Stress tensors, Riemannian metrics and the alternative descriptions in elasticity

Marsden, J. E. and J. C. Simo

Springer LNP, 195, (1984), 369-383

Abstract:

Our purpose is first to discuss the remarkable duality existing between these alternative descriptions. A key role in describing this duality is played by the spatial formula connecting the spatial metric g and the Cauchy stress tensor: $\sigma$ = $\rho$ $\partial$$\psi$/$\partial$g , due to Doyle & Ericksen [1956]; and its material counter part connecting the material metric tensor G and the rotated stress tensor: $\sum$ = $\rho$$\partial$$\Psi$/$\partial$g , due to Simo & Marsden [1984]. These formulae illustrate the face that regardless of the description employed, the stress tensor in that description is obtained by varying the corresponding metric tensor.