Statistical homogenization of random porous media

In recent times, the scientific community paid great attention to the influence of inherent uncertainties on system behavior and recognize the importance of stochastic and statistical approaches to engineering problems [21]. In particular, statistical computational methods may be useful to the constitutive characterization of complex materials, such as composite materials characterized by non-periodic internal micro-structure. Random porous media exhibit a microstructure made of randomly distributed pores embedded into a continuous matrix. They can be modelled as a bi-material system in which circular soft inclusions (pores) with random distribution and variable diameters are dispersed in a stiffer matrix. A key aspect, recently investigated by many researchers, is the evaluation of appropriate mechanical properties to be adopted for the study of their behaviour. Differently from classical homogenization approaches, in the case of materials with random microstructure it is not possible to 'a-priori' define a Representative Volume Element (RVE), this being an unknown of the problem. Statistical homogenization procedures may be adopted for the definition of equivalent moduli able to take into account at the macroscale the material properties emerging from the internal microstructure with random distribution [26]. Here, a Fast Statistical Homogenization Procedure (FSHP) based on Virtual Element Method (VEM) approach for the numerical solution-previously developed by some of the authors [13] has been adopted for the definition of the Representative Volume Element (RVE) and of the related equivalent elastic moduli of random porous media with different volume fraction, defined as the ratio between mechanical properties of inclusions and matrix. In particular, FSHP with virtual Elements of degree 1 [2] for modelling the inclusions provides reliable results for materials with low contrast.