Ultrasonics 54 (2014) 1553–1558
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Nonlinear guided waves in plates: A numerical perspective Vamshi Krishna Chillara ⇑, Cliff J. Lissenden Department of Engineering Science and Mechanics, The Pennsylvania State University, 16802 PA, United States
a r t i c l e
i n f o
Article history: Received 19 February 2014 Accepted 6 April 2014 Available online 16 April 2014 Keywords: Nonlinear ultrasound Nonlinear guided waves Higher harmonic generation Ultrasonics
a b s t r a c t Harmonic generation from non-cumulative fundamental symmetric (S0 ) and antisymmetric (A0 ) modes in plate is studied from a numerical standpoint. The contribution to harmonic generation from material nonlinearity is shown to be larger than that from geometric nonlinearity. Also, increasing the magnitude of the higher order elastic constants increases the amplitude of second harmonics. Second harmonic generation from non-phase-matched modes illustrates that group velocity matching is not a necessary condition for harmonic generation. Additionally, harmonic generation from primary mode is continuous and once generated, higher harmonics propagate independently. Lastly, the phenomenon of mode-interaction to generate sum and difference frequencies is demonstrated. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Use of nonlinear ultrasound for characterizing microstructure of structural materials, especially metals, has been a topic of interest for several decades. Initial investigations by Breazeale and Thompson [1] and Hikata et al. [2] put forth elastic material nonlinearity and dislocations as predominant causes for nonlinear ultrasonic behavior i.e., higher harmonic generation. These results motivated the use of higher harmonic generation for investigating nonlinear behavior and hence material degradation in structures. Cantrell and Yost [3] investigated fatigued microstructures using nonlinear ultrasound. Cantrell [4] presented a comprehensive approach to relate the nonlinearity parameter (b) to the dislocation substructures in metals. For polycrystalline nickel, a monotonic increase in b with the fatigue cycles was predicted. The above theoretical/ experimental investigations employed bulk waves (which travel in unbounded media) to characterize nonlinearity. On the other hand, guided waves (which travel in bounded structures) offer several advantages from an inspection standpoint in that long-range inspection can be carried out from a single location. Hence these are more amenable for structural health monitoring applications. Nonlinear guided waves combine the penetration power of guided waves with the early damage detection capabilities of nonlinear ultrasound. Hence they have emerged as an attractive alternative for detecting microstructural changes preceding macro-scale damage in the structures.
⇑ Corresponding author. Tel.: +1 814 954 2291. E-mail address: [email protected] (V.K. Chillara). http://dx.doi.org/10.1016/j.ultras.2014.04.009 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.
Deng [5,6] analyzed second harmonic generation from guided waves in plates. De Lima and Hamilton [7] presented a perturbation based approach to analyze second harmonic generation using the normal mode expansion technique [8] and arrived at two conditions necessary for cumulative second harmonic generation, namely phase matching and non-zero power flux. While bulk waves satisfy the above criterion for all frequencies of excitation, only specific guided wave modes satisfy them. These were identified [9,10] from the dispersion relations governing Rayleigh Lamb modes in the plate and experimental investigations [11–14] corroborated theoretical predictions. While the above investigations dealt with second harmonic generation, theoretical investigations were also carried out for predicting the nature of higher harmonic generation from guided waves in plates. Srivastava and Lanza di Scalea [15] concluded that cumulative even harmonics exist only as symmetric modes while odd harmonics can exist either as symmetric or antisymmetric modes. Chillara and Lissenden [16] presented a generalized theory to study the nonlinear interaction of guided wave modes. They concluded that the interaction of guided wave modes of the same nature generate symmetric modes while those of opposite nature generate antisymmetric modes. They also proposed a procedure to predict the nature of higher harmonics from the theory of mode interaction developed. While a comprehensive theoretical framework is now available to study higher harmonic generation from guided waves in plates, some issues still need to be addressed. These stem from the following issues: 1. The theoretical analysis is carried out for time-harmonic (single-frequency continuous wave) excitations while the experiments employ transducers with finite band-width.
1554
V.K. Chillara, C.J. Lissenden / Ultrasonics 54 (2014) 1553–1558
2. Group velocity matching in addition to the phase velocity matching between the primary mode and other higher harmonics was assumed to be necessary for cumulative harmonic generation [17]. However, Deng et al. [18] presented an analysis and an experimental investigation confirming that it is not a necessary condition. In general, all the primary Rayleigh-Lamb modes capable of cumulative second harmonic generation in plates satisfy group velocity matching in addition to phase velocity matching [9]. However, with finite wave packets, it would be interesting to examine if the higher harmonic modes can propagate without group velocity matching as the nonzero-power-flux criterion which requires power flow from primary to higher harmonic modes may not be satisfied once the primary and higher modes separate due to difference in group velocity. 3. The analysis is generally carried out using a perturbation approach as no closed form solution is available. 4. The capability of the existing constitutive model to capture the effect of different kinds of nonlinearities has not been established.
where k; l are Lame’s constants and A; B; C are higher order elastic constants. Another equivalent version of the above model is the Murnaghan model given by
WðEÞ ¼
1 1 kðtrðEÞÞ2 þ ltrðE2 Þ þ ðl þ 2mÞðtrðEÞÞ3 mtrðEÞ 2 3 ððtrðEÞÞ2 trðE2 ÞÞ þ ndetðEÞ
ð5Þ
where l; m and n are Murnaghan constants and trðÞ and detðÞ denote trace and determinant of the tensor respectively. The relation between A; B; C and l; m; n are given in [21]; l ¼ B þ C; m ¼ 12 A þ B and n ¼ A. The second Piola–Kirchhoff stress tensor is obtained using
TRR ¼
@WðEÞ @E
ð6Þ
The first Piola Kirchhoff stress tensor S and Cauchy stress tensor T are given by
S ¼ FTRR
ð7Þ
and In this article, we address some of the above issues using results from numerical simulations carried out in COMSOL Multiphysics 4.3, a commercial finite element software. Specifically, we consider the harmonic generation from the primary Symmetric (S0 ) and Antisymmetric (A0 ) modes in a plate. While neither of the above modes are exactly phase matched, they are deliberately chosen to highlight some of the aspects to be discussed later. As part of the study, we also comment on some aspects of the constitutive model like the contribution of material and geometric nonlinearities to higher harmonic generation. The content of the article is organized as follows. Section 2 presents some continuum mechanics preliminaries. Section 3 then presents the results obtained from numerical simulations. Finally, conclusions are drawn in Section 4. 2. Preliminaries In this section, we present some preliminaries intended to enhance understanding of the results to be presented. The discussion is brief so we refer the reader to [19] for more details. We denote the deformation gradient by F and the Lagrangian strain by E. Also, the displacement gradient is denoted by H and the following relations exist between the above quantities.
F¼IþH E¼
1 T 1 ðF F IÞ ¼ ðH þ HT þ HT HÞ 2 2
ð1Þ ð2Þ
We denote the linearized strain by El , which is related to the displacement gradient (H) by
El ¼
1 ðH þ HT Þ: 2
ð3Þ
Note that the difference between the Langrangian strain and the linearized strain is that the second order term involving HT H is dropped for the linearized strain. Including geometric nonlinearity means we consider Langrangian strain (full strain) as opposed to the linearized strain in the analysis. A widely used constitutive model used for studying higher harmonic generation was proposed by Landau and Lifshitz [20]. The corresponding strain energy function is given by
1 1 1 WðEÞ ¼ kðtrðEÞÞ2 þ ltrðE2 Þ þ CðtrðEÞÞ3 þ BtrðEÞtrðE2 Þ þ AðtrðE3 ÞÞ: 2 3 3 ð4Þ
T ¼ FSFT :
ð8Þ
While the first Piola–Kirchhoff stress tensor is used for a formulation in the reference configuration, Cauchy stress is used for a formulation in the current configuration. 3. Numerical simulations In this section, we present results obtained from numerical simulations performed in COMSOL 4.3. All the simulations were carried out using the Murnaghan model (Eq. (5)) for Aluminum, the elastic constants of which are tabulated in Table 1. As mentioned earlier, fundamental symmetric and antisymmetric modes are used in the simulation. The schematic of the 2D model used for the simulation is shown in Fig. 1. The thickness of the plate is chosen to be 1 mm and the length of the plate is assumed to be 100 mm unless otherwise specified. Triangular elements are used to discretize the plate with a maximum element size of 0.1 mm and a minimum element size of 0.03 mm. The resulting mesh is then scaled by a factor of 1.5 (both along the length and the thickness) to obtain a finer discretization. A maximum time step of 0.01 ls is used for the simulation. Appropriate displacement boundary conditions are enforced on the left end of the plate to excite the intended modes. The x-component of the displacement is denoted by ‘u’ and the y-component is denoted by ‘v’ where the axes are shown in Fig. 1. The dispersion curves for the plate are shown in Fig. 2 and the primary modes used to study harmonic generation in the plate are indicated. None of the primary modes selected are phase matched to the secondary modes, so the second harmonics are not known as ‘cumulative’. However, it is clear from both the theory [7] and the results that second harmonics are generated. 3.1. S0 mode at 0.5 MHz In this section, we present the results obtained for the S0 mode at 0.5 MHz. This mode is almost phase matched to the second harmonic as the phase speed of the primary mode is 5.34 mm/ls
Table 1 Elastic constants in GPa used for simulation. k
l
l
m
n
51
26
250
333
350
1555
V.K. Chillara, C.J. Lissenden / Ultrasonics 54 (2014) 1553–1558
x=50mm
−7
x 10
1
LE NL NG
0.8 0.6 Fig. 1. Schematic of the model used for simulations.
0.4
9
0 −0.2
8
−0.4
7
cp (mm/ µs)
0.2
u (m)
10
6
−0.6
S0
5
−0.8
4
−1
0
5
10
15
2
A0
25
30
Fig. 3. S0 mode at 0.5 MHz – time domain signals for LE, NL and NG constitutive models.
1 0
20
t (µs)
3
0
0.5
1
2
3
4
5
fd (MHz−mm)
x=50mm
−8
10 Fig. 2. Dispersion curves for the plate showing the primary modes used in the study.
LE NL NG
−9
10
1. Linear elastic material (LE) – No material or geometric nonlinearities are included i.e., l ¼ m ¼ n ¼ 0 and linearized strain El is used as a strain measure. 2. Nonlinear (NL) – Both material and geometric nonlinearities are included i.e., l – 0; m – 0; n – 0 and Lagrangian strain (E) is used as a strain measure. 3. Geometrically nonlinear (NG) – Only geometric nonlinearity is included i.e., l ¼ m ¼ n ¼ 0 and Lagrangian strain (E) is used as a strain measure. The displacement amplitude for the boundary condition is set to 1E7 m for the results presented in Figs. 3–5. Fig. 3 shows the time domain signal obtained for the displacement ‘u’ at 50 mm from the left end of the plate. It shows very little difference for the three cases, although the pulse in the ‘NL’ case arrives slightly later in time (i.e., travels at a lower speed) although this is barely visible. But in the frequency domain, as shown in the FFT (log-scale) in Fig. 4, nonlinearity is evident from the presence of static (or zero-frequency) and second harmonic components in the displacement. Notice also that a short toneburst results in a reasonably wide frequency bandwidth. While the ‘LE’ case shows no second harmonic (or static) component as expected, the other two cases have considerable second harmonic (and static) components. Also, the amplitude of the second harmonic
Amplitude
−10
10
−11
10
−12
10
−13
10
−14
10
0
0.5
1
1.5
2
frequency (MHz) Fig. 4. S0 mode at 0.5 MHz – FFT’s for the signals obtained using LE, NL and NG constitutive models.
−8
10
40mm 50mm 60mm
−9
10
−10
10
Amplitude
and that of the second harmonic is 5.27 mm/ls. It also satisfies the non-zero power flux criterion for generation of a cumulative second harmonic. We first discuss the contribution of material and geometric nonlinearities to the second harmonic generation in the plate. The amplitude of the excitation specified as the displacement boundary condition is chosen to be 1E7 m. The amplitude of 1E7 corresponds to a stress wave amplitude of a few MPa which is typical of an ultrasonic wave propagating in a solid. In other cases that follow, results for higher amplitude are presented to emphasize some of the effects not easily decipherable at low amplitudes. The results for three simulations corresponding to three variants of the constitutive model (Eq. (5)) are presented.
−11
10
−12
10
−13
10
−14
10
0.2
0.4 0.5 0.6
0.8
1
1.2
1.4
1.6
1.8
frequency (MHz) Fig. 5. S0 mode at 0.5 MHz – FFT’s for the signals at 40, 50 and 60 mm from the left end of the plate.
