SUMMARYThis 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.

A novel formulation for multiscale finite element analysis of multi-phase solids undergoing large strains is proposed in this paper. Within the described homogenization technique no constitutive assumptions are made at the macrolevel. A crucial aspects of the approach is the modelling of antiperiodic traction on the boundary of the representative volume element, condensation technique and the formulation performed on a deformation-driven context whereby the macroscopic deformation gradient is prescribed. Numerical tests on solids with voids demonstrated the robustness of the technique.

This paper presents a multiscale finite element homogenization technique (MFEH) for modelling nonlinear deformation of multi-phase materials. A novel condensation technique to relate force variations acting on the representative volume element (RVE) -involving antiperiodicity of traction forces at RVE corners-and displacement variations on boundary-nodes is proposed. The formulation to accommodate the condensation technique and overall tangent modulus is presented in detail. In this context, the effective homogenized tangent modulus is computed as a function of microstructure stiffness matrix which, in turn, depends upon the material properties and, geometrical distribution of the micro-constituents. Numerical tests concerning plastic materials with different voids distributions are presented to show the robustness of the proposed MFEH.