An implicit incrementally objective formulation for the solution of elastoplastic problems at finite strain by the F.E.M.

The mechanical formulation presented in this paper is based on an incremental updated Lagrange procedure using the principle of virtual work at the end of each load increment and an implicit incremental flow rule obtained by an approximate time integration of the objective rate constitutive equations. The approximate time integration is carried out along a particular path in the deformation and rotation space. This path ensures the incremental objectivity and minimizes the equivalent strain over the increment among all the possible paths, consequently avoiding an artificial increase of the plastic equivalent strain during the interpolation. The mechanical formulation presented leads to a fixed set of nonlinear equations, whose unknowns are the nodal displacements of the structure. A numerical algorithm based on a quasi Newton-Raphson method is then proposed to solve this system. The separation of the mechanical formulation from the resolution algorithm ensures the path independence. Numerical tests are carried out for a material obeying an isotropic with work-hardening von Mises criterion and associated flow rule. Single element tests show that this approach gives a very accurate solution even when the strain increment reaches twenty times the elastic strain up to yield. A structural test on a beam measures the influence of the incremental objectivity on the displacements, the equivalent plastic strain and the stresses.