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![Dimensionless effective shear viscosity of a fiber suspension in the e1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{e}_1$$\end{document}-e2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{e}_2$$\end{document}-plane: Results regarding classical mean-field models shown in (a) and results regarding PCW shown in (b) (orientation state and settings given in Sect. 7.1)](publication/363760607/figure/fig3/AS:11431281094872080@1667616809948/Dimensionless-effective-shear-viscosity-of-a-fiber-suspension-in-the.png)
Dimensionless effective shear viscosity of a fiber suspension in the e1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{e}_1$$\end{document}-e2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{e}_2$$\end{document}-plane: Results regarding classical mean-field models shown in (a) and results regarding PCW shown in (b) (orientation state and settings given in Sect. 7.1)
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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 homogenizat...
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... For this purpose, analytical and computational homogenization methods are valuable tools to provide viscosity estimates for molding simulations. However, significant challenges arise for the holistic analytical modelling of the suspension viscosity, because the suspension viscosity depends on the local microstructure [13], the fiber volume fraction [14], and the fiber orientation state [15]. Additionally, the flow field [16,17] and the melt temperature [18] influence the suspension viscosity as well. ...
We extend the laminate based framework of direct Deep Material Networks (DMNs) to treat suspensions of rigid fibers in a non-Newtonian solvent. To do so, we derive two-phase homogenization blocks that are capable of treating incompressible fluid phases and infinite material contrast. In particular, we leverage existing results for linear elastic laminates to identify closed form expressions for the linear homogenization functions of two-phase layered emulsions. To treat infinite material contrast, we rely on the repeated layering of two-phase layered emulsions in the form of coated layered materials. We derive necessary and sufficient conditions which ensure that the effective properties of coated layered materials with incompressible phases are non-singular, even if one of the phases is rigid. With the derived homogenization blocks and non-singularity conditions at hand, we present a novel DMN architecture, which we name the Flexible DMN (FDMN) architecture. We build and train FDMNs to predict the effective stress response of shear-thinning fiber suspensions with a Cross-type matrix material. For 31 fiber orientation states, six load cases, and over a wide range of shear rates relevant to engineering processes, the FDMNs achieve validation errors below 4.31% when compared to direct numerical simulations with Fast-Fourier-Transform based computational techniques. Compared to a conventional machine learning approach introduced previously by the consortium of authors, FDMNs offer better accuracy at an increased computational cost for the considered material and flow scenarios.
... However, finding analytical models for the suspension viscosity is a challenging task, partly because of the locally inhomogeneous flow field inside the suspension, as well as the hydrodynamic interactions [12] and mechanical contacts [13] between the fibers. Furthermore, the suspension viscosity also depends strongly on the local microstructure, i.e., the fiber geometry [14], the fiber volume fraction [15], and the fiber orientation state [16]. As far as external influences are concerned, the loading direction, the shear rate [17,18], and the melt temperature [19] also affect the suspension viscosity. ...
In this article, we combine a Fast Fourier Transform based computational approach and a supervised machine learning strategy to discover models for the anisotropic effective viscosity of shear-thinning fiber suspensions. Using the Fast Fourier Transform based computational approach, we study the effects of the fiber orientation state and the imposed macroscopic shear rate tensor on the effective viscosity for a broad range of shear rates of engineering process interest. We visualize the effective viscosity in three dimensions and find that the anisotropy of the effective viscosity and its shear rate dependence vary strongly with the fiber orientation state. Combining the results of this work with insights from literature, we formulate four requirements a model of the effective viscosity should satisfy for shear-thinning fiber suspensions with a Cross-type matrix fluid. Furthermore, we introduce four model candidates with differing numbers of parameters and different theoretical motivations, and use supervised machine learning techniques for non-convex optimization to identify parameter sets for the model candidates. By doing so, we leverage the flexibility of automatic differentiation and the robustness of gradient based, supervised machine learning. Finally, we identify the most suitable model by comparing the prediction accuracy of the model candidates on the fiber orientation triangle, and find that multiple models predict the anisotropic shear-thinning behavior to engineering accuracy over a broad range of shear rates.
... Traxl et al. [55] formulated equations for four different matrix fluids and three different inclusion geometries, where both rigid particles and pores were considered. Karl and Böhlke [56] addressed various mean-field homogenization models in parallel for the estimation of effective viscous and elastic properties of fiber suspensions and composites. In the same study, the Hill-Mandel condition [57,58] for viscous suspensions was reconsidered in detail with respect to singular surfaces, complementing the work of, e.g., Adams and Field [59] and Traxl et al. [55]. ...