1556
V.K. Chillara, C.J. Lissenden / Ultrasonics 54 (2014) 1553–1558
generated in the ‘NL’ case is 4–5 times greater than that generated in the ‘NG’ case. This shows that the second harmonic generation is dominated by the material nonlinearity as opposed to geometric nonlinearity. Signals were received at 40, 50 and 60 mm from the left end of the plate and Fig. 5 shows their corresponding FFT’s on a log-scale. As predicted by theory [7], the amplitude of the second harmonic increases with the propagation distance although it is not strictly cumulative. In the following example, we discuss the effect of scaling higher order elastic constants on the harmonic generation. The displacement amplitude for the boundary condition is set to 1E7 m. Simulations are run for the ‘NL’ case using three different sets of higher order elastic constants obtained by scaling l; m and n (Table 1) by factors of 1, 2 and 4. Fig. 6 shows the FFT’s of the time domain signals on a log-scale. As can be seen, the FFT’s show increasing zero-frequency and second harmonic components with the scaling. The amplitude of the second harmonic is almost proportional to the scaling factor used. Furthermore, the third harmonic is apparent for scaling factors of 2 and 4.
Fig. 7. S0 mode at 1 MHz – time domain signals at 50 mm from the left end of the plate for ‘LE’ and ‘NL’ cases.
3.2. S0 mode 1 MHz
1 2 4
−9
10
LE NL
−8
10
−9
10
−10
10
−11
10
−12
10
−13
10
0
0.5
1
1.5
2
2.5
3
frequency (MHz) Fig. 8. S0 mode at 1 MHz – FFT’s of the signal at 50 mm from the left end of the plate for ‘LE’ and ‘NL’ cases.
the second harmonic. This is because the static component, unlike second-harmonic, is always cumulative [23]. Also, observe that the FFT shows a small dip around 2 MHz. This is due to the presence of the second harmonic tail in the time domain window (27 ls) used for taking the FFT.
x=50mm
−8
10
x=50mm
−7
10
Amplitude
Here, we present the results obtained for second harmonic generation from the S0 mode at 1 MHz. The phase velocity of the mode is 5.27 mm/ls and that of the second harmonic (i.e., S0 mode at 2 MHz) is 4.71 mm/ls. These modes are not phase matched and hence the second harmonic is not cumulative. The amplitude of the displacement for the boundary condition is set to 1E6 m. Fig. 7 shows the time domain signal for ‘LE’ and ‘NL’ cases. As indicated in the inset, the second harmonic signal almost separates from the primary mode. Also, the time domain signal for the ‘NL’ case shows asymmetry about the zero. This arises due to the static or the zero-frequency component in the signal as evident from the FFT shown in Fig. 8. Furthermore, note that the time domain pulse for the ‘NL’ case overlaps with that of the ‘LE’ case to a greater extent during the times when ‘u’ is increasing i.e., particle velocity is positive than when ‘u’ is decreasing i.e., when particle velocity is negative. This is due to the tension–compression asymmetry exhibited by the constitutive model in Eq. (5). Detailed discussion on the above and other relevant aspects of the above constitutive model can be found in [22]. Fig. 8 shows the FFT’s of the corresponding time-domain signals in Fig. 7. As can be seen, the static component has a much larger amplitude as compared to that of
Amplitude
−10
10
3.3. A0 mode at 0.5 MHz −11
10
−12
10
−13
10
−14
10
0
0.5
1
1.5
2
frequency (MHz) Fig. 6. S0 mode at 0.5 MHz – FFT’s for the signal obtained by scaling higher order elastic constants by 1, 2 and 4 respectively.
In this section, we present results obtained for second harmonic generation from the A0 mode at 0.5 MHz. The length of the waveguide (plate) is increased to 200 mm to allow for a longer propagation distance. The other discretization parameters were unaltered from the previous set of simulations for the S0 mode. The amplitude of the displacement boundary condition (‘v’) is increased to 1E5 m. Fig. 9 shows a zoomed-in view of time domain signals at 40, 80 and 120 mm. The second harmonic (S0 mode) is the smaller wave packet that arrives before the larger wave packet (A0 mode). Fig. 10 shows the through thickness profiles for ‘v’ displacement at ‘x’ = 120 mm for t = 28–35 ls. The following observations can be made:
1557
V.K. Chillara, C.J. Lissenden / Ultrasonics 54 (2014) 1553–1558 −7
x=40,80,120mm
−8
2
x 10
40mm 80mm 120mm
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
10
20
30
40
50
40mm 60mm
1.5
u (m)
v (m)
1.5
−2
x 10
2
−2
60
0
5
10
15
t (µs) Fig. 9. A0 mode at 0.5 MHz – time domain signals at 40, 80 and 120 mm from the left end of the plate.
x 10
30
−8
10
0.9
40mm 60mm
−9
10
0.8 0.7
−10
Amplitude
Thickness (m)
25
Fig. 11. Mode interaction (S0 (0.4 MHz) S0 (1.1 MHz)) – time domain signals at 40 and 60 mm from the left end of the plate.
x=120mm
−3
1
20
t (µs)
0.6 0.5 0.4
10
−11
10
−12
10
0.3 0.2
−13
10
0.1 −14
0 −8
10
−6
−4
−2
0
v (m)
2
4
6
8
0
x 10
Fig. 10. A0 mode at 0.5 MHz – through thickness profiles for ‘v’ displacement at 120 mm from the left end of the plate for various times t = 28–35 ls.
1. The second harmonic pulse clearly separates from the primary mode and propagates independently. This demonstrates that group velocity matching as described in [17] is not necessary for second harmonic generation. 2. The second harmonic mode is S0 as opposed to A0 , which is in agreement with that predicted by the perturbation approach. This can be inferred from Fig. 10 which shows that the through thickness profile for ‘v’ is antisymmetric. 3. In fact, it can be concluded that the second harmonic is generated continuously from the primary mode and both propagate independently once they separate. This can be inferred with the following line of reasoning. As the primary and secondary modes separate, the primary mode again generates a second harmonic as it propagates independently. This means that this process continuously takes place and hence second harmonics are continuously generated. When sufficient time has elapsed, these pulses separate and the process continues. Also, the generated harmonics can further generate higher harmonics, albeit very small in amplitude. A procedure to assess the nature of such harmonic generation is presented in [16].
0.4 0.5
0.7
1 1.1
1.5
2
frequency (MHz)
−9
Fig. 12. Mode interaction (S0 (0.4 MHz) S0 (1.1 MHz)) – FFT’s for the signals at 40 and 60 mm from the left end of the plate.
