Effect of Platelet Length and Stochastic Morphology on Flexural Effect of Platelet Length and Stochastic Morphology on Flexural Behavior of Prepreg Platelet Molded Composites Behavior of Prepreg Platelet Molded Composites

Prepreg platelet molding compound (PPMC) can be used to create structural grade material with a heterogeneous mesoscale morphology. The present work considered various platelet lengths of the prepreg system IM7/8552 to study the effect of platelet length on the flexural behavior of PPMC composite. A progressive failure finite-element analysis was used to understand competing failure modes in PPMC with the different platelet length. The interlaminar and in-plane damage mechanisms were employed to describe complex failure modes within the mesostructure of PPMCs. Experimental results of the flexural tests of the PPMC with different platelet length sizes were used to validate the modeling prediction. The experimental and modeling results revealed complex behavior of the flexural mechanical properties (modulus and strength) on the platelet length. The experimental results indicate that PPMC composites processed with a plate length of 12.7 mm have a higher flexural modulus and strength than 25.4 and 6.35 mm. The platelet length effect on the flexural mechanical behavior was attributed to interactions between various damage mechanisms and the stochastic fiber orientation distribution variability in the material.


| INTRODUCTION
[3] PPMC are fabricated by cutting the prepreg into platelets with a prescribed length and then compression molding the platelets into a required geometry (Figure 1). [3]Figure 1 shows three prescribed prepreg platelet lengths (l p ) and a molded PPMC panel.The collimated fibers of the same length within a platelet after compression molding result in enhanced mechanical properties and sufficient formability of PPMCs compared to the composites processed with sheet molding compound (SMC) or bulk molding compound (BMC) with 10%-30% FVF. [4,5]PPMCs and other long discontinuous fiber composites are suitable for high manufacturability in terms of production rates and molding of components with complex geometry, which along with the structural grade mechanical properties meet the design requirements. [6,7]PMC as a hierarchal composite system can be presented in two levels of length scale: (i) a micro-scale where individual fibers are distinguishable in a platelet, and (ii) a mesoscale, which captures the stochastic platelet distribution. [8]The local stress transfer between the interacting platelets significantly affects the mechanical response of PPMCs.Therefore, PPMC effective mechanical properties depend on the platelet geometry (e.g., length, thickness, aspect ratio between the two), effective properties of the parent tape, platelet orientation state and its uniformity. [3]The interactions between platelets determine the progressive damage and macroscopic properties of PPMC component.
The mesoscale morphology of PPMCs is characterized by the dimension of platelets, platelet orientation and overlapping between the platelets, while the uniformity of orientation state is also affected by the number of platelets within the component. [9]The global orientation state of PPMC describes how the platelets are oriented and is an important morphology descriptor in the PPMC mechanical variability. [2]The orientation distribution of the platelets can be stochastic with the irregular arrangement or deterministic with the regular arrangement. [8]A stochastic distribution of PPMC is achieved when the orientation of the platelets results from the uncontrolled initial deposition of platelets in the mold and further anisotropic flow of heterogeneous compound causing reorientation and relocation of platelets. [10]The platelets in PPMC are arranged at various angles relative to one another, where most platelets have 2D planar orientation.However, platelets may bend to form out-of-plane waviness during the molding process.Therefore, the local structural orientations of platelets have a major role in the failure mechanisms.Stochastic morphology translates into the variability in effective stiffness and strength measured in typical tensile coupons due to variability in the local platelet orientation. [11]10] The effective mechanical properties of PPMC showed enhanced tensile strength and stiffness with increasing platelet length-to-thickness ratio (platelet aspect ratio), [4,12,13] while longer platelets for the given geometry lead to higher non-uniformity of the prepreg platelet orientation. [3] greater degree of platelet alignment results in the increased tensile properties of PPMCs in that direction. [3]The mesoscale failures that develop within the PPMC have a complex fracture mechanism including platelet transverse damage (platelet splitting), platelet debonding (delamination), and fiber breakage. [14,15]A complex mesostructure of PPMC with a large number of randomly oriented platelets provides an internal structural redundancy resulting in the load redistribution to undamaged regions and damage progression with alternative paths during loading. [16]omputational progressive failure analysis (PFA) of PPMCs allows to understand internal structure-property relationship by considering the effect of mesoscale parameters on the material variability and failure behavior. [17]Several authors studied the effect of PPMC sample size and the platelet geometry on the failure mode in tension.Feraboli et al. [18] proposed a stochastic laminate analogy for the discontinuous carbon fiber epoxy composites based on the experimental observation and Classical Laminated Plate Theory (CLPT), to account for modulus variability.An evident limitation of the modeling developed by Feraboli [18] is that it did not account for the 3D stress transfer between the platelets and, therefore, was not capable of capturing platelet length or thickness effects on the strength.Kravchenko et al. [6,13] performed a parametric FEA analysis for randomly oriented PPMC to study the effect of the orientation and the aspect ratio of the platelets on the effective tensile stiffness and strength by employing a continuum damage mechanics (CDM) and a cohesive interfacial behavior to simulate the propagation of the damage in platelets.The results showed the high dependency on the effective mechanical properties and failure mechanism on the platelet aspect ratio and orientation distribution of platelets.Increasing platelet aspect ratio in PPMC under uniform tensile loading is known to result in changes of failure mode: from platelet disbonding to fiber failure, resulting in increase of both modulus and strength. [3]he main objective of this work was to understand the variability of effective flexural modulus and strength of a stochastic PPMC and study the effect of the prepreg platelet length on flexural mechanical properties through the interaction of different failure modes.To study the effect of the platelet length and the number of platelets (platelet count) on the flexural behavior of PPMC, various platelet lengths, namely 6.35, 12.7, and 25.4 mm of the prepreg were considered with sample thickness of 2.5 mm.The experimental studies showed competing contribution of the platelet aspect ratio and platelet count on the mechanical flexural behavior, specifically higher average flexural strength and modulus were found for PPMCs with 12.7 mm platelet length.The PFA of PPMCs included CDM of the platelet and a cohesive interfacial behavior between the platelets.PFA of virtual PPMC samples allowed to study the changes in the failure modes interactions under flexural loading and determine the dominant failure modes.The failure mechanisms in the simulations varied from one coupon to another due to the local variability in fiber orientation distribution (FOD) of PPMC.Two different methods used to generate virtual morphology of 2D random PPMC coupons, namely directly generated coupons and individual virtual samples that were extracted from full size plaques, did not provide statistically different results.Therefore, the proposed methodology for PFA is supported by the experimental observations and can be used to investigate the structure-property relationship in PPMCs.

