Anisotropic effective higher-order response of heterogeneous Cauchy elastic materials

Abstract

The homogenization results obtained by Bacca et al. (Homogenization of heterogeneous Cauchy-elastic materials leads to Mindlin second-gradient elasticity. Part I: Closed form expression for the effective higher-order constitutive tensor. http://arxiv.org/abs/1305.2365 Submitted, 2013), to define effective second-gradient elastic materials from heterogeneous Cauchy elastic solids, are extended here to the case of phases having non-isotropic tensors of inertia. It is shown that the nonlocal constitutive tensor for the homogenized material depends on both the inertia properties of the RVE and the difference between the effective and the matrix local elastic tensors. Results show that: (i.) a composite material can be designed to result locally isotropic but nonlocally orthotropic; (ii.) orthotropic nonlocal effects are introduced when a dilute distribution of aligned elliptical holes and, in the limit case, of cracks is homogenized.

Isotoxal star-shaped polygonal voids and rigid inclusions in nonuniform antiplane shear fields. Part I: Formulation and full-field solution

“…The investigation of the perturbation induced by an inclusion (a void, or a crack, or a stiff insert) in an ambient stress field loading a linear elastic infinite space is a fundamental problem in solid mechanics, whose importance need not be emphasized. Usually this problem is analyzed with respect to uniform ambient stress fields [1,5,8,13,23,25], although inhomogeneous, self-equilibrated stresses have also been considered [4,6,12,32,34,39,40]. The interplay between stress inhomogeneities and singularities generated at inclusion corners is important in the design of ultra-resistant composites, as stress singularities are known to be detrimental to strength.…”

Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

“…In this relation we should define the averaged strain gradient in the inclusion ε lm,n 2 and the averaged moment of Cauchy stresses in the composite τ ij x k that arise under prescribed QBC (9). The former can be found based on the relation (23) and dilute solution for the strain gradient concentration tensor (39). Using standard self-consistent approximation (Fig.…”

“…1b), i.e. using the effective properties instead of matrix properties in (39), we obtain the following relation for the strain gradient concentration tensor that takes into account the interactions between inclusions and can be used for the arbitrary volume fractions:…”

“…Dilute approximation solution (for the case of small volume fraction of inclusions) have been proposed in Ref. [37] and used for evaluation of the effective gradient properties of the composites with different types of inclusions [38,39]. Homogenization problem for the bounded domain with cylindrical inclusion have been considered in Ref.…”

Analytical solution with validity analysis for an elliptical void and a rigid inclusion under uniform or nonuniform anti-plane loading

“…Most often, the anti-plane problem of an inclusion has been investigated with respect to uniform boundary conditions [17,18,19,20,21,22,23]. Nevertheless, nonuniform loadings have also been taken into account by [24,25,26,27,28,29,30,31,32]. Recently, the problem of a circular void and a rigid inclusion embedded in a bounded domain (annulus) subject to uniform and nonuniform anti-plane shear has been solved analytically in [16].…”