Fibre orientation in short-fibre-reinforced thermoplastics II. Quantitative measurements by image analysis

Composites Science and Technology 59 (1999) 547±560

Fibre orientation in short-®bre-reinforced thermoplastics II. Quantitative measurements by image analysis B. Mlekusch Institute for Designing Plastics and Composite Materials, University of Leoben, 8700 Leoben, Austria Received 15 September 1997; received in revised form 16 April 1998; accepted 14 May 1998

Abstract To determine the ®bre orientation state in short-®bre-reinforced thermoplastics, polished cross-sections are examined. If a circular cross-section is assumed for the ®bres, they are pictured as ellipses on the intercepting plane. The picture evaluation is done by image analysis which uses an analysis program specially developed for this problem. User-de®ned parameters allow an adjustment of the evaluation algorithm, so that fully automated picture processing is possible. To describe the ®bre-orientation state orientation tensors are used. Under some assumptions about the ®bre orientation state, which are well met in regular plastic components, all tensor components can be obtained from one inclined polished cross section. Measurements on experimental components are made by using the described analysis procedure and con®rm the validity of the assumptions. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Short-®bre composites; A. Glass ®bres; E. Injection moulding

1. Introduction Because of the market-imposed pressure to shorten development times, computer-aided simulation analysis has acquired signi®cant importance in recent years. These simulations include the production process as well as the part behaviour in service. For short-®brereinforced thermoplastics (SFRT), which are widely used in technical applications, this means the calculation of the ®bre orientation (FO) from ¯ow conditions during the injection-moulding process [1±3]. The resulting FO state is the basis for calculation models, to determine the anisotropic part characteristics, like sti€ness [4,5], thermal expansion [6] and strength [7]. Knowledge of the real FO in the component is of great importance as an interface for the model calculations mentioned. The results of the FO calculations from mould-®lling simulations, which are often part of commercial injection-moulding simulation software [8,9] can be veri®ed. On the other hand, micromechanical models Ð and their assumptions Ð for predicting the part behaviour can only be checked with correct, measured input parameters. In addition to the injection-moulding process, knowledge of the FO state is of essential interest for every process, where ®bre reinforcement is used. This also includes all manufacturing techniques processing

long ®bres, like the sheet-moulding compound (SMC) or resin-transfer moulding (RTM) processes. The FO state can be used in de®ning quality-control criteria as well as the input parameters Ð through micromechanical models Ð for ®nite-element calculations. As a measuring technique for determining the quantitative FO state, the evaluation of polished cross sections is employed. Therefore a freely programmable image-analysis software (analySIS 2.1, SIS-Soft-Imaging Software Gmbh., D-48153 MuÈnster) is used. The cross sections are either magni®ed by light microscopy, whereby a special contrasting process is used, and digitised by a video camera or they are directly taken to the image analysis from the scanning electron microscope [10]. Since many images are necessary for the analysis of one cross section, one has to use a controllable specimen-®xing device together Ð in case of the light microscope Ð with an automatic focus system. With the help of the algorithm in Section 3 a complete automation of the analysis is possible. 2. Fundamentals In accordance with the theory of ®bre orientation in ¯ow and with the ideas of Ref. [11] all investigations done concerning this work showed a constant ®bre

0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0266 -3 538(98)00101 -8

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concentration in the whole part as well as in the thickness direction. The only exception is the edge zone. This spatially very narrow region is marked by a lower ®bre concentration [12]. 2.1. Fibre-orientation distribution function

ˆ …1 ; 1 † sin 1 dd':

…1†

Instead of using a spherical co-ordinate system with the co-ordinates  and ' a description of the probability function with the help of a unit co-ordinate vector can be done, …; '† ˆ …p†. In this form the orientation in space is given by the co-ordinates of the unit vector, which are connected to the spherical co-ordinates (Fig. 1) in the following way: 2 3 p1 ˆ sin  cos '; p1 …2† p ˆ p…; '† ˆ 4 p2 5; with p2 ˆ sin  sin '; p3 p3 ˆ cos : Because two ®bres orientated at angles …; '† und … ÿ ; ' ‡ † cannot be distinguished from one another, must be de®ned only in one half-space, or we have to demand the following symmetry …; '† ˆ … ÿ ; ' ‡ †

or

…p† ˆ …ÿp†:

Fig. 1. Orientation angles of a single ®bre in space.

… 2 …

‡ …; † sin  d d' ˆ

dp ˆ 1:

…4†

ˆ0'ˆ0

If only ®bres with a circular cross section and a unique diameter are considered, the most universal means of describing the microstructure is a combined orientation and length distribution function, OL [3]. Because the ®bre length distribution cannot be determined by image analysis (unless we assume geometrically perfect ®bre ends [13]), a constant ®bre length has to be assumed. This assumption is considered in the error-estimating calculations. The FO distribution function, , which describes the probability of a ®bre being orientated between the spacial angles (Fig. 1) 1 and …1 ‡ d† and '1 and …'1 ‡ d'†, is de®ned by P…1    1 ‡ d; '1  '  '1 ‡ d'†

Because is the density of a probability function, also a normalisation condition can be given,

…3†

2.2. Orientation tensors Although the orientation-distribution function is the most universal description form, in practical use it is dicult to handle. The numerical form is combined with a high level of information. It is therefore common to calculate distribution parameters, which characterise the FO state. Such parameters are always de®ned such that the limiting orientation states lead to distinct and simple parameters. In the literature di€erent de®nitions for orientation parameters are given [14±16], which often appear quite arbitrarily chosen and are limited to planar orientation states only. A more universal possibility to describe the FO state is given through the so-called orientation tensors [3]. They allow the representation of the whole, threedimensional distribution function. The quality of the description is given by the order of the tensors. Orientation tensors are de®ned as the moments of the orientation-distribution function. Moments, dyadic products respectively, of arbitrary order can be calculated. Because of the symmetry condition (equation 3) all uneven moments are equal zero. The ®rst two non-vanishing orientation tensors, the second- and the fourthrank tensor, are therefore ‡ …5† a ˆ aij ˆ pi pj …p† dp;

