Comparative modelling results between a separable and a non-separable form of principal stretches–based strain energy functions for a variety of isotropic incompressible soft solids: Ogden model compared with a parent model

Abstract

This paper presents a comparative analysis on the application of two models from the family of principal stretches–based strain energy functions to a wide range of incompressible rubber-like materials, including hydrogels, foams, silicone elastomers, filled rubbers and biomaterials. The two models of concern here are the seminal Ogden model of the separable form, and a more general non-separable form parent model proposed by the author. To this end, the three-term Ogden model, with corresponding six model parameters, and the one-term parent model with four model parameters, are applied to exemplar extant deformation datasets of the aforementioned rubber-like materials, and the ensuing modelling results are then compared. This comparison encompasses the quality of the obtained fits as measured by the resulting relative errors, and the mechanistic validity of the ensuing strain energy function as assessed by the convexity of the iso-energy plots in the principal stretches λ1,λ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left({\\lambda }_{1},{\\lambda }_{2}\\right)$$\\end{document} plane. The studied modes of deformation involve a wide range of deformation types including uniaxial, biaxial, simple shear and inflation. It is shown that for the considered range of materials and deformations, the optimal fits obtained by the Ogden model either (i) possess higher relative errors; or (ii) offer a marginal improvement but with the loss of convexity. This loss of convexity is further exemplified on considering an additional soft tissue dataset. Finally, an important implication of the separable versus non-separable functional forms of the two models in application to the deformation of elastomers will be demonstrated, where a separable function proves to be an inappropriate choice. In all these applications, the proposed parent model, by contrast, will be shown to provide equally good or better fits, with fewer model parameters, while remaining free of the foregoing undesirable modelling outcomes.