Effective flow surface of porous materials with two populations of voids under internal pressure: I. A GTN model

This study is devoted to the effective plastic flow surface of a bi-porous material saturated by a fluid. The material under consideration exhibits two populations of voids. The smaller voids are spherical voids whereas the larger ones are spheroidal and randomly oriented inside the material. These two populations of voids are subjected to internal pressures due to the presence of gases. Approximate models for the effective plastic flow surface of such a bi-porous saturated material have previously been proposed in Vincent et al. (2009), where a three-scale homogenization procedure has been performed: first, smearing out all the small spherical bubbles using a Gurson-like matrix, and second, smearing out the intergranular ellipsoidal bubbles. Our objective here is to derive a simple analytical expression of the effective flow surface, starting from one of these previous models, obtained by generalizing the approach of Gologanu et al. (1994) to compressible materials. The main contributions of the present paper are: (1) an expression for the average dilatation-rate in the matrix, (2) an approximation of the effective flow surface in the form of a Gurson–Tvergaard–Needleman criterion. The accuracy of this new model is assessed in a companion paper by comparison with full field numerical simulations.