A thermodynamically consistent time integration scheme for non-linear thermo-electro-mechanics

The aim of this paper is the design of a new one-step implicit and thermodynamically consistentEnergy-Momentum (EM) preserving time integration scheme for the simulation of thermoelectro-elastic processes undergoing large deformations. The time integration scheme takes advantageof the notion of polyconvexity and of a new tensor cross product algebra. These twoingredients are shown to be crucial for the design of so-called discrete derivatives fundamental forthe calculation of the second Piola-Kirchhoff stress tensor, the entropy and the electric field. Inparticular, the exploitation of polyconvexity and the tensor cross product, enable the derivationof comparatively simple formulas for the discrete derivatives. This is in sharp contrast to muchmore elaborate discrete derivatives which are one of the main downsides of classical EM timeintegration schemes. The newly proposed scheme inherits the advantages of EM schemes recentlypublished in the context of thermo-elasticity and electro-mechanics, whilst extending to the moregeneric case of nonlinear thermo-electro-mechanics. Furthermore, the manuscript delves intosuitable convexity/concavity restrictions that thermo-electro-mechanical strain energy functionsmust comply with in order to yield physically and mathematically admissible solutions. Finally, aseries of numerical examples will be presented in order to demonstrate robustness and numericalstability properties of the new EM scheme