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Fiber orientation tensors (FOT) are widely used to approximate statistical orientation distributions of fibers within fiber-reinforced polymers. The design process of components made of such fiber-reinforced composites is usually accompanied by a virtual process chain. In this virtual process chain, process-induced FOT are computed in a flow simulation and transferred to the structural simulation. Within the structural simulation, effective macroscopic properties are identified based on the averaged information contained in the FOT. Solving the field equations in flow simulations as well as homogenization of effective stiffnesses necessitates the application of a closure scheme, computing higher-order statistical moments based on assumptions. Additionally, non-congruent spatial discretizations require an intermediate mapping operation. This mapping operation is required, if the discretization, i.e., mesh, of the flow simulation differs from the discretization of the structural simulation. The main objective of this work is to give an answer to the question: Does the sequence of closure and mapping influence the achieved results? It will turn out, that the order influences the result, raising the consecutive question: Which order is beneficial? Both questions are addressed by deriving a quantification of the closure-related uncertainty. The two possible sequences, mapping followed by closure and closure followed by mapping, yield strongly different results, with the magnitude of the deviation even exceeding the magnitude of a reference result. Graphical consideration reveals that for both transversely isotropic and planar FOT-input, invalid results occur if the mapping takes place prior to closure. This issue is retrieved by orientation averaging stiffness tensors. As a by-product, we explicitly define for the first time the admissible parameter space of orthotropic fourth-order fiber orientation tensors and define a distance measure in this parameter space. [ABSTRACT FROM AUTHOR]

The anisotropic elastic properties of injection molded composites are fundamentally coupled to the flow of the fiber suspension during mold-filling. Regarding the modeling of mold-filling processes, both a decoupled and a flow–fiber coupled approach are possible. In the latter, the fiber-induced viscous anisotropy is considered in the computation of the flow field. This in turn influences the evolution of the fiber orientation compared to the decoupled case. This study investigates how flow–fiber coupling in mold-filling simulation affects the stress field in the solid composite under load based on the final elastic properties after fluid–solid transition. Furthermore, the effects of Newtonian and non-Newtonian polymer matrix behavior are investigated and compared. The entire process is modeled micromechanically unified based on mean-field homogenization, both for the fiber suspension and for the solid composite. Different numerical stabilization methods of the mold-filling simulation are discussed in detail. Short glass fibers with a typical aspect ratio of 20 and a volume fraction of 20% are considered, embedded in polypropylene matrix material. The results show that the flow–fiber coupling has a large effect on the fiber orientation tensor in the range of over ± 30% with respect to the decoupled simulation. As a consequence, the flow–fiber coupling affects the stress field in the solid composite under load in the range of over ± 10%. In addition, the predictions based on a non-Newtonian modeling of the matrix fluid differ significantly from the Newtonian setup and thus the necessity to consider the shear-thinning behavior is justified in a quantifiable manner. [ABSTRACT FROM AUTHOR]

Fiber orientation tensors represent averaged measures of fiber orientations inside a microstructure. Although, orientation-dependent material models are commonly used to describe the mechanical properties of representative microstructure, the influence of changing or differing microstructure on the material response is rarely investigated systematically for directional measures which are more precise than second-order fiber orientation tensors. For the special case of planar orientation distributions, a set of admissible fiber orientation tensors of fourth-order is known. Fiber orientation distributions reconstructed from given orientation tensors are of interest both for numerical averaging schemes in material models and visualization of the directional information itself. Focusing on the special case of planar orientations, this paper draws the geometric picture of fiber orientation distribution functions reconstructed from fourth-order fiber orientation tensors. The developed methodology can be adopted to study the dependence of material models on planar fourth-order fiber orientation tensors. Within the set of admissible fiber orientation tensors, a subset of distinct tensors is identified. Advantages and disadvantages of the description of planar orientation states in two- or three-dimensional tensor frameworks are demonstrated. Reconstruction of fiber orientation distributions is performed by truncated Fourier series and additionally by deploying a maximum entropy method. The combination of the set of admissible and distinct fiber orientation tensors and reconstruction methods leads to the variety of reconstructed fiber orientation distributions. This variety is visualized by arrangements of polar plots within the parameter space of fiber orientation tensors. This visualization shows the influence of averaged orientation measures on reconstructed orientation distributions and can be used to study any simulation method or quantity which is defined as a function of planar fourth-order fiber orientation tensors. [ABSTRACT FROM AUTHOR]

