A space-time formulation for time-dependent behaviors at small or finite strains

A general formalism is proposed, based on the definition of a space-time potential, for developing space-time formulations adapted to nonlinear and time dependent behaviors. The focus is given to the case of standard generalized materials that are expressed from the knowledge of two potentials, a strain energy and a dissipation potential in a convex framework with the help of internal variables. Viscoplasticity with isotropic hardening and nonlinear finite viscoelasticity are investigated. Starting from the definition of an appropriate space-time potential, time discontinuous Galerkin forms are developed for use in the case of time singularities (in particular with regard to time integration of internal variables). Furthermore, NURBS approximation are used, such as to propose Space-Time Isogeometric Analysis models. Numerical examples allow to compare the obtained isogeometric space-time models with standard finite-element models (that are based on standard time integration procedures: radial return for viscoplasticity and backward euler for viscosity) and allow to illustrate the new possibilities offered with the proposed space-time formulations.