On Eshelby Tensors, Thermodynamics and Calculus of Variations

摘要

The connections between the notion of Eshelby tensor and the variation of Hamiltonian like action integrals are investigated, in connection with the thermodynamics of continuous open bodies exchanging mass, heat and work with their surrounding. Considering first a homogeneous representative volume element (RVE), it is shown that a possible choice of the Lagrangian density is the material derivative of a suitable thermodynamic potential. The Euler equations of the so built action integral are the state laws written in rate form. As the consequence of the optimality conditions of the resulting Jacobi action, the vanishing of the surface contribution resulting from the general variation of this Hamiltonian action leads to the well-known Gibbs-Duhem condition. A general three-field variational principle describing the equilibrium of heterogeneous systems is next written, based on the zero potential, the stationnarity of which delivers a balance law for a generalized Eshelby tensor in a thermodynamic context. Adopting the rate of the grand potential as the lagrangian density, a generalized Gibbs-Duhem condition is obtained as the transversality condition of the thermodynamic action integral, considering a solid body with a movable boundary.