In this contribution, selected results from [1–3] are presented in a compact and simplified way. In addition, the variety of fiber orientation tensors is used to determine a maximum deviation of the direction‐dependent Young's modulus, which can arise if only second‐order directional information is included in a specific meanfield homogenization. Focusing on the special case of planar fiber distributions, the variety of fiber orientation tensors identified in [1] is considered as a design space. This design space is completely explored for the orientation‐averaging homogenization following [4], fixed material parameters and fixed fiber volume content. The possible directional dependence of the resulting effective stiffnesses is graphically presented using polar plots of the direction‐dependent Young's modulus. These polar plots are arranged on two‐dimensional slices within the parameter space of planar fourth‐order fiber orientation tensors. This gives a complete representation of the influence of the orientation tensor on the anisotropic stiffness tensor. Consequences of closure approximations, i.e., restriction to second‐order directional information, are demonstrated and motivate measurement of fourth‐order fiber orientation tensors. [ABSTRACT FROM AUTHOR]

Mean-field homogenization is an established and computationally efficient method estimating the effective linear elastic behavior of composites. In view of short-fiber reinforced materials, it is important to homogenize consistently during process simulation. This paper aims to comprehensively reflect and expand the basics of mean-field homogenization of anisotropic linear viscous properties and to show the parallelism to the anisotropic linear elastic properties. In particular, the Hill–Mandel condition, which is generally independent of a specific material behavior, is revisited in the context of boundary conditions for viscous suspensions. This study is limited to isothermal conditions, linear viscous and incompressible fiber suspensions and to linear elastic solid composites, both of which consisting of isotropic phases with phase-wise constant properties. In the context of homogenization of viscous properties, the fibers are considered as rigid bodies. Based on a chosen fiber orientation state, different mean-field models are compared with each other, all of which are formulated with respect to orientation averaging. Within a consistent mean-field modeling for both fluid suspensions and solid composites, all considered methods can be recommended to be applied for fiber volume fractions up to 10%. With respect to larger, industrial-relevant, fiber volume fractions up to 20%, the (two-step) Mori–Tanaka model and the lower Hashin–Shtrikman bound are well suited. [ABSTRACT FROM AUTHOR]

Fiber orientation tensors are established descriptors of fiber orientation states in (thermo-)mechanical material models for fiber-reinforced composites. In this paper, the variety of fourth-order orientation tensors is analyzed and specified by parameterizations and admissible parameter ranges. The combination of parameterizations and admissible parameter ranges allows for studies on the mechanical response of different fiber architectures. Linear invariant decomposition with focus on index symmetry leads to a novel compact hierarchical parameterization, which highlights the central role of the isotropic state. Deviation from the isotropic state is given by a triclinic harmonic tensor with simplified structure in the orientation coordinate system, which is spanned by the second-order orientation tensor. Material symmetries reduce the number of independent parameters. The requirement of positive-semi-definiteness defines admissible ranges of independent parameters. Admissible parameter ranges for transversely isotropic and planar cases are given in a compact closed form and the orthotropic variety is visualized and discussed in detail. Sets of discrete unit vectors, leading to selected orientation states, are given. [ABSTRACT FROM AUTHOR]

Anisotropic fiber-reinforced composites are used in lightweight construction, which is of great industrial relevance. During mold filling of fiber suspensions, the microstructural evolution of the local fiber arrangement and orientation distribution is determined by the local velocity gradient. Based on the Folgar–Tucker equation, which describes the evolution of the second-order fiber orientation tensor in terms of the velocity gradient, the present study addresses selected states of deformation rates that can locally occur in complex flow fields. For such homogeneous flows, exact solutions for the asymptotic fiber orientation states are derived and discussed based on the quadratic closure. In contrast to the existing literature, the derived exact solutions take into account the fiber-fiber interaction. The analysis of the asymptotic solutions relying upon the common quadratic closure shows disadvantages with respect to the predicted material symmetry, namely, the anisotropy is overestimated for strong fiber-fiber interaction. This motivates us to suggest a novel normalized fully symmetric quadratic closure. Two versions of this new closure are derived regarding the prediction of anisotropic properties and the fiber orientation evolution. The fiber orientation states determined with the new closure approach show an improved prediction of anisotropy in both effective viscous and elastic composite behaviors. In addition, the symmetrized quadratic closure has a simple structure that reduces the effort in numerical implementation compared to more elaborated closure schemes. [ABSTRACT FROM AUTHOR]

