Title : Behavior of Nonlinear Higher Harmonics in Plate and Rod Guided Waves

The study of nonlinear elastic wave propagation has been of considerable interest for the last four decades. This, in part, is due to the fact that nonlinear parameters are, in general, more sensitive to structural defects than linear parameters. Although guided waves combine the sensitivity of nonlinear parameters with large inspection ranges there are very few studies of guided nonlinear elastic waves due to the mathematical complexity of the problem. This paper presents a theoretical study into nonlinear Lamb waves and cylindrical waves. It has been known that the nonlinearity driven double harmonic in Lamb waves does not support antisymmetric motion. However the proof of this has not been obvious. Moreover, little is known on nonlinearity driven higher harmonics for either Lamb waves or cylindrical waves. These gaps are here studied by the method of perturbation coupled with wavemode orthogonality and forced response. For Lamb waves it is shown that antisymmetric motion is prohibited at all the higherorder even harmonics, whereas all the higher order odd harmonics allow both symmetric and antisymmetric motions. Similarly, in the case of cylindrical waves, it is shown that the nature of the primary generating mode severely restricts the families of modes that are generated at the higher harmonics. Finally, experimental results on measuring prestress levels in 7-wire PS tendons using nonlinear ultrasonic guided waves in the 100 kHz – 2 MHz range are presented. _____________ Ankit Srivastava, Graduate Student, NDE & SHM Laboratory, Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093-0085 Ivan Bartoli, Ph.D., NDE & SHM Laboratory, Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093-0085 Salvatore Salamone, Ph.D., NDE & SHM Laboratory, Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093-0085 Francesco Lanza di Scalea (corresponding author), Professor, NDE & SHM Laboratory, Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093-0085 INTRODUCTION The study of nonlinear elastic wave propagation has been of considerable interest for the last four decades. This, in part, is due to the fact that nonlinear parameters are, in general, more sensitive to structural defects than linear parameters [1]. Guided waves combine the sensitivity of nonlinear parameters with large inspection ranges [2]. Therefore, their application to nondestructive evaluation and structural health monitoring has drawn considerable research interest [3] and [4]. There are very few studies of guided nonlinear elastic waves due to the mathematical complexity of the problem. A recent investigation pertaining to the second harmonic generation in guided Lamb waves was reported by Deng [5], [6], [7] and [8] but the physical manifestations were obscured due to the complexity of their formulation. de Lima and Hamilton [9], [10] and subsequently Deng [8] analyzed the problem of nonlinear guided waves in isotropic plates and rods by using normal mode decomposition and forced response as suggested by Auld [11]. The authors used their formulation to explain the generation of the double harmonic and the cumulative growth of a phase-matched higher harmonic guided mode. However, their conclusions are limited to the double harmonics. Srivastava and Lanza di Scalea [12] extended the analysis to higher harmonics to prove that Rayleigh–Lamb antisymmetric motion is never allowed at even harmonics (2ω, 4ω,...) , whereas both symmetric and antisymmetric Rayleigh–Lamb motions can exist at the odd harmonics (3ω, 5ω,...). They also concluded that, in the case of cylindrical waves, the nature of the primary generating mode severely restricts the families of modes that are generated at the higher harmonics [13]. Multi-wire steel strands are used in civil engineering as the tensioning components of pre-stressed concrete structures and in cable systems of cable-stayed and suspension bridges. As documented in several studies [14], [15], [16], [17], the presence of defects or the tendon breakage can induce serious consequences for these structures. Recently, the attention has been focused on the behavior of nonlinear ultrasonic guided waves for the monitoring of pre-stress levels in the strands. In fact, it has been observed that the nonlinear features are more sensitive than the linear features to the prestress levels. The ultrasonic nonlinearities in the strands manifest themselves by the generation of higher harmonics (2ω, 3ω,..) and/or sub-harmonics (0.5ω) when the excitation is at a primary frequency of ω. Experimental studies will be presented showing that the nonlinearity increases with decreasing inter-wire stresses (i.e. decreasing pre-stress level). STATEMENT OF THE NONLINEAR PROBLEM The equation of motion for nonlinear elasticity in a waveguide is given by (Fig. 1)