Contact problem of pressing a stamp into an elastic half-plane with a thin reinforcing covering: PMM vol. 39, n≗5, 1975, pp. 857–875

Abstract

The contact problem is investigated when a stamp of sufficiently general configuration is pressed, under the effect of an arbitrary system of forces, into an elastic half-plane with a thin reinforcing layer. It is assumed that the half-plane is in a state of plane strain or generalized plane stress. Relative to the thin reinforcing layer, it is assumed that it bends as an ordinary beam in the vertical direction, while it stretches or is compressed as a bar in the horizontal direction. In other words, the beam bending model, in combination with the model of the uniaxial state of stress of a bar is considered valid for the reinforcing layer. In conformity with the results for thin plates [1, 2], this model has a sufficiently broad range of applicability and the possible error in the magnitudes of the stress which it will admit is ordinarily on the order of the ratio h 2 l 2 in the case of a layer of finite length l, where h is the height of the layer. Let us discuss some papers bordering on the problem being investigated and based on the model mentioned. Let us note that if the reinforcing layer is so flexible that its bending stiffness can be neglected, then this layer will be deformed as a bar in the uniaxial stress state. The uniaxial state of stress model of a bar has been proposed in [3] for application to problems on load transmission from elastic bracing in the form of small thickness gussets (stringers) to massive bodies which are important for engineering practice. On the other hand, if the deformation of the reinforcing layer as a bar is neglected by considering that it bends only as a beam, then a model is obtained which is well known in problems of the theory of bending of beams and slabs on an elastic basis. Without discussing the results and papers from this area of elasticity theory, let us just note that its fundamental achievements have been examined with sufficient completeness in the monographs [4–7] and detailed survey papers [8, 9]. Furthermore, contact problems when the stamp is pressed into a plate resting freely on an elastic base have been examined in [10], Finally, let us mention the paper [11] in which substantially the same problem as in the present paper was examined in a somewhat different formulation. The influence of the reinforcing covering was introduced in the boundary condition for the half-plane, and the solution of the original problem was reduced to the solution of a Fredholm integral equation of the first kind for the unknown pressure under the stamp. An explicit expression for the exact or approximate solution of this equation was not constructed. According to certain results relative to the behavior of the contact stresses near the ends of the stamp, the forces applied to the stamp are transmitted to the base by means of concentrated forces and moments applied to the ends of the contact section, which contradicts the properties of solutions of boundary value problems of elasticity theory and the results of singular integral equations theory. Moreover, the known solutions of classical contact problems for a half-plane with inherent singularities at the ends of the stamp are not obtained from the results in [11] in the absence of the reinforcing covering. In summary, the results obtained in the paper mentioned are false because of the incorrect application of the appropriate mathematical apparatus. An exact solution of the problem posed is constructed in this paper on the basis of the assumptions presented above. Determination of the distribution laws for the contact stresses under the stamp and under the reinforcing covering reduces to solving integral or integro-differential equations with a Cauchy kernel. These solutions are represented outside the stamp and under the reinforcing coating by using power series, and under the stamp by series in the classical Chebyshev and Jacobi orthogonal polynomials. Numerical results are presented.