Three-phase plane composites of minimal elastic stress energy: High-porosity structures

Abstract

The paper establishes exact lower bound on the effective elastic energy of two-dimensional, three-material composite subjected to the homogeneous, anisotropic stress. It is assumed that the materials are mixed with given volume fractions and that one of the phases is degenerated to void, i.e. the effective composite is porous. Explicit formula for the energy bound is obtained using the translation method enhanced with additional inequality expressing certain property of stresses. Sufficient optimality conditions of the energy bound are used to set the requirements which have to be met by the stress fields in each phase of optimal effective material regardless of the complexity of its microstructural geometry. We show that these requirements are fulfilled in a special class of microgeometries, so-called laminates of a rank. Their optimality is elaborated in detail for structures with significant amount of void, also referred to as high-porosity structures. It is shown that geometrical parameters of optimal multi-rank, high-porosity laminates are different in various ranges of volume fractions and anisotropy level of external stress. Non-laminate, three-phase microstructures introduced by other authors and their optimality in high-porosity regions is also discussed by means of the sufficient conditions technique. Conjectures regarding lowporosity regions are presented, but full treatment of this issue is postponed to a separate publication. The corresponding "G-closure problem" of a three-phase isotropic composite is also addressed and exact bounds on effective isotropic properties are explicitly determined in these regions where

Phase transformations surfaces and exact energy lower bounds

“…We do not find translator parameters by straight maximization in (58). Instead, following approach suggested in [41,42] and used in [1,16], we assume that the lower bound is exact and attained by laminates which can be referred to as trial microstructures at this stage. We use compatibility conditions (23) and (24) and the properties of the strains ε + in optimal laminates to find translator parameters in various domains of strain space.…”

“…2 q + and finally, for rank-n laminates, come to the formula (16) and then to the formulae (12)- (15) and (18). Note that M This provides the existence of the inverse tensors (…”

“…This way, translation bound and optimal laminates' energy are coupled and allow for the definition of optimal translation parameters and optimal parameters of laminates. Here we follow the approach developed in [21,16] for optimal structures of multimaterial mixtures.…”

A note on optimal design of multiphase elastic structures

“…Geometric parameters of optimal laminates are different in each subregion of R(α A ). They are found by the technique used previously in (Albin et al, 2007;Cherkaev and Zhang, 2011) and (Cherkaev and Dzierżanowski, 2013). Roughly speaking, two types of equations are involved in the calculations: (i) formulae for average stress in two neighboring phases, and (ii) continuity of normal stress component on the interface between these phases.…”

“…Optimal multiphase composites are much less investigated; notice the pioneering contributions of Gibiansky and Sigmund (2000) and Sigmund (2000), see also (Albin et al, 2007;Cherkaev and Zhang, 2011;Cherkaev and Dzierżanowski, 2013) for continuation and extensions.…”