Initial stress symmetry and its applications in elasticity

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 Ψ in terms of initial stress τ and the elastic deformation gradient F, so we write Ψ=Ψ(F,τ). 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 condition. First, we examine how ISS restricts the possible choices of free energy densities Ψ=Ψ(F,τ) and present two examples of Ψ that satisfy the 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, which links nicely with the notion of homeostasis.