| EXPERIMENTAL CHARACTERIZATION AND METHOD
The PPMC material system used for this study was a unidirectional Hexcel carbon fiber thermoset prepreg tape (IM7/8552).Three groups of coupons were fabricated with platelet length of 6.35, 12.7, and 25.4 mm to study the effect of the platelet length on the distribution of the effective flexural properties of stochastic PPMC.The platelet width (w p ) and thickness (t p ) were 6.35 and 0.13 mm, respectively.The thickness and width of the compression molded sample were 25 and 2.50 mm, respectively.In the following sections, the microscopy analysis to calculate the platelet orientation state and the mechanical test procedures are described.

| Fabrication and mechanical testing
The flexural tests were performed to analyze the impact of the platelet length within the PPMC on the flexural properties (strength and modulus).Compression molding was performed using scattered prepreg platelets with prescribed length in a 127 mm Â 127 mm stainless-steel mold.Orientation state of the platelets within the PPMC depends on the manufacturing conditions during the molding process.A low-flow process resulting from the full mold coverage, which had initial random platelet orientation, produces 2D random stochastic orientation in the molded plaque.The mold surfaces were coated with high-temperature release wax before placing the charge into the mold to prevent the sample damage during the extraction.The depth of the mold was 25.4 mm with no cavity in between the top and bottom parts.After placing the charge, the mold was heated in the oven at 108 C for 35 min, and then transferred into the hot press for the curing process.Molding pressure of 15 MPa was used with two stage isothermal cure cycle: 30 min at 108 C followed by temperature increase to 180 C for 2 h.Once cured, press was opened and heated platens were turned off and sample was allowed to cool down to room temperature.Specimens with dimensions of 127 mm Â 25 mm were cut by a water jet machine to produce five specimens for every group of PPMC coupons.The specimens were tested under three-point bending with using the MTS test machine with a 10 kN load cell.The crosshead displacement rate of 2 mm/min was used with the frequency of the data acquisition for load and displacement at 2 Hz.Digital image correlation (DIC) was used to evaluate the strain distribution field on the top surface of the specimens, which was subjected to tensile stresses in x 1 direction.The black and white speckle pattern was spray-painted on the surface of the samples.ARAMIS GOM Correlate software was used to analyze strain distribution.Figure 2 shows three-point bending setup with 90 mm span for a PPMC sample F I G U R E 2 Three-point bending setup for a PPMC sample with 90 mm span captured with DIC camera.The effective flexural modulus and strength of stochastic PPMC coupons were measured.Flexural stress and strain were calculated using Equations ( 1) and ( 2), respectively, where, F is the applied load, L = 90 mm is the span length, b is the width of the specimen, t is the thickness and d is the deflection.

