Contact problems of elasticity theory for wedge-shaped regions under conditions of friction and adhesion

Abstract

A wedge-shaped punch with included angle close to π is pressed onto an elastic half-plane by a centrally applied vertical force P; the contact area, divided into a frictional region and an adhesive region, is either known in advance (problem 1a) or has to be determined (problem 1b). Two-dimensional contact is investigated for an elastic wedge-shaped punch pressed down by a vertical force P, a horizontal force T and a couple of moment M (problem 2); the punch extends beyond the apex of the wedge and is flat-faced; the contact area is divided into an inner adhesive region and two outer regions of Coulomb friction. An analytical solution, accurate to within any prescribed limits, will be presented for these problems, thus generalizing the solution described in [1]; the method used is that employed in [2], where the problem is reduced to a Riemann vector problem for two pairs of functions (problems 1a, 1b) or three pairs (problem 2), which is then solved. The boundaries of the adhesive and frictional regions will be determined, and in problem 1b the contact area also. Formulae will be developed for the contact stresses. It will be shown that the stresses are continuous across the common boundary of the adhesive and frictional regions. The statement made in [3]that when the punch is pressed symmetrically onto the half-plane the ratio λ of the length 2 b of the adhesive region to the length 2 a of the contact region is the same for a flat-faced punch and a punch whose profile is described by the function f(x) = Λ¦x¦ n (n⩾ 1) will be disproved. It will be proved that if the punch profile is smooth in the vicinity of the point a, then λ is uniquely defined by Poisson's ratio v, the coefficient of friction μ and the exponent n; it is independent of the coefficient A and the force P (in particular, λ in problem 1b is independent of the included angle of the punch). The introduction of the regions of friction in the contact area for problem 2, enables one not only to eliminate oscillation of the contact stresses near the ends of the punch, but also to construct an analytic solution of the contact problem for a wedge when the contact shear and normal stresses are unknown (such a solution has not been obtained when the punch is fully adhesive). The problem of two wedge-shaped elastic bodies in contact with no shear stresses was solved in [4].