S. Towfighi T. Kundu Fellow ASME M. Ehsani Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ 85721 Elastic Wave Propagation in Circumferential Direction in Anisotropic Cylindrical Curved Plates Ultrasonic nondestructive inspection of large-diameter pipes is important for health monitoring of ailing infrastructure. Longitudinal stress-corrosion cracks are detected more efficiently by inducing circumferential waves; hence, the study of elastic wave propagation in the circumferential direction in a pipe wall is essential. The current state of knowledge lacks a complete solution of this problem. Only when the pipe material is isotropic a solution of the wave propagation problem in the circumferential direction exists. Ultrasonic inspections of reinforced concrete pipes and pipes retrofitted by fiber composites necessitate the development of a new theoretical solution for elastic wave propagation in anisotropic curved plates in the circumferential direction. Mathematical modeling of the problem to obtain dispersion curves for curved anisotropic plates leads to coupled differential equations. Unlike isotropic materials for which the Stokes-Helmholtz decomposition technique simplifies the problem, in anisotropic case no such general de- composition technique works. These coupled differential equations are solved in this paper. Dispersion curves for anisotropic curved plates of different curvatures have been computed and presented. Some numerical results computed by the new technique have been compared with those available in the literature. DOI: 10.1115/1.1464872 Introduction Mathematical modeling of wave propagation in the axial direc- tion of a cylinder has been studied extensively. However, for wave propagation in the circumferential direction, which is essential for nondestructive testing NDT of large diameter pipes, literature shows fewer investigations. Viktorov’s work 1 establishes the fundamental mathematical modeling of the problem for isotropic material properties. He has introduced the angular wave number concept and has derived, decomposed and solved the governing differential equations. He has considered only one curved surface; in other words, he has found the solution for convex and concave cylindrical surfaces. In order to obtain the results for curved plates Qu et al. 2 have added the boundary conditions for the second surface and solved the problem of guided wave propagation in isotropic curved plates. Different aspects of the circumferential direction wave propagation along one or multiple curved surfaces have been analyzed by Grace and Goodman 3, Brekhovskikh 4, Cerv 5, Liu and Qu 6,7 and Valle, Qu, and Jacobs 8. In all these works the material has been modeled as isotropic elastic material. Many investigators have solved elastic wave propagation prob- lem in homogeneous and multilayered anisotropic solids. How- ever, all those works have been limited to the flat-plate case 9 or for waves propagating in the axial direction of a cylinder 10. Wave propagation in the circumferential direction of an aniso- tropic curved plate has not been analyzed earlier, and solved for the first time in this paper. Unlike isotropic materials for which the Stokes-Helmholtz de- composition technique simplifies the problem, for anisotropic case no such general decomposition technique works. The differential equations remain coupled and require a more general solution technique. The new technique, presented in this paper, solves coupled set of differential equations without attempting to decouple the equa- tions. Hence it removes the obstacle arising from not being able to decouple the equations. Consequently it provides a systematic and unifying solution method, which is capable of solving a set of coupled differential equations, and can be utilized to solve a va- riety of wave propagation problems. Fundamental Equations The formulation presented here is for the wave propagation in a cylindrical curved plate in the direction of the curvature as shown in Fig. 1. We will interchangeably call the wave carrier a ‘‘curved plate,’’ ‘‘cylinder,’’ ‘‘pipe segment,’’ or simply ‘‘pipe’’ all meaning the same thing. What we are interested in is analyzing the disper- sive waves in the curved plate for waves propagating from section T to R see Fig. 1. This analysis does not include the reflected guided waves from the plate boundary. The problem geometry can be a segment of a cylinder or a complete cylinder. Wave propagation in circumferential direction in pipes with iso- tropic material properties is usually modeled as a plane strain problem; i.e., the displacement component along the longitudinal axis of the pipe is set equal to zero. For a few other types of anisotropy this situation remains valid. However, for general an- isotropy the longitudinal component of displacement must be con- sidered in the mathematical modeling. The symmetry of both ge- ometry and material properties is required for plane-strain idealization. In absence of such symmetry a three-dimensional mathematical modeling is necessary. In cylindrical coordinates, strain components in terms of dis- placements can be written as Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED ME- CHANICS. Manuscript received by the ASME Applied Mechanics Division, April 5, 2001; final revision, November 1, 2001. Associate Editor: A. K. Mal. Discussion on the paper should be addressed to the Editor, Prof. Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS. Copyright © 2002 by ASME Journal of Applied Mechanics MAY 2002, Vol. 69 Õ 283