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Harmonic generation of an obliquely incident ultrasonic wave in solid–solid contact interfaces
Ultrasonics 52 (2012) 778–783
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Harmonic generation of an obliquely incident ultrasonic wave in solid–solid contact interfaces Taehyung Nam a, Taehun Lee b, Chungseok Kim a, Kyung-Young Jhang c,⇑, Nohyu Kim d a
Department of Automotive Engineering, Hanyang Univ., Seoul 133-791, Republic of Korea Plant Support Engineering Center, KHNP-CRI, Daejeon 305-343, Republic of Korea c School of Mechanical Engineering, Hanyang Univ., Seoul 133-791, Republic of Korea d Department of Mechanical Engineering, Korea Univ. of Technology and Education, Chunan 330-860, Republic of Korea b
a r t i c l e
i n f o
Article history: Received 27 December 2011 Accepted 13 February 2012 Available online 23 February 2012 Keywords: Contact acoustic nonlinearity Harmonic generation Contact interface Contact stiffness Nonlinear contact stiffness
a b s t r a c t The conventional acoustic nonlinear technique to evaluate the contact acoustic nonlinearity (CAN) at solid–solid contact interfaces (e.g., closed cracks), which uses the through-transmission of normally incident bulk waves, is limited in that access to both the inner and outer surfaces of structures for attaching pulsing and receiving transducers is difficult. The angle beam incidence and reflection technique, where both the pulsing and receiving transducers are located on the same side of the target, may allow the above problem to be overcome. However, in the angle incidence technique, mode-conversion at the contact interfaces as well as the normal and tangential interface stiffness should be taken into account. Based on the linear and nonlinear contact stiffness, we propose a theoretical model for the reflection of an ultrasonic wave angularly incident on contact interfaces. In addition, the magnitude of the CAN-induced second harmonic wave in the reflected ultrasonic wave is predicted. Experimental results obtained for the contact interfaces of A16061-T6 alloy specimens at various loading pressures showed good agreement with theoretical predictions. Such agreement proves the validity of the suggested oblique incidence model. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction An understanding of the contact condition of solids is important for the tribology and design of mechanical components with contacting interfaces. Typical examples of contacting solids found in solid mechanics are fracture surfaces, which are often closed and contacting under loading. Many researchers have studied the micro-mechanical behavior of contacting surfaces [1–7]. However, the micro-mechanical behavior of a contacting surface is quite complicated due to nonlinearity, which is hardly understood by examining the micro-scale properties of solids. The micro-mechanical behavior of contacting surfaces is effectively explained on the macro-scale by the normal and tangential interfacial stiffnesses, which are employed to model the stress–displacement relation of the contact surfaces as a simple spring-force–displacement system. The interfacial stiffnesses are known to offer useful information on the nature of a contact interface [8–12]. Due to its penetration power and sensitivity to discontinuity, ultrasound is an attractive tool for the non-invasive evaluation of the contact condition between solid components. However, it is nearly impossible to detect the location and size of closed interfaces with conventional nondestructive testing methods that use ⇑ Corresponding author. Tel.: +82 2 2220 0434; fax: +82 2 2299 7207. E-mail address: [email protected] (K.-Y. Jhang). 0041-624X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2012.02.008
linear ultrasound because the reflection coefficient of such techniques is little [13]. In order to overcome this obstacle, pioneering researchers have suggested the application of the contact acoustic nonlinearity (CAN) at contact interfaces. A well-known acoustical manifestation of nonlinear behavior is the generation of harmonics [14–16]. Harmonic generation due to the non-linear stress–displacement relation of a contact interface can be explained by the fact that at an interface between two solids in contact, the load is supported by surface asperities. With an increase in the load, more asperities come into contact and each asperity undergoes flattening deformation. Due to the change in the contact asperity configuration with a variation in the contact pressure, the mechanical response of a contact interface includes a certain degree of nonlinear behavior. Therefore, second-order or higher-order harmonics are generated when a wave interacts with a contact interface. The spring model of a contact surface has been used to theoretically analyze the CAN mechanism. In this model, the nonlinear stress–displacement relation is modeled by linear stiffness and nonlinear stiffness terms. Several investigators have shown that the nonlinear interfacial stiffness can quantitatively represent the contact acoustic nonlinearity at a contacting surface [17,18]. Based on such research, the practical implementation of the second harmonic method has often been attempted to assess the contact state or to detect cracks in nondestructive
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evaluation (NDE) applications [19–31]. In particular, the measurement of ultrasonic transmission/reflection has been considered as one of the most attractive approaches, and the transmission and reflection characteristics at contacting surfaces have been the subject of extensive research related to the evaluation of contact interfaces and integrity monitoring in nondestructive testing (NDT) [32–34]. In terms of experimental methodologies, most studies have been limited to the use of the normal through-transmission test. This is because the through-transmission test is a convenient way to detect the second harmonic wave. In this technique, a narrowband detector with high sensitivity at the frequency of the second harmonic is used to detect the second harmonic wave; a separate transmitter of the fundamental incidence frequency is also employed. Unfortunately, this method is difficult to apply in the field since two transducers should be attached on both sides of the target material. On the other hand, the normal pulse-echo method using a single transducer is convenient for field applications, but detection of the second harmonic wave with a high level of sensitivity is difficult. Given the limitations of the aforementioned methods, we are interested in the implementation of an oblique incidence and reflection method that uses both pulsing and receiving transducers on the same side of target. Such a method will be very convenient for field applications. The technique allows for the sensitive detection of the second harmonic and involves the use of a detector with high sensitivity at the frequency of the second harmonic. However, we must take into account the mode conversion of the incident ultrasonic wave at the contacting surfaces and four interfacial stiffnesses: linear normal, nonlinear normal, linear tangential and nonlinear tangential. There have been previous works on the interaction of obliquely incident ultrasound waves with contacting interfaces. Baltazar et al. extended the spring model by assuming that the contacting interface consisted of normal and tangential springs [9]. The model was used to predict both the normal incidence and oblique incidence reflection coefficients from a rough contacting interface. Liaptsis et al. studied the interaction of an obliquely incident wave with contacting interfaces. However, the researchers reported that only the linear normal and tangential stiffnesses could be evaluated from the reflectivity and transmittance of an ultrasonic wave at the contact interface [35]. Pecorari analyzed the normal and tangential interfacial stiffnesses based on micro-mechanical theory and obtained the longitudinal and shear second harmonic amplitudes generated due to the scattering of longitudinal and shear incidence waves as a function of the incidence angle. However, the work provides only theoretical suggestions without experimental verification, although it may useful in identifying the mechanisms behind the experimental observations of nonlinear material responses to an incident ultrasonic wave [28]. In this paper, a theoretical model for an ultrasonic wave obliquely incident on contact interfaces is proposed and the reflection characteristics with four interfacial stiffnesses are analyzed. The developed model was used to predict the magnitude of the second harmonic wave in the received wave reflected at the interface. To verify the theoretical analysis, a solid–solid interface was constructed using aluminum blocks and inspected by a specially constructed experimental system. Oblique incidence tests and normal incidence longitudinal and shear echo tests were performed to determine the linear normal and tangential interfacial stiffnesses; a hydraulic press was employed to control the contact pressure at the interface. Experiments were carried out at two different incidence angles, 22.5° and 45°, as well as at normal incidence. The experimental results are discussed so as to verify the proposed method.
2. Theory 2.1. Characteristics of reflection and transmission A two-dimensional model for analyzing the characteristics of reflection and transmission for an obliquely incident longitudinal ultrasonic wave on solid–solid contact interfaces is shown in Fig. 1. In the figure, P(0) is the incident longitudinal wave with a single frequency, h0 is the angle of incidence, P(1), P(2), P(3), and P(4) are the reflected longitudinal, reflected shear, transmitted longitudinal, and transmitted shear waves (the shear waves are derived from the mode conversion), respectively, and h1, h2, h3, and h4 are the angles of reflection and transmission for each wave. The angles of reflection and transmission all follow Snell’s law. According to the classical linear and nonlinear spring model of a contact interface, the pressure–displacement behavior at the contacting interfaces can be approximated as follows;
@pðuÞ 1 þ @u u¼u0 2 ! @ 2 pðuÞ @u2
rðu0 þ DuÞ rðu0 Þ þ Du ðDuÞ2
ð1Þ
u¼u0
where u = uxi + uyj, ux and uy are the displacements in the x and y direction, respectively, and u0 denotes the static displacement. From Eq. (1), the linear interfacial stiffness K and the nonlinear interfacial stiffness Kn can be defined as follows;
Kx ¼
@pðux Þ ; @ux
K xn ¼
@ 2 pðux Þ @u2x
Ky ¼
@pðuy Þ ; @uy
K yn ¼
@ 2 pðuy Þ @u2y
ð2Þ
where Kx and Kxn are the linear and nonlinear tangential interfacial stiffnesses in the x direction, and Ky and Kyn are the linear and nonlinear normal interfacial stiffnesses in the y direction, respectively. The reflection and transmission of a normally incident ultrasonic wave across a contact interface have previously been investigated and are well explained using the linear and nonlinear spring model of a contact interface [10]. The reflection and transmission of an obliquely incident ultrasonic wave across a contact
y p (4)
θ4 θ3
p (3)
K x ,K y
upper
x
K nx ,K ny
lower
p (0)
θ1
θ0 θ2
p (1) p (2)
Fig. 1. Two-dimensional model for analyzing the characteristics of reflection and transmission for an obliquely incident longitudinal ultrasonic wave.
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interface can be obtained in a similar manner. The resulting amplitudes of the reflected longitudinal, reflected shear, transmitted longitudinal, and transmitted shear waves can be obtained as follows;
2 6 6 6 4
k1 ð2l cos2 h1 þ kÞ k2 l sin 2h2
k3 ð2l cos2 h3 þ kÞ
k1 sin 2h1
k2 cos 2h2
k3 sin 2h3
K y cos h1
K y sin h2
ik3 ð2l cos2 h3 þ kÞ þ K y cos h3
K x sin h1
K x cos h2
ik3 l sin 2h3 þ K x sin h3
6 6 6 4
k1 ð2l cos2 h1 þ kÞ k2 l sin2h2 k1 sin2h1 K y cosh1
k2 cos2h2 K y sinh2
K x sinh1
K x cosh2
8 9 38 9 A1 > k0 ð2l cos2 h0 þ kÞ > > > > > > > > > > > < k sin 2h = 7< A = k4 cos 2h4 2 0 0 7 ¼ A0 7 > A3 > > > > ik4 l sin 2h4 þ K y sin h4 5> K y cos h0 > > > > > : > : ; ; A4 ik4 l cos 2h4 K x cos h4 K x sin h0 k4 l sin 2h4
where A0 is the amplitude of the incident longitudinal wave, A1, A2, A3, and A4 are the amplitudes of the reflected longitudinal, reflected shear, transmitted longitudinal, and transmitted shear waves, respectively, at the fundamental frequency (which is identical to the incidence frequency), k0, k1, k2, k3, and k4 are the wave numbers of each wave, and k and l are Lame constants. The amplitudes of the second harmonic waves generated by contact acoustic nonlinearity can be obtained as follows,
2
transmitted longitudinal wave (denoted as the transmittance coefficient Tl).
( A1 ) A0 A3 A0
8 9 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 8 1 9 > > > 4K 2 > 2K y > > > > y > > > > > 1þq2 c2 x2 > = R < 1ixqcl = < l l 2K y ¼ ¼ ¼ 2K y xqcl > > > T r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi l > > > > > > > 2K y ; : 1ixq > 4K 2 > > cl : qcl x 1þq2 c2yx2 > ;
ð3Þ
ð6Þ
l
Rl and Tl are a function of the linear normal interfacial stiffness Ky. Therefore, Ky can be expressed as Eq. (7), which includes only the reflection coefficient of the longitudinal wave, Rl.
