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

The formula of Doyle and Ericksen (1956) (DEF) for the Cauchy stress tensor is investigated analytically within the framework of continuum mechanics. The basic relations are reviewed; the material version of the DEF for the rotated stress tensor is derived; a rotated elasticity tensor is obtained; and the rate form of the hyperelastic constitutive equations given by Marsden and Hughes (1983) is established materially. Spatial, material, and convected versions of the DEF for hyperelasticity are shown to yield the principle of virtual work directly from the covariant formulation of energy balance.