| Fiber orientation measurement using microscopy analysis
The stochastic mesostructure of PPMCs is achieved by a random arrangement of the platelets in the mold following by an anisotropic flow of heterogeneous compound, results in a relocation and reorientation of the platelets. [3]A highresolution microscopy analysis was used to measure the platelet orientation state in the cross-section of PPMC specimens.The original micrograph contained 8-bit grayscale images with 256 different intensity levels.The highresolution micrographs with 20Â magnification and visible and recognizable fiber boundaries were used in this study for adaptive binarization and the threshold was selected to ensure accurate representation of the fiber boundaries while choosing the threshold intensity during FOD measurements.The micrograph at the cross-section of the PPMC with the dimension of 2.5 mm Â 5 mm was used to measure the platelet orientation.Figure 3A shows an individual platelet that was binarized using ImageJ software with an adaptive threshold and 256 color levels were reduced to two levels of the black and white scheme referred to as the matrix and fiber region, respectively.Ellipse fitting on the recognized fibers within the platelet was performed to measure the major and minor diagonals, shown in Figure 3B for a single fiber, where k i and h i is the measured major and minor diagonals, respectively.The spatial orientation of fibers within the platelets was defined as the 2nd order symmetric tensor (Equation 3) calculated as the dyadic product of the fiber orientation vector p i (Equations 4 and 5) based on the angle ψ i between the plane's normal and the cylinder's longitudinal axis of the fiber.The angle of ψ i is calculated by Equation 6. Equation ( 7) was used to calculate the individual platelet orientation by taking the average over the calculated fibers orientation in the platelet, where n f is the number of the fibers within the platelet.The global platelet orientation was measured by taking the average over the calculated individual platelet orientations (Equation 8) where n p is the number of the platelets. [19,20]fiber ¼ Schematic of orientations for a fiber

| MODELING PROCEDURE FOR PROGRESSIVE FAILURE ANALYSIS
Three-point bending simulation of PPMC coupons was done using 3D finite element (FE) analysis with CDM in Abaqus implicit.The mesostructure of virtual PPMC was generated in Digimat FE using 100% of platelet volume fraction and Monte Carlo approach. [21]The stochastic platelets distribution was achieved by the random placement algorithm, where platelets were placed sequentially by generating their center points and angles according to the assigned probability distribution functions (PDFs).This Monte-Carlo probabilistic approach for creating virtual sample morphologies allowed to study the variation and uncertainty of stochastic PPMC properties caused by the variability in the mesostructure.The PFA simulation results for strength and modulus, as well as, experimentally measured mechanical properties, were assumed to follow normal distribution.Both distributions were compared for statistical significance using analysis of variance by the two-sample t-test.The following sections outline the mesostructure generation and development of the computational model to predict the PPMC macroscopic flexural propertied and the effect of platelet length on the mechanical behavior from the composite mesostructure and platelet material properties.

