Frictionless contact problem for a layer on an elastic half plane loaded by means of two dissimilar rigid punches

Abstract

The contact problem for an elastic layer resting on an elastic half plane is considered according to the theory of elasticity with integral transformation technique. External loads P and Q are transmitted to the layer by means of two dissimilar rigid flat punches. Widths of punches are different and the thickness of the layer is h. All surfaces are frictionless and it is assumed that the layer is subjected to uniform vertical body force due to effect of gravity. The contact along the interface between elastic layer and half plane will be continuous, if the value of load factor, <TEX>${\lambda}$</TEX>, is less than a critical value, <TEX>${\lambda}_{cr}$</TEX>. However, if tensile tractions are not allowed on the interface, for <TEX>${\lambda}$</TEX> > <TEX>${\lambda}_{cr}$</TEX> the layer separates from the interface along a certain finite region. First the continuous contact problem is reduced to singular integral equations and solved numerically using appropriate Gauss-Chebyshev integration formulas. Initial separation loads, <TEX>${\lambda}_{cr}$</TEX>, initial separation points, <TEX>$x_{cr}$</TEX>, are determined. Also the required distance between the punches to avoid any separation between the punches and the layer is studied and the limit distance between punches that ends interaction of punches, is investigated. Then discontinuous contact problem is formulated in terms of singular integral equations. The numerical results for initial and end points of the separation region, displacements of the region and the contact stress distribution along the interface between elastic layer and half plane is determined for various dimensionless quantities.