Asymptotic fiber orientation states of the quadratically closed Folgar–Tucker equation and a subsequent closure improvement

Karl, Tobias; Gatti, Davide; Böhlke, Thomas

doi:10.1122/8.0000245

Cited by 14 publications

(8 citation statements)

References 54 publications

(144 reference statements)

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“…Concerning the definition of the orientation tensors of the second and third kind, the reader is referred to Kanatani [98]. These tensors are given in Karl et al [99] consistent with the notation of this manuscript.…”

Section: Description Of Fibrous Microstructuresmentioning

confidence: 97%

Unified mean-field modeling of viscous short-fiber suspensions and solid short-fiber reinforced composites

Karl

Böhlke

2022

Arch Appl Mech

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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 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.

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Section: Description Of Fibrous Microstructuresmentioning

confidence: 97%

Unified mean-field modeling of viscous short-fiber suspensions and solid short-fiber reinforced composites

Karl

Böhlke

2022

Arch Appl Mech

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“…A scientific topic which is intimately connected to the question on the phase space of fourth-order fiber-orientation tensors, is fiber-orientation closure approximations. Such closure approximations [22,37,[56][57][58][59][60][61] are tensor-valued functions which postulate a functional relationship between a given second-order fiber-orientation tensor and an unknown fourth-order fiber-orientation tensor [19]. Identifying the phase space of fourth-order fiberorientation tensors is essential to solve a fundamental problem of closure approximationswhich fourth-order fiber-orientation tensors can be realized?…”

Section: State Of the Artmentioning

confidence: 99%

On the Phase Space of Fourth-Order Fiber-Orientation Tensors

Bauer

Schneider

Böhlke

2023

J Elast

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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.

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“…A scientific topic which is intimately connected to the question on the phase space of fourth-order fiberorientation tensors, is fiber-orientation closure approximations. Such closure approximations [22,37,[56][57][58][59][60][61][62] are tensor-valued functions which postulate a functional relationship between a given second-order fiber-orientation tensor and an unknown fourth-order fiber-orientation tensor [19]. Identifying the phase space of fourth-order fiber-orientation tensors is essential to solve a fundamental problem of closure approximations -which fourth-order fiber-orientation tensors can be realized?…”

Section: State Of the Artmentioning

confidence: 99%

On the phase space of fourth-order fiber-orientation tensors

Bauer¹,

Schneider²,

Böhlke³

2022

Preprint

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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 fiberorientation 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.

show abstract

Asymptotic fiber orientation states of the quadratically closed Folgar–Tucker equation and a subsequent closure improvement

Cited by 14 publications

References 54 publications

Unified mean-field modeling of viscous short-fiber suspensions and solid short-fiber reinforced composites

Unified mean-field modeling of viscous short-fiber suspensions and solid short-fiber reinforced composites

On the Phase Space of Fourth-Order Fiber-Orientation Tensors

On the phase space of fourth-order fiber-orientation tensors

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