Further developments in the constitutive theory of the family of models with higher order rational approximant response functions for application to isotropic compressible soft solids

Abstract

In a series of prior papers, various members of a new family of incompressible constitutive models whose response function(s), namely $W_1$ and $W_2$, are of higher order rational approximants in invariants $I_1$ and $I_2$ were devised for application to the finite deformation of isotropic rubber-like materials. The extension of the models at the bottom of the hierarchy of this family; i.e., with $W_1$ and $W_2$ of orders [1/1] and [0/1], respectively, to the compressible case has been presented previously (Anssari-Benam & Horgan, 2022a). The current work is concerned with developing the compressible forms of the recently developed incompressible models at the top of the hierarchy of this family, where $W_1$ and $W_2$ are of further generalised orders; e.g., [$\beta$/1] and [1/1], respectively. The improvement in the accuracy of the modelling results will be demonstrated, on using extant multiaxial and uniaxial experimental datasets of a wide variety of compressible soft solids, ranging from polyethylene foams to (hydro)gels and biological materials. The presented developments here complete hitherto the extension of the incompressible forms of this family of models to the compressible case, and provide more accurate constitutive models for application to the large deformation of compressible soft materials.