xc, yc, a, b, �) are given in Appendix A. Inorder to solve the above equation for the object peri-meter, a constraint must be applied in order to avoid thetrivial solution where all the coefficients are equal tozero. A number of different constraints have been sug-gested, although in this particular application an ellipse-specific constraint is the obvious choice [12].

The LST also requires a splitting algorithm whenellipses are fitted to touching fibres or truncated cross-sections. A splitting algorithm is employed which tracesan object’s perimeter and the local curvature is calcu-lated at each point on the perimeter. At any perimeterpoint where the curve is concave or strongly convex, theperimeter is split. In this way an object consisting oftouching fibres or a truncated ellipse will have its peri-meter split into sections. If after analysis of the perimeter,two are found to give very similar ellipses, they are re-grouped. In Fig. 3(ii) the performance of the LST iscompared directly with that of the SMT by fitting ellip-ses to the same set of data. The LST gives a much betterfitted ellipse for truncated cross-sections, and a slightimprovement for irregular cross-sections.

The quality of a fitted curve using the LST can beassessed by calculating the mean deviation of the peri-meter pixels from the fitted ellipse:

d ¼

P�2i

nð8Þ

where, �i is the deviation of a ith perimeter pixel fromthe fitted ellipse and n is the total number of perimeterpixels. With this parameter, the data produced by theLST can be filtered in much the same way as the SMT,using a fit factor and minor axis length filters.

3.4. Errors and uncertainties

To clarify the terminology used in this paper, theword specimen is used to describe a single cross-sectionthrough a composite part, whereas the word sample is

Fig. 3. (i) A number of fibre cross-sections identified by the second

moments technique and (ii) the same fibres identified by the least

squares technique. The small circles that can be seen on the cross-sec-

tion perimeters in (ii) indicate where the object perimeter has been split

up into sections. The two techniques perform quite differently for

partial fibre cross sections, as illustrated by object A. The second

moments technique fits an ellipse to the entire object, whereas the least

squares technique splits the perimeter and fits an ellipse to each sec-

tion, ellipses that are too large or small are rejected.


used to describe a collection of one or more fibre orien-tation measurements from a single specimen. The errorsassociated with the measurement of fibre orientationmay be classified as either: (i) measurement error, thevariation in results as an experiment is repeated underidentical conditions for the same specimen; (ii) sys-tematic error, a non-random error which may not beeliminated or reduced by increasing the specimen size;(iii) sampling error, the variation between differentFOD measurements (i.e. samples) from the same speci-men.

Previous studies of the uncertainties in FODmeasurement by the SMT have revealed that themeasurement error increases as � tends to zero [10] andthat it can be reduced by increasing the image magnifi-cation [11]. However, a thorough investigation of theerrors in the SMT has not yet been undertaken.

4. Quantifying the different types of error in the 2D

technique

4.1. Overview of the 3D confocal technique

In a previous paper, a new technique for automatedFOD measurement by confocal laser scanning micro-scopy (CLSM) was presented [13]. A brief description ofthis technique will be given here but for further detailsreaders are referred to the earlier publication.

The confocal technique enables the non-destructiveoptical sectioning of semi-transparent materials. Withthis technique it is possible to follow individual glassfibres for a small distance (typically �z=20 mm) intothe matrix of SFRT. The orientation of individual fibresis calculated from the displacement of their centre, (�x,�y), over a known distance, �z. The process of locatingand following fibre centres has been fully-automatedenabling the collection of large datasets (mm�mm)within a few hours. A study of the uncertainties in fibrecentre location has shown that this technique givesFOD measurements which are accurate in out-of-planeangle, � to within �� <1.5� for all � and in-plane angle,� to within ��<3� for �>10�. For a more detaileddiscussion of the errors in the confocal technique read-ers are again referred to the previous paper [13]. There-fore, it is assumed that the confocal technique gives anaccurate measurement of the orientation of individualfibres:

�CLSM � �

�CLSM � � ð9Þ

and, hence, the subscript (CLSM) will be not be used inthe equations and text that follows.

4.2. Methodology

The confocal technique not only provides sub-surfaceimages of a specimen but also reflection mode images atthe specimen surface. These images are very similar tothose produced by reflected light microscopy, althougha small increase in lateral resolution can be expected[14]. Therefore the surface images can be analysed usingboth the SMT and the LST techniques and the resultscompared to those obtained by CLSM fibre centre fol-lowing technique.