9 8 9 >0 > 38 > > > A12 > > > > > > > > > 0 > > > > = 7< A = < k3 sin2h3 k4 cos2h4 22 7 K ½A cosh þ A sinh A cosh ¼ yn 3 3 4 4 0 0 7 > > > A ik3 ð2l cos2 h3 þ kÞ þ K y cosh3 ik4 l sin2h4 þ K y sinh4 5> 32 > > > > 2 > > > ; > > : þA1 cosh1 A2 sinh2 K xn ½A3 sinh3 A4 cosh4 A0 sinh0 > > > > > A42 ik3 l sin2h3 þ K x sinh3 ik4 l cos2h4 K x cosh4 ; : 2 A1 sinh1 A2 cosh2 k3 ð2l cos2 h3 þ kÞ
k4 l sin2h4
ð4Þ
where A12, A22, A32, and A42 are the amplitudes of the second harmonic wave in the reflected longitudinal, reflected shear, transmitted longitudinal, and transmitted shear waves, respectively. We can see that the fundamental frequency amplitudes are dependent on the linear interfacial stiffness only, while the harmonic amplitudes are dependent on both the linear and nonlinear interfacial stiffnesses.
Ky ¼
The interfacial stiffness can be experimentally measured because it depends on surface conditions such as the roughness of the contact interfaces and the contact pressure. To obtain the linear normal interfacial stiffness of the contact interfaces, the reflection coefficient of the normally incident waves is used since Eq. (3) can be simplified by applying only the incidence angle h0 = 0 (as in Eq. (5)) with the interfacial stiffness, the amplitude of the transmitted and reflected wave, and the amplitude of the incident wave,
kð2l þ kÞ
kð2l þ kÞ
Ky
ikð2l þ kÞ þ K y
A1 A3
¼ A0
kð2l þ kÞ Ky
ð5Þ
where kðk þ 2lÞ ¼ qcl x ðk ¼ wave number; q ¼ density, cl ¼ wave speed; x ¼ angular frequencyÞ. It should be noted that the wave numbers of the incident wave, reflection wave, and transmitted wave are equal and thus, they were replaced with k. The solution of Eq. (5) is obtained as Eq. (6), where A1/A0 is the amplitude ratio of the normally incident longitudinal wave to the normally reflected longitudinal wave (denoted as the reflection coefficient Rl), and A3/A0 is the amplitude ratio of the normally incident longitudinal wave to the normally
ð7Þ
Likewise, the linear tangential interfacial stiffness Kx can be obtained with the normal incidence transverse wave model as follows
Kx ¼ 2.2. Measurement of interfacial stiffness
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 R2l
qc l x
qc t x 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 R2t
ð8Þ
These results are in good agreement with the findings of other studies where a one-dimensional normal incidence model was employed [17]. From Eqs. (7) and (8), the linear interfacial stiffnesses Ky and Kx can be experimentally obtained by measuring the reflection coefficients Rl and Rt. The amplitudes of the reflected and transmitted longitudinal waves are theoretically obtained by applying these parameters to Eq. (3) at any oblique incidence angle. Meanwhile, to obtain the generated second harmonic using Eq. (4), the nonlinear interfacial stiffnesses Kyn and Kxn are required. These parameters can be obtained through the use of the powerlaw model [17], which may be expressed as a function of the interfacial contact pressure p as follows
K ¼ Cpm ;
Kn ¼
1 mC 2 p2m1 2
ð9Þ
where K is the linear interfacial stiffness, Kn is the nonlinear interfacial stiffness, and C and m are experimental constants. According to Eq. (9), the contact interfacial stiffness K increases as the contact pressure p increases, which mean that the contact interfaces are being closed. The acoustic reflection coefficient also decreases, as predicted from Eqs. (7) and (8). Thus, if we measure the reflection coefficients at different contact pressures, we can
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3. Experimental procedure 3.1. Experimental system To measure the reflection coefficient of longitudinal and transverse waves at normal incidence as well as those of obliquely incident waves, the experimental system shown in Fig. 2 was employed. In this system, a hydraulic press was used to apply loads to the contact interfaces. The loads were measured with a load cell attached at the bottom of the specimen. The contact pressure applied at the contact interfaces was calculated as the mean value divided by the area of the contact interfaces. A tone-burst signal of 2.25 MHz was generated with a high-power gated amplifier (Ritec, RAM-5000 SNAP). Longitudinal and transverse wave transducers with a center frequency of 2.25 MHz were used for measurement of the reflection coefficient of the normally incident waves. In the case of oblique incidence, a 2.25 MHz longitudinal wave transducer was used as a transmitter and a 5 MHz longitudinal wave transducer was employed as a receiver. The received signal was monitored with a digital oscilloscope (LECROY WS452) and a fast Fourier transform (FFT) was performed to obtain the amplitude of the fundamental and second harmonic waves. By measuring the reflection coefficient of the normally incident waves, the linear interface stiffnesses Ky and Kx were calculated using Eqs. (7) and (8). Experiments were performed after a loading/unloading process was repeated 10 times so as to eliminate the hysteresis effect in the contact interfaces [34]. The reflection coefficient and the amplitude of the second harmonic wave of the reflected obliquely incident longitudinal waves were measured experimentally and compared with the theoretical analysis.
Two incident angles, 22.5° and 45°, were employed in the tests. The contact interface was polished using #600 sand paper.
4. Experimental results The experimentally obtained reflection coefficients for the normally incident longitudinal and transverse waves as a function of the applied nominal contact pressure are shown in Fig. 3. The reflection coefficients were normalized using the initial coefficient value obtained at a no-load condition. With the obtained results, the linear interfacial stiffnesses Ky and Kx were calculated from Eqs. (7) and (8); the results are shown in Fig. 4. As the applied contact pressure increased, the reflection coefficient decreased and the linear interfacial stiffness increased. The interfacial stiffness has a nonlinear relationship with the nominal contact pressure. This relationship was fitted to the power-law model in Eq. (9) so that the constants C and m could be obtained. Finally, the representation of the linear normal and tangential interfacial stiffnesses, as well as the nonlinear normal and tangential interfacial stiffnesses as a function of the nominal contact pressure was complete.
