The problem of a wedge-shaped punch on the face of an elastic wedge

Specific features of the stress behavior in an incompressible half-space with internal V-shaped crack lying in a plane perpendicular to the surface of the half-space with a tip reaching the surface

“…(1,2) (μ) and b s (1,2) (μ) are unknown even functions of μ , s is an unknown parameter, which, for now, is assumed to be a purely imaginary number, and Γ(ξ) is the Euler gamma function [4,6]. Note that, for this choice of the unknown functions, conditions (4) are already satisfied, and it follows from condition (5) that…”

“…where the unknown functions σ s (1,2) are responsible for the distribution of normal stresses at ϕ = 0 , but beyond the crack boundaries, the unknown function u s determines normal deviations of the crack walls, and, in this case,…”

“…where 2) (μ) are unknown functions of the complex variable μ , and F(a, b, c, ξ) is the hypergeometric function [4,6]. Note that the functions X s (1) (μ) and X s (2) (μ) are not independent because they are determined via the same unknown function u s (θ).…”

“…Among these works, we should mention [18,19,23], from which, obviously, the extensive investigation of problems for bodies with subsurface cracks began. It is also necessary to dwell on [1,3,5,15,16,[20][21][22], which are devoted to the problems of an elastic half-space and a space wedge with internal and subsurface cracks, and on [2,17], in which contact problems for a space wedge with a V -shaped contact zone, similar in their statement, are considered. A more detailed analysis of works devoted to these problems can be found in [12,24].…”

Problems of a cut in a three-dimensional elastic wedge

“…Here we obtain ␥ * = 0.500 for 2␤ = 180 •7 and ␥ * = 0.815 for 2␤ = 90 • ; 17 at other crack angles, the index ␥ * corresponds to the singularity ␥ * − 1 of the contact pressures at the tip of a V-shaped punch with angle 2 − 2␤ on an elastic half-space. 8,9 Table 2 gives values of ␥ * for the three problems with ␤ * = 0 (the axis of symmetry of the cut is perpendicular to the edge of the wedge), = 0.3, and different angles of the crack and the elastic wedge. The influence of the boundary conditions on the faces of the wedge on the asymptotic forms required falls as the aperture angle 2␣ of the elastic wedge increases.…”