In Doyle and Ericksen [1956, p. 77] it is observed that the Cauchy stress tensor
can be derived by varying the internal energy
e
with respect to the Riemannian metric on space:
= 2
e /gab
. Their formula has gone virtually unnoticed in the elasticity literature.
In this lecture we shall explain some of the reasons why this formula is, in fact, of fundamental significance. Some additional reasons for its importance follow. First of all, it allows for a rational derivation of the Duhamel-Neumann hypothesis on a decomposition of the rate of deformation tensor, which is useful in the identification problem for constitutive functions. This derivation, due to Hughes, Marsden and Pister, is described in Marsden and Hughes [1983, p. 204-207]. Second, it is used in extending the Noll-Green-Naghdi-Rivlin balance of energy principal (using invariance under rigid body motions) to a covariant theory which allows arbitrary mappings. This is described in §2.4 of Marsden and Hughes [1983] and is closely related to the discussion herein. Finally, in classical relativistic field theory, it has been standard since the pioneering work of Belinfante [1940] and Rosenfeld [1940] to regard the stress-energy-momentum tensor as the derivative of the Lagrangian density with respect to the spacetime (Lorentz) metric. This modern point of view has largely replaced the construction of "canonical stress-energy-momentum tensors". Thus, for the lagrangian formulation of elasticity (relativistic or not) the Doyle-Ericksen formulation plays the same role as the Belinfante-Rosenfeld formula and brings it into line with developments in other areas of classical field theory.
