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Cover page Title: Behavior of Nonlinear Higher Harmonics in Plate and Rod Guided Waves Authors: Ankit Srivastava Ivan Bartoli Salvatore Salamone Francesco Lanza di Scalea ABSTRACT 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) 2u 2 u u f 0 2 t (1) with stress free boundary conditions on the surface: S L u S u nr 0 (2) z Ω Γ Figure 1. Schematic of a stress free waveguide. where u is the particle displacement, , and are the Lame constants, 0 is the initial density of the body, f is the body force, nr is the unit vector normal to the surface of the waveguide , S L , and S are the linear and nonlinear parts of the second Piola-Kirchoff stress tensor, respectively. Energy is written in Murnaghan potentials [18]: E 2 3 4 ... (3) where n corresponds to the set of terms in the energy expression which are of degree n in strain multiples. SOLUTION TO THE NONLINEAR PROBLEM Following Auld [11] and de Lima and Hamilton [10] and using method of perturbation, the first order nonlinear solution is written as a linear combination of the existing guided wavemodes at 2 : v r , z , t .5 Am ( z )v m (r )e i 2t m 1 (4) where v m is the particle velocity of the mth mode at 2 , and Am is the higher order modal amplitude given by: * Am ( z ) Am ( z )ei 2 z Am (0)ei 2 n z (5a) where, Am ( z ) i Am f vol n f nsurf 4 Pmn 2 * n f ( z) i Pmn .25 vol n f nsurf 4 Pmn ; n* 2 ; * n 2 vn* Sm vm Sn* .nz d (5b) (5c) (5d) f nsurf ( z ) vn* S nr d (5e) f nvol ( z ) vn* . f d (5f) is the wavenumber of the primary generating mode, n is the wavenumber of the wave that is not orthogonal to the mth mode at the higher harmonic. S m is the stress tensor for the mth mode, nz is the unit vector in the wave propagation direction. S and f are the nonlinear surface traction and body force, respectively. NONLINEAR HIGHER HARMONICS IN PLATES Srivastava and Lanza di Scalea [12] proved that in the case of Rayleigh-Lamb dispersion relation, the nonlinear solution (Eqs. 4, 5) is such that antisymmetric modes are nonexistent at even harmonics (2ω, 4ω,...), whereas odd harmonics (3ω, 5ω,...) support both symmetric and antisymmetric modes. Two experiments were carried out to test the theoretical result. In the first experiment, one Pico transducer (Physical Acoustics Corporation, 0.1–1 MHz, central frequency 0.543 MHz) was used to generate Lamb waves in an aluminum plate of thickness 2.54 mm. The response was measured at a distance of 25 cm by a Pinducer sensor (Valpey Fisher VP-1093). The plate was loaded quasistatiscally to a level large enough to induce measurable nonlinearity driven higher harmonics of the primary Rayleigh–Lamb wave. Both the Pico and the Pinducer work by exciting and sensing out of plane displacements, therefore, generate and receive predominantly antisymmetric motion. The excitation was driven at a monochromatic frequency of 320 kHz. Figure 2. Joint time–frequency analysis of the antisymmetric excitation and detection in the plate: (a) time history, (b) wavelet scalogram of the signal in the DC-1 MHz range, (c) zoomed view of the wavelet scalogram. White lines are the theoretical arrival times from the Rayleigh–Lamb formulation. Fig. 2(b) shows the wavelet scalogram applied to the time signal depicted in Fig. 2(a). Fig 2(c) shows a zoomed in view of the scalogram in the frequency range of 0.5–1 MHz. The white lines are the theoretical arrival times of the pertinent modes according to Rayleigh–Lamb theory. It can be seen that while a strong antisymmetric mode is present at the generation frequency (320 kHz), there is no antisymmetric mode present at the double harmonic (640 kHz). The antisymmetric mode is, instead, present at the triple harmonic (960 kHz), as predicted by the theoretical formulations. In the second experiment, two macro-fiber composite (MFC) transducers (Smart Materials Corporation, M2814P1) were used for both excitation and detection. MFCs work by generating and detecting in-plane strains and hence they are preferentially sensitive to symmetric waves. Figure 3. Joint time–frequency analysis of the symmetric excitation and detection in the plate: (a) time history, (b) wavelet scalogram, (c) zoomed view of the wavelet scalogram. White lines are theoretical arrival times from Rayleigh– Lamb formulation. Fig. 3 shows the continuous wavelet scalogram of the received signal in the antisymmetric experiment, along with the theoretical Rayleigh–Lamb curves of the pertinent modes. It can be seen from Fig. 3(b) that while the primary harmonic (320 kHz) consists of both symmetric and antisymmetric modes, the energy at the double harmonic (640 kHz) corresponds exclusively to the fundamental symmetric mode (Fig. 3c). The energy at the triple harmonic (960 kHz), as expected, consists of a combination of the fundamental antisymmetric, the fundamental symmetric, and the first-order antisymmetric modes. Hence the experiments confirm that the antisymmetric modes can only exist at odd harmonics, whereas symmetric modes can exist at both odd and even harmonics. NONLINEAR HIGHER HARMONICS IN RODS Srivastava and Lanza di Scalea [13] studied the problem of nonlinear higher harmonics in rods by using wavemode orthogonality and perturbation techniques. The following conclusions, as applicable to rod waves, were reached: A primary generating mode in a rod with an angular order p will generate a rod mode with an angular order l at the nth higher harmonic if and only if l kp for some values of k where: 1. k spans all odd numbers from 1 to n when n is odd (odd harmonics). 2. k spans all even numbers from 0 to n when n is even (even harmonics). Therefore, for odd harmonics: 1. A longitudinal primary generating mode will not produce any flexural modes. A longitudinal primary generating mode can only produce longitudinal modes. 2. Only selected modes can be generated by flexural primary generating modes. For example, a first order flexural mode ( p 1 ) at the triple harmonic ( n 3 ) can only generate the first order ( l 1 ) and third order ( l 3 ) flexural modes. In general, a p th order flexural mode can only generate at the nth harmonic flexural modes of orders equal to odd multiples of p , up to np . For even harmonics: 1. Longitudinal modes can be generated irrespective of whether the primary generating mode is longitudinal or flexural. Moreover, a longitudinal primary generating mode does not produce flexural modes. 2. As in the case of odd harmonics, only selected modes can be generated by flexural primary generating modes. For example, a first order flexural mode ( p 1 ) at the double harmonic can only generate the longitudinal ( l 0 ) and second order ( l 2 ) flexural modes. In general, a p th order flexural mode can only generate at the nth harmonic the longitudinal mode and flexural modes of orders equal to even multiples of p , up to np . PRE-STRESS LEVEL MONITORING IN SEVEN-WIRE STRAND (a) (b) SEVEN-WIRE STRAND PZT5 D 182cm PZT4 D PZT3 D PZT2 D PZT1 PICO CENT PICO PER Figure 4. (a) The seven-wire strand on the SATEC machine. (b) Ultrasonic transducer lay-out and pictures. Tests were performed at UCSD’s Powell Labs on the SATEC M600XWHVL, 600 kip capacity testing machine (Figure 4a). A 0.6-in, 1.82m long, seven-wire strand (grade 270, U.T.S. 1.86 GPa270ksi) was subjected to load-unload cycles. The loading cycles were performed between zero stress and a maximum stress of 70% U.T.S. and divided in 6 intervals. A number of ultrasonic transducers were placed on the strand. The most successful configuration used ultrasonic generation by transducer PZT1 and ultrasonic reception by transducers PZT2 through PZT5 (Figure 4b). A special holder was designed to keep two additional ultra-mini broadband sensors of the Physical Acoustic Corporation (PICO type) in place at the strand’s free end. A 30-cycle ultrasonic toneburst, with center frequency controllable by software, was used to create a narrowband excitation at the primary frequency of ω. The nonlinearity was quantified by the socalled “beta parameter”, β, defined as: Amplitude of the 2 nd Harmonic at 2 Amplitude of the primary excitation at (6) Load = 0 % 0.05 Amplitude Amplitude 0.04 0.03 0.02 0.01 0 0 500 1000 1500 Frequency [kHz] 2000 0 Load = 40 % 500 1000 1500 Frequency [kHz] 2000 2500 2000 2500 2000 2500 Load = 60 % 0.04 Amplitude Amplitude 0 0.05 0.03 0.02 0.01 0.03 0.02 0.01 0 500 1000 1500 Frequency [kHz] 2000 0 2500 Load = 80 % 0.05 0 500 1000 1500 Frequency [kHz] Load = 100 % 0.05 0.04 Amplitude 0.04 Amplitude 0.02 2500 0.04 0.03 0.02 0.01 0 0.03 0.01 0.05 0 Load = 20 % 0.05 0.04 0.03 0.02 0.01 0 500 1000 1500 Frequency [kHz] 2000 0 2500 0 500 1000 1500 Frequency [kHz] nd Non-linear Parameter referred to the 2 Harmonic versus Load Level Loading Phase - Frequency Input = 550 kHz 0.014 0.013 0.012 0.011 0.01 0.009 0.008 0% 20% 40% 60% Load Level 80% 100% Figure 5. (a) FFT spectra of signals detected by PZT5 for a 30 cycle toneburst excitation centered at 550 kHz. (b) Nonlinear parameter β from 2nd harmonic as a function of load applied to the strand. Figure 