Measured Reflection Coefficient R 1 Normal Incidence of Longitudinal Wave Normal Incidence of Shere Wave
0.9
Reflection Coefficient R
obtain the linear interfacial stiffness as a function of the contact pressure; the constants C and m in Eq. (9) can be determined by curve fitting. The nonlinear interfacial stiffness Kn can also be calculated from Eq. (9). Finally, the amplitude of the reflected wave and the second harmonic generation for oblique incidence can be determined from Eqs. (3) and (4).
0.8 0.7 0.6 0.5 0.4
3.2. Specimens
0
5
10
15
20
25
30
Nominal Contact Pressure [MPa] A pair of aluminum 6061-T6 blocks put into contact with an external normal load was used as the test specimen. The contacting area has a length of 100 mm and a width of 40 mm. The test specimen consisted of upper and lower blocks; the upper block was machined as shown in Fig. 2 so as establish oblique incidence.
Fig. 3. Reflection coefficient of a normally incident longitudinal wave and a shear wave.
14
2.5
x 10
Measured Linear Interfacial Stiffness Kx,Ky
Linear Interfacial Stiffness Kx,Ky [N/m]
Normal Interfacial Stiffness Ky Tangential Interfacial Stiffness Kx
2
1.5
1
0.5
0 0
5
10
15
20
25
30
Nominal Contact Pressure [MPa]
Fig. 2. Schematic diagram of the experimental setup.
Fig. 4. The measured linear interfacial stiffnesses Kx and Ky as a function of the nominal contact pressure.
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waves are shown in Fig. 6. The theoretical results were calculated from Eq. (4). The amplitudes of the second harmonic waves are normalized by the peak amplitude from the 22.5° result. The second harmonic component was 20% lower at a 22.5° oblique incidence and 55% lower at a 45° oblique incidence when compared to the normal incidence case. However, the trends in second harmonic generation with respect to an increase in the contact pressure were the same for each case. In addition, the experimental results are in good agreement with the theoretical predictions. The normalized amplitude of the second harmonic wave, A2, increases sharply for each incidence angle when the contact pressure is below 7 MPa. The amplitude reaches a maximum value around 7 MPa and then decreases monotonically since the interfaces of the aluminum blocks become fully closed. Such findings appropriately reflect the nonlinear characteristics of contact interfaces and thus, it is possible to use the oblique incidence technique suggested in this paper for an evaluation of contact interfaces.
Reflection Coefficient
Reflection Coefficient R
1
0.8
0.6
0.4 Simulation Data of Normal Incidnece (0 deg) Simulation Data of Oblique Incidence (22.5 deg) Simulation Data of Oblique Incidence (45 deg) Experimental Data of Oblique Incidence (22.5 deg) Experimental Data of Oblique Incidence (45 deg)
0.2
0 0
0.5
1
1.5
2
2.5 14
Normal Linear Interfacial Stiffness Ky [N/m] x 10
Fig. 5. Comparison of the theoretical and experimental ultrasonic reflection coefficients as a function of the nominal linear interfacial stiffness.
Normalized Amplitude of 2nd Harmonic wave A
2
Shown in Fig. 5 are the longitudinal reflection coefficients theoretically predicted by Eq. (3) for oblique incidence angles of 22.5° and 45° using the linear normal and tangential interfacial stiffnesses obtained as described above. The normal incidence data were used as a reference and the experimental results are included in Fig. 5. The experimental findings exhibit good agreement with theoretical expectations. Thus, we can confirm the validity of our model for the amplitude of the reflected longitudinal wave based on the interfacial stiffnesses. The reflection coefficients of the 22.5° obliquely incident longitudinal waves are almost identical to those obtained at normal incidence because the reflection characteristics of oblique incidence at small angles are similar to those of normal incidence cases. On the other hand, the reflection coefficients of the 45° obliquely incident longitudinal waves are about 20% higher than those of the normally incident waves because the characteristics of waves obliquely incident at large angles are affected by both the normal interfacial stiffness and the tangential interfacial stiffness. The theoretical and experimental results for the second harmonic generation of normally and obliquely incident longitudinal
Normalized Amplitude of 2nd Harmonic Wave 1.4
1.2
5. Conclusions In this work, a theoretical model for the reflection of an ultrasonic wave obliquely incident on contact interfaces was proposed by introducing the parameters of linear and nonlinear contact stiffness. Using this model, we can predict the magnitude of a CAN-induced second harmonic wave in a reflected ultrasonic wave. The experimental results obtained from the contact interfaces of A16061-T6 alloy specimens at various loading pressures showed good agreement with the theoretical predictions. Such agreement proves the validity of the suggested oblique incidence model. The findings of this study may be summarized as follows: (1) The experimentally obtained reflection coefficients for the 22.5° and 45° obliquely incidence longitudinal waves are in good agreement with theoretical predictions. (2) The reflection coefficients of the 22.5° obliquely incident longitudinal waves are similar to those of the normally incident longitudinal waves because the characteristics of reflection are mainly influenced by the normal interfacial stiffness. On the other hand, the reflection coefficients of the 45° obliquely incident longitudinal waves are higher than those of the normally incident longitudinal waves because the coefficients are influenced by both the normal and tangential interfacial stiffnesses. (3) The amplitude of the second harmonic wave of the obliquely incident longitudinal wave is lower than that of the normally incident longitudinal wave. Nevertheless, the variation in the amplitude of the second harmonic waves with respect to the contact pressure exhibited similar trends.
1
Acknowledgment 0.8
This work was financially supported by the National Research Foundation of Korea (NRF) (2008-2003505).
0.6
References
0.4 Simulation Data of Normal Incidence (0 deg) Simulation Data of Oblique Incidence (22.5 deg) Simulation Data of Oblique Incidence (45 deg) Experimental Data of Oblique Incidence (22.5 deg) Experimental Data of Oblique Incidence (45 deg)
0.2
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Nominal Contact Pressure [MPa] Fig. 6. Normalized amplitude of the second harmonic wave as a function of the nominal contact pressure at each incidence angle.
