Constitutive models for power-law viscous solids containing spherical voids

In the analysis of non-linear porous solids, it is commonplace to employ a spherical unit cell owing to the simplicity it affords. The macroscopic constitutive response of the material is then predicted based upon either uniform traction or linear displacement/velocity boundary conditions applied to the outer surface of the cell. In this investigation, we carry out a careful computational analysis of the effect of these two types of boundary conditions on the macroscopic response of the (idealized) porous solid and in particular, we explore the sensitivity of the predicted response to the macroscopic stress, void volume fraction and material non-linearity. The numerical results are then used as a basis for establishing an approximate constitutive model that is expressed in a compact, explicit form. The study is carried out in the context of an incompressible, isotropic power-law viscous matrix material, and the computational analysis is focused on axisymmetric deformation of the unit cell. While the macroscopic strain-rate potential is found to exhibit a dependence on the third invariant of the macroscopic stress deviator, this dependence is slight (particularly for the linear displacement/velocity boundary condition) and, toward developing an approximate strain-rate potential applicable to general macroscopic stress states, a simple averaging scheme is employed to suppress the role of this quantity. Guided by the numerical results as well as by various previously proposed constitutive relations, an approximate generalized elliptic form for the macroscopic strain-rate potential is then proposed. The constitutive potential which is ultimately developed involves a fairly simple dependence upon the void volume fraction and the properties of the matrix material, yet it gives rise to predictions that agree well with the detailed unit cell calculations over the full range of properties and macroscopic stress states considered.