Numerical and experimental study of the nonlinear interaction between a shear wave and a frictional interface Philippe Blanloeuila) University Bordeaux, CNRS, I2M, UMR 5295, 351 Cours de la Liberation, 33405 Talence Cedex, France
Anthony J. Croxford Department of Mechanical Engineering, University of Bristol, Queens Building, University Walk, Bristol BS8 1TR, United Kingdom
Anissa Meziane University Bordeaux, CNRS, I2M, UMR 5295, 351 Cours de la Liberation, 33405 Talence Cedex, France
(Received 13 March 2012; revised 3 May 2013; accepted 3 March 2014) The nonlinear interaction of shear waves with a frictional interface are presented and modeled using simple Coulomb friction. Analytical and finite difference implementations are proposed with both in agreement and showing a unique trend in terms of the generated nonlinearity. A dimensionless parameter n is proposed to uniquely quantify the nonlinearity produced. The trends produced in the numerical study are then validated with good agreement experimentally. This is carried out loading an interface between two steel blocks and exciting this interface with different amplitude normal incidence shear waves. The experimental results are in good agreement with the numerical results, suggesting the simple friction model does a reasonable job of capturing the fundamental physics. The resulting approach offers a potential way to characterize a contacting interface; however, the difficulty in activating that interface may ultimately limit its applicability. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4868402] V PACS number(s): 43.25.Dc [ROC]
Pages: 1709–1716
I. INTRODUCTION
The evaluation of damage at an early stage is of key importance in many safety critical fields, such as nuclear power plants and aerospace structures, where the nature and expense of inspections contributes significantly to operating costs. Ultrasonic methods based on linear wave scattering are efficient for detecting defects on the scale of the wavelength but are less sensitive to micro or closed cracks. Exploiting the nonlinear behavior of these defects, nonlinear ultrasonic techniques such as nonlinear resonance,1 sub- and higherharmonic generation,2 and frequency-modulation3 have been shown to be sensitive to these difficult to detect damage types (for an overview of nonlinear acoustics applications see Refs. 4–6). When an ultrasonic wave with large amplitude is incident on contacting surfaces with a frictional interface (e.g., closed cracks), higher harmonics appear in the frequency spectrum of transmitted and reflected waves. This effect, called Contact Acoustic Nonlinearity,2 is of increasing interest for characterization of closed cracks or imperfectly bonded interfaces and it is this that is investigated here. Both longitudinal and shear wave propagation through a rough surface were investigated in Ref. 7 using an interface contact model based on Hertz theory for a time harmonic incident wave. Beyond the extension of a spring model to tangential excitation, they studied the influence of the tangential and normal stiffnesses on the generated harmonics. The partial contact model was subsequently applied to model scattering from a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 135 (4), April 2014
surface breaking cracks,8 and numerical simulations indicated that such cracks would produce second harmonics. Another approach introduces the interface stiffness to account for quantitative transmission and reflection of waves and harmonic evolution.7,9,10 Time domain studies have concentrated on numerical implementations, such as a boundary element modeling formulation of Shear Horizontal (SH) slip motion at an arbitrary interface.11 Using a generalization of this method to include in-plane motion, it was shown in Refs. 12 and 13 that the amplitudes of the higher harmonics of the scattered far-fields can be useful in determining both the pre-stress and friction coefficient. Measurements of second harmonic generation from normal incidence longitudinal waves on a contacting interface between aluminum blocks have been reported in Refs. 9 and 14. These experiments indicate that the amplitude of the second harmonic decreases rapidly with applied normal contact pressure initially, and then falls off in magnitude at a lesser rate. These findings are in agreement with experimental measurements on contacting adhesive bonds.15 The literature suggests that none of the simulations or experiments to date have considered the explicit dependence of the friction effects at the contact interface. A one-dimensional (1D) model was proposed in Ref. 16 for the nonlinear interaction of an SH wave normally incident on a frictional contact interface. This analytical model predicts that only odd harmonics are generated by time harmonic incident wave motion. Moreover, the third harmonic is characterized by a maximum of amplitude when the normal load is changed. More recently, a similar analytical analysis and a numerical model were proposed in Ref. 17. The authors pointed out the relation between the energy
