A variational multiscale approach to strain localization – formulation for multidimensional problems

Abstract

The multiscale approach to strain localization problems developed earlier, is generalized to higher dimensions. The key idea is the identification of the fine scale field with a component of the displacement that has a large gradient. The difference between the total displacement and the fine scale is the coarse scale field. The formulation is variationally based, with the weak form as the point of departure. With suitable assumptions on the space of fine scale fields, the weak form is separable into coarse and fine scale parts. The weak form of the fine scale problem is used to eliminate the fine scales from the formulation. This is achieved by way of a projection onto the fine scale space and allows a reformulation of the full problem in terms of coarse scale components alone. In this framework, fine scale interpolations are identified that ensure sharp resolution of localized displacement irrespective of the underlying finite element mesh. It is demonstrated that by correctly accounting for the “microstructural” fine scale behavior, the formulation results in solutions devoid of pathological mesh sensitivity. For problems wherein the detailed structure of the localized displacement is of interest, recovery of the fine scale components can be performed to reconstruct the entire displacement field. Several numerical examples are presented that demonstrate the robustness of the formulation.