Computational homogenisation for non-linear heterogeneous solids

This chapter presents a computational homogenization strategy, which provides a rigorous approach to determine the macroscopic response of heterogeneous materials with accurate account for microstructural characteristics and evolution. When using this micro-macro strategy there is no necessity to define homogenized macroscopic constitutive equations, which in case of large deformations and complex microstructures, would be generally a hardly feasible task. Instead, the constitutive behaviour at macroscopic integration points isdetermined by averaging the response of the deforming microstructure. This enables a straight forward application of the method to geometrically and physically non-linear problems, making it a particularly valuable tool for the modelling of evolving non-linear heterogeneous microstructures under complex macroscopic loading paths. In this chapter, the underlying concepts and the details of the computational homogenization technique are given.Formulation of the microscopic boundary value problem and the consistent micro-macro coupling in a geometrically and physically nonlinear framework are elaborated. The implementation of the computational homogenization scheme in a finite element framework is discussed. Some recent extensions of the computational homogenization schemes are summarized.