Dispersive waves in functionally graded plates

The dispersion waves propagating in anisotropic functionally graded (FG) plates with arbitrary transverse heterogeneity and arbitrary elastic monoclinic anisotropy are analysed within a recently developed sixdimensional formalism. The dispersion relation is obtained for all modes of dispersive harmonic waves propagating in an unbounded plate.


Introduction
Functionally graded (FG) materials with transverse inhomogeneity can considerably change material acoustic properties. That may result in filtering specific types of acoustic signals at some frequencies [1 -13]. These properties of wave propagation in FG plates are of particular interest in various NDT applications [14 -16]. However, until now there were no closed form analytical solutions for dispersion equations suitable for Lamb waves propagating in FG plates with arbitrary elastic (monoclinic) anisotropy and arbitrary transverse inhomogeneity.
The developed methodology relies on a previously developed six-dimensional formalism, known as Cauchy formalism [17 -21] for deriving the dispersion equations for stratified plates or stratified halfplanes with homogeneous anisotropic layers of arbitrary elastic monoclinic anisotropy and the transverse inhomogeneity.
The modification for the FG transverse inhomogeneity utilizes a specially constructed matrix exponential solution to the matrix six-dimensional ODE with the non-constant coefficients. It is assumed that within the particular layer the inhomogeneity is continuously differentiable with respect to the transverse variable.

Basic equations
Herein, the main equations for constructing the solution for Lamb waves in a functionally graded plate are derived.

Equations of motion
The linear equations of motion for anisotropic inhomogeneous material can be written in a form div ( ) r is the wave number having dimension 1 l  and 1 i ; see Fig. 1. Origin of the global coordinate system belongs to the median plane   . The following wave representation for Lamb wave is adopted m is (unknown) variation of the vector wave amplitude across thickness of the plate; n is the unit vector indicating direction of propagation; c is the independent of x phase velocity of Lamb wave; t is time.
Substituting representation (1.4) into equations of motion, yields In (1.6) I denotes unit diagonal matrix. Note, that strongly ellipticity condition (1.2) ensures where 3  is the smallest eigenvalue of () x  n C n . Hereafter, it is assumed that the condition (1.9) is satisfied. (1.10) yields equations of motion in terms of two unknown vector functions m and w :

Formalism Cauchy
where it is assumed that acoustical tensor allows us to rewrite Eqs. (1.11) in the following form ( ) Equation (

Matrix solution
Applying matrix function analysis [14]

Basic equations
Substituting the general solution at x irh  into representation (1.21) with account of (1.12), yields 3) The following 66  matrix is needed for the subsequent analysis 41 0 () ( ) ( ) Useful for computation purposes formula flows out from analyzing composition

Conclusions
Propagation of harmonic Lamb waves in plates made of functionally graded materials (FGM) with transverse inhomogeneity is analyzed by applying and modifying Cauchy sixdimensional formalism previously developed for studying Lamb wave propagation in homogeneous or stratified anisotropic plates with arbitrary elastic anisotropy [13,14]. For FG plates with arbitrary transverse inhomogeneity closed form implicit solution to dispersion equation is derived and analyzed.
Closed form implicit solutions for anisotropic FG plates with exponential inhomogeneity are constructed and compared with the corresponding implicit solutions for homogeneous plates, revealing their resemblance.