3.4. Mode interaction-S0 mode at 0.4 MHz and S0 mode at 1.1 MHz In this section, we demonstrate the sum and difference harmonic frequency generation for the ‘NL’ case by mixing two modes, namely, S0 mode at 0.4 MHz and S0 mode at 1.1 MHz propagating in the same direction. The amplitude of the displacement boundary condition is 1E7 for both modes. Fig. 11 shows the time domain signals at 40 and 60 mm. Fig. 12 shows the FFT (log-scale) for the time domain signals in Fig. 11. As can be seen, it shows the presence of sum and difference frequency components i.e., 1.1 0.4 = 0.7 MHz and 1.1 + 0.4 = 1.5 MHz. 4. Conclusions In this article we investigated the behavior of nonlinear guided waves in plates from a numerical perspective. Simulations of nonlinear wave propagation were carried out in COMSOL using the Murnaghan model of hyper-elasticity. Harmonic generation from the fundamental guided wave modes S0 and A0 is studied. Solutions from ‘linear’, ‘nonlinear’ and ‘geometrically nonlinear’ variants of
1558
V.K. Chillara, C.J. Lissenden / Ultrasonics 54 (2014) 1553–1558
the Murnaghan model are compared for S0 mode at 0.5 MHz. While time domain signals show very little difference, harmonic generation is quite evident in the frequency domain. Also, the contribution of material nonlinearity to the harmonic generation is higher when compared to the geometric nonlinearity. As the material damage state/ microstructure is typically inferred from the b parameter, which in turn depends on the higher order elastic constants, it is essential that one understands the behavior of harmonic generation with respect to the changes in higher order elastic constants. To that end, the effect of scaling the higher order elastic constants on the harmonic generation is studied. It is found that harmonic generation is proportional to the scaling factor. Second harmonic generation from the S0 mode at 1 MHz suggested that second harmonics can propagate independent of the primary mode. To confirm this, simulations were carried out for the A0 mode at 0.5 MHz. It was concluded that the second harmonic can propagate independent of the primary mode and hence group velocity matching is not necessary for higher harmonic generation although it may simplify experimental data analysis. Also, it was found that the primary mode continuously generates second harmonics. Generation of sum and difference frequency components via mode-interaction is demonstrated for the S0 mode at 0.4 MHz and the S0 mode at 1.1 MHz. Higher harmonic generation via mode interaction [16] seems an attractive alternative which is yet to be explored from a practical standpoint. Even though the results presented here are for plates, similar conclusions are expected for nonlinear guided waves in pipes/cylindrical waveguides as was illustrated in [24] using an asymptotic approximation.
Acknowledgements This material is based upon work supported by the Nuclear Energy Universities Program under Award number 00102946 and the National Science Foundation under Award number 1300562.
References [1] M.A. Breazeale, D.O. Thompson, Finite-amplitude ultrasonic waves in aluminum, Appl. Phys. Lett. 3 (1963) 77. [2] A. Hikata, B.B. Chick, C. Elbaum, Dislocation contribution to the second harmonic generation of ultrasonic waves, J. Appl. Phys. 36 (1965) 229. [3] J.H. Cantrell, W.T. Yost, Nonlinear ultrasonic characterization of fatigue microstructures, Int. J. Fatigue 23 (2001) 487–490.
[4] J.H. Cantrell, Substructural organization, dislocation plasticity and harmonic generation in cyclically stressed wavy metals, Proc. Roy. Soc. London A 460 (2004) 757–780. [5] M. Deng, Cumulative second-harmonic generation accompanying nonlinear shear horizontal mode propagation in a solid plate, J. Appl. Phys. 84 (7) (1998) 3500–3505. [6] M. Deng, Cumulative second-harmonic generation of Lamb-mode propagation in a solid plate, J. Appl. Phys. 85 (6) (1999) 3051–3058. [7] W.J.N. De Lima, M.F. Hamilton, Finite-amplitude waves in isotropic elastic plates, J. Sound Vib. 265 (4) (2003) 819–839. [8] B.A. Auld, Acoustic Fields and Waves in Solids, Robert E. Krieger Publishing Company, 1990. [9] N. Matsuda, S. Biwa, Phase and group velocity matching for cumulative harmonic generation in Lamb waves, J. Appl. Phys. 109 (9) (2011). 094903– 094903. [10] V.K. Chillara, C.J. Lissenden, Higher harmonic guided waves in isotropic weakly non-linear elastic plates, in: AIP Conference Proceedings, vol. 1511, 2013, p. 145–150. [11] C. Bermes, J.Y. Kim, J. Qu, L.J. Jacobs, Experimental characterization of material nonlinearity using Lamb waves, Appl. Phys. Lett. 90 (2) (2007). 021901– 021901. [12] C. Pruell, J.Y. Kim, J. Qu, L.J. Jacobs, Evaluation of plasticity driven material damage using Lamb waves, Appl. Phys. Lett. 91 (23) (2007). 231911–231911. [13] Y. Liu, J.Y. Kim, L.J. Jacobs, J. Qu, Z. Li, Experimental investigation of symmetry properties of second harmonic Lamb waves, J. Appl. Phys. 111 (5) (2012). 053511-053511. [14] Y. Liu, V.K. Chillara, C.J. Lissenden, On selection of primary modes for generation of strong internally resonant second harmonics in plates, J. Sound Vib. 33 (19) (2013) 4517–4528. [15] A. Srivastava, F. Lanza di Scalea, On the existence of antisymmetric or symmetric Lamb waves at nonlinear higher harmonics, J. Sound Vib. 323 (3) (2009) 932–943. [16] V.K. Chillara, C.J. Lissenden, Interaction of guided wave modes in isotropic weakly nonlinear elastic plates: higher harmonic generation, J. Appl. Phys. 111 (12) (2012). 124909-124909. [17] M.F. Müller, J.Y. Kim, J. Qu, L.J. Jacobs, Characteristics of second harmonic generation of Lamb waves in nonlinear elastic plates, J. Acoust. Soc. Am. 127 (2010) 2141. [18] M. Deng, Y. Xiang, L. Liu, Time-domain analysis and experimental examination of cumulative second-harmonic generation by primary Lamb wave propagation, J. Appl. Phys. 109 (11) (2011) 113525. [19] R.M. Bowen, Introduction to Continuum Mechanics for Engineers, Plenum, New York, 1989. [20] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, second ed., Permagon Press, New York, 1970. [21] M. Destrade, R.W. Ogden, On the third-and fourth-order constants of incompressible isotropic elasticity, J. Acoust. Soc. Am. 128 (2010) 3334. [22] V.K. Chillara, C.J. Lissenden, On some microstructural aspects of elastic strain energy function used for studying ultrasonic higher harmonic generation, submitted for publication. [23] K.T. Narasimha, E. Kannan, K. Balasubramaniam, Issues on the pulse-width dependence and the shape of acoustic radiation induced static displacement pulses in solids, J. Appl. Phys. 105 (7) (2009). 073506-073506. [24] V.K. Chillara, C.J. Lissenden, Analysis of second harmonic guided waves in pipes using a large radius asymptotic approximation for axis-symmetric longitudinal modes, Ultrasonics 53 (4) (2013) 862–869.