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C 2014 Acoustical Society of America V
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dissipated by friction and the amplitude of higher harmonics. Two contact laws have been used: The first is a classic Coulomb’s law whereas the second is a slip-weakening law in order to investigate the impact of the friction law on the third harmonic. The study points out a small shift of the third harmonic maximum. However, these results are purely theoretical without experimental validation. The purpose of the present paper is to examine the third harmonic generation resulting from shear wave interaction with a frictional interface. The friction at the interface leads to nonlinearity due to switching between the sticking and sliding states of the interfacial contact. To provide insight into the physical phenomena involved, a finite difference (FD) model is used to describe the interaction between a shear wave propagating in a semi-infinite media and a rigid plane. A classic Coulomb’s law is used to model the frictional interface and the solution is evaluated in the time domain. This model is obviously simpler than the finite element (FE) model used in Ref. 17 but it captures what is believed to be the dominant physical processes. The results are in agreement with those of the FE model when investigating the effect of parameters such as the normal load or the input wave amplitude, confirming the model reasonably presents the physical reality. The governing equations and the numerical implementation of this model are introduced in Sec. II. Energy transfers are analyzed in order to show that the model is energy conservative. The results regarding the third harmonic complete Sec. II. An experimental setup was built to validate the numerical observation which we believe is the first such confirmation of the nonlinear behavior of a frictional interface with an incident shear wave. The developed experimental system and the procedure are presented in Sec. III. II. NUMERICAL STUDY OF A WAVE REFLECTED FROM A FRICTIONAL INTERFACE
The numerical model is defined for a homogeneous, isotropic elastic half-space defined as X and assumed to be in perfect contact with a rigid wall dXs at x ¼ 0. The half space and the rigid plane are brought into contact under a given normal stress. A vertically polarized plane shear wave is input to the system at x ¼ L, normal to the contact interface. Here, a transparent boundary dXt is defined in order to model an infinite medium. The system is presented in Fig. 1. A Coulomb’s law with a constant friction coefficient is used at the interface. With these assumptions, a 1D model of the wave propagation and reflection from the frictional interface is built. Assuming a large enough incident amplitude, alternative stick-slip motion will occur and produce higher harmonics in the reflected wave. The wave equation governs the wave propagation in X;
(1)
where c refers to wave velocity (approximately 3000 m s1 for a shear wave in steel) and uðt; xÞ is the displacement in y. The latter can be seen as the sum of the incident and reflected waves uþ and u , respectively, 1710
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uðt; xÞ ¼ uþ ðt x=cÞ þ u ðt þ x=cÞ:
(2)
The medium is initially at rest with mechanical properties, Young modulus E ¼ 200 GPa, Poisson’s ratio ¼ 0.33, and shear modulus G ¼ 75 GPa. A static normal stress r0 is applied to the solid. Let s be the shear stress and s the parti_ tÞ is the relative speed between the cle velocity. Then, uð0; solid and the interface. Finally, l ¼ 0:3 is the friction coefficient at the interface dXs . The Coulomb’s law is defined as follows 8 _ tÞ ¼ 0 < if jsð0; tÞj < lr0 ) sticking: uð0; in dXs : : _ tÞ 0 if jsð0; tÞj ¼ lr0 ) sliding: s:uð0; (3) The problem is tackled with a stress-velocity formulation. Using this formulation with Eq. (2) we obtain the shear stress: @u G ¼ ðu0þ ðt x=cÞ þ u0 ðt þ x=cÞÞ @x c ¼ sþ ðt x=cÞ þ s ðt þ x=cÞ;
sð x; tÞ ¼ G
A. Governing equations
@2u @2u c2 2 ¼ 0; 2 @t @x
FIG. 1. System used to study the interaction between a normal incidence shear wave and a contacting interface.
(4)
with the following expressions for both tangential stresses 8 G > > < sþ ðt x=cÞ ¼ u0þ ðt x=cÞ c (5) G > > : s ðt þ x=cÞ ¼ u0 ðt þ x=cÞ: c The derivatives are defined as 8 1 c > > _ tÞ sðt; xÞÞ < u0þ ðt x=cÞ ¼ ðuðx; 2 G 1 c > > 0 : u ðt þ x=cÞ ¼ ðuðx; _ tÞ þ sðt; xÞÞ: 2 G
(6)
As done in Ref. 16, the solution at the interface can easily be obtained analytically, since two criteria define the transition between sticking and sliding. This analytical solution consists of finding the instant of switch between the two modes, slipping and sticking. Consequently, it is a piecewise implementation that could be time consuming when the number of Blanloeuil et al.: Shear wave and frictional interface
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FIG. 2. Numerical procedure to solve the contact problem.
