Abstract
The problem of the effect of an absolutely rigid stamp with a wedge planform on an elastic space is considered. There is assumed to be no friction in the domain of contact between the stamp and the half-space. Galin first considered this problem in [1]. The effect of the stamp on the half-space was accompanied, in that paper, by the effect of some loading outside i A characteristic singularity of this solution is the fact that the contact pressures p ( τ, ϑ) have a r −1 singularity at the wedge apex. Later, Rvachev attempted to solve the mentioned problem without the outside loading [2]. He reduced it to an eigenvalue problem for a certain differential equation on a sphere and utilized the Galerkin method. The Rvachev solution has a r γ−1 singularity at the wedge apex, where 0 < γ ( α) < 1, and 2α is the wedge angle. In this paper the problem of a wedge-shaped stamp with an arbitrary base is apparently successfully solved analytically for the first time by utilizing the asymptotic “method of large λ” [3], and the singularity in the contact pressure at the wedge apex is isolated exactly for sufficiently small α. It hence turns out that in the general case the function p ( r, ϑ) behaves as r − 3 2 cos (θ In r) in the neighborhood of the point r = 0, where θ = 0 ( α). The r −1 and r γ−1 singularities also hold, but are contained in the following members of the asymptotic of the function p ( r, ϑ) as r → 0. The question of the construction of an asymptotic solution of the problem under consideration for wedge angles near 2π is also posed herein.
Published Version
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