Equivalent states and the concept of symmetry in a kinematic theory of continuously defective elastic crystals.

This article focuses on the equivalence of kinematic states and the symmetry of continuously defective elastic crystals, particularly those with uniform defectiveness characterized by a constant dislocation density tensor. It develops a geometric framework where a kinematic state is defined by three linearly independent smooth vector fields forming a lattice, whose defectiveness is measured by the dislocation density tensor and related structure functions. Uniformly defective states correspond to Lie groups acting on the crystal body, making these states self-equivalent with symmetry groups isomorphic to the associated Lie group. The paper further explores the concept of equivalence via elastic deformations preserving these invariants and introduces the notion of symmetry groups as collections of self-equivalences, laying groundwork for a phenomenological theory of material uniformity and symmetry in defective elastic crystals.