virtual tensile testing of prepreg platelet composite ...

VIRTUAL TENSILE TESTING OF PREPREG PLATELET COMPOSITE MOLDED WITH STOCHASTIC MORPHOLOGY Sergii G. Kravchenko, R. Byron Pipes Composites Manufacturing & Simulation Center, Purdue University, 1105 Challenger Avenue, Suite 100, West Lafayette IN 47901

ABSTRACT Continuous fiber prepreg tape can be slit and cut into platelets, which are further used as an advanced compression-molding compound. The resulting platelet-based discontinuous long-fiber composite has a meso-scale heterogeneous structure, which means that it is the stress concentrations resulting from interaction between individual platelets and semi-laminated composition of the system govern the composite effective mechanical properties. One form of prepreg platelet composite meso-scale morphology is stochastic. It occurs when the orientation and arrangement disorder are determined by the uncontrolled deposition of platelets into the mold and further molding conditions. The understanding of composite process-structure-property relationship is essential for the application of prepreg platelet molded composite form. Progressive failure analysis is herein used to study the damaged deformation up to ultimate failure in plateletbased tensile coupons with stochastic morphology. Computational damage mechanics approaches (continuum and discrete) are utilized for constitutive modelling and addressing complex interacting damage mechanisms. The developed failure analysis allows for understanding of how the composite structure details, meaning the platelet geometry and system morphology (geometrical arrangement and orientation distribution of platelets), define the effective properties of a platelet-molded composite system, its stiffness, strength and variability in properties.

1. INTRODUCTION Previously, the discontinuous long-fiber composite systems molded from prepreg platelets were shown [1] to over perform the systems manufactured from traditional sheet molding compounds [2]. Prepreg platelets are obtained by transforming precursor continuous fiber (CF) prepreg tape into the discontinuous pieces of assigned length (L p ) and width (w p ) [3], while the platelet thickness (t p ) is the prepreg tape thickness (Figure 1). Prepreg platelet molding compounds exist in the form of unconfined/loose platelets (typically, thermoplastic based prepreg) or mats/sheets of stacked platelets (typically, thermoset-based prepreg). Compression or transfer molding of prepreg platelets result in a meso-scale, i.e. platelet level heterogeneous, semi-laminated composite system with efficiently packed fibers, which combines good formability characteristics [4] [5] and mechanical properties [6]. The competitive advantage of the prepreg platelet molding compound over the traditional sheet and bulk molding compounds (SCM and BMC) of dispersed fibers is in obtaining the discontinuous composite system with (i) controlled fiber length and (ii) collimation of fibers within the platelet for achieving high fiber volume fraction in a molded part to meet structural requirements.

Copyright 2018. Used by the Society of the Advancement of Material and Process Engineering with permission. SAMPE Conference Proceedings. Long Beach, CA, May 21-24, 2018. Society for the Advancement of Material and Process Engineering – North America.

Figure 1. Schematics of prepreg platelet molded composite with stochastic meso-structure (left) and determinants of composite structure (right)

Prepreg platelet based composite material system has two levels of recognized scale, namely a micro-scale where individual fibers are distinguishable and a meso-scale, meaning the scale where individual platelets are distinguishable. Effective/macroscopic mechanical properties of prepreg platelet molded composite are controlled by the meso-scale heterogeneities, as they provide for dominant stress concentrations. Three-dimensional platelet-to-platelet stress transfer depends on the meso-structural descriptors of composite system and along with the properties of individual platelets governs the composite macroscopic stiffness and strength. A meso-structure of prepreg platelet based composite is defined by the system morphology and platelet geometrical parameters [7], while morphology is determined by the geometrical accommodation of platelets, meaning their arrangement and orientation, as schematically shown in Figure 1. An ability to design the desired combinations of properties of prepreg platelet composite comes with increased understanding of the origins of properties from the composite meso-structure, which is determined by the manufacturing specification. The study of structure-property relationship therefore is fundamental for development of material systems compatible with intended applications. A stochastic meso-structure of a platelet molded composite system is achieved when the orientation and arrangement disorder result from the uncontrolled deposition of platelets into the