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Nonlinear guided waves in plates: A numerical perspective Vamshi Krishna Chillara ⇑, Cliff J. Lissenden Department of Engineering Science and Mechanics, The Pennsylvania State University, 16802 PA, United States
a r t i c l e
i n f o
Article history: Received 19 February 2014 Accepted 6 April 2014 Available online 16 April 2014 Keywords: Nonlinear ultrasound Nonlinear guided waves Higher harmonic generation Ultrasonics
a b s t r a c t Harmonic generation from non-cumulative fundamental symmetric (S0 ) and antisymmetric (A0 ) modes in plate is studied from a numerical standpoint. The contribution to harmonic generation from material nonlinearity is shown to be larger than that from geometric nonlinearity. Also, increasing the magnitude of the higher order elastic constants increases the amplitude of second harmonics. Second harmonic generation from non-phase-matched modes illustrates that group velocity matching is not a necessary condition for harmonic generation. Additionally, harmonic generation from primary mode is continuous and once generated, higher harmonics propagate independently. Lastly, the phenomenon of mode-interaction to generate sum and difference frequencies is demonstrated. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Use of nonlinear ultrasound for characterizing microstructure of structural materials, especially metals, has been a topic of interest for several decades. Initial investigations by Breazeale and Thompson [1] and Hikata et al. [2] put forth elastic material nonlinearity and dislocations as predominant causes for nonlinear ultrasonic behavior i.e., higher harmonic generation. These results motivated the use of higher harmonic generation for investigating nonlinear behavior and hence material degradation in structures. Cantrell and Yost [3] investigated fatigued microstructures using nonlinear ultrasound. Cantrell [4] presented a comprehensive approach to relate the nonlinearity parameter (b) to the dislocation substructures in metals. For polycrystalline nickel, a monotonic increase in b with the fatigue cycles was predicted. The above theoretical/ experimental investigations employed bulk waves (which travel in unbounded media) to characterize nonlinearity. On the other hand, guided waves (which travel in bounded structures) offer several advantages from an inspection standpoint in that long-range inspection can be carried out from a single location. Hence these are more amenable for structural health monitoring applications. Nonlinear guided waves combine the penetration power of guided waves with the early damage detection capabilities of nonlinear ultrasound. Hence they have emerged as an attractive alternative for detecting microstructural changes preceding macro-scale damage in the structures.
⇑ Corresponding author. Tel.: +1 814 954 2291. E-mail address: [email protected] (V.K. Chillara). http://dx.doi.org/10.1016/j.ultras.2014.04.009 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.
Deng [5,6] analyzed second harmonic generation from guided waves in plates. De Lima and Hamilton [7] presented a perturbation based approach to analyze second harmonic generation using the normal mode expansion technique [8] and arrived at two conditions necessary for cumulative second harmonic generation, namely phase matching and non-zero power flux. While bulk waves satisfy the above criterion for all frequencies of excitation, only specific guided wave modes satisfy them. These were identified [9,10] from the dispersion relations governing Rayleigh Lamb modes in the plate and experimental investigations [11–14] corroborated theoretical predictions. While the above investigations dealt with second harmonic generation, theoretical investigations were also carried out for predicting the nature of higher harmonic generation from guided waves in plates. Srivastava and Lanza di Scalea [15] concluded that cumulative even harmonics exist only as symmetric modes while odd harmonics can exist either as symmetric or antisymmetric modes. Chillara and Lissenden [16] presented a generalized theory to study the nonlinear interaction of guided wave modes. They concluded that the interaction of guided wave modes of the same nature generate symmetric modes while those of opposite nature generate antisymmetric modes. They also proposed a procedure to predict the nature of higher harmonics from the theory of mode interaction developed. While a comprehensive theoretical framework is now available to study higher harmonic generation from guided waves in plates, some issues still need to be addressed. These stem from the following issues: 1. The theoretical analysis is carried out for time-harmonic (single-frequency continuous wave) excitations while the experiments employ transducers with finite band-width.
1554
V.K. Chillara, C.J. Lissenden / Ultrasonics 54 (2014) 1553–1558
2. Group velocity matching in addition to the phase velocity matching between the primary mode and other higher harmonics was assumed to be necessary for cumulative harmonic generation [17]. However, Deng et al. [18] presented an analysis and an experimental investigation confirming that it is not a necessary condition. In general, all the primary Rayleigh-Lamb modes capable of cumulative second harmonic generation in plates satisfy group velocity matching in addition to phase velocity matching [9]. However, with finite wave packets, it would be interesting to examine if the higher harmonic modes can propagate without group velocity matching as the nonzero-power-flux criterion which requires power flow from primary to higher harmonic modes may not be satisfied once the primary and higher modes separate due to difference in group velocity. 3. The analysis is generally carried out using a perturbation approach as no closed form solution is available. 4. The capability of the existing constitutive model to capture the effect of different kinds of nonlinearities has not been established.
where k; l are Lame’s constants and A; B; C are higher order elastic constants. Another equivalent version of the above model is the Murnaghan model given by
WðEÞ ¼
1 1 kðtrðEÞÞ2 þ ltrðE2 Þ þ ðl þ 2mÞðtrðEÞÞ3 mtrðEÞ 2 3 ððtrðEÞÞ2 trðE2 ÞÞ þ ndetðEÞ
ð5Þ
where l; m and n are Murnaghan constants and trðÞ and detðÞ denote trace and determinant of the tensor respectively. The relation between A; B; C and l; m; n are given in [21]; l ¼ B þ C; m ¼ 12 A þ B and n ¼ A. The second Piola–Kirchhoff stress tensor is obtained using
TRR ¼
@WðEÞ @E
ð6Þ
The first Piola Kirchhoff stress tensor S and Cauchy stress tensor T are given by
S ¼ FTRR
ð7Þ
and In this article, we address some of the above issues using results from numerical simulations carried out in COMSOL Multiphysics 4.3, a commercial finite element software. Specifically, we consider the harmonic generation from the primary Symmetric (S0 ) and Antisymmetric (A0 ) modes in a plate. While neither of the above modes are exactly phase matched, they are deliberately chosen to highlight some of the aspects to be discussed later. As part of the study, we also comment on some aspects of the constitutive model like the contribution of material and geometric nonlinearities to higher harmonic generation. The content of the article is organized as follows. Section 2 presents some continuum mechanics preliminaries. Section 3 then presents the results obtained from numerical simulations. Finally, conclusions are drawn in Section 4. 2. Preliminaries In this section, we present some preliminaries intended to enhance understanding of the results to be presented. The discussion is brief so we refer the reader to [19] for more details. We denote the deformation gradient by F and the Lagrangian strain by E. Also, the displacement gradient is denoted by H and the following relations exist between the above quantities.
