Stresses and energy-stresses

The traction at the boundary of a continuum body leads almost straightforwardly to the Cauchy stress tensor. By contrast, forces acting on a ‘defect’ in a continuum rather address to the introduction of an energy-stress tensor. In elasticity, such a tensor, also known as the Eshelby stress tensor, plays a fundamental role in the theory of inhomogeneous and heterogeneous (i.e. non-uniform) materials. Surprisingly, the Cauchy stress tensor turns out to be eventually expressible in the form of an energy-stress tensor. Eshelby and Cauchy tensors play distinct physical roles and, in order to avoid misunderstandings, a discriminating criterion is desirable. In elasticity, one may note that the Cauchy stress is defined in the current framework. Thus, it is a full Eulerian tensor. By contrast, the Eshelby stress is a full Lagrangian one as it is completely defined in the referential framework. Unfortunately, such a criterion fails in certain circumstances. This is the case of liquid crystals whose mechanical behaviour is entirely described in the actual framework. As is known, the equilibrium of line-singularities in liquid crystals is governed by an energy-stress tensor: the so-called Ericksen tensor. The question may arise whether such a tensor is Cauchy-like or Eshelby-like; or, also, whether the two coincide in this and other specific cases. It will be shown that the distinct physical role of the two tensors is preserved in any case. Such distinctness can be ascertained by a novel criterion, which appeals to dynamics.