Stress 1m!! Riemannian Metrics in Nonlinear Elasticity

U. Introduction In Doyle and Bricksen [1956. p. 77] it is observed that the Cauchy stress tensor 0 can be derived by varying the internal energy e with respect to the Riemannian metric on space: oab =Zpae/agab. 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 (see Sokolnikoff [1956, p. 359]), 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 NoU-Green-Naghdi-Rivlin balance of energy principle (using invariance under rigid body motions) to a covariant theory which allows arbitrary mappings. This is described in Section 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 [1939] and Rosenfeld [1940] to regard the stress-energy-momentum tensor as the derivative of the Lagrangian density with respect to the spacetime (Lorentz) metric; see for example, Hawking and Bllis [1973, Sect. 3.3] and Misner, Thorne and Wheeler [1973, Sect. 21.3]. This modern point of view has largely replaced the construction of "canonical •Research partially supported by DOE Contract DE-AT03-82BR12097. Lecture given in Chern's Mathematical Sciences Research Institute PDE Seminar, April 4, ]983.