A nonlinear shell finite element formulation for shape memory alloy applications

In the last decades, the development of efficient computational models for the nonlinear analysis of structures made of shape memory alloys (SMA) has been one of the most important research activities. The shape memory alloys (SMA) represent one of the most interesting smart material for their ability to recover large strains during mechanical patterns, the “pseudo elastic effect”, and to recover residual deformations through mechanical-thermal cycles, the “shape memory effect”. In fact, under loading-unloading cycles, even up to 10-15% strains, the material shows distinct plateaux during the loading and unloading branches, hysteretic response and no permanent deformations. The present work presents a finite element model for the analysis of shell structures constituted of shape memory alloy material considering finite strains. A three dimensional constitutive model [1] for shape memory alloys in the framework of finite strains which is capable of describing the typical macroscopic effects of SMA, as the pseudo-elasticity and the shape memory effect is adopted. The structural model is formulated with a 2D shell theory where the midsurface and the covariant components of kinematic quantities are approximated element-wise with the standard isoparametric approach [2]. The displacement field assumption is based on the classical expansion in thickness direction in terms of increasing powers of the transverse coordinate and leads to an analogous form for the deformation gradient. The equilibrium statement is formulated considering the Virtual Work Principle in the total Lagrangian format. The proposed formulation is suitable for the simple derivation of high-order elements in a fully compatible fashion. The treatment of locking phenomena is then discussed. A set of numerical examples are presented, showing the accuracy and robustness of the proposed computational strategy and its capability of describing the structural response of shape memory alloy devices of technical interest.