Don Robbins, Andrew Morrison, Rick Dalgarno Autodesk, Inc., Laramie, Wyoming. Abstract

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1 PROGRESSIVE FAILURE SIMULATION OF AS-MANUFACTURED SHORT FIBER FILLED INJECTION MOLDED PARTS: VALIDATION FOR COMPLEX GEOMETRIES AND COMBINED LOAD CONDITIONS Don Robbins, Andrew Morrison, Rick Dalgarno Autodesk, Inc., Laramie, Wyoming Abstract Short fiber filled injection molded plastic parts are widely used in industrial applications due to their enhanced stiffness-to-weight and strength-to-weight ratios compared to homogeneous plastics and metals. Injection molding simulation software packages can be used to predict the as-manufactured configuration for such parts which includes the distribution of the fiber orientation tensor and fiber volume fraction throughout the part, in addition to the warped shape of the ejected, room-temperature part. In order to facilitate subsequent nonlinear (progressive failure) structural simulation of the as-manufactured, short fiber filled part, Autodesk has developed new software to seamlessly link the results of injection molding simulation with nonlinear structural response simulation that features a multiscale progressive failure model for short fiber filled plastics and explicitly accounts for the spatial distribution of fiber orientation and fiber volume fraction. The theoretical foundations and capabilities of the new software are described in a companion paper. The present paper describes the process of validating the computation methodology against novel biaxial tensile data obtained with cruciform specimens. Introduction The use of short fiber reinforcing fillers has become common place in an effort to achieve higher stiffness-to-weight and higher strength-to-weight ratios for injection molded plastic parts. Modern software tools such as Moldflow efficiently and accurately predict the orientation of the reinforcing fibers throughout the molded part, in addition to predicting the warped shape of the room temperature part after ejection from the mold. However, to produce optimal designs for injection molded parts, the designer must often consider the in-service thermo-mechanical performance characteristics of the part. For injection molded plastic parts that contain short fiber reinforcing fillers, prediction of the mechanical response is complicated by the fact that the elastic, plastic, and rupture responses of the composite material are highly anisotropic due to the local orientation of the reinforcing fibers [1], and these local fiber directions can vary throughout the injection molded part due to spatial variation of flow conditions during the injection molding process [2]. Thus an accurate simulation of the mechanical response of a fiber-filled, injection molded part requires a model that can 1) accurately represent the anisotropic elastic, plastic and rupture response of the composite material as influenced by the local fiber direction, and 2) accurately account for the variation of local fiber direction throughout the part [1]. To facilitate nonlinear structural analysis of the as-manufactured configuration of short fiber filled injection molded parts, Autodesk is currently developing software that provides a seamless transition from Moldflow s injection molding simulation to the nonlinear structural response simulation provided by Autodesk Helius PFA (Progressive Failure Analysis). The key features of this simulation methodology include: Page 1

