On the limit equilibrium of a stamp with a rounded base on the boundary of an elastic semiplane

Abstract

In this study we consider the problem of limit equilibrium of a stamp with a rounded base on the boundary of an elastic semiplane in the case of vertical force P applied to the middle of the stamp, horizontal force ϱP (ϱ is the coefficient of friction between the stamp and the boundary of the semiplane) and moment M causing translational motion of the stamp. We assume that τ( x) = ϱp( x), where p( x) is the pressure and τ( x) is the shear stress under the stamp. The problem is solved by means of Fourier integral transforms and the Wiener-Hopf method. It is shown that in the case of a variable interval of contact between the stamp and the base there exists such a state of limit equilibrium of the stamp (in contrast to the case of finite pressure at the ends of the contact interval) where pressure has singularity at one of the ends of the contact interval. Besides, when the contact interval is constant, the pressure under the stamp is determined in closed form. Also, the values of moment M (depending on the friction coefficient) at which stamp rotation does not take place are found. The problems of limit equilibrium of a stamp on the boundary of an elastic semiplane were considered by virtue of the method of complex potentials in [Galin, L. A., The Contact Problems of the Theory of Elasticity and Viscoelasticity; Nauka, Moscow (1980)] and [Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity; Nauka, Moscow (1966)]. In Muskhelishvili's book, the problems of limit equilibrium of a stamp with a rectilinear base are investigated in detail. In Galin's boot, along with the problem for a stamp with a plane and rectilinear base, the problem on the limit equilibrium of a stamp with a rounded base is discussed. There, proceeding from the assumption that the stamp base is a plane, it is stated that the pressure under the stamp must be finite under both of its ends, as it takes place in the case of absence of frictional forces. This comes down to a condition fully determining the location of the contact interval.