A variational multiscale method for inelasticity: Application to superelasticity in shape memory alloys

This paper presents a variational multiscale method for developing stabilized finite element formulations for small strain inelasticity. The multiscale method arises from a decomposition of the displacement field into coarse (resolved) and fine (unresolved) scales. The resulting finite element formulation allows arbitrary combinations of interpolation functions for the displacement and the pressure fields, and thus yields a family of stable and convergent elements. Specifically, equal order interpolations that are easy to implement but violate the celebrated Babuska-Brezzi condition, become stable and convergent. An important feature of the present method is that it does not lock in the incompressible limit. A nonlinear constitutive model for the superelastic behavior of shape memory alloys is integrated in the multiscale formulation. Numerical tests of the performance of the elements are presented and representative simulations of the superelastic behavior of shape memory alloys are shown.