Complex dispersion solutions for guided waves and properties of non-propagating waves in a piezoelectric spherical plate

The vibration modes of an elastic plate are usually divided into propagating and non-propagating (evanescent) kinds. Non-propagating wave modes are very important for guided wave inspection of defect size and shape. But it is difficult to obtain the complex solutions of the transcendental dispersion equation, corresponding to the non-propagating wave. In this article, we present an improved Legendre polynomial method to calculate the complex-valued dispersion and study properties of the non-propagating wave in a piezoelectric spherical plate. Comparisons with other related studies are conducted to validate the correctness of the presented method. The complete dispersion and attenuation curves are plotted in three-dimensional frequency-complex wave number space. The influences of material piezoelectricity and radius–thickness ratio on non-propagating waves in piezoelectric spherical plates are investigated. The amplitude distributions of the electric potential and displacement are also discussed in detail. All the results presented in this work can provide theoretical guidance for ultrasonic nondestructive evaluation and are promising to be applied to improve the resolution of piezoelectric transducers.

The Bending Stress in a Cracked Plate on an Elastic Foundation

Classical Kirchhoff bending solutions for a normally loaded elastically supported flat plate containing a semi-infinite straight crack are obtained using an integral equation formulation. Because the effects of initial spherical plate curvature are related to those of an elastic foundation, the solution can be applied to the problem of a crack in an initially curved unsupported plate as well. The explicit nature of the stresses near the crack point is found to depend upon the inverse half power of the nondimensional distance from the point, r/(D/k)1/4, where D is the flexural rigidity of the plate and k the foundation modulus. The particular case of an infinite strip containing the crack along the negative x-axis and loaded by constant moments M* along y = ±y* is presented. The inverse half-power decay of stress is additionally damped by an exponential factor of the form exp(−λy*/2).