| Generation of PPMC virtual specimen
This section describes the computational FE model to predict macroscopic flexural properties and the failure behavior of PPMC.The individual platelets were modeled as transversely isotropic homogenized material with properties of the prepreg tape. [9]PFA was based on the stiffness reduction scheme for the constitutive tensor of the platelet material. [8]The macroscopic behavior of a PPMC specimens was defined by the interactions between the individual platelets.The voxels within a platelet boundary were identified with orientation of the platelet.Platelets were modeled as 3D elements (C3D8) The orientation probability distribution of platelets was generated from the assigned orientation tensor (OT).The platelets were generated using voxel mesh (Figure 4B,C) with inplane dimensions of 0.68 mm Â 0.68 mm and thickness of 0.13 mm.The average number of generated platelets in PPMCs, N, with 25.4, 12.7, and 6.35 mm platelet length were 362, 621, and 988, respectively.Virtual stochastic PPMC was modeled in the global coordinate system (x 1 ,x 2 ,x 3 ) with assigned local coordinate system of every platelet (1 i , 2 i ,3 i ) to represent stacked platelets.Figure 4B shows two platelets along with their local material coordinate systems where p 1 is the local platelet fiber direction and p 2 is the local transverse direction.shows a schematic of a platelet orientation vector in the global coordinate system.The platelet fiber orientation vector and global x 1 x 2 -plane are coplanar in the 2D random platelet distribution (φ = 90 ) and θ defines the angle of platelet with respect to x 1 direction.The discrete form of OT for N platelets with the 2D random orientation is given by a ij as shown in Equation ( 9).
where, p i ð Þ k indicates the vector of kth platelet fiber direction (Figure 4C).
The first component of the OT a 11 is representative of the platelet degree of alignment in the global x 1 direction.The random distribution of the platelets within a PPMC coupon results in the non-uniform local orientation within the volume of PPMC specimen.The 2D-random OT is given by a 11 ¼ 0:5.
Figure 4D shows overlapping platelets including a cohesive interface and the schematic for the idealized stress components.The cohesive elements were placed at the platelet interface to define the normal and shear interactions between the neighboring platelets.The FVF of the platelets is equal to that of the prepreg tape and was included in the model by using previously characterized material properties. [22]Two approaches were used to generate the virtual morphology of the specimens: (i) individual virtual coupons generated with the 2D random platelet orientation in a full-size flexural test specimen (127 Â 25 Â 2.5 mm), and (ii) virtually cut coupons from a generated 127 mm Â 127 mm plaque with the random platelet orientation (Figure 5).The latter option allowed to consider the actual molded plaque geometry, which was used to prepare physical PPMC samples.The two approaches for generating PPMC samples were used to study the effect of the sample size and the finite number of platelets in the volume of specimen on the predicted flexural mechanical properties.Eight coupons were fabricated for every batch of samples with prescribed platelet length.After generation of the virtual specimens, the model was imported into Abaqus to apply three-point bending boundary conditions. [23]The discrete rigid bottom pins were fixed and located according to the span size.The top pin was positioned at the midspan location with the assigned displacement in the x 3 direction.The frictionless surface to surface contact was assigned as the interaction between the rigid pins and specimen.Figure 4E shows a deformed state of 3-point bending model of a PPMC sample with 25.4 mm prepreg platelet length.

| Constitutive modeling of PPMCs
A stiffness reduction scheme was used to describe the damaged response of the platelets using user subroutine UMAT [24] using (Equation 10).The damaged stiffness matrix (C ij ) depends on the undamaged stiffness components (C 0 ij ) and the state of the damage variables in the fiber direction (d 1 ) and transverse to the fiber direction (d 2 ) (Equation 11).
The in-plane failure criterion was based on a strainbased continuum damage formulation.The failure initiation criterion was predicted by applying the damage initiation functions f 1 and f 2 for damage variable d 1 and d 2 , respectively (Equations 12 and 13). [10] where, ε f ,t ii and ε f ,c ii , (i = 1, 2), are the failure strains in the in-plane principal directions in tension and compression, respectively; ε f 12 , is the failure strain in shear.The damage variables d 1 and d 2 are computed by Equation ( 14), where the damage variable evolves from 0 to 1 and l Ã is the element length associated with the material, and G f i is the fracture energy.
The CDM with orthotropic damage for platelets was combined with a CZM at the interfaces between platelets to capture debonding using a traction separation law.Equation (15) shows traction separation law where interlaminar σ ij stress, δ j is the relative separations between the elements, d is the damage variable, and k 0 i ¼ 1 Â 10 6 MPa = mm is the initial stiffness. [25]13 The failure initiation criterion of the disbonding between the platelets are given by Equation ( 16) where N max ¼ 50 MPa and S max ¼ T max ¼ 80 MPa are the cohesive strength. [2]33 The propagation of the debonding is defined by a linear fracture mechanism of power law through Equation ( 17). [26] where, G I ,G II ,G III are the work done by the tractions and their corresponding displacements in the normal and shear directions, and G IC ,G IIC ,G IIIC are the critical strain energy release rates corresponding to the fracture mode

