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Polyamides serve as matrix material for fiber reinforced composites and are widely applied in many different engineering applications. In this context, they are exposed to various environmental influences ranging from temperature to humidity. Thus, the influence of these environmental conditions on the mechanical behavior and the associated implications on the performance of the material is of utmost importance. In this work, the thermoviscoelastic behavior of polyamide 6 (PA 6) for two equilibrium moisture contents is investigated. To this end, dynamic mechanical analysis tests with and without humidity control of the environmental chamber were performed. In terms of relaxation tests, the experimental results reveal drying effects and increased diffusion activities when the sample's equilibrium moisture content differs from the ambient humidity level within the testing chamber. Temperature-frequency tests quantify the humidity-induced shift of the glass transition temperature. The linear generalized Maxwell model (GMM) and time-temperature superposition are used to analyze the hydrothermal effects on the linear viscoelastic material properties and the onset of mechanical nonlinearity. Based on these investigations and findings, insight is gained on the humidity influence on the material properties and the limitations of linear thermoviscoelastic modeling. Furthermore, the computational construction of master curves and the parameter identification for a generalized Maxwell model are described in detail. [ABSTRACT FROM AUTHOR]

Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation tensor is obtained via a closure approximation. Unfortunately, such a description limits the predictive capabilities of the modeling process significantly, because the wealth of possible fourth-order fiber-orientation tensors is not exploited by such closures, and the restriction to second-order fiber-orientation tensors implies artifacts. Closures based on the second-order fiber-orientation tensor face a fundamental problem – which fourth-order fiber-orientation tensors can be realized? In the literature, only necessary conditions for a fiber-orientation tensor to be connected to a fiber-orientation distribution are found. In this article, we show that the typically considered necessary conditions, positive semidefiniteness and a trace condition, are also sufficient for being a fourth-order fiber-orientation tensor in the physically relevant case of two and three spatial dimensions. Moreover, we show that these conditions are not sufficient in higher dimensions. The argument is based on convex duality and a celebrated theorem of D. Hilbert (1888) on the decomposability of positive and homogeneous polynomials of degree four. The result has numerous implications for modeling the flow and the resulting microstructures of fiber-reinforced composites, in particular for the effective elastic constants of such materials. Based on our findings, we show how to connect optimization problems on fourth-order fiber-orientation tensors to semi-definite programming. The proposed formulation permits to encode symmetries of the fiber-orientation tensor naturally. As an application, we look at the differences between orthotropic and general, i.e., triclinic, fiber-orientation tensors of fourth order in two and three spatial dimensions, revealing the severe limitations inherent to orthotropic closure approximations. [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]

Classical solution methods in fast Fourier transform‐based computational micromechanics operate on, either, compatible strain fields or equilibrated stress fields. By contrast, polarization schemes are primal‐dual methods whose iterates are neither compatible nor equilibrated. Recently, it was demonstrated that polarization schemes may outperform the classical methods. Unfortunately, their computational power critically depends on a judicious choice of numerical parameters. In this work, we investigate the extension of polarization methods by Anderson acceleration and demonstrate that this combination leads to robust and fast general‐purpose solvers for computational micromechanics. We discuss the (theoretically) optimum parameter choice for polarization methods, describe how Anderson acceleration fits into the picture, and exhibit the characteristics of the newly designed methods for problems of industrial scale and interest. [ABSTRACT FROM AUTHOR]

Beyond a certain loading threshold, glass-fiber-reinforced polymer composites with vinylester matrix can show considerably nonlinear deformation behavior due to damage of the matrix material. To model such nonlinear deformation properties, we follow the approach of a fully anisotropic damage model suggested by Govindjee et al. (Govindjee S, Kay GJ and Simo JC. Anisotropic modelling and numerical simulation of brittle damage in concrete. International Journal for Numerical Methods in Engineering 1995; 38(21): 3611–3633).In addition, the inter-fiber fracture criterion introduced by Puck (Puck A. Festigkeitsanalyse von Faser-Matrix-Laminaten: Modelle für die Praxis. München, Germany: Hanser Fachbuchverlag, 1996) is used for the damage function that defines the damage initiation and the material strength. A particular procedure is chosen to identify the material properties to validate the model: three glass-fiber-reinforced polymer laminates with different layups are tested in tension under different loading orientations, assuming laminate theory to be reasonably well fulfilled. Finally, further experiments are compared with corresponding simulation results to demonstrate the performance of the model. [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]

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]

