%0 Journal Article
%T Cauchy Six-Dimensional Formalism for Lamb Waves in Multilayered Plates
%A Sergey V. Kuznetsov
%J ISRN Mechanical Engineering
%D 2013
%R 10.1155/2013/698706
%X Propagation of Lamb waves in multilayered elastic anisotropic plates is studied in the framework of combination of the six-dimensional Cauchy formalism and the transfer matrix method. The closed form secular equations for dispersion curves of Lamb waves propagating in multilayered plates with arbitrary elastic anisotropy are obtained. 1. Introduction Herein, a brief introduction to the theory of Lamb waves and a review of some of the most important works on this matter are presented. 1.1. Lamb Waves in a Homogeneous Isotropic Plate The first works [1, 2] on waves propagating in an infinite isotropic homogeneous plate with the traction-free boundary surfaces were done at the assumption that the wavelength is much longer than the plate thickness. The complete theory of harmonic Lamb waves free from the long wavelength limit assumption was presented in [3]. The starting point of the Lamb theory is considering the equation of motion in the form where is the displacement field and and are velocities of the longitudinal and transverse bulk waves, respectively: In (2), and are Lam¨¦ constants and is the material density. Then, the displacement field was represented in terms of scalar ( ) and vector ( ) potentials The potentials were assumed to be harmonic in time: Substituting representation (4) into (1) yields two independent Helmholtz equations: To define the spatial periodicity and to simplify the analysis, the splitting spatial argument is needed: where is the unit wave vector, is the unit normal to the median plane of the plate, and . Remark 1. For the considered waves, it was further assumed that the displacement field does not depend upon argument . That allowed Lamb [3] to consider scalar potentials and in (4) instead of vector ones (actually, Lamb considered the vector potential composed of one nonvanishing component ). The further assumption relates to the periodicity of the potentials in the direction of propagation where the dimensionless complex coordinates and are In (8), and is the wave number related to the wavelength by Substituting representations (7) into (5) results in the decoupled ordinary differential equations where the phase speed relates to the frequency and the wave number by the following relation: The general solution of (10) can be written in the form where The unknown coefficients in (12) are defined (up to a multiplier) from the following boundary conditions on the free surfaces: where is the depth of the plate. Substitution representation (3) into boundary conditions (14) yields boundary conditions written in terms of potentials
%U http://www.hindawi.com/journals/isrn.mechanical.engineering/2013/698706/