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Experimental and theoretical study of harmonic generation at contacting interface
Ultrasonics 44 (2006) e1319–e1322 www.elsevier.com/locate/ultras
Experimental and theoretical study of harmonic generation at contacting interface S. Biwa *, S. Hiraiwa, E. Matsumoto Department of Energy Conversion Science, Graduate School of Energy Science, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan Available online 2 June 2006
Abstract The second harmonic generation behavior of a contacting interface has been evaluated experimentally and discussed theoretically in the light of a nonlinear interface model. Two aluminum blocks were mated together to constitute a contact interface and subjected to normal compressive loading. A 5 MHz longitudinal toneburst wave was sent to the interface in the normal direction and the transmitted wave was recorded, from which the fundamental and the second harmonic components were extracted. A nonlinearity parameter was obtained as the ratio of the second harmonic amplitude to the squared fundamental amplitude. From the measured contact pressure dependence of the transmitted fundamental amplitude, the linear and the second-order interfacial stiffness parameters were identified, which enabled the evaluation of the nonlinearity parameter based on the theoretical model. The theoretical contact pressure dependence of the nonlinearity parameter was found to be in good qualitative agreement with the experimental results. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Nonlinear ultrasonics; Contact interface; Harmonic generation; Interfacial stiffness
1. Introduction When an ultrasonic wave with sufficiently high amplitude interacts with a contacting interface, it gives rise to higher harmonic components in the transmitted and reflected waves. This effect of so-called contact acoustic nonlinearity [1–3] is recently attracting increasing attention regarding the characterization of closed defects or imperfect bond interfaces. Foregoing studies [4–7] have revealed that when the ultrasonic wavelength is much greater than the roughness scales of the contacting interface, it can be modeled as a spring interface with equivalent stiffness properties. The interfacial stiffness changes sensitively with the contact pressure, which is believed to be a source of the nonlinear acoustic effect. The nonlinear elastic wave transmission/reflection at a contacting interface was studied theoretically in Refs. [8,9] based on simplistic models. Recently, this problem was *
Corresponding author. Fax: +81 75 753 5897. E-mail address: [email protected] (S. Biwa).
0041-624X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2006.05.010
investigated [10,11] based on a nonlinear interface model where the interfacial stiffness was assumed to be a function of the contact pressure. In these works, the second harmonic amplitude was derived in terms of the stiffness parameters of the contacting interface. As pointed out in Ref. [11], however, there seem to be yet no coordinated experimental studies in open literatures that link the nonlinear interfacial stiffness to the harmonic generation behavior. The aim of the present study is then to evaluate the contact pressure dependence of both the interfacial stiffness and the second harmonic amplitude for aluminum–aluminum contacting surfaces. Using the measured pressure-stiffness relation, the pressure dependence of the second harmonic generation behavior is examined theoretically based on the above nonlinear interface model, and compared to the experimental data. 2. Theoretical relations The main theoretical results are outlined from Ref. [11] regarding the longitudinal wave propagation across a
e1320
S. Biwa et al. / Ultrasonics 44 (2006) e1319–e1322
contacting interface between two similar elastic solids, based on a nonlinear interface model. The contact pressure p is given by a nonlinear function of the gap distance h(t) = h0 + u(X+, t) u(X, t), where the wave displacement is denoted by u(x, t) referring to Fig. 1. The initial gap distance h0 = X+ X corresponds to the applied static pressure p0, i.e., p0 = p(h0). This relation can be expanded around h0 as its Taylor series, and used here with up to the second-order term, 2
pðhÞ ¼ p0 K 1 ðh h0 Þ þ K 2 ðh h0 Þ ; dp 1 d2 p K1 ; K2 ; dh 2 dh2 h¼h0
ð1Þ ð2Þ
h¼h0
