- Email: [email protected]
epoxy composite
Composites Science and Technology 60 (2000) 535±541
Fiber packing and elastic properties of a transversely random unidirectional glass/epoxy composite Andrei A. Guseva,*, Peter J. Hineb, Ian M. Wardb a
Department of Materials, Institute of Polymers, ETH-Zentrum, CH-8092 Zurich, Switzerland b IRC in Polymer Science and Technology, University of Leeds, Leeds LS2 9JT, UK
Received 8 March 1999; received in revised form 23 August 1999; accepted 14 October 1999
Abstract Image analysis was used to characterize the microstructure of a unidirectional glass/epoxy composite which was found to be transversely randomly packed. Starting from a measured distribution of ®ber diameter, a Monte Carlo procedure was employed to generate periodic computer models with unit cells comprising of random dispersion of a hundred non-overlapping parallel ®bers of dierent diameter. The morphology generated in this way showed excellent agreement with that of the actual composite studied. An ultrasonic velocity method was used to measure a complete set of composite elastic constants and those of the epoxy matrix. On the basis of periodic three-dimensional meshes, the composite elastic constants of the Monte Carlo models were calculated numerically. Numerical and measured elastic constants were in good agreement. It was shown numerically that the randomness of the composite microstructure had a signi®cant in¯uence on the transverse composite elastic constants while the eect of ®ber diameter distribution was small and unimportant. The predictive potential of the Halpin-Tsai and some other models commonly employed for predicting the elastic behavior of unidirectional composites was also assessed. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Polymer-matrix composites; C. Elastic properties; C. Statistics; C. Computational simulation; D. Ultrasonics
1. Introduction Reinforcing inclusions in actual composites are usually of dierent size and shape and they form an in®nite variety of local packing arrangements. In a suf®ciently large laboratory sample, all important packing arrangements are present and their statistics are practically indistinguishable from that in an in®nitely large sample. As a consequence, laboratory samples behave homogeneously (i.e. sample-independent) and one can employ the constitutive approach for describing their behavior. In this approach one does not directly consider the microstructural peculiarities of the composite but operates instead with homogeneous composite quantities such as composite elastic constants, composite thermal expansion coecients, etc. Assuming simpli®ed, ad hoc arrangements of composite phases, numerous models have been proposed for predicting the properties of composite materials. Some of them, often employed for unidirectional composites, can be found in Refs. [1±12]. However, none of the * Corresponding author. Tel.: +41-1-632-3035; fax: +41-1-632-1096. E-mail address: [email protected] (A.A. Gusev).
models takes explicit account of either distribution of inclusions shapes and sizes or their random packing. Instead the models operate with simpli®ed one-inclusion problems allowing closed form solutions. This has always led to questions about the adequacy of the simplifying assumptions. In particular, how important is the scatter of the inclusion size and shape? How signi®cant is the randomness of the composite microstructure? The goal of this study is to use a combination of experimental and numerical techniques for illuminating the contribution of these factors in a ®ber-reinforced composite. The material used in these investigations was moulded from a commercial prepreg comprising unidirectionally arranged continuous glass ®bers in an epoxy resin matrix. Measurements showed that the ®bers were arranged randomly in the transverse plane. 2. Experimental 2.1. Characterization of the composite microstructure Panels of a unidirectional E glass ®ber/913 epoxy resin composite were laminated from a commercial
0266-3538/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(99)00152-9
536
A.A. Gusev et al. / Composites Science and Technology 60 (2000) 535±541
prepreg supplied by Ciba SC using the recommended curing schedule. The chosen technique, to characterize the composite microstructure, was image analysis of polished transverse sections taken from the composite, using a transputer controlled image analyzer built by the Instrumentation Group at Leeds University [13±15]. The sample is scanned in a raster fashion in a series of partially overlapping frames. Fig. 1 shows a typical image frame together with the axis de®nition used throughout this paper. The ®ber direction was de®ned as the 3 axis, the width of the sample the 1 axis, and the thickness of the sample as the 2 axis. The image analysis system, which was designed for measuring the threedimensional orientation of ®bers, works by analysing the elliptical footprint that each ®ber makes with the section plane. A threshold value is set such that every pixel above that value is recognized as in a `®ber' while every pixel below that value is in the matrix: in this way the captured grey scale pixel data is reduced to either on (®ber) or o (matrix). It then remains to determine the elliptical parameters of each group of pixels that are recognized as a ®ber. The method adopted for this task is to calculate the moments of the ®ber pixel distributions, up to the second moment. This method has been found [16,17] to give the most accurate determination of the ellipse axes from the discrete pixel distributions. For this study we have determined the distribution of the minor axis of the ®ber array and the transverse coordinates of the center of each ®ber, and hence the distribution of the nearest neighbor center to center distance. One can see from Fig. 1 that the glass ®bers have generally dierent diameters and show a large variety of dierent local packing arrangements. The average ®ber diameter is approximately 15 mm. A typical area scanned
