The Convexity Properties of a Class of Constitutive Models for Biological Soft Issues

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Abstract
During the last three decades, the theory of nonlinear elasticity has been used extensively to model biological soft issues. Although this research has generated many different models to describe the constitutive response of these issues, surprisingly little work has been done to analyze the mathematical features of the various models proposed. Here we carefully examine the convexity properties, specifically the Legendre-Hadamard and the strong ellipticity conditions, for a class of soft-issue models related to the classical Fung model. For various versions of models within this class, we discover necessary and sufficient conditions that must be satisfied for a model to have one or both of these convexity properties. We interpret our results mechanically and discuss the related implications for constitutive modeling. Also, we explore some more general points on how convexity can be studied and on the relation among different convexity properties.