A new restriction for initially stressed elastic solids

TL;DR

This paper introduces a fundamental restriction called initial stress reference independence (ISRI) for strain energy functions in initially stressed elastic solids, ensuring physical consistency and reducing unknowns in stress modeling.

Contribution

It establishes the ISRI restriction, demonstrates its importance, shows many existing models violate it, and proposes new strain energy functions satisfying this restriction.

Findings

Most existing strain energy functions violate ISRI.

New strain energy functions satisfying ISRI are derived.

ISRI reduces the unknowns in linear stress tensors, aiding stress measurement.

Abstract

We introduce a fundamental restriction on the strain energy function and stress tensor for initially stressed elastic solids. The restriction applies to strain energy functions $W$ that are explicit functions of the elastic deformation gradient $\mathbf{F}$ and initial stress $\boldsymbol \tau$, i.e. $W:= W(\mathbf F, \boldsymbol \tau)$. The restriction is a consequence of energy conservation and ensures that the predicted stress and strain energy do not depend upon an arbitrary choice of reference configuration. We call this restriction: initial stress reference independence (ISRI). It transpires that almost all strain energy functions found in the literature do not satisfy ISRI, and may therefore lead to unphysical behaviour, which we illustrate via a simple example. To remedy this shortcoming we derive three strain energy functions that do satisfy the restriction. We also show that using initial strain (often from a virtual configuration) to model initial stress leads to strain energy functions that automatically satisfy ISRI. Finally, we reach the following important result: ISRI reduces the number of unknowns of the linear stress tensor of initially stressed solids. This new way of reducing the linear stress may open new pathways for the non-destructive determination of initial stresses via ultrasonic experiments, among others.