A geometric framework for the kinematics of crystals with defects

Abstract

Presented is a general theoretical framework capable of describing the finite deformation kinematics of several classes of defects prevalent in metallic crystals. Our treatment relies upon powerful tools from differential geometry, including linear connections and covariant differentiation, torsion, curvature and anholonomic spaces. A length scale dependent, three-term multiplicative decomposition of the deformation gradient is suggested, with terms representing recoverable elasticity, residual lattice deformation due to defect fields, and plastic deformation resulting from defect fluxes. Also proposed is an additional micromorphic variable representing additional degrees-of-freedom associated with rotational lattice defects (i.e. disclinations), point defects, and most generally, Somigliana dislocations. We illustrate how particular implementations of our general framework encompass notable theories from the literature and classify particular versions of the framework via geometric terminology.