Frequency response of the axisymmetrically finite pre-stretched slab from incompressible functionally graded material on a rigid foundation

Within the framework of the three-dimensional linearized theory of elastic waves in the initially stressed bodies (TLTEWISB) the axisymmetric frequency response problem of the pre-stretched slab from the incompressible functionally graded material (FGM) resting on the rigid foundation is studied. The elasticity relations of the slab material are given through the Treloar potential. It is assumed that the time-harmonic point located normal force acts on a free face plane of the slab. The approach for the investigation of the problem is developed. According to this approach, the problem for the slab from the continuously inhomogeneous material (FGM) is reduced to the corresponding problem for the slab from the multi-layered, i.e. from the piecewise-homogeneous material. The number of the layers in the reduced problem is determined by the convergence requirement of the numerical results. This replacement allows us to use so-called discrete-analytical approach. By employing this approach the concrete numerical results are presented for the case where the properties of the slab’s material change continuously along the thickness of the slab. In particular, it is established that the “resonance” values of the amplitude increase (decrease), but the “resonance” values of the frequency decrease (increase) with an increase in the slab’s material stiffness and density along the thickness of that.