Space-time variational material modeling: a new paradigm demonstrated for thermo-mechanically coupled wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage

We formulate variational material modeling in a space-time context. The starting point is the description of the space-time cylinder and the definition of a thermodynamically consistent Hamilton functional which accounts for all boundary conditions on the cylinder surface. From the mechanical perspective, the Hamilton principle then yields thermo-mechanically coupled models by evaluation of the stationarity conditions for all thermodynamic state variables which are displacements, internal variables, and temperature. Exemplary, we investigate in this contribution elastic wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage. Therein, one key novel aspect are initial and end time velocity conditions for the wave equation, replacing classical initial conditions for the displacements and the velocities. The motivation is intensively discussed and illustrated with the help of a prototype numerical simulation. From the mathematical perspective, the space-time formulations are formulated within suitable function spaces and convex sets. The unified presentation merges engineering and applied mathematics due to their mutual interactions. Specifically, the chosen models are of high interest in many state-of-the art developments in modeling and we show the impact of this holistic physical description on space-time Galerkin finite element discretization schemes. Finally, we study a specific discrete realization and show that the resulting system using initial and end time conditions is well-posed.