CONTACT PROBLEMS OF ELASTICITY THEORY FOR WEDGE-SHAPED REGIONS UNDER CONDITIONS OF FRICTION AND ADHESION

A wedge-shaped punch with included angle close to pi is pressed onto a n elastic half-plane by a centrally applied vertical force P; the cont act area, divided into a frictional region and an adhesive region, is either known in advance (problem la) or has to be determined (problem 1b). Two-dimensional contact is investigated for an elastic wedge-shap ed 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 inne r adhesive region and two outer regions of Coulomb friction. An analyt ical solution, accurate to within any prescribed limits, will be prese nted for these problems, thus generalizing the solution described in [ 1]; the method used is that employed in [2], where the problem is redu ced to a Riemann vector problem for two pairs of functions (problems l a, lb) or three pairs (problem 2), which is then solved. The boundarie s of the adhesive and frictional regions will be determined, and in pr oblem lb the contact area also. Formulae will be developed for the con tact stresses. It will be shown that the stresses are continuous acros s the common boundary of the adhesive and frictional regions. The stat ement made in [3] that when the punch is pressed symmetrically onto th e half-plane the ratio lambda of the length 2b of the adhesive region to the length 2a of the contact region is the same for a flat-faced pu nch and a punch whose profile is described by the function f(x) = LAMB DA Absolute value of x n (n greater-than-or-equal-to 1) will be dispro ved. It will be proved that if the punch profile is smooth in the vici nity of the point a, then lambda is uniquely defined by Poisson's rati o v, the coefficient of friction mu and the exponent n; it is independ ent of the coefficient LAMBDA and the force P (in particular, lambda i n problem lb is independent of the included angle of the punch). The i ntroduction of the regions of friction in the contact area for problem 2, enables one not only to eliminate oscillation of the contact stres ses near the ends of the punch, but also to construct an analytic solu tion of the contact problem for a wedge when the contact shear and nor mal stresses are unknown (such a solution has not been obtained when t he punch is fully adhesive). The problem of two wedge-shaped elastic b odies in contact with no shear stresses was solved in [4].