ˆ

‡ a ˆ aijkl ˆ pi pj pk pl …p† dp:

ˆ ˆ

…6†

From these definitions the full symmetry of the tensors is evident. Furthermore the normalisation condition (4) requires, that the sum of the main diagonal components is equal to unity. The orientation distribution function, , can be expanded with the help of the orientation tensors. The representation is more accurate, the higher the order of the tensors used [3]. For the second-rank tensor there are, because of the symmetry, a maximum of six di€erent components, ®ve of them as a result of the normalisation condition independence. Graphically one can interpret the secondorder orientation tensor as the tensor of inertia of the unit sphere surface, if the ®bre-orientation distribution, , is used for the surface density (Fig. 2). From this the

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small in comparison to the overall precision of the method. To weight the measurement data of a single ®bre, the data has to be multiplied with the inverse of the appearance function. Therefore the weighting function, F, follows: ) F…; L† ˆ L cos1 …† for  < krit F…; L† ˆ d1 krit

Fig. 2. Picture of the ®bre orientation distribution function as density of the unit sphere. The points on the sphere surface symbolise individual ®bres.

tensorial character of the description form is evident as well as the meaning of the individual components. The main diagonal components describe the amount of orientation in the directions of the used reference frame. The o€-diagonal components show the rotation of the main orientation axis with respect to the co-ordinate system used. Because of this, we may calculate the main orientations and their values from the eigenvalue problem det jaij ÿ lij j ˆ 0:

…7†

2.3. Calculation of the tensor components From a sample of N ®bres taken at random the tensor components of the distribution function are given as [17] PN nˆ1 Fn …aij †n : …8† aij ˆ P N nˆ1 Fn The function Fn describes the weighting function for the nth ®bre. This function is needed because the probability of being intercepted by the cutting plane is dependent on the inclination angle. If the 1-2 plane is chosen as the cutting plane, the probability of being hit is proportional to the projected ®bre length on the third axis. Therefore the probability function, A, depends on the orientation angle, , and the ®bre length, L, as  A…; L† ˆ L cos …† for  < krit A…; L† ˆ L cos …krit † ˆ d for krit <  < 90   d : …9† krit ˆ arccos L The cut-o€ of the appearance function, A, in the region  > krit is an approximation, a more accurate investigation of this interval, which, for usual ®bre length-todiameter ratios of 300 m=10 m is very narrow, will not be done. In [13] the ®bre diameter is also taken into account, when de®ning the appearance function. The error caused by the approximation used in this work is

for   d ˆ arccos : L

krit <  < 90 …10†

3. Image evaluation For determining the orientation of a ®bre in space, which is given by two orientation angles (Fig. 1), a circular cross-section is assumed. The ®bres are therefore pictured as ellipses on the cutting plane. The orientation of the major axis is identical to the angle '. The other orientation angle, , can easily be calculated from the ratio of the minor axis, b, to the major axis, a, and from the associated angular relation in the following way   b : …11†  ˆ arc cos a Only the sign of the angle stays undetermined, because a cylinder orientated ‡ and ÿ yields the same cutting ellipse. The consequences of this fact on the orientation state will be discussed in detail in Section 4. The assumption of a circular cross section for glass ®bres is very well ful®lled [18]. The analysis program developed in this work calculates a regression coecient for all cutting ellipses (see Section 3.1), which is a measure of the quality of the contour. For a completely pictured cutting ellipse the investigations showed a regression coecient, from which a nearly perfect circular form can be concluded. 3.1. Image analysis Image-analysis systems convert grey value pictures into binary ones by using a threshold. This means black or white, in accordance with the threshold, is assigned to every pixel of the picture. The picture analysis operations are mainly done on these binary pictures, the following strategy being used: The black or white parts of the binary picture are detected as objects and from these objects several parameters are determined. Four commercial image-analysis systems have been tested. In general two major problems concerning the evaluation of the SFRT cross sections were met, viz. the separation of the single objects and the calculation of the parameters of the ellipses in a proper way. Usual ®bre volume fractions cause many ®bres touch each

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other and therefore they have to be separated into single objects. For this separation many di€erent algorithms are available which give good results for nearly circular pictured ®bres, or for small  angles. Some of them misrepresent the original picture. If the orientation angle increases and the ellipses become more stretched, these ellipses are often also broken into pieces. The calculation of the ellipse parameters in commercial systems is always done by an evaluation of the object area. The minor and major axis are either determined from the minimum and maximum object diameter or they are calculated from the moments of inertia. Both methods [19,20] have in common, that they do not solve the problem in an optimal manner, because they are general picture analysis functions, and that break outs of the ®bres on the cross section cause severe measurement errors. Furthermore all ®bres possess nearly the same ®bre diameter, which is equal to the length of the minor axis of the cutting ellipse. This information should be used in the image analysis. The complete information, which is needed to calculate the ellipse parameters, is stored in the contour of the object. Instead of evaluating the object area, only the contour will be used in the following analysis. No arti®cial separation lines will be drawn between the objects, which would e€ect the results. The evaluation strategy can be summarised in the following way: . separation of the object contour in single segments; . calculation of the ellipse parameters for every segment; . putting together the segments, belonging to one ®bre; . re-calculation of the ellipse parameters with all segments from one ®bre. By this procedure it is possible to eliminate the mis®t parts of the contour, e.g. break outs, scratches and so on, as well as objects, which do not represent ®bres, e.g. glass dust. The main advantage is given by the fact, that the whole information of the contour is used, leading to minor measurement errors (see Section 3.2). Other positive aspects are the mentioned ®ltering of mis®t contours and contour parts. It is also possible to evaluate correctly ellipses which pass through a scratch and therefore are pictured as two separate objects. 3.1.1. Separation algorithm Following the object contour, along an ellipse only a distinct, convex curvature is possible. A concave curvature as well as a very strong convex curvature, e.g. from a break out, are not possible. To determine the curvature of the contour regardless of the co-ordinate system used, in every contour point, P, the angle between a secant forward, V, and backward, R, is calculated (Fig. 3). Local noise will be suppressed and only the trend of the curve is calculated, when several points of the contour are used to determine the secant. If the