where the linear interfacial stiffness K1 and the second-order stiffness parameter K2 both depend on the initial gap h0, or equivalently, the applied pressure p0. For the monochromatic incident wave with angular frequency x, a simple perturbation analysis gives the transmitted and reflected waves containing the static (DC), fundamental and second harmonic components, which are explicitly given in Ref. [11]. Denoting the density and the wave velocity of the contacting solids by q and c, respectively, the amplitude transmission and reflection coefficients of the fundamental component are given by ~1 2K T ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ~ 21 1 þ 4K
1 R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ~ 21 1 þ 4K
~ 1 ¼ K1 : K qcx
ð3Þ
When the absolute displacement amplitudes of the fundamental and the second harmonic components in the transmitted wave are denoted by A1 and A2, respectively, their ratio is given by A2 ¼ A1
K 2A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ~ 21 1 þ K ~ 21 2K 1 1 þ 4K
ð4Þ
These expressions show that the second harmonic amplitude depends on the applied contact pressure. 3. Experimental procedure Two aluminum blocks were mated and pressed to each other to make a dry contact interface (Fig. 2). The contacting surfaces were a square with the side-length 30 mm. The length along the loading direction was also 30 mm for each block. The contacting surfaces were polished with the #1000 paper and lapped with alumina powders. Their roughness values (arithmetical mean deviation) were found to be typically less than 0.1 lm. A 5 MHz longitudinal toneburst wave was sent to the interface from the end of one block, using a signal generator (Agilent Technologies 33220A) and a gated amplifier (RITEC RAM-5000) to excite a piezoelectric transducer (Panametrics) with 5 MHz nominal frequency. The transmitted wave was detected at the end of the other block by a wideband transducer with 10 MHz nominal frequency. The transducers were pushed against the aluminum blocks, with glycerin as a couplant medium, by rubber pads inside metal spacers in order to maintain a constant coupling force irrespective of the compression loading level applied to the contacting blocks. The time interval of the incident toneburst was fixed as 6 ls (30 periods). The excitation voltage (peak-to-peak value) applied to the input transducer was varied at three levels of (a) 374 V, (b) 486 V and (c) 600 V. The transmitted waveforms were stored via a digital oscilloscope (Tektronix TDS340AP), in a loading/unloading cycle of the contacting interface up to about 5 MPa nominal contact pressure. Prior to the measurements, the mated interface was subjected to several loading/unloading cycles in order to avoid the significant hysteresis due to the inelastic flattening during the first contact loading [5].
where A is the amplitude of the incident wave [11]. In the present study, the nonlinearity parameter b is defined by the second harmonic amplitude divided by the square of the fundamental amplitude, which is given by b¼
A2 qcxK 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ 2 A1 4K 2 1 þ K ~ 21 1
Signal generator
ð5Þ
Transmitter (5 MHz)
Rubber pad
Gated amplifier
Spacer p0
p0
Aluminum blocks Incident wave
Sinusoidal waveform
Contacting interface
Transmitted wave Digital oscilloscope
x Spacer
Reflected wave
Receiver (10 MHz)
Rubber pad
Waveform stored
Load cell Gap distance h X+ + u (X+,t) X– + u (X–,t) Fig. 1. Schematic model of a contacting interface.
PC
Fig. 2. Experimental setup.
S. Biwa et al. / Ultrasonics 44 (2006) e1319–e1322
4. Results and discussion The measured variation of the transmission coefficient of the fundamental (5 MHz) component with the nominal contact pressure is shown in Fig. 3 for three levels of excitation voltage, (a) 374 V, (b) 486 V and (c) 600 V. Different excitation voltages imply different incident wave amplitudes, although the latter quantities could not be quantified in the present experiment. Fig. 3 shows that the transmission coefficient is independent of the excitation voltage, indicating that this is essentially a linear phenomenon. Slight hysteresis can be observed for loading and unloading. The measured transmission coefficient can be converted to the linear stiffness K1 from Eq. (3) using the acoustic properties of aluminum, q = 2700 [kg/m3] and c = 6420 [m/s]. The pressure dependence of the linear stiffness is also shown in Fig. 3.
Exp.(c)
0 0 1 2 3 4 50 1 2 3 4 50 1 2 3 4 5 Contact pressure[MPa]
Fig. 4. Variation of the normalized second harmonic amplitude with the applied contact pressure, for the transducer excitation (peak-to-peak) voltages (a) 374 V, (b) 486 V and (c) 600 V.