A polyarylamide SFRT with a fibre volume fractionof 30% was analysed. A region measuring 1.6�1.8 mmwas scanned and five subsurface images covering a totaldepth of �z=20 mm were acquired. The orientation ofeach individual fibre was calculated using the fibre cen-tre displacement technique and compared to SMT andLST. Differences between the three techniques occa-sionally resulted in a fibre being correctly identified (i.e.parameters within the filtering bounds) by one or two ofthe techniques but not all three. For example, a fibremay be detectable on the surface by both the SMT andLST techniques but if the fibre end occurs at a depth of10 mm, it will not be fully traced by the confocal tech-nique and hence a measurement cannot be made. Simi-larly, either the SMT or LST may fail to split a pair oftouching fibres, which may be identified by the CLSMtechnique. In the following analysis fibres were onlyconsidered if they were successfully detected by all threetechniques.

4.3. Results

The 1.6mm�1.8mm�20 mm region was scanned inapproximately 2 hours, and a total of 2876 fibres wereidentified by all three techniques. The position, orientation,and relevant fit factors as calculated by each techniquewere stored for each fibre. All the data were filtered byminor axis length, with a maximum allowed deviationfrom the mean fibre diameter of 25%.

Fig. 4 shows the error in determining the out-of-planeangle �SMT, ��SMT by the SMT, which is calculated foreach fibre as follows,

D�SMT ¼ �SMT � � ð10Þ

The results were collated into ‘bins, 0 < � < 10,10 < � < 20, etc. with the mean and standard deviationof ��SMT calculated for each ‘bin’. The deviation of themean angular error, ��SMTh i from the x axis as �approaches zero indicates the presence of a systematicerror in the SMT. The out-of-plane angle, � is calculatedfrom the ratio of the ellipse axes [Eq. (4)] which leads tothe non-linear behaviour seen in this figure. For nearperpendicular fibres, i.e. � � 0�, small errors in themeasurement of ellipse axes will result in a large error


in the measured value of �. The error decreases when amore strict fit factor is applied, although the number offibres in the dataset is approximately halved in order toobtain this improvement. However, such a strict filter-ing may be undesirable because highly elliptical fibrecross-sections occupy a greater area on the section planeand are hence more prone to defects. Therefore a strictfit factor will inevitably introduce an angular bias to theresults. The ‘error’ bars on this graph indicate the stan-dard deviation about the mean, indicating the randommeasurement error associated with the SMT.

Fig. 5 shows the corresponding plot for the LST andagain both measurement and systematic errors areobserved. A strict fit factor has been chosen thatremoves a comparable number of fibres from the data-set, halving the number of fibres present. This time, verylittle improvement is observed which suggests that best-fit ellipses obtained by the LST may be less affected bysmall defects in the fibre cross-sections when comparedto the SMT. This will indeed be the case if the perimeterabout the defect is removed during the splitting process.

The error in angle �SMT, ��SMT by the SMT isdetermined using the same method as for ��SMT,

��SMT ¼ �SMT � � ð11Þ

Fig. 6 shows the mean and variance of the error indetermining �. The mean error, ��SMTh i has bothpositive and negative values, i.e. little evidence of a sys-tematic error is present in this plot. The mean error��SMTh i, increases considerably as � tends to zero.

However, when calculating the orientation tensor, thesignificance of the true value of � decreases accordingly.

To date, research groups have determined the error ofboth the SMT and LST by either simulation (Mlekusch[11]) or estimation (Davidson et al. [4]). Those findingsare plotted in Fig. 7 for comparison. Care must be takenwhen making a comparison between these resultsbecause it is not possible to ensure that the 2D imageanalysis systems all have the same spatial resolution.Furthermore, differences between the specimens studiedmay affect the accuracy of the FOD measurement.However, it would appear that the error predicted byMlekusch, who used the LST, is an underestimate and

Fig. 4. The error in determining the out-of-plane angle of a fibre, �SMT

by the second moments technique for two different fit factor thresh-

olds. The true value of � has been determined by the confocal techni-

que. The graph deviates significantly from ��SMT=0 signifying the

presence of a systematic error, the deviation about the mean illustrated

by the error bars gives the respective measurement error.

