On the Rotated Stress Tensor and the
Material Version of the Doyle-Ericksen Formula

Simo, J. C. and J. E. Marsden

ARMA, 86, 213-231

Dedicated to J. L. Ericksen

Abstract:

DOYLE & ERICKSEN [1956, p.77] observed that the Cauchy stress tensor $\sigma$ can be derived by varying the internal free energy $\psi$ with respect to the Riemannian metric g on the ambient space: $\sigma$ = 2g$\partial$$\psi$/$\partial$g. Their formula has gone virtually unnoticed in the literature of continuum mechanics. In this paper we shall establish the material version of this formula: $\Sigma$ - 2g$\partial$$\bar{T}$/$\partial$G for the rotated stress tensor $\Sigma$, and shall addresss some of the reasons why these formulae are of fundamental significance. Making use of these formulae one can derive elasticity tensors and establish rate forms of the hyperelastic constitutive equations for the Cauchy stress tensor and the rotated stress tensor, as discussed in Sections 4 and 5.