#### Vol. 10, No. 3, 2015

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Dynamic conservation integrals as dissipative mechanisms in the evolution of inhomogeneities

### Xanthippi Markenscoff and Shailendra Pal Veer Singh

Vol. 10 (2015), No. 3, 331–353
DOI: 10.2140/jomms.2015.10.331
##### 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 $M$-integral, 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.

##### Keywords
elastodynamics, conservation laws, Noether's theorem, dissipative mechanism, dynamic $J$-integral, dynamic $L$-integral, dynamic $M$-integral, inhomogeneity