Fig. 3. Determination of the curvature along the contour.

measured angle lies above a certain positive value, vex (to strong convex curvature), or below a negative one, kav (concave curvature), the contour is separated in this point. As a criterion the condition ÿkav <  < vex ;

…12†

is used. The object contour therefore is broken into pieces, if separation points are found along its contour. 3.1.2. Ellipse regression For each group of points, which consists either of the whole contour or only a segment, a regression calculation is used to determine the best-®t ellipse. An ellipse in a general position is characterised by ®ve parameters, namely the major axis, a, the minor axis, b, the coordinates of the midpoint, (MX, MY), and the orientation angle, '. The direct determination of the ®ve parameters from the ellipse equation causes a system of non-linear equations. To avoid this, an indirect procedure is used. The mathematical details of the ellipse regression and the following main axis transformation are given in Appendix A. As a measure of the quality of the curve ®tting and therefore of the quality of the segment a mean regression coecient,
, is de®ned (Eq. (A.7)). The advantage of this indirect regression calculation is the simple way in which it can be done, but there are two disadvantages. At ®rst, the regression seeks for the best-®t curve of second order. Besides ellipses this includes mainly hyperbolas. Especially if the ellipse segment is short and originates from the area of the minor axis, the ellipse conditions are not ful®lled. To avoid losing the information of these contour parts a special algorithm is used (Section 3.1.3). Secondly, it is not possible to ®t an ellipse with a certain minor axis. Once again for short segments, this leads to the fact, that very small

B. Mlekusch / Composites Science and Technology 59 (1999) 547±560

or large ellipses can result from the curve ®tting. In this case the same procedure as that mentioned above is used. 3.1.3. Special subroutines Checking for overlap: This sub-program was developed to determine whether or not two ellipses overlap. With this information a decision can be made if two ellipses obtained from the regression calculation come from the same ®bre. Because this has to be checked very often, a fast and through user-guided-parameters algorithm is used. If an overlap is detected, a ellipse regression calculation has to be done again with the group of points consisting of both segments. To be sure that both segments are part of the ellipse contour, the newly calculated regression coecient (A.7) is checked. Arti®cial circles: This subroutine is used, to ®t circles with a prede®ned radius to those segments, for which no or mis®tted ellipses are calculated in the regression. If a ``useful'' circle is ®tted to these segments, the information they include is still used through the indirect way of overlap checking. As a radius for this arti®cial circle the ®bre diameter is employed. If no overlapping is detected, the circles are rejected. Filters: For the evaluation procedure there is possibility of rejecting ellipses which are not well ®tted by ®lters. A limit can be set for the regression coecient as well as an allowable interval for the minor axis of the ellipses.

551

3.2. Error estimates The following error estimates are mainly concerned with the measurement error, which is caused by the analysing scheme described. First, the errors in determining the orientation angles of a single ®bre have to be investigated and then their in¯uence on the summation (Eq. (8)) of the tensor components has to be calculated.

3.1.4. Evaluation example In Fig. 4 an evaluation example is pictured showing 28 ®bres detected by the analysing system. The calculated ®ve ellipse parameters, the orientation angle and the regression coecient are stored. From this ®gure it can be seen, that the separation in individual objects is done correctly. The picture was made by an electron microscope using the material contrast, the contrast is not changed or corrected by the image-analysis system.

3.2.1. Measurement error in determining the orientation angles of a single ®bre Both orientation angles, ' and , of a ®bre can only be determined with a certain degree of precision. The amount of error is strongly dependent on the resolution used for digitization of the images, because the continuous contour is converted into discrete pixels. The measurement error of the orientation angles is itself dependent on the orientation angles, whereas the dependence from the inclination angle, , dominates. The  orientation is calculated from the ratio of the ellipse axises and the arccos function (11), causing highly non-linear behaviour. Therefore small spacial angles of , ie. ®bres oriented nearly perpendicular to the cutting plane, possess a high gradient. A small change in the axis length leads to large deviations in the results. In this region a high measurement error can be expected. Also the ' orientation can be determined more precisely, the more stretched the object is. On the other hand, the demand for high resolution, and therefore magni®cation, runs counter to the minimizing of computing e€ort. The standard deviation in determining the angle  is shown in Fig. 5.  is plotted in the radial direction, ' in the circumferential. The data are generated in the region 04475 and 04'445 (for symmetry reasons) by evaluating 30 ellipses for each orientation with the arbitrary mid-point co-ordinates from the region MX

Fig. 4. Polished cross section of a SFRT-sample and cutting ellipses, which where detected by the image analysing system.

Fig. 5. Standard deviation
 when determining the orientation angle of a ®bre with the help of a regression calculation in dependence of the orientation angles  from [0,75] and ' from [0,45].