Fig. 5. The error in determining the orientation of a fibre, �LST by the

least squares technique for two mean perimeter pixel deviations. The

true value of � has been determined by the confocal technique. The

graph deviates significantly from ��LST=0 signifying the presence of a

systematic error, the deviation about the mean illustrated by the error

bars gives the respective measurement error.

Fig. 6. The error in determining the orientation of a fibre, �SMT by the

second moments technique. The true value of � has been determined

by the confocal technique. Although there is no clear systematic error,

the random measurement error is large for small �.


that of Davidson, who used the SMT, is an over-estimate.

It is expected that the resolution of the surface imageand its subsequent analysis will have an effect on themeasurement error. In order to measure this effect thedata were re-analysed using the SMT with a simulatedreduction in resolution, where a limited number ofimage pixels were used in the moments calculations.Fig. 8 shows the resulting increase in systematic error ofdetermining �, as the resolution is successively halved.Each time the resolution is halved the number of pixelsused in the second moments calculation is reduced by afactor of four. Despite this, there is very little observabledifference between the results for hDi ¼ 67 and hDi ¼ 33pixels. These results imply that a limit has been reachedwhich is not related to the image resolution. This limitin accuracy could be due to the resolution limit of themicroscope itself or the fibres having slightly non-cir-

cular cross-sections. A calibration slide with a largenumber of circles with a diameter of 25 mm was ana-lysed using the SMT. Even at the highest resolutionavailable where hDi ¼ 130 pixels, a systematic error,h�i ¼ 6� was observed. Ref. [15] measured the cross-section of a number of fibres within a specimen anddeduced that the upper limit to the intrinsic ellipticity is,e ¼ 0:011, which implies that a fibre when perpendicularto the section plane would appear to be at an angle of8�. These results demonstrate that, although an increasein resolution will increase the FOD measurement accu-racy, the systematic error which is inherent in 2D imageanalysis techniques cannot be reduced.

5. Treatment of errors

5.1. Measurement errors

If the orientation tensor of a group of fibres isrequired, the measurement error contribution for eachindividual fibre must be combined to give the measure-ment error of this group. The error in determining theorientation tensor of an individual fibre is as follows,

�akij

2

¼@akij@�k

��k

!2

þ@akij@�k

��k

!2

ð12Þ

This equation is formed under the assumption thatthe error in � and the error in � are not correlated.From analyses of the fibre orientation data obtainedusing the confocal technique, the correlation coefficient,�� was found to be of the order 10�2, hence it will beneglected. The error in determining the orientation of afibre, �� and ��, has been measured in the previoussection. The measurement error may be described by thefollowing functions, fm

� �ð Þ and fm� �ð Þ which are derived

from fitting curves to the experimental data:

�� fm�ð�Þ

�� fm�ð�Þ ð13Þ

If the effect of angular biasing is ignored, the orienta-tion tensor of a group of fibres, given by Eq. (1), willhave the following standard error:

s2aij ¼1

n

Xnk¼1

�akij

2

¼1

n

Xnk¼1

@akij@�k

fm� �k� � !2

þ@akij@�k

fm� �k� � !2

0@

1A ð14Þ

The partial derivatives that are required to evaluate

Fig. 7. A comparison of the experimentally determined error in mea-

suring the out-of-plane orientation, � of a single fibre and the results of

Mlekusch [11] and Davidson et al. [4].

Fig. 8. The effect of resolution on the error in determining the orien-

tation of a fibre using the second moments technique. The true value

of � has been determined by the confocal technique.


this equation are provided in Appendix B.1. The aboveequation is simply a sum of the error components forindividual fibres and is straightforward to evaluate.

In order to correct the effects of angular bias, theweighted orientation tensor, given in Eq. (3), isrequired. The weighting function, Fk, is dependant onthe fibre orientation, �k, resulting in a much more com-plicated expression for the standard error of the weigh-ted orientation tensor. The formulation of this expressionis given in Appendix B.2, following the method describedby Mlekusch [11], which has been modified in order toincorporate a different weighting function.

