Coupled thermomechanical response of gradient plasticity

The coupled thermo-mechanical strain gradient plasticity theory that accounts for micro-structure based size effects is outlined within this work. This incorporates spatial gradients of selected micro-structural fields based on length-scales that describe the evolving dissipative mechanisms. In the mechanical part, the model problem of von Mises plasticity with gradient-extended hardening/softening response is considered as discussed in Miehe et al. (2013, 2014a). In the thermal part, we follow the investigations of Simó and Miehe (1992) that demonstrate the effect of temperature on the mechanical fields resulting in a thermal expansion. To this end, two classes of solution schemes for the coupled problem are considered: (i) Global product formula algorithm arising from operator split which leads to a two step solution procedure, and (ii) an implicit coupled algorithm which employs simultaneous solution of the coupled system of equations. In the product formula algorithm, the mechanical and thermal problems are solved separately, resulting in a symmetric problem. However, in the implicit coupled algorithm, a simultaneous solution of the coupled system of equations for gradient thermo-plasticity is employed. A noteworthy drawback of this solution scheme arises from the high computational efforts in comparison with the product formula algorithm. From the computational viewpoint, the standard Galerkin finite element method fails in the context of isochoric plastic flow due to the over-constrained pressure field. To circumvent these difficulties, we extend the well-known Q1P0-type and MINI-type mixed finite elements design of gradient plasticity to account for thermal effects. The performance of the formulation is demonstrated by means of some representative examples.