2 1. Automated mapping of the injection molding simulation predicted fiber orientation distribution and fiber volume fraction distribution onto the finite element mesh that will be used for the nonlinear structural response simulation, 2. Enhancement of the structural response simulation with a multiscale, progressive failure, constitutive model for short fiber filled plastic materials that accounts for plasticity and rupture of the matrix constituent material, resulting in a composite material that exhibits an anisotropic, nonlinear response, and 3. A robust material characterization process that uses relatively simple, measured experimental data of the short fiber filled plastic material to fit the parameters of the multiscale, progressive failure, constitutive model. The capabilities, limitations and theoretical foundations of the new software are fully described in a companion paper by Kenik et al. [3] along with a discussion of the method required to use the software. The present paper describes the process of validating the methodology against novel biaxial tension data obtained with cruciform specimens that are machined from short fiber filled injection molded plaques. Sequence for Simulating the As-Manufactured Configuration The Moldflow software package is used to simulate the injection molding process for the short fiber filled plastic part of interest. In particular, the injection molding simulation is used to predict the spatial distribution of the fiber orientation tensor in the short fiber filled plastic part. The 2 nd order fiber orientation tensor at a point essentially provides a statistical description (in the continuum sense) of the orientation of fibers that lie in the immediate neighborhood of the point in question [4] and thus exerts a profound influence on the structural properties of the composite material. After simulating the actual injection molding process for a particular specimen, the predicted fiber orientation tensor distribution is mapped onto the finite element mesh that will be used to simulate the mechanical response of the specimen. During the structural response simulation, the fiber orientation tensor is used to operate on the constitutive matrix of a comparable idealized composite material that contains perfectly aligned fibers in order to compute the anisotropic stiffness matrix of the actual composite material with the specified fiber orientation distribution (a process referred to as fiber orientation averaging [2]), and this process has been validated by Gustev et al. [5]. Multiscale Plasticity and Rupture of the Short Fiber Filled Plastic Material Under mechanical loading, short fiber filled injection molded plastic parts typically exhibit a significant amount of plasticity prior to final rupture. However, both the degree of plasticity exhibited by the material and the final rupture load become strongly directionally dependent as the degree of fiber alignment increases from a random fiber orientation [6]. In this case, the term directionally dependent refers to the fact that the material response depends on the direction of the loading relative to the average direction of the reinforcing fibers. Furthermore, since the reinforcing fibers are short, the filled plastic material is able to rupture without actually breaking any of the reinforcing fibers; i.e., rupture occurs primarily by tearing of the plastic matrix material with some degree of short fiber pull-out [7,8]. Based on the preceding description of the response characteristics of the short fiber filled plastic material, a multiscale material model was developed. The companion paper by Kenik et al. [3] provides a complete mathematical description of the material model which will not be repeated here for brevity sake. However, it is useful to list the assumptions and constraints that were employed in developing the model: Page 2

3 The short reinforcing fibers do not exhibit any plasticity or rupture, rather the fibers exhibit a simple linear elastic response, The plastic matrix constituent exhibits both plasticity and rupture, The idealized model s matrix plasticity and matrix rupture are intended to also account for any fiber/matrix debonding that occurs in the real material, All nonlinearity exhibited by the composite material is due to nonlinearity (plasticity and rupture) in the plastic matrix material, Plasticity and rupture of the plastic matrix constituent are driven by stress in the plastic matrix constituent as opposed to being driven by the homogenized stress in the composite material. Material Characterization In order to use the multiscale material model that was discussed in the previous section, we must first determine the value of the model s coefficients by fitting the model to a collection of experimental data for the material in question. Ideally, to allow for a robust, definitive fitting of the model s coefficients, the collection of experimental data should cover the full range of behaviors that can be exhibited by the material. However, from a practical point of view, it is highly desirable to limit both the number of different test types that have to be conducted and the complexity of the tests that have to be conducted. For the present model, good fits can be obtained by using uniaxial tensile tests that are conducted to complete rupture. The tensile tests are performed using ASTM Type I tensile specimens that are cut from rectangular injection molded plaques. In order to obtain a sufficiently broad range of material response, uniaxial tensile tests are performed on specimens that are cut at three different orientations relative to the flow direction in the injection molded plaque, namely, 0 (flow direction), 90 (cross-flow direction), and 45 relative to the flow and cross-flow directions. Figure 1 shows measured uniaxial tensile test-to-failure data for the Extron 3019 HS material (30% glass fiber filled) that is used in this study. ASTM Type I tensile test specimens were cut from rectangular injection molded plaques that are 3mm thick and exhibit a well defined flow direction and cross-flow direction that governs the orientation of the short glass fibers in the plaque. The tensile test coupons are cut at three different orientations relative to the injection flow direction, namely, 0 (flow direction), 90 (cross-flow direction), and 45 relative to the flow and cross-flow directions. The tensile test data in Figure 1 was taken at an imposed uniaxial strain rate of 0.05 (mm/mm)/min and the last data point in each curve was taken just prior to rupture of the specimen. Note that all three load directions show significant levels of plastic response prior to final rupture. Figure 1 also shows that the stiffness and strength of the fiber filled material are highly dependent on the direction of loading relative to the dominant fiber direction. Further, it should be emphasized that this particular short fiber filled plastic is somewhat unusual in that the strain to failure for loading in the flow (0 ) direction is actually larger than the strain to failure for loading in the cross-flow (90 ) direction. Page 3