mold and further flow conditions, which also provide for the non-deterministic geometrical dimensions of processed platelets. During deposition and molding, platelets are structurally arranged (stacked) at various orientations relative to each other to produce complex irregular morphology. This morphology is semi-laminated, meaning the majority of platelets are planardistributed, though platelets bend around their neighbors to produce wavy compliant overlaps through the molded volume of composite system. Orientation distribution of platelets is an important factor that influences the mechanical performance of a composite since orthotropic mechanical properties of a platelet are a function of orientation. This composite form was first patented in 1989 [8], and subsequently commercialized by several dominant material manufacturers. Stochastic discontinuous composite systems of molded prepreg platelets are typically referred to as having “random distributed” orientations in attempt to yield quasi-isotropic elastic and strength properties, but show a marked variability in both effective modulus and strength. Different research groups [6] [9] [10] [11] previously reported a high level of variability in effective tensile properties of prepreg platelet composite systems attributed to the stochastic nature of their uncontrolled morphology. The primary conclusions from the aforementioned experimental studies was that the strength of a platelet-based stochastic composite increased with platelet length (for constant tape thickness), while longer platelets provided for increased variability in effective properties. Next, it has been experimentally observed that thinner tape provided for reduced variability and increased averages of effective properties [10] [12]. As pointed out by Yamashita et al. [10], although some theoretical studies have suggested various predictive models to estimate effective stiffness and strength of stochastic platelet-based composite systems, their dependence on the platelet length was not sufficiently reproduced. The objective of this work is to develop a mechanistic finite-element based computational model to predict the effective tensile properties of a discontinuous prepreg platelet composite system considering the spatial stochasticity of its meso-scale morphology. The study involves conducting a progressive failure analysis in a full-sized virtual tensile coupon using Monte-Carlo simulations. The effective response of a prepreg platelet composite coupon is deduced from the known input properties of a platelet material (UD CF tape) and explicitly modeled meso-scale stochastic morphology. This allows to tailor the effective composite properties from the precursor tape properties in response to the changes in composite meso-structure and therefore to study the structure-property relationship and understand potential property limits of the composite. Specifically, the influence of the platelet dimensions (length and thickness) on the variability of effective quasi-static tensile properties are discussed.

2. VIRTUAL SIMULATION OF A UNIAXIAL TENSILE TEST The section describes the development and application of a computational model to predict stiffness and strength of a stochastic prepreg platelet composite system by simulation of a quasistatic uniaxial tensile test of a coupon with semi-laminated stochastic meso-scale morphology and deterministic platelets size. “Computational window” size of a composite meso-structure sample for virtual testing is selected to have the typical dimensions of a full-sized tensile coupon. It is because (i) the “computational window” must represent a substantially large geometric volume such that it contains enough characteristic discontinuities and heterogeneities to be representative of the irregular morphology at a given level of observation, which is macro-scale and (ii) there are no repeating unit cells in the meso-structure of a physical tensile coupon. The results of a virtual tensile test of a full-sized coupon are amenable to elementary homogenization when