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= [ÿpixel/2; pixel/2] and MY = [ÿpixel/2; pixel/2], as well as a minor axis radius out of b=[14,5 pixel; 15,5 pixel]. For these ellipses discrete contours have been calculated and an evaluation was done with the described procedure. The minor axis radius of 15 pixel is a compromise between a minimum of computing e€ort and measurement precision, and is used in all the following measurement examples. Discrete ellipse-contour regression coecients are also obtained from the analysis program. The magnitudes of these coecients are equal to those of ®bre images on high-quality polished cross sections. It can therefore be concluded, that the assumption of circular ®bres is very well ful®lled. Fig. 5 shows that the measurement error decreases with increasing  and is nearly independent of '. The same is true for the ' deviation. For the following considerations it is sucient to determine error limitations as analytical functions of , which are larger in every point than the maximum deviation. This means that for both angles one has to de®ne functions in the following way: 
 ˆ f…; '†  f …† : …13†
' ˆ f…; '†  f' …† Assuming an equally distributed minor axis in the region 14.5±15.5 [Pixel] e.g. the following limiting equation for the deviation of the  orientation can be given:
 ˆ f …† ˆ 5:4 eÿ0:17 ‡ 2:5 eÿ0:043  :

…14†

In this formula the angles are in degrees, and the equation was obtained by bounding the errors in a semilogarithmic diagram. The same can be done for the ' orientation. Fig. 6 shows the limiting curve together with all deviation points over . Furthermore the measurement error caused by the regression calculation is compared with the error given by a conventional method from [22]. The improvement brought about by the use of the whole contour points can be seen clearly.

3.2.2. Measurement errors of the tensor components and statistical errors The calculation of the measurement errors of the tensor components follows the procedure outlined in [17], although it has to be signi®cantly modi®ed to the analysing procedure used. The mathematical background is given in detail in Appendix A. Both error sources Ð the measurement and the statistical error Ð decrease with increasing numbers of ®bres. The decrease in the statistical error is slower and, depending on the orientation state and the desired precision, a minimum of 200 ®bres should be evaluated for each measurement. 4. Usage of inclined polished samples From the cutting ellipse the orientation of a ®bre is not uniquely determined: two states are possible. Because of this, not all tensor components can be determined from one polished cross section. Which components remain undetermined is discussed for the second-order tensor. Therefore a cutting ellipse in the 1-2 plane is considered. The realisation point of the ®bre on the unit sphere can be above or below the 1-2 plane. Therefore the sign of the third component remains undetermined. From the analogy given for the inertia tensor it can be seen that there is no e€ect on the a12 component of a single ®bre, but the signs of the a13 and a23 components remain undetermined. As a consequence one cannot calculate these components for the whole FO state given by the mean values of a number of ®bres (8). So it may be concluded, that from one cross section the amount of orientation in the three co-ordinate directions, or, which amounts to the same thing, the main diagonal components, can be measured. For the o€-diagonal components the in-plane rotation, a12 , can be calculated, while measurement of the out-of-plane rotations, a13 and a23 , is not possible. The whole orientation state is therefore not fully described by the information from one cross section. In the following considerations a global co-ordinate system (Fig. 7) is used, which is aligned to the part in the way, that: (a) the 1-2 plane coincides with the plane of the components extension, (b) the 1 axis if possible with the ¯ow direction, (c) the 3 axis with the thickness direction.

Fig. 6. (a) Upper error limitation function and standard deviation
 plotted against  using a regression calculation and a middle ®bre radius r ˆ 15 pixel. (b) (c) Standard deviation
 in determining the orientation angle using a conventional method due to [22] for a ®bre radius of r ˆ 10 pixel and r ˆ 20 pixels, respectively.

In SFRT components the variation of the orientation state in the thickness direction is of special interest, and one would therefore suggest polished cross sections in the 1-3 or 2-3 planes. In so doing, two problem areas are caused: (1) As mentioned above, not all o€-diagonal components can be measured. For cuts in the 1-3 or 2-3 planes

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and some assumptions are made for the ®bre orientation state. The co-ordinate system rotated by an angle  is called the x, y, z system and is shown in Fig. 7. In the following the x, y plane in this local co-ordinate system is chosen as the cutting plane. 4.1. Assumptions Fig. 7. SFRT-sample with an inclined cutting surface and the used coordinate systems.

this means that the orientation component, a12 , is not determined in either case. Because of this neither the main orientation in the 1-2 plane nor the direction of the main orientation are given. In injection moulded parts the ®bres are mainly aligned in the 1-2 plane, in¯uencing the in-plane component properties severely. That the determination of a12 is not possible, is therefore a severe restriction. To overcome this limitation, cuts in the 1-2 plane can be fabricated. For practical use this results in a great increase in preparation cost, because for every measurement point across the part thickness a new polished sample has to be fabricated. (2) The second problem is concerned with the precision of the analysing method. Fibres crossing the cutting plane nearly perpendicularly cause a large measurement error. In the edge zones the ®bres are oriented mainly in ¯ow direction (1 axis) and in the middle they align across (2 axis). By using samples in the 1-3 and the 2-3 planes the measurement errors are high in either the edge zones or in the middle. Both of these problems can be overcome, if the cutting plane is taken under a certain angle to the 3 axis

Some assumptions are necessary in the measurement region to determine the whole orientation state from one single inclined cross-section. Either a symmetric ®bre orientation with respect to the 1-2 plane or a ``nearly planar'' orientation state has to be assumed. The validity of these assumptions is discussed in the following: they can be justi®ed as follows. 4.1.1. Through measurement results Looking at polished cross sections from injectionmoulded components one can easily recognise, that the ®bres are orientated mainly in the 1-2 plane. Of course this is only true for the component areas, where no ¯ow occurs in the third direction, e.g. as caused by an abrupt change in thickness. This visible impression was con®rmed by many measurements done in the 1-3 and 2-3 planes. As an example, a measurement in the 1-2 plane is shown in Fig. 8, the tensor components being plotted against the component thickness. From Fig. 8 it can be seen that the assumptions are well ful®lled. Along the whole thickness the orientation state is symmetrical with respect to the 1-2 plane, no out-of-plane rotation being shown …a13  0†. Furthermore the orientation state is nearly planar, because a33 is also very small in the whole plot.

Fig. 8. Measurement of the tensor components a11 ; a22 ; a33 and a23 against the part thickness as well as a picture from the polished cross section.