would have a cross-section of about 1 cm2 and contain, therefore, in the order of 106 ®bers. The diameter and axis-to-axis distance to the nearestneighbor ®ber was measured for approximately 1.5.104 ®bers. The ®ber volume fraction, f 0:54 0:01, was determined from the results of image analysis, by measuring the area fraction occupied by the ®bers. This value was con®rmed by density measurement. 2.2. Composite and phase elastic moduli The experimental elastic constants of the glass ®ber/ epoxy composite were measured by the ultrasonic velocity method developed at Leeds [18], based on the original work of the National Physics Laboratory [19]. The advantage of the technique is that it enables a full set of elastic constants of a material to be quickly and accurately measured and, in consequence, has been used extensively at Leeds for the development of modeling schemes. The sample to be tested is placed between an ultrasonic transmitter and receiver (2.25 MHz) in a water bath set to a temperature of 25 C and measurements are made of the time for an ultrasonic pulse to travel between the transducers. The unique aspect of the technique is that for non-normal incidence, mode conversion at the front face of the sample causes the incident longitudinal wave to split into a longitudinal wave and a transverse wave inside the sample. Measuring the transit time of these waves over a range of incidence angles, allows the variations of the velocities of the two waves with angle of refraction (i.e. the angle of travel in the sample) to be determined. Two equations relate the shape of these two velocity/angle curves to the four stiness constants in the plane of wave propagation. For example if the wave propagates in the 13 plane of the sample then the analysis yields the stiness constants C33, C11, C44 and C13. In this paper, the 3 axis is de®ned as the ®ber direction. As the samples in this study were measured as being isotropic in the 12 plane, only ®ve independent elastic constants are required for a full description of the composite elastic anisotropy. Therefore, rotating the sample so that sound propagates in the 12 plane would yield the other outstanding constant C12. The accuracy of ultrasonics measurements is typically a few per cent. The elastic properties of a sample of pure epoxy were also measured using this ultrasonic technique, at the same frequency, and gave a Young's modulus of E 5:32 GPa and a Poisson's ratio of 0:365. 3. Microstructural statistics
Fig. 1. A typical image frame of a transverse cross-section of a glass/ epoxy unidirectional composite with a ®ber volume fraction of 0.54. The ®bers are shown in black and the matrix in grey.
We have guessed that hard-disk statistics can be used for describing the microstructure of the glass/epoxy unidirectional composite studied (see Fig. 1). This is
A.A. Gusev et al. / Composites Science and Technology 60 (2000) 535±541
equivalent to surmising that all possible packing arrangements without ®ber overlap have an equal statistical weight. To validate this premise, we generated 150 computer models made up of a periodic unit cell containing a random dispersion of 100 nonoverlapping ®bers and checked to see whether the resulting nearestneighbor distribution was similar to the measured one. The ®ber diameters in the computer models were assigned based on the measured cumulative distribution of ®ber diameter (see Fig. 2). To specify diameter d of a given ®ber, a random number was drawn from the uniform distribution in the interval (0,1). Repeating the procedure a hundred times, we assigned the diameters of all ®bers in a particular realization. The ®bers were placed without overlap on a regular square grid (see Fig. 3). Periodic boundary conditions de®ned by two continuation vectors A and B were imposed. The two vectors A and B were orthogonal and had an equal magnitude which was selected to avoid the overlaps between the ®bers. A total of 150 dierent initial con®gurations was generated. The ®ber volume fraction in the initial con®gurations was typically a factor of two smaller than the measured one. To generate samples at the desired volume fraction f 0:54, variable-box Monte Carlo (MC) runs were started from the initial con®gurations under periodic boundary conditions. At each MC step, an attempt was made to alter the coordinates of a randomly selected ®ber and to change the box size towards the value appropriate for the measured ®ber volume fraction f 0:54. The new con®guration was accepted if no overlaps between the ®bers were detected. The amplitudes
Fig. 2. Assigning ®bers diameters. Solid circles give measured cumulative distribution of ®ber diameter. The data-points were evaluated based on the measurements of the diameter of ca. 15 thousand individual ®bers. Note that the ®ber images in Fig. 1 appear to be non-circular. Probably, this occurs because the transverse cross-sections studied were not perfectly perpendicular to the ®bres. We have chosen to work with the minor ®ber diameters.
537
of attempted coordinate and box changes were chosen to give an acceptance ratio of about 50%. After the MC runs, all computer models had a ®ber volume fraction of f 0:54. Fig. 4 indicates that, within the statistics available, measured and model distributions of ®ber diameter are indistinguishable. This result validates the implementation
Fig. 3. Generation of periodic Monte Carlo realizations with unit cells comprising a random dispersion of nonoveralpping ®bers of dierent diameter.
Fig. 4. Measured and numerical distributions of ®ber diameter and nearest-neighbour axis-to-axis distance. The distributions were evaluated based on equivalent statistics of 1.5.104 data points.
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A.A. Gusev et al. / Composites Science and Technology 60 (2000) 535±541
of the modeling procedure employed for generating the initial sets of ®bers of variable diameter. Fig. 4 also shows that measured and model distributions of axis-toaxis distance are in good agreement, thus validating the premise that the glass/epoxy unidirectional composite studied has a random microstructure obeying hard-disk statistics and con®rming that the computer models can be used for determining comparative composite elastic constants. 4. Numerical 4.1. Mesh generation To mesh a Monte Carlo model, we ®rst specify a set of N nodal points xa in the plane de®ned by the unit cell's transverse continuation vectors A and B (see Fig. 3). Fiber nodal points are placed on concentric circular layers inside the ®bers while matrix nodal points are placed on a grid in Cartesian coordinates into the sections not occupied by the ®bers. This plane set of nodal points is then replicated twice in the C-direction to yield three equidistant parallel sets of nodal points. Tessellation into a periodic three-dimensional network of Delaunay tetrahedra with the vertices at the nodal points is carried out next. The Delaunay tessellation is unique and yields 6 nonoverlapping tetrahedra covering space without holes. By construction, the meshes contain no tetrahedra extending through the interfaces between the matrix and the ®bers. As a consequence, the material inside each tetrahedron is always homogeneous and the nodal points at the interfaces are common vertices of the tetrahedra belonging to the dierent phases. This corresponds to perfect adhesion at the interfaces between the matrix and inclusions.