transitions increases. Therefore, a numerical solution of the problem is preferred in the present work and the implemented FD approach is presented in Sec. II B. B. Numerical implementation
The wave equation is solved in time using the classic second order FD approach. Coulomb’s law governs the contact as defined in Eq. (3). A transparent boundary condition enables the size of the computational domain to be limited and is defined at dXt as u0þ ðt þ L=cÞ ¼ f ðtÞ;
in dXt ;
(7)
where u0þ is given in Eq. (6) and f ðtÞ is the source term, equivalent to an imposed particle velocity. This equation defines the incident wave and is transparent to the reflected wave. This condition is rewritten into a strain formulation _ @u 2f ðtÞ uðL; tÞ ðL; tÞ ; on dXt : ¼ @x c c
(8)
This equation is discretized as follows: m m 3um 1 umþ1 um 2f ðtÞ 0 þ 4u1 u2 0 0 ¼ ; 2dx dt c c
on dXt ; (9)
where dx and dt are, respectively, the space step and the time step. The wave equation [Eq. (1)] is discretized in X using the classic Euler FD scheme, with the indices “m” and “k” denoting time and space discretization, respectively, m m1 um 2um umþ1 2um k þ uk1 k þ uk k c2 kþ1 ¼ 0; 2 2 dt dx 8k 2 ½2; n 2; in X:
(10)
This second order general scheme is conditionally stable under the classic Courant-Friedrichs-Lewy (CFL) condition: cðdt=dxÞ < 1. Using Eqs. (8), (10), and (3) we can build two J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
matrices, Ast for the sticking condition and Asl for the sliding condition. The next time solution is computed by a matrixvector product like U mþ1 ¼ Ast;sl U m þ B, where B contains the excitation source term. The algorithm representing the numerical implementation is shown in Fig. 2. C. Numerical results
The convergence through grid size and time step is ensured by comparing the numerical solution at the interface with the analytical solution described in Sec. II A at dXs . In the following results, the simulation is run at 0.7 CFL and the space discretization is dx ¼ k3x =24, where k3x is the wavelength of the third harmonic. These values of the numerical parameters ensure the convergence of the model. The incident wave is a five cycle sinusoidal signal modulated by a Gaussian window. Figure 3(a) illustrates the solution at x ¼ L, showing the incident and the reflected wave. The reflected wave is then windowed and transformed to the frequency domain. Its spectrum is shown in Fig. 3(b). These curves are normalized using the value of incident harmonic Ax . The reflected wave spectrum contains odd higher harmonics, A3x and A5x , as expected. Since the nonlinearity appears when sliding with friction occurs, we define a dimensionless parameter n from the Coulomb’s law [as given in Eq. (3)] to easily compare the results n¼
lr0 2ksincident k1
:
(11)
Therefore, when n ¼ 0, there is no friction and no dissipation of energy. When n 1 the interface remains stuck and no motion occurs. Thus the nonlinear frictional behavior will only occur within these bounds. Having determined the basic numerical procedure, the influence of the normal load and the incident amplitude on the evolution of the third harmonic is studied. The harmonic amplitudes are obtained by applying bandpass filters at the first and third harmonic in post processing. The time signal Blanloeuil et al.: Shear wave and frictional interface
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envelope of these time signals give the value of each harmonic. The third harmonic is normalized using the incident harmonic size, At3 ¼ At3x =Atxincident . Figure 4 shows the evolution of the third harmonic in the time domain for three different input amplitudes. The curves are plotted for different magnitudes of incident wave as a function of n. The result is that the response is plotted for each input amplitude over a range of normal stresses. The same peak of At3 at n ’ 0:45 is observed irrespective of the input amplitude, confirming the relevance of the dimensionless parameter chosen. This maximum value of A3 occurs at the peak of nonlinearity and is directly related to the friction coefficient at the interface. This result is in agreement with Ref. 17 and suggests that measurements of the third harmonic can be used to quantify friction and dissipation effects at a sliding interface. We can define the energy transported by a wave as ð tfin _ tÞdt; sðx; tÞ:uðx; (12) EðxÞ ¼ tini
FIG. 3. (a) Time solution at x ¼ L and (b) the associated spectrum for the incident and reflected wave. The spectrum is normalized using the incident harmonic amplitude Ax . The fundamental frequency fx ¼ 1 MHz, normal stress r0 ¼ 2 MPa, incident amplitude A ¼ 2.5 nm, friction coefficient l ¼ 0:3 and 1200 space points are used at 0.7 CFL.
is converted in the frequency domain using a FFT (Fast Fourier Transform). The parts of the spectrum corresponding respectively to the first and the third harmonic are then separately transformed back to the time domain using an IFFT (Inverse Fast Fourier Transform). The amplitudes of the
FIG. 4. Evolution of At3 relative to n for different incident amplitudes. The normal load varies from 0.5 to 4 MPa. Fundamental frequency fx ¼ 1 MHz and the friction coefficient l ¼ 0:3. 1712
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where s is the shear stress and u_ is the speed at point x. For x ¼ 0, this energy corresponds to the dissipation of energy by friction. Figure 5 shows the evolution of the dissipated energy against n for the same three configurations as described in Fig. 4. These curves are normalized by the incident energy defined by Eq. (12). Obviously, the bigger the input amplitude is, the bigger the dissipation of energy by the frictional interface is. However, the relative change remains constant as shown in the normalized data of Fig. 5. For each case, the peak value is around n ’ 0:33. This result suggests the existence of a link between the third harmonic A3x and the amount of energy dissipated by friction. However the exact relationship is unclear at this point. Finally, we look at the energy balance between the incident and the reflected wave. Figure 6 shows the evolution of the dissipated energy, the total reflected energy, and the reflected energy at the first, third, and fifth harmonics. As expected, the maximum dissipated energy corresponds with the minimum of reflected energy. Moreover we can verify that the model is conservative since
FIG. 5. Evolution of dissipated energy by friction for three input amplitudes. Blanloeuil et al.: Shear wave and frictional interface
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FIG. 6. Transfer of energy from the incident wave to the reflected wave and dissipation by friction.
