Conservation laws in non-homogeneous electro-magneto-elastic materials

Noetherʹs theorem on invariant variational principles is applied for linear transversely isotropic non-homogeneous electro-magneto-elastic material. Invariant condition of the electromagnetic enthalpy is derived and expressed in original manner. Usually, inserting infinitesimal symmetry-transformations relating to material space of homogeneous materials into the invariant condition, one can obtain some expressions for the non-homogeneous material. These expressions indicate the relations between material inhomogeneity force and energy-momentum tensor. In order to obtain conservation laws of the non-homogeneous material, it follows from the invariant condition that the material coefficients have to satisfy a set of first-order linear partial differential equations with which there is a possible mathematical form of infinitesimal symmetry-transformation relating to material space. Satisfaction of these partial differential equations not only assures the existence of infinitesimal symmetry-transformations but also determines the number of them. Several independent and non-trivial conservation laws relating to material space with different non-homogeneous material coefficients are given. Some results of the path-independent integrals emanating from the conservation laws calculated around a crack tip are presented.