On conservation laws in geometrically nonlinear elasto-dynamic field of non-homogenous materials

By applying Noether’s theorem to the Lagrangian density of non-homogenous elastic materials in the so-called Lagrangian framework, conservation laws in geometrically nonlinear elasto-dynamic field have been studied, and a clear picture of relations between the conservation laws in material space and the material balance laws is given. It is found that the mass density and Lamé’s moduli have to satisfy a set of first-order linear partial differential equations, which contain all the symmetry-transformations of space–time based on Newtonian viewpoint of mechanics. The existence and existent forms of conservation laws in material space are governed by these equations. Especially, translation and rotation of coordinates are symmetry-transformations of the Lagrangian density for obtaining both the conservation laws of homogenous material and the material balance laws of non-homogenous material, but change of coordinate scale is not. However, if the mass density and Lamé’s moduli satisfy special equations simplified from those partial differential equations, change of coordinate scale becomes a symmetry-transformation of the Lagrangian density from which a conservation law follows, whereas the associated material balance law does not exist still. An insight into the usability of those equations for constructing conservation laws is presented, and all the non-trivial conservation laws of the functionally graded material (FGM) layer bonded to a substrate are given for mechanical analysis.