Unified treatment of motion of an extended defect and a crack via Legendre's equation

• Published in: The Journal of physics and chemistry of solids 1999-02, Vol.60 (2), p.285-287
• Main Authors: March, N.H, Razavy, M, Paranjape, B.V
• Format: Article
• Language: English
• Subjects: Condensed matter: structure, mechanical and thermal properties; D. Defects; D. Lattice dynamics; Defects and impurities in crystals; microstructure; Exact sciences and technology; Fatigue, brittleness, fracture, and cracks; Linear defects: dislocations, disclinations; Mechanical and acoustical properties of condensed matter; Mechanical properties of solids; Physics; Structure of solids and liquids; crystallography
• ISSN: 0022-3697; 1879-2553
• DOI: 10.1016/S0022-3697(98)00277-7

The motion of a straight screw dislocation in the lattice snapping model of Celli and Flytzanis (V. Celli, N. Flytzanis, Journal of Applied Physics 41 (1970) 4443) is first treated in the almost continuum limit and shown to lead to a differential equation for the displacement which is Legendre's equation, but with pure imaginary argument. The intimate relation between this problem and the continuum limit of the brittle–elastic bond model of Slepyan is then demonstrated. Again, a related, though different, form of Legendre's equation emerges in the continuum limit. Different boundary conditions from the dislocation propagation require another solution of Legendre's equation, again for pure imaginary argument. Finally the displacements of the crack problem and the screw dislocation in these breaking-bond models are shown to be precisely related by an integral equation, which is formulated here.