The Einsteinian T(3)-Gauge Approach and the Stress Tensor of the Screw Dislocation in the Second Order: Avoiding the Cut-off at the Core

Abstract

A translational gauge approach of the Einstein type is proposed for obtaining the stresses that are due to non-singular screw dislocation. The stress distribution of second order around the screw dislocation is classically known for the hollow circular cylinder with traction-free external and internal boundaries. The inner boundary surrounds the dislocation's core, which is not captured by the conventional solution. The present gauge approach enables us to continue the classically known quadratic stresses inside the core. The gauge equation is chosen in the Hilbert--Einstein form, and it plays the role of non-conventional incompatibility law. The stress function method is used, and it leads to the modified stress potential given by two constituents: the conventional one, say, the `background' and a short-ranged gauge contribution. The latter just causes additional stresses, which are localized. The asymptotic properties of the resulting stresses are studied. Since the gauge contributions are short-ranged, the background stress field dominates sufficiently far from the core. The outer cylinder's boundary is traction-free. At sufficiently moderate distances, the second order stresses acquire regular continuation within the core region, and the cut-off at the core does not occur. Expressions for the asymptotically far stresses provide self-consistently new length scales dependent on the elastic parameters. These lengths could characterize an exteriority of the dislocation core region.

arXiv comment34 pages, LaTeX

References in corpus (12)

• Metric-Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance
• Geometric Theory of Defects
• On Dislocations in a Special Class of Generalized Elasticity
• An elastoplastic theory of dislocations as a physical field theory with torsion
• Nonholonomic Mapping Principle for Classical Mechanics in Spaces with Curvature and Torsion. New Covariant Conservation Law for Energy-Momentum Tensor
• A nonsingular solution of the edge dislocation in the gauge theory of dislocations
• Dislocation theory as a 3-dimensional translation gauge theory
• Scattering on Dislocations and Cosmic Strings in the Geometric Theory of Defects
• The Dislocation Stress Functions From the Double Curl T(3)-Gauge Equation: Linearity and a Look Beyond
• Screw dislocations in the field theory of elastoplasticity
• Disclination vortices in elastic media
• A Modified Screw Dislocation With Non-Singular Core of Finite Radius From Einstein-Like Gauge Equation (Non-Linear Approach)

Cited by in corpus (4)

• The gauge theory of dislocations: static solutions of screw and edge dislocations
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• A gauge theoretic approach to elasticity with microrotations
• Non-singular screw dislocations as the Coulomb gas with smoothed out coupling and the renormalization of the shear modulus