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4.1.2. From equations for determining the ®bre orientation state Considering a rigid particle immersed in a Newtonian ¯uid, the motion of this single, undisturbed ellipsoidal particle in the ¯ow can be described by the theory of Je€ery [1]. Because the main orientation of the ®bres is of special interest, the ®bre/®bre interaction, which causes a randomising e€ect [2], will be neglected. For a general three-dimensional homogeneous ¯ow an analytical solution of Je€rey's equation for one particle is possible [23]. The actual orientation, pi , of a particle with the initial condition p0i can be calculated. If the initial orientation state, 0 , is known, the tensor components can be determined from their de®nitions [24]. In the following it will be shown, that the components a13 and a33 of the second rank orientation tensor have to vanish in a one-dimensional shear ¯ow in the 1-3 plane. Initially, a condition of equally distributed ®bres will be assumed, 0 …p† ˆ 1=4. is the shear deformation, experienced by a particle, and this is therefore a quantity steadily increasing along the ¯ow path. For the tensor components we derive " ‡ # 1 …p1 ‡ p3 †p3 dp ˆ 0; a13; !1 ˆ lim !1 4 …p1 ‡ p3 †2 ‡ p22 ‡ p23 …15† as well as " a33; !1 ˆ lim !1

# ‡ 1 p23 dp ˆ 0: 4 …p1 ‡ p3 †2 ‡ p22 ‡ p23

4.2. Schemes of analysis Under the given assumptions of a symmetrical or nearly planar FO state two schemes of analysis are discussed in the following. 4.2.1. Tensor transformation In the global 1,2,3 co-ordinate system assumptions can be made referring to the FO state. In the local x, y, z system, where the cutting plane is de®ned and the primary measurement takes place, the orientation state cannot be completely determined. With the rules of tensor transformation [25] both co-ordinate systems can be linked together in a way, that complete information is obtained. 4.2.1.1. Second-order tensor. Using the co-ordinate systems (CO) de®ned in Fig. 8 one gains for the secondorder tensors 2 3 axx axy x 6 7 local CO : 4 axy ayy x 5; x x azz 2 3 a11 a12 0 6 7 global CO : 4 a12 a22 0 5: 0 0 a33

…17†

For every undetermined component in the local CO system a condition is given by the symmetry of the ®bre orientation. By using the transformation rules are obtain:

…16†

ˆ 0 ˆ A1k A31 a…xyz† a…123† 13 kl ;

How fast the investigated components decrease can easily be analysed by choosing appropriate values. But for this estimate there is the problem of ®nding the correct initial condition, when the equal distribution is valid. For the injection-moulding process this cannot be the gate location, because the areas of very high shear levels Ð the runner system and the nozzle Ð are then neglected. Because of the fact that the a33 component also vanishes for this simple model, the orientation in this direction can only be of the magnitude of the ®bre/ ®bre interaction. Therefore for shear ¯ow, which is very dominant in injection moulding, a symmetrical and nearly planar ®bre orientation are the consequence. In [24] the orientation ®eld of a centre-gated disk is investigated. In addition to the shear ¯ow, elongational ¯ow also appears during mould ®lling, causing a more general ¯ow pro®le than the one discussed above. Also in this case the components a13 and a33 decrease rapidly along the streamlines, and results are presented for an initial equal distribution at the gate and a Newtonian ¯ow.

ˆ 0 ˆ A2k A31 a…xyz† a…123† 23 kl ;

…18†

where Ai;j are direction cosines. From this linear system (18) all components can be calculated. 4.2.1.2. Fourth-order tensor. An analogue procedure can be given for the fourth-order tensors. Instead of two equations, the linear equation system contains six transformation equations. These are given as …xyz† a…123† 1333 ˆ 0 ˆ A1k A11 A1m A3n aklmn ; …xyz† a…123† 1333 ˆ 0 ˆ A1k A31 A3m A3n aklmn ; …xyz† a…123† 1223 ˆ 0 ˆ A1k A21 A2m A3n aklmn ; …xyz† a…123† 2223 ˆ 0 ˆ A2k A21 A2m A3n aklmn ; …xyz† a…123† 2333 ˆ 0 ˆ A2k A31 A3m A3n aklmn ; …xyz† a…123† 1123 ˆ 0 ˆ A1k A11 A2m A3n aklmn :

…19†

B. Mlekusch / Composites Science and Technology 59 (1999) 547±560

This scheme can of course be extended to orientation tensors of arbitrary order. Therefore in the global 1,2,3 system all tensor components containing an uneven number of 3 indices are set to zero leading to exactly one equation for every undetermined component in the local system. As a restriction on this method of analysis it must be emphasised, that the ®bre orientation distribution has to ful®l the symmetry assumption exactly. Otherwise physically not useful, negative diagonal elements are calculated for the global system. In this case instead of the direct evaluation of the systems (18) and (19) an optimisation problem has to be formulated with the restriction of positive main diagonal components. 4.2.2. Direct angle determination Because not only a symmetric but also a planar orientation can be assumed, the use of this information when determining the tensor components of a single ®bre suggests itself. For one cutting ellipse of a ®bre in the local system, two  orientations are possible. In the following the  orientation is used, which is more closely aligned to the planar 1-2 orientation. The magnitude of the inclination angle of the cutting plane determines the region, in which no error is caused by this analysis scheme. From Fig. 9 we note that an angle of 45 is the best choice. The region, in which no ®bre should be orientated for a correct evaluation, is limited to an opening angle of 90 symmetrically about the 3 axis. 5. Measurements The application of the discussed analysing schema is shown on two examples, where the FO state of a SFRT component is investigated. The component itself and both measurement points are shown in Fig. 10. The component was specially designed for this research project [26]. 5.1. Orientation measurement I At ®rst a measurement point in the middle of the component is chosen, where the ¯ow conditions should cause a symmetrical ®bre orientation with respect to the 1-3 plane. With this assumption, the main orientation direction is given a priori and the location is best suited to compare FO results from conventional, perpendicular specimens with inclined ones. All samples are taken from components processed under the same moulding conditions. To verify the validity of the chosen weighting function two conventional cross sections in the 1-3 and the 2-3 planes are investigated to beginwith. From both measurements the main diagonal components of the

555

Fig. 9. Area where a correct analyse can be done, for nearly planar orientations. The orientation of each single ®bre is determined directly, the ®bre aligned closer to the 1±2 plane is chosen.

