A study of strain localization in a multiple scale framework—The one-dimensional problem Original Research Article

This work approaches strain localization by recognizing the multiple scales inherent in the problem. A component associated with the region of localized strain (typically, one with high gradient) is referred to as the fine scale. It represents the microstructure. The field obtained by removing the fine scale from the total solution is referred to as the coarse scale. The aim of the multiple scale method advanced here is to derive a model for the coarse scale field that accounts for the finite scale. This process eliminates the fine scale from the problem, yet retains its effect. In applying this framework to the nonlinear problems with which localized strains are associated, a crucial step is a first-order approximation of the relevant relations. The fully nonlinear problem is solved by an iterative scheme. By accounting directly for the microstructure the multiple scale model recovers the regularizing effects of various alternative formulations for softening strain localization. It thus presents itself as a unifying framework for such models. Numerical solutions are shown to be invariant with respect to the discretization. Furthermore, for cases in which the displacements assume a distinct profile within the localization band, the multiple scale model provides a resolution that can be made as accurate as desired even with the coarsest mesh possible. The model is applied to strain localization problems that arise in inviscid and viscoplastic solids. Numerical simulations are presented that demonstrate the efficacy of the approach.