On the rotated stress tensor and the material version of the Doyle-Ericksen formula

Doyle & Ericksen [1956, p. 77] observed that the Cauchy stress tensor σ can be derived by varying the internal free energy ψ with respect to the Rie- mannian metric g on the ambient space: σ=2ϱ ∂ψ/∂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: \( \sum { = 2\varrho \partial \bar{\Psi }} /\partial G \) for the rotated stress tensor Σ, and shall address 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. The role of the rotated stress tensor in the formulation of continuum theories has been noted by Green & Naghdi [1965]. Generalizations of hypoelasticity based on the use of the rotated stress tensor have been considered by Green & McInnis [1967].