Fig. 10. Experimental part (2 mm thick) with measurement locations and coordinate systems.

2nd order orientation tensor can be determined directly. The ®rst three plots of Fig. 11 show the components a11 , a22 and a33 . To present the course of the components, the thickness is divided into eight equally spaced intervals. All three components coincide well within the error estimates. In this region the component shows a threelayer structure, typical of SFRTs, where the ®bres in the boundary zones are aligned in the ¯ow direction, because of shearing e€ects, and the middle layer is orientated transversely. The latter is caused by elongational ¯ow e€ects. Furthermore, as another consequence of the ¯ow ®eld, the orientation state is almost planar …a33  0†. From each of the perpendicular cross sections one o€-diagonal component can be determined. The corresponding curves a13 and a23 are shown in the last plot of Fig. 11. These tensor components describe rotations of the main orientation directions out of the 1-2 plane. As concluded by the theoretical considerations, no rotation out of the 1-2 plane is visible. To compare the conventionally received results with measurements from an inclined specimen, a cross section at an angle of 30 is taken and polished. With the help of the suggested scheme of analysis, all six tensor components can be derived from this measurement. As stated in Fig. 12, where the main diagonal components of the conventional and the inclined samples are compared, only small deviations appear between these measurements. The shown, conventional curve consists of the measured ®bres from both, the 1-3 and the 2-3 evaluation.

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The last plot of Fig. 12 shows all three o€-diagonal components of the inclined evaluation. Since these components tend to zero throughout the whole measurement, the given main diagonal components can be interpreted as the main orientation components aI , aII and aIII . As a consequence of symmetry, the measurement reference frame coincides with the main ®bre orientation directions.

5.2. Orientation measurement II This measurement is done at a point in the component, where the main ®bre orientation directions are not immediately given as a result of symmetry. This position is well suited to show the advantages of the inclined cross-section concept.

Fig. 11. Plots for the tensor components and the error limitations comparing two conventional, perpendicular cross sections in the 1±3 (m13) and the 2±3 plane (m23) at location I. The ®rst measurement allows the determination of the components a11 ; a22 ; a33 ; and a13 whilst the second gives a11 ; a22 ; a33 and a23 .

Fig. 12. Comparison of the plots for the tensor components at location I between the perpendicular evaluation (measurement conventional) (all ®bres for the measurements in Fig. 11 are used) and the inclined evaluation (measurement inclined). Under some assumptions all tensor components can be received using an inclined cutting plane.

B. Mlekusch / Composites Science and Technology 59 (1999) 547±560

557

Fig. 13. Comparison of the tensor component plots at location II between a perpendicular evaluation (mc) in the 1±3 plane and an inclined evaluation (mi).

For comparison, once again specimens with an inclined and a perpendicular cross section are taken. For the former the 1-3 plane is chosen. As results of these two measurements the main diagonal components are shown in the ®rst three plots of Fig. 13. As can be seen, the curves match quite well. The small tolerances can be contributed either to a small change in the processing conditions or a slight change of the evaluation position. As an unsurprising result, the ®bre orientation is again nearly planar, …a33  0† through the thickness direction. The ®rst impression, when looking at the curves of a11 and a22 , is that it seems as if the three-layer structure is not as well developed as at position I. The o€-diagonal components, calculated from the inclined specimen measurement, are shown in the last diagram of Fig. 13. It can be seen, that there is no out-of-plane rotation …a13 ˆ 0 and a23 ˆ 0† of the main orientation directions, but there is clearly an in-plane rotation …a12 6ˆ 0†. Solving the corresponding eigenvalue problem (Eq. (7), we can transform the orientation tensors into the main systems and determine the magnitude of the main orientations aI , aII and aIII as well as their directions. The orientation angles ' and  of the main directions are shown in Fig. 14 above, the magnitude of the orientation below. ' designates the in-plane orientation angle,  ˆ 90 ÿ  the calculated out-o€ plane rotation. From these diagrams we see that the boundary layers are orientated at about ÿ30 with respect to the reference frame, the middle layer is orientated at nearly 90 . The magnitude of the orientation is as high as at position I and a fully developed three-layer structure, with a slightly larger middle layer, can be concluded. It is also

apparent from this ®gure, that the alignment of the ®bres' transverse ¯ow in the middle layer is as high as the alignment in the ¯ow direction in the boundary layers. The measurement at position II clearly shows the necessity of knowing the full information about the inplane ®bre rotation, a12 , to judge the orientation state

Fig. 14. Main orientation angles ' in the ¯ow plane as well as  ˆ 90 ÿ  out of this plane and magnitude of the main orientations aI ; aII ; and aIII . The values can easily be determined from the measurement values using an inclined evaluation.

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correctly. As shown, the use of inclined cross section is a very suitable concept for determining this information.

