On the spatial discretization of mechanical equilibrium

An updated Lagrangian formulation, taking into account the deformation history, is used to formulate the mechanical equilibrium during material forming. The strategy for numerically solving the equations of the coupled thermo-elastoplastic problem involves spatial discretization by finite elements and temporal discretization by the incremental loading procedure. This leads to a highly non-linear system of partial differential equations associated with initial and boundary conditions, and first-order ordinary differential equations (constitutive equations of damageable elastoplastic behavior).

The damageable deformable domain Ω is assumed to be subjected to volume forces $\overline{f}($