On micro‐to‐macro transitions for multi‐scale analysis of non‐linear heterogeneous materials: unified variational basis and finite element implementation

Abstract This work describes a homogenization‐based multi‐scale procedure required for the computation of the material response of non‐linear microstructures undergoing small strains. Such procedures are important for computer modelling of heterogeneous materials when the length‐scale of heterogeneities is small compared to the dimensions of the body. The described multi‐scale procedure relies on a unified variational basis which, apart from the continuum‐based variational formulation at both micro‐ and macroscales of the problem, also includes the variational formulation governing micro‐to‐macro transitions. This unified variational basis leads naturally to a generic finite element‐based framework for homogenization‐based multi‐scale analysis of heterogenous solids. In addition, the unified variational formulation provides clear axiomatic basis and hierarchy related to the choice of boundary conditions at the microscale. Classical kinematical constraints are considered over the representative volume element: (i) Taylor, (ii) linear boundary displacements, (iii) periodic boundary displacement fluctuations and (iv) minimal constraint, also known as uniform boundary tractions. In this context the Hill‐Mandel averaging requirement, which links microscopic and macroscopic stress power, plays a fundamental role in defining the microscopic forces compatible with the assumed kinematics. Numerical examples of both microscale and two‐scale finite element simulations of elasto‐plastic material with microcavities are presented to illustrate the main features and scope of the described computational strategy. Copyright © 2010 John Wiley & Sons, Ltd.