On the use of various equations of the n order plate theory in problems of normal wave dispersion in an elastic layer | Mekhanika | kompozitsionnykh | materialov i konstruktsii
> Volume 25 > №4 / 2020 / Pages: 595-613

On the use of various equations of the n order plate theory in problems of normal wave dispersion in an elastic layer


The dispersion of normal waves in a plane elastic layer is considered. The approximate solution of this problem is obtained on the background of various formulations of the quasi-3D plate theory of N order. The plate model is based on the Lagrangian formalism of analytical dynamics of constrained continuum systems; it is defined within the configuration space with the set of field variables, the density of Lagrangian, and the constraint equations following from the boundary conditions shifted from the faces onto the base plane. The general variational formulation of the extended theory of heterogeneous anisotropic plates allows one to satisfy the boundary conditions exactly, at the same time it is covariant and allows one to use different base functions such as orthogonal polynomials or finite functions corresponding to the finite element discretization of a plate across its thickness. The equations of dynamics for an isotropic transversally heterogeneous plate are derived by the Lagrange multiplier method, and the dynamic equation with eliminated multipliers are considered; these equations are analogous to the Voronets equations in the analytical dynamics of constrained discrete systems. It is shown that the dispersion problem based on the extended plate theory leads to a singular generalized eigenvalue problem. The locking frequencies for propagating modes are computed, and the solutions based on the extended plate theory and the constraint-free one are computed; it is shown that accounting for the constraints allows one to reduce the locking effect. The solutions given by the elementary theory based on the Legendre polynomials and on the piecewise linear finite element basis (e. g. the spectral element solution) are compared; it is shown that the solution based on the orthogonal polynomials leads to faster convergence to the exact solution of Rayleigh-Lamb as compared with the spectral element solution using linear shape functions.