calculated the composite elastic constants. Periodic meshes of about 6.105 tetrahedra were used in calculations. A typical minimization run took about 10 min on a Pentium 233. More numerical and validation details can be found elsewhere [21]. 5. Results and discussion 5.1. Comparison of measured and numerical elastic constants It was found that the elastic constants calculated with individual MC realizations of 100 ®bers were within a percent of the average values. To check to see if the models were representative, we repeated calculation of the composite elastic constants assuming 25 and 49 ®bers, in the periodic unit cell. The results diered by less than 1% from those obtained assuming 100 ®bers. We, therefore, concluded that the computer models with 100 ®bers were large enough to provide accurate estimates for the composite elastic constants of the transversely-random unidirectional composite studied. Fig. 5 shows that measured and calculated elastic constants, Cij , are generally in excellent agreement. Previous work [15] has shown that the C31 is often overestimated by the ultrasonic technique and it is, therefore, likely that the numerical value is more accurate. It is more common to describe the elastic properties of materials in terms of the engineering elastic constants, Young's moduli E, Poisson's ratios , and the shear moduli G. These are directly related to the elastic constants Cij and have, therefore, been used for the comparisons in the rest of this paper.
4.2. Calculation of the composite elastic constants We have used a linear version of a constant-strain-tetrahedra displacement-based ®nite element code with an iterative conjugate-gradient solver [21]. This in-house code has been speci®cally developed for the modeling of the mechanical behavior of random composites under periodic boundary conditions. The elastic constants of the epoxy matrix were as described at the end of Section 2. E-glass ®ber data, taken from the literature, were E 72:5 GPa and 0:20. The composite elastic constants Cij were calculated numerically based on periodic Monte Carlo realizations with unit cells comprising a random dispersion of 100 non-overlapping ®bers (see Section 3 and Fig. 3). To calculate the composite elastic constants of a given realization, we applied six dierent strains, minimized the total elastic strain energy with respect to the nodal displacements, and, from the stresses at the minima,
Fig. 5. Measured and numerical elastic constants.
A.A. Gusev et al. / Composites Science and Technology 60 (2000) 535±541
5.2. Eect of ®ber diameter distribution and random microstructure
539
Table 2 Predictions of commonly used engineering models (3 axis-®ber direction)
To study the eect of ®ber diameter distribution, periodic MC realizations comprising a random dispersion of 100 nonoverlapping identical ®bers were generated. The ®ber volume fraction was f 0:54, i.e. the same as in the above MC realizations with the ®bers of variable diameter. Comparison of the results indicated that the eect of ®ber diameter distribution on the composite elastic constants is very small (see Table 1). It is apparently the ®ber volume fraction which predominantly determines the composite elastic constants of the glass/ epoxy unidirectional composite studied. Table 1 also shows composite engineering elastic constants calculated by similar procedures to those described earlier, assuming square and hexagonally packed identical ®bers. One can see that numerical predictions obtained with randomly packed arrays are in the closest agreement with the measured ones. All packing arrays give identical predictions for the ®ber direction properties E33 , 13 , and G13 but predictions obtained assuming regular square and hexagonal packed arrays of identical ®bers are considerably less accurate for the transverse properties E11 , 12 and G12 . We conclude that the most accurate prediction of the elastic properties of a unidirectional ®ber reinforced composite must take into account the random nature of the ®ber packing. 5.3. Comparison with commonly used models Table 2 shows predictions of three commonly used micromechanical models which do not take into account the random character of the transverse packing. The ®rst two models, namely those of Halpin±Tsai [5] and Tandon and Weng [20], are the most commonly used micromechanical models, while the model of Wilczynski [9], which utilizes a single unit composed of a cylindrical ®ber surrounded by a cylindrical sheath of matrix, has been used with some success by these
E11 [GPa] 12 G12 [GPa] E33 [GPa] 13 G13 [GPa]
Halpin±Tsai
Tandon±Weng
Wilczynski
17.7 0.788 4.94 41.6 0.276 5.47
14.5 0.454 4.97 41.6 0.268 5.47
19.2 0.280 7.50 41.6 0.266 5.47
authors. Tables 1 and 2 indicate that all the models predict accurate values for the longitudinal E33 , 13 , and G13 . However, these models provide considerably less accurate predictions for the transverse E11 , 12 and G12 . As an indication, a normalized percentage dierence has been determined for the transverse constants E11 , 12 and G12 . The calculated average dierences were 6% for the numerical predictions presented here, 35% for the Halpin-Tsai predictions dominated by the poor prediction of 12 , 17% for the Tandon and Weng theory, and 22% for the Wilczynski theory. The Halpin-Tsai model [5] has most commonly been used in industry so we decided to compare its predictions with numerical results calculated over a wide range of ®ber fraction. For this, we repeated numerical calculations using a set of variable-®ber-fraction Monte Carlo realizations made up of a random dispersion of 100 identical parallel ®bers. For a unidirectional composite the Halpin-Tsai model provides closed form analytical expressions for ®ve independent composite elastic moduli, namely E11 , G12 , E33 , G13 , and 13 . Figs. 6 and 7 show that for the three longitudinal elastic moduli E33 , G13 , and 13 the Halpin-Tsai predictions are acceptable while for the transverse moduli E11 and G12 the values predicted are somewhat less accurate. In particular, in a technologically interesting range of ®ber volume fractions of about 0.6, the predicted G12 values are too low by about 30%.
