A multilevel computational strategy for handling microscopic and macroscopic instabilities

This paper presents a numerical technique to deal with instability phenomena in the context of heterogeneous materials where buckling may occur at both macroscopic and/or microscopic scales. We limit ourselves to elastic materials but geometrical nonlinearity is taken into account at both scales. The proposed approach combines the multilevel finite element analysis  (FE2)(FE2) and the asymptotic numerical method   (ANM). In that framework, the unknown nonlinear constitutive relationship at the macroscale is found by solving a local finite element problem at the microscale. In contrast with FE2FE2, the use of the asymptotic development allows to transform the nonlinear microscopic problems into a sequence of linear problems. Thus, a direct analogy with classical linear homogenization can be made to construct a localization tensor at each step of the asymptotic development, and an explicit macroscopic constitutive relationship can be constructed at each step. Furthermore, the salient features of the ANM allow treating instabilities and limit points in a very simple way at both scales. The method is tested and illustrated through numerical examples involving local instabilities which have significant influence on the macroscopic behaviour.