The effect of measurement errors on the orientationtensor has been examined experimentally. Fig. 9 showsthe diagonal components of the orientation tensor foran injection moulded specimen where the familiar skin-core-skin configuration is visible. The plot has beengenerated from the 33,000 fibres which remained afterstrict filtering criteria. The following functions, whichenvelope the errors observed in Figs. 4 and 6, have beenchosen in order to describe the measurement error in �and � using the SMT,

fm� ¼ 19:6e�=27:6 ð15Þ

fm� ¼ 74:5e�=26:5 ð16Þ

where � is measured in degrees. The error bars denotethe standard error, saii , of the respective orientationtensor. Despite the large measurement error in deter-mining the orientation of a single fibre, the largeamount of data available results in a surprisingly smallerror in the average orientation tensor. Therefore, if themeasurement errors are considered and the systematicerrors ignored, the derived error in the orientation ten-sor will be under-estimated and hence unrealistic.

5.2. Systematic errors

Note that the error of the orientation tensor illu-strated in Fig. 9 has been generated by consideration ofthe random measurement error only. While there is littleevidence of systematic error in the measurement of �,there is a sizeable systematic error in the measurementof �, as illustrated in Figs. 4–7.

The standard error of the orientation tensor may berewritten to include a systematic component, as follows:

s2aij ¼ s2aij

mþ s2aij

s

ð17Þ

where the subscripts m and s denote measurement andsystematic errors, respectively. The systematic error canbe estimated from the orientation data acquired by theconfocal technique enabling the evaluation of theexpression above. Alternatively, knowledge of the sys-tematic error could be used to correct the measuredfibre orientation of individual fibres, resulting in a moreaccurate measurement of the orientation tensor.

From Fig. 4 it can be seen that the systematic error isnot the same for every fibre, it is dependant on the qual-ity of each fibre’s cross-section. Typically, the majority offibres will have a fit factor in the range 0:99 < s < 0:999and hence the systematic error of fibres in this range willbe considered. Fig. 10 shows the relationship between�SMT, and the true value for � as measured by the con-focal technique. The following exponential equation hasbeen fitted to the data:

�SMT ¼ yþ Ae�=t ð18Þ

with the following values for the constants, y ¼ �21,A ¼ 36 and t ¼ 73, where � is measured in degrees. Thiscan be re-arranged as follows:

� ¼ tln�SMT � y

A

� �ð19Þ

Fig. 9. The diagonal components of the measured orientation tensor

of an injection moulded component. The error bars denote the

measurement error as derived using Eq. (14).

Fig. 10. The systematic error in determining � by the second moments

technique.


enabling the correction of the systematic error in �SMT.One problem with the above equation is that it will givenegative values for � when �SMT < y� A. This situationcan be prevented by imposing the restriction that0 < � < 90. An alternative method for correcting thesystematic errors, which has not been adopted for rea-sons of computation time, could be developed by con-sidering the distribution of angles within each ‘bin’ ofFig. 4. This method would result in a probability dis-tribution of possible values of �, from a single measure-ment of �SMT.

The relationship between � and �SMT has been derivedfor a restricted sample of fibres from a single specimen,hence its applicability to other specimens is question-able. It would be expected that specimens where fibreshave a similar mean diameter that have been analysed atthe same resolution should have a similar systematicerror. In order to test this assumption, a second speci-men was analysed using the confocal technique. Theorientation measured by the SMT was corrected usingEq. (19) and the results are illustrated in Fig. 11. Therehas clearly been an increase in the accuracy of theorientation measurement. The slight positive trend inthe error is probably indicative of a different fit factordistribution for this particular specimen. Fig. 12 com-pares the diagonal components of the orientation tensorcalculated from the corrected SMT and the confocaltechnique. The error bars have been calculated usingEq. (15), with the following formulae for the measure-ment and systematic errors:

fm� ¼ 8:2 � 0:08�

fs� ¼ 1:5

fm� ¼ 74:5e�=26:5 ð20Þ

where � is measured in degrees. The measurement errorin fm

�, has been derived from a linear fit to the standarddeviation of the corrected orientation, ��SMT illustratedin Fig. 11. The systematic error fs

�, is a constant whichbounds the mean error in the corrected orientation,��SMTh i. The error in �, is the same as that given in Eq.

