Singularities of elastic stresses and of harmonic functions at conical notches or inclusions

The nature of singularities at the vertex of conical notches and inclusions is found for problems of potential theory and for elastostatic problems of torsion and of axisymmetric stress. A solution in terms of spherical harmonics and a general numerical solution based upon the field equations are used to determine the power dependence of the field quantities upon the distance from the apex of the cone. Eigenvalues representing the exponent are computed for various values of cone angle and for various Poisson ratios. Whereas singularities in two-dimensiona problems of potential theory and of elasticity have already been thoroughly examined, Iittfe is known of the singularities in three-dimen- sional problems. Even the singularities at conical notches and inclusions, the simplest of the three-dimensional problems, seem to have escaped attention and will, therefore, be examined in this study. Spherical coordinates will be employed and will be separated in a manner which is analogous to that introduced by Knein(l) (upon suggestion of van K&-man), and later indep~ndentIy developed by WilliamsfZ), and Karp and Karal(3). Attention will be restricted to problems that either are axisymmetric or are reducible to ane-dimensional problems. The axisymmetric elasticity problems for a cone has recently been investigated by Thompson and Little(4), but they considered only cones of acute vertex angle (less than rr) in which no singularities arise. Nevertheless, it is easy to check that their general solution applies for all angles, and so it can be used herein. Two variants of soIution will be used. One will be based on spherical harmonics(4). The other will be a genera1 numerical method which, after some additional refinements(5), is applicable to three-dimensional singular problems in generaI. A demonstration of this numerical method is a second objective of this study. Abstract-The nature of singularities at the vertex of conical notches and inclusions is found for problems of potential theory and for elastostatic problems of torsion and of axisymmetric stress. A solution in terms of spherical harmonics and a general numerical solution based upon the field equations are used to determine the power dependence of the field quantities upon the distance from the apex of the cone. Eigenvalues representing the exponent are computed for various values of cone angJe and for various Poisson ratios. I. INTRODUCTION Whereas singularities in two-dimensional problems of potential theory and of elasticity have already been thoroughly examined, little is known of the singularities in three-dimen­ sional problems. Even the singularities at conical notches and inclusions, the simplest of the three-dimensional problems, seem to have escaped attention and will, therefore, be examined in this study. Spherical coordinates will be employed and will be separated in a manner which is analogous to that introduced by Knein(l J (upon suggestion of von Karman), and later independently developed by WiJliams(2), and Karp and Karal(3J. Attention will be restricted to problems that either are axisymmetric or are reducible to one-dimensional problems. The axisymmetric elasticity problems for a cone has recently been investigated by Thompson and Little(4), but they considered only cones of acute vertex angle (less than rr) in which no singularities arise. Nevertheless, it is easy to check that their general solution applies for all angles, and so it can be used herein. Two variants of solution will be used. One will be based on spherical harmonics(41. The other will be a general numerical method which, after some additional refinements(5), is applicable to three-dimensional singular problems in general. A demonstration of thi~ numerical method is a second objective of this study. 2. POTENTIAL THEORY Consider a three-dimensional harmonic function u(O, ¢, r) in spherical coordinates e, ¢, r which are centered in the apex of a cone whose axis coincides with the pole. In analogy with the method of Knein(l), Williams(2), and Karp and Karal(3), the radial coordinate r will be separated by assuming the harmonic function in the form