4 flow direction cross-flow direction Uniaxial Stress (Mpa) degree - Measured 45 degree - Measured 90 degree - Measured Uniaxial Strain (mm/mm) Figure 1. Collection of measured tensile test-to-failure data that is used to fit the coefficients of the multiscale material model. The material characterization process is carried out in three steps. The first step is to determine the elastic coefficients of the fiber and matrix constituent materials. Specifically, we determine the matrix and fiber moduli (denoted E m and E f respectively) and the matrix and fiber Poisson ratios (denoted µ m and µ f respectively) that cause the material model to accurately match the first few data points of all three measured material response curves (0, 45 and 90 ). Once the elastic coefficients are determined, the second step is to determine the matrix constituent s four plasticity coefficients (σ o, n, α, β, see Eqs. 1-8 in the companion paper [3]) that cause the multiscale material model to accurately represent the full response history of all three tensile tests (0, 45 and 90 ). The final phase of the material characterization process is to determine the effective strength S eff of the matrix constituent material (see Eq. 9 in the companion paper [3]) that causes the matrix rupture criterion to be triggered at the rupture loads that were measured in the three tensile tests. Page 4

5 Figure 2 shows the results of fitting the multiscale material model to the 0, 90, and 45 tensile test data for the Extron 3019 HS (30% glass filled) material. As seen in Figure 2, the fitted material model closely matches the elastoplastic response and the rupture load for all three load orientations (0, 90, and 45 ). Table 1 lists the fitted coefficients for the Extron 3019 HS (30% glass fiber filled) material. As seen in Table 1, the fiber modulus of 22 GPa is rather low compared to the expected modulus of glass fibers which typically fall in the range of GPa. However, the constituent properties shown in Table 1 are in situ properties that cause the micromechanical model to reproduce the measured properties of the composite material. It should be noted that the micromechanical model always represents certain simplifications of the real composite material, e.g., the current micromechanical model assumes perfect bonding between the short fibers and the plastic matrix material. Consequently, the in situ constituent properties must be different from bulk constituent properties in order to compensate for any simplifications or inaccuracies that are inherent in the micromechanical model degree - Measured 0 degree - Predicted 45 degree - Measured 45 degree - Predicted 90 degree - Measured 90 degree - Predicted 70 Uniaxial Stress (Mpa) Uniaxial Strain (mm/mm) Figure 2. Extron 3019 HS (30% glass fiber filled) Comparison of measured and predicted responses for tensile tests to failure at three different load orientations. Page 5