Table 1 Comparison of measured and calculated engineering constants (3 axis-®ber direction). They are de®ned as follows: E11 1=S11 , 12 ÿS12 =S22 , G12 1=S66 , E33 1=S33 , 13 ÿS13 =S33 , and G13 1=S55 , where Sik are the composite elastic compliances, which are related to the elastic constants Cij
E11 [GPa] 12 G12 [GPa] E33 [GPa] 13 G13 [GPa]
Measured
Random dierent d
Random identical d
Hexagonal identical d
Square identical d
17.1 0.391 6.07 41.5 0.316 5.63
16.0 0.413 5.63 41.6 0.265 5.75
16.0 0.410 5.61 41.6 0.265 5.75
15.1 0.430 5.25 41.6 0.265 5.75
18.2 0.551 4.23 41.6 0.265 5.75
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A.A. Gusev et al. / Composites Science and Technology 60 (2000) 535±541
Fig. 6. Comparison of measured and numerical transverse elastic moduli with the Halpin±Tsai predictions. Open circles show measured data, ®lled squares numerical results, and solid lines the Halpin±Tsai predictions.
Fig. 7. Comparison of measured and numerical longitudinal elastic moduli with the Halpin±Tsai predictions. The symbol speci®cation is the same as in Fig. 6.
The unidirectional composite studied is transversely isotropic and has, therefore, only ®ve independent composite elastic moduli. As a consequence, any other modulus can be expressed through these ®ve elastic moduli. For example, the transverse Poisson's ratio can be written as 12 E11 =2G12 ÿ 1. As we have seen, the Halpin±Tsai model gives more or less satisfactory predictions for E11 and G12 so one might think that the prediction for 12 should also be satisfactory. However, it turns out that the predictions are very poor. In particular, in a technologically interesting range of ®ber volume fractions about 0.6, the predictions for 12 are a factor of 2±3 too high.
6. Conclusions We experimentally measured and numerically predicted the composite elastic constants of an industrial unidirectional glass/epoxy composite with a transversely random microstructure. Measured and numerical results were in excellent agreement. It appeared that the randomness of the composite microstructure had a signi®cant in¯uence on the transverse composite elastic constants while the eect of ®ber diameter distribution was small and unimportant. Numerical and measured results were compared with predictions of some most commonly used models, including the Halpin±Tsai
A.A. Gusev et al. / Composites Science and Technology 60 (2000) 535±541
model which is presently viewed as a kind of industrial standard. It turns out that none of the models could simultaneously provide accurate predictions for all the composite elastic constants. Realistic numerical predictions can readily be obtained not only for the unidirectional composite studied in this paper but also for an arbitrary composite with anisotropic phases of any morphology [21]. One should, therefore, favorably consider the use of numerical methods to complement and possibly replace phenomenological models commonly used nowadays for predicting elastic properties of composite materials. Acknowledgements AAG would like to thank Professor Ulrich W. Suter for stimulating discussions and important comments, and PJH and IMW would like to thank Dr. Alan Duckett for his helpful comments during this program. References [1] Voigt W. Ueber die Beziehung zwischen den beiden ElastizitaÈtskonstanten isotroper KoÈrper. Annalen der Physik 1889;38:573±87. [2] Reuss A. Berechnung der Fliessgrenze von Mischkristallen auf grund der PlastizitaÈtsbedingung fuÈr Einkristalle. Zeitschrift fuÈr angewandte Mathematik und Mechanik 1929;9:49±58. [3] Hill R. Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 1963;11:357±72. [4] Brody H, Ward IM. Modulus of short carbon and glass ®bre reinforced composites. Polymer Engineering and Science 1971;11:129±41. [5] Halpin JC, Kardos JL. The Halpin±Tsai equations: a review. Polymer Engineering and Science 1976;16:344±52.
