Three-dimensional harmonic functions near termination or intersection of gradient singularity lines: A general numerical method

Abstract

The harmonic function u near point 0 from which a single singularity ray emanates is assumed to be dominated by the term rλρρU where r = distance from point 0, p = known constant and ρ = chosen function of angular spherical coordinates θ, θ{symbol}, for which a partial differential equation with boundary conditions, especially those at the singularity rays, and a variational principle, are derived. Because grad U is nonsingular, a 1numerical solution is possible, using, e.g. the finite difference or finite element methods. This reduces the problem to finding λ of the smallest real part satisfying the equation Det (Aij) = 0 where Aij is a large matrix whose coefficients depend linearly on μ = λ(λ + 1). In general λ and Aij are complex. Solutions can be obtained either by reduction to a standard matrix eigenvalue problem for μ, or by successive conversions to nonhomogeneous linear equation systems. Computer studies have confirmed the feasibility of the method and have shown that highly accurate results can be obtained. Solutions for cracks and notches ending at a plane or conical surface, and for cracks ending obliquely at a halfspace surface, are presented. In these cases, λ is real and the singularity is always weaker (λ > p) than on the singularity line and may even disappear (λ > 1). Furthermore, elastic stresses under a wedge-shaped rigid sliding stamp or at a corner of a crack edge, and also harmonic functions at three-sided pyramidal notches, have been analyzed. Here λ < p was found to occur. A simple analytical solution for one class of special cases has also been found and used to check some of the numerical results. © 1974.