F¼IþH E¼
1 T 1 ðF F IÞ ¼ ðH þ HT þ HT HÞ 2 2
ð1Þ ð2Þ
We denote the linearized strain by El , which is related to the displacement gradient (H) by
El ¼
1 ðH þ HT Þ: 2
ð3Þ
Note that the difference between the Langrangian strain and the linearized strain is that the second order term involving HT H is dropped for the linearized strain. Including geometric nonlinearity means we consider Langrangian strain (full strain) as opposed to the linearized strain in the analysis. A widely used constitutive model used for studying higher harmonic generation was proposed by Landau and Lifshitz [20]. The corresponding strain energy function is given by
1 1 1 WðEÞ ¼ kðtrðEÞÞ2 þ ltrðE2 Þ þ CðtrðEÞÞ3 þ BtrðEÞtrðE2 Þ þ AðtrðE3 ÞÞ: 2 3 3 ð4Þ
T ¼ FSFT :
ð8Þ
While the first Piola–Kirchhoff stress tensor is used for a formulation in the reference configuration, Cauchy stress is used for a formulation in the current configuration. 3. Numerical simulations In this section, we present results obtained from numerical simulations performed in COMSOL 4.3. All the simulations were carried out using the Murnaghan model (Eq. (5)) for Aluminum, the elastic constants of which are tabulated in Table 1. As mentioned earlier, fundamental symmetric and antisymmetric modes are used in the simulation. The schematic of the 2D model used for the simulation is shown in Fig. 1. The thickness of the plate is chosen to be 1 mm and the length of the plate is assumed to be 100 mm unless otherwise specified. Triangular elements are used to discretize the plate with a maximum element size of 0.1 mm and a minimum element size of 0.03 mm. The resulting mesh is then scaled by a factor of 1.5 (both along the length and the thickness) to obtain a finer discretization. A maximum time step of 0.01 ls is used for the simulation. Appropriate displacement boundary conditions are enforced on the left end of the plate to excite the intended modes. The x-component of the displacement is denoted by ‘u’ and the y-component is denoted by ‘v’ where the axes are shown in Fig. 1. The dispersion curves for the plate are shown in Fig. 2 and the primary modes used to study harmonic generation in the plate are indicated. None of the primary modes selected are phase matched to the secondary modes, so the second harmonics are not known as ‘cumulative’. However, it is clear from both the theory [7] and the results that second harmonics are generated. 3.1. S0 mode at 0.5 MHz In this section, we present the results obtained for the S0 mode at 0.5 MHz. This mode is almost phase matched to the second harmonic as the phase speed of the primary mode is 5.34 mm/ls
Table 1 Elastic constants in GPa used for simulation. k
l
l
m
n
51
26
250
333
350
1555
V.K. Chillara, C.J. Lissenden / Ultrasonics 54 (2014) 1553–1558
x=50mm
−7
x 10
1
LE NL NG
0.8 0.6 Fig. 1. Schematic of the model used for simulations.
0.4
9
0 −0.2
8
−0.4
7
cp (mm/ µs)
0.2
u (m)
10
6
−0.6
S0
5
−0.8
4
−1
0
5
10
15
2
A0
25
30
Fig. 3. S0 mode at 0.5 MHz – time domain signals for LE, NL and NG constitutive models.
1 0
20
t (µs)
3
0
0.5
1
2
3
4
5
fd (MHz−mm)
x=50mm
−8
10 Fig. 2. Dispersion curves for the plate showing the primary modes used in the study.
LE NL NG
−9
10
1. Linear elastic material (LE) – No material or geometric nonlinearities are included i.e., l ¼ m ¼ n ¼ 0 and linearized strain El is used as a strain measure. 2. Nonlinear (NL) – Both material and geometric nonlinearities are included i.e., l – 0; m – 0; n – 0 and Lagrangian strain (E) is used as a strain measure. 3. Geometrically nonlinear (NG) – Only geometric nonlinearity is included i.e., l ¼ m ¼ n ¼ 0 and Lagrangian strain (E) is used as a strain measure. The displacement amplitude for the boundary condition is set to 1E7 m for the results presented in Figs. 3–5. Fig. 3 shows the time domain signal obtained for the displacement ‘u’ at 50 mm from the left end of the plate. It shows very little difference for the three cases, although the pulse in the ‘NL’ case arrives slightly later in time (i.e., travels at a lower speed) although this is barely visible. But in the frequency domain, as shown in the FFT (log-scale) in Fig. 4, nonlinearity is evident from the presence of static (or zero-frequency) and second harmonic components in the displacement. Notice also that a short toneburst results in a reasonably wide frequency bandwidth. While the ‘LE’ case shows no second harmonic (or static) component as expected, the other two cases have considerable second harmonic (and static) components. Also, the amplitude of the second harmonic
Amplitude
−10
10
−11
10
−12
10
−13
10
−14
10
0
0.5
1
1.5
2
frequency (MHz) Fig. 4. S0 mode at 0.5 MHz – FFT’s for the signals obtained using LE, NL and NG constitutive models.
−8
10
40mm 50mm 60mm
−9
10
−10
10
Amplitude
and that of the second harmonic is 5.27 mm/ls. It also satisfies the non-zero power flux criterion for generation of a cumulative second harmonic. We first discuss the contribution of material and geometric nonlinearities to the second harmonic generation in the plate. The amplitude of the excitation specified as the displacement boundary condition is chosen to be 1E7 m. The amplitude of 1E7 corresponds to a stress wave amplitude of a few MPa which is typical of an ultrasonic wave propagating in a solid. In other cases that follow, results for higher amplitude are presented to emphasize some of the effects not easily decipherable at low amplitudes. The results for three simulations corresponding to three variants of the constitutive model (Eq. (5)) are presented.
−11
10
−12
10
−13
10
−14
10
0.2
0.4 0.5 0.6
0.8
1
1.2
1.4
1.6
1.8
frequency (MHz) Fig. 5. S0 mode at 0.5 MHz – FFT’s for the signals at 40, 50 and 60 mm from the left end of the plate.
1556
V.K. Chillara, C.J. Lissenden / Ultrasonics 54 (2014) 1553–1558
generated in the ‘NL’ case is 4–5 times greater than that generated in the ‘NG’ case. This shows that the second harmonic generation is dominated by the material nonlinearity as opposed to geometric nonlinearity. Signals were received at 40, 50 and 60 mm from the left end of the plate and Fig. 5 shows their corresponding FFT’s on a log-scale. As predicted by theory [7], the amplitude of the second harmonic increases with the propagation distance although it is not strictly cumulative. In the following example, we discuss the effect of scaling higher order elastic constants on the harmonic generation. The displacement amplitude for the boundary condition is set to 1E7 m. Simulations are run for the ‘NL’ case using three different sets of higher order elastic constants obtained by scaling l; m and n (Table 1) by factors of 1, 2 and 4. Fig. 6 shows the FFT’s of the time domain signals on a log-scale. As can be seen, the FFT’s show increasing zero-frequency and second harmonic components with the scaling. The amplitude of the second harmonic is almost proportional to the scaling factor used. Furthermore, the third harmonic is apparent for scaling factors of 2 and 4.
Fig. 7. S0 mode at 1 MHz – time domain signals at 50 mm from the left end of the plate for ‘LE’ and ‘NL’ cases.
3.2. S0 mode 1 MHz
1 2 4
−9
10
LE NL
−8
10
−9
10
−10
10
−11
10
−12
10
−13
10
0
0.5
1
1.5
2
2.5
3
frequency (MHz) Fig. 8. S0 mode at 1 MHz – FFT’s of the signal at 50 mm from the left end of the plate for ‘LE’ and ‘NL’ cases.
the second harmonic. This is because the static component, unlike second-harmonic, is always cumulative [23]. Also, observe that the FFT shows a small dip around 2 MHz. This is due to the presence of the second harmonic tail in the time domain window (27 ls) used for taking the FFT.
x=50mm
−8
10
x=50mm
−7
10
Amplitude
Here, we present the results obtained for second harmonic generation from the S0 mode at 1 MHz. The phase velocity of the mode is 5.27 mm/ls and that of the second harmonic (i.e., S0 mode at 2 MHz) is 4.71 mm/ls. These modes are not phase matched and hence the second harmonic is not cumulative. The amplitude of the displacement for the boundary condition is set to 1E6 m. Fig. 7 shows the time domain signal for ‘LE’ and ‘NL’ cases. As indicated in the inset, the second harmonic signal almost separates from the primary mode. Also, the time domain signal for the ‘NL’ case shows asymmetry about the zero. This arises due to the static or the zero-frequency component in the signal as evident from the FFT shown in Fig. 8. Furthermore, note that the time domain pulse for the ‘NL’ case overlaps with that of the ‘LE’ case to a greater extent during the times when ‘u’ is increasing i.e., particle velocity is positive than when ‘u’ is decreasing i.e., when particle velocity is negative. This is due to the tension–compression asymmetry exhibited by the constitutive model in Eq. (5). Detailed discussion on the above and other relevant aspects of the above constitutive model can be found in [22]. Fig. 8 shows the FFT’s of the corresponding time-domain signals in Fig. 7. As can be seen, the static component has a much larger amplitude as compared to that of
Amplitude
−10
10
3.3. A0 mode at 0.5 MHz −11
10
−12
10
−13
10
−14
10
0
0.5
1
1.5
2
frequency (MHz) Fig. 6. S0 mode at 0.5 MHz – FFT’s for the signal obtained by scaling higher order elastic constants by 1, 2 and 4 respectively.
In this section, we present results obtained for second harmonic generation from the A0 mode at 0.5 MHz. The length of the waveguide (plate) is increased to 200 mm to allow for a longer propagation distance. The other discretization parameters were unaltered from the previous set of simulations for the S0 mode. The amplitude of the displacement boundary condition (‘v’) is increased to 1E5 m. Fig. 9 shows a zoomed-in view of time domain signals at 40, 80 and 120 mm. The second harmonic (S0 mode) is the smaller wave packet that arrives before the larger wave packet (A0 mode). Fig. 10 shows the through thickness profiles for ‘v’ displacement at ‘x’ = 120 mm for t = 28–35 ls. The following observations can be made:
1557
V.K. Chillara, C.J. Lissenden / Ultrasonics 54 (2014) 1553–1558 −7
x=40,80,120mm
−8
2
x 10
40mm 80mm 120mm
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
10
20
30
40
50
40mm 60mm
1.5
u (m)
v (m)
1.5
−2
x 10
2
−2
60
0
5
10
15
t (µs) Fig. 9. A0 mode at 0.5 MHz – time domain signals at 40, 80 and 120 mm from the left end of the plate.
x 10
30
−8
10
0.9
40mm 60mm
−9
10
0.8 0.7
−10
Amplitude
Thickness (m)
25
Fig. 11. Mode interaction (S0 (0.4 MHz) S0 (1.1 MHz)) – time domain signals at 40 and 60 mm from the left end of the plate.
x=120mm
−3
1
20
t (µs)
0.6 0.5 0.4
10
−11
10
−12
10
0.3 0.2
−13
10
0.1 −14
0 −8
10
−6
−4
−2
0
v (m)
2
4
6
8
0
x 10
Fig. 10. A0 mode at 0.5 MHz – through thickness profiles for ‘v’ displacement at 120 mm from the left end of the plate for various times t = 28–35 ls.
1. The second harmonic pulse clearly separates from the primary mode and propagates independently. This demonstrates that group velocity matching as described in [17] is not necessary for second harmonic generation. 2. The second harmonic mode is S0 as opposed to A0 , which is in agreement with that predicted by the perturbation approach. This can be inferred from Fig. 10 which shows that the through thickness profile for ‘v’ is antisymmetric. 3. In fact, it can be concluded that the second harmonic is generated continuously from the primary mode and both propagate independently once they separate. This can be inferred with the following line of reasoning. As the primary and secondary modes separate, the primary mode again generates a second harmonic as it propagates independently. This means that this process continuously takes place and hence second harmonics are continuously generated. When sufficient time has elapsed, these pulses separate and the process continues. Also, the generated harmonics can further generate higher harmonics, albeit very small in amplitude. A procedure to assess the nature of such harmonic generation is presented in [16].
0.4 0.5
0.7
1 1.1
1.5
2
frequency (MHz)
−9
Fig. 12. Mode interaction (S0 (0.4 MHz) S0 (1.1 MHz)) – FFT’s for the signals at 40 and 60 mm from the left end of the plate.