macroscopic/effective composite stress-strain curve is obtained by dividing the reaction force by coupon cross-sectional area and coupon elongation by its original length, respectively, to compare the theoretical analysis results to the experimental data. The modeling strategy is to assess the variation of effective tensile stiffness and strength caused by composite meso-structure through the explicit modeling of composite meso-scale morphology and utilizing progressive failure analysis. The realistic morphological features (complex platelets arrangement and variable platelet orientation) are reflected in a virtual material architecture to lend the reliable predictions of mechanical properties. Meso-scale computational approach considers platelets as composite constituents (with homogeneous orthotropic material properties) and their interactions to predict the overall (macroscopic) mechanical behavior of the composite coupon. Meso-sclae damage is considered as a permanent weakling of the material due to micro-scale failure mechanisms that occurs in a progressive manner prior to macroscopic rupture. Computational damage mechanics is used to treat progressive failure by reducing (damaging) the material elastic constants via scalar damage variables. Platelet volume fraction in a tensile coupon is assumed 100%, as in reality a physical coupon may have up to 10% of neat resin pockets. Platelet thickness variation in a physical platelet-molded composite is not addressed in the simulation, meaning the platelet size is considered deterministic in a given virtual coupon. Thermal-residual stresses are not herein considered. Homogenized properties of the prepreg tape (parent platelet material) are assumed deterministic, meaning that the local fiber volume fraction variability in a platelet and the dependence of in-situ lamina strength on CF UD ply thickness are not considered. 2.1 Computational Finite-Element Model for Virtual Uniaxial Tensile Test of a Coupon with Deterministic Platelet Size and Stochastic Meso-Scale Morphology The composite system is assumed as a collection of platelets and interfaces between platelets. The domain occupied by a tensile coupon consists of sub-domains occupied by misaligned individual platelets with defined boundaries, as shown in Figure 2, which create local geometrical and stiffness discontinuities that control meso-scale interactions between platelets and, therefore, their collective mechanical response. The scale of a single composite phase, a platelet, is not separable from the scale of a laboratory test piece such as tensile coupon for the platelet sizes used in practice. For this reason, the morphological details of a composite structure (a coupon) is explicitly incorporated into the computational model for studying the structure/property relationship. Geometry and mesh generation are combined in a voxel-based process. This means that geometry is discretized with a regular pattern of eight-node brick finite elements. This pattern is known as a voxel mesh. The voxelized morphology is output as mesh for subsequent finite-element based progressive failure analysis. All platelet elements are assigned the same homogenized material properties of prepreg tape, and the local orientation is mapped from geometry to the elements. Such voxel mesh is illustrated in Figure 2. Neighboring elements with similar local material orientation effectively represent a single platelet. Figure 2 shows two platelets (i, j) along with their local material coordinate systems (1 i 2 i , 1 j 2 j ), where “1” is local platelet fiber direction and “2” is local transverse direction. Single parallelipiped finite element (voxel) thickness is a required platelet thickness, while its length and width are 0.7mm. The geometry of a tensile coupon is generated in DIGIMAT FE.

Figure 2. Voxel-based finite element mesh of a virtual tensile coupon with prepreg platelet mesostructure

The coupon global length, width and thickness are labeled as L, w, and t, respectively. The boundary conditions on a virtual coupon are as follows: one edge (x 1 =0) is fixed while the opposite edge (x 1 =L) is uniformly pulled along the x 1 -direction with displacement u* and clamped againts other displacements. The non-linear finite element analysis is carried out with a commercial finite element code ABAQUS/Standard 6.14-1. The platelets are modeled with solid three-dimensional stress elements (C3D8), while the interfaces in the overlap of platelets are modeled with cohesive elements (COH3D8). The schematics of stress carrying capacity of abovementioned finite elements is summarized in Figure 2. 2.2 Approximation of Stochastic Meso-Scale Morphology in a Finite-Element Model The virtual material samples (coupons) are constructed with sequential adsorption algorithm. It is an efficient and flexible computer simulation capable of capturing some of the important details of complex morphology of platelet composite. It means that stochastic structural arrangement and orientation of platelets in a virtual system is approximated by the random placement algorithm,

