The asymptotic homogenization elasticity tensor properties for composites with material discontinuities

Penta, R. and Gerisch, A. (2017) The asymptotic homogenization elasticity tensor properties for composites with material discontinuities. Continuum Mechanics and Thermodynamics, 29(1), pp. 187-206. (doi: 10.1007/s00161-016-0526-x)

Abstract

The classical asymptotic homogenization approach for linear elastic composites with discontinuous material properties is considered as a starting point. The sharp length scale separation between the fine periodic structure and the whole material formally leads to anisotropic elastic-type balance equations on the coarse scale, where the arising fourth rank operator is to be computed solving single periodic cell problems on the fine scale. After revisiting the derivation of the problem, which here explicitly points out how the discontinuity in the individual constituents’ elastic coefficients translates into stress jump interface conditions for the cell problems, we prove that the gradient of the cell problem solution is minor symmetric and that its cell average is zero. This property holds for perfect interfaces only (i.e., when the elastic displacement is continuous across the composite’s interface) and can be used to assess the accuracy of the computed numerical solutions. These facts are further exploited, together with the individual constituents’ elastic coefficients and the specific form of the cell problems, to prove a theorem that characterizes the fourth rank operator appearing in the coarse-scale elastic-type balance equations as a composite material effective elasticity tensor. We both recover known facts, such as minor and major symmetries and positive definiteness, and establish new facts concerning the Voigt and Reuss bounds. The latter are shown for the first time without assuming any equivalence between coarse and fine-scale energies (Hill’s condition), which, in contrast to the case of representative volume elements, does not identically hold in the context of asymptotic homogenization. We conclude with instructive three-dimensional numerical simulations of a soft elastic matrix with an embedded cubic stiffer inclusion to show the profile of the physically relevant elastic moduli (Young’s and shear moduli) and Poisson’s ratio at increasing (up to 100 %) inclusion’s volume fraction, thus providing a proxy for the design of artificial elastic composites.

• Item Type: Articles
• Additional Information: This work was supported by the DFG priority program SPP1420, Project GE 1894/3 and RA 1380/7, Multiscale structure-functional modeling of musculoskeletal mineralized tissues, PIs Alf Gerisch and Kay Raum.We acknowledge Eli Duenisch for programming support
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Penta, Dr Raimondo
• Authors: Penta, R., and Gerisch, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Continuum Mechanics and Thermodynamics
• Publisher: Springer
• ISSN: 0935-1175
• ISSN (Online): 1432-0959
• Published Online: 18 August 2016
• Copyright Holders: Copyright © 2016 Springer-Verlag Berlin Heidelberg
• First Published: First published in Continuum Mechanics and Thermodynamics 29(1):187-206
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher.