It was noted that, in order to use the Volterra theory of dislocations to describe mechanical phenomena in the dislocation core, it was necessary to take account of non-linear effects via the application of the non-linear theory of elasticity. By using the examples of wedge disclinations and screw dislocations, it was shown here that making allowance for physical and geometrical non linearity in problems concerning the equilibrium of an elastic body containing an isolated defect could produce qualitatively new results as compared with the linear theory. One result was the possible existence of so-called singular solutions that described the formation of a cavity along the defect axis in a continuous body. An integral relationship was formulated which was useful for analyzing the possible existence of a singular solution, as well as for determining the dependence of the radius of the resultant cavity upon the defect characteristics. The parameter ranges for which a singular solution existed were determined, and the sizes of the resultant cavities were calculated for particular families of non-linear elastic potentials. The singular solution was energetically more profitable, and its existence was not a consequence of a violation of the Hadamard inequality.

Singular Solutions of the Problems of the Non-Linear Theory of Elastic Dislocations. M.I.Karyakin, O.G.Pustovalova: Journal of Applied Mechanics and Technical Physics, 1995, 36[5], 789-95