6 Table 1. Fitted material model coefficients for Extron 3019 HS (30% glass fiber filled) Elasticity coefficients for the matrix constituent material: E m = 3251 MPa, µ m = Elasticity coefficients for the fiber constituent material: E f = MPa, µ f = Plasticity coefficients for the matrix constituent material: n = 8.24, σ o = 38.2 MPa, α = 1.43, β = 1.03 λ m,i = 0.85 Effective strength of the matrix constituent material: S eff = 43.8 MPa After the multiscale material model s coefficients have been determined by fitting the model to the simple 0, 90, and 45 uniaxial tensile test data, the resulting material model is validated by using it to simulate the failure of more complex cruciform specimens that are loaded in biaxial tension. Figure 3 shows the in-plane geometry of the biaxial cruciform specimen and the applied loading. The in-plane loads Fx and Fy can be applied at different ratios to create an entire range of biaxial tensile load scenarios. Figure 4 shows the thickness dimension of the biaxial cruciform specimen. Note that the gauge section thickness is 1 mm, while the thickness of the load arms is 3mm. Each biaxial cruciform specimen is cut from a 3mm thick, rectangular plaque that is injection molded. The central gauge section of the biaxial cruciform specimen is then machined down to a thickness of 1mm by removing equal amounts of material from the top and bottom surfaces of the injection molded plaque. Figure 5 shows the finite element mesh used to simulate the progressive failure of the biaxially loaded cruciform specimen. In this study, 8-node, 3-D hexahedral elements are used throughout the model. The finite element model is used to simulate six different Fx/Fy load ratios in order to define the biaxial failure surface of the Extron 3019 HS (30% glass fiber filled) material. Note that during the injection molding process, the orientation of the short glass fibers will vary through the thickness of the plaque. Near the surface of the plaque, the fibers tend to be strongly aligned in the flow direction, while the inner core of the part tends to exhibit less fiber alignment (i.e., a more random distribution of fiber orientation). Consequently, it is critical to accurately map the predicted fiber orientation tensor from the injection molding simulation mesh of the rectangular plaque to the structural response simulation mesh of the cruciform specimen. Page 6

7 F y Y X L G = 24mm L = 109mm F x L = 109mm Figure 3. Geometry of the biaxially loaded cruciform specimen, showing the overall specimen length, gauge section length, tensile load arms, fillet regions and coordinate system. Z 3 mm 1 mm X Closeup of mesh density in the tapered region L G = 24mm L = 109mm Figure 4. Thickness geometry of the biaxially loaded cruciform specimen, showing the 1mm thick gauge section and 3mm thick load arms. Page 7

8 closeup of fillet region Figure 5. Finite element mesh of the biaxially loaded cruciform specimen showing a close-up view of the fillet region. 8-node, 3-D hexahedral elements are used throughout the mesh. The characterized multiscale material model is used in a progressive failure finite element simulation of the cruciform specimen for six different biaxial load ratios. In each case, the tensile loads were applied as imposed displacement increments at the ends of the load arms. The load increment size was chosen so that the specimen could sustain approximately fifty load increments before global fracture of the specimen occurred. Qualitatively speaking, the predicted response of each of the biaxially loaded cruciform specimens was quite similar. In each simulation, the matrix constituent material undergoes considerable plastic deformation within those regions of the specimen that are most highly stressed (e.g., the filleted corners and the thin square gauge section). As local plastic deformation evolves, the stiffness of the matrix constituent decreases, and consequently the stiffness of the composite material decreases, causing localized load redistribution to occur in the finite element model. As the applied loads continue to increase, the stress state in the matrix constituent will eventually satisfy the matrix rupture criterion at some location within the model, at which time the stiffness of the ruptured composite material is reduced to a very low level. For the biaxially loaded cruciform specimens, the predicted fracture process is quite sudden, i.e., once local rupture occurs, the continuing fracture process is unstable and the fracture surface very rapidly spans the specimen resulting in complete global failure. This agrees with the actual experimental specimens where the fracture process appeared to be instantaneous. Page 8

9 It should be emphasized that the set of six biaxially loaded cruciform specimens exhibit several challenging characteristics for progressive failure simulation validation. First, the biaxial cruciform specimens are subjected to an entire range of different global (Fx/Fy) load ratios that lead to complex local stress states dominated by various combinations of in-plane stress components σ xx, σ yy and σ xy. Second, the biaxial cruciform specimens are geometrically complex. Specifically, the biaxial cruciform specimens contain both in-plane and out-of-plane fillet regions that produce a non-homogeneous stress and strain field with moderate stress concentrations, regardless whether the loading on the specimen is uniaxial or biaxial. To illustrate the non-homogeneous stress field exhibited by the cruciform specimen, Figure 6 shows the distribution of von Mises stress predicted in a cruciform specimen when subjected to a simple uniaxial load case (Fy>0, Fx=0). As seen in Figure 6, the stress field is quite complex, and there are nine different local maxima that are clearly identifiable. Figure 6. Distribution of von Mises stress predicted in cruciform specimen when subjected to the simple load case Fy>0, Fx=0. Nine different local maxima are clearly identifiable. Figure 7 shows the predicted rupture loads for the biaxial cruciform models computed at six different biaxial (Fx/Fy) load ratios. Also shown in Figure 6 are the measured rupture loads for the actual biaxial cruciform specimens at five different biaxial (Fx/Fy) load ratios labeled A through E. Note that the measured results contain two or three replicates at each load ratio to show the amount of scatter inherent in the test data. As seen in Figure 7, the predicted biaxial failure surface very closely matches both the size and shape of the measured biaxial failure surface. In particular, note that the model captures the strengthening effect that is observed when some level of flow direction (Y) loading accompanies a high level of cross-flow direction (X) loading. Page 9

