###
An application of the nth order extended plate theory in the wave dispersion problem for a functionally graded layer

Zhavoronok Sergey I.

#### Abstract:

A solution of the wave dispersion problem for a functionally graded plane layer is based on the extended plate theory of I.N. Vekua – A.A. Amosov type satisfying the boundary conditions on the faces exactly within an arbitrary approximation order. A variational problem’s statement corresponding to the heterogeneous plate theory of Nth order is is given by a set of field variables of the first kind being the displacement expansion factors with respect to biorthogonal function system, the surface Lagrangian density, and the non-holonomic constraints following from the boundary conditions on the faces. The generalized Lagrange equations of the second kind for a 2D continuum are obtained. The spectral problem for normal waves in the functionally graded layer is formulated as a constrained stationary problem for two quadratic forms and solved by the Golub approach. The phase locking frequencies and wave forms for the asymmetric layer with power gradation law are computed, as well as the corresponding stress distributions across the thickness. The convergence of approximate locking frequencies is analyzed for different gradation laws. The minimum order of theory required to secure the convergence correspond to the homogeneous layer if the stiffer phase prevails; the waveforms are close to the ones in the homogeneous layer. The softer phase prevailing leads to minimum orders exceeding the ones of the homogeneous layer for some modes; the wave forms for higher frequencies differ significantly from the ones of homogeneous layer. The stress distributions across the thickness are significantly asymmetric, especially for higher frequencies.

#### Keywords:

BOUNDARY CONDITIONS,

constraint equations,

dispersion problems,

extended Lagrange equations,

extended plate theory,

functionally graded heterogeneous plates,

normal waves,

spectra,

волны нормальные,

задачи дисперсионные,

Лагранжа уравнения обобщенные,

пластины градиентно-неоднородные,

связей уравнения,

спектры,

теория пластин расширенная,

условия краевые
0