where objects (platelets) are placed sequentially by generating their center points and angles according to the probability distribution functions by means of a pseudo-random number generator (PRNG). The packing of virtual platelets is essentially the packing of tetris-like items, i.e. clusters of mutually orthogonal parallelepipeds inside a given domain (which is a “coupon”). The full three-dimensional character of the platelet-stacked system (semi-laminated morphology) is preserved in a digital (virtual) tensile coupon since the virtual platelet placement proceeds throughout the thickness of the tensile coupon, allowing for multiple and complex-structured overlaps of platelets. Virtual coupons are generated in DIGIMAT, where sampling of pseudo-random numbers is from the uniform probability density function for platelet centroids, while the orientation probability density function is extracted from the requested second-order orientation tensor, a ij [13]. The recovery of the orientation distribution function from the a ij is achieved by using the fourth-order orientation tensor a ijkl , which is constructed by the fitted orthotropic closure approximation [14]. In the present investigation, the nominal “2D-random” second-order orientation distribution tensor is used, 𝒂𝒂𝑖𝑖𝑖𝑖 = [[0.5 0] [0 0.5]] to generate individual virtual coupons, which is equivalent to using the uniform probability density function. The described morphology generation is a simplified modeling of physical morphology which results from platelets kinematic and interaction during the deposition (into the mold, prior to compaction and molding) along with “human”/”process” factors which are also responsible for introducing the local morphological variability, as well as re-orientation of platelets during local flow in molding process. 2.3 Constitutive Models for Damageable Platelet Material and Platelet Interfaces Ultimate macroscopic failure of a meso-scale heterogeneous prepreg platelet composite system is preceded by the successive evolution and interaction of local damage in platelets and platelet-toplatelet interfaces, and occurs after enough sub-critical damage is gradually accumulated for the system to lose its load carrying capacity. The assumed primary meso-scale failures that develop within a prepreg platelet based system include platelet transverse damage (platelet splitting along the fibers), debonding between adjacent platelets (delamination) and failure transverse to the platelet fiber direction (fiber fracture). The meso-scale platelet damage modes and the damageable interfaces between the platelets are treated in the simulation with different approaches of computational damage mechanics, both based on stiffness reduction scheme when local progressive failures are represented by reducing the corresponding material elastic constants. The elastic-brittle continuum damage mechanics (CDM) model with orthotropic damage for platelets is coupled with discrete damage theory with isotropic damage (cohesive zone modeling) to capture disbonding between platelets and thereby address the two competing failure mechanisms in the system. The damage in a platelet is idealized as damage in a unidirectional (UD) CF ply at the meso-scale. The modeling of dis-bonding between the platelets is similar to previous modeling of the interlaminar damage (delamination) between the individual plies of a laminate. The platelet constitutive model herein utilized is adopted from Linde et al. [15] and employed as the commercially available user subroutine UMAT described therein [16] for ABAQUS/Standard (Implicit). The platelet is considered as an orthotropic homogenized continuum in which in-plane damage is the result of the tri-axial state of stress [15]:

{𝜎𝜎11 , 𝜎𝜎22 , 𝜎𝜎33 , 𝜎𝜎12 , 𝜎𝜎13 , 𝜎𝜎23 }𝑇𝑇 = 𝑪𝑪(𝒅𝒅) {𝜖𝜖11 , 𝜖𝜖22 , 𝜖𝜖33 , 2𝜖𝜖12 , 2𝜖𝜖13 , 2𝜖𝜖23 }𝑇𝑇

[1]

where 𝑪𝑪(𝒅𝒅) is the damaged stiffness matrix given in Equation [2], which depends on the virgin (undamaged) stiffness components of prepreg tape and two damage variables, with d 1 and d 2 reflecting the state of damage in the fiber and transverse to the fiber directions, respectively. Damage variables effectively lump the material sub-scale heterogeneous degradation. The virgin stiffness components, C ij , are expressed in terms of elastic moduli of the platelet’s parent unidirectional continuous fiber tape. 𝑪𝑪(𝒅𝒅) = (1 − 𝑑𝑑1 )𝐶𝐶11 ⎡ 𝑠𝑠𝑠𝑠𝑠𝑠 ⎢ 𝑠𝑠𝑠𝑠𝑠𝑠 =⎢ 0 ⎢ ⎢ 0 ⎣ 0