10 Flow Stress - Sy (MPa) E Measured Predicted Cross-Flow Stress - Sx (MPa) Figure 7. Comparison of predicted and measured biaxial rupture loads for cruciform specimens made from Extron 3019 HS (30% glass fiber filled) injection molded plaques. D C A B Figure 8 shows the predicted net load vs. imposed displacement for specimen A (biaxial load ratio Fx>0, Fy=0). The nonlinear response seen in Figure 7 is typical of all six simulated specimens and clearly shows significant and continual softening of the specimen prior to final rupture. The specimen softening that occurs prior to final rupture is caused by plasticity in the matrix constituent material which is fairly localized in the most highly stressed regions of the specimen (e.g., the filleted corners and the thin square gauge section). Figure 9 contains closeup views of the gauge section of specimen A (biaxial load ratio Fx>0, Fy=0) showing the predicted evolution of effective plastic strain in the matrix constituent at points 1-6 labeled on the load/displacement curve in Figure 8. As seen in Figure 9, the effective plastic strain exceeds 3% in the filleted corners prior to specimen rupture, while an extensive portion of the thin square gauge section exceeds 2% effective plastic strain prior to specimen rupture. Page 10

11 Net Specimen Load (N) specimen rupture Imposed Axial Elongation (mm) Figure 8. Predicted net load vs. imposed displacement for specimen A (Fx>0, Fy=0) showing nonlinear response due to localized plasticity and global rupture of the specimen. 1 4 effective plastic strain Figure 9. Close-up views of the gauge section of specimen A (Fx>0, Fy=0) showing the predicted evolution of effective plastic strain in the matrix constituent at points 1-6 labeled on the load/displacement curve in Figure 7. Page 11

