Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme

TLDR

The authors propose a gradient‑enhanced computational homogenization procedure that models microstructural size effects within a general nonlinear framework. The method imposes the macroscopic deformation gradient and its gradient on a representative volume element, solves the micro‑scale boundary‑value problem with classical continuum constituents, and derives the macroscopic stress and higher‑order stress tensors via an extended Hill–Mandel condition. This approach automatically yields a microstructurally based constitutive response for the higher‑order macro continuum, naturally accounting for size effects, as demonstrated in several illustrative examples. © 2002 John Wiley & Sons, Ltd.

Abstract

Abstract A gradient‐enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general non‐linear framework. In this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non‐uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher‐order stress tensor are derived based on an extension of the Hill–Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher‐order macro continuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects. Copyright © 2002 John Wiley & Sons, Ltd.