Energy-Momentum Tensors in Nonsimple Elastic Dielectrics

We use Noether's theorem to derive energy-momentum tensors for a simple elastic material, a nonsimple elastic material of grade two, a simple elastic dielectric and a nonsimple elastic dielectric of grade two. The technique is easily extendable to a nonsimple elastic dielectric of any grade. Introduction The concept of the force acting on a defect, e.g., an impurity, vacant lattice site, dislocation, inclusion, void or a crack, is related, in a broad sense, to the notion of an inhomogeneity. Eshelby [1] showed that the force on a defect in an elastic body equals the integral of an energy-momentum tensor over a closed surface enclosing only this defect; subsequently he [2] derived the energy-momentum tensor for a second-grade elastic material which is nonsimple according to Noll [3]. Recently Maugin and Trimarco [4] used a general variational principle and the concept of pseudomomentum to derive the energy-momentum tensor for a second-grade elastic material. By invoking the ideas of a basic reference configuration, Maugin and Epstein [5], Epstein and Maugin [6], Maugin and Trimarco [4] and Maugin et al. [7] have derived energy-momentum tensors for simple electromagnetic elastic materials. Here we use Noether's theorem [8] to derive the energy-momentum tensor for simple and nonsimple elastic dielectrics. This approach requires considerably less work as compared to the techniques employed previously and is easily extendable to electromagnetic materials of grade N. We note that Noether's theorem ~as been used by Knowles and Sternberg [9] to derive conservation laws in linearized and finite elastostatics, by Golebiewska-Herrmann [10] to obtain a unified formulation leading to all conservation laws of continuum mechanics, by Pak and Herrmann [11] to obtain conservation laws and the material momentum tensor for an elastic dielectric, and by Maugin [12] to obtain pseudo-momentum and Eshelby's material tensor in electromagneto-mechanical framework. Ma,ugin [12] noted that the work can be extended to nonsimple hyperelastic solids but did not provide any results. Maugin and Trimarco [13] have applied Noether's theorem to study Eshelby's tf'.n!:or for nf'.m~ti~ 1imlici ~rv!:t~1!: 276 YU-NING HUANG AND R.C. BATRA Noether's Theorem We state a version of Noether's theorem [8] appropriate for our work; according to Soper [14] earlier versions of the theorem were given by Hamel [15]. For fields 4>J(X), J = 1,2,..., N, depending upon coordinates xcx, a = 1,2,..., M, the Lagrangian £, in general will be a function of X, c/> and derivatives of c/> up to some finite order. That is 1:, = 1:,(4>J, lJa4>J, lJalJf;J4>J,...; Xa), (1) where 8cx8f3c/>J = 82c/>J/8xcx8Xf3. 8cx4>J = 84>J / 8Xcx, (2) The variation of the Hamiltonian action A is given by