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Harmonic generation of an obliquely incident ultrasonic wave in solid–solid contact interfaces Taehyung Nam a, Taehun Lee b, Chungseok Kim a, Kyung-Young Jhang c,⇑, Nohyu Kim d a
Department of Automotive Engineering, Hanyang Univ., Seoul 133-791, Republic of Korea Plant Support Engineering Center, KHNP-CRI, Daejeon 305-343, Republic of Korea c School of Mechanical Engineering, Hanyang Univ., Seoul 133-791, Republic of Korea d Department of Mechanical Engineering, Korea Univ. of Technology and Education, Chunan 330-860, Republic of Korea b
a r t i c l e
i n f o
Article history: Received 27 December 2011 Accepted 13 February 2012 Available online 23 February 2012 Keywords: Contact acoustic nonlinearity Harmonic generation Contact interface Contact stiffness Nonlinear contact stiffness
a b s t r a c t The conventional acoustic nonlinear technique to evaluate the contact acoustic nonlinearity (CAN) at solid–solid contact interfaces (e.g., closed cracks), which uses the through-transmission of normally incident bulk waves, is limited in that access to both the inner and outer surfaces of structures for attaching pulsing and receiving transducers is difficult. The angle beam incidence and reflection technique, where both the pulsing and receiving transducers are located on the same side of the target, may allow the above problem to be overcome. However, in the angle incidence technique, mode-conversion at the contact interfaces as well as the normal and tangential interface stiffness should be taken into account. Based on the linear and nonlinear contact stiffness, we propose a theoretical model for the reflection of an ultrasonic wave angularly incident on contact interfaces. In addition, the magnitude of the CAN-induced second harmonic wave in the reflected ultrasonic wave is predicted. Experimental results obtained for the contact interfaces of A16061-T6 alloy specimens at various loading pressures showed good agreement with theoretical predictions. Such agreement proves the validity of the suggested oblique incidence model. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction An understanding of the contact condition of solids is important for the tribology and design of mechanical components with contacting interfaces. Typical examples of contacting solids found in solid mechanics are fracture surfaces, which are often closed and contacting under loading. Many researchers have studied the micro-mechanical behavior of contacting surfaces [1–7]. However, the micro-mechanical behavior of a contacting surface is quite complicated due to nonlinearity, which is hardly understood by examining the micro-scale properties of solids. The micro-mechanical behavior of contacting surfaces is effectively explained on the macro-scale by the normal and tangential interfacial stiffnesses, which are employed to model the stress–displacement relation of the contact surfaces as a simple spring-force–displacement system. The interfacial stiffnesses are known to offer useful information on the nature of a contact interface [8–12]. Due to its penetration power and sensitivity to discontinuity, ultrasound is an attractive tool for the non-invasive evaluation of the contact condition between solid components. However, it is nearly impossible to detect the location and size of closed interfaces with conventional nondestructive testing methods that use ⇑ Corresponding author. Tel.: +82 2 2220 0434; fax: +82 2 2299 7207. E-mail address: [email protected] (K.-Y. Jhang). 0041-624X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2012.02.008
linear ultrasound because the reflection coefficient of such techniques is little [13]. In order to overcome this obstacle, pioneering researchers have suggested the application of the contact acoustic nonlinearity (CAN) at contact interfaces. A well-known acoustical manifestation of nonlinear behavior is the generation of harmonics [14–16]. Harmonic generation due to the non-linear stress–displacement relation of a contact interface can be explained by the fact that at an interface between two solids in contact, the load is supported by surface asperities. With an increase in the load, more asperities come into contact and each asperity undergoes flattening deformation. Due to the change in the contact asperity configuration with a variation in the contact pressure, the mechanical response of a contact interface includes a certain degree of nonlinear behavior. Therefore, second-order or higher-order harmonics are generated when a wave interacts with a contact interface. The spring model of a contact surface has been used to theoretically analyze the CAN mechanism. In this model, the nonlinear stress–displacement relation is modeled by linear stiffness and nonlinear stiffness terms. Several investigators have shown that the nonlinear interfacial stiffness can quantitatively represent the contact acoustic nonlinearity at a contacting surface [17,18]. Based on such research, the practical implementation of the second harmonic method has often been attempted to assess the contact state or to detect cracks in nondestructive
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evaluation (NDE) applications [19–31]. In particular, the measurement of ultrasonic transmission/reflection has been considered as one of the most attractive approaches, and the transmission and reflection characteristics at contacting surfaces have been the subject of extensive research related to the evaluation of contact interfaces and integrity monitoring in nondestructive testing (NDT) [32–34]. In terms of experimental methodologies, most studies have been limited to the use of the normal through-transmission test. This is because the through-transmission test is a convenient way to detect the second harmonic wave. In this technique, a narrowband detector with high sensitivity at the frequency of the second harmonic is used to detect the second harmonic wave; a separate transmitter of the fundamental incidence frequency is also employed. Unfortunately, this method is difficult to apply in the field since two transducers should be attached on both sides of the target material. On the other hand, the normal pulse-echo method using a single transducer is convenient for field applications, but detection of the second harmonic wave with a high level of sensitivity is difficult. Given the limitations of the aforementioned methods, we are interested in the implementation of an oblique incidence and reflection method that uses both pulsing and receiving transducers on the same side of target. Such a method will be very convenient for field applications. The technique allows for the sensitive detection of the second harmonic and involves the use of a detector with high sensitivity at the frequency of the second harmonic. However, we must take into account the mode conversion of the incident ultrasonic wave at the contacting surfaces and four interfacial stiffnesses: linear normal, nonlinear normal, linear tangential and nonlinear tangential. There have been previous works on the interaction of obliquely incident ultrasound waves with contacting interfaces. Baltazar et al. extended the spring model by assuming that the contacting interface consisted of normal and tangential springs [9]. The model was used to predict both the normal incidence and oblique incidence reflection coefficients from a rough contacting interface. Liaptsis et al. studied the interaction of an obliquely incident wave with contacting interfaces. However, the researchers reported that only the linear normal and tangential stiffnesses could be evaluated from the reflectivity and transmittance of an ultrasonic wave at the contact interface [35]. Pecorari analyzed the normal and tangential interfacial stiffnesses based on micro-mechanical theory and obtained the longitudinal and shear second harmonic amplitudes generated due to the scattering of longitudinal and shear incidence waves as a function of the incidence angle. However, the work provides only theoretical suggestions without experimental verification, although it may useful in identifying the mechanisms behind the experimental observations of nonlinear material responses to an incident ultrasonic wave [28]. In this paper, a theoretical model for an ultrasonic wave obliquely incident on contact interfaces is proposed and the reflection characteristics with four interfacial stiffnesses are analyzed. The developed model was used to predict the magnitude of the second harmonic wave in the received wave reflected at the interface. To verify the theoretical analysis, a solid–solid interface was constructed using aluminum blocks and inspected by a specially constructed experimental system. Oblique incidence tests and normal incidence longitudinal and shear echo tests were performed to determine the linear normal and tangential interfacial stiffnesses; a hydraulic press was employed to control the contact pressure at the interface. Experiments were carried out at two different incidence angles, 22.5° and 45°, as well as at normal incidence. The experimental results are discussed so as to verify the proposed method.
