Contributions on the theory and computation of mono‐ and poly‐crystalline cyclic martensitic phase transformations

Abstract

AbstractIn this article new contributions to the theory and computation of cyclic martensitic phase transformations (PT) in mono‐ and poly‐crystalline metallic shape memory alloys are presented. The PT models of the non‐convex variational problem are based on the Cauchy‐Born hypothesis and Bain's principle. A quasi‐convexified C1‐continuous thermo‐mechanical micro‐macro constitutive model for metallic monocrystals is developed which is represented together with the phase transformation constraints by a unified Lagrangian variational functional including phase evolution equations with mass conservation. The unified setting presented here includes poly‐crystalline shape memory alloys whose microstructure is modeled using lattice variants. A pre‐averaging scheme for randomly distributed poly‐crystalline variants of transformation strains is used to transform them into those of a fictitious monocrystal. Thus, the incremental integration in process time and the spatial integration algorithms of the discrete variational problems for both mono‐ and poly‐crystalline phase transformations can be implemented into a unified algorithm with branching for mono‐ and poly‐crystalline phase transformations. Furthermore, an error‐controlled adaptive 3D finite element method in space is presented for phase transformation problems using explicit error indicator with gradient smoothing and mesh refinements via new mesh generation in each adaptive step. Computations of informative examples with convergence studies, and comparisons with published experimental results are presented using 3D finite elements.