Nonlinear feature of phase matched Lamb waves in solid plate

Applied Acoustics 160 (2020) 107124

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Nonlinear feature of phase matched Lamb waves in solid plate Weibin Li a, Bingyao Chen a, Younho Cho b,⇑ a b

School of Aerospace Engineering, Xiamen University, South Siming Road, 361005 Xiamen, China School of Mechanical Engineering, Pusan National University, 10511, San 30, Jangjeon-dong, Geumjeong-gu, 609-735 Busan, South Korea

a r t i c l e

i n f o

Article history: Received 12 January 2019 Received in revised form 2 July 2019 Accepted 29 October 2019

Keywords: Lamb waves Phase matching Nonlinear parameters Second harmonic generation

a b s t r a c t The measurement of second harmonic Lamb waves is investigated as a promising feature extraction technique to characterize material nonlinearity in plate-like structures. In general, the effect of second harmonic generation can be weak, due to the dispersive nature of Lamb waves. The phase matched Lamb modes are explored to generate the second harmonics with a cumulative growth effect. Among all phase matched Lamb modes, it is of particular interest to find the most suitable Lamb modes for improving the efficiency of second harmonic generation and reception. Analytical expressions and experimental observations of second harmonic generation of Lamb waves in an isotropic plate with weak nonlinearity, are presented in this paper. Nonlinear parameters of Lamb waves are derived in this paper for the first time as a function of signal frequency, wave mode and geometric parameters of waveguides. Nonlinear features of various phase matched Lamb modes are discussed for the comparison of efficiency of second harmonic generation. The experimental results are found to be consistent with the theoretical predictions in a qualitative manner. This study shows that physics based feature selection is essential for efficient generation and reception of second harmonic Lamb waves. Ó 2019 Published by Elsevier Ltd.

1. Introduction Early stage detection of micro-defects to identify sub-structural material damage prior to the crack initiation [1], is becoming increasingly important for some critical structures such as nuclear power plants and aircrafts. However, reliable experimental methods to detect material damages before the appearance of microcracks in structural component are rarely available. The use of nonlinear ultrasonic waves has been found to be one of the most promising methods for evaluating material micro-structural changes in their early stages [2–8]. Furthermore, ultrasonic guided wave based techniques are becoming the increasingly popular nondestructive evaluation (NDE) methods. Possibility to inspect inaccessible or hidden areas and great cost-effectiveness are the key advantages of guided wave-based damage detection approaches [9,10]. Because of the high sensitivity of the nonlinear ultrasonic approaches and the great advantages of guided wave based techniques, the nonlinear ultrasonic guided waves have drawn significant attentions for material characterization and micro-damage detection [11–15]. One popular nonlinear ultrasonic technique is the generation of second harmonics. Compared to bulk waves, the second harmonic

⇑ Corresponding author. E-mail address: [email protected] (Y. Cho). https://doi.org/10.1016/j.apacoust.2019.107124 0003-682X/Ó 2019 Published by Elsevier Ltd.

fields of Lamb waves are much more complex because of dispersion and multi-mode nature of propagating Lamb waves. In general, the effect of second harmonic generation is often very small and can easily be overlooked due to the dispersive nature of Lamb waves. Consequently, proper mode tuning with physically based feature is essential to enhance the efficiency of second harmonic Lamb wave generation and reception. The second harmonic generation of the Lamb wave propagation in an isotropic plate has been theoretically investigated using perturbation method and normal modal analysis technique [16,17]. Srivastava et al. [18] reported the possibility of the existence of anti-symmetric or symmetric Lamb modes at higher harmonics. As revealed in these works, the second harmonic modes of Lamb waves display an accumulative effect when phase matching and non-zero power transfer from the fundamental mode to the second harmonic one conditions are satisfied. Group velocities of primary wave modes and second harmonic wave modes should also be equal to achieve the cumulative harmonic generation as discussed in [19,20], while this idea is still in dispute [21]. Recently, Liu et al. [22] investigated on the selection of primary modes for generation of cumulative second harmonic waves in plate. In experimental work, it is critical to find the second harmonic generation with cumulative effect to enable measurements of the nonlinear effects with sufficient signal-to-noise ratio. Deng [23] used the S2 Lamb mode under internal resonant condition to