2. Theory 2.1. Characteristics of reflection and transmission A two-dimensional model for analyzing the characteristics of reflection and transmission for an obliquely incident longitudinal ultrasonic wave on solid–solid contact interfaces is shown in Fig. 1. In the figure, P(0) is the incident longitudinal wave with a single frequency, h0 is the angle of incidence, P(1), P(2), P(3), and P(4) are the reflected longitudinal, reflected shear, transmitted longitudinal, and transmitted shear waves (the shear waves are derived from the mode conversion), respectively, and h1, h2, h3, and h4 are the angles of reflection and transmission for each wave. The angles of reflection and transmission all follow Snell’s law. According to the classical linear and nonlinear spring model of a contact interface, the pressure–displacement behavior at the contacting interfaces can be approximated as follows;
@pðuÞ 1 þ @u u¼u0 2 ! @ 2 pðuÞ @u2
rðu0 þ DuÞ rðu0 Þ þ Du ðDuÞ2
ð1Þ
u¼u0
where u = uxi + uyj, ux and uy are the displacements in the x and y direction, respectively, and u0 denotes the static displacement. From Eq. (1), the linear interfacial stiffness K and the nonlinear interfacial stiffness Kn can be defined as follows;
Kx ¼
@pðux Þ ; @ux
K xn ¼
@ 2 pðux Þ @u2x
Ky ¼
@pðuy Þ ; @uy
K yn ¼
@ 2 pðuy Þ @u2y
ð2Þ
where Kx and Kxn are the linear and nonlinear tangential interfacial stiffnesses in the x direction, and Ky and Kyn are the linear and nonlinear normal interfacial stiffnesses in the y direction, respectively. The reflection and transmission of a normally incident ultrasonic wave across a contact interface have previously been investigated and are well explained using the linear and nonlinear spring model of a contact interface [10]. The reflection and transmission of an obliquely incident ultrasonic wave across a contact
y p (4)
θ4 θ3
p (3)
K x ,K y
upper
x
K nx ,K ny
lower
p (0)
θ1
θ0 θ2
p (1) p (2)
Fig. 1. Two-dimensional model for analyzing the characteristics of reflection and transmission for an obliquely incident longitudinal ultrasonic wave.
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interface can be obtained in a similar manner. The resulting amplitudes of the reflected longitudinal, reflected shear, transmitted longitudinal, and transmitted shear waves can be obtained as follows;
2 6 6 6 4
k1 ð2l cos2 h1 þ kÞ k2 l sin 2h2
k3 ð2l cos2 h3 þ kÞ
k1 sin 2h1
k2 cos 2h2
k3 sin 2h3
K y cos h1
K y sin h2
ik3 ð2l cos2 h3 þ kÞ þ K y cos h3
K x sin h1
K x cos h2
ik3 l sin 2h3 þ K x sin h3
6 6 6 4
k1 ð2l cos2 h1 þ kÞ k2 l sin2h2 k1 sin2h1 K y cosh1
k2 cos2h2 K y sinh2
K x sinh1
K x cosh2
8 9 38 9 A1 > k0 ð2l cos2 h0 þ kÞ > > > > > > > > > > > < k sin 2h = 7< A = k4 cos 2h4 2 0 0 7 ¼ A0 7 > A3 > > > > ik4 l sin 2h4 þ K y sin h4 5> K y cos h0 > > > > > : > : ; ; A4 ik4 l cos 2h4 K x cos h4 K x sin h0 k4 l sin 2h4
where A0 is the amplitude of the incident longitudinal wave, A1, A2, A3, and A4 are the amplitudes of the reflected longitudinal, reflected shear, transmitted longitudinal, and transmitted shear waves, respectively, at the fundamental frequency (which is identical to the incidence frequency), k0, k1, k2, k3, and k4 are the wave numbers of each wave, and k and l are Lame constants. The amplitudes of the second harmonic waves generated by contact acoustic nonlinearity can be obtained as follows,
2
transmitted longitudinal wave (denoted as the transmittance coefficient Tl).
( A1 ) A0 A3 A0
8 9 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 8 1 9 > > > 4K 2 > 2K y > > > > y > > > > > 1þq2 c2 x2 > = R < 1ixqcl = < l l 2K y ¼ ¼ ¼ 2K y xqcl > > > T r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi l > > > > > > > 2K y ; : 1ixq > 4K 2 > > cl : qcl x 1þq2 c2yx2 > ;
ð3Þ
ð6Þ
l
Rl and Tl are a function of the linear normal interfacial stiffness Ky. Therefore, Ky can be expressed as Eq. (7), which includes only the reflection coefficient of the longitudinal wave, Rl.