0.6
0.4
0.4
0.2
K1, exp.(a) K1, exp.(b) K1, exp.(c)
1
2 3 Contact pressure[MPa]
4
0.2 0 5
Fig. 3. Variation of the transmission coefficient at 5 MHz and the linear interfacial stiffness with the applied contact pressure, for the transducer excitation (peak-to-peak) voltages (a) 374 V, (b) 486 V and (c) 600 V.
0.03
parameter K2 [MPa/nm2]
0.8
0.6
0 0
Exp. (b)
0.05
Second-order stiffness
0.8
Exp. (a) 0.1
1 T, exp.(a) T, exp.(b) T, exp.(c)
Contact stiffness K1 [MPa/nm]
Transmission coefficient T
1
Fig. 4 shows the variations of A2V/A1V with the applied contact pressure for the three excitation voltage levels. It is seen that this quantity exhibits more profound hysteresis at loading/unloading, although its reason is unclear yet. The ratio A2V/A1V appears to attain a maximum at a very low contact pressure, and decrease monotonically as the contact pressure increases. This ratio increases more or less proportionally with the excitation voltage, in qualitative correspondence with the theoretical result that A2/A1 is proportional to the incident wave amplitude. In order to evaluate the magnitude of the second harmonic component due to the factors other than the contact nonlinearity, a single aluminum block with the same crosssection and the doubled propagation length (60 mm) was used in the same setup instead of the paired aluminum blocks. This measurement revealed that the harmonic amplitude corresponding to about A2V/A1V = 0.03 was present in the absence of the contact interface. This value appears to be close to the values of A2V/A1V in Fig. 5 when the contact pressure becomes sufficiently high. Each measured relation between the contact pressure (p0) and the linear stiffness (K1) in Fig. 3 was P fitted by a 5th-order polynomial expression K 1 ðp0 Þ ¼ 5k¼0 ak pk0 , with ak being constants. Then, from the definitions in Eq. (2),
Normalized second harmonic amplitude A2V/A1V
From the recorded waveforms, the magnitude of the fundamental (5 MHz) and the second harmonic (10 MHz) components, denoted as A1V and A2V, respectively, were extracted (the subscript V indicates that these quantities correspond to the signal voltages and not to the absolute displacements). The normalized second harmonic amplitude A2V/A1V and the nonlinearity parameter bV ¼ A2V =A21V were calculated for the three excitation voltages and for different contact pressures. A caution should be paid, however, to the fact that these values were obtained by a transducer with specific spectral characteristics and are not to be directly compared to A2/A1 and b in the previous section: it is the contact pressure dependence of these values that is of present interest. When converting the amplitude of the transmitted fundamental component to the transmission coefficient T, the same transducer (5 MHz) was used in a pitch-catch mode to measure reflection coefficients R at the contacting interface for different pressures by aid of the spectrum analysis for the broadband echo [6]. Using the formula T2 + R2 = 1, the transmitted fundamental amplitude was calibrated to obtain the transmission coefficient at 5 MHz.
e1321
0.02
0.01 Experiment(a) 374 V (p-p) Loading Unloading
0 0
1 2 Contact pressure [MPa]
3
Fig. 5. Variation of the second-order contact stiffness parameter with the applied contact pressure for the excitation (peak-to-peak) voltage (a) 374 V identified from the measured pressure-linear stiffness relation.
S. Biwa et al. / Ultrasonics 44 (2006) e1319–e1322
0.8 0.6 0.4 0.2
0
1 2 Contact pressure [MPa]
0 3
Fig. 6. Variation of the nonlinearity parameter with the applied contact pressure (experimental and theoretical results), for the transducer excitation (peak-to-peak) voltages (a) 374 V, (b) 486 V and (c) 600 V.
