Effective behavior of porous elastomers containing aligned spheroidal voids

The theoretical need to recognize the link between the basic microstructure of nonlinear porous materials and their macroscopic mechanical behavior is continuously rising owing to the existing engineering applications. In this regard, a semi-analytical homogenization model is proposed to establish an overall, continuum-level constitutive law for nonlinear elastic materials containing prolate/oblate spheroidal voids undergoing finite axisymmetric deformations. The microgeometry of the porous materials is taken to be voided spheroid assemblage consisting of confocally voided spheroids of all sizes having the same orientation. Following a kinematically admissible deformation field for a confocally voided spheroid, which is the basic constituent of the microstructure, we make use of an energy-averaging procedure to obtain a constitutive relation between the macroscopic nominal stress and deformation gradient. In this work, both prolate and oblate voids are considered. As a numerical example, we study macroscopic nominal stress components for a hyperelastic porous material consisting of a neo-Hookean matrix and prolate/oblate voids subjected to 3-D and plane strain dilatational loadings. In this numerical study, the relation between the relevant microstructural variables (i.e.; initial porosity and void aspect ratio) for a rather large range of applied stretch is put into evidence for two types of loading. Finally, a finite element (FE) simulation is presented, and the homogenization model is assessed through comparison of its predictions with the corresponding FE results. The illustrated agreement between the results demonstrates a good accuracy of the model up to rather large deformations.