The present work discusses the impact of the back coupling of the fiber orientation distribution on the base flow and on the fiber orientation itself during mold filling simulations. Flows through a channel and over a backward-facing step are investigated. Different closure approximations are considered for modeling the flow-induced evolution of anisotropy. The results corresponding to the decoupled approach, in which the effect of fibers on local fluid properties is neglected, build the basis of comparison. The modeling is limited to a laminar, incompressible, and isothermal flow of a fiber suspension consisting of rigid short fibers suspended in an isotropic Newtonian matrix fluid. A linear, anisotropic constitutive law is used in combination with a uniform fiber volume fraction of 10% and an aspect ratio of 10. To evaluate the impact of back coupling and of different closure methods in view of the manufactured solid composite the resulting anisotropic elastic properties are investigated based on the Mori–Tanaka method combined with an orientation average scheme. Relative to the range [0, 1] the pointwise difference in fiber orientation between the decoupled and the coupled approach is found to be ± 5 % in the channel and ± 30 % in the backward-facing step, respectively. The viscosity and the elasticity tensor show both significant flow-induced anisotropies as well as a strong dependence on closure and coupling. [ABSTRACT FROM AUTHOR]

Fiber orientation tensors are widely used to efficiently describe the fiber orientation evolution of anisotropic fiber suspensions during mold-filling. Real viscous fiber suspensions show fiber-induced anisotropic behavior. It is an open question how to correctly account for the anisotropic microstructure of fiber suspensions in general in the evolution equation of the fiber orientation tensors. In this study, a generalized evolution equation for the fiber orientation tensor of arbitrary even order is formulated, which takes into account the microstructural anisotropy of the fiber suspension. This formulation is based on a linear homogenization approach which allows the application of arbitrary mean-field models. The derived interaction term, representing the anisotropic environment of a single fiber, is discussed as a micromechanical convergence criterion of the underlying integral operator. It is shown and discussed in detail how the special cases of the Jeffery equation and the Folgar–Tucker equation follow from this general formulation. In addition, the generalized evolution equation is specified for selected mean-field models. In this context, an equation is presented to describe the evolution of the fiber orientation tensor depending on the spatial distribution of the fibers. For simple shear flow, all models describing the orientation dynamics are numerically investigated and compared. The results for the second-order orientation tensor as well as for a single fiber show that the well-known periodic behavior is present and strongly depends on the volume fraction and the chosen mean-field model. Considering the spatial distribution of the fibers has a significant effect on the orientation evolution and strongly affects the periodic reorientation. [Display omitted] • The evolution equation of orientation tensors of arbitrary even order is derived. • Fiber-induced anisotropic viscosity effects are covered in the orientation evolution. • Anisotropic effects are shown to be a micromechanical convergence criterion. • The chosen mean-field model greatly influences the fiber orientation dynamics. • The spatial fiber distribution strongly affects the periodic fiber reorientation. [ABSTRACT FROM AUTHOR]