(16). The orientation tensor derived from the SMTwhere the systematic error has been corrected agreesvery well with the results from the confocal technique.The confocal technique is assumed to give an accurateand unbiased measure of fibre orientation (see Section4.1) and hence an accurate measure of the orientationtensor. In the majority of cases, the orientation tensorcomponents given by the confocal technique fall withinthe error bars of the respective measurement using thecorrected SMT. Hence, the error estimates provided byEqs. (20) provide a realistic estimate of the error in thederived orientation tensor.

5.3. Sampling error

A single cross-section of a SFRT specimen typicallyintersects many 1000s of fibres. It would be expectedthat once the systematic error has been reduced, a verygood estimate of the orientation tensor would be possi-ble from such a large sample size. However, if the fibreorientation varies significantly over the cross-section itmay be more informative to determine the orientationof smaller sub-areas. For example, in order to visualisethe familiar skin-core-skin configuration illustrated inFig. 12, the dataset must be divided into strips, reducingthe number of fibres included in each orientation tensorcalculation. The number of fibres is further reduced if a2D area plot of fibre orientation is required. As thenumber of fibres included in an orientation tensor cal-culation is reduced, the sampling error increases. With

Fig. 11. The error in fibre orientation measurement in a second spe-

cimen with the systematic error corrected by Eq. (19) in the text. The

dotted line shows the mean measurement error before correction.

Fig. 12. A comparison of orientation tensor measurements by the

confocal technique and the second moments technique where the error

in �SMT has been corrected using Eq. (19).


knowledge of the sampling error it is possible to placeconfidence limits on the orientation tensor components.It is impossible to make meaningful comparisonsbetween different samples, specimens and numericalsimulations if one has no knowledge of the samplingerror.

Mattfeldt et al. [16] describe a method for testing fibreorientation measurements from hydrostatically extru-ded composites for directional randomness. A sample ofN fibres is tested by comparing it to a Monte Carlosimulation of samples of size N, from a directionallyrandom population. Confidence limits are placedaround the resulting Monte Carlo simulations, in orderto test the hypothesis of directional randomness for theparticular sample. The Monte Carlo method can beadapted to test a sample for any FOD described by theorientation tensor, as shown below.

The probability distribution function (PDF) (p),which describes the orientation state at a point in spacecan be recovered from the second-order orientationtensor as follows [5],

pð Þ ¼1

4�þ

15

8�bij fij pð Þ

bij ¼ aij �1

3�ij

fijðpÞ ¼ pipj �1

3�ij ð21Þ

where bij is a traceless orientation tensor, fij (p) defines abasis function and �ij is the unit tensor. The MonteCarlo simulation of a randomly selected fibre from apopulation with orientation tensor, aij is a two stepprocess. Firstly, the � component of the fibre orientationis calculated and then its � component. The probabilityof a single fibre being drawn from the population hav-ing an angle �, is given by:

P �ð Þ ¼

ð�¼2�

�¼0

�; �ð Þsin�d� ð22Þ

Integrating this function over all possible values of �,gives the cumulative distribution function (CDF),

F �ð Þ ¼

ð�i¼��i¼0

P �ið Þd�i ð23Þ

A random number chosen in the range [F(0)=0, F(p/2)=1] can be used to evaluate the inverse CDF, F�1(� )giving a random value of � for a single fibre. The prob-ability of this fibre having orientation � is given by:

P� �ð Þ ¼ �; �ð Þ ð24Þ

with the following CDF,

F� �ð Þ ¼

ð�i¼��i¼0

P� �ið Þd�i ð25Þ

A random number chosen in the range [F�(0), F�(2p)]gives the � component of the fibre. It is possible to findan exact solution for the Eqs. (22)–(25) for any fibreorientation tensor aij, however a numerical approach tothe integration is a much simpler solution.

Fig. 13 illustrates a number of Monte Carlo simula-tions of samples drawn from a population with orien-tation tensor axx ¼ 1 and aij ¼ 0 for all othercomponents. As the sample size increases the derivedorientation tensor more closely resembles the orienta-tion tensor of the population, i.e. Fig. 13(i).

