A variational constitutive model for the distribution and interactions of multi-sized voids

The evolution of defects or voids, generally recognized as the basic failure mechanism in most metals and alloys, has been intensively studied. Most investigations have been limited to spatially periodic cases with non-random distributions of the radii of the voids. In this study, we use a new form of the incompressibility of the matrix to propose the formula for the volumetric plastic energy of a void inside a porous medium. As a consequence, we are able to account for the weakening effect of the surrounding voids and to propose a general model for the distribution and interactions of multi-sized voids. We found that the single parameter in classical Gurson-type models, namely void volume fraction is not sufficient for the model. The relative growth rates of voids of different sizes, which can in principle be obtained through physical or numerical experiments, are required. To demonstrate the feasibility of the model, we analyze two cases. The first case represents exactly the same assumption hidden in the classical Gurson's model, while the second embodies the competitive mechanism due to void size differences despite in a much simpler manner than the general case. Coalescence is implemented by allowing an accelerated void growth after an empirical critical porosity in a way that is the same as the Gurson-Tvergaard-Needleman model. The constitutive model presented here is validated through good agreements with experimental data. Its capacity for reproducing realistic failure patterns is shown by simulating a tensile test on a notched round bar.