Contact problems for a two-layer elastic wedge

Abstract

We obtain integral equations for plane contact problems for a two-layer wedge (composite) under three types of boundary conditions on one of its sides (absence of stresses, sliding, or rigid fixation). The composite consists of two wedges completely linked with each other, which have different opening angles and elasticity parameters. Using the symbols (Mellin transforms) of the kernels of integral equations for the two-layer wedge, one can derive the symbols of the kernels of integral equations for symmetric problems about a crack in a three-layer wedge or a three-layer strip and for contact problems for a two-layer strip (by passing to the limit in a special way). The complex zeros of the Mellin transform determine the asymptotics of the normal contact pressure at the corner point of the composite as the contact region approaches this point. It is important that this asymptotics is also preserved in three-dimensional contact problems as the die enters the edge of a two-layer wedge (outside the corner points of the die itself). Taking into account this asymptotics, we obtain solutions of the contact problems as the die enters the vertex of the composite. We show that by appropriately choosing the materials and the internal angle of the two-layer wedge one can avoid contact pressure oscillations at the vertex, which occur in the case of a homogeneous wedge and result in loss of contact. The contact pressure at the wedge vertex can be made zero for a composite, while for a homogeneous wedge with the same opening angle it increases unboundedly. We construct asymptotic solutions of the contact problems for a plane die located relatively close or to the vertex of a two-layer wedge or relatively far from the vertex.