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detect accumulative second harmonics. Pruell et al. [19,20] evaluated the material nonlinearity using S1 Lamb mode as the primary wave with phase matching condition in aluminum specimens. Although the well-phase matched Lamb modes can generate cumulative second harmonic waves, there have been few studies on the comparison of the sensitivity of these modes for proper mode tuning and selection. It is important to understand the nonlinear features of various Lamb modes for the efficient data acquisition of second harmonic generation. The characteristics of various phase matched Lamb modes were first discussed by Müller et al. [24] and Matlack et al. [25], who presented the investigations of the second harmonic generation of different phase matched Lamb modes in nonlinear elastic plates. The frequencies and corresponding phase velocities of different Lamb modes that can be used to generate cumulative higher harmonic waves were also discussed by Matsuda et al. [26]. Two different mode pairs of circumferential guided waves which satisfy the phase matching condition were studied numerically and experimentally in a circular tube [27]. However, the guideline for choosing the better phase matched Lamb modes has not been reported in detail yet. The physical insight of nonlinear feature among different Lamb modes is not well-understood and still obscure. Selecting certain modes from all phase matched ones can play a significant role to improve the sensitivity of the detection of material nonlinearity. In this paper, analytical expressions and experimental observations of second harmonic generation of various Lamb modes in an isotropic plate are presented. Nonlinear parameters of Lamb waves are derived for the first time as a function of wave mode, geometric information of waveguides and frequency. Nonlinear features of various phase matched Lamb modes are discussed for comparing efficiency of second harmonic generation. The experimental results are found to be consistent with the theoretical predictions. This study shows that physics based feature selection is essential for efficient generation of second harmonic Lamb waves. 2. Second harmonic generation of Lamb waves in an isotropic plate 2.1. Solution of second harmonic Lamb modes The equation of motion for elastic waves in an isotropic, homogeneous, nonlinear elastic infinite plate with stress-free boundary condition is given by [16,17]

ðk þ 2lÞrðr  uÞ  lr  ðr  uÞ þ f ¼ q0

2

@ u @t 2

ð1Þ

where u is the displacement field, k and l are the elastic constants, q0 is the initial density of the body, and f is the nonlinear term. Perturbation method is used to solve the nonlinear wave equations. The solution is taken as the sum of the primary wave u1 and the second harmonic u2:

u ¼ u1 þ u2

ð2Þ

where u2 is assumed to be much smaller than u1, and the primary wave has the frequency-wave number ðx; kÞ, and is written as follows:

u1 ¼ uðzÞeiðkxxtÞ

ð3Þ

Constructing the second order solution using modal expansion [28], the total second harmonic field can be written as

u2 ¼

1 X



Am ðxÞu2 ðzÞei2xt ;

ð4Þ

m¼1

where u2 is the displacement field function, x is the wave propagation direction as shown in Fig. 1. 2h is the thickness of the

Fig. 1. Schematic of a stress free plate.

waveguide, and Am ðxÞis the corresponding expansion coefficient, which is the second order modal amplitude to be determined. Each secondary mode m enters the sum with its own amplitude coefficient Am ðxÞ making certain second harmonic modes more or less excitable by the primary wave mode. Thus, the amplitude coefficients Am ðxÞ for the second harmonic modes provided information and can be expressed by the solution of the following ordinary differential equation:

 4pmn

   d  surf v ol þ ikn Am ðxÞ ¼ f n þ f n ei2kx ; dx

ð5Þ

where

pmn ¼  surf

Z

1 4

X

Z

f n ðxÞ ¼ v ol

f n ðxÞ ¼



C

Z X

vn  sm þ v m  sn



 nx dX;

ð6Þ

v n  s2w  nx dC;

ð7Þ

vn  f 2w dX;

ð8Þ

X is the waveguide cross-sectional area and C is the curve enclosing  X per unit length. kn is the wavenumber of mth second harmonic surf

v ol

Lamb wave. The termsf n and f n are defined as the complex exter2w

nal power due to the surface traction s2w and the volume force f , respectively. The expansion coefficient Am ðxÞ in Eq. (5) can be formally expressed as,

  surf v ol fn þ fn   sinðDn xÞ Am ðxÞ ¼ exp iðkn þ Dn Þx ; Dn 4pmn 

Dn ¼ ð2k  kn Þ=2;

Am ðxÞ ¼

  surf v ol fn þ fn x 4pmn

ð9Þ ð10Þ

  exp iðkn þ Dn Þx ;



kn ¼ 2k:

ð11Þ

As shown in Eqs. (9), (10) and (11), it is found that the magnitude of the mth second harmonic Lamb wave Am ðxÞ is linearly pro portional to propagation distance x,at the condition of kn ¼ 2k (second harmonic Lamb mode has the same phase velocity as that of primary one). However, if the phase velocity of generated second  harmonic mode does not equal that of primary one (kn –2k), the magnitude of second harmonic Lamb wave Am ðxÞis a sine function of propagation distance x, which means that magnitude of generated second harmonic Lamb modes under the mismatching condition, will vanish at a certain propagation distance. Thus, interest is focused on the second harmonic generation with the cumulative effect since the cumulative second harmonic dominates the second harmonic field after a certain propagating distance. 2.2. Phase matched Lamb modes Considering that single primary mode propagation in the waveguide generates multiple secondary modes, the second harmonic field of Lamb wave propagation can be assumed as

