On an internal symmetry for arbitrary dislocations in solids

Abstract

We discuss an arbitrary distribution of dislocations moving in an anisotropic finite linear elastic solid. The field equations for theelastic strain tensor e are decomposed into two independent systems of equations, the equations for acompatible elastic displacement fields and the equations for anincompatible elastic strain ɛ v . This can be done in such a way thats contains the full information on anisotropy, external forces, and boundaries, whereas ɛ v contains only a single material constant μ, which is related to a signal velocity\(c_T = \sqrt {\mu /\rho } \), wherep is the mass density. In order to understand the symmetries of the ɛ v -field equations we introduce ac T -relativistic space-time. As a consequence of certain hypothesis concerning the balance of eigenstresses for moving dislocations the Lorentz group becomes the symmetry group for the ɛ v -field equations. We call this aninternal symmetry. Thematerial symmetry of the field equations for the elastic displacement vectors which is defined by Hooke's tensor breaks this Lorentz symmetry for the complete elastic strain e. Some conclusions for the dynamics of dislocations are discussed. It is found that Seeger's theory of kinks on dislocations describes elementary processes of this dynamics. Within the limits of the continuum model plasticity becomes a field theory with broken Lorentz symmetry.