Repeated Monte Carlo samples of the same sizedrawn from the population provide different values forthe expected population orientation tensor. The stan-dard deviation of these values decreases as the samplesize increases. A confidence limit for each tensor com-ponent is defined as follows:

aij � zc aij� �

ð26Þ

where aij� �

is the standard deviation of a large numberof simulated samples of size N and zc=1.96, zc=2.58given 95 and 99% confidence limits, respectively. Fig. 14illustrates the 95% confidence limits derived fromMonte Carlo simulations of samples of various sizes, i.e.there is a 95% probability that the components of thederived orientation tensor for a sample of 100 fibresdrawn from this population (axx ¼ 1 and aij ¼ 0 for allother components) will fall within the ranges:

axx ¼ 1:0 � 0:17

ayy ¼ azz ¼ 0:0 � 0:10 ð27Þ

The confidence limits placed around the tensor com-ponents provided by the Monte Carlo technique canprovide meaningful comparisons of two different sam-ples, or between a sample and its numerical simulation.For example, if a comparison is made between twosamples of fibre orientation, a1 and a2 taken either fromtwo points on the same specimen or different specimens,the Monte Carlo technique provides confidence limitsabout the tensor components of each sample, consider-ing sample 1,

a1ij � zc a1

ij

ð28Þ

where a1ij

is derived from a number of Monte Carlo

simulations of the same sample size, N1. If the tensorcomponent a2

ij from sample 2 falls within the confidencelimits of the respective components from sample 1, a1

ij

then it could conceivably be a sample drawn from the


same population as sample 1. Hence any observed dif-ference between the mean of the two samples is not sig-nificant. This same test could be performed by comparinga tensor component from sample 1 against the confidencelimits for the data from sample 2, however in practicesimilar results are obtained in both cases. When compar-ing an experimentally derived orientation tensor with anumerical simulation, this technique will only give con-fidence limits about the experimental results. Therefore, ifthe results of the simulation fall within the confidencelimits of the experimentally derived tensor components,the results are in agreement and the simulation can besaid to provide an accurate estimate of the FOD.

To illustrate the use of the confidence limits, Fig. 15shows the through thickness variation of the orientationtensor at two strips separated by �1 mm on the same

specimen. The orientation distribution of strip A showsa change in orientation approximately half way throughthe thickness of the sample which is not present on stripB. Approximately 50 fibres were present in each orien-tation tensor calculation, therefore in order to simplifythe sampling error calculations a sample size of N ¼ 50was assumed for all samples. At two points in Fig. 15,the orientation tensor of strip B falls outside of the con-fidence limits around strip A. Therefore, the hypothesisthat there is a significant difference in fibre orientationbetween strips A and B at these points is accepted with aprobability of error of 5%.

The method of evaluating the sampling error descri-bed above is dependant on the assumption that the sec-ond order orientation tensor gives a good approximationof the PDF. From inspection of Fig. 13 it can be seen

Fig. 13. (i) The PDF described by the tensor where all components are equal to zero apart from, axx=1. The following plots show Monte Carlo

simulations of samples taken from this population, where sample sizes are (ii) one fibre, (iii) 10 fibres and (iv) 100 fibres. The resulting PDF described

by the orientation tensor for each sample is plotted and the small spots on the surface of the PDF denote individual fibres.


that with a second-order tensor approximation of thePDF it is not possible to generate a ‘sharp’ peak in thedistribution, which may be observed in regions of uni-directional fibre alignment. In this case the above estimateof the sampling error would over-estimate correspond-ingly. However, the technique may be readily adaptedto higher order tensors.

6. Conclusions

The recent development of a fully automated confocaltechnique for the determination of fibre orientations hasenabled rapid and accurate fibre orientation measure-ments. In this paper, the confocal technique has beenused as an independent means of verifying fibre orien-

tation measurements taken by the 2D image analysistechnique.

Two different techniques for determining the ellipticalparameters and hence fibre orientation from fibre cross-sections, namely second moments and least squares, wereinvestigated. It was found that both techniques gave asimilar performance and the importance of strict filteringin order to remove poor ‘fits’ was demonstrated.

