Initial Stress Symmetry and Applications in Elasticity

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 $\Psi =\Psi (\mathbf{F},\mathbf{\tau })$ in terms of initial stress $\mathbf{\tau }$ and the elastic deformation gradient $\mathbf{F}$. 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 $\Psi =\Psi (\mathbf{F},\mathbf{\tau })$ and present two examples of $\Psi (\mathbf{F},\mathbf{\tau })$ 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.

25 pages, 10 figures

References in corpus (2)

• Initial stresses in elastic solids: Constitutive laws and acoustoelasticity
• On stress-dependent elastic moduli and wave speeds

Cited by in corpus (8)

• Morphology of residually-stressed tubular tissues: Beyond the elastic multiplicative decomposition
• Influence of initial residual stress on growth and pattern creation for a layered aorta
• Shape transitions in a soft incompressible sphere with residual stresses
• On the existence of elastic minimizers for initially stressed materials
• The Föppl-von Kármán equations of elastic plates with initial stress
• Basis functions for residual stresses
• Responses of any arbitrary initially stressed reference and the stress-free reference
• Modelling of initially stressed solids: structure of the energy density in the incompressible limit