Vibration of an Elastic Circular Plate on an Elastic Half Space—A Direct Approach

The axisymmetric problem of a vibrating elastic plate on an elastic half space is solved by a direct method, in which the contact stresses and the normal displacements of the plate are taken as the unknown functions. First, the influence functions that give the displacements in terms of the stresses are determined for the half space and the plate. Displacement continuity then takes the form of an integral equation. Due to the half space the kernel is weakly singular, and a special solution technique that accounts for this is employed. The solution implies a direct matrix relation between the expansion coefficients of the contact stresses and plate deformations. The solution technique is valid for all frequencies and avoids asympototic expansion in terms of the frequency. The plate is represented by the theory of Reissner and Mindlin, which imposes physical limitations for high frequencies, but the method is easily extended to more general plate theories as well as nonsymmetric oscillations. The results include displacement and phase curves for rigid disks, power input for elastic plates, and typical stress and deformation distributions at selected phase angles. The results show considerable influence from the elastic properties of the plate.