Oldroyd's convected derivatives derived via the variational action principle and their corresponding stress tensors

A general form of Oldroyd's convective derivative is derived that is valid for an arbitrary microstructural variable, Phi(F), which is an unspecified but physically relevant function of the deformation gradient tensor, F, irrespective of its order or tensorial character, directly from the variational Principle of Least Action. The method is applied to adiabatic deformation of an ideal elastic medium following a six-step procedure, by which it is apparent that the variational derivation also requires a specific form of the corresponding thermodynamic pressure and extra stress tensor for a particular choice of Phi. A coupled set of evolution equations for all spatial variables necessary to describe the physical state of the material is derived that governs the reversible dynamics of an ideal elastic material under deformation. Furthermore, the method is shown to be readily extendable to inhomogeneous materials where stresses due to gradients in elasticity play a role in the dynamical response of the medium.