On the dependence of orientation averaging mean field homogenization on planar fourth-order fiber orientation tensors

Bauer, Julian Karl; Böhlke, Thomas

doi:10.1016/j.mechmat.2022.104307

Cited by 13 publications

(9 citation statements)

References 60 publications

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“…Thinking of fiber orientation tensors as a design space in combination with the known variety of planar fourth-order fiber orientation tensors enables a complete visualization of the directional dependence of effective Young's modulus for a given homogenization method. The proposed visualization technique can be applied to any homogenization method, see [3]. The…”

Section: Results and Conclusionmentioning

confidence: 99%

“…with R(λ 1 ) = (λ 1 − λ 2 1 )/2 and d1 = d 1 − R(λ 1 ) + 4/35 following [3], combined with the algebraic requirement of positive semi-definiteness of N planar in the Kelvin-Mandel space, leads to the body of admissible and structurally distinct planar fourth-order fiber orientation tensors visualized as the wire-frame body in Figure 9. Algebraic expression for this body are given in [3,Equations (11) and (12)]. Two tensors are structurally distinct, if they differ by more than just a rotation, see also [2,Equation (18)].…”

Section: Planar Subspacementioning

confidence: 99%

“…Each point in the body of admissible planar fourth-order fiber orientation tensors in Figure 9 with vanishing value parameter d 8 represents an orthotropic fiber orientation tensor. In Figure 12, ten points in this orthotropic plane within the admissible body are selected and highlighted following [3]. Each of these points represents a fourth-order fiber orientation tensor.…”

Section: Directional Variety Of Effective Stiffnessesmentioning

confidence: 99%

“…This homogenization method is selected due to its popularity. Details on this method can be found in [3,4] and an open-source implementation in [23]. The fiber volume content and material properties of fiber and matrix are fixed to SMC-specific values given in [3, Table 1], originating from [8, Table 2].…”

Section: Orientation-averaging Meanfield Homogenizationmentioning

confidence: 99%

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Variety of Planar Fourth‐Order Fiber Orientation Tensors and Implications on Effective Elastic Stiffnesses

Bauer

Seelig

Hrymak

et al. 2023

Proc Appl Math and Mech

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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.

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Section: Results and Conclusionmentioning

confidence: 99%

Section: Planar Subspacementioning

confidence: 99%

Section: Directional Variety Of Effective Stiffnessesmentioning

confidence: 99%

Section: Orientation-averaging Meanfield Homogenizationmentioning

confidence: 99%

See 2 more Smart Citations

Variety of Planar Fourth‐Order Fiber Orientation Tensors and Implications on Effective Elastic Stiffnesses

Bauer

Seelig

Hrymak

et al. 2023

Proc Appl Math and Mech

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“…This variety is presented for special cases motivated by material symmetry. The case of planar fourth-order fiber-orientation tensors and derived quantities is studied in successive papers [54,55]. However, the necessary condition of positive semidefiniteness of the completely symmetric tensor is assumed to be sufficient.…”

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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Generating microstructures of long fiber reinforced composites by the fused sequential addition and migration method

Lauff,

Schneider,

Montesano

et al. 2024

Numerical Meth Engineering

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We introduce the fused sequential addition and migration (fSAM) algorithm for generating microstructures of fiber composites with long, flexible, nonoverlapping fibers and industrial volume fractions. The proposed algorithm is based on modeling the fibers as polygonal chains and enforcing, on the one hand, the nonoverlapping constraints by an optimization framework. The connectivity constraints, on the other hand, are treated via constrained mechanical systems of d'Alembert type. In case of straight, that is, nonflexible, fibers, the proposed algorithm reduces to the SAM (Comput. Mech., 59, 247–263, 2017) algorithm, a well‐established method for generating short fiber‐reinforced composites. We provide a detailed discussion of the equations governing the motion of a flexible fiber and discuss the efficient numerical treatment. We elaborate on the integration into an existing SAM code and explain the selection of the numerical parameters. To capture the fiber length distributions of long fiber reinforced composites, we sample the fiber lengths from the Gamma distribution and introduce a strategy to incorporate extremely long fibers. We study the microstructure generation capabilities of the proposed algorithm. The computational examples demonstrate the superiority of the novel microstructure‐generation technology over the state of the art, realizing large fiber aspect ratios (up to 2800) and high fiber volume fractions (up to for an aspect ratio of 150) for experimentally measured fiber orientation tensors.

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On the dependence of orientation averaging mean field homogenization on planar fourth-order fiber orientation tensors

Cited by 13 publications

References 60 publications

Variety of Planar Fourth‐Order Fiber Orientation Tensors and Implications on Effective Elastic Stiffnesses

Variety of Planar Fourth‐Order Fiber Orientation Tensors and Implications on Effective Elastic Stiffnesses

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

Generating microstructures of long fiber reinforced composites by the fused sequential addition and migration method

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