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W. Li et al. / Applied Acoustics 160 (2020) 107124

super-positions of a series of double frequency Lamb wave components. However, we focus on the component that has the same phase velocity as the fundamental wave mode since only the wave mode with proper phase matching can remain in the field after it propagates a certain distance, while all others can decay due to destructive interference with each other. For this reason, the accumulative effect of this double frequency component will dominate the second harmonic field, after the fundamental wave mode has propagated over a certain distance till it also starts to decay due to material attenuation. The contributions of other components having different phase velocity values than the fundamental mode can be neglected in the second harmonic calculation, since they will be out of phase and the energy of the fundamental wave cannot be efficiently transferred to them. Consequently, the primary Lamb mode and the phase matched secondary wave mode can be taken as the mode pair. From the analysis of Eq. (11), it is found that the second harmonic amplitude linearly grows with the propagation distance, 

surf

v ol

when kn ¼ 2k (synchronism) and f n þ f n –0 (nonzero power flux). If the mode chosen satisfies these two conditions, the second harmonic amplitude will be accumulative. Even a series of double frequency wave components will be generated by the driving sources of nonlinearity. In practice, interest is focused on the second harmonic generation with the cumulative effect since the cumulative second harmonic dominates the second harmonic field after a certain propagating distance. From the dispersion curves, the phase matched guided wave modes can be found. The fundamental Lamb modes and double frequency second harmonic Lamb modes should have the same phase velocity for the reason stated above. All of the modes satisfy the phase matching conditions and have potential to generate accumulative second harmonic waves. The key concept of nonlinear Lamb wave test is to use those modes with a good ‘‘phase matching” associated with significant energy transmission from the fundamental frequency wave to the second harmonic one.

tudinal wave and transverse wave, respectively. h is half of the thickness of the waveguide. A is a constant that represents the displacement amplitude to be determined. For Lamb waves, neither the in-plane nor the out-of-plane displacement is constant across the thickness of the plate, both of them vary across the thickness, even for a specific Lamb mode at certain frequency. From practical view points, the displacement profile on the surface is of more concern than that inside the specimen. In this work we use the displacement (either in-plane or out-of-plane displacement) on the surface of the plate to simplify the expressions. Therefore, on the surface (z ¼ h), if only the in-plane displacement is considered, the displacement of the primary wave and second harmonic Lamb waves can be expressed as:

"

Uðf Þ ¼ AcoshðphÞ 1 

# 2 ðk þ q2 Þ

" Uð2f Þ ¼ Dcoshð2phÞ 1 

2k

2

2

eiðkxxtÞ

ðk þ q2 Þ 2k

2

ð16Þ

# ei2ðkxxtÞ

ð17Þ

The phase matched second order component will dominate the second harmonic field, so, the contribution from other components can be ignored. The partial wave technique and normal modal expanding method are used to analyze the amplitude relation of fundamental wave and second harmonic wave displacement. On the surface, the Lamb wave can be decomposed into the crossinteraction and self-interaction of two partial longitudinal and shear wave components [30]. The nonlinear interaction of the two partial bulk waves includes the self-interaction of each longitudinal wave and shear wave, as well as the cross-interaction between one longitudinal wave and one shear wave. Therefore, three nonlinear components represented by displacement of second harmonic Lamb wave field on the surface can be approximately decomposed into three components as follows [31–33]:

D¼

3 X

ðnÞ

C ðnÞ eik

x

;

ð18Þ

n¼1

3. Nonlinear parameters for Lamb waves

where 3.1. Derivation of nonlinear parameters for Lamb waves The displacement field for Lamb wave propagation in an isotropic plate with traction free boundary conditions can be decomposed into in-plane (U) and out-of-plane (W) contributions, and each contribution can be divided into symmetric and antisymmetric parts as following [29]:

U ¼ Us þ Ua ;

ð12Þ

W ¼ W s þ W a;

ð13Þ

"

" # 2 p ðk þ q2 Þ coshðphÞsinhðqzÞ iðkxxtÞ e sinhðpzÞ  Ws ¼ A ik 2pq coshðqhÞ

 2  3 qcl þ C 166 kt A2 C ð2Þ ¼  sin½ðkt  kl Þx; 2 2 2 4qcl ðkt  kl Þ

C

where U s ; W s and U a ; W a represent the symmetric and antisymmetric motions respectively. The acoustic nonlinear parameter relates to the ratio of the fundamental and second harmonic displacement amplitudes. For example, for a symmetric Lamb mode, the displacement can be expressed as:

# 2 ðk þ q2 Þ coshðphÞcoshðqzÞ iðkxxtÞ e U s ¼ A coshðpzÞ  2 coshðqhÞ 2k

  3qc2l þ C 111 2 2 ð1Þ kl A x; k ¼ 2kl C ð1Þ ¼  8qc2l

ð14Þ

ð15Þ

where k is the wave number of nth Lamb mode, x is the wave propagation distance. sinh and coshare hyperbolic functions, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 p ¼ k  kl and q ¼ k  kt ; kl and kt are wave numbers of longi-

k

ð2Þ

¼ kl þ kt

ð20Þ

  2 2 2    2  qcl þ C 166 kl kt þ kt kl A kt  kl x ; ¼ sin 2 2 2 qc2t ðkl þ kt Þ  4kt

ð3Þ

ð3Þ

k

ð19Þ

¼

3kt þ kl 2

ð21Þ

It is found that both C ð2Þ and C ð3Þ are the sine functions of propagation distance, which means that the value of C ð2Þ and C ð3Þ will vanish at a certain propagation distance. However, it is also clearly found that the value of C ð1Þ is linearly proportional to the propagation distance. So, if the wave propagation distance is sufficiently long, second harmonic field could be mainly attributed to C ð1Þ . From this analysis, the relationship between D and A on the surface can be approximately taken as: 2

D¼

bl kl A2 x ; 8

where bl ¼ 

ð22Þ 3qc2l þC 111

qc2l

is the acoustic nonlinearity parameter for lon-

gitudinal wave. The analysis in an isotropic material reveals that the

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W. Li et al. / Applied Acoustics 160 (2020) 107124

quadratic nonlinearity for shear waves is zero [33,34]. The absence of transverse harmonics can be simply understood in terms of the isotropic material symmetry, which forbids quadratic terms for shear deformation. The same results can also be found in [32,35] for Rayleigh surface waves. Let us first consider the case of inplane displacement for symmetric Lamb mode. The ratio of the displacement amplitude of primary and second harmonic waves can be represented as: 2

2

U 2 ðf Þ 8 cosh ðphÞ k þ q2 ¼ 2 1 2 Uð2f Þ bkl x coshð2phÞ 2k

! ð23Þ

the plate. It has been shown that thick plates (h >> k), the total displacements of Lamb waves in the upper and lower halves of the plate are similar to that of Rayleigh waves [32,35]. It is shown in Eqs. (26) and (27) that the definitions of the nonlinear parameter of Lamb waves include geometric information of the waveguide. In this work, the acoustic nonlinear parameter of Rayleigh surface wave is compared with that of Lamb wave in a plate with infinite thickness. The limitation of acoustic nonlinear parameters for Lamb wave with infinite thickness as follows: 2

lim bL:s

h!1

Therefore, the nonlinear parameter for symmetric Lamb modes can be represented by in-plane displacement on the surface as: 2

bL:s

2

A2 8 cosh ðphÞ k þ q2 1 ¼ 2 2 2 coshð2phÞ A1 kl x 2k

! ð24Þ

where A2 and A1 are the in-plate displacement amplitude for the second harmonic mode and primary symmetric wave on the surface of the specimen. The nonlinear parameter for anti-symmetric modes can also be derived following the same procedure as: 2

bL:a

2

A2 8 sinh ðphÞ k þ q2 1 ¼ 2 2 2 A1 kl x coshð2phÞ 2k

! ð25Þ

Using the same method, the nonlinear parameters of Lamb wave can be represented by the out-of-plane displacement on the surface as

! 2 2 A2 i8 p sinh ðphÞ 2k ; 1  2 A21 k2l x k sinhð2phÞ k þ q2

ð26Þ

! 2 2 A2 i8 p cosh ðphÞ 2k : 1 2 ¼ 2 2 A1 kl x k sinhð2phÞ k þ q2

ð27Þ

bL:s ¼

bL:a

where bL:s and bL:a are nonlinear parameters for symmetric and antisymmetric Lamb modes respectively. It is important to note that the cumulative second harmonic filed is symmetric even when the primary wave mode is anti-symmetric [16,19–25]. The formulae of the nonlinear parameter for a Lamb mode show that, the acoustic nonlinear parameter for Lamb wave is a function of frequency, material properties and geometric dimensions of the waveguide. Therefore, nonlinear feature of a Lamb mode can be affected by the mode type, frequency of the incident signal, the material properties and the geometry of the waveguide. 3.2. Comparison of nonlinear parameters for Rayleigh surface waves and Lamb waves Considering the relation between Lamb and Rayleigh surface waves, the acoustic nonlinear parameters of Lamb waves are compared with that of Rayleigh surface waves to check the validity of Lamb wave nonlinear parameter in this study. As reported by Herrmann et al. [35], the acoustic nonlinear parameter for a Rayleigh surface wave near the free surface of a half-space can be represented by the out-of-plate displacement as