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[6] Christensen RM, Lo KH. Solution for eective shear properties in three phase sphere and cylinder models. Journal of the Mechanics and Physics of Solids 1979;27:315±30. [7] Tsai SW, Hahn HT. Introduction to composite materials. Westport: Connecticut: Technomic Publishing Company, 1980. [8] McCartney LN. New theoretical model of stress transfer between ®bre and matrix in a uniaxially reinforced composite. Proceedings of the Royal Society of London 1989;A425:215±44. [9] Wilczynski AP. A basic theory of reinforcement for unidirectional ®brous composites. Composites Science and Technology 1990;38:327±41. [10] Dyer SRA, Lord D, Hutchinson IJ, Ward IM, Duckett RA. Elastic anisotropy in unidirectional ®bre reinforced composites. Journal of Physics D: Applied Physics 1992;25:66±73. [11] Hine PJ, Duckett RA, Ward IM. Modelling the elastic properties of ®bre-reinforced composites: II. Theoretical predictions. Composites Science and Technology 1993;49:13±21. [12] Shkoller S, Maewa LA. A model for defective ®brous composites. Journal of the Mechanics and Physics of Solids 1996;44:1929±51. [13] Clarke AR, Davidson N, Archenhold G. A multi-transputer image analyser for 3D ®ber orientation studies in composites. Transactions of the Royal Microscopial Society 1990;1:305. [14] Clarke AR, Davidson N, Archenhold G. A large area high resolution image analyser for polymer research. Proceedings of Transputing '91, IOS Press, 1991. p. 1±17. [15] Clarke JF, Duckett RA, Hine PJ, Hutchinson IJ, Ward IM. Negative Poisson's ratios in angle ply laminates: theory and experiment. Composites 1994;25:863. [16] Stobie RS. Analysis of astronomical images using moments. Journal of British Interplanetary Society 1980;33:323±35. [17] Stobie RS. The COSMOS image analyser. Pattern Recognition Letters 1986;4:317±22. [18] Lord D. Ph.D. thesis University of Leeds, UK, 1989. [19] Read BE, Dean GD. The determination of the dynamic properties of polymers and composites. Bristol, UK: Adam Hilger, 1978. [20] Tandon GP, Weng W. The eect of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites. Polymer Composites 1984;5:327±34. [21] Gusev AA. Representative volume element size in elastic composites: a numerical study. Journal of the Mechanics and Physics of Solids 1997;45:1449±59.
Fiber packing and elastic properties of a transversely random unidirectional glass/epoxy composite Andrei A. Guseva,*, Peter J. Hineb, Ian M. Wardb a
Department of Materials, Institute of Polymers, ETH-Zentrum, CH-8092 Zurich, Switzerland b IRC in Polymer Science and Technology, University of Leeds, Leeds LS2 9JT, UK
Received 8 March 1999; received in revised form 23 August 1999; accepted 14 October 1999
Abstract Image analysis was used to characterize the microstructure of a unidirectional glass/epoxy composite which was found to be transversely randomly packed. Starting from a measured distribution of ®ber diameter, a Monte Carlo procedure was employed to generate periodic computer models with unit cells comprising of random dispersion of a hundred non-overlapping parallel ®bers of dierent diameter. The morphology generated in this way showed excellent agreement with that of the actual composite studied. An ultrasonic velocity method was used to measure a complete set of composite elastic constants and those of the epoxy matrix. On the basis of periodic three-dimensional meshes, the composite elastic constants of the Monte Carlo models were calculated numerically. Numerical and measured elastic constants were in good agreement. It was shown numerically that the randomness of the composite microstructure had a signi®cant in¯uence on the transverse composite elastic constants while the eect of ®ber diameter distribution was small and unimportant. The predictive potential of the Halpin-Tsai and some other models commonly employed for predicting the elastic behavior of unidirectional composites was also assessed. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Polymer-matrix composites; C. Elastic properties; C. Statistics; C. Computational simulation; D. Ultrasonics
1. Introduction Reinforcing inclusions in actual composites are usually of dierent size and shape and they form an in®nite variety of local packing arrangements. In a suf®ciently large laboratory sample, all important packing arrangements are present and their statistics are practically indistinguishable from that in an in®nitely large sample. As a consequence, laboratory samples behave homogeneously (i.e. sample-independent) and one can employ the constitutive approach for describing their behavior. In this approach one does not directly consider the microstructural peculiarities of the composite but operates instead with homogeneous composite quantities such as composite elastic constants, composite thermal expansion coecients, etc. Assuming simpli®ed, ad hoc arrangements of composite phases, numerous models have been proposed for predicting the properties of composite materials. Some of them, often employed for unidirectional composites, can be found in Refs. [1±12]. However, none of the * Corresponding author. Tel.: +41-1-632-3035; fax: +41-1-632-1096. E-mail address: [email protected] (A.A. Gusev).
