Computational homogenization for multi scale finite element simulation.

This work presents a general formulation of small and large strain multiscale solid constitutive models based on the volume averaging of the microscopic strain (deformation gradient under large strain) and stress fields over a locally attached microstructure Representative Volume Element (RVE). Both elasto-plastic and hyperelastic behaviour are considered in the modelling of the RVE. A multiscale first-order computational homogenization method for modelling nonlinear deformation processes of evolving multi-phase materials is developed based on the Finite Element discretisation of both macro- and micro-structure. The approach consist of suitably imposing the macroscopic strain on the RVE and then computing the macroscopic stress as the volume average of the microscopic stress field obtained by solving numerically the local (initial) boundary value problem. In this context, the effective (homogenized) tangent modulus is obtained as a function of microstructure stiffness matrix which, in turn, depends upon the material properties and geometrical distribution of the micro-constituents in the RVE. The multiscale material presented here is restricted to two-dimensional problems, however we remark that the extension to three dimensions is trivial. The effectiveness of the proposed strategies is is demonstrated by means of numerical examples