1 3 Nonlinear guided waves in plates: A numerical perspective 4 5 6 Vamshi Krishna Chillara Q1 ⇑ , Cliff J. Lissenden 7 Department of Engineering Science and Mechanics, The Pennsylvania State University, 16802 PA, United States Q2 8 9 11 article info 12 Article history: 13 Received 19 February 2014 14 Accepted 6 April 2014 15 Available online xxxx 16 Keywords: 17 Nonlinear ultrasound 18 Nonlinear guided waves 19 Higher harmonic generation 20 Ultrasonics 21 22 abstract 23 Harmonic generation from non-cumulative fundamental symmetric (S 0 ) and antisymmetric (A 0 ) modes 24 in plate is studied from a numerical standpoint. The contribution to harmonic generation from material 25 nonlinearity is shown to be larger than that from geometric nonlinearity. Also, increasing the magnitude 26 of the higher order elastic constants increases the amplitude of second harmonics. Second harmonic gen- 27 eration from non-phase-matched modes illustrates that group velocity matching is not a necessary con- 28 dition for harmonic generation. Additionally, harmonic generation from primary mode is continuous and 29 once generated, higher harmonics propagate independently. Lastly, the phenomenon of mode-interaction 30 to generate sum and difference frequencies is demonstrated. 31 Ó 2014 Published by Elsevier B.V. 32 33 34 35 1. Introduction 36 Use of nonlinear ultrasound for characterizing microstructure of 37 structural materials, especially metals, has been a topic of interest 38 for several decades. Initial investigations by Breazeale and Thomp- 39 son [1] and Hikata et al. [2] put forth elastic material nonlinearity 40 and dislocations as predominant causes for nonlinear ultrasonic 41 behavior i.e., higher harmonic generation. These results motivated 42 the use of higher harmonic generation for investigating nonlinear 43 behavior and hence material degradation in structures. Cantrell 44 and Yost [3] investigated fatigued microstructures using nonlinear 45 ultrasound. Cantrell [4] presented a comprehensive approach to 46 relate the nonlinearity parameter (b) to the dislocation substruc- 47 tures in metals. For polycrystalline nickel, a monotonic increase 48 in b with the fatigue cycles was predicted. The above theoretical/ 49 experimental investigations employed bulk waves (which travel 50 in unbounded media) to characterize nonlinearity. On the other 51 hand, guided waves (which travel in bounded structures) offer sev- 52 eral advantages from an inspection standpoint in that long-range 53 inspection can be carried out from a single location. Hence these 54 are more amenable for structural health monitoring applications. 55 Nonlinear guided waves combine the penetration power of guided 56 waves with the early damage detection capabilities of nonlinear 57 ultrasound. Hence they have emerged as an attractive alternative 58 for detecting microstructural changes preceding macro-scale dam- 59 age in the structures. 60 Deng [5,6] analyzed second harmonic generation from guided 61 waves in plates. De Lima and Hamilton [7] presented a perturbation 62 based approach to analyze second harmonic generation using the 63 normal mode expansion technique [8] and arrived at two condi- 64 tions necessary for cumulative second harmonic generation 65 namely, phase matching and non-zero power flux. While bulk 66 waves satisfy the above criterion for all frequencies of excitation, 67 only specific guided wave modes satisfy them. These were identi- 68 fied [9,10] from the dispersion relations governing Rayleigh Lamb 69 modes in the plate and experimental investigations [11–14] corrob- 70 orated theoretical predictions. While the above investigations dealt 71 with second harmonic generation, theoretical investigations were 72 also carried out for predicting the nature of higher harmonic gener- 73 ation from guided waves in plates. Srivastava and Lanza di Scalea 74 [15] concluded that cumulative even harmonics exist only as sym- 75 metric modes while odd harmonics can exist either as symmetric or 76 antisymmetric modes. Chillara and Lissenden [16] presented a gen- 77 eralized theory to study the nonlinear interaction of guided wave 78 modes. They concluded that the interaction of guided wave modes 79 of the same nature generate symmetric modes while those of oppo- 80 site nature generate antisymmetric modes. They also proposed a 81 procedure to predict the nature of higher harmonics from the the- 82 ory of mode interaction developed. 83 While a comprehensive theoretical framework is now available 84 to study higher harmonic generation from guided waves in plates, 85 some issues still need to be addressed. These stem from the follow- 86 ing issues: 87 1. The theoretical analysis is carried out for time-harmonic (sin- 88 gle-frequency continuous wave) excitations while the experi- 89 ments employ transducers with finite band-width. http://dx.doi.org/10.1016/j.ultras.2014.04.009 0041-624X/Ó 2014 Published by Elsevier B.V. ⇑ Corresponding author. Tel.: +1 814 954 2291. E-mail address: vkc5017@psu.edu (V.K. Chillara Q1 ). Ultrasonics xxx (2014) xxx–xxx Contents lists available at ScienceDirect Ultrasonics journal homepage: www.elsevier.com/locate/ultras ULTRAS 4808 No. of Pages 6, Model 5G 18 April 2014 Please cite this article in press as: V.K. Chillara Q1 , C.J. Lissenden, Nonlinear guided waves in plates: A numerical perspective, Ultrasonics (2014), http:// dx.doi.org/10.1016/j.ultras.2014.04.009