Concentration tensors preserving elastic symmetry of multiphase composites

Segura, Nabor Jiménez; Pichler, Bernhard; Hellmich, Christian

doi:10.1016/j.mechmat.2023.104555

Cited by 9 publications

(3 citation statements)

References 49 publications

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“…Nevertheless, as regards classical composite microstructures, well-known homogenization methods based on the Eshelby-Laws matrix-inhomogeneity problem, such as suitable generalizations of the Mori-Tanaka method (Benveniste, 1987), accounting for symmetrization strategies if required (Sevostianov and Kachanov, 2014;Jiménez Segura et al, 2023), may turn out as more efficient than an approach starting from the general expression given in Eq. 48 and Eq.…”

Section: Discussionmentioning

confidence: 99%

“…The strains in the latter are approximated from the solutions of Eshelby's matrixinhomogeneity problem (Eshelby, 1957), and combination of these solutions with the strain average rule specified for a finite number of phases leads to the well-known Mori-Tanaka or self-consistent models (Kröner, 1958;Mori and Tanaka, 1973;Benveniste, 1987;Benveniste et al, 1991), with many applications in a variety of disciplines, including construction and biomedical engineering (Bernard et al, 2003;Hellmich et al, 2004;Hellmich and Mang, 2005;Hofstetter et al, 2005;Fritsch and Hellmich, 2007). In this context, we note that composites with inclusions of different shapes and/or orientations may require additional symmetrization steps guaranteeing the existence of an elastic potential (Sevostianov and Kachanov, 2014;Jiménez Segura et al, 2023). Besides this classical approach, it proved useful to introduce, within an RVE, infinitely many (non-spherical) phases, being associated with infinitely many space directions quantified through longitudinal and latitudinal Euler angles, and to associate infinitely many Eshelby problems to each of these directions (Fritsch et al, 2006).…”

Section: Introduction-motivation and Scopementioning

confidence: 99%

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A Green’s function-based approach to the concentration tensor fields in arbitrary elastic microstructures

2023

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Computational homogenization based on FEM models is the gold standard when it comes to homogenization over a representative volume element (RVE), of so-called complex material microstructures, i.e., such which cannot be satisfactorily represented by an assemblage of homogeneous subdomains called phases. As a complement to the aforementioned models, which depend on the boundary conditions applied to the representative volume element and which, as a rule, do not give direct access to the macro-micro-relations in terms of concentration tensors, we here introduce a Green’s function-based homogenization method for arbitrary inhomogeneous microstructures: Inspired by the ideas underlying traditional phase-based homogenization schemes, such as the Mori-Tanaka or the self-consistent model, the new method rests on mapping, through the strain average rule, the microscopic strain fields associated with an auxiliary problem to the macroscopic strains subjected to the RVE. Thereby, the auxiliary problem is defined on a homogeneous infinite matrix subjected to homogeneous auxiliary strains and to inhomogeneous (fluctuating) polarization stresses representing the fluctuations of the microstiffness field, i.e., the complex microstructure within the RVE. The corresponding microscopic strains appear as the solution of a Fredholm integral equation, delivering a multilinear operator linking the homogeneous auxiliary strains to the microscopic strains. This operator, together with the aforementioned mapping, eventually allows for completing the model in terms of concentration tensor and homogenized stiffness quantification. This is illustrated by example of a sinusoidally fluctuating microstructure, whereby the corresponding singular convolution integrals are analytically evaluated from the solution of the Poisson’s equation, and this evaluation strategy is then analytically verified through a Cauchy principal value analysis, and numerically validated by a state-of-the-art FFT homogenization procedure. For the given example, the novel analytical method is several thousand times faster than an FTT-based computational homogenization procedure.

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Section: Discussionmentioning

confidence: 99%

Section: Introduction-motivation and Scopementioning

confidence: 99%

A Green’s function-based approach to the concentration tensor fields in arbitrary elastic microstructures

2023

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“…A comprehensive study on the effects of the orientation averaging on homogenization procedures can be found in . To circumvent effective stiffness tensors, which are not majorsymmetric and therefore violate thermodynamical principles, the algorithm in Jiménez Segura et al (2023) was implemented and can be activated by a flag parameter to ensure symmetric stiffnesses. Alternatively, the shear-lag modified Halpin-Tsai approach (cf.…”

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confidence: 99%