12 As seen earlier in Figure 7, the rupture loads were predicted quite accurately across the entire range of biaxial load ratios. As mentioned earlier, the actual specimen rupture process (or fracture process) is unstable; once localized tearing initiates within the specimen, it immediately proceeds to grow across the specimen, resulting in global fracture. Consequently, the entire fracture process is predicted to occur within a single load increment. Figures 10 through 14 show a comparison of the predicted and observed rupture trajectories (fracture surfaces) for the cruciform specimens at five different biaxial load ratios that were labelled in Figure 7 as points A,B,C,D,E respectively. Figure 10 shows a comparison of the predicted and observed rupture trajectory (fracture surface) for biaxial load ratio A (i.e., the case Fx>0, Fy=0, or loading only in the cross-flow direction). In the image of the finite element model seen in Figure 10, the red region indicates the location of ruptured material, while the blue region indicates un-ruptured material. Note that the model correctly predicts that the fracture surface runs from fillet to fillet, effectively tearing one of the load arms off at the attachment point. Figure 11 shows a comparison of the predicted and observed rupture trajectories (fracture surfaces) for biaxial load ratio B (i.e., the case Fx/Fy = 2.3). Again, the model correctly identifies the fracture surface observed in the experimental specimens, namely, the fracture surface runs from fillet to fillet, effectively tearing the cross-flow direction load arm off at the attachment point. Figure 12 shows a comparison of the predicted and observed rupture trajectories (fracture surfaces) for biaxial load ratio C (i.e., the case Fx/Fy = 0.8). Note that for load ratio C, the two experimental specimens shown in Figure 12 exhibit different fracture trajectories, possibly suggesting that the load ratio Fx/Fy=1.2 is near the transition between a diagonal fracture and a fracture that simply tears one of the horizontal load arms off. The finite element model predicts that the dominant fracture trajectory simply tears one of the horizontal load arms off (similar to the fracture shown in the experimental specimen in the upper left corner of Figure 12); however, notice that the finite element model also shows very localized, isolated zones of rupture at three of the four fillets, suggesting that the model senses that the specimen also has a tendency toward a diagonal fracture. Figure 13 shows a comparison of the predicted and observed rupture trajectories (fracture surfaces) for biaxial load ratio D (i.e., the case Fx/Fy = 1.7). Note that for load ratio D, the two experimental specimens shown in Figure 13 exhibit different fracture trajectories, possibly suggesting that the load ratio Fx/Fy=0.6 is near the transition between a diagonal fracture and a fracture that simply tears one of the vertical load arms off. The finite element model predicts that the dominant fracture trajectory simply tears one of the vertical load arms off (similar to the fracture shown in the experimental specimen in the upper right corner of Figure 13); however, notice that the finite element model also shows two very small diagonal fractures (one from each of the lower fillets), suggesting that the model senses that the specimen also has a tendency toward a diagonal fracture. Page 12

13 Figure 14 shows a comparison of the predicted and observed rupture trajectories (fracture surfaces) for biaxial load ratio E (i.e., the case of loading only in the flow direction, or Y direction). The single experimental specimen shown in Figure 14 is representative of the fracture observed in all three replicates of this load ratio where one of the vertical load arms is simply torn off at the attachment point. However, the finite element model for specimen E incorrectly predicted a diagonal fracture (as shown in the upper right hand corner of Figure 14). The relatively coarse load incrementation scheme was suspected to be the cause of the incorrect fracture trajectory predicted by the model, so the specimen was simulated a second time using a more refined load incrementation scheme (i.e., the new load increment size was 1/10 the original load increment size). As seen in the lower right corner of Figure 14, reducing the load increment size resulted in the correct fracture path being predicted, without any significant change to the predicted fracture load level. This result prompted the authors to retest several of the biaxial load ratios using smaller load increments. In all cases tested, the reduced load increment size did not significantly change the rupture load level or the fracture trajectory. A Y X Figure 10. Comparison of predicted and observed rupture trajectories at the biaxial load ratio identified as point A in Figure 7. Flow direction is parallel with the global Y direction Page 13

14 B Figure 11. Comparison of predicted and observed rupture trajectories at the biaxial load ratio identified as point B in Figure 7. Flow direction is parallel with the global Y direction Page 14

15 C Figure 12. Comparison of predicted and observed rupture trajectories at the biaxial load ratio identified as point C in Figure 7. Flow direction is parallel with the global Y direction D Figure 13. Comparison of predicted and observed rupture trajectories at the biaxial load ratio identified as point D in Figure 7. Flow direction is parallel with the global Y direction Page 15

16 E Y X Original coarse load incrementation F y > 0 F x = 0 Correct rupture trajectory via refined load incrementation Figure 14. Comparison of predicted and observed rupture trajectories at the biaxial load ratio identified as point E in Figure 7. Flow direction is parallel with the global Y direction Finally, let us consider the change in fracture trajectory that is observed in the actual cruciform test specimens as the biaxial load ratio is changed. The lower half of Figure 15 shows the complete collection of measured rupture loads and simulated rupture loads. Based solely on the fracture trajectories observed in the actual test specimens, one can divide the biaxial load spectrum into five different fracture trajectory sections shown in red in the lower half of Figure 15. Notice that none of the experimentally tested load ratios produced a consistent diagonal fracture pattern, thus the middle section (labeled diagonal fracture ) is void of any experimental data points. However, one of the simulated load ratios does fall clearly in the middle of the diagonal fracture region. The upper half of Figure 15 shows the fracture trajectory predicted in the finite element model that was biaxially loaded at a ratio of Fx/Fy=1.2, and the predicted fracture process is dominated by a diagonal fracture path consistent with expectation. Thus it can be concluded that the finite element model is successful in discerning the change in final fracture path as a function of biaxial load ratio. Page 16