| Analysis of local bending and coupling stiffness in virtual PPMC morphologies
To study the variation of the local bending stiffness within a PPMC coupon, classical laminated plate expressions were used. [4,15]Figure 6 represents the cross-sectional view of the virtual PPMC sample.The cross-section was made of 12 voxels through the thickness (each voxel represents a platelet) and 50 voxels in the width.The cross-sectional stiffness matrix for PPMC sample changes along x 1 direction, unlike in classical laminates, due to discontinuous platelet geometry and as a result of random orientation states.The formulation of laminate stiffness matrix is given in Equation ( 18) below: where, N are the resultant forces; M are the resultant moments; ε are the midplane strains and κ are midplane curvatures.
Stiffness matrix consists of the three sub matrices A, B, and D with the size of 3 Â 3. A represents the extensional stiffness matrix, B the coupling stiffness, and D corresponds to the bending stiffness matrix.The presence of nonzero elements in the coupling matrix B indicates that the application of in-plane traction produces curvature or warping of the plate, while the applied bending moment generates an extensional strain.To study the effect of stochastic platelet orientation on mechanical behavior of virtual PPMC, as predicted by FEA, the B 11 and D 11 components were calculated along the length of the coupon (x 1 direction).B 11 component represents coupling between the normal strain due to bending, while D 11 shows the relationship between bending moment and the curvature.Equations ( 19) and ( 20) were used to calculate B 11 and D 11 of every 12 considered platelet layers along the specimen span.
where, Q k 11 is the first component of transformed stiffness matrix in element layer k considering the average angle (θ) of the voxels within a layer which is the angle between the fiber orientation vector of voxels with respect to the global x 1 direction; z k and z kÀ1 were the location of the top and bottom of each equivalent layer with reference to the mid-surface, respectively.

| Characterization of local platelet orientation and stiffness
The increased platelet length in PPMC coupon of the same size with random orientation state, resulted in the decrease of the platelet count.Fewer platelets in a stochastic PPMC system implies fewer possible orientation states, which, as discussed later, lead to increased variation in local orientation states, both within a coupon and between different coupons.Figure 7A shows three polished cross-sectional images along the span of the different samples with 25.4, 12.7, and 6.35 mm platelet length.The platelet orientation state in the cross-section was measured using Equations ( 3)-( 8) by calculating an average a 11 for the entire cross-section (Figure 7), which captured the degree of alignments in the span direction at that particular x 1 location of the sample.with 6.35 mm platelet length showed visibly more platelets in the width direction, which contributed to a more 2D random local platelet orientation (a 11 ¼ 0:48) compared with PPMCs with longer platelet length, where a 11 was lower.
Similar trend was observed in synthetic morphologies.Local variation of fiber alignment in the span direction was analyzed in the virtual samples by considering the thickness average a 11 (Figure 7B) and global distribution of a 11 (Figure 7C).The fiber orientation vectors in each voxel of the virtual PPMC samples were used for analyzing fiber alignment in span direction for the samples with different platelet lengths.Specifically, the local through-thickness a 11 was calculated along the A-A line (shown in Figure 7B) and indicated increased variability in a 11 for longer platelet length.The same observation was made when considering a 11 distribution in the entire virtual PPMC specimen with different platelet F I G U R E 6 Considered layers within a virtual PPMC specimen length (Figure 7C).The global FOD showed more narrow distribution as the number of platelets increased by reducing the platelet length given by standard deviation of 0.090 for PPMC-25.4 mm, 0.033 for PPMC-12.7 mm, and 0.054 for PPMC-6.35mm.

