Dynamic contact problem for a visco-elastic half-plane: PMM vol. 41, n≗ 5, 1977, pp. 926–934

Abstract

Steady oscillations of a rigid stamp at the boundary of a visco-elastic half-plane are considered. No tractions exist outside the region of contact, and the absence of friction, Coulomb friction or coupling of the stamp to the half-plane all prevail within the region. A system of integral equations of the first kind obtained with the help of a Fourier transformation is reduced, by differentiating with respect to the coordinate and separating the kernal singularities, to a system of singular integral equations. At zero oscillation frequency the latter system coincides with the equations of the analogous static problem of the theory of elasticity. The exact solutions of the static problem are utilized for regularization of the system according to Carleman-Vekua when the oscillation frequency is not zero. The low frequency asymptotic of the system kernals is investigated using the contour integration, and the asymptotic properties of the Laplace transformation. The solution of the system is constructed in first approximation for low frequency oscillations. Oscillations of a stamp on an elastic half-plane were studied in [1, 2], while the results of the problem with coupling were presented at the All- Union Winter School (∗) (∗) Zlatina I.N. and Zil'bergleit A.S. Use of dual integral equations in the dynamic contact problem of oscillations of a rigid stamp on an elastic half-space. In “ Contact Problems of the Mechanics of Deformable Solids ”. Erevan, 1976. .