(1 − 𝑑𝑑1 )(1 − 𝑑𝑑2 )𝐶𝐶12 (1 − 𝑑𝑑2 )𝐶𝐶22 𝑠𝑠𝑠𝑠𝑠𝑠 0 0 0

(1 − 𝑑𝑑1 )𝐶𝐶13 (1 − 𝑑𝑑2 )𝐶𝐶23 𝐶𝐶33 0 0 0

0 0 0 (1 − 𝑑𝑑1 )(1 − 𝑑𝑑2 )𝐶𝐶44 0 0

0 0 0 0 𝐶𝐶55 0

0 ⎤ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 𝐶𝐶66 ⎦

[2]

Given the complex state of stress/strain acting at a material point, the initial failure is predicted by applying the damage initiation criteria. The damage initiation functions f 1 and f 2 (for internal damage variables d 1 and d 2 , respectively) are given by Equation [3]. (𝜖𝜖11 )2 1 1 𝑓𝑓12 = 𝑓𝑓,𝑡𝑡 𝑓𝑓,𝑐𝑐 + � 𝑓𝑓,𝑡𝑡 − 𝑓𝑓,𝑐𝑐 � 𝜖𝜖11 = 1 𝜖𝜖11 𝜖𝜖11 𝜖𝜖11 𝜖𝜖11 [3] 2 2 (𝜖𝜖 ) 1 1 1 22 𝑓𝑓22 = 𝑓𝑓,𝑡𝑡 𝑓𝑓,𝑐𝑐 + � 𝑓𝑓,𝑡𝑡 − 𝑓𝑓,𝑐𝑐 � 𝜖𝜖22 + � 𝑓𝑓 � (𝜖𝜖12 )2 = 1 𝜖𝜖22 𝜖𝜖22 𝜖𝜖22 𝜖𝜖22 𝜖𝜖12 𝑓𝑓,𝑡𝑡

𝑓𝑓,𝑐𝑐

where 𝜖𝜖11 , 𝜖𝜖11 are the failure strains in fiber direction in tension and compression, respectively; 𝑓𝑓,𝑡𝑡 𝑓𝑓,𝑐𝑐 𝜖𝜖22 , 𝜖𝜖22 are the failure strains perpendicular to the fiber direction in tension and compression, 𝑓𝑓 respectively; 𝜖𝜖12 is the failure strain in shear. The failure strains are computed from the dividing the corresponding strengths over stiffness (assuming linear-brittle fracture). Damage variables evolve monotonically in the range 0 ≤ 𝑑𝑑𝑖𝑖 ≤ 1. The damage evolution laws are based on strains according to Equation [4], where 𝑑𝑑𝑖𝑖 = 0 corresponds to the initial undamaged state and 𝑑𝑑𝑖𝑖 = 1 is the state of complete loss of integrity. 𝑑𝑑i = 1 −

𝑓𝑓,𝑡𝑡

𝑓𝑓,𝑡𝑡

𝑓𝑓,𝑡𝑡 −𝐶𝐶ii 𝜖𝜖ii �𝑓𝑓i − 𝜖𝜖ii �𝐿𝐿𝑐𝑐 𝜖𝜖ii 𝑒𝑒𝑒𝑒𝑒𝑒 � � , 𝑖𝑖 = 1,2 𝑓𝑓i 𝐺𝐺i

[4]

Delamination of platelets is modeled with cohesive zone scheme. The interfacial (cohesive) element [16] is characterized by a constitutive equation called a traction-separation law, which ����), relates the interlaminar stress components, σ i3 , to the relative separations, δ i =Δu i (𝑖𝑖 = 1,3 between the top and bottom surfaces of the element:

(1 − 𝑑𝑑)𝑘𝑘10 𝜎𝜎13 �𝜎𝜎23 � = � 0 𝜎𝜎33 0 MPa

0 (1 − 𝑑𝑑)𝑘𝑘20 0

0 𝛿𝛿1 𝛿𝛿 � � 0 2� 0 (1 − 𝑑𝑑)𝑘𝑘3 𝛿𝛿3

[5]

In Eq.(5), 𝑘𝑘𝑖𝑖0 = 1(106 ) mm (𝑖𝑖 = 1 − 3) is the initial stiffness and d is the isotropic damage variable. A stress-based quadratic criterion is selected for the initiation of dis-bonding between platelets, Equation [6]: 2

〈𝜎𝜎33 〉 𝜎𝜎13 2 𝜎𝜎23 2 � � +� � +� � =1 𝑁𝑁𝑚𝑚𝑚𝑚𝑚𝑚 𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚

[6]

The numerical values of cohesive strengths used in the simulations were: N max =80MPa, S max =T max =100MPa. For the propagation of delamination, a power-law fracture mechanics-based criterion is used: 𝐺𝐺𝐼𝐼 𝐺𝐺𝐼𝐼𝐼𝐼 𝐺𝐺𝐼𝐼𝐼𝐼𝐼𝐼 + + =1 𝐺𝐺𝐼𝐼𝐼𝐼 𝐺𝐺𝐼𝐼𝐼𝐼𝐼𝐼 𝐺𝐺𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼

[9]

where G I , G II , and G III are the work done by the tractions and their corresponding displacements in the normal and shear directions, respectively. Quantities with subscript “C” denote the critical strain energy release rates corresponding to each fracture mode. The orthotropic platelets are assumed to be made of AS4/PEEK CF UD tape with deterministic elastic and failure properties given in [17], and deterministic fracture properties given in [18]. 2.4 Progressive Failure Analysis in a Virtual Tensile Coupon The simulated effective/macroscopic response of a tensile coupon is characterized with engineering strain, 𝜖𝜖̅11, and stress, 𝜎𝜎�11 , defined as the fraction of loading displacement (i.e. coupon elongation) to initial length and reaction force to the cross-sectional area of a coupon, respectively: 𝜖𝜖̅11

𝑢𝑢1∗ 𝐹𝐹1 = ; 𝜎𝜎�11 = 𝐿𝐿 𝑤𝑤𝑤𝑤

[10]

The typical simulated evolution of stress as a function of applied strain, 𝜎𝜎�11 (𝜖𝜖̅11 ), is shown in Figure 3. Prior to damage initiation, the effective stress-strain response is linear-elastic. After damage initiation, the platelet meso-structure starts developing the damage process zone (DPZ), which is an accumulation of growing and coalescing damaged local regions. Path of damage progress is represented by the distribution of internal state (damage) variables, which reflect the state of degradation of local mechanical properties (stiffness) responsible for local loss of stress carrying capacity and thus representing “smeared” local cracks. A coupon with platelet mesostructure is a complex and highly redundant structural system able to redistribute the applied load to undamaged elements when localized sub-scale failures develop, and to form multiple precursory damage sites and develop alternative paths of damage progression. With the incipience of the DPZ, the load-carrying capacity of the composite is reduced and the coupon stress-strain relationship exhibits non-linear behavior.

Figure 3 Results of a virtual uniaxial tensile test of a coupon with prepreg platelet stochastic morphology