It was demonstrated that the 2D image analysis tech-nique provides fibre orientation measurements whichexhibit a significant systematic error in � . This error canbe reduced to a certain extent by increasing the resolu-tion, ensuring that the sample is well-polished and bystrict filtering. However, the systematic error cannot bereduced beyond �� ¼ 10� for fibres approximately per-pendicular to the section plane, i.e. � � 0. This is due tothe non-linear relationship between the measurement ofellipse axis lengths and the orientation angle, �. Amethod for calibrating and correcting the systematicerror was described and good results were achieved. Thesame calibration curves should be applicable to a widerange of samples although it would be wise to re-checkthe systematic errors using the confocal technique forsamples with very different fibre diameters or a largenumber of fractured fibres.

A technique for determining the sampling error of afibre orientation measurement was presented, enablingthe calculation of confidence limits about the derivedorientation tensor components. This technique can beapplied in the case where two samples are being com-pared (from different specimens or different regions fromthe same specimen) or when a sample is being comparedto data from a numerical simulation. The assignment ofa confidence level for the tensor components should beadvantageous to all future fibre orientation studies and3D model simulations.

Fig. 14. The confidence limits of the tensor components derived from samples of a population where axx ¼ 1 and aij ¼ 0 elsewhere.

Fig. 15. A comparison of two different cross sections through a SFRT

sample. The 95% confidence limits for the sample taken along strip A

are plotted.


Acknowledgements

C.E. is supported by a joint EPSRC MoD grant GR/N/18413, ‘3D characterisation of fibre-reinforced com-posites and foams by confocal microscopy’. The tworesearch groups at Leeds and Nice have collaborated onan ‘Alliance’ research grant. NIU/TQ/SF would like toacknowledge the financial aid and technical supportfrom Moldflow, Bosch, Solvay and Schneider Electric.We would like to thank both Peter Hine and AlanDuckett for years of discussions. Finally, we would like tothank the referee for his useful comments on this paper.

Appendix A

A curve of second order can be defined as follows:

Ax2 þ Bxþ CxyþDyþ Ey2 þ F ¼ 0

if the parameters A–F define a curve which is an ellipse,the following equations can be used to determine itscentre (xc, yc), major and minor axis lengths (a, b) andorientation � as follows:

xc ¼BE� 2CD

4AC� B2

yc ¼BD� 2AE

4AC� B2

a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

Aþ C� �

s

b ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

Aþ Cþ �

s

� ¼ �1

2atan

B

A-C

� �where

¼ F� Ax2c � Bycxc � Cy2

c

� ¼ F

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 þ ðA� CÞ2

q

Appendix B

B1. Measurement error of the orientation tensor

In order to calculate the measurement error of anorientation tensor calculated from a group of n fibres, asgiven by Eq. (15), the following partial derivatives arerequired.

@ak11

@�k¼ 2cos2�ksin�kcos�k

@ak11

@�k¼ �2sin2�ksin�kcos�k

@ak22

@�k¼ 2sin2�ksin�kcos�k

@ak22

@�k¼ 2sin2�ksin�kcos�k

@ak33

@�k¼ �2sin�kcos�k

@ak33

@�k¼ 0

B2. Measurement error of the weighted orientationtensor

The weighted orientation tensor calculated from agroup of n fibres is given below:

aij ¼G

H¼

Pnk¼1

Fkakij

Pnk¼1

Fk

with the following standard error:

s2aij ¼Xnk¼1

@aij@�k

��k þ@aij@�k

��k� �

Note that in this case we are evaluating the partialderivatives of the average orientation tensor, i.e. aij asopposed to the non-weighted case (B1) where partialderivatives of the individual fibre’s orientation tensors,akij are required.

The partial derivatives of a11 are calculated as follows:

@a11

@�k¼

@G

H@�k

¼1

H2

@G

@�kH� G

@H

@�k

� �

where

@H

@�k¼@Pk

@�k¼ Lcos� þDsin�;

@G

@�k¼

@

@�kFkak11

� �¼

1

Pk@ak11

@�kPk �

@Pk

@�kak11

� �

where Pk is the inverse of Fk as given in Eq. (2) and thepartial derivative of ak11 with respect to �k is given insection B1.

@a11

@�k¼

@G

H@�k

¼1

H2

@G

@�kH� G

@H

@�k

� �


since

@H

@�k¼@Pk

@�k¼ 0

@a11

@�k¼

Fk@ak11

@�k

H

where the partial derivative of ak11 with respect to �k isgiven in section B1. The partial derivatives of theremaining components of aij may be determined usingthe same method.

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