the second-order contact stiffness parameter K2 can be calculated as a function of the applied pressure, 1 dK 1 K 2 ðp0 Þ ¼ K 1 ðp0 Þ 2 dp0 ! ! 5 5 X X 1 k m1 ¼ ak p 0 mam p0 : 2 k¼0 m¼1
5. Conclusion
-1
Nonlinearity parameter β V [a.u.] (experimental)
β V, exp. (a) β V, exp. (b) β V, exp. (c) β , calculated from (a) β , calculated from (b) β , calculated from (c)
Nonlinearity parameter β [nm ] (theoretical)
e1322
ð6Þ
As an example, Fig. 5 illustrates the variation of K2 with the contact pressure for the loading and unloading processes in the case of excitation voltage 374 V. Since the pressure-linear stiffness relation was basically insensitive to the excitation level (Fig. 3), so was the pressure dependence of K2. Using the obtained values of K1 and K2, Eq. (5) can be used to compute the nonlinearity parameter b at different contact pressures. The experimental results for the nonlinearity parameter bV are plotted in Fig. 6 against the contact pressure. It is remarkable that bV appears to be independent of the excitation voltage for the entire range of the applied pressure. This is in good agreement with the theoretical result given in Eq. (5). The parameter bV is found to decrease monotonically as the applied pressure increases, and the associated hysteresis is almost negligible. This feature is in agreement with the classical experiment [12]. The theoretical curves for b are also included in Fig. 6 where different curves corresponding to different excitation voltages appear to overlap onto a single curve. It is noted, however, that the direct quantitative comparison between the theoretically calculated b and the corresponding experimental values is not possible due to the arbitrariness of the scale of the measured wave amplitude. In spite of this, as seen in Fig. 6 the qualitative agreement between the theoretical calculations and the experimental data is quite satisfactory in terms of their dependence on the applied contact pressure.
The second harmonic generation at a contacting aluminum–aluminum interface has been studied from both experimental and theoretical points of view. From the measured contact pressure dependence of the linear interfacial stiffness, the second-order contact stiffness parameter was identified. The nonlinearity parameter was calculated using the so-obtained stiffness parameters according to the theoretical results based on a nonlinear interface model. It has been shown that the computed contact pressure dependence of the nonlinearity parameter is in qualitative agreement with the experimental results. Acknowledgement This study has been supported in part by the Grant-inAid for Young Scientists (B) (No.16760066) from MEXT, as well as the IVNET Development Project ‘‘Advancement of Maintenance Quality for Nuclear Power Plant’’ from METI, which are acknowledged gratefully. References [1] I.Y. Solodov, Ultrasonics of non-linear contacts: propagation reflection and NDE-applications, Ultrasonics 36 (1998) 383–390. [2] D. Donskoy, A. Sutin, A. Ekimov, Nonlinear acoustic interaction on contact interfaces and its use for nondestructive testing, NDT & E Int. 34 (2001) 231–238. [3] S. Hirsekorn, Nonlinear transfer of ultrasound by adhesive joints a theoretical description, Ultrasonics 39 (2001) 57–68. [4] P.B. Nagy, Ultrasonic classification of imperfect interfaces, J. Nondestruct. Eval. 11 (1992) 127–139. [5] B.W. Drinkwater, R.S. Dwyer-Joyce, P. Cawley, A study of the interaction between ultrasound and a partially contacting solid–solid interface, Proc. R. Soc. London Ser. A 452 (1996) 2613–2628. [6] S. Biwa, A. Suzuki, N. Ohno, Evaluation of interface velocity reflection coefficients and interfacial stiffnesses of contacting surfaces, Ultrasonics 43 (2005) 495–502. [7] A. Suzuki, S. Biwa, N. Ohno, Numerical and experimental evaluation of ultrasonic wave propagation characteristics at contact interface, JSME Int. J. Ser. A 48 (2005) 20–26. [8] J.M. Richardson, Harmonic generation at an unbonded interface: I. planar interface between semi-infinite elastic media, Int. J. Eng. Sci. 17 (1979) 73–85. [9] O.V. Rudenko, Chin An Vu, Nonlinear acoustic properties of a rough surface contact and acoustodiagnostics of a roughness height distribution, Acoust. Phys. 40 (1994) 593–596. [10] C. Pecorari, Nonlinear interaction of plane ultrasonic waves with an interface between rough surfaces in contact, J. Acoust. Soc. Am. 113 (2003) 3065–3072. [11] S. Biwa, S. Nakajima, N. Ohno, On the acoustic nonlinearity of solidsolid contact with pressure-dependent interface stiffness, Trans. ASME J. Appl. Mech. 71 (2004) 508–515. [12] O. Buck, W.L. Morris, J.M. Richardson, Acoustic harmonic generation at unbonded interfaces and fatigue cracks, Appl. Phys. Lett. 33 (1978) 371–373.
