The elastic sphere under arbitrary concentrated surface loads

Abstract

The elastostatic problem of a sphere subject to a concentrated surface load of arbitrary direction which is equilibrated in a very simple manner by a distribution of surface tractions is solved. Particular attention is placed in the analysis of the singularity at the point of application of the concentrated load.The solution obtained provides a means for reducing problems pertaining to a sphere under arbitrary concentrated and distributed loads to a regular second boundary-value problem for the sphere. For illustration the problem of a sphere acted by two equal, opposite and collinear loads applied at two arbitrary surface points is treated.The paper also contains an exposition and an essential extension of an integration scheme developed by A1-mansi. Thus, an explicit integral representation of the displacements in an elastic sphere in terms of a vector valued harmonic potential which coincides on the surface with the tractions is obtained. RESUMI~On prdsente la rdsolution du probldme 61astostatique d'une sphdre soumise en surface h une charge ponctuelle de direction arbitraire 6quilibr6e par une distribution trds simple de contraintes superficielles. On s'attache plus spdcialement ~t l'analyse de la singularit6 au point d'application de la charge ponctuelle.La solution obtenue permet de rdduire les probldmes d'une sphdre soumise de manidre quelconque h des charges ponctuelles et rdparties, ~t un probl~me avec conditions aux limites rdgulidres. A titre d'exemple, on traite le probldme d'une sphdre soumise h deux forces ponctuelles, co lindaires, 6gales et de sens opposd, appliqudes en deux points arbitraires de la surface.L'article contient 6galement une exposition et une extension d'un schdma d'intdgration ddvelopp6 par A1-mansi. On obtient ainsi une reprdsentation explicite, sous forme d'intdgrale, des ddplacements dans une sphdre 61astique, en termes de potentiel vecteur qui coincide, ~ la surface, avec le champ de contraintes.

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

“…Sternberg and Rosenthal showed that the quickly convergent residual field retains a weaker singularity of logarithmic form, a result that is also evident in the solution developed here. The singular solutions obtained by Sternberg and Rosenthal were extended to arbitrarily oriented point forces by Guerrero et al [3]. Our interest here is in developing an analogous separation of the Green's function (circular ring loading).…”

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“…Sternberg and Rosenthal showed that the quickly convergent residual field retains a weaker singularity of logarithmic form, a result that is also evident in the solution developed here. The singular solutions obtained by Sternberg and Rosenthal were extended to arbitrarily oriented point forces in [Guerrero and Turteltaub 1972]. Our interest here is in developing an analogous separation of the Green's function (circular ring loading).…”

“…ALEXEY S. TITOVICH AND ANDREW N. NORRIS A series of products of Legendre functions is given by Equations 6.11.3.1 and 6.11.3.2 of [Brychkov 2008]:…”