Mathematical Modeling of Interaction of a Circular Plate with an Elastic Inhomogeneous Layer

Axisymmetric contact problem about bending of a plate lying on inhomogeneous foundation with complicated structure is considered. The plate is bent under the action of distributed load and elastic response from a foundation. The foundation consists of elastic inhomogeneous soft interlayer and elastic homogeneous half-space. Lame parameters of interlayer vary with depth by arbitrary law. Both continuously inhomogeneous and stratified interlayers are considered. Also case when layer is significantly softer than an underlying half-space is considered. Bilateral asymptotic method was used to construct an analytical solution of the problem. Analytical expressions for contact stresses and deflection of the plate are provided. The obtained solution is bilaterally asymptotically exact both for large and small ratio of layer thickness to plate radius. Both flexible and stiff plates can be modeled. Numerical results are given which shows that found approximations for kernel transform of integral equation of the problem allow us to construct analytical solution that is effective in the whole range of values of inhomogeneous layer thickness and plate stiffness.