The response of an elastic three-dimensional half-space to random correlated displacement perturbations on the boundary

The response of an elastic half-space to random excitations of displacements on the boundary under the condition of no shearing forces is studied. We analyze the white noise excitations and general random fluctuations of displacements prescribed on the boundary. We consider the case of partially ordered defects on the boundary whose positions are governed by an exponential-cosine-type correlation function. The analysis is based on a Poisson-type integral formula which we derive here for the case of zero shearing forces on the boundary. We obtain exact representations for the displacement correlation tensor and the Karhunen-Loève expansion for the solution of the Lamé equation itself, and analyze some features of the correlation structure of the displacements. The Monte Carlo technique developed can be applied to a wide class of differential equations with random boundary conditions.