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

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.Comment: 34 pages, LaTe

Non-singular screw dislocations as the Coulomb gas with smoothed out coupling and the renormalization of the shear modulus

A field theory is developed for a thermodynamical description of array of parallel non-singular screw dislocations in elastic cylinder. The partition function of the system is considered in the functional integral form. Self-energy of the dislocation cores is chosen in the form suggested by the gauge-translational model of non-singular screw dislocation. It is shown that the system of the dislocations is equivalent to the two-dimensional Coulomb gas. The coupling potential is prevented from a short-distance divergency since the core energies are taken into account. Two-point correlation functions of the stress components are obtained. Renormalization of the shear modulus caused by the presence of the dislocations is studied in the approximation of non-interacting dislocation dipoles. It is demonstrated that the finite size of the dislocation cores results in a modification of the renormalization law.Comment: 20 pages, LaTe