| Comparison of experimental and modeling prediction
The effective flexural modulus and strength of the PPMCs were measured through the experiments and compared with PFA simulation results.Figure 8 shows an example of predicted macroscopic stress-strain curves related to the virtual coupons with prepreg length of 25.4, 12.7, and 6.35 mm compared to the stress-strain response from the three-point bending tests.The initial linear flexural stress-strain response is followed by initiation of damage and progressive degradation, resulting in nonlinear stress-strain behavior before the ultimate strength was reached.A similar gradual decrease of stress was observed in both simulations and experiments after reaching the maximum stress.As discussed previously, two approaches were considered to generate the virtual PPMC coupons with the 2D random orientation.Approach 1 generated individual coupons with the full size of specimen, while Approach 2 generated a virtual 127 mm Â 127 mm plaque that was then used to create five coupons by partitioning them from the plate (Figure 5).In each approach the assigned global FOD was 2D random.The goal of using two approaches was to understand if the accurate representation of the PPMC mechanical variability is affected by the finite dimensions of the plaque and the sample, when considering the platelet size.The statistical analysis was performed to analyze the effect of modeling approach to generate virtual PPMC morphology.Results of two simulation approaches were compared with experimental results by the using two-sample t-test, wherein the null F I G U R E 7 (A) Cross-section micrographs of PPMCs, (B) distribution of the through thickness a 11 within the virtual coupons with prescribed platelet length, and (C) a 11 distribution through the thickness along the A-A line hypothesis stated that there is no difference between the distributions at the certain significance level.The difference between the sample distribution would be statistically significant if the probability of the phenomenon (pvalue) is less than 5%. [28]The calculated p-value of the two-sample t-test between the predicted flexural strength and modulus obtained from simulations and the flexural strength and modulus measured from the experiments are greater than 0.05.Therefore, the two sample distributions are not significantly different at 95% confidence interval, implying that the simulation results for both approaches can be considered reliable.This result indicates that to predict flexural behavior of thin 2D random PPMCs plates (2-3 mm plate thickness and 15-20 The variations in the flexural modulus and strength observed in Figure 9 and Table 1 result from the variability of the local platelet orientation and the nonsymmetric FOD.The results indicate that PPMCs with longer platelet length mm) have similar flexural strength and modulus compared with PPMCs of 12.7 mm prepreg platelet length.Experimental flexural strength and modulus was marginally higher by 9.5% and 16%, respectively, when comparing 25.4 mm PPMC and 12.7 mm PPMC samples.As shown in the previous section, fewer number of platelets in a 25.4 mm PPMC specimen lead to a more non-uniform local platelet orientation when compared to 12.7 mm PPMC.Therefore, even though the aspect ratio was doubled in PPMC 25.4 mm, PPMC-12.7 mm had the same level of effective flexural modulus and strength.It can be expected that this result is affected by smaller number of platelets through the thickness (15 layers).The effect of number of platelets in the coupon was previously discussed for tensile loading of stochastic PPMC system. [3]However, further reducing the platelet length from 12.7 to 6.35 mm in PPMCs resulted in significantly decreased flexural modulus and strength by 38% and 23%, respectively.This behavior was attributed due to the reduced stress sharing between the platelets, as a result of reducing platelet aspect ratio.Therefore, both platelet aspect ratio and non-uniformity in FOD are affecting the effective flexural mechanical behavior.