! 2 A2 i8 p 2k : 1 2 bS ¼ 2 2 A1 kl x k k þ q2

ð28Þ

Rayleigh surface waves are free waves on the surface of a simiinfinite solid, which is placed on one free boundary surface. Surface traction must vanish on the boundary, and the waves must decay with depth [36,37]. Lamb waves are waves of plane strain that occur in a free solid plate, which has two boundary surfaces. The surface traction must vanish on the upper and lower surfaces of

2

lim bL:a ¼ lim

h!1

2

A2 i8 p sinh ðphÞ 2k 1 2 ¼ lim 2 2 h!1 A k x k sinhð2phÞ k þ q2 1 l

h!1

2

!

A2 i8 p cosh ðphÞ 2k 1 2 A21 k2l x k sinhð2phÞ k þ q2

¼

1 b; 2 S

ð29Þ

¼

1 b: 2 S

ð30Þ

!

The physical reason for the coefficient of 1=2 is that the whole displacement field of Lamb wave is divided into symmetric and anti-symmetric parts. As the thickness of the layer increases to infinity, the total displacement field with symmetric and antisymmetric parts is similar to the Rayleigh surface wave displacement field, as shown below:

lim ðbL:s þ bL:a Þ ¼ bS :

h!1

ð31Þ

It is shown that the definition of the acoustic nonlinear parameter for Lamb waves includes geometric information of the waveguide. These verifications validate the nonlinear parameter of the Lamb wave in an isotropic solid layer. 3.3. Nonlinear parameters adjusted for attenuation Generally, there are two main mechanisms for amplitude decay, which are ‘‘material damping or attenuation” and ‘‘wave packet spreading” corresponding to the wave dispersion effect. The physical nature of dispersion is that, the wave velocity is changed along with frequency change. To generate an accumulative second harmonic wave, we focus on ‘‘phase matching”, i. e., modes whose phase velocity is the same at the fundamental frequency and the double frequency. The dispersive effect is insignificant in this study, for the reason that the Lamb modes selected are all satisfied the phase matching condition, the amplitude decay is assumed to be mainly due to the material attenuation. Taking the attenuation factor into consideration, the acoustic nonlinear parameter can be corrected by the attenuation correction factor as [38]:

Da ¼



m  ; 1  em

m ¼ ða2  2a1 Þx;

ð32Þ

where a1 and a2 are the attenuation coefficients of the fundamental wave and the second harmonic one, respectively. Thus, the nonlinear parameters of Lamb wave can be modified as:

b¼

A2 8Da A21 k2 x

F:

ð33Þ

So, the definition of the acoustic nonlinear parameter for Lamb waves can be divided into three parts: the basic part (A2 =A21 ), the attenuation term (Da =x) and a feature function (F). It is important to note that Eq. (33) is only shown that the generation of second harmonic Lamb waves in a determined specimen, can be influenced by the feature function, which is the function of thickness, material properties of waveguide, frequency, wave number and symmetric nature of selected Lamb mode. Meanwhile, the attenuation factors of primary mode and second harmonic one can also affect the generation of second harmonic waves. We can theoretically predict the correlation of A2 =A21 versus the propagation distance, at the condition of that we know all of the parameters

