No abstract

An interative approach is proposed for the numerical analysis of elastic-plastic continua. This approach gives after convergence an implicit scheme of integration of the evolution problem, and is concerned with elastic-perfectly plastic materials and with hardening standard materials. Under a generalized assumption of positive hardening, the proof of convergence of the iterative solutions is given. Some numerical examples by the finite element method are also discussed.

The standard gradient models have been intensively studied in the literature, cf. Fremond (1985) or Gurtin (1991) for various applications in plasticity, damage mechanics and phase change analysis. The governing equations for a solid have been introduced essentially from an extended version of the virtual equation. It is shown here first that these equations can also be derived from the formalism of energy and dissipation potentials and appear as a generalized Biot equation for the solid. In this spirit, the governing equations for higher gradient models can be straightforwardly given. The interest of gradient models is then discussed in the context of damage mechanics and crack simulation. The phenomenon of strain localization in a time-dependent or time-independent process of damage is explored as a convenient numerical method to simulate the propagation of cracks, in relation with some recent works of theliterature, cf. Bourdin & Marigo [3], Lorentz & al  [5], Henry & al [12].