Background: The commonly used flow models for Fiber Reinforced Polymers (FRPs) often neglect the flow-induced anisotropy of the suspension, but with increasing fiber volume fraction, this plays an important role. There exist already some models which count on this effect. They are, however, phenomenological and need a fitted model parameter. In this paper, a micromechanically-based constitutive law is proposed which considers the flow-induced anisotropic viscosity of the fiber suspension. Methods: The introduced viscosity tensor can handle arbitrary anisotropy of the fluid-fiber suspension which depends on the actual fiber orientation distribution. Assuming incompressible material behaviour, a homogenization method for unidirectional structures in contribution with orientation averaging is used to determine the effective viscosity tensor. The motion of rigid ellipsoidal fibers induced by the flow of the matrix material is described based on Jeffery's equation. The reorientation of the fibers is modeled in two ways: by describing them with fiber orientation vectors, and by fiber orientation tensors. A numerical implementation of the introduced model is applied to representative flow modes. Results: The predicted effective stress values depending on the actual fiber orientation distribution through the anisotropic viscosity are analyzed in transient and stationary flow cases. In the case of the assumed incompressibility, they show similar effective viscous material behaviour as the results obtained by the use of the Dinh-Armstrong constitutive law. Conclusions: The introduced model is a possible way to describe the flow-induced anisotropic viscosity of a fluid-fiber suspension based on the mean field theory. [ABSTRACT FROM AUTHOR]

The main objective of this work is to give an answer to the question: is it sufficient to consider only the second-order fiber orientation tensor as microstructure variable describing the orientation distribution of short-fiber reinforced composites (SFRCs) in the prediction of effective elastic properties? This question is addressed in the context of SFRCs on the one hand with an overall transversal symmetric orientation distribution of fibers and, hence, effective transversally isotropic properties, and on the other hand with experimentally determined microstructure data using micro-computed tomography. Applying the maximum entropy principle, it is shown, how the fiber orientation distribution function (FODF) can be estimated by relying on the second and/or the fourth-order orientation tensor, only. Both estimates are used within the self-consistent and the interaction direct derivate approach to calculate the effective linear elastic properties. It is shown, that the predicted stiffness tensors significantly depend on the estimation of the FODF. Relative deviations of up to 20% in terms of stiffnesses and up to 46% in terms of Young's modulus are observed. For the experimentally determined microstructure, small deviations of up to 4.3% are found. [ABSTRACT FROM AUTHOR]

A comprehensive study on the influence of planar fourth-order fiber orientation tensors on effective linear elastic stiffnesses predicted by orientation averaging mean field homogenization is given. Fiber orientation states of sheet molding compound (SMC) are identified to be in most cases approximately planar. In the planar case, all possible fourth-order fiber orientation tensors are given by a minimal invariant set of structurally differing planar fourth-order fiber orientation tensors. This set defines a three-dimensional body and forms the basis for a comprehensive study on the influence of a fiber orientation distribution in terms of a fourth-order tensor on homogenized stiffnesses. The methodology of this study is the main contribution of this work and can be adopted to analyze the orientation dependence of any quantity which is a function of a planar fourth-order fiber orientation tensor. At specific points inside the set of planar fiber orientation tensors, effective stiffnesses are calculated with selected mean field homogenization schemes. These schemes are based on orientation averaging of transversely isotropic elasticity tensors following Advani and Tucker (1987), which is explicitly recast as linear invariant composition in the fiber orientation tensors of second and fourth order of Kanatani third kind. A maximum entropy reconstruction of a fiber orientation distribution function based on leading fiber orientation tensors, enables a new numerical formulation of the Advani and Tucker average for the special planar case. Polar plots of Young's modulus and generalized bulk modulus obtained by selected homogenization schemes are arranged on two-dimensional slices within the body of admissible fiber orientation tensors, visualizing the influence of the orientation tensor on the stiffness tensor. The orientation-dependence of the generalized bulk modulus differs significantly between selected homogenizations. Restrictions on the effective anisotropic material response caused by orthotropy of closure approximations are discussed. [Display omitted] • Fiber orientation distributions of sheet molding compound are approximately planar. • Orientation averaging mean field: Complete orientation dependence for the planar case. • Explicit linear representation of the Advani–Tucker orientation average. • New numeric formulation of the Advani–Tucker orientation average for the planar case. • Orientation averaging in stiffness or compliance domain affects generalized bulk modulus. [ABSTRACT FROM AUTHOR]

This contribution investigates the thermodynamic consistency of a meanfield damage model published by the authors [1]. The model is based on an empirical description of the fiber orientation distribution function (FODF). Due to a combination of matrix damage and fiber‐matrix interface debonding, the model is able to predict the inhomogeneous and anisotropic evolution of the overall behavior of the considered Sheet Molding Compound (SMC) composite. [ABSTRACT FROM AUTHOR]

