Bounds and estimates for the effective yield surface of porous media with a uniform or a nonuniform distribution of voids

This paper aims at studying the effects of a nonuniform distribution of voids on the macroscopic yield response of porous media with a rigid-perfectly plastic matrix. For this purpose, a semi-analytical model, recently proposed by Bilger et al. [Bilger, N., Auslender, F., Bornert, M., Masson, R., 2002. New bounds and estimates for porous media with rigid perfectly plastic matrix. C. R. Mecanique 330, 127–132], is extended to more general situations where the local porosity can fluctuate. The microstructure is described by a generalized Hashin-type assemblage of hollow spheres and the distribution of the local porosity is obtained from a three-dimensional simulated microstructure. The matrix layer around the voids is discretized into concentric sub-layers so as to take better into account the plasticity gradient along the radial direction. Classical homogenization techniques then provide new self-consistent estimates and upper bounds for the macroscopic yield surface. These results are compared first to the predictions of the Gurson model and its extensions and then to numerical results derived from three-dimensional Fast Fourier Transform (FFT) calculations carried out with the same material porosity distribution. A good agreement is obtained with the three-dimensional FFT calculations and with Gurson–Tvergaard's predictions even for high triaxiality and without fitting any parameter. Nevertheless, when the heterogeneous distribution of voids tends to form clusters, the proposed model fails to capture the properties of the macroscopic yield surface for large triaxiality factors.