the sum of the reflected and dissipated energy equals the incident energy. It is worth noting that the simple model of friction employed here is responsible for this complex behavior. In order to determine if this model is sufficient, experimental validation of the behavior is essential. To conclude, a normal stress r0 2 ½0:4; 4 MPa and an input amplitude A 2 ½1; 8 nm should provide a large enough parameter space to study the evolution of A3 with a friction coefficient of 0.3. This simple numerical model can provide quantitative evaluation as regards to third harmonic generation and dissipation of energy. An experimental study was carried out to validate this behavior, informed by the range of values determined here. III. EXPERIMENTAL INVESTIGATION
The experimental setup is illustrated schematically in Fig. 7. Two steel blocks are brought into contact and loaded
in compression. The force is applied using a threaded rod and nuts to ensure a constant load. This force is monitored through a load-cell situated between the screw thread and the samples. In practice the system is mounted horizontally to allow the application of loads lower than the block weight. A 25.4 mm (1 in.) broadband shear transducer of center frequency 1 MHz (V152, Panametrics, Waltham, MA) sends wave pulses into the system at location T1. A 12.7 mm (1/2 in.) transducer of center frequency 5 MHz (V155, Panametrics, Waltham, MA) is used as a receiver at location T2. A personal computer controls the process of excitation and reception. The input signal is generated by a wave generator (33120A, Agilent Tec., Santa Clara, CA) and the received signal is measured on an oscilloscope (LT224, Lecroy, Chestnut Ridge, NY) with both communicating with the computer over a GPIB interface. The excitation is a 15 cycle pulse which is amplified (GA-2500A, Ritec Inc., Warwick, RI) and then sent to the transducer. A longer signal than in the numerical modeling is used to ensure the best chance of measuring the generated nonlinearity. The two steel blocks used have dimensions of 46 46 55 mm3 and were machined to ensure parallel faces, with a dimensional tolerance estimated to be 0.4 mm. The contacting faces were then polished with P80 and P180 paper to produce an interface of known quality and with an average roughness of Ra ¼ 0:4 lm. The amplitude of the incident shear wave at the interface, effectively giving shear stress, is of prime importance when using the dimensionless parameter n defined in Eq. (11). This was measured by transmitting an input wave into one of the steel blocks. The response was then measured on the parallel face using a laser vibrometer (OFV 353, Polytec, Waldbronn, Germany). The measurement was made at an angle of 45 to ensure sensitivity to the largely in plane shear mode and the measured displacement corrected to give the true amplitude. This was repeated for both input and receiving transducers at the frequencies of interest, namely input, and third harmonic. This process ensures that the values that are measured in the later load experiments can be directly converted to the amplitude at the interface, without having to specifically correct for the effects of attenuation or beam spreading, increasing the confidence in the results. The same procedure was repeated to calibrate the input amplitude that the amplifier would produce when set using the amplifiers arbitrary amplitude scale. The resulting amplitudes are shown in Table I. Although the surfaces are polished, there still remain asperities that could interfere in the contact mechanism when the two blocks are first put into contact. In order to ensure the reproducibility of the measurements made, the contacting blocks are subjected to three loading cycles prior to ultrasonic measurements being made. The laboratory TABLE I. Input displacement A at the interface against arbitrary amplifier amplitude scale. The measure is done on a single block with a laser vibrometer measurement for each graduation of the amplifier scale.
FIG. 7. (Color online) Experimental setup for bulk transverse wave transmission through a plane interface. J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
Ritec A (nm) Ritec A (nm)
3 0.12 6.5 2.10
3.5 0.14 7 3.39
4 0.17 7.5 5.17
4.5 0.25 8 7.09
5 0.47 8.5 9.02
5.5 0.72 9 10.9
6 1.23
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environment is constant on the scale of the measurement time so a high degree of signal averaging was performed (1000 averages) in order to maximize the signal to random noise level. The evolution of the third harmonic as a function of the n parameter was then measured, paying special care to ensure a good range of measurements was made between 0 and 1 where the nonlinearity is expected. This was done using the parameters generated from the earlier numerical model. The experimental procedure consists first in increasing the normal load while the input amplitude is at its highest level; that is 10.9 nm at the interface. For this amplitude, the load was increased from 0.6 to 9.5 kN in 24 steps, which results in a normal stress r0 2[0.3;4.5] MPa, assuming plane strain. The peak load of 9.5 kN force is then maintained and the input amplitude is decreased from 10.9 to 0.12 nm in the 13 steps shown in Table I. In doing so, 36 increasing values of n have been obtained. For each of them, the amplitudes of the first and third harmonic have been evaluated, leading to 36 independent values. For the record, the definition of n is n ¼ lr0 =j2rInc j. Figure 8 shows an example of a received time signal with three different wave arrivals clearly visible. The first wave is a longitudinal wave. The second one corresponds to a longitudinal wave converted from the incident shear wave at the interface. The third is the shear wave that is the primary interest of this paper. Each measure is processed as described in Sec. II to extract the third harmonic. The amplitudes obtained outside of the temporal window give the noise level associated with each harmonic. Finally, these data are converted from Voltage (V) to displacement (m) using the calibration values obtained as described above. In order to use the nondimensional parameter defined in Eq. (11) the static friction coefficient at the interface must be measured. This was done by measuring at what angle relative to the horizontal the two blocks began to slip and resolving the forces to determine the friction coefficient. Though primitive this was repeated ten times and found to give the same slip angle within 1 . The resulting coefficient of friction used was 0.3. For each measurement of load and amplitude at interface, the n value is computed. Normal stress is defined as the ratio of the applied force over the surface of contact.
