An Approximate Theory of Linear Waves in an Elastic Layer and Its Relation to Microstructured Solids

For the dynamics of an infinite elastic layer or plate , an approximate model is introduced. The governing equations are obtained by Lagrange’s method with suitable restrictions imposed to the transverse deformation of the layer. It turns out that the resulting equations, in the one-dimensional case, are of the same kind as those introduced by Engelbrecht and Pastrone (Proc. Estonian Acad. Sci. Phys. Math. 52(1), 12–20 (2003)) for the dynamics of a microstructured solid in the sense of Mindlin. These equations cover a wide range of possible applications, but usually, there is no validation as to what extent they describe the actual behaviour in reality. The approximate layer model is governed by the same system of equations. It may serve as a test that allows to compare the approximation with the known results of the exact theory. To this end, the propagation of harmonic waves in the layer is considered. The dispersion curves of the approximate and exact theories are compared for some values of Poisson’s ratio . It is shown that the main, acoustical branch fits well for wavelengths above three times the plate thickness. The optical branches of the approximate model deviate from their exact counterparts but exhibit qualitatively the same behaviour.