Homogenization of Ductile Porous Materials by Limit and Shakedown Analysis

Abstract

This paper is a survey of recent trends in the poroplasticity combined with Direct methods. Using the hollow sphere model as Reference Elementary Volume (REV) with a matrix obeying von Mises microscopic plastic yield criterion, a stress variational model (SVM), dual of Gurson’s one, has been proposed to find by the Limit Analysis a macroscopic criterion depending on the porosity. Remarkably, it depends on the third invariant $J_3$ but only through its sign. Applying the normality law to the macroscopic criterion, the evolution of porosity with respect to the stress triaxiality exhibit clear discrepancies with Gurson’s one which is known to overestimate the variation of the porosity. Some extensions has been proposed to obtain a continuous dependence with respect to $J_3$ through Lode’s angle, to improve the strength value for the pure deviatoric loading. Thanks to the bipotential formulation, a macroscopic yield criterion was also proposed for a non associated Drucker-Prager matrix. Using the Shakedown Analysis, the method has been extended to the repeated variable loadings to obtain a fatigue criterion for the porous materials. It depends on the porosity but also strongly on Poisson’s coefficient. The general case involving shear effects with any cyclic load fluctuations ranging from the pulsating load to the alternating one is considered. The macroscopic criteria depend on the first and second macroscopic stress invariants and the sign of the third one.