| Analysis of damage states in PPMC
The non-uniform mesostructure of PPMC with random orientation of platelets form multiple simultaneous damage sites and provided alternative damage progression paths during the three-point bending test.A primary damage path under three-point bending was observed away from the central bending line, indicating the role of non-uniform FOD on damage initiation and propagation along the different locations.The location of damage site varied from one sample to another and was affected by platelet length.Figure 10A shows a strain field on the surface of 25.5 and 12.7 mm PPMC specimens at the failure with localized strain away from the center location.The strain concentration in x 1 direction shown in Figure 10 was due to the damage propagation.It can be observed that PPMC-25.4 mm sample had a more localized damage, whereas 12.7 mm sample showed more spread out, distributed damage.Figure 10C shows the complex pattern of failure modes at the ultimate strength of virtual three-point bending tests at the Damage Process Zone (DPZ) confined to the region of 40 mm around the middle roller.The local dissimilar FOD state in a stochastic meso-structure lead to the localized material softening and the damage propagation before ultimate strength was reached.The distribution of the damage variables in DPZ showed multiple local failures in the matrix direction and through the thickness damage attributed to the debonding of the platelets.
Figure 11A compares the cumulative relative frequency of the damage variables in DPZ at the maximum stress for the weakest and strongest virtual coupons in every group of PPMC samples.The cumulative relative frequency of damage variables was calculated by considering the damaged elements as a fraction of total number of elements in DPZ.Stronger samples show higher accumulated damage variable compared to the weaker ones.Virtual PPMC coupons with l p = 6.35 mm showed the lowest accumulated fiber damage due to the shorter platelet length, while the transverse and through thickness damage variables were about the same level for all specimens with l p ¼ 12:7; 25:4 mm.Strength values vary from one coupon to another due to inherent morphological dissimilarities.To understand the effect of platelet alignment on predicted strength, the predicted strength for each simulation was plotted against the average degree of alignment of a 11 in the DPZ (Figure 11B).Similarly, the uncertainty in platelet alignment within DPZ was evaluated by the coefficient of variance (CV) of a 11 which was used to plot against predicted strength (Figure 11C).All three groups of samples, PPMC-6.35,12.7, and 25.4 mm, showed a

F
I G U R E 4 (A) A virtual coupon with prepreg platelet meso-structure; (B) schematic of the overlapping of the platelets with cohesive element; (C) fiber direction vectors in the global coordinate system; (D) stress components on a platelet and cohesive element; (E) deformed state of 3-point bending model σ

Figure 9
Figure 9 summarizes the results of predicted effective flexural modulus and strength of PPMCs with platelet length of 25.4, 12.7, and 6.35 mm compared with the experiments.The simulation results agreed with the experimental test results and showed a decreasing trend of average flexural strength and modulus with reducing the platelet length from 25.4 to 6.35 mm.As discussed previously, two approaches were considered to generate the virtual PPMC coupons with the 2D random orientation.Approach 1 generated individual coupons with the full size of specimen, while Approach 2 generated a virtual 127 mm Â 127 mm plaque that was then used to create five coupons by partitioning them from the plate (Figure5).In each approach the assigned global FOD was 2D random.The goal of using two approaches was to understand if the accurate representation of the PPMC mechanical variability is affected by the finite dimensions of the plaque and the sample, when considering the platelet size.The statistical analysis was performed to analyze the effect of modeling approach to generate virtual PPMC morphology.Results of two simulation approaches were compared with experimental results by the using two-sample t-test, wherein the null

F I G U R E 8
Flexural stress strain behavior of the experiment and modeling of the PPMCs with (A) 25.4, (B) 12.7, and (C) 6.35 mm prepreg platelet length F I G U R E 9 Variability of effective flexural modulus and strength in a stochastic PPMC made of (A) 25.4, (B) 12.7, and (C) 6.35 mm platelet length in simulations and experiments platelets through the thickness) it is possible to generate individual coupons, as well as, use random distribution to create individual PPMC samples.

F
I G U R E 1 0 (A) DIC image of PPMC three point bending test at maximum stress, (B) identified damage process zone, and (C) distribution of the damage variables in three-point bending simulations of PPMCs with 25.4, 12.7, and 6.35 mm prepreg platelet length F I G U R E 1 1 (A) Cumulative frequency of the damage variables in three-point simulations of PPMCs, (B) variability of a 11 , and (C) CV of a 11 within virtual PPMC coupons with respect to the predicted strength similar spread of a 11 within the DPZ ranging between 0.46 and 0.52.When linear regression was used to fit to modeling predictions, no correlation between a 11 and strength was found for PPMC-12.7 mm, which can be seen in linear regression curve fit with extremely low R 2 value.In contrary, PPMC-6.35 and 25.4 mm, showed some correlation between a 11 and simulated strength (Figure 11B).When standard deviation of FOD within DPZ was considered through coefficient of variance, CV, in a 11 it showed general insensitivity on the predictedF I G U R E 1 2 Variation of a 11 and B 11 and the distribution of matrix damage variable within the virtual PPMC specimens at DPZ T A B L E 1 Variation of PPMC tensile properties from experiments and simulations