... The Hill-Mandel condition [57,58] for viscous suspensions and the underlying Stokes flow condition are assumed to be valid and only isotropic phases (fibers and matrix) with phase-wise uniform mechanical properties are considered [57]. In each homogenization step, the microstructure of the fiber suspension is seen as constant in time [45,56]. Based on the approach for linear elastic solids [48,[82][83][84][85], a homogeneous reference viscosity V 0 is introduced and the corresponding viscous stress polarization and the constitutive law read ...
... The rigid fiber assumption V → ∞ or, in other words, a vanishing fluidity V −1 [51,56] leads to the following fiber spin tensor and to an equation for the rate=̄− P 0 ...
... For dilute and semidilute suspensions of slender rods in Newtonian fluids, models were proposed among others by Batchelor [10,11], Dinh and Armstrong [12], as well as Shaqfeh and Frederickson [13]. For an overview of anisotropic meanfield homogenization models of particle suspensions in Newtonian fluids we refer the reader to Karl and Böhlke [14]. Various other models are detailed in Petrie [15]. ...
... Typically, it is not feasible to take the large amount of information contained in the fiber orientation distribution function into account at every spatial element of component-scale simulations. To reduce the necessary computational effort in treating fiber orientation states, it is common to use the fiber orientation tensors of second and fourth order [67,68] With these two descriptors at hand, a large variety of anisotropic meanfield models has been built to model material behavior [14]. In the case of a transversely isotropic fluid with constant viscosity, a convenient approximation of the popular Mori-Tanaka model is available with [69] V = 2 ...
... Further classical mean field approaches such as Mori-Tanaka [40], differential scheme [44], the self-consistent method [19,31] or Hashin-Shtrikman type formulations [17,18], have long been established in solid mechanics and provide the basis of (semi-)analytical, numerical or hybrid calculations of heterogeneous structures on different scales. Contemporary contributions focusing on ferroelectrics are found, e.g., in [22,51,55], while [29] presents a general survey on modern mean field theories. ...
... and the electric displacement is obtained from Eqs. (26) and (29) according to ...
... Interactions between grains, inducing residual stresses and electric displacements, are involved in Eqs. (29) and (30) by imposing constraints upon strains and electric fields of grains in order to satisfy Eqs. (23) and (24). ...
Constitutive modeling of ferroelectrics is a challenging task, spanning physical processes on different scales from unit cell switching and domain wall motion to polycrystalline behavior. The condensed method (CM) is a semi-analytical approach, which has been efficiently applied to various problems in this context, ranging from self-heating and damage evolution to energy harvesting. Engineering applications, however, inevitably require the solution of arbitrary boundary value problems, including the complex multiphysical constitutive behavior, in order to analyze multifunctional devices with integrated ferroelectric components. The well-established finite element method (FEM) is commonly used for this purpose, allowing sufficient flexibility in model design to successfully handle most tasks. A restricting aspect, especially if many calculations are required within, e.g., an optimization process, is the computational cost which can be considerable if two or even more scales are involved. The FEM–CM approach, where a numerical discretization scheme for the macroscale is merged with a semi-analytical methodology targeting at material-related scales, proves to be very efficient in this respect.
... In order to realize the flow-fiber coupling, the effective viscosity tensorV of the fiber suspension is formulated in terms of fiber orientation tensors N and N. Throughout this study, the effective Cauchy stress tensorσ refers to a generalized incompressible Newtonian fluid. Furthermore, the Mori-Tanaka model [26,[38][39][40] is used readinḡ ...
... The polarization tensor P 0 contains information about the fiber geometry and the matrix material and can be computed analytically for spheroidal fibers. In this work the expressions given in Ponte Castañeda and Willis [41] are adapted for viscosity and simplified for incompressible matrix behavior following Karl and Böhlke [40]. The inversion P −1 0 is defined on the symmetric deviatoric subset (symDev) for the considered case of incompressible suspensions with P −1 0 P 0 = P 2 representing the identity on symDev for second-order tensors. ...
... In Eq. (3) the matrix viscosity V M is considered both Newtonian and non-Newtonian in the context of a generalized Newtonian fluid. Within the present study, the shear-thinning Cross-model [45] for the polymer matrix is implemented as follows to be compared against the Newtonian behavior [40,46] ...
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.