The commonly used flow models for fiber reinforced polymers often neglect the flow induced mechanical anisotropy of the suspension. With an increasing fiber volume fraction, this plays, however, an important role. There are some models which count on this effect, they are, however, phenomenological and require a fitted model parameter. In this paper, a micromechanically based constitutive law is proposed which considers the flow induced anisotropic viscosity of the fiber suspension. The introduced viscosity tensor can handle arbitrary anisotropy of the fluid-fiber mixture depending on the actual fiber orientation distribution. A homogenization method for unidirectional structures in contribution with orientation averaging is used to determine the effective viscosity tensor. The motion of rigid ellipsoidal fibers induced by the flow of the matrix material is described by Jeffery's equation. A numerical implementation of the introduced model is applied to representative flow modes. The calculated stress values are analyzed in transient and stationary flow cases. They show a less pronounced anisotropic viscous behaviour in every investigated case compared to the results obtained by the use of the Dinh-Armstrong constitutive law. The reason for the qualitative difference is that the presented model depends on the complete orientation information, while the other one is linear in the fourth-order fiber orientation tensor. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

A novel closure approximation method for fiber orientation tensors is proposed namely the fully symmetric implicit closure. Besides the full index symmetry, implicitly formulated closures based on the contraction condition fulfill the trace condition and the trace-preserving property of the Folgar–Tucker equation. As a first example, the fully symmetric implicit quadratic closure is considered as a simple modification of a recently proposed symmetric quadratic closure. It is shown that this closure can be realized by a fiber orientation distribution function. Secondly, the fully symmetric implicit hybrid closure is proposed as a counter-example of a closure not being based on a orientation distribution function. Both closures are compared against classical approximations in view of orientation evolution in a simple shear flow. Furthermore, the capability of predicting the effective viscous and elastic behavior of fiber suspensions and solid composites is investigated for a given fiber orientation state. The results show that the proposed implicit closures can be used to approximate the maximum entropy closure. Thereby, both the quadratic and the hybrid approach alleviate the high computational burden of the maximum entropy closure, as their evaluation requires solving a one-dimensional problem only. In addition, the predicted effective behavior based on the implicit closures shows an overall good agreement with predictions based on measured orientation data. • A novel implicit closure approach for fiber orientation tensors is proposed. • By considering full index symmetry, all necessary algebraic properties are fulfilled. • The considered closures can be reduced to an efficiently solvable 1D formulation. • The closures show a reliable prediction of effective viscous and elastic properties. • The closures considered are efficient maximum entropy approximations. [ABSTRACT FROM AUTHOR]

In this work, we propose a fully coupled multiscale strategy for components made from short fiber reinforced composites, where each Gauss point of the macroscopic finite element model is equipped with a deep material network (DMN) which covers the different fiber orientation states varying within the component. These DMNs need to be identified by linear elastic precomputations on representative volume elements, and serve as high-fidelity surrogates for full-field simulations on microstructures with inelastic constituents. We discuss how to extend direct DMNs to account for varying fiber orientation, and propose a simplified sampling strategy which significantly speeds up the training process. To enable concurrent multiscale simulations, evaluating the DMNs efficiently is crucial. We discuss dedicated techniques for exploiting sparsity and high-performance linear algebra modules, and demonstrate the power of the proposed approach on an injection molded quadcopter frame as a benchmark component. Indeed, the DMN is capable of accelerating two-scale simulations significantly, providing possible speed-ups of several magnitudes. [ABSTRACT FROM AUTHOR]

Understanding the effective viscosity of fiber-filled polymer melts is essential for predicting the local fiber orientation of injection molded short-fiber reinforced components. To circumvent the intrinsic difficulties of experimentally determining the strongly anisotropic viscosity of such particle-reinforced melts, an FFT-based computational homogenization method for computing the effective quasi-static viscosity of a particle-suspended Newton fluid is introduced, based on a dual formulation. The dependence of the effective viscosity on microstructural parameters, like fiber volume fraction, fiber aspect ratio and fiber orientation are investigated by computational experiments on representative volume elements, and the results are compared to an analytical model for the effective viscosity. Based on the computational findings, a closed-form approximation for the effective viscosity is provided. [ABSTRACT FROM AUTHOR]

