Effect of Friction on Edge Singularities in Slip Bands

Resistance to dislocation glide in slip bands due to the presence of other dislocations may be represented by an additional term in the integral equation formulation of the equilibrium state. As a consequence, the asymptotic behaviour of dislocation density at the edge of interval is modified from the conventional square root singularity. We propose an efficient and fast numerical method that is suitable for solving the resulting singular equations of the second kind based on the Gauss-Jacobi quadrature. The quadrature formulae involve fixed nodal points and provide exact results for polynomials up to degree 2n-1, where n is the number of nodes. A numerical example of the application of the Gauss-Jacobi rule to a slip band problem is provided as a demonstration of the validity of the method.