5(a) shows the Fast-Fourier Transform (FFT) of the signal measured by PZT5 under an excitation frequency of 550 kHz for different levels of load applied to the strand. It is apparent that the FFT spectra reveal the generation of higher-harmonics, beyond the 550 kHz primary excitation, indicating nonlinear behavior. Figure 5(b) shows that the nonlinear parameter β indeed decreases with increasing load level from zero stress to 70% Ultimate Tensile Stress (U.T.S.). It can also be seen that the trend of β vs. Load is reasonably linear, indicating the suitability for this parameter to provide a direct indication of the level of pre-stress in the strand once the slope of the β vs. Load line is known. CONCLUSIONS The nonlinear Rayleigh–Lamb guided wave problem was studied using the method of perturbation coupled with wavemode orthogonality and forced response. It was shown that nonlinearity induced even harmonics do not support antisymmetric motion whereas symmetric Rayleigh–Lamb waves are allowed at all (odd or even) harmonics. For the case of rod waves it was concluded that the condition of existence/nonexistence of nonlinearity-driven higher harmonics of longitudinal and flexural waves in rods depends upon an angular order-based constraint. Finally, experimental results were presented which indicated that the nonlinear ultrasonic parameter β, measuring the ratio between higher order harmonics and the fundamental generated harmonic is a suitable feature for monitoring prestress levels in free and embedded strands. ACKNOWLEDGMENTS This work was funded by Air Force Office of Scientific Research contract no. FA9550-07-1-0016 (Dr. Victor Giurgiutiu, Program Manager), Office of Naval Research contract no. N00014-08-10973 (Drs. Paul Hess and Liming Salvino, Program Managers), and by the California Department of Transportation under contract # 59A0538 (Dr. C. Sikorsky, Program Manager). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. Dace G.E., Thompson R.B. and Brash L.J.H., "Nonlinear acoustics, a technique to determine microstructural changes in materials." In: D.O. Thompson and D.E. Chimenti, Editors, Review of Progress in Quantitative Nondestructive Evaluation Vol. 10B, Plenum Press, New York (1991). Bermes C., Kim J.Y., Qu J. and Jacobs L.J., "Experimental characterization of material nonlinearity using Lamb waves", Applied Physics Letters 90 (2007), p. 021901-1. Zaitsev V.Y., Sutin A.M., Belyaeva I.Y. and Nazarov V.E., "Nonlinear interaction of acoustical waves due to cracks and its possible usage for cracks detection", Journal of Vibration and Control 1 (1995), p. 335. Ekimov A.E., Didenkulov I.N. and Kazakov V.V., "Modulation of torsional waves in a rod with a crack," Journal of the Acoustical Society of America 106 (1999), p. 1289. Deng M., "Second-harmonic properties of horizontally polarized shear modes," Japan Journal of Applied Physics 35 (1996), p. 4004. Deng M., "Cumulative second-harmonic generation accompanying nonlinear shear horizontal mode propagation in a solid plate," Journal of Applied Physics 84 (1998), p. 3500. Deng M., "Cumulative second-harmonic generation of Lamb-mode propagation in a solid plate," Journal of Applied Physics 85 (1999), p. 3051. Deng M., "Analysis of second-harmonic generation of Lamb modes using a modal analysis approach," Journal of Applied Physics 94 (2003), p. 4152. de Lima W.J.N. and Hamilton M.F., "Finite-amplitude waves in isotropic elastic plates," Journal of Sound and Vibration 265 (2003), p. 819. de Lima W.J.N. and Hamilton M.F., "Finite amplitude waves in isotropic elastic waveguides with arbitrary constant cross-sectional area," Wave Motion, 41, (2005), pp. 1-11 Auld A., Acoustic Fields and Waves in Solids, Wiley, New York (1973). Srivastava A., Lanza di Scalea F., " On the existence of antisymmetric or symmetric Lamb waves at nonlinear higher harmonics," Journal of Sound and Vibration, 323, (2009), pp. 932-943 Srivastava A., Lanza di Scalea F., " On the existence of longitudinal or flexural waves in rods at nonlinear higher harmonics," submitted to Journal of Sound and Vibration Watson S.C. and Stafford D., “Cables in Trouble,” Civil Engineering 58, 38-41 (1988). Woodward R.J., “Collapse of Ynys-y-Gwas Bridge, West Glamorgan,” Proceedings - Institution of Civil Engineers 84, 635-669 (1988). Parker D., “Tropical overload,” New Civil Engineer 18-21, (1996). Chase S.B., “Smarter bridges, why and how?,” Smart Materials Bulletin 2, 9-13 (2001). F.D. Murnaghan, Finite deformation of an elastic solid, Wiley, New York, (1951).