In this work, we investigate the fiber orientation dependent viscosity of fiber suspensions using a computational homogenization method. To improve computational prediction capabilities for the viscosity of fiber suspensions, we extend an existing, Fast Fourier Transform based computational approach for fiber suspensions with linear viscous, i.e., Newtonian, matrix behavior to nonlinear viscous matrix behavior. Specifically, a Cross-type shear-thinning rheology is assumed for the matrix fluid. We employ composite voxels to lower resolution requirements and find through resolution studies that the resolution error decreases for certain shear rates. Furthermore, we conduct a volume element study and find that the representative volume element sizes for engineering considerations in a specific Newtonian case and the investigated Cross-type case are highly similar. For shear rates of engineering process interest we visualize the effective suspension viscosity in three dimensions and study the effects of the fiber volume fraction and the imposed macroscopic shear rate tensor on the suspension viscosity. We find that the elongational viscosity and the degree of anisotropy of the suspension viscosity vary stronger with the shear rate for higher fiber volume fractions. In a comparison with an analytical mean-field model for the suspension viscosity, the deviations between the computational and analytical results turn out to be substantial. • Complete 3D visualization of the shear rate dependent suspension viscosity • Degree of anisotropy of the suspension viscosity varies with the shear rate • Shear rate influence on the suspension viscosity grows with the fiber volume fraction [ABSTRACT FROM AUTHOR]

We present an algorithm for generating short fiber-reinforced microstructures with almost planar fiber orientation. The orientation corrected shaking (OCS) method achieves a high accuracy regarding the volume fraction, fiber length distribution and fiber orientation state. Additionally, the algorithm is capable of generating microstructures for industrial materials, e.g., for a PA66GF35 material with a volume fraction of 19.3% and an aspect ratio of 33. For typical manufacturing processes, short fiber-reinforced composites show a mainly planar fiber arrangement. Therefore, we extend the two-step shaking algorithm of Li et al. [J. Ind. Text. 51(1), pp. 506–530, 2022] for a user-selected rectangular size of the unit cell and periodic boundary conditions. Additionally, the hidden closure structure of the algorithm is uncovered and a precise realization of the fiber orientation state achieved. We examine the representative volume element size for the OCS method, observing representative errors below 2% even for unit cells with edge lengths smaller than the mean fiber length. Additionally, the influence of different closure approximations on the stiffness is investigated. When applied to an industrial PA66GF35 material with sandwich structure, the OCS method demonstrates differences below 2% and 9% for the computed directional Young's moduli E 1 and E 2 compared to experimental data. [ABSTRACT FROM AUTHOR]

This paper presents a micromechanical high cycle fatigue damage model for short fiber reinforced composite materials. Because the damage processes within such materials are influenced strongly by their microstructure, we use a mean field homogenization framework to compute the macroscopic behavior of the composite as well as the microscopic stresses and strains in the constituent phases. This allows us to account for damage phenomena related to both the fiber phase as well as in the matrix material. Fiber and fiber-interface damage is modeled using a Tsai–Wu and Weibull-based approach and the progressive damage due to cyclic loading is described by a progressive matrix damage model. The latter includes a novel coupling term in which the matrix damage progression is linked to the damage state of the reinforcing fibers. A cycle-based numerical formulation is used to overcome the computational limits of such load cases in the time domain. While the approaches are in principle applicable to different types of fiber–matrix composite, a short glass fiber reinforced polyamide 6.6 is used as an example material, for which microstructural analyses as well as tensile and fatigue tests are reported. The model's capabilities with regard to complex fiber orientations and different fiber fractions are studied using different grades of this material class. Furthermore, the model is analyzed via benchmarks of the numerical schemes and by parameter sensitivity studies. The results show that the approach is capable of modeling the complex and microstructure-dependent fatigue damage and fatigue limits for the different material grades with limitations only becoming visible at very high fiber fractions. [ABSTRACT FROM AUTHOR]

