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.

Finsler differential geometry in continuum mechanics: Fundamental concepts, history, and renewed application to ferromagnetic solids

“…The following decomposition of G AB into a Riemannian part G ¯ AC and a director-dependent part G ^ B C has been proposed and used often for solving boundary value problems in prior work [61, 62, 91, 94]:…”

“…The latter two prescriptions are standard among phase field theories of fracture [103, 104]. A conformal Weyl-type transformation [94, 97, 105] is chosen for the dependence of G on ξ . When measured according to such a generalized Finsler metric, a material element expands with increasing ξ , physically associated with cavitation, void growth, and/or bulking that arise during progression of local degradation processes.…”

“…A double-well potential [107, 108] is used for f 0 , where the transformation energy barrier is A 16 at ξ = 1 2 . A conformal transformation [94, 97, 105] is used for G ( ξ ) . The material is assumed to expand in atomically disordered zones in the vicinity of phase boundaries, as has been likewise suggested for dislocations, point defects, stacking faults, and grain boundaries arising from anharmonicity [25, 109].…”

Connection Structures of Topological Singularity in Micromechanics from a Viewpoint of Generalized Finsler Space

“…The classical elasticity theory is based on the infinitesimal transformation [12][13][14] and the boson transformation. [2,3,15] On the other hand, in the Finslerian continuum mechanics, the geometric framework of finite deformation in nonlinear mechanics has been introduced, [33][34][35][36][37][38] and boundary value problems on grain boundary, twin boundaries, failure interfaces have been discussed. In this case, a reference state is given by a material coordinate X A , and an internal state variable D A is attached to each point.…”

“…Moreover, when a finite characteristic size in continuum is considered, a finite deformation of nonlinear elasticity is characterized by the non-locality in the Finsler space. [33][34][35][36][37][38] In this case, an internal state vector attached to material points has been introduced to formulate the finite deformation. Then, various boundary value problems on such as grain boundary, twin boundaries and failure interfaces have been discussed.…”