Exact Solution of a Nonstationary Problem for the Elastic Layer with Rigid Cylindrical Inclusion

Abstract

We present the exact solution of a nonstationary problem for the infinite elastic layer containing a rigid cylindrical inclusion with conditions of smooth contact imposed on the cylindrical surface. One surface of the layer is subjected to the action of an axisymmetric normal nonstationary compressive load. The other surface either is perfectly coupled with an absolutely rigid foundation or is supported by a smooth foundation without friction. To construct the fields of displacements and stresses in the layer, we successively apply the Laplace and Weber integral transformations to the axisymmetric equations of motion. This gives an inhomogeneous vector boundary-value problem for the unknown transforms of displacements. The problem is solved with the help of the matrix differential calculus. The normal stresses are analyzed on the cylindrical surface of the inclusion and on the bottom surface of the elastic layer. The obtained solution is studied in detail for the case of quasistatic vibration.