Generalized Elastic-Plastic Decomposition in Defective Crystals

Abstract

I outline ideas which allow one to prove a rigorous type of elasticplastic decomposition between related crystal states. The context of the work is a theory of defective crystals where the microstructure is represented by fields of lattice vectors, and the construction of the “generalized elastic-plastic decomposition” is a geometrical procedure which does not involve any notion of stress. The decomposition of the change of state has the form F ^e₁ F ^p F ^e₂ , where F ^e₁ and F ^e₂ are piecewise elastic deformations and F ^p is a rearrangement of “unit cells” of a related crystal structure. The relevant unit cells do not derive from any supposed perfect lattice-like properties (e.g., translational symmetry) of the crystal states (no such presumption is appropriate since the crystals, and corresponding unit cells, support continuously distributed defects) but are constructed by means of the piecewise elastic deformations using a self-similarity property of the crystal states. Further, each individual unit cell, with its attendant distribution of lattice vector fields and defect densities, is translated to its new position by the rearrangement F ^p, so that the corresponding mapping of points of the body represents a “change of shape” of the crystal domain