FIG. 8. Received signal for a normal load of 6.5 kN and displacement at the interface of 10.9 nm. 1714
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Incident shear stress of the wave is computed using the calibrated incident amplitude, and the measured coefficient of friction was used. Repeat measurements were made at the high and low end of the load and interface displacement amplitude with a range of 5%. A value that will be seen in Sec. IV is sufficient to ensure that the measured results are clear of the noise floor. The degree of nonlinearity input to the system increases with amplitude as expected. To prevent this effect masking the changes in nonlinearity at the interface, the input nonlinearity must be removed. Therefore, for the data corresponding to high values of n ðn > 2:5Þ where the system is not exhibiting nonlinear contact behavior, we determine a scale and factor a that minimizes the gap between A1 =Amax 1 for the measured data. In this case, a is such that, A3 =Amax 3 A3 A3 A1 ¼ ¼ a : (13) Amax Amax Amax 3 3 1 n>2:5 n>2:5 n>2:5 |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} measured
system
Thus, the final normalized value of the third harmonic due to contact nonlinearity is A3 A1 corrected ¼ a max Amax (14) A3 3 : Amax A1 3 Since the nonlinearity generated at the interface is of a different origin to that produced by the amplifier this will not be suppressed. For each starting input amplitude, this analysis is applied to the data. When the amplitude is small (0.47 nm, for example), there is no difference between A3 and A1 as shown in Fig. 9. Therefore, the corrected A3 is zero. The nonlinearity is not active at the interface since the input amplitude is not large enough to trigger sliding. The same treatment is applied to the data that start at the highest input amplitude, 10.9 nm. For high values of n, the trend of A3 is exactly the same as A1 , as shown in Figure 10. However, for low n between 0 and 1, a difference exists. Without removing the effects of the input nonlinearity, the true change in the region of interest is impossible to see.
FIG. 9. Correction of A3 by subtraction of A1 trend for an initial interface amplitude of 0.47 nm. The amplitude of the incident wave is not large enough to trigger sliding. Blanloeuil et al.: Shear wave and frictional interface
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FIG. 10. Correction of A3 by subtraction of A1 trend for an initial interface amplitude of 10.9 nm.
After subtraction of the linear trend there is still a clear increase in the harmonic value between 0 n 1, indicating that the third harmonic is greater than the inherent system nonlinearity (electronics and material), and is therefore due to the nonlinear effect at the contacting interface. The corrected A3 defined by Eq. (14) for a large interface amplitude (10.9 nm) is shown in Fig. 11. Included as a solid line is a comparison with the numerical result described earlier. In this example the parameters used in the numerical model are the same as those measured experimentally for the real system. The numerical results show a good qualitative match with the experimental results. The nonlinearity is clearly evident for n 2 ½0; 1 and the maximum of the third harmonic amplitude is observed at n ¼ 0.4, both for the experimental and numerical results. IV. DISCUSSION AND CONCLUSION
This study presents a numerical and experimental analysis of the propagation of a shear wave through a frictional interface with the express goal of using the generated nonlinearity to characterize the interface. A simple FD model is developed that describes the case of a classical Coulomb friction law and
a plane wave. This model gives the third harmonic behavior in the dimensionless parameter n space, with a maximum at n ¼ 0.4. This response corresponds to a specific signature of sliding with friction resulting in contact nonlinearity. Third harmonic behavior shows an explicit dependence on frictioninduced dissipated energy. The results are in accordance with previous studies16,17 and are used to determine suitable parameters for the experimental investigation. An experimental procedure to validate these numerical results was developed. Good agreement between the experimental and numerical results is demonstrated leading to a twofold conclusion. First, classical Coulomb’s law included in the model provides a sufficient description of friction during sliding at the contact interface. Therefore, this law can be used in more complex FE models like solids, including cracks. Second, an estimation of the friction coefficient can be made by fitting numerical curves to the experimental data through adjusting the numerical parameter n. Finally, when an acoustic wave with oblique incidence interacts with a crack, the sliding can be activated even with longitudinal waves. This work provides an experimental justification for the use of a Coulomb law to describe friction. The specific behavior of the third harmonic due to frictional effects may be useful to detect and identify closed-cracks or characterize interfaces exhibiting sliding with friction. Although potentially offering a unique way to measure the contact nonlinearity and the coefficient of friction the approach does present significant difficulties. Very high input amplitudes are necessary to activate the sliding mechanism and without these, no contact nonlinearity could be measured. This was made more complex in this specific case where the testing frame was unable to support much higher loads. The primary advantage of this approach relative to say conventional generation of second harmonics at a clapping interface is in the ability to remove the effects of input nonlinearity; that is, a measurement of nonlinearity can essentially be made at the same time as the interface nonlinearity. This is not possible with a conventional second harmonic generation approach. In comparison to the second harmonic approach the method presented here does not have the same sensitivity when a crack is open; however, when closed (by a load say) it may have greater sensitivity. This is, however, complicated by the potentially high coefficients of friction in such a scenario preventing the interface nonlinearity activating. Further work is necessary to determine the usefulness of this novel approach in these cases. ACKNOWLEDGMENTS
The authors would like to thank Christine Biateau and Philippe Malerne for their technical support and valuable comments.
