A consistent multiscale mechanical formulation for media with randomly distributed voids

Abstract

In this work we are interested in constructing a multiscale framework for solid media featuring a random distribution of voids. The reliability of standard lower bound multiscale models is questionable when voids reach the boundary, and the problem becomes critical when they are randomly arranged over the micro-cell (MC) boundary, such that periodic boundary conditions are not an option to be considered. This work presents a novel multiscale mechanical formulation to model the material behavior of media featuring such kind of heterogeneous constitution. The multiscale model is built within the framework posited by the Method of Multiscale Virtual Power. The original contribution lies at the core of the gradient homogenization formula, which generalizes existing strategies. Such generalized homogenization approach results in new Minimally Constrained Kinematical Multiscale Model (MCKMM). Moreover, alternative – kinematically more constrained with respect to the MCKMM – boundary conditions are proposed in order to obtain a wide family of multiscale sub-models (characterized by corresponding sub-spaces of the space associated with the minimally constrained model), which can be useful in practical applications. Numerical experiments are reported that provide support to the concepts introduced in this work.