Generalized pseudo-Finsler geometry applied to the nonlinear mechanics of torsion of crystalline solids

Abstract

A continuum theory of the mechanical behavior of solid materials is presented wherein fundamental geometric quantities such as the metric tensor and connection coefficients can depend on one or more director vectors, also called internal state vectors. This theory, referred to as generalized pseudo-Finsler geometric continuum mechanics, enables depiction of a very broad class of physical phenomena in deformable solid bodies. The general nonlinear theory is reported first, primarily summarizing prior work by the author. Next, a new application of the theory to torsional deformation of solids is presented, whereby a cylindrical sample of material may be simultaneously subjected to twisting, extension or compression along its axis, as well as possible radial confinement or contraction. The internal state variable, when normalized by a regularization length, is identified with an order parameter associated with inelastic deformation that may include slippage, fracture, and/or structural collapse. Evolution of the internal state follows a generalized Ginzburg–Landau type of kinetic equation. For axially homogeneous fields, a coupled system of nonlinear partial differential equations is obtained that can be integrated numerically. Results are first documented for generic solids representative of many crystals that exist in nature. Then solutions corresponding to realistic properties of crystals of boron carbide ceramic and ice are reported. Results for boron carbide predict a dominant effect of shearing over compression on the structural transformation process, in agreement with observations from atomistic simulations. Results for ice demonstrate periods of steady plastic flow under constant applied average shear stress as well as torsional rigidity varying with sample size. Existence of both phenomena agrees with experimental observations.