On Isolated Point Singularities in Classical Elasticity

Abstract

This paper is concerned with isolated point singularities in an otherwise smooth solution to the mixed boundary value problem of classical linear elasticity on bounded regions. The singularity, which may be located either on the surface of the region or at an interior póint, is due to the application of point loads, material defects, non-smoothness of the surface or some other effect. The precise cause of the singularity is irrelevant to the purpose of the present treatment, which aims at determining a lower bound for the order of magnitude of all possible isolated point singularities that can occur in the solution. The results obtained are widely applicable but this same generality implies that the results are unlikely to be the best possible for any individual singularity e.g. that produced by point loads. Nevertheless, the present investigation may be regarded as complementing previous studies of singularities arising in elasticity or elliptic systems of differential equations, where the emphasis has tended towards establishing removability of singularities rather than estimating possible order of magnitudes. (Compare, for example, Serrin [3,11], Aviles [1]; Oleinik et al [7,8,9] and Grisvard [4,5] are amongst those who have examined order of magnitudes. See also Castellani Rizzonelli [2].)