3.4. Mode interaction-S0 mode at 0.4 MHz and S0 mode at 1.1 MHz In this section, we demonstrate the sum and difference harmonic frequency generation for the ‘NL’ case by mixing two modes, namely, S0 mode at 0.4 MHz and S0 mode at 1.1 MHz propagating in the same direction. The amplitude of the displacement boundary condition is 1E7 for both modes. Fig. 11 shows the time domain signals at 40 and 60 mm. Fig. 12 shows the FFT (log-scale) for the time domain signals in Fig. 11. As can be seen, it shows the presence of sum and difference frequency components i.e., 1.1 0.4 = 0.7 MHz and 1.1 + 0.4 = 1.5 MHz. 4. Conclusions In this article we investigated the behavior of nonlinear guided waves in plates from a numerical perspective. Simulations of nonlinear wave propagation were carried out in COMSOL using the Murnaghan model of hyper-elasticity. Harmonic generation from the fundamental guided wave modes S0 and A0 is studied. Solutions from ‘linear’, ‘nonlinear’ and ‘geometrically nonlinear’ variants of
1558
V.K. Chillara, C.J. Lissenden / Ultrasonics 54 (2014) 1553–1558
the Murnaghan model are compared for S0 mode at 0.5 MHz. While time domain signals show very little difference, harmonic generation is quite evident in the frequency domain. Also, the contribution of material nonlinearity to the harmonic generation is higher when compared to the geometric nonlinearity. As the material damage state/ microstructure is typically inferred from the b parameter, which in turn depends on the higher order elastic constants, it is essential that one understands the behavior of harmonic generation with respect to the changes in higher order elastic constants. To that end, the effect of scaling the higher order elastic constants on the harmonic generation is studied. It is found that harmonic generation is proportional to the scaling factor. Second harmonic generation from the S0 mode at 1 MHz suggested that second harmonics can propagate independent of the primary mode. To confirm this, simulations were carried out for the A0 mode at 0.5 MHz. It was concluded that the second harmonic can propagate independent of the primary mode and hence group velocity matching is not necessary for higher harmonic generation although it may simplify experimental data analysis. Also, it was found that the primary mode continuously generates second harmonics. Generation of sum and difference frequency components via mode-interaction is demonstrated for the S0 mode at 0.4 MHz and the S0 mode at 1.1 MHz. Higher harmonic generation via mode interaction [16] seems an attractive alternative which is yet to be explored from a practical standpoint. Even though the results presented here are for plates, similar conclusions are expected for nonlinear guided waves in pipes/cylindrical waveguides as was illustrated in [24] using an asymptotic approximation.
Acknowledgements This material is based upon work supported by the Nuclear Energy Universities Program under Award number 00102946 and the National Science Foundation under Award number 1300562.
References [1] M.A. Breazeale, D.O. Thompson, Finite-amplitude ultrasonic waves in aluminum, Appl. Phys. Lett. 3 (1963) 77. [2] A. Hikata, B.B. Chick, C. Elbaum, Dislocation contribution to the second harmonic generation of ultrasonic waves, J. Appl. Phys. 36 (1965) 229. [3] J.H. Cantrell, W.T. Yost, Nonlinear ultrasonic characterization of fatigue microstructures, Int. J. Fatigue 23 (2001) 487–490.
[4] J.H. Cantrell, Substructural organization, dislocation plasticity and harmonic generation in cyclically stressed wavy metals, Proc. Roy. Soc. London A 460 (2004) 757–780. [5] M. Deng, Cumulative second-harmonic generation accompanying nonlinear shear horizontal mode propagation in a solid plate, J. Appl. Phys. 84 (7) (1998) 3500–3505. [6] M. Deng, Cumulative second-harmonic generation of Lamb-mode propagation in a solid plate, J. Appl. Phys. 85 (6) (1999) 3051–3058. [7] W.J.N. De Lima, M.F. Hamilton, Finite-amplitude waves in isotropic elastic plates, J. Sound Vib. 265 (4) (2003) 819–839. [8] B.A. Auld, Acoustic Fields and Waves in Solids, Robert E. Krieger Publishing Company, 1990. [9] N. Matsuda, S. Biwa, Phase and group velocity matching for cumulative harmonic generation in Lamb waves, J. Appl. Phys. 109 (9) (2011). 094903– 094903. [10] V.K. Chillara, C.J. Lissenden, Higher harmonic guided waves in isotropic weakly non-linear elastic plates, in: AIP Conference Proceedings, vol. 1511, 2013, p. 145–150. [11] C. Bermes, J.Y. Kim, J. Qu, L.J. Jacobs, Experimental characterization of material nonlinearity using Lamb waves, Appl. Phys. Lett. 90 (2) (2007). 021901– 021901. [12] C. Pruell, J.Y. Kim, J. Qu, L.J. Jacobs, Evaluation of plasticity driven material damage using Lamb waves, Appl. Phys. Lett. 91 (23) (2007). 231911–231911. [13] Y. Liu, J.Y. Kim, L.J. Jacobs, J. Qu, Z. Li, Experimental investigation of symmetry properties of second harmonic Lamb waves, J. Appl. Phys. 111 (5) (2012). 053511-053511. [14] Y. Liu, V.K. Chillara, C.J. Lissenden, On selection of primary modes for generation of strong internally resonant second harmonics in plates, J. Sound Vib. 33 (19) (2013) 4517–4528. [15] A. Srivastava, F. Lanza di Scalea, On the existence of antisymmetric or symmetric Lamb waves at nonlinear higher harmonics, J. Sound Vib. 323 (3) (2009) 932–943. [16] V.K. Chillara, C.J. Lissenden, Interaction of guided wave modes in isotropic weakly nonlinear elastic plates: higher harmonic generation, J. Appl. Phys. 111 (12) (2012). 124909-124909. [17] M.F. Müller, J.Y. Kim, J. Qu, L.J. Jacobs, Characteristics of second harmonic generation of Lamb waves in nonlinear elastic plates, J. Acoust. Soc. Am. 127 (2010) 2141. [18] M. Deng, Y. Xiang, L. Liu, Time-domain analysis and experimental examination of cumulative second-harmonic generation by primary Lamb wave propagation, J. Appl. Phys. 109 (11) (2011) 113525. [19] R.M. Bowen, Introduction to Continuum Mechanics for Engineers, Plenum, New York, 1989. [20] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, second ed., Permagon Press, New York, 1970. [21] M. Destrade, R.W. Ogden, On the third-and fourth-order constants of incompressible isotropic elasticity, J. Acoust. Soc. Am. 128 (2010) 3334. [22] V.K. Chillara, C.J. Lissenden, On some microstructural aspects of elastic strain energy function used for studying ultrasonic higher harmonic generation, submitted for publication. [23] K.T. Narasimha, E. Kannan, K. Balasubramaniam, Issues on the pulse-width dependence and the shape of acoustic radiation induced static displacement pulses in solids, J. Appl. Phys. 105 (7) (2009). 073506-073506. [24] V.K. Chillara, C.J. Lissenden, Analysis of second harmonic guided waves in pipes using a large radius asymptotic approximation for axis-symmetric longitudinal modes, Ultrasonics 53 (4) (2013) 862–869.