Experimental and theoretical study of harmonic generation at contacting interface S. Biwa *, S. Hiraiwa, E. Matsumoto Department of Energy Conversion Science, Graduate School of Energy Science, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan Available online 2 June 2006
Abstract The second harmonic generation behavior of a contacting interface has been evaluated experimentally and discussed theoretically in the light of a nonlinear interface model. Two aluminum blocks were mated together to constitute a contact interface and subjected to normal compressive loading. A 5 MHz longitudinal toneburst wave was sent to the interface in the normal direction and the transmitted wave was recorded, from which the fundamental and the second harmonic components were extracted. A nonlinearity parameter was obtained as the ratio of the second harmonic amplitude to the squared fundamental amplitude. From the measured contact pressure dependence of the transmitted fundamental amplitude, the linear and the second-order interfacial stiffness parameters were identified, which enabled the evaluation of the nonlinearity parameter based on the theoretical model. The theoretical contact pressure dependence of the nonlinearity parameter was found to be in good qualitative agreement with the experimental results. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Nonlinear ultrasonics; Contact interface; Harmonic generation; Interfacial stiffness
1. Introduction When an ultrasonic wave with sufficiently high amplitude interacts with a contacting interface, it gives rise to higher harmonic components in the transmitted and reflected waves. This effect of so-called contact acoustic nonlinearity [1–3] is recently attracting increasing attention regarding the characterization of closed defects or imperfect bond interfaces. Foregoing studies [4–7] have revealed that when the ultrasonic wavelength is much greater than the roughness scales of the contacting interface, it can be modeled as a spring interface with equivalent stiffness properties. The interfacial stiffness changes sensitively with the contact pressure, which is believed to be a source of the nonlinear acoustic effect. The nonlinear elastic wave transmission/reflection at a contacting interface was studied theoretically in Refs. [8,9] based on simplistic models. Recently, this problem was *
Corresponding author. Fax: +81 75 753 5897. E-mail address: [email protected] (S. Biwa).
0041-624X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2006.05.010
investigated [10,11] based on a nonlinear interface model where the interfacial stiffness was assumed to be a function of the contact pressure. In these works, the second harmonic amplitude was derived in terms of the stiffness parameters of the contacting interface. As pointed out in Ref. [11], however, there seem to be yet no coordinated experimental studies in open literatures that link the nonlinear interfacial stiffness to the harmonic generation behavior. The aim of the present study is then to evaluate the contact pressure dependence of both the interfacial stiffness and the second harmonic amplitude for aluminum–aluminum contacting surfaces. Using the measured pressure-stiffness relation, the pressure dependence of the second harmonic generation behavior is examined theoretically based on the above nonlinear interface model, and compared to the experimental data. 2. Theoretical relations The main theoretical results are outlined from Ref. [11] regarding the longitudinal wave propagation across a
e1320
S. Biwa et al. / Ultrasonics 44 (2006) e1319–e1322
contacting interface between two similar elastic solids, based on a nonlinear interface model. The contact pressure p is given by a nonlinear function of the gap distance h(t) = h0 + u(X+, t) u(X, t), where the wave displacement is denoted by u(x, t) referring to Fig. 1. The initial gap distance h0 = X+ X corresponds to the applied static pressure p0, i.e., p0 = p(h0). This relation can be expanded around h0 as its Taylor series, and used here with up to the second-order term, 2
pðhÞ ¼ p0 K 1 ðh h0 Þ þ K 2 ðh h0 Þ ; dp 1 d2 p K1 ; K2 ; dh 2 dh2 h¼h0
ð1Þ ð2Þ
h¼h0
where the linear interfacial stiffness K1 and the second-order stiffness parameter K2 both depend on the initial gap h0, or equivalently, the applied pressure p0. For the monochromatic incident wave with angular frequency x, a simple perturbation analysis gives the transmitted and reflected waves containing the static (DC), fundamental and second harmonic components, which are explicitly given in Ref. [11]. Denoting the density and the wave velocity of the contacting solids by q and c, respectively, the amplitude transmission and reflection coefficients of the fundamental component are given by ~1 2K T ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ~ 21 1 þ 4K
