A CONTACT PROBLEM WITH FRICTION AND ADHESION FOR AN ELASTIC LAYER WITH STIFFENERS

The contact of an elastic layer with an infinite stiffener to which a uniform constant normal load and a concentrated tangential force are a pplied, is considered. In the neighbourhood of the point of applicatio n 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 f riction is taken into account. Outside this segment the stiffener and the layer are under conditions of complete adhesion. The problem is re duced to a Prandtl-type integro-differential equation specified on two semi-infinite segments, for whose solution an analytical method is pr oposed. The method is based on reducing the equation to a vectorial Ri emann problem and then to an algebraic Poincare-Koch system. The latte r admits of an explicit solution and also inversion through recurrent relations that are effective when using numerical computations. The le ngth of the Coulomb friction zone and the contact tangential stresses in the adhesion zone are determined. Unlike Melan's problem [1] the co ntact stresses have no logarithmic singularity and are continuous ever ywhere in the contact area. The solution of the problem of the contact of a layer with a finite stiffener subject to a uniform pressure alon g the whole length and to an extension by forces concentrated at the t ips is also obtained. The contact area is divided into an intermediate zone of adhesion and two zones of coulomb friction. The problem is re duced to a Prandtl-type integro-differential equation specified on the segment, and it is solved by analogy with the solution of the equatio n of the first problem. Such a formulation of the problem implies that the contact tangential stresses are bounded at the tips of the stiffe ner and are continuous at the points of the boundary between the zones of adhesion and Coulomb friction. When adhesion occurs along the enti re line of contact the tangential stresses, in general, have a root si ngularity [2]. In the problem of the contact of a plane punch with a h alf-plane under conditions of friction and adhesion, the contact stres ses at the tips of the punch have 131 a power singularity (that differ s from a root one).