models takes explicit account of either distribution of inclusions shapes and sizes or their random packing. Instead the models operate with simpli®ed one-inclusion problems allowing closed form solutions. This has always led to questions about the adequacy of the simplifying assumptions. In particular, how important is the scatter of the inclusion size and shape? How signi®cant is the randomness of the composite microstructure? The goal of this study is to use a combination of experimental and numerical techniques for illuminating the contribution of these factors in a ®ber-reinforced composite. The material used in these investigations was moulded from a commercial prepreg comprising unidirectionally arranged continuous glass ®bers in an epoxy resin matrix. Measurements showed that the ®bers were arranged randomly in the transverse plane. 2. Experimental 2.1. Characterization of the composite microstructure Panels of a unidirectional E glass ®ber/913 epoxy resin composite were laminated from a commercial
0266-3538/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(99)00152-9
536
A.A. Gusev et al. / Composites Science and Technology 60 (2000) 535±541
prepreg supplied by Ciba SC using the recommended curing schedule. The chosen technique, to characterize the composite microstructure, was image analysis of polished transverse sections taken from the composite, using a transputer controlled image analyzer built by the Instrumentation Group at Leeds University [13±15]. The sample is scanned in a raster fashion in a series of partially overlapping frames. Fig. 1 shows a typical image frame together with the axis de®nition used throughout this paper. The ®ber direction was de®ned as the 3 axis, the width of the sample the 1 axis, and the thickness of the sample as the 2 axis. The image analysis system, which was designed for measuring the threedimensional orientation of ®bers, works by analysing the elliptical footprint that each ®ber makes with the section plane. A threshold value is set such that every pixel above that value is recognized as in a `®ber' while every pixel below that value is in the matrix: in this way the captured grey scale pixel data is reduced to either on (®ber) or o (matrix). It then remains to determine the elliptical parameters of each group of pixels that are recognized as a ®ber. The method adopted for this task is to calculate the moments of the ®ber pixel distributions, up to the second moment. This method has been found [16,17] to give the most accurate determination of the ellipse axes from the discrete pixel distributions. For this study we have determined the distribution of the minor axis of the ®ber array and the transverse coordinates of the center of each ®ber, and hence the distribution of the nearest neighbor center to center distance. One can see from Fig. 1 that the glass ®bers have generally dierent diameters and show a large variety of dierent local packing arrangements. The average ®ber diameter is approximately 15 mm. A typical area scanned
would have a cross-section of about 1 cm2 and contain, therefore, in the order of 106 ®bers. The diameter and axis-to-axis distance to the nearestneighbor ®ber was measured for approximately 1.5.104 ®bers. The ®ber volume fraction, f 0:54 0:01, was determined from the results of image analysis, by measuring the area fraction occupied by the ®bers. This value was con®rmed by density measurement. 2.2. Composite and phase elastic moduli The experimental elastic constants of the glass ®ber/ epoxy composite were measured by the ultrasonic velocity method developed at Leeds [18], based on the original work of the National Physics Laboratory [19]. The advantage of the technique is that it enables a full set of elastic constants of a material to be quickly and accurately measured and, in consequence, has been used extensively at Leeds for the development of modeling schemes. The sample to be tested is placed between an ultrasonic transmitter and receiver (2.25 MHz) in a water bath set to a temperature of 25 C and measurements are made of the time for an ultrasonic pulse to travel between the transducers. The unique aspect of the technique is that for non-normal incidence, mode conversion at the front face of the sample causes the incident longitudinal wave to split into a longitudinal wave and a transverse wave inside the sample. Measuring the transit time of these waves over a range of incidence angles, allows the variations of the velocities of the two waves with angle of refraction (i.e. the angle of travel in the sample) to be determined. Two equations relate the shape of these two velocity/angle curves to the four stiness constants in the plane of wave propagation. For example if the wave propagates in the 13 plane of the sample then the analysis yields the stiness constants C33, C11, C44 and C13. In this paper, the 3 axis is de®ned as the ®ber direction. As the samples in this study were measured as being isotropic in the 12 plane, only ®ve independent elastic constants are required for a full description of the composite elastic anisotropy. Therefore, rotating the sample so that sound propagates in the 12 plane would yield the other outstanding constant C12. The accuracy of ultrasonics measurements is typically a few per cent. The elastic properties of a sample of pure epoxy were also measured using this ultrasonic technique, at the same frequency, and gave a Young's modulus of E 5:32 GPa and a Poisson's ratio of 0:365. 3. Microstructural statistics
Fig. 1. A typical image frame of a transverse cross-section of a glass/ epoxy unidirectional composite with a ®ber volume fraction of 0.54. The ®bers are shown in black and the matrix in grey.
We have guessed that hard-disk statistics can be used for describing the microstructure of the glass/epoxy unidirectional composite studied (see Fig. 1). This is
A.A. Gusev et al. / Composites Science and Technology 60 (2000) 535±541
equivalent to surmising that all possible packing arrangements without ®ber overlap have an equal statistical weight. To validate this premise, we generated 150 computer models made up of a periodic unit cell containing a random dispersion of 100 nonoverlapping ®bers and checked to see whether the resulting nearestneighbor distribution was similar to the measured one. The ®ber diameters in the computer models were assigned based on the measured cumulative distribution of ®ber diameter (see Fig. 2). To specify diameter d of a given ®ber, a random number was drawn from the uniform distribution in the interval (0,1). Repeating the procedure a hundred times, we assigned the diameters of all ®bers in a particular realization. The ®bers were placed without overlap on a regular square grid (see Fig. 3). Periodic boundary conditions de®ned by two continuation vectors A and B were imposed. The two vectors A and B were orthogonal and had an equal magnitude which was selected to avoid the overlaps between the ®bers. A total of 150 dierent initial con®gurations was generated. The ®ber volume fraction in the initial con®gurations was typically a factor of two smaller than the measured one. To generate samples at the desired volume fraction f 0:54, variable-box Monte Carlo (MC) runs were started from the initial con®gurations under periodic boundary conditions. At each MC step, an attempt was made to alter the coordinates of a randomly selected ®ber and to change the box size towards the value appropriate for the measured ®ber volume fraction f 0:54. The new con®guration was accepted if no overlaps between the ®bers were detected. The amplitudes
Fig. 2. Assigning ®bers diameters. Solid circles give measured cumulative distribution of ®ber diameter. The data-points were evaluated based on the measurements of the diameter of ca. 15 thousand individual ®bers. Note that the ®ber images in Fig. 1 appear to be non-circular. Probably, this occurs because the transverse cross-sections studied were not perfectly perpendicular to the ®bres. We have chosen to work with the minor ®ber diameters.