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FIG. 11. Evolution of the third harmonic with increasing n measured experimentally and calculated numerically. The amplitude of the numerical result is fitted to the experimental data. J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
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Blanloeuil et al.: Shear wave and frictional interface
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Anthony J. Croxford Department of Mechanical Engineering, University of Bristol, Queens Building, University Walk, Bristol BS8 1TR, United Kingdom
Anissa Meziane University Bordeaux, CNRS, I2M, UMR 5295, 351 Cours de la Liberation, 33405 Talence Cedex, France
(Received 13 March 2012; revised 3 May 2013; accepted 3 March 2014) The nonlinear interaction of shear waves with a frictional interface are presented and modeled using simple Coulomb friction. Analytical and finite difference implementations are proposed with both in agreement and showing a unique trend in terms of the generated nonlinearity. A dimensionless parameter n is proposed to uniquely quantify the nonlinearity produced. The trends produced in the numerical study are then validated with good agreement experimentally. This is carried out loading an interface between two steel blocks and exciting this interface with different amplitude normal incidence shear waves. The experimental results are in good agreement with the numerical results, suggesting the simple friction model does a reasonable job of capturing the fundamental physics. The resulting approach offers a potential way to characterize a contacting interface; however, the difficulty in activating that interface may ultimately limit its applicability. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4868402] V PACS number(s): 43.25.Dc [ROC]
Pages: 1709–1716
I. INTRODUCTION
The evaluation of damage at an early stage is of key importance in many safety critical fields, such as nuclear power plants and aerospace structures, where the nature and expense of inspections contributes significantly to operating costs. Ultrasonic methods based on linear wave scattering are efficient for detecting defects on the scale of the wavelength but are less sensitive to micro or closed cracks. Exploiting the nonlinear behavior of these defects, nonlinear ultrasonic techniques such as nonlinear resonance,1 sub- and higherharmonic generation,2 and frequency-modulation3 have been shown to be sensitive to these difficult to detect damage types (for an overview of nonlinear acoustics applications see Refs. 4–6). When an ultrasonic wave with large amplitude is incident on contacting surfaces with a frictional interface (e.g., closed cracks), higher harmonics appear in the frequency spectrum of transmitted and reflected waves. This effect, called Contact Acoustic Nonlinearity,2 is of increasing interest for characterization of closed cracks or imperfectly bonded interfaces and it is this that is investigated here. Both longitudinal and shear wave propagation through a rough surface were investigated in Ref. 7 using an interface contact model based on Hertz theory for a time harmonic incident wave. Beyond the extension of a spring model to tangential excitation, they studied the influence of the tangential and normal stiffnesses on the generated harmonics. The partial contact model was subsequently applied to model scattering from a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 135 (4), April 2014
surface breaking cracks,8 and numerical simulations indicated that such cracks would produce second harmonics. Another approach introduces the interface stiffness to account for quantitative transmission and reflection of waves and harmonic evolution.7,9,10 Time domain studies have concentrated on numerical implementations, such as a boundary element modeling formulation of Shear Horizontal (SH) slip motion at an arbitrary interface.11 Using a generalization of this method to include in-plane motion, it was shown in Refs. 12 and 13 that the amplitudes of the higher harmonics of the scattered far-fields can be useful in determining both the pre-stress and friction coefficient. Measurements of second harmonic generation from normal incidence longitudinal waves on a contacting interface between aluminum blocks have been reported in Refs. 9 and 14. These experiments indicate that the amplitude of the second harmonic decreases rapidly with applied normal contact pressure initially, and then falls off in magnitude at a lesser rate. These findings are in agreement with experimental measurements on contacting adhesive bonds.15 The literature suggests that none of the simulations or experiments to date have considered the explicit dependence of the friction effects at the contact interface. A one-dimensional (1D) model was proposed in Ref. 16 for the nonlinear interaction of an SH wave normally incident on a frictional contact interface. This analytical model predicts that only odd harmonics are generated by time harmonic incident wave motion. Moreover, the third harmonic is characterized by a maximum of amplitude when the normal load is changed. More recently, a similar analytical analysis and a numerical model were proposed in Ref. 17. The authors pointed out the relation between the energy
