The Elastic Sphere Under Concentrated Loads

Abstract

This paper contains an exact solution for the stress distribution in an elastic sphere under two equal and opposite concentrated loads, applied at the end points of a diameter. The solution is based on the Boussinesq stress-function approach to axisymmetric problems and is represented as a sum of two solutions: A singular solution in closed form, and a series solution corresponding to surface tractions which are finite and continuous throughout the surface of the sphere. It is shown that the singularity at the points of application of the loads is not identical with that arising at a concentrated load acting normal to a plane boundary. The existence of pseudosolutions to the problem under consideration, as well as to concentrated-force problems in general, is illustrated, and the validity of the present solution is confirmed through a limit process applied to the case of tractions uniformly distributed over a portion of the boundary. The numerical results for the normal stress on the plane of symmetry perpendicular to the load axis, are compared with the corresponding stress values obtained in a recent three-dimensional photoelastic investigation of the same problem by Frocht and Guernsey (1).

Controlling the adhesive pull-off force via the change of contact geometry

“…Considering the problem of controlling the sphere geometry independently of the local contact deformation, we restrict ourselves to the case of lateral self-equilibrated loading. Moreover, to make our consideration more illustrative, we examine the case of an elastic sphere under two equal and diametrically opposite concentrated loads [59].…”

Green’s function for symmetric loading of an elastic sphere with application to contact problems

“…Equations ( 8), (9a) and ( 12) indicate the expected Boussinesq-like θ −1 singularity as well as the weaker log θ singularity first described by Sternberg and Rosenthal [2]. The logarithmic singularities in S j (θ), j = 0, 1, 2 can be compared to the potential functions [D 1 ], [D 2 ], and [D 3 ] in equation ( 17) of [2], which provide a logarithmic singularity. In the present notation these are, respectively (using capital Φs so as not to be confused with the angle φ, and making the…”

“…Sternberg and Rosenthal [2] present an in-depth study of the nature of the singularities on elastic sphere loaded by two opposing concentrated point forces. As expected, the dominant inverse square singularity in the stress components can be removed by subtraction of an appropriate multiple of Boussinesq's solution for a point load at the surface of a half space.…”

“…No direct integration is required. The methodology for determining the analytical functions is motivated by the simple example of a point load on a sphere, for which we derive a solution similar in spirit to that of [2], but using a fundamentally different approach: partial summation of infinite series as compared with a functional ansatz. The present methods allows us to readily generalize the point load solution to arbitrary symmetric normal loading.…”

The elastic sphere under concentrated torques

“…This stress function corresponds to a singular self-equilibrated system of forces at B. Superposing (2.4) and (4.2) we obtain a function which fulfills all conditions of problem (P) except boundedness. In the terminology of E. Sternberg and F. Rosenthal [5], who treated the case of concentrated loads, it is a "pseudosolution" of the physical problem.…”