A virtual process chain for sheet molding compound (SMC) composites is established and validated by means of experimental investigations on a demonstrator structure. The flow in the compression molding step is simulated via a Coupled-Eulerian-Lagrangian approach using an anisotropic non-Newtonian fluid flow model. Evolution of the fiber orientation distribution (FOD) is described by Jeffery's equation. The predicted FOD is mapped to structural simulations employing a neutral data format. A mean-field anisotropic damage model is used to predict the damage evolution in the demonstrator. Simulated FOD at the end of the compression molding is validated by computer tomography. Structural simulations are validated by means of a cyclic four-point bending test on the demonstrator. The predicted results show increased accuracy with the experiments by transferring FOD data within the virtual process chain. Critical points of high damage concentrations leading to failure agree with the experimental observations. [ABSTRACT FROM AUTHOR]

The mechanical properties of long fiber reinforced thermoplastics (LFT), which highly depend on the fiber orientation induced through manufacturing on a direct LFT line, are predicted for compression molded rectangular plates. Therefore, three two-scale structural mechanical simulation schemes are applied and discussed: a two-step approach, the Mori-Tanaka scheme and the self-consistent method. Fiber orientation tensors based on measured micro computed tomography data of selected samples as well as on filling simulations are used for the determination of mechanical properties, as e.g. the storage modulus. The results have been compared with dynamic mechanical analysis measurements of tensile specimens. The influence of the initial strand position on the effective mechanical properties of the plate and the variation of those are examined. [ABSTRACT FROM AUTHOR]

• Sheet molding compound composites exhibit an inherent three-scale structure. • Exact closure approximation allows for accurate microstructure reconstruction. • Quasi-random orientation sampling improves accuracy of generated microstructures. • Fast and robust generation of unit cells based on random sequential addition. • Sensitivity analyses of effective composite stiffness w.r.t. material parameters. Sheet molding compound (SMC) composites combine high lightweight potential with excellent formability and are frequently used in industrial applications. To reduce safety factors in dimensioning SMC parts, the influence of processing parameters and stochastic variation of microstructural and physical properties needs to be quantified accurately. Taking into account the inherent three-scale structure of SMC, we improve the microstructure generator of Chen et al. [Compos. Struct. 188, pp. 25–38, 2018] in various respects. Firstly, we consistently rely upon state-of-the-art closure approximations for the fourth order fiber orientation tensor. More precisely, we show that for a planar fiber orientation state, there is an explicit formula for the fast exact closure approximation of Montgomery-Smith et al. [J. Fluid Mech. 680, pp. 321–335, 2011]. Secondly, we exploit the use of quasi-random numbers in sampling the fiber orientation distribution, leading to dramatic improvements in accuracy compared to pseudo-random Monte Carlo sampling. Last but not least, we rely upon fast Fourier transform based methods for rapid computational homogenization. With these methodological improvements at hand, we thoroughly investigate the influence of the mechanical and microstructural parameters on the effective elastic properties of SMC composites, and compare the results to direct numerical simulations on large scale digital volume images and mean-field estimates. [ABSTRACT FROM AUTHOR]

As a result of the injection molding process, short glass fiber reinforced thermoplastics (SFRT) exhibit complex microstructures with fibers of different length and orientation as well as spatially varying fiber volume fractions. Their mechanical and functional performance can therefore only be predicted and ensured based on precise knowledge of said microstructure. To that end, x-ray micro-computed tomography (μCT) is commonly employed, with analysis software most-often yielding the fiber orientation tensors but lacking more detailed results. In order to increase the quality of the microstructural data, more accurate analysis techniques are necessary that include single fiber information for detailed correlations of microstructural parameters. In this work, a novel algorithm is suggested that relies on an iterative single fiber segmentation and merging procedure to obtain the fiber characteristics: orientation, location, radius and length. The algorithm is implemented in an efficient manner in Python, making heavy use of parallelization techniques. It is then applied to μCT scans and artificially generated 3D data of short glass fiber reinforced polyamide 6.6 with fiber mass fractions of 35 %. The algorithm has been validated using reference data, commercial software and experimental fiber length data from an incinerated specimen and was shown to be both robust and accurate. [ABSTRACT FROM AUTHOR]