The role of the symmetry group in the non-uniqueness of a uniform reference. Case study: An isotropic solid body

Abstract

According to the theory of materially uniform but inhomogeneous bodies, when the symmetry group of a material is continuous it induces a non-uniqueness to the uniform reference. Therefore, it is possible, that by manipulating the symmetry group the inhomogeneity of the material – namely, the dislocations – may cancel out. A solid mathematical framework is constructed in order to describe this situation using the language of exterior calculus. We set down a system of exterior differential equations which when solved render the totality of the uniform references that may be healed by a given symmetry group. From a mathematical point of view these equations have the form of Cartan’s equations of structure. We present the generic set of solutions of these equations and then specialize to the particular case of an isotropic solid body.