W. Li et al. / Applied Acoustics 160 (2020) 107124

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above. However, the formula is clearly indicated the unique nonlinear feature of different Lamb modes and proper mode selection and frequency tuning with physically based feature is useful to increase the efficiency of second harmonic generation. But in experimental test, efficiency of second harmonic Lamb wave generation is more complicated. For example, second harmonic amplitude could be bigger of selected mode at higher frequency based on the theoretical prediction of Eq. (33). But the big value of excitation frequency of signal will cause a drastic increase of attenuation. Thus, amplitude of double frequency secondary mode will decay significantly in experimental test. The derived Eq. (33) just provides an analyzable formula for theoretical prediction of second harmonic generation of various Lamb modes in a determined specimen. 4. Results and discussion For a given specimen, to test the material nonlinearity using Lamb waves, the modal-based nonlinear guided wave technique is necessary for higher efficiency data acquisition of the second harmonics. It has been recognized that the phase matched modes have the potential to generate accumulative second harmonic Lamb waves. However, from a large number of phase matched mode pairs, it is important to find out the most efficient pair for the second harmonic generation. In order to make a comparison of second harmonic generations among different phase matched Lamb modes, the theoretical and experimental investigations of the efficiency of second harmonic generations in a given specimen are presented below. The material properties for computing the dispersion curves (see Fig. 2) of the propagating Lamb modes in the specimen are shown in Table 1. Second harmonic Lamb mode at double frequency of primary Lamb mode, which is driven by the fundamental mode, is depended on primary Lamb wave. We take the fundamental mode and the same phase velocity second harmonic Lamb mode as one mode pair. For example, the propagated S1 mode packet carries the fundamental and double frequency second harmonic waves. In this study, the fundamental mode (S1) and second harmonic mode (S2) have the same phase velocity, this is analogous to the phase condition in resonance vibration (internal resonance), which can be taken as one mode pair. Three different phase matched Lamb modes (S1 mode at frequency of 3.58 MHz, S2 mode at 7.16 MHz and A2 mode at frequency of 5.05 MHz, see Fig. 2) are compared in this study. The information of phase matched Lamb modes is list in the Table 2. 4.1. Theoretical prediction Considering the unique nonlinear feature of different Lamb modes, the analysis of nonlinear feature of different modes is favorable for higher efficient data acquisition of second harmonics. The efficiency of second harmonic generation can be theoretically predicted for a fixed Lamb mode with certain frequency in a given specimen. In the above theoretical analysis, it is shown that phase matching and non-zero power transfer are the necessary conditions for the cumulative second harmonic Lamb wave generation, while group velocity matching condition is not involved into the theoretical definition. More detailed discussion on the cumulative second harmonic generation of guided waves with group velocity mis-matching can be found in [40]. As reported by Li et al. [41], the amplitude ratio [the second harmonic amplitude divided by the square of the fundamental wave (A2 =A21 )] has a correlation with the wave propagation distance (x). The slope of the amplitude ratio (A2 =A21 ) plotted against the wave propagation distance can be used as a measure of the efficiency

Fig. 2. Phase matched Lamb modes indicated by square symbols, in (a) phase velocity dispersion curve and (b) group velocity dispersion curve.

of second harmonic generation. The variation of the amplitudes ratio (A2 =A21 ) with the wave propagation distance is calculated as shown in Fig. 3. It is found that the amplitudes ratio grows with the propagation distance because of the accumulative effect up to a certain point until the material attenuation starts to dominant. To ensure that the nonlinearity measurements are not arising from the measurement system nonlinearity, but it is really due to the damage-induced nonlinearity, the demonstration of this cumulative effect is essential. 4.2. Experimental verification Piezoelectric transducers (PZT) and wedges are used here for the generation of different Lamb modes. The angle of incidence for generating Lamb modes is given by Snell’s law (sinðhÞ ¼ cl:w =cp ), where cp is the phase velocity of Lamb wave and cl:w is the longitudinal wave velocity in the wedge material. The phase velocities of Lamb modes can be found from the dispersion curve shown in Fig. 2. The schematic diagram of the experimental setup is shown in Fig. 4. The wedge transducer TX, consisting of the plexiglas wedge with an oblique angle and the narrow band longitudinal wave piezoelectric transducer (manufactured by Panametrics Inc.), is used to generate the desired primary mode. The angle is determined by Snell’s law (sinðhÞ ¼ cl:w =cp ), where cp is the phase velocity of Lamb wave and cl:w is the longitudinal wave velocity in the wedge material. While, the wedge trans-

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W. Li et al. / Applied Acoustics 160 (2020) 107124

Table 1 Material properties and dimensions of the specimen.

q (g/cm3)

Thickness (mm)

C l (km/s)

C t (km/s)

Material inherent nonlinearity

2.72

1

6.35

3.1

5.67 [39]

Table 2 Information of phase matched Lamb mode. Mode pair

Frequency of primary mode (MHz)

Phase velocity of primary mode (km/s)

Group velocity of primary mode (km/s)

S1-S2 A2-S4 S2-S4

3.58 5.05 7.16

6.35 8.46 6.35

4.33 2.11 4.33

Fig. 3. Theoretical prediction of amplitudes ratios of different phase matched Lamb modes.

ducer TR consisting of the plexiglas wedge and the broad band longitudinal wave piezoelectric transducer (manufactured by Panametrics Inc.), is initially designed and used to receive the second harmonic Lamb mode at the double frequency of primary one. The oblique angle is also determined by Snell law. The longitudinal wave piezoelectric transducers can readily be assembled onto or removed from the respective plexiglas wedge with bolts. Light lubrication oil is used to couple the transducers to the wedges. Consistent pressure is applied to the pair of transducers through mechanical grips to ensure uniform contact conditions.

Fig. 4. Schematic diagram of experimental setup.