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of attempted coordinate and box changes were chosen to give an acceptance ratio of about 50%. After the MC runs, all computer models had a ®ber volume fraction of f 0:54. Fig. 4 indicates that, within the statistics available, measured and model distributions of ®ber diameter are indistinguishable. This result validates the implementation
Fig. 3. Generation of periodic Monte Carlo realizations with unit cells comprising a random dispersion of nonoveralpping ®bers of dierent diameter.
Fig. 4. Measured and numerical distributions of ®ber diameter and nearest-neighbour axis-to-axis distance. The distributions were evaluated based on equivalent statistics of 1.5.104 data points.
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of the modeling procedure employed for generating the initial sets of ®bers of variable diameter. Fig. 4 also shows that measured and model distributions of axis-toaxis distance are in good agreement, thus validating the premise that the glass/epoxy unidirectional composite studied has a random microstructure obeying hard-disk statistics and con®rming that the computer models can be used for determining comparative composite elastic constants. 4. Numerical 4.1. Mesh generation To mesh a Monte Carlo model, we ®rst specify a set of N nodal points xa in the plane de®ned by the unit cell's transverse continuation vectors A and B (see Fig. 3). Fiber nodal points are placed on concentric circular layers inside the ®bers while matrix nodal points are placed on a grid in Cartesian coordinates into the sections not occupied by the ®bers. This plane set of nodal points is then replicated twice in the C-direction to yield three equidistant parallel sets of nodal points. Tessellation into a periodic three-dimensional network of Delaunay tetrahedra with the vertices at the nodal points is carried out next. The Delaunay tessellation is unique and yields 6 nonoverlapping tetrahedra covering space without holes. By construction, the meshes contain no tetrahedra extending through the interfaces between the matrix and the ®bers. As a consequence, the material inside each tetrahedron is always homogeneous and the nodal points at the interfaces are common vertices of the tetrahedra belonging to the dierent phases. This corresponds to perfect adhesion at the interfaces between the matrix and inclusions.
calculated the composite elastic constants. Periodic meshes of about 6.105 tetrahedra were used in calculations. A typical minimization run took about 10 min on a Pentium 233. More numerical and validation details can be found elsewhere [21]. 5. Results and discussion 5.1. Comparison of measured and numerical elastic constants It was found that the elastic constants calculated with individual MC realizations of 100 ®bers were within a percent of the average values. To check to see if the models were representative, we repeated calculation of the composite elastic constants assuming 25 and 49 ®bers, in the periodic unit cell. The results diered by less than 1% from those obtained assuming 100 ®bers. We, therefore, concluded that the computer models with 100 ®bers were large enough to provide accurate estimates for the composite elastic constants of the transversely-random unidirectional composite studied. Fig. 5 shows that measured and calculated elastic constants, Cij , are generally in excellent agreement. Previous work [15] has shown that the C31 is often overestimated by the ultrasonic technique and it is, therefore, likely that the numerical value is more accurate. It is more common to describe the elastic properties of materials in terms of the engineering elastic constants, Young's moduli E, Poisson's ratios , and the shear moduli G. These are directly related to the elastic constants Cij and have, therefore, been used for the comparisons in the rest of this paper.
4.2. Calculation of the composite elastic constants We have used a linear version of a constant-strain-tetrahedra displacement-based ®nite element code with an iterative conjugate-gradient solver [21]. This in-house code has been speci®cally developed for the modeling of the mechanical behavior of random composites under periodic boundary conditions. The elastic constants of the epoxy matrix were as described at the end of Section 2. E-glass ®ber data, taken from the literature, were E 72:5 GPa and 0:20. The composite elastic constants Cij were calculated numerically based on periodic Monte Carlo realizations with unit cells comprising a random dispersion of 100 non-overlapping ®bers (see Section 3 and Fig. 3). To calculate the composite elastic constants of a given realization, we applied six dierent strains, minimized the total elastic strain energy with respect to the nodal displacements, and, from the stresses at the minima,
Fig. 5. Measured and numerical elastic constants.