Local damage mechanisms contribute gradually to a platelet degradation, while degradation of many platelets leads to the overall composite macroscopic failure. As the loading is increased, the damage expands more over the coupon volume. At the macroscopic scale, failure of a coupon is associated with its separation in two pieces by a macro-crack and is preceded by the formation of a band from local damage precursors where most of the damage activity accumulates. In Figure 3, the undamaged and damaged elements are identified by the light gray (d=0, no crack) and solid black (d=1, local crack) colors, respectively (d being the maximum damage index). The model is able to capture the variations of the strain field 𝜖𝜖11 (𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 ) resulting from the underlying morphology. The strain field is heterogeneous, with high local strains at locations with platelets less aligned with the loading direction (x 1 ). There are regions of high and low strains, having size and shape relevant to the local orientation state and global deformation of a coupon. The regions of high strain are the regions where damage develops in the first place. The model provides the description of interactions of stress concentrations which lead to complex damage pattern due to influence of irregular and complex internal characteristics of stochastic morphology. Mechanical coupling between different modes of deformation in a tensile coupon of stacked platelets is caused by non-symmetric distribution of geometrical and material properties through the coupon thickness over its entire volume. Variable local stiffness in a tensile coupon makes a position of the shear center and a non-coincident centroid vary between the cross sections along the coupon length. Thus, the uniform tension applied on a coupon results in its bending-twisting-extension coupled deformation. It is illustrated by Figure 3, where out-of-plane displacements are purposely exaggerated to show the coupled deformation of a coupon in tension. The distribution of damage index is also shown in the same figure on a deformed configuration in post-mortem. It seems that the major and most of the minor failure sites are located where the coupling between deformation

modes is the most prominent, close to the convex and concave surfaces of a coupon. Additionally, the damage tends to initiate near the free edges of a coupon close to those locations.

3. THEORETICAL STUDY OF PLATELET SIZE EFFECTS ON COMPOSITE STOCHASTIC TENSILE PROPERTIES Monte-Carlo simulation approach to statistical characterization of composite stochastic stiffness and strength is realized through the progressive failure analysis of virtual (digital) full-sized tensile coupons with modified meso-structure descriptors. The effects of material morphology and modified geometrical characteristics of a platelet on variability of composite stiffness and strength are considered. There are two mechanisms controlling the effective stiffness and strength in a composite system of many platelets with stochastic morphology. First, it is the platelet size (L p /t p -ratio), which determines the stress transfer efficiency between the neighboring platelets [19], and, therefore, governs the effective/macroscopic response of a tensile coupon. Second, it is the number of platelets in the composite system, which (i) correlates with the degree of structural redundancy and number of orientations in the composite system (thus influencing the variability of properties) and (ii) depends on the platelet size under the given volume of a coupon. In general, increased platelet L p /t p -ratio improves composite effective properties, while increased platelets count reduces the property variability between coupons. 3.1 Effect of Platelet Length on Composite Stochastic Tensile Stiffness and Strength The platelet length (L p ) is one of the basic parameters, which determine the composite mesostructure. The effective mechanical properties and several phenomena occurring in the composite system depend on it. Figure 4 summarizes the results of simulated effective tensile stiffness and strength of coupons with several platelet sizes. The virtual coupon length (L), width (w), and thickness (t) were 152.4mm, 25.4mm, 2.52mm, respectively. Ten virtual coupons for three platelet lengths, L p = 12.7, 25.4, and 50.8mm were considered. Platelets were 0.140 mm-thick, and platelet length-to-width ratio, L p /w p , was equal to four. Within each batch, macroscopic strength and modulus vary from one coupon to another due to the dissimilarities of their meso-scale stochastic morphology. With the platelet length increased (platelet thickness is constant), the average stiffness and strength also increase, but so does the variability. The improved effective properties are explained by enhanced stress sharing between longer platelets. The increased variability is explained by the interference of scales between a platelet and a coupon: longer platelet means a smaller number of platelets in a coupon, so morphologies of individual coupons of longer platelets are statistically “more different” from one another. Consider statistics of apparent global alignment of platelets in analyzed virtual coupons with different platelet length to support the above statement. Average coupon scale degree of platelets alignment with the loading direction, a 11 , between ten coupons with L p =12.7mm is 0.493 with cov. 1.73%, while average coupon scale a 11 between coupons with L p =50.8mm is 0.506 with cov. 5.50%. Next, the average number of platelets in coupons was 1285, 504, and 234 for platelet lengths of 12.7mm, 25.4mm, and 50.8mm, respectively. Platelets count is important since there is a limited number of platelets and orientations within a given volume. Fewer platelets in a stochastic system implies a fewer possible orientation states, which increases the probability of very different local orientation states to occur,

both within a coupon and between several different coupons. This is the implication of how a platelet size alters the orientation state/morphology of the entire system.

