Elastic potentials on piecewise smooth surfaces

Abstract

Elastic potentials are functions which describe the influence of surface layers on points in the elastic continuum. These functions are smooth almost everywhere. The exceptional points are the points on the surface where the layer is defined. There some potentials or their tractions have a discontinuity, i.e., on traversing the surface the function jumps. A closer study shows that the behaviour depends on the character of the point on the surface, specifically, whether it is a smooth or a non-smooth point. The behaviour of the potentials at smooth points is well known. In this paper we study their behaviour at non-smooth points. We find that the potential of the first kind is a continuous function at any non-smooth point, but its traction becomes singular at such points. The potential of the second kind behaves at such points as at smooth points, i.e., it jumps. The factor 1/2 which characterizes its jump term at smooth points becomes at non-smooth points a (3×3)-matrix whose elements depend on the solid angle of the point and on Poisson's ratio v.