A contact problem with friction and adhesion for an elastic layer with stiffeners

Abstract

The contact of an elastic layer with an infinite stiffener to which a uniform constant normal load and a concentrated tangential force are applied, is considered. In the neighbourhood of the point of application of this force, on the line of the contact of the stiffener with the layer a segment is separated out, on which the effect of the Coulomb friction is taken into account. Outside this segment the stiffener and the layer are under conditions of complete adhesion. The problem is reduced to a Prandtl-type integro-differential equation specified on two semi-infinite segments, for whose solution an analytical method is proposed. The method is based on reducing the equation to a vectorial Riemann problem and then to an algebraic Poincaré-Koch system. The latter admits of an explicit solution and also inversion through recurrent relations that are effective when using numerical computations. The length of the Coulomb friction zone and the contact tangential stresses in the adhesion zone are determined. Unlike Melan's problem [1] the contact stresses have no logarithmic singularity and are continuous everywhere in the contact area. The solution of the problem of the contact of a layer with a finite stiffener subject to a uniform pressure along the whole length and to an extension by forces concentrated at the tips is also obtained. The contact area is divided into an intermediate zone of adhesion and two zones of coulomb friction. The problem is reduced to a Prandtl-type integro-differential equation specified on the segment, and it is solved by analogy with the solution of the equation of the first problem. Such a formulation of the problem implies that the contact tangential stresses are bounded at the tips of the stiffener and are continuous at the points of the boundary between the zones of adhesion and Coulomb friction. When adhesion occurs along the entire line of contact the tangential stresses, in general, have a root singularity [2]. In the problem of the contact of a plane punch with a half-plane under conditions of friction and adhesion, the contact stresses at the tips of the punch have [3] a power singularity (that differs from a root one).