Conservation laws in nonlinear elastodynamics

Abstract

The paper proposes a derivation of the full set of basic balance laws of nonlinear elastodynamics within the framework of Cartan's exterior calculus of forms. To that purpose a set of appropriate exterior differential forms is defined on the proper manifold. These forms are annulled by the solution surfaces of the field equations. They also generate a closed ideal in the ring of forms. Conservation laws are found as exact forms within this ideal. Fundamental equations that give the conservation law structure of the field equations are found. Their explicit solution is obtained for an arbitrary strain energy function. The paper bases on previous works by \textit{H. D. Wahlquist} and \textit{F. B. Estabrook} [e.g.: J. Math. Phys. 16, 1-7 (1975; Zbl 0298.35012); Classical quantum gravity 6, No.3, 263-274 (1989; Zbl 0672.53035)] and \textit{P. J. Olver} [e.g.: Arch. Ration. Mech. Anal. 85, 111-129 (1984; Zbl 0559.73019), and 131-160 (1984; Zbl 0582.73024); Bull. Am. Math. Soc., New Ser. 18, No.1, 21-26 (1988; Zbl 0643.49030)]. In addition to the classical balance laws of nonlinear elasticity, the conservation of canonical (material) momentum involving Eshelby's energy-momentum tensor is obtained. However, in the present case, this could have been deduced as a consequence of the other balance laws (especially the balance of physical momentum), a fact not noticed by the author.