A second-order reduced asymptotic homogenization approach for nonlinear periodic heterogeneous materials

An efficient second-order reduced asymptotic homogenization approach is developed for nonlinear heterogeneous media with large periodic microstructure. The two salient features of the proposed approach are (i) an asymptotic higher-order nonlinear homogenization that does not require higher-order continuity of the coarse-scale solution and (ii) an efficient model reduction scheme for solving higher-order nonlinear unit cell problems at a fraction of computational cost in comparison to the direct computational homogenization. The former is a consequence of a sequential solution of increasing order solutions, which permits evaluation of higher-order coarse-scale derivatives by postprocessing from the zeroth-order solution. The efficiency and accuracy of the formulation in comparison to the classical zeroth-order homogenization and direct numerical simulations are assessed on hyperelastic and elastoplastic periodic structures.