In the experimental examination, the wedge transducer is driven by the sinusoidal tone-burst voltages (provided by the output channel of the Ritec 5000 SNAP system, Ritec Inc.). To suppress the transient behavior from the amplifier, the amplified tone-burst voltages pass through the specific attenuator, which is set to purify the incident signal for a high signal-to-noise ratio (SNR). Meanwhile, band-pass filter with certain cut frequencies are used to suppress the unexpected frequency components in the amplified tone-burst voltages. When the tone-burst voltages mentioned above is, applied to the longitudinal transducer, the refraction of the bulk longitudinal wave from the plexiglas wedge into the aluminum sheet can induce the formation of the desired primary Lamb modes (i.e., S1 at f = 3.58 MHz, S2 at f = 7.16 MHz and A2 at f = 5.05 MHz). To generate S1 mode at frequency 3.58 MHz, the tone burst system generates an input electrical signal with 3.58 MHz frequency and the signal is fed into the narrowband transducer with center frequency of 3.5 MHz. The receiving transducer simultaneously detects the primary and second harmonic wave amplitudes with the center frequency of 7.5 MHz. A Hanning window is imposed on the steady state part of the signal and signals are digitally processed using fast Fourier transform (FFT) to obtain the amplitudes A1 for the fundamental frequency and A2 for the second-harmonic double frequency. The second harmonic generation for two other phase matched Lamb modes - the S2 mode at frequency 7.16 MHz and the A2 mode at frequency 5.05 MHz – were done following similar technique. It is important to note that when we generate the A2 mode at 5.05 MHz, it is inevitable to generate also a S2 mode since they are very close in the phase velocity dispersion curve plot. This is sometimes unavoidable because of the multimodal nature of the Lamb wave propagation. However, the group velocity of the S2 mode is quite different from that of the A2 mode. As shown in Fig. 2(b), the group velocity of S2 mode is distinctly faster than that of A2 mode. Therefore, after the multi-modal signal propagates a certain distance, the S2 and A2 wave-packages will eventually separate. Fig. 5 shows the typical measured guided wave propagation in the plate at the deriving frequency of 5 MHz. It is well known that interaction of propagating guided waves with material nonlinearity in the specimen will cause the waveform distortion. Thus the formation of a second harmonic double the frequency of the fundamental input frequency will come out, which results from waveform distortion in the time domain. The propagated guided wave mode packet carries the fundamental and double frequency second harmonic wave. In time domain spectrum, the second harmonic wave at double frequency can be extracted effectively form the primary wave through the high-pass digital filtering processing, as shown in Fig. 5(a). The spectrums of fundamental at frequency of 5 MHz and second harmonic mode at frequency of 10 MHz are clearly separated in frequency domain, by performing the fast Fourier transformation on the measured time-domain signals. We could check their amplitude in frequency spectrum as shown in Fig. 5(b). In experimental measurement, the received propagated wave packet carries both the primary wave and second harmonic one. Actually, the primary wave and second harmonic wave has the same phase and group velocity as shown in Table 2 (for this example, primary wave is A2 mode and second harmonic one is the S4 mode.). The phase and group velocity matching of primary and

W. Li et al. / Applied Acoustics 160 (2020) 107124

7

Fig. 5. Example of measured signals (a) time domain signals, (b) amplitude-frequency of primary wave and second harmonic wave.

second harmonic wave ensure the consistency of energy transfer from primary wave to second harmonic one during the propagation in the specimen. We can intentionally extract the second harmonic wave from the primary wave by high-pass digital filtering processing as shown in Fig. 5(a). But, it shows that the second harmonic wave arrived a litter bit later than the primary wave after filtering. The reason for this phenomenon can be attributed to the factor that the second harmonic wave, which driven by the primary wave, is depended on the interactions of propagating primary wave with waveguide. The generation of second harmonic wave is due to the waveform distortion caused by material nonlinearity of primary wave after a certain propagation distance. Similar phenomenon can also be found in other publications [42,43]. It is important to note that the wave-structure of the phasematched modes is used to illustrate the displacement (in-plane or out-of plane displacement) flux across the section of waveguide. If the in-plane displacement of the selected mode is dominant on the surface of the plate, the formula derived by in-plane displacement will be chosen to increase the comparability of the experimental study, otherwise, the out-of-plane displacement should be used. In this investigation, the in-plane displacements of the chosen modes are dominant on the surface. It is known that the horizontal displacement domination at the surface displays a disadvantage for experimental measurement. However, the horizontal displacement component at the surface displays the only measurable quantity, for the reason that the normal component vanishes at the surface. In this investigation, experimental setup is modified by coupling the wedge to aluminum plate with consistent solid-state couplant to detect the in-plane displacement efficiently at the surface. The transducers are mounted on the surface of the elastic aluminum plate with a consistent solid couplant to efficiently generate and measure the phase matched modes with only in-plane displacement at the surface. Fig. 6 shows that the measured amplitude ratio (A2 =A21 ) grows with the propagation distance due to the accumulative effect up to a certain distance until the material attenuation starts to dominate for all phase matched mode pairs. The increase of the measured amplitude ratio with the propagation distance is a characteristics of phase matched Lamb modes. As mentioned in Section 2.2, this cumulative effect of the second harmonic amplitude is very important in experimental work to measure the nonlinear effect with acceptable signal-to-noise ratio. Second harmonic generation of Lamb wave propagation in the specimen is the typical acoustic nonlinear response, which can be used for material characterization. The measureable value of amplitude ratio (A2 =A21 ) can be qualitatively used to characterize the nonlinearity. Cumulative effect of second harmonic generation