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C 2014 Acoustical Society of America V
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dissipated by friction and the amplitude of higher harmonics. Two contact laws have been used: The first is a classic Coulomb’s law whereas the second is a slip-weakening law in order to investigate the impact of the friction law on the third harmonic. The study points out a small shift of the third harmonic maximum. However, these results are purely theoretical without experimental validation. The purpose of the present paper is to examine the third harmonic generation resulting from shear wave interaction with a frictional interface. The friction at the interface leads to nonlinearity due to switching between the sticking and sliding states of the interfacial contact. To provide insight into the physical phenomena involved, a finite difference (FD) model is used to describe the interaction between a shear wave propagating in a semi-infinite media and a rigid plane. A classic Coulomb’s law is used to model the frictional interface and the solution is evaluated in the time domain. This model is obviously simpler than the finite element (FE) model used in Ref. 17 but it captures what is believed to be the dominant physical processes. The results are in agreement with those of the FE model when investigating the effect of parameters such as the normal load or the input wave amplitude, confirming the model reasonably presents the physical reality. The governing equations and the numerical implementation of this model are introduced in Sec. II. Energy transfers are analyzed in order to show that the model is energy conservative. The results regarding the third harmonic complete Sec. II. An experimental setup was built to validate the numerical observation which we believe is the first such confirmation of the nonlinear behavior of a frictional interface with an incident shear wave. The developed experimental system and the procedure are presented in Sec. III. II. NUMERICAL STUDY OF A WAVE REFLECTED FROM A FRICTIONAL INTERFACE
The numerical model is defined for a homogeneous, isotropic elastic half-space defined as X and assumed to be in perfect contact with a rigid wall dXs at x ¼ 0. The half space and the rigid plane are brought into contact under a given normal stress. A vertically polarized plane shear wave is input to the system at x ¼ L, normal to the contact interface. Here, a transparent boundary dXt is defined in order to model an infinite medium. The system is presented in Fig. 1. A Coulomb’s law with a constant friction coefficient is used at the interface. With these assumptions, a 1D model of the wave propagation and reflection from the frictional interface is built. Assuming a large enough incident amplitude, alternative stick-slip motion will occur and produce higher harmonics in the reflected wave. The wave equation governs the wave propagation in X;
(1)
where c refers to wave velocity (approximately 3000 m s1 for a shear wave in steel) and uðt; xÞ is the displacement in y. The latter can be seen as the sum of the incident and reflected waves uþ and u , respectively, 1710
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uðt; xÞ ¼ uþ ðt x=cÞ þ u ðt þ x=cÞ:
(2)
The medium is initially at rest with mechanical properties, Young modulus E ¼ 200 GPa, Poisson’s ratio ¼ 0.33, and shear modulus G ¼ 75 GPa. A static normal stress r0 is applied to the solid. Let s be the shear stress and s the parti_ tÞ is the relative speed between the cle velocity. Then, uð0; solid and the interface. Finally, l ¼ 0:3 is the friction coefficient at the interface dXs . The Coulomb’s law is defined as follows 8 _ tÞ ¼ 0 < if jsð0; tÞj < lr0 ) sticking: uð0; in dXs : : _ tÞ 0 if jsð0; tÞj ¼ lr0 ) sliding: s:uð0; (3) The problem is tackled with a stress-velocity formulation. Using this formulation with Eq. (2) we obtain the shear stress: @u G ¼ ðu0þ ðt x=cÞ þ u0 ðt þ x=cÞÞ @x c ¼ sþ ðt x=cÞ þ s ðt þ x=cÞ;
sð x; tÞ ¼ G
A. Governing equations
@2u @2u c2 2 ¼ 0; 2 @t @x
FIG. 1. System used to study the interaction between a normal incidence shear wave and a contacting interface.
