In this paper, we analyze the microstructural effects on non-linear elastic and composites within the framework of asymptotic homogenization. We assume that the constitutive laws of individual constituents derive from strain potentials. The microstructural effects are incorporated by considering the higher order terms which come from the asymptotic series expansion. The complete solution at any order requires the resolution of a chain of cell problems in which the source terms depend on the solution at the lower order. The influence of these terms on the macroscopic response of the non-linear composite is evaluated in the particular case of a stratified microstructure. The analytic solutions of the cell problems at the first and second order are provided for arbitrarily local strain-stress laws which derive from potentials. As classically, the non-linear dependence on the applied macroscopic strain is retrieved for the solution at the first order. It is proved that the second order term in the expansion series also exhibits a non-linear dependence with the macroscopic strain, but linearly depends on the gradient of macroscopic strain. As a consequence, the macroscopic potential obtained by homogenization is a quadratic function of the macroscopic strain gradient when the series is truncated at the second order. This model generalizes the well known first strain gradient elasticity theory to the case of non-linear elastic materials. The influence of the non local correctors on the macroscopic potential is investigated in the case of power law elasticity under macroscopic plane strain or antiplane conditions.