Fig. 6. Experimental verification of amplitudes ratio for different phase matched Lamb modes.

of Lamb wave versus propagation distance is due to the consistent energy transfer from primary wave to second harmonic one. Among all these phase matched Lamb modes, different Lamb modes must have different rates of energy transfer efficiency from the primary mode to second harmonic one. Bigger normalized second harmonic wave amplitude (A2 =A21 ) corresponds to higher energy transfer efficiency for a given propagation distance. In addition, it is experimentally easier to measure the bigger normalized second harmonic amplitude with a higher signal-to-noise ratio. Thus the efficiency of second harmonic generation of different mode pair can be represented by the slope ratio of normalized second harmonic wave amplitude against propagation distance. It is shown that this slope for different Lamb modes are quite different. It should be noted that the experimental results shown in Fig. 6 and the theoretical predictions shown in Fig. 3 are in good agreement, both indicated that among the three modes investigated here S2 mode is the most efficient and A2 is the least efficient mode for second harmonic signal generation and detection. Therefore, althrough all phase matched Lamb modes can be used to generate the accumulative second harmonic waves, certain Lamb modes are more sensitive than others to material nonlinearity and should be used for such investigation. The theoretical analysis shown in Fig. 3 indicates the efficiency of second harmonic generation of various phase matched Lamb modes. While the experimental nonlinear responses shown in

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W. Li et al. / Applied Acoustics 160 (2020) 107124

Fig. 6 of different Lamb modes are attributed to efficiency of second harmonic generation, frequency characteristics of exciting/receiving transducers, the wave-structure of Lamb modes and symmetry of excitation mode, which can affect the experimental measurement. However, it is also meaningful to conduct the comparison of efficiency of second harmonic generation and the experimental response sensitivity. The change rate of second harmonic generation of Lamb waves versus propagation distance in a given specimen is corresponding to the differences of acoustic nonlinear response of various Lamb modes with same material nonlinearity. The results obtained can be used to provide a reference of potential mode selection combinations that the cumulative effect of wave propagation and potentially with a higher signal-to-noise ratio. The measured nonlinear response enables the qualitatively characterization of efficiency of second harmonic generation. This investigation of detecting nonlinear features using Lamb modes provides more insight into choosing the proper mode for detecting material nonlinearity. It shows that some Lamb modes are more sensitive than others for material nonlinearity detection. It is important to note that the efficiency of second harmonic generation is not the only factor that should be considered for nonlinear ultrasonic tests. Some other factors like displacement of the mode at the surface of the specimen, frequency and group velocity of the mode can also affect the excitation and measurement of second harmonics and should be considered while choosing the mode. 5. Conclusions Efficiency of second harmonic generation of various phase matched Lamb modes in an isotropic plate is investigated. Second order solutions of the nonlinear wave equation in a plate with traction free condition on the surfaces are newly explored. Nonlinear parameters of Lamb waves are derived as a function of wave mode, geometric parameters of waveguides and frequency. Nonlinear features of various phase matched Lamb modes are discussed for comparison of efficiency of second harmonic generation. Theoretical prediction and experimental verification of the efficiency of second harmonic generation of various phase matched Lamb modes are conducted in this work. It turns out that certain Lamb modes are more sensitive to material nonlinearity than others. This study of the nonlinear feature extraction from the Lamb modes provides more insight into choosing the proper mode for better sensitivity in detecting material nonlinearity. Conflicts of interest The authors declared that they have no conflicts of interest to this work. Acknowledgement This research was supported by National Natural Science Foundation of China (11774295), and also supported by Korea Evaluation Institute of Industrial Technology (KEIT) grant funded by the Korea government (MOTIE) (No.10085576). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.apacoust.2019.107124. References [1] Bermes C, Kim JY, Qu J, Jacobs LJ. Nonlinear Lamb waves for the detection of material nonlinearity. Mech Syst Signal Pr 2008;22:638–46.

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