BIRS Workshop 10w5075 Rate-independent systems: Modeling, Analysis, and Computations

A nonlinear shell finite element formulation for shape memory alloy applications Edoardo Artioli (Roma Tor Vergata) 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 pseudoelasticity 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. [1] Evangelista V., Marfia S., Sacco E., A 3D SMA constitutive model in the framework of finite strain, International Journal for Numerical Methods in Engineering, DOI: 10.1002/nme, 2009. [2] Arciniega R.A., Reddy J.N., Tensor-based finite element formulation for geometrically nonlinear analysis of shell structures, Computer Methods in Applied Mechanics and Engineering, 196, 1048-1073, 2007. A variational model for adhesive contact with friction Elena Bonetti (Pavia) We discuss a rate independent frictional law combined with unilateral conditions for a model describing contact with adhesion. In the framework of continuum mechanics, we derive the constitutive laws by a generalization of the priciple of virtual powers in which internal constraints, as well as the unilateral and the rate independent frictional conditions, are included in the balance of the energy and dissipation. The resulting PDE system is highly nonlinear: the main analytical diculties are related to the presence of multivalued operators rendering the Signorini conditions, the Coulomb law (accounting for friction) and the physical constraints on the variables, and to the nonlinear coupling between the equations themselves. We obtain a global-in-time existence result. These results have been obtained in a joint work with Riccarda Rossi and Giovanna Bonfanti. The role of hardening in models in crystal plasticity Georg Dolzmann (Regensburg) We investigate Gamma limits of elastoplastic models in the limit of large elastic constants. The surprising result is that models with dissipation only are predicted to have a very soft behaviour while models with hardening converge in the sense of Gamma convergence to models with rigid elasticity. This is joint work with Sergio Conti (Bonn) and Carolin Kreisbeck (Regensburg)