A global-discretized semi-analytical formulation in polar coordinate system for the wave characteristics in multi-layer cylindrical waveguides

Abstract

Ultrasonic guided waves are widely applied in health monitoring of slender structures. Being different from the well-known semi-analytical finite element method (SAFE), the global-discretized semi-analytical formulation (GDSA) exactly satisfies all the continuity and boundary conditions accurately while has improved computational efficiency, but is only applicable to the plate-like problems described in the Cartesian coordinate system currently, which is not applicable to the cylindrical waveguide. In the present work, the polar coordinate system is therefore introduced into the GDSA formulation to improve the computational efficiency of calculating the dispersion relation in a multi-layer cylindrical waveguide without loss of accuracy. The characteristic equations of the unit layer are derived from the principle of virtual work. The involved matrices are explicitly derived in the form of Kronecker product to reduce the dimension of the matrices to be evaluated and a reduced Boolean matrix is introduced to avoid the singularity problem caused by the trivial radial displacement of the central point. The dispersion curves of a steel wire are firstly analyzed and are verified in comparison with the analytical solutions solved from the Pochhammer-Chree equations. Taking the steel wire having a surface corrosion as a two-layer case example, the dispersion curves are obtained based on the quadratic eigenvalue equation. It is found that the cut-off frequency of the F(1,2) mode is sensitive to corrosion, having potential to detect corrosion of hidden cable wires.