Group velocity and characteristic wave curves of Lamb waves in composites: Modeling and experiments Lei Wang, F.G. Yuan * Department of Mechanical and Aerospace Engineering, Campus box 7921, North Carolina State University, Raleigh, NC 27695-7921, USA Received 11 June 2006; received in revised form 10 September 2006; accepted 20 September 2006 Available online 14 December 2006 Abstract The propagation characteristics of Lamb waves in composites, with emphasis on group velocity and characteristic wave curves, are investigated theoretically and experimentally. In particular, the experimental study focuses on the existence of multiple higher-order Lamb wave modes that can be observed from piezoelectric sensors by the excitation of ultrasonic frequencies. Using three-dimensional (3-D) elasticity theory, the exact dispersion relations governed by transcendental equations are numerically solved for an infinite number of possible wave modes. For symmetric laminates, a robust method by imposing boundary conditions on the mid-plane and top surface is proposed to separate symmetric and anti-symmetric wave modes. A new semi-exact method is developed to calculate group velocities of Lamb waves in composites. Meanwhile, three characteristic wave curves: velocity, slowness, and wave curves are adopted to analyze the angular dependency of Lamb wave propagation. The dispersive and anisotropic behavior of Lamb waves in a two different types of sym- metric laminates is studied in detail theoretically. In the experimental study, two surface-mounted piezoelectric actuators are excited either symmetric or anti-symmetric wave modes with narrowband signals, and a Gabor wavelet transform is used to extract group veloc- ities from arrival times of Lamb wave received by a piezoelectric sensor. In comparison with the results from the theory and experiment, it is confirmed that multiple higher-order Lamb waves can be excited from piezoelectric actuators and the measured group velocities agree well with those from 3-D elasticity theory. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Lamb waves; Dispersion relation; Phase and Group velocity; Slowness; Structural health monitoring 1. Introduction The integrity and degradation of composite structures have been traditionally evaluated by nondestructive evalu- ation (NDE) [1–3], now being potentially assessed by struc- tural health monitoring (SHM) [4], to assure the performance of these structures. For the active diagnosis that utilizes ultrasonic transient waves for damage detec- tion, localization and then assessment of the damage, understanding the wave propagation characteristics in composites is essential for successful application of these techniques. The wave propagation in composites is com- plex due to the nature of heterogeneity of the constituents, inherent material anisotropy, and the multi-layered con- struction, which leads to the velocity of wave mode being macroscopically dependent on the laminate layup, the direction of wave propagation, frequency, and interface conditions. When elastic waves propagate in isotropic plate-like structures, they would experience repeated reflections at the top and bottom surfaces alternately and the resulting wave propagation from their mutual interference is guided by the plate surfaces. The guided wave can be modeled by imposing surface boundary conditions on the equations of motion. However, this approach introduces the dispersion phenomenon; that is, the velocity of propagation of the guided wave along the plate being a function of frequency, 0266-3538/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2006.09.023 * Corresponding author. Tel.: +1 919 515 5947; fax: +1 919 515 5934. E-mail address: yuan@ncsu.edu (F.G. Yuan). www.elsevier.com/locate/compscitech Composites Science and Technology 67 (2007) 1370–1384 COMPOSITES SCIENCE AND TECHNOLOGY