Generalization of the stress tensor to nonuniform fluids and solids and its relation to Saint-Venant’s strain compatibility conditions

Abstract

Combining some results from the classical theory of elasticity with the modern functional derivative approach to nonuniform systems, we obtain a well-defined stress tensor for nonuniform equilibrium fluids and solids. This stress tensor is symmetric and satisfies the force-balance equation, so it provides an unambiguous route to quantities such as the surface free energy. The ambiguities associated with earlier stress-tensor definitions are traced back to their failure to take account of Saint-Venant’s strain compatibility conditions.