70

450 avg. 337 MPa

avg. 51.6 GPa cov. 7.6%

cov. 13.9%

400

60 avg. 43 GPa

350

cov. 6.3%

avg. 253 MPa

50

cov. 11.2%

300

avg. 37.8 GPa

avg. 217 MPa

cov. 4% 40

250

cov. 7.5%

200

30

150 20 100 10 50

0

0 12.7mm

25.4mm

Platelet length,

50.8mm

L

12.7mm

25.4mm

Platelet length, p

50.8mm

L p

Figure 4 Simulated platelet length effect on tensile properties of a coupons with stochastic morphology (platelet thickness, t p = 0.14mm)

QI CF

1

0.8 McGill experimental data [20] McGill simulation [20] Present investigation

0.6

0.4

0.2

0 12.7mm

25.4mm

Platelet length,

50.8mm

L

p

Figure 5 Platelet length effect on the strength of AS4/PEEK UD prepreg platelet based stochastic composite system: experimental and theoretical. Effective strength of prepreg platelet composite is normalized by the AS4/PEEK UD CF quasi-isotropic (QI) [00/±450/900] s laminate

Recently, a research group from McGill University (Montreal, Canada) carried out the experimental and theoretical study of the platelet length effect on platelet composite strength (platelets were obtained from UD CF AS4/PEEK). Figure 5 compares the theoretical results of platelet length effect on tensile strength versus the experimental data developed by McGill University group [20]. Data in Figure 5 is normalized to the theoretical tensile strength of a quasiisotropic CF laminate. A reasonable agreement between the theoretical and experimental trends in strength variability can be observed from the data in Figure 5. 3.2 Effect of Platelet Thickness on Composite Stochastic Tensile Stiffness and Strength Herein, we keep platelet length constant as L p =50.8mm (2in) and vary the platelet thickness (t p ). Improved effective stiffness and strength and reduced variability is seen for coupons with thinner platelets from data in Figure 6. This is due to the increased platelet (L p /t p )-ratio and increased platelets count in the coupons with platelet thickness decreased. Increased platelet aspect ratio improves stress transfer between platelets, and a greater number of platelets allows for the more uniform orientation state over the tensile coupon (reduced variability of local morphology) and, consequently, reduced variability in effective properties.

500

70 avg. 52.1 GPa 60

cov. 3%

450

avg. 51.8 GPa

avg. 404 MPa cov. 3.1%

cov. 7.1%

avg. 45.6 GPa

avg. 320 MPa cov. 10.6%

400

cov. 5.9% 50

350 avg. 215 MPa

300 40

cov. 19.3% 250

30 200 150

20

100 10 50 0

0

0.050mm

0.100mm

Platelet thickness,

0.250mm

0.050mm

0.100mm

Platelet thickness,

t p

0.250mm

t p

Figure 6 Simulated platelet thickness effect on tensile properties of a coupons with stochastic morphology

4. CONCLUSIONS Progressive failure analysis is herein used to study the structure-property relationship in a composite system with stochastic morphology made of pepreg platelets. Damaged deformation up to ultimate failure in virtual tensile coupons is analyzed with computational damage mechanics approaches (continuum and discrete), which are utilized for constitutive modelling and addressing complex interacting/competing damage mechanisms. The developed failure analysis allows for virtual characterization of how the composite meso-structure details, meaning the platelet geometry and system morphology (geometrical arrangement and orientation distribution of platelets), define the effective properties of a platelet-molded composite system, its stiffness, strength and variability in properties. The simulated results are shown to correlate with the experimental data previously reported by other researchers.

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