Conservation laws in nonlinear elasticity, I.: one-dimensional elastodynamics
Introduction. The conservation laws satisfied by the solutions of a previously given system of differential equations play a basic role in various problems of continuum physics: conservation laws such as the conservation laws of energy, momentum and others that one uses for the analysis of theoretical properties of differential equation solutions (a priori estimates, theorems for existence and uniqueness, stability, etc.), as well as the numerical discretizations—the method of integral relations, conservative difference schemes and finite elements. In elasticity, in addition to the general applications mentioned above, the conservation laws (or, rather their path-independent integral forms) are of key importance in the study of cracks and dislocations [1, 2], Instructive review of the conservation laws in elasticity and their numerous applications is presented by Olver in [3], The results of Knowles, Sternberg [4], and Fletcher [5] obtained by variational principles being invariant at infinitesimal transformations are generalized for viscoelasticity in the paper [6], The traditional method for obtaining conservation laws is based on Noether's theorem and its generalizations [7, 8]. For its application it is necessary to analyze in advance the group properties of the initial differential system of equations and to obtain the functional for which Euler's equations appear as equations to be analysed. Moreover, if the functional is degenerate (this is possible in elasticity [9]) one loses the one-to-one map between the groups which admit the initial equations and the conservation laws [7, 8]. Thus the investigation of conservation laws by previous determination of the groups of corresponding functionals sometimes leads to tedious computations. In the present work we seek the conservation laws without using group analysis and a variational principle. In the case of an adiabatic, an elastic solid for the density, and the flux of an arbitrary conservation law, an overdetermined partial differential equations system of the first order is obtained, and the presence of the entropy permits one to perform a direct analysis for compatibility. Briefly, the main result that we obtain is the following. If the internal energy function does not satisfy a system of two partial differential equations of the second order with constant coefficients