The motion of a straight screw dislocation in the Celli-Flytzanis lattice-snapping model was first treated in the almost-continuum limit. This led to a differential equation for the displacement which was essentially Legendre's equation; but with a purely imaginary argument. An intimate relationship between this problem and the continuum limit of the Slepyan brittle-elastic bond model was demonstrated. A related but different form of Legendre's equation again emerged in the continuum limit. Different boundary conditions to dislocation propagation required another solution of Legendre's equation, again for a purely imaginary argument.

Unified Treatment of Motion of an Extended Defect and a Crack via Legendre's Equation. N.H.March, M.Razavy, B.V.Paranjape: Journal of the Physics and Chemistry of Solids, 1999, 60[2], 285-7