Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior

A computational homogenization method to determine the effective parameters of Mindlin's Strain Gradient Elasticity (SGE) model from a local heterogeneous Cauchy linear material is developed. The devised method, which is an extension of the classical one based on the use of Quadratic Boundary Conditions, intents to correct the well-known non-physical problem of persistent gradient effects when the Representative Volume Element (RVE) is homogeneous. Those spurious effects are eliminated by introducing a microstructure-dependent body force field in the homogenization scheme together with alternative definitions of the localization tensors. With these modifications, and by a simple application of the superposition principle, the higher-order stiffness tensors of SGE are computed from elementary numerical calculations on RVE. Within this new framework, the convergence of SGE effective properties is investigated with respect to the size of the RVE. Finally, a C-1-FEM procedure for simulating the behavior of the effective material at the macro scale is developed. We show that the proposed model is consistent with the solutions arising from asymptotic analysis and that the computed effective tensors verify the expected invariance properties for several classes of anisotropy. We also point out an issue that the present model shares with asymptotic-based solutions in the case of soft inclusions. Applications to anisotropic effective strain-gradient materials are provided, as well as comparisons between fully meshed structures and equivalent homogeneous models. (C) 2020 Elsevier Ltd. All rights reserved.