9 8 9 >0 > 38 > > > A12 > > > > > > > > > 0 > > > > = 7< A = < k3 sin2h3 k4 cos2h4 22 7 K ½A cosh þ A sinh A cosh ¼ yn 3 3 4 4 0 0 7 > > > A ik3 ð2l cos2 h3 þ kÞ þ K y cosh3 ik4 l sin2h4 þ K y sinh4 5> 32 > > > > 2 > > > ; > > : þA1 cosh1 A2 sinh2 K xn ½A3 sinh3 A4 cosh4 A0 sinh0 > > > > > A42 ik3 l sin2h3 þ K x sinh3 ik4 l cos2h4 K x cosh4 ; : 2 A1 sinh1 A2 cosh2 k3 ð2l cos2 h3 þ kÞ
k4 l sin2h4
ð4Þ
where A12, A22, A32, and A42 are the amplitudes of the second harmonic wave in the reflected longitudinal, reflected shear, transmitted longitudinal, and transmitted shear waves, respectively. We can see that the fundamental frequency amplitudes are dependent on the linear interfacial stiffness only, while the harmonic amplitudes are dependent on both the linear and nonlinear interfacial stiffnesses.
Ky ¼
The interfacial stiffness can be experimentally measured because it depends on surface conditions such as the roughness of the contact interfaces and the contact pressure. To obtain the linear normal interfacial stiffness of the contact interfaces, the reflection coefficient of the normally incident waves is used since Eq. (3) can be simplified by applying only the incidence angle h0 = 0 (as in Eq. (5)) with the interfacial stiffness, the amplitude of the transmitted and reflected wave, and the amplitude of the incident wave,
kð2l þ kÞ
kð2l þ kÞ
Ky
ikð2l þ kÞ þ K y
A1 A3
¼ A0
kð2l þ kÞ Ky
ð5Þ
where kðk þ 2lÞ ¼ qcl x ðk ¼ wave number; q ¼ density, cl ¼ wave speed; x ¼ angular frequencyÞ. It should be noted that the wave numbers of the incident wave, reflection wave, and transmitted wave are equal and thus, they were replaced with k. The solution of Eq. (5) is obtained as Eq. (6), where A1/A0 is the amplitude ratio of the normally incident longitudinal wave to the normally reflected longitudinal wave (denoted as the reflection coefficient Rl), and A3/A0 is the amplitude ratio of the normally incident longitudinal wave to the normally
ð7Þ
Likewise, the linear tangential interfacial stiffness Kx can be obtained with the normal incidence transverse wave model as follows
Kx ¼ 2.2. Measurement of interfacial stiffness
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 R2l
qc l x
qc t x 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 R2t
ð8Þ
These results are in good agreement with the findings of other studies where a one-dimensional normal incidence model was employed [17]. From Eqs. (7) and (8), the linear interfacial stiffnesses Ky and Kx can be experimentally obtained by measuring the reflection coefficients Rl and Rt. The amplitudes of the reflected and transmitted longitudinal waves are theoretically obtained by applying these parameters to Eq. (3) at any oblique incidence angle. Meanwhile, to obtain the generated second harmonic using Eq. (4), the nonlinear interfacial stiffnesses Kyn and Kxn are required. These parameters can be obtained through the use of the powerlaw model [17], which may be expressed as a function of the interfacial contact pressure p as follows
K ¼ Cpm ;
Kn ¼
1 mC 2 p2m1 2
ð9Þ
where K is the linear interfacial stiffness, Kn is the nonlinear interfacial stiffness, and C and m are experimental constants. According to Eq. (9), the contact interfacial stiffness K increases as the contact pressure p increases, which mean that the contact interfaces are being closed. The acoustic reflection coefficient also decreases, as predicted from Eqs. (7) and (8). Thus, if we measure the reflection coefficients at different contact pressures, we can
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3. Experimental procedure 3.1. Experimental system To measure the reflection coefficient of longitudinal and transverse waves at normal incidence as well as those of obliquely incident waves, the experimental system shown in Fig. 2 was employed. In this system, a hydraulic press was used to apply loads to the contact interfaces. The loads were measured with a load cell attached at the bottom of the specimen. The contact pressure applied at the contact interfaces was calculated as the mean value divided by the area of the contact interfaces. A tone-burst signal of 2.25 MHz was generated with a high-power gated amplifier (Ritec, RAM-5000 SNAP). Longitudinal and transverse wave transducers with a center frequency of 2.25 MHz were used for measurement of the reflection coefficient of the normally incident waves. In the case of oblique incidence, a 2.25 MHz longitudinal wave transducer was used as a transmitter and a 5 MHz longitudinal wave transducer was employed as a receiver. The received signal was monitored with a digital oscilloscope (LECROY WS452) and a fast Fourier transform (FFT) was performed to obtain the amplitude of the fundamental and second harmonic waves. By measuring the reflection coefficient of the normally incident waves, the linear interface stiffnesses Ky and Kx were calculated using Eqs. (7) and (8). Experiments were performed after a loading/unloading process was repeated 10 times so as to eliminate the hysteresis effect in the contact interfaces [34]. The reflection coefficient and the amplitude of the second harmonic wave of the reflected obliquely incident longitudinal waves were measured experimentally and compared with the theoretical analysis.
Two incident angles, 22.5° and 45°, were employed in the tests. The contact interface was polished using #600 sand paper.
4. Experimental results The experimentally obtained reflection coefficients for the normally incident longitudinal and transverse waves as a function of the applied nominal contact pressure are shown in Fig. 3. The reflection coefficients were normalized using the initial coefficient value obtained at a no-load condition. With the obtained results, the linear interfacial stiffnesses Ky and Kx were calculated from Eqs. (7) and (8); the results are shown in Fig. 4. As the applied contact pressure increased, the reflection coefficient decreased and the linear interfacial stiffness increased. The interfacial stiffness has a nonlinear relationship with the nominal contact pressure. This relationship was fitted to the power-law model in Eq. (9) so that the constants C and m could be obtained. Finally, the representation of the linear normal and tangential interfacial stiffnesses, as well as the nonlinear normal and tangential interfacial stiffnesses as a function of the nominal contact pressure was complete.
