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.

Acknowledgements

JDC thanks the Weapons and Materials Research Directorate of the US Army Research Laboratory. DJB is grateful for support of Sandia National Laboratories under US Department of Energy contract no. DE-AC04-94AL85000. DLM acknowledges support of AFOSR (MURI (1606U81) and F49620-01-1-0034).

Notes

† A non-Riemannian metric tensor would not be positive-definite. Such metrics arise in more generalized geometries such as Finlser spaces, an example of which is Minkowski's spacetime (cf. Rund 60). Metrics are symmetric by definition.

‡ In the present work we focus on linear (also called affine) connections, which strictly obey (1). Nonlinear connections arise in more generalized spaces such as Finsler manifolds 60 and will not be dealt with here.