1 R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ~ 21 1 þ 4K
~ 1 ¼ K1 : K qcx
ð3Þ
When the absolute displacement amplitudes of the fundamental and the second harmonic components in the transmitted wave are denoted by A1 and A2, respectively, their ratio is given by A2 ¼ A1
K 2A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ~ 21 1 þ K ~ 21 2K 1 1 þ 4K
ð4Þ
These expressions show that the second harmonic amplitude depends on the applied contact pressure. 3. Experimental procedure Two aluminum blocks were mated and pressed to each other to make a dry contact interface (Fig. 2). The contacting surfaces were a square with the side-length 30 mm. The length along the loading direction was also 30 mm for each block. The contacting surfaces were polished with the #1000 paper and lapped with alumina powders. Their roughness values (arithmetical mean deviation) were found to be typically less than 0.1 lm. A 5 MHz longitudinal toneburst wave was sent to the interface from the end of one block, using a signal generator (Agilent Technologies 33220A) and a gated amplifier (RITEC RAM-5000) to excite a piezoelectric transducer (Panametrics) with 5 MHz nominal frequency. The transmitted wave was detected at the end of the other block by a wideband transducer with 10 MHz nominal frequency. The transducers were pushed against the aluminum blocks, with glycerin as a couplant medium, by rubber pads inside metal spacers in order to maintain a constant coupling force irrespective of the compression loading level applied to the contacting blocks. The time interval of the incident toneburst was fixed as 6 ls (30 periods). The excitation voltage (peak-to-peak value) applied to the input transducer was varied at three levels of (a) 374 V, (b) 486 V and (c) 600 V. The transmitted waveforms were stored via a digital oscilloscope (Tektronix TDS340AP), in a loading/unloading cycle of the contacting interface up to about 5 MPa nominal contact pressure. Prior to the measurements, the mated interface was subjected to several loading/unloading cycles in order to avoid the significant hysteresis due to the inelastic flattening during the first contact loading [5].
where A is the amplitude of the incident wave [11]. In the present study, the nonlinearity parameter b is defined by the second harmonic amplitude divided by the square of the fundamental amplitude, which is given by b¼
A2 qcxK 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ 2 A1 4K 2 1 þ K ~ 21 1
Signal generator
ð5Þ
Transmitter (5 MHz)
Rubber pad
Gated amplifier
Spacer p0
p0
Aluminum blocks Incident wave
Sinusoidal waveform
Contacting interface
Transmitted wave Digital oscilloscope
x Spacer
Reflected wave
Receiver (10 MHz)
Rubber pad
Waveform stored
Load cell Gap distance h X+ + u (X+,t) X– + u (X–,t) Fig. 1. Schematic model of a contacting interface.
PC
Fig. 2. Experimental setup.
S. Biwa et al. / Ultrasonics 44 (2006) e1319–e1322
4. Results and discussion The measured variation of the transmission coefficient of the fundamental (5 MHz) component with the nominal contact pressure is shown in Fig. 3 for three levels of excitation voltage, (a) 374 V, (b) 486 V and (c) 600 V. Different excitation voltages imply different incident wave amplitudes, although the latter quantities could not be quantified in the present experiment. Fig. 3 shows that the transmission coefficient is independent of the excitation voltage, indicating that this is essentially a linear phenomenon. Slight hysteresis can be observed for loading and unloading. The measured transmission coefficient can be converted to the linear stiffness K1 from Eq. (3) using the acoustic properties of aluminum, q = 2700 [kg/m3] and c = 6420 [m/s]. The pressure dependence of the linear stiffness is also shown in Fig. 3.
Exp.(c)
0 0 1 2 3 4 50 1 2 3 4 50 1 2 3 4 5 Contact pressure[MPa]
Fig. 4. Variation of the normalized second harmonic amplitude with the applied contact pressure, for the transducer excitation (peak-to-peak) voltages (a) 374 V, (b) 486 V and (c) 600 V.
0.6
0.4
0.4
0.2
K1, exp.(a) K1, exp.(b) K1, exp.(c)
1
2 3 Contact pressure[MPa]
4
0.2 0 5
Fig. 3. Variation of the transmission coefficient at 5 MHz and the linear interfacial stiffness with the applied contact pressure, for the transducer excitation (peak-to-peak) voltages (a) 374 V, (b) 486 V and (c) 600 V.
