Initial Stress Symmetry and Applications in Elasticity

TL;DR

This paper introduces the initial stress symmetry (ISS), a new constitutive condition for initially stressed solids, and explores its implications for free energy formulation and residual stress estimation, with applications in biomechanics.

Contribution

The paper proposes the initial stress symmetry (ISS) as a novel constitutive condition and demonstrates its use in deriving free energy density and residual stresses in initially stressed materials.

Findings

ISS restricts the form of free energy density in stressed solids.

Two example free energy functions satisfying ISS are presented.

ISS enables derivation of residual stress from Cauchy stress and deformation.

Abstract

An initial stress within a solid can arise to support external loads or from processes such as thermal expansion in inert matter or growth and remodelling in living materials. For this reason it is useful to develop a mechanical framework of initially stressed solids irrespective of how this stress formed. An ideal way to do this is to write the free energy density $\Psi= \Psi(\boldsymbol F, \boldsymbol {\tau})$ in terms of initial stress $\boldsymbol \tau$ and the elastic deformation gradient $\boldsymbol F$. In this paper we present a new constitutive condition for initially stressed materials, which we call the initial stress symmetry (ISS). We focus on two consequences of this symmetry. First we examine how ISS restricts the free energy density $\Psi = \Psi (\boldsymbol F, \boldsymbol \tau) $ and present two examples of $\Psi (\boldsymbol F, \boldsymbol \tau)$ that satisfy ISS. Second we show that the initial stress can be derived from the Cauchy stress and the elastic deformation gradient. To illustrate we take an example from biomechanics and calculate the optimal Cauchy stress within an artery subjected to internal pressure. We then use ISS to derive the optimal target residual stress for the material to achieve after remodelling.