17 vertical arm torn off E D C A B horizontal arm torn off Figure 15. Comparison of predicted and observed rupture trajectories at the biaxial load ratio identified as point E in Figure 7. Flow direction is parallel with the global Y direction Page 17

18 Conclusions Autodesk has developed software for short fiber filled, injection molded plastic parts that provides a seamless transition from the injection molding simulation to the nonlinear structural response simulation. Specifically, the software provides a seamless link between Autodesk Simulation Moldflow Insight (ASMI) and Autodesk Helius PFA. The key features of this software include: Automated mapping of the Moldflow-predicted fiber orientation distribution and fiber volume fraction distribution onto the finite element mesh that will be used for the Helius PFA nonlinear structural response simulation, Enhancement of Helius PFA with a multiscale, progressive failure, constitutive model for short fiber filled plastic materials that accounts for plasticity and rupture of the matrix constituent material, resulting in a composite material that exhibits an anisotropic, nonlinear response, and A robust material characterization process that requires only relatively simple, measured uniaxial tensile data of the short fiber filled plastic material to fit the parameters of the multiscale, progressive failure, constitutive model. The multiscale, progressive failure, elastoplastic material model was characterized for Extron 3019 HS material (30% glass fiber filled) using uniaxial tensile test data that was taken at three different orientations relative to the flow direction. The characterized material model was then used to predict the progressive failure response of biaxially loaded cruciform specimens that were made from the same Extron 3019 HS material. The finite element simulation of the biaxially loaded cruciform specimens was shown to accurately predict the rupture loads and fracture trajectories for an entire range of biaxial load ratios. Bibliography 1. Nguyen, B.N, Bapanapalli, S.K., Holbery, J.D., Smith, M.T., V. Kunc, V., Frame, B.J., Phelps, J.H., and Tucker, C.L. III, (2008) Fiber Length and Orientation Distributions in Long-Fiber Injection-Molded Thermoplastics Part I: Modeling of Microstructure and Elastic Properties, Journal of Composite Materials, 42: (1994) Flow and Rheology in Polymer Manufacturing, Ed: S.G. Advani, Elsevier Science B.V., Amsterdam, The Netherlands. 3. Kenik, D., Robbins, D., Morrison, A., and Gies, J., Bridging The Gap: As-Manufactured Structural Simulation Of Injection Molded Plastics, Society of Plastics Engineers, Automotive Composites Conference, Sept. 9-11, 2015, Novi, MI. 4. Advani, S. and Tucker, C.L. III, (1987) The Use of Tensors to Describe and Predict Fiber Orientation in Short-Fiber Composites, Journal of Rheology, 31: Gusev, A., Heggli, M., Lusti, H.R. and Hine, P.J. (2002) Orientation Averaging for Stiffness and Thermal Expansion of Short Fiber Composites, Advanced Engineering Materials, Vol. 4, No. 12, pp Yang, Q.S. and Qin, Q.H. (2001) Fiber Interactions and Effective Elastic-Plastic Properties of Short Fiber Composites, Composite Structures, 54: Meraghni, F. and Benzeggagh, M.L. (1995) Micromechanical modeling of matrix degradation in randomly discontinuous-fibre composites. Composite Science and Technology, 55: Meraghni, F., Blakeman, C.J., Benzeggagh, M.L. (1996) Effect of interfacial decohesion on stiffness reduction in a random discontinuous-fibre composite containing matrix microcracks. Comp. Sci. and Tech., 56: Page 18

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