Measured Reflection Coefficient R 1 Normal Incidence of Longitudinal Wave Normal Incidence of Shere Wave
0.9
Reflection Coefficient R
obtain the linear interfacial stiffness as a function of the contact pressure; the constants C and m in Eq. (9) can be determined by curve fitting. The nonlinear interfacial stiffness Kn can also be calculated from Eq. (9). Finally, the amplitude of the reflected wave and the second harmonic generation for oblique incidence can be determined from Eqs. (3) and (4).
0.8 0.7 0.6 0.5 0.4
3.2. Specimens
0
5
10
15
20
25
30
Nominal Contact Pressure [MPa] A pair of aluminum 6061-T6 blocks put into contact with an external normal load was used as the test specimen. The contacting area has a length of 100 mm and a width of 40 mm. The test specimen consisted of upper and lower blocks; the upper block was machined as shown in Fig. 2 so as establish oblique incidence.
Fig. 3. Reflection coefficient of a normally incident longitudinal wave and a shear wave.
14
2.5
x 10
Measured Linear Interfacial Stiffness Kx,Ky
Linear Interfacial Stiffness Kx,Ky [N/m]
Normal Interfacial Stiffness Ky Tangential Interfacial Stiffness Kx
2
1.5
1
0.5
0 0
5
10
15
20
25
30
Nominal Contact Pressure [MPa]
Fig. 2. Schematic diagram of the experimental setup.
Fig. 4. The measured linear interfacial stiffnesses Kx and Ky as a function of the nominal contact pressure.
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waves are shown in Fig. 6. The theoretical results were calculated from Eq. (4). The amplitudes of the second harmonic waves are normalized by the peak amplitude from the 22.5° result. The second harmonic component was 20% lower at a 22.5° oblique incidence and 55% lower at a 45° oblique incidence when compared to the normal incidence case. However, the trends in second harmonic generation with respect to an increase in the contact pressure were the same for each case. In addition, the experimental results are in good agreement with the theoretical predictions. The normalized amplitude of the second harmonic wave, A2, increases sharply for each incidence angle when the contact pressure is below 7 MPa. The amplitude reaches a maximum value around 7 MPa and then decreases monotonically since the interfaces of the aluminum blocks become fully closed. Such findings appropriately reflect the nonlinear characteristics of contact interfaces and thus, it is possible to use the oblique incidence technique suggested in this paper for an evaluation of contact interfaces.
Reflection Coefficient
Reflection Coefficient R
1
0.8
0.6
0.4 Simulation Data of Normal Incidnece (0 deg) Simulation Data of Oblique Incidence (22.5 deg) Simulation Data of Oblique Incidence (45 deg) Experimental Data of Oblique Incidence (22.5 deg) Experimental Data of Oblique Incidence (45 deg)
0.2
0 0
0.5
1
1.5
2
2.5 14
Normal Linear Interfacial Stiffness Ky [N/m] x 10
Fig. 5. Comparison of the theoretical and experimental ultrasonic reflection coefficients as a function of the nominal linear interfacial stiffness.
Normalized Amplitude of 2nd Harmonic wave A
2
Shown in Fig. 5 are the longitudinal reflection coefficients theoretically predicted by Eq. (3) for oblique incidence angles of 22.5° and 45° using the linear normal and tangential interfacial stiffnesses obtained as described above. The normal incidence data were used as a reference and the experimental results are included in Fig. 5. The experimental findings exhibit good agreement with theoretical expectations. Thus, we can confirm the validity of our model for the amplitude of the reflected longitudinal wave based on the interfacial stiffnesses. The reflection coefficients of the 22.5° obliquely incident longitudinal waves are almost identical to those obtained at normal incidence because the reflection characteristics of oblique incidence at small angles are similar to those of normal incidence cases. On the other hand, the reflection coefficients of the 45° obliquely incident longitudinal waves are about 20% higher than those of the normally incident waves because the characteristics of waves obliquely incident at large angles are affected by both the normal interfacial stiffness and the tangential interfacial stiffness. The theoretical and experimental results for the second harmonic generation of normally and obliquely incident longitudinal
Normalized Amplitude of 2nd Harmonic Wave 1.4
1.2
5. Conclusions In this work, a theoretical model for the reflection of an ultrasonic wave obliquely incident on contact interfaces was proposed by introducing the parameters of linear and nonlinear contact stiffness. Using this model, we can predict the magnitude of a CAN-induced second harmonic wave in a reflected ultrasonic wave. The experimental results obtained from the contact interfaces of A16061-T6 alloy specimens at various loading pressures showed good agreement with the theoretical predictions. Such agreement proves the validity of the suggested oblique incidence model. The findings of this study may be summarized as follows: (1) The experimentally obtained reflection coefficients for the 22.5° and 45° obliquely incidence longitudinal waves are in good agreement with theoretical predictions. (2) The reflection coefficients of the 22.5° obliquely incident longitudinal waves are similar to those of the normally incident longitudinal waves because the characteristics of reflection are mainly influenced by the normal interfacial stiffness. On the other hand, the reflection coefficients of the 45° obliquely incident longitudinal waves are higher than those of the normally incident longitudinal waves because the coefficients are influenced by both the normal and tangential interfacial stiffnesses. (3) The amplitude of the second harmonic wave of the obliquely incident longitudinal wave is lower than that of the normally incident longitudinal wave. Nevertheless, the variation in the amplitude of the second harmonic waves with respect to the contact pressure exhibited similar trends.
1
Acknowledgment 0.8
This work was financially supported by the National Research Foundation of Korea (NRF) (2008-2003505).
0.6
References
0.4 Simulation Data of Normal Incidence (0 deg) Simulation Data of Oblique Incidence (22.5 deg) Simulation Data of Oblique Incidence (45 deg) Experimental Data of Oblique Incidence (22.5 deg) Experimental Data of Oblique Incidence (45 deg)
0.2
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Nominal Contact Pressure [MPa] Fig. 6. Normalized amplitude of the second harmonic wave as a function of the nominal contact pressure at each incidence angle.
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