Approximate theory of a laminated anisotropic plate

A. K. Belyaev ; A. V. Zelinskaya ; D. N. Ivanov ; N. F. Morozov ; N. V. Naumova ; P. E. Tovstik ; T. P. Tovstik

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 4, pp. 397-411

Résumé

The multi-layered plate vibration is investigated. A two-dimensional asymptotic model of the second order accuracy with respect to the small thickness parameter is proposed with account for the transverse shear and the normal fibre extension. The model is appropriate for a monoclinic plate described by 13 elastic moduli which is heterogeneous in the thickness direction. In particular, the model can be applied to a multi-layered plate consisting of orthotropic layers of arbitrary orientation. In this case the elastic moduli are piece-wise constant functions. The elastic and inertia properties of plate are assumed to be constant in the tangential directions. The main achievement of this work is derivation of the equivalent constant coefficients of 2D system of partial differential equations of the second order accuracy. In the first approximation these coefficients can be found based on the Kirchhoff–Love hypotheses on the straight normal, while a more complex asymptotic algorithm is used for second approximation. For a multi-layered plate the influence of transverse shear with alternating hard and soft layers is discussed. More attention is given to a plate which is infinite in the tangential directions. The solution is shown to be essentially simplified since no boundary condition is needed and the solution can be expressed in terms of functions which are harmonic in the tangential directions. For this solution the error of 2D model is estimated by comparison with the numerical solution of the three-dimensional problem of elasticity theory, since for harmonic case it is reduced to one-dimensional equations in the thickness direction. Free and forced bending vibration and long-length bending wave propagation are investigated under harmonic approximation. In general case two different natural frequencies are shown to correspond to a fixed bending mode. The dependence of wave velocity on the wave propagation direction is found out.