DYNAMIC CONSERVATION INTEGRALS AS DISSIPATIVE MECHANISMS IN THE EVOLUTION OF INHOMOGENEITIES

Abstract

By the application of Noether’s theorem, conservation laws in linear elastodynamics are derived by invariance of the Lagrangian functional under a class of infinitesimal transformations. The recent work of Gupta and Markenscoff (2012) providing a physical meaning to the dynamic J -integral as the variation of the Hamiltonian of the system due to an infinitesimal translation of the inhomogeneity if linear momentum is conserved in the domain, is extended here to the dynamic M- and L-integrals in terms of the “if” conditions. The variation of the Lagrangian is shown to be equal to the negative of the variation of the Hamiltonian under the above transformations for inhomogeneities, which provides a physical meaning to the dynamic J -, L- and M-integrals as dissipative mechanisms in elastodynamics. We prove that if linear momentum is conserved in the domain, then the total energy loss of the system per unit scaling under the infinitesimal scaling transformation of the inhomogeneity is equal to the dynamic Mintegral, and if linear and angular momenta are conserved then the total energy loss of the system per unit rotation under the infinitesimal rotational transformation is equal to the dynamic L-integral. Conservation laws can be expressed as dissipative mechanisms related to the variation of the energy of the system due to infinitesimal configurational variations in the inhomogeneities. Eshelby [1951] used the energy momentum tensor to define the force on an elastic singularity as a variation of the total energy of the body due to the infinitesimal displacement of the defect. Furthermore, he provided additional insights by extending this idea in a series of papers [Eshelby 1956; 1970; 1975] through his ingenious cutting and rewelding thought experiment. Rice [1968] independently discovered the two-dimensional path-independent J -integral for a crack. Gunther [1962] and Knowles and Sternberg [1972] derived two additional nontrivial conservation laws (M- and L-integrals) by applying Noether’s theorem [Noether 1918] in linear elastostatics. Rice and Drucker [1967] calculated the energy changes during the growth of voids and cracks. Budiansky and Rice [1973] interpreted these new laws as energy release rates associated with the expansion and the rotation rates of a cavity or a crack. Rice [1985] provided further applications of these integrals to the defects. Fletcher [1976] extended the application of Noether’s theorem to derive the conservation laws in linear elastodynamics, and established the completeness of the corresponding conservation laws under a certain group of the infinitesimal transformations. Hermann [1981; 1982] presented a unified formulation to recover the conservations laws by employing different vector calculus operations on the Lagrangian