The effective elastic material properties of short fiber reinforced polypropylen are determined by means of the self-consistent (SC) method and the interaction direct derivative (IDD) method. In order to account for thermoelastic effective material properties, a Hashin-Shtrikman (HS) based two-step homogenization method with variable reference stiffness is used. The influence of the reference stiffness, dependent on a scalar parameter is investigated. Information on the microstructure are derived by computed tomography scans (µCT) and considered within the homogenization schemes. Thermomechanical properties of a long fiber reinforced polymer (LFRP) and a short fiber reinforced polymer (SFRP) are obtained by means of dynamic mechanical analysis (DMA). Simulation results for SFRP are compared to experimental results. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Since the microstructure of short fiber reinforced composites is inhomogeneous, the application of micromechanical models is useful, that take into account their characteristics like the fiber orientation and the aspect ratio of fibers. Two different methods are considered in this work: A two-step approach is utilized to get approximately the upper and lower bounds of the elastic properties. Furthermore, an approximation for the elastic properties is calculated by the self-consistence method. Both methods use discretely microstructural information including the length, the diameter and the orientation of each single fiber. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Carbon/carbon composites produced by chemical vapor infiltration consist of carbon fibers embedded in a matrix of pyrolytic carbon with anisotropic mechanical properties. The matrix around fibers consists of cylindrically shaped pyrolytic carbon layers of coating, which may have different textures with different mechanical properties in the axial, radial and circumferential directions. The goal of the present numerical study is to investigate the influence of the coating microstructure on stress concentrations and possible modes of failure in the carbon composite. Numerical modeling was performed on two length scales. First, the material properties of the differently textured pyrolytic carbon layers were determined on the nanometer scale using methodology based on the Eshelby theory for continuously distributed inclusions. Then, the obtained material parameters for each layer were used as input for the finite element models on the micrometer scale. The numerical simulations were conducted for three basic loading scenarios: uniaxial tension, shear loading and hydrostatic compression. The calculated stress distributions show zones of maximum stress concentrations and provide information on the possible failure regions for each material under all considered loading cases. The numerical results demonstrate good correspondence with experimentally identified failure regions. Copyright © 2008 John Wiley & Sons, Ltd. [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]

We study fast and memory-efficient FFT-based implicit solution methods for small-strain phase-field crack problems for microstructured brittle materials. A fully implicit first order formulation of the problem coupling elasticity and damage permits using comparatively few, but large, time steps compared to semi-explicit schemes. We investigate memory-efficient FFT-based solution techniques, and identify the heavy ball scheme as particularly powerful. We discuss the memory-efficient implementation and present demonstrative numerical examples. • Memory-efficient FFT-based solution methods for implicit phase field damage. • A novel fast gradient method, staggered and memory-efficient solver. • Demonstration of advantages compared to semi-explicit approaches. • Application to fiber reinforced composite microstructures of large scale. [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]

Analytic shear-lag models enjoy great popularity for assessing and interpreting microstructure dependent stationary creep in fibrous metal composites, especially the formulation of Kelly-Street (Kelly and Street, 1972 [2]). Beyond the original model's scope, i.e. large aspect ratios of inclusions, predictions are highly inaccurate, which was recently pointed out by Wicht et al. (Wicht et al., 2022 [10]), by comparing model predictions to micromechanical Fast-Fourier-Transform-based simulations. In this study we therefore modify basic Kelly-Street model assumptions, concerning effective creep rate, stress transfer and inclusion spacing to arrive at a modified model with an extended scope. To validate the modified model, we benchmark the model against Fast-Fourier-Transform-based micromechanical simulations. We show, that the proposed modifications are successful in extending the model's scope to inclusions with small aspect ratios ≤20. Thus, the reformulated model is a powerful tool to describe and interpret microstructure dependent creep. [ABSTRACT FROM AUTHOR]

The main contribution of this work lies in the detailed comparison of the predictions of linear elastic properties of mean field homogenization approaches and full field, voxel-based homogenization methods for short-fiber reinforced materials. In the former case, the self-consistent, the interaction direct derivative and a two-step-bounding approach, applying the Hashin–Shtrikman bounds, are used. In the latter case, the boundary value problem for representative volume elements is solved using fast Fourier transformation. Model microstructures with unidirectional aligned and two misaligned fiber configurations are considered exemplarily. Fiber volume fractions of 13%, 17% and 21% and phase contrasts of 44, 100 and 1000 in the elastic moduli have been taken into account. The different homogenization schemes are compared by means of effective directional dependent Young’s modulus. This detailed comparison shows that mean field and full field solutions deliver similar results for moderate phase contrasts and volume fractions. Especially in the range of realistic phase contrasts like 44 for a composite of polypropylene and glass, the mean field approaches pose reliable alternatives for full field solution. Large phase contrasts result in relative deviations of up to 68%. [ABSTRACT FROM AUTHOR]

This paper revisits the topic of mean field homogenization for short fiber reinforced composite materials. A short glass fiber reinforced thermoplastic polyamide 6.6 with a fiber mass fraction of 35 % is used as an example material for which microstructural analyses and experimental tests were performed. We cover a set of common models: the Self-Consistent (SC), Mori–Tanaka (MT), Ponte Castañeda-Willis (PCW), Interaction Direct Derivative (IDD) and Two-Step (TS) schemes. We first recast them into a unified formulation that permits a thorough theoretical comparison, including the topics of loss of major symmetry as well as connections between the models. We are able to show that the MT, PCW and IDD schemes can be expressed in a surprisingly similar form. This extends to the equations for prediction of effective stiffness and compliance tensors as well as the strain and stress localization tensors. Secondly, we address the resolution of the material microstructure within mean field homogenization schemes, comparing classical and more recent and efficient methods. Last, we benchmark the different mean field schemes and modeling approaches, including comparisons to numerical homogenization by Fast Fourier Transformation (FFT) and experimental data from tensile tests. These show that the MT and TS schemes are both capable of accurately modeling the composite material for complex orientations and homogeneous fiber phases. For possibly inhomogeneous fiber phases only the TS scheme yields both accurate and physically reasonable results. Other models such as the IDD or PCW are of great theoretical importance, but cannot be generally applied for the given material class. [ABSTRACT FROM AUTHOR]