State-vector formalism and the Legendre polynomial solution for modelling guided waves in anisotropic plates

- The hybrid method solved the dispersion curves of a generally anisotropic plate.
- The orthogonality and recurrence relation of Legendre polynomial were used to deduce analytically all the matrices.
- Comparison of results obtained by the hybrid method and global matrix method.
- The hybrid method could not miss any root or trace a wrong mode.

We presented a numerical method to solve phase dispersion curve in general anisotropic plates. This approach involves an exact solution to the problem in the form of the Legendre polynomial of multiple integrals, which we substituted into the state-vector formalism. In order to improve the efficiency of the proposed method, we made a special effort to demonstrate the analytical methodology. Furthermore, we analyzed the algebraic symmetries of the matrices in the state-vector formalism for anisotropic plates. The basic feature of the proposed method was the expansion of field quantities by Legendre polynomials. The Legendre polynomial method avoid to solve the transcendental dispersion equation, which can only be solved numerically. This state-vector formalism combined with Legendre polynomial expansion distinguished the adjacent dispersion mode clearly, even when the modes were very close. We then illustrated the theoretical solutions of the dispersion curves by this method for isotropic and anisotropic plates. Finally, we compared the proposed method with the global matrix method (GMM), which shows excellent agreement.