0.03
parameter K2 [MPa/nm2]
0.8
0.6
0 0
Exp. (b)
0.05
Second-order stiffness
0.8
Exp. (a) 0.1
1 T, exp.(a) T, exp.(b) T, exp.(c)
Contact stiffness K1 [MPa/nm]
Transmission coefficient T
1
Fig. 4 shows the variations of A2V/A1V with the applied contact pressure for the three excitation voltage levels. It is seen that this quantity exhibits more profound hysteresis at loading/unloading, although its reason is unclear yet. The ratio A2V/A1V appears to attain a maximum at a very low contact pressure, and decrease monotonically as the contact pressure increases. This ratio increases more or less proportionally with the excitation voltage, in qualitative correspondence with the theoretical result that A2/A1 is proportional to the incident wave amplitude. In order to evaluate the magnitude of the second harmonic component due to the factors other than the contact nonlinearity, a single aluminum block with the same crosssection and the doubled propagation length (60 mm) was used in the same setup instead of the paired aluminum blocks. This measurement revealed that the harmonic amplitude corresponding to about A2V/A1V = 0.03 was present in the absence of the contact interface. This value appears to be close to the values of A2V/A1V in Fig. 5 when the contact pressure becomes sufficiently high. Each measured relation between the contact pressure (p0) and the linear stiffness (K1) in Fig. 3 was P fitted by a 5th-order polynomial expression K 1 ðp0 Þ ¼ 5k¼0 ak pk0 , with ak being constants. Then, from the definitions in Eq. (2),
Normalized second harmonic amplitude A2V/A1V
From the recorded waveforms, the magnitude of the fundamental (5 MHz) and the second harmonic (10 MHz) components, denoted as A1V and A2V, respectively, were extracted (the subscript V indicates that these quantities correspond to the signal voltages and not to the absolute displacements). The normalized second harmonic amplitude A2V/A1V and the nonlinearity parameter bV ¼ A2V =A21V were calculated for the three excitation voltages and for different contact pressures. A caution should be paid, however, to the fact that these values were obtained by a transducer with specific spectral characteristics and are not to be directly compared to A2/A1 and b in the previous section: it is the contact pressure dependence of these values that is of present interest. When converting the amplitude of the transmitted fundamental component to the transmission coefficient T, the same transducer (5 MHz) was used in a pitch-catch mode to measure reflection coefficients R at the contacting interface for different pressures by aid of the spectrum analysis for the broadband echo [6]. Using the formula T2 + R2 = 1, the transmitted fundamental amplitude was calibrated to obtain the transmission coefficient at 5 MHz.
e1321
0.02
0.01 Experiment(a) 374 V (p-p) Loading Unloading
0 0
1 2 Contact pressure [MPa]
3
Fig. 5. Variation of the second-order contact stiffness parameter with the applied contact pressure for the excitation (peak-to-peak) voltage (a) 374 V identified from the measured pressure-linear stiffness relation.
S. Biwa et al. / Ultrasonics 44 (2006) e1319–e1322
0.8 0.6 0.4 0.2
0
1 2 Contact pressure [MPa]
0 3
Fig. 6. Variation of the nonlinearity parameter with the applied contact pressure (experimental and theoretical results), for the transducer excitation (peak-to-peak) voltages (a) 374 V, (b) 486 V and (c) 600 V.
