| This thesis presents a multiscale variational method for developing stabilized finite element formulations for application in nonlinear solid mechanics. The multiscale method arises from a decomposition of the displacement field into coarse (resolved) and fine (unsolved) scales. The resulting finite element formulation allows arbitrary combinations of interpolation functions for the displacement and stress fields, and thus yields a family of stable and convergent elements. Specially, equal order interpolations that are easy to implement but violate the celebrated B-B condition, become stable and convergent. A phenomenological constitutive model for the superelastic behavior of shape memory alloys is proposed and integrated in this multiscale/stabilized finite element framework. Numerical tests of the performance of the elements are presented and representative simulations of the shape memory behavior are shown. The model has been extended to finite deformation regime. Issues related to consistent linearization and objective stress updates have been addressed in detail. Representative numerical simulations of finite deformation of some beams are also presented. |
