Mathematical modeling of layered elastic structures and asymptotic solution methods

In this thesis, the anti-plane motion of inhomogeneous dynamic layers with unst- ressed surfaces is investigated. The dispersion relationship is obtained for the three layered structure under free boundary conditions. Considering the long wave-low frequency case, the dispersion relation is expressed in polynomial form. After the analysis of the dispersion relation in the general case, the problem is handled as two contrast cases. The first corresponds to a plate with a stiff-thick outer layer and a soft inner layer, the second corresponds to a traditional sandwich-type plate with a thin-stiff skin and a soft core layer. Considering the contrast situations arising from the material parameters in the layers, the dispersion equations around the cut- off frequency are analyzed for both cases and the dispersion equation is expressed as a polynomial in a shorter form. After obtaining the aforementioned shortened dispersion relations, an asymptotic approach is applied to the equations of motion expressed in terms of stresses, boundary condition and continuity conditions, and a 1-dimensional differential equation is derived around the cut off frequency for two contrast cases. It has been observed that the solutions of these equations give the same result as the shortened dispersion relations. Then, the three layered elastic la- minate with homogeneous surfaces is considered on the restricted to a semi-infinite plane. By applying the Laplace transform to the equations of motion, boundary and continuity conditions and using Saint Venant's principle, boundary conditions are obtained asymptotically for the structure with a small but non-zero cut off frequency at the x1 = 0 boundary for the first setup. Keywords: Wave, Layered Structure, Cut off frequency, Contrast, Asymptotic