the second-order contact stiffness parameter K2 can be calculated as a function of the applied pressure, 1 dK 1 K 2 ðp0 Þ ¼ K 1 ðp0 Þ 2 dp0 ! ! 5 5 X X 1 k m1 ¼ ak p 0 mam p0 : 2 k¼0 m¼1
5. Conclusion
-1
Nonlinearity parameter β V [a.u.] (experimental)
β V, exp. (a) β V, exp. (b) β V, exp. (c) β , calculated from (a) β , calculated from (b) β , calculated from (c)
Nonlinearity parameter β [nm ] (theoretical)
e1322
ð6Þ
As an example, Fig. 5 illustrates the variation of K2 with the contact pressure for the loading and unloading processes in the case of excitation voltage 374 V. Since the pressure-linear stiffness relation was basically insensitive to the excitation level (Fig. 3), so was the pressure dependence of K2. Using the obtained values of K1 and K2, Eq. (5) can be used to compute the nonlinearity parameter b at different contact pressures. The experimental results for the nonlinearity parameter bV are plotted in Fig. 6 against the contact pressure. It is remarkable that bV appears to be independent of the excitation voltage for the entire range of the applied pressure. This is in good agreement with the theoretical result given in Eq. (5). The parameter bV is found to decrease monotonically as the applied pressure increases, and the associated hysteresis is almost negligible. This feature is in agreement with the classical experiment [12]. The theoretical curves for b are also included in Fig. 6 where different curves corresponding to different excitation voltages appear to overlap onto a single curve. It is noted, however, that the direct quantitative comparison between the theoretically calculated b and the corresponding experimental values is not possible due to the arbitrariness of the scale of the measured wave amplitude. In spite of this, as seen in Fig. 6 the qualitative agreement between the theoretical calculations and the experimental data is quite satisfactory in terms of their dependence on the applied contact pressure.
The second harmonic generation at a contacting aluminum–aluminum interface has been studied from both experimental and theoretical points of view. From the measured contact pressure dependence of the linear interfacial stiffness, the second-order contact stiffness parameter was identified. The nonlinearity parameter was calculated using the so-obtained stiffness parameters according to the theoretical results based on a nonlinear interface model. It has been shown that the computed contact pressure dependence of the nonlinearity parameter is in qualitative agreement with the experimental results. Acknowledgement This study has been supported in part by the Grant-inAid for Young Scientists (B) (No.16760066) from MEXT, as well as the IVNET Development Project ‘‘Advancement of Maintenance Quality for Nuclear Power Plant’’ from METI, which are acknowledged gratefully. References [1] I.Y. Solodov, Ultrasonics of non-linear contacts: propagation reflection and NDE-applications, Ultrasonics 36 (1998) 383–390. [2] D. Donskoy, A. Sutin, A. Ekimov, Nonlinear acoustic interaction on contact interfaces and its use for nondestructive testing, NDT & E Int. 34 (2001) 231–238. [3] S. Hirsekorn, Nonlinear transfer of ultrasound by adhesive joints a theoretical description, Ultrasonics 39 (2001) 57–68. [4] P.B. Nagy, Ultrasonic classification of imperfect interfaces, J. Nondestruct. Eval. 11 (1992) 127–139. [5] B.W. Drinkwater, R.S. Dwyer-Joyce, P. Cawley, A study of the interaction between ultrasound and a partially contacting solid–solid interface, Proc. R. Soc. London Ser. A 452 (1996) 2613–2628. [6] S. Biwa, A. Suzuki, N. Ohno, Evaluation of interface velocity reflection coefficients and interfacial stiffnesses of contacting surfaces, Ultrasonics 43 (2005) 495–502. [7] A. Suzuki, S. Biwa, N. Ohno, Numerical and experimental evaluation of ultrasonic wave propagation characteristics at contact interface, JSME Int. J. Ser. A 48 (2005) 20–26. [8] J.M. Richardson, Harmonic generation at an unbonded interface: I. planar interface between semi-infinite elastic media, Int. J. Eng. Sci. 17 (1979) 73–85. [9] O.V. Rudenko, Chin An Vu, Nonlinear acoustic properties of a rough surface contact and acoustodiagnostics of a roughness height distribution, Acoust. Phys. 40 (1994) 593–596. [10] C. Pecorari, Nonlinear interaction of plane ultrasonic waves with an interface between rough surfaces in contact, J. Acoust. Soc. Am. 113 (2003) 3065–3072. [11] S. Biwa, S. Nakajima, N. Ohno, On the acoustic nonlinearity of solidsolid contact with pressure-dependent interface stiffness, Trans. ASME J. Appl. Mech. 71 (2004) 508–515. [12] O. Buck, W.L. Morris, J.M. Richardson, Acoustic harmonic generation at unbonded interfaces and fatigue cracks, Appl. Phys. Lett. 33 (1978) 371–373.