On calculating dispersion curves of waves in a functionally graded elastic plate

The dispersion behavior of waves in an elastic plate with material properties varying along the thickness direction is studied. First, the inhomogeneous plate is approximated by a laminate plate model consisting of a sufficiently large number of layers, each of which is therefore regarded as homogeneous. Then, two local coordinates are set up for any layer according to the newly developed reverberation matrix method. Phase relations are derived between two solutions corresponding to the two coordinates. Scattering relations are also obtained by considering the continuity conditions at the interfaces between any two adjacent layers as well as the boundary conditions at the upper and lower surfaces. A reverberation matrix is finally constructed and the characteristic equation governing the dispersion of waves is presented. It is noted that no element of the reverberation matrix contains exponential function with positive index, hence avoiding the numerical instability induced by the small difference between two large numbers. Numerical examples are presented and comparison is made with the traditional displacement method and the state-space method (both formulations are presented in Appendix A for completeness). It is shown that the reverberation matrix method behaves well for all values of wave numbers except when the frequency is very low, while the state-space method is very efficient as well as numerically stable at small wave numbers. It is recommended that the two methods be combined together for the purpose of calculating dispersion curves over a wide range of wave numbers.