Appendix

6. Conclusions

All curves of second order in the plane (e.g. ellipses, hyperbolas...) are given trough points, which suce the equation [21]  ‡ 2cxy  ‡ ey  2 ‡ 2bx  ‡ 2dy  2 ‡ f ˆ 0: ax …A:1†

To describe the orientation distribution in SFRTs orientation tensors are used which represent a compact, general and short formulation. For the measurement of the ®bre orientation in SFRT components polished cross sections are evaluated. An image-analysis system is used to measure the ellipses, and a specially developed evaluation algorithm is employed. The major advantages compared to commercial software are improved object detection and a signi®cant reduction in measurement errors through a best-®t regression. Because for each ellipse two orientations of the ®bre are possible, the ®bre orientation state cannot be determined completely from one cross section. Therefore cuts are made at an inclined angle and a symmetrical or nearly planar orientation state is assumed. That these assumptions are valid in large regions of injection moulded parts, is shown from measurements and ¯ow kinematical considerations. By using the outlined scheme of analysis the major problems in determining the ®bre orientation state from one cross section can be solved. The magnitude of the main orientation and the direction in the plane in which the component is extended, can be determined. Furthermore an analysis with a small measurement error across the thickness direction is possible. A minor disadvantage apart from the assumptions about the orientation state, can be seen in the increase in computing costs because of the larger cross section. By contrast with the use of perpendicular cuts through the sample the visual interpretation of the inclined cross section is very dicult. Only after evaluating the orientation data can the information be interpreted. On the other hand, the visual impression of perpendicular cross sections is also strongly in¯uenced by the fact that the probability of cutting a ®bre is dependent on the ®bre orientation. As the measurements show, the assumptions concerning the ®bre-orientation state are valid and an excellent characterisation of the ®bre orientation state in an SFRT using this simple, fully automated method is possible. The information about the in-plane orientation is obtained from one cross-section, and input data for further use in other calculations, e.g. for micromechanical models, is therefore provided. The method itself is not restricted to SFRTs and the assumptions about the FO state should also hold for manufacturing processes using long ®bres, like SMC or RTM. For long ®bres, if the ®bre length is above the component thickness, the ®bres have to be oriented in-plane for obvious geometrical reasons.

A.1. Ellipse regression and main axis transformation

Conditions for the coecients in (A.1) can be given, so that the equation describes an ellipse. Linear regression: Because of the simple form of Eq. (A.1), the parameters a ÿ f can be determined with a linear regression calculation. As mentioned above, an ellipse is given by ®ve independent parameters. Therefore the number of unknowns in (A.1) has to be reduced leading to 2c 2d a 2 2b e x ‡ x ‡ xy ‡ y ‡ y2 ‡ 1 ˆ 0 f f f f …A:2† f ) x2 ‡ x ‡ xy ‡ y ‡ "y2 ‡ 1 ˆ 0: In this normalised form the deviation caused by a single point P…xi ; yi † is
i ˆ x2i ‡ xi ‡ xi yi ‡ yi ‡ "y2i ‡ 1:

…A:3†

For a group of n points the sum of the squared errors can be written as n X
2i : …A:4†
ˆ
ges ˆ iˆ1

After di€erentiating this sum by the single parameters and setting the results equal to zero a linear system of equations is derived,from which the extreme values for the best ®tted ellipse follow. The derivative by  is n n n X X X @
ˆ 2  x2i ‡ xi ‡ xi yi @ iˆ1 iˆ1 iˆ1

! n n n X X X 2 ‡  yi ‡ " yi ‡ 1 x2i ˆ 0: iˆ1

iˆ1

…A:5†

iˆ1

If this is done the same way for ; ;  and  the following equation system is received: 



X

X

x4i ‡ ‡" x3i ‡ ‡"

X X X X

x3i ‡ x2i y2i ‡ x2i ‡ xi y2i

‡

X X X X

x3i yi ‡  x2i ˆ 0; x2i yi ‡ 

X

X

x2i yi

xi yi

xi ˆ 0; X X X X x2i yi ‡ x2i y2i ‡  xi y2i  x3i yi ‡ X X ‡" xi y3i ‡ xi yi ˆ 0;

…A:6†

B. Mlekusch / Composites Science and Technology 59 (1999) 547±560

X X X x2i yi ‡ xi yi ‡ xi y2i ‡  y2i X X ‡" y3i ‡ yi ˆ 0; X X X X xi y2i ‡ xi y3i ‡  y3i  x2i y2i ‡ X X ‡" y4i ‡ y2i ˆ 0: 

X

559

Squaring this expression and substituting the
0 s by the standard deviations of the single error sources, we obtain S2haiN 11

For a given group of points P…xi ; yi † the sums in (A.6) have to be calculated and in the following it is an easy task to solve the linear equation system. If the sum of the squared errors
is divided by the number of points n,  ˆ
…A:7†
n is a measure for the quality of the curve ®tting. Main axis transformation: If the coecients a-e of the regression calculation ful®l the conditions for an ellipse, the parameters of this ellipse in a general position can be calculated using the vector description of the ellipse and a main axis transformation [21]. A.2. Measurement error for the tensor components Additionally to the measurement error caused by both orientation angles, the error in¯uence of the assumption of a constant ®bre length has to be taken into account. The mean ®bre length and deviation is determined by incineration of a sample and measuring the ®bre length in the image analysis. The measurement error on the components of the second rank orientation tensor is investigated. For a single ®bre (index (1)) the in-plane components are: ÿ …1†
2 ÿ
2 cos '…1† ; a…1† 11 ˆ sin  ÿ …1†
2 ÿ
ÿ
…A:8† sin '…1† cos '…1† ; a…1† 12 ˆ sin  ÿ
ÿ
2 2 …1† sin '…1† : a…1† 22 ˆ sin  The following derivations are limited to a11 , a consideration for a12 and a22 is straight foreword. From Eq. (8) the orientation tensor component calculated from N ®bres is GN F1 a111 ‡ F2 a211 ‡ F2 a211


FN aN 11 ˆ : …A:9† HN F1 ‡ F2 ‡ F3


FN Expanding a11 into a linear series, three times N derivatives have to be calculated, haiN 11 ˆ