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5.2. Eect of ®ber diameter distribution and random microstructure
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Table 2 Predictions of commonly used engineering models (3 axis-®ber direction)
To study the eect of ®ber diameter distribution, periodic MC realizations comprising a random dispersion of 100 nonoverlapping identical ®bers were generated. The ®ber volume fraction was f 0:54, i.e. the same as in the above MC realizations with the ®bers of variable diameter. Comparison of the results indicated that the eect of ®ber diameter distribution on the composite elastic constants is very small (see Table 1). It is apparently the ®ber volume fraction which predominantly determines the composite elastic constants of the glass/ epoxy unidirectional composite studied. Table 1 also shows composite engineering elastic constants calculated by similar procedures to those described earlier, assuming square and hexagonally packed identical ®bers. One can see that numerical predictions obtained with randomly packed arrays are in the closest agreement with the measured ones. All packing arrays give identical predictions for the ®ber direction properties E33 , 13 , and G13 but predictions obtained assuming regular square and hexagonal packed arrays of identical ®bers are considerably less accurate for the transverse properties E11 , 12 and G12 . We conclude that the most accurate prediction of the elastic properties of a unidirectional ®ber reinforced composite must take into account the random nature of the ®ber packing. 5.3. Comparison with commonly used models Table 2 shows predictions of three commonly used micromechanical models which do not take into account the random character of the transverse packing. The ®rst two models, namely those of Halpin±Tsai [5] and Tandon and Weng [20], are the most commonly used micromechanical models, while the model of Wilczynski [9], which utilizes a single unit composed of a cylindrical ®ber surrounded by a cylindrical sheath of matrix, has been used with some success by these
E11 [GPa] 12 G12 [GPa] E33 [GPa] 13 G13 [GPa]
Halpin±Tsai
Tandon±Weng
Wilczynski
17.7 0.788 4.94 41.6 0.276 5.47
14.5 0.454 4.97 41.6 0.268 5.47
19.2 0.280 7.50 41.6 0.266 5.47
authors. Tables 1 and 2 indicate that all the models predict accurate values for the longitudinal E33 , 13 , and G13 . However, these models provide considerably less accurate predictions for the transverse E11 , 12 and G12 . As an indication, a normalized percentage dierence has been determined for the transverse constants E11 , 12 and G12 . The calculated average dierences were 6% for the numerical predictions presented here, 35% for the Halpin-Tsai predictions dominated by the poor prediction of 12 , 17% for the Tandon and Weng theory, and 22% for the Wilczynski theory. The Halpin-Tsai model [5] has most commonly been used in industry so we decided to compare its predictions with numerical results calculated over a wide range of ®ber fraction. For this, we repeated numerical calculations using a set of variable-®ber-fraction Monte Carlo realizations made up of a random dispersion of 100 identical parallel ®bers. For a unidirectional composite the Halpin-Tsai model provides closed form analytical expressions for ®ve independent composite elastic moduli, namely E11 , G12 , E33 , G13 , and 13 . Figs. 6 and 7 show that for the three longitudinal elastic moduli E33 , G13 , and 13 the Halpin-Tsai predictions are acceptable while for the transverse moduli E11 and G12 the values predicted are somewhat less accurate. In particular, in a technologically interesting range of ®ber volume fractions of about 0.6, the predicted G12 values are too low by about 30%.
Table 1 Comparison of measured and calculated engineering constants (3 axis-®ber direction). They are de®ned as follows: E11 1=S11 , 12 ÿS12 =S22 , G12 1=S66 , E33 1=S33 , 13 ÿS13 =S33 , and G13 1=S55 , where Sik are the composite elastic compliances, which are related to the elastic constants Cij
E11 [GPa] 12 G12 [GPa] E33 [GPa] 13 G13 [GPa]
Measured
Random dierent d
Random identical d
Hexagonal identical d
Square identical d
17.1 0.391 6.07 41.5 0.316 5.63
16.0 0.413 5.63 41.6 0.265 5.75
16.0 0.410 5.61 41.6 0.265 5.75
15.1 0.430 5.25 41.6 0.265 5.75
18.2 0.551 4.23 41.6 0.265 5.75
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Fig. 6. Comparison of measured and numerical transverse elastic moduli with the Halpin±Tsai predictions. Open circles show measured data, ®lled squares numerical results, and solid lines the Halpin±Tsai predictions.
Fig. 7. Comparison of measured and numerical longitudinal elastic moduli with the Halpin±Tsai predictions. The symbol speci®cation is the same as in Fig. 6.
The unidirectional composite studied is transversely isotropic and has, therefore, only ®ve independent composite elastic moduli. As a consequence, any other modulus can be expressed through these ®ve elastic moduli. For example, the transverse Poisson's ratio can be written as 12 E11 =2G12 ÿ 1. As we have seen, the Halpin±Tsai model gives more or less satisfactory predictions for E11 and G12 so one might think that the prediction for 12 should also be satisfactory. However, it turns out that the predictions are very poor. In particular, in a technologically interesting range of ®ber volume fractions about 0.6, the predictions for 12 are a factor of 2±3 too high.
6. Conclusions We experimentally measured and numerically predicted the composite elastic constants of an industrial unidirectional glass/epoxy composite with a transversely random microstructure. Measured and numerical results were in excellent agreement. It appeared that the randomness of the composite microstructure had a signi®cant in¯uence on the transverse composite elastic constants while the eect of ®ber diameter distribution was small and unimportant. Numerical and measured results were compared with predictions of some most commonly used models, including the Halpin±Tsai
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model which is presently viewed as a kind of industrial standard. It turns out that none of the models could simultaneously provide accurate predictions for all the composite elastic constants. Realistic numerical predictions can readily be obtained not only for the unidirectional composite studied in this paper but also for an arbitrary composite with anisotropic phases of any morphology [21]. One should, therefore, favorably consider the use of numerical methods to complement and possibly replace phenomenological models commonly used nowadays for predicting elastic properties of composite materials. Acknowledgements AAG would like to thank Professor Ulrich W. Suter for stimulating discussions and important comments, and PJH and IMW would like to thank Dr. Alan Duckett for his helpful comments during this program. References [1] Voigt W. Ueber die Beziehung zwischen den beiden ElastizitaÈtskonstanten isotroper KoÈrper. Annalen der Physik 1889;38:573±87. [2] Reuss A. Berechnung der Fliessgrenze von Mischkristallen auf grund der PlastizitaÈtsbedingung fuÈr Einkristalle. Zeitschrift fuÈr angewandte Mathematik und Mechanik 1929;9:49±58. [3] Hill R. Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 1963;11:357±72. [4] Brody H, Ward IM. Modulus of short carbon and glass ®bre reinforced composites. Polymer Engineering and Science 1971;11:129±41. [5] Halpin JC, Kardos JL. The Halpin±Tsai equations: a review. Polymer Engineering and Science 1976;16:344±52.
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