@haiN @haiN 11 11
1 ‡
'1 1 @ @'1 @haiN @haiN 11 11 ‡
L1 ‡
2


1 @L @2  N  X @haiN @haiN @haiN n n n 11 11 11
 ‡
' ‡
L ˆ : @n @'n @Ln nˆ1


haiN 11 ˆ

…A:10†

 2  2 N X @haiN @haiN  n 2 11 11 ˆ f … † ‡ f' …n †2 n n @ @' nˆ1  2 @haiN 11 ‡ S…L†2 : @L

…A:11†

The mixed terms resulting from the squaring disappear, because the individual error sources are independent from one another. This it true although for each ®bre the error functions for  and ' are both dependent on the same variables, but e.g. a larger angle in determining  does not in¯uence the error in '. It remains to determine the derivatives in Eq. (A.11). Starting with ,
@G @haiN 1 ÿ 11 ˆ H1 ˆ 2 G01 H ÿ GH01 ; 1 @ @ H

…A:12†

whereas G01 ˆ

@G cos2 …'1 † ˆ 1 @ L



sin …1 † ‡

 tan …1 † ; cos …1 †

when 1 < krit ; G01 ˆ


cos2 …'1 † ÿ 2 sin …1 † cos …1 † ; d

when 1 > krit ; …A:13†

and H01 ˆ

@H 1 sin …1 † ; ˆ @1 L cos2 …1 †

H01 ˆ 0;

when 1 < krit ;

…A:14†

when 1 > krit ;

is. The derivative by '1 is given by @haiN 2F1 sin2 …† sin …'† cos …'† 11 : ˆ ÿ H @'1

…A:15†

Finally the derivative by L is calculated, @haiN 1 11 ˆ 2 …G0L1 H ÿ GH0L1 † 1 @L H   1 G ÿ a111 H ; ˆ 2 2 H L cos …† @haiN 11 ˆ 0; @L1

1 < krit ;

…A:16†

1 > krit :

For every single measured ®bre three error terms have to be calculated and then summed, as given by (A.11), to the ®nal overall error for the tensor component. The

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procedure seems quite extensive, but is easy to program. Similar relations for the other tensor components can be derived. A.2.1. Statistical error Statistical errors depend on the magnitude and the way, the sample is chosen. Only under the crude assumption that the orientation of each ®bre is independent of all others, an easy error analysis of the statistical error is possible. The variance of the tensor component a11 calculated from n ®bres is given by V…ha11 i† ˆ

 n  1 X ha11 iH 2 i i F a ÿ : 11 H2 iˆ1 n

…A:17†

Because the distribution function for the tensor component is calculated from the sum of distribution functions, a 95% con®dence interval for this normal distribution is given by p p ha11 i ÿ 2 V…ha11 i†4ha11 i4ha11 i ‡ 2 V…ha11 i†: …A:18†

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[8] Randall Curtis S, Chiang HH. Application of ®bre orientation analysis in injection moulding of ®bre-®lled composites. ANTEC'94. [9] Henry E, Kjeldsen S, Kennedy P. Fibre orientation and the mechanical properties of SFRP parts. ANTEC'94. [10] Mlekusch B, Lehner EA, Geymayer W. Fibre orientation of short-®bre-reinforced-thermoplastics I: contrast enhancement for image analysis. Composites Science and Technology 1999;59:543. [11] Kamal MR, Singh P. The distribution of ®bre phase properties in injection moulded short glass ®bre composites. ANTEC'89. [12] Hegler RP. Faserorientierung beim verarbeiten kurzfaserverstaÈrkter thermoplaste. Kunststo€e 1984;74. [13] Zhu YT, Blumenthal WR, Lowe TC. Determination of nonsymmetric 3-D ®bre-orientation distribution and average ®bre length in short-®bre composites. Journal of Composite Materials 1997;31:13. [14] Hegler RP, AltstaÈdt V, Ehrenstein GW, Mennig G, Scharschmidt J, Weber G. Elin¯uû sto‚icher parameter auf die faserorientierung beim verarbeiten kurzfaserverstaÈrkter thermoplaste. Kunststo€e 1986;76. [15] Vincent M, Agassant JF. A study of glass ®bre orientation in ¯ows of polymer composites. Interrelation between Processing Structure and Properties of Polymeric Materials, 1984. [16] Singh P, Kamal MR. The e€ect of processing variables on microstructure of injection moulded short ®bre reinforced composites. Polymer Composites 1989;10(5). [17] Bay RS, Tucker CL. Stereological measurement and error estimates for three dimensional ®bre orientation. Polymer Engineering and Science 1992;32(4). [18] Fischer G, Eyerer P. Measuring spatial orientation of short ®bre reinforced thermoplastics by image analysis. Polymer Composites 1988;9. [19] Clarke AR, Archenhold G, Davidson NC. A large area, high resolution image analyser for polymer research. Transputing'91, IOS press, 1991. [20] Fakirov S, Fakirov C. Direct determination of the orientation of short glass ®bres in an injection-moulded PET-system. Polymer Composites 1985;6(1). [21] Bronstein IN, Semendjajew KA. Taschenbuch der mathematik. B.G. Teubner Verlagsgesellschaft Stuttgard. [22] O'Connell PA, Duckett RA. Measurement of ®bre orientation in short-®bre-reinforced thermoplastics. Composite Science and Technology 1991;42. [23] Bretherton FP. The motion of rigid particles in a shear ¯ow at low reynolds number. Fluid Mechanics 1962;14. [24] Altan MC, Rao BN. Closed-form solution for the orientation ®eld in a centre-gated disk. Journal of Rheology 1995;39(3). [25] Betten J. Tensorrechnung fuÈr ingenieure. B.G. Teubner Stuttgard, 1987. [26] Mlekusch B. Short ®bre reinforced thermoplastics Ð characterisation and measuring of the ®bre orientation, thermo-elastic properties as well as shrinkage and warpage. Dissertation, Leoben, Austria, 1997.