(4)
with the following expressions for both tangential stresses 8 G > > < sþ ðt x=cÞ ¼ u0þ ðt x=cÞ c (5) G > > : s ðt þ x=cÞ ¼ u0 ðt þ x=cÞ: c The derivatives are defined as 8 1 c > > _ tÞ sðt; xÞÞ < u0þ ðt x=cÞ ¼ ðuðx; 2 G 1 c > > 0 : u ðt þ x=cÞ ¼ ðuðx; _ tÞ þ sðt; xÞÞ: 2 G
(6)
As done in Ref. 16, the solution at the interface can easily be obtained analytically, since two criteria define the transition between sticking and sliding. This analytical solution consists of finding the instant of switch between the two modes, slipping and sticking. Consequently, it is a piecewise implementation that could be time consuming when the number of Blanloeuil et al.: Shear wave and frictional interface
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FIG. 2. Numerical procedure to solve the contact problem.
transitions increases. Therefore, a numerical solution of the problem is preferred in the present work and the implemented FD approach is presented in Sec. II B. B. Numerical implementation
The wave equation is solved in time using the classic second order FD approach. Coulomb’s law governs the contact as defined in Eq. (3). A transparent boundary condition enables the size of the computational domain to be limited and is defined at dXt as u0þ ðt þ L=cÞ ¼ f ðtÞ;
in dXt ;
(7)
where u0þ is given in Eq. (6) and f ðtÞ is the source term, equivalent to an imposed particle velocity. This equation defines the incident wave and is transparent to the reflected wave. This condition is rewritten into a strain formulation _ @u 2f ðtÞ uðL; tÞ ðL; tÞ ; on dXt : ¼ @x c c
(8)
This equation is discretized as follows: m m 3um 1 umþ1 um 2f ðtÞ 0 þ 4u1 u2 0 0 ¼ ; 2dx dt c c
on dXt ; (9)
where dx and dt are, respectively, the space step and the time step. The wave equation [Eq. (1)] is discretized in X using the classic Euler FD scheme, with the indices “m” and “k” denoting time and space discretization, respectively, m m1 um 2um umþ1 2um k þ uk1 k þ uk k c2 kþ1 ¼ 0; 2 2 dt dx 8k 2 ½2; n 2; in X:
(10)
This second order general scheme is conditionally stable under the classic Courant-Friedrichs-Lewy (CFL) condition: cðdt=dxÞ < 1. Using Eqs. (8), (10), and (3) we can build two J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
matrices, Ast for the sticking condition and Asl for the sliding condition. The next time solution is computed by a matrixvector product like U mþ1 ¼ Ast;sl U m þ B, where B contains the excitation source term. The algorithm representing the numerical implementation is shown in Fig. 2. C. Numerical results
The convergence through grid size and time step is ensured by comparing the numerical solution at the interface with the analytical solution described in Sec. II A at dXs . In the following results, the simulation is run at 0.7 CFL and the space discretization is dx ¼ k3x =24, where k3x is the wavelength of the third harmonic. These values of the numerical parameters ensure the convergence of the model. The incident wave is a five cycle sinusoidal signal modulated by a Gaussian window. Figure 3(a) illustrates the solution at x ¼ L, showing the incident and the reflected wave. The reflected wave is then windowed and transformed to the frequency domain. Its spectrum is shown in Fig. 3(b). These curves are normalized using the value of incident harmonic Ax . The reflected wave spectrum contains odd higher harmonics, A3x and A5x , as expected. Since the nonlinearity appears when sliding with friction occurs, we define a dimensionless parameter n from the Coulomb’s law [as given in Eq. (3)] to easily compare the results n¼
lr0 2ksincident k1
:
(11)
Therefore, when n ¼ 0, there is no friction and no dissipation of energy. When n 1 the interface remains stuck and no motion occurs. Thus the nonlinear frictional behavior will only occur within these bounds. Having determined the basic numerical procedure, the influence of the normal load and the incident amplitude on the evolution of the third harmonic is studied. The harmonic amplitudes are obtained by applying bandpass filters at the first and third harmonic in post processing. The time signal Blanloeuil et al.: Shear wave and frictional interface
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envelope of these time signals give the value of each harmonic. The third harmonic is normalized using the incident harmonic size, At3 ¼ At3x =Atxincident . Figure 4 shows the evolution of the third harmonic in the time domain for three different input amplitudes. The curves are plotted for different magnitudes of incident wave as a function of n. The result is that the response is plotted for each input amplitude over a range of normal stresses. The same peak of At3 at n ’ 0:45 is observed irrespective of the input amplitude, confirming the relevance of the dimensionless parameter chosen. This maximum value of A3 occurs at the peak of nonlinearity and is directly related to the friction coefficient at the interface. This result is in agreement with Ref. 17 and suggests that measurements of the third harmonic can be used to quantify friction and dissipation effects at a sliding interface. We can define the energy transported by a wave as ð tfin _ tÞdt; sðx; tÞ:uðx; (12) EðxÞ ¼ tini
FIG. 3. (a) Time solution at x ¼ L and (b) the associated spectrum for the incident and reflected wave. The spectrum is normalized using the incident harmonic amplitude Ax . The fundamental frequency fx ¼ 1 MHz, normal stress r0 ¼ 2 MPa, incident amplitude A ¼ 2.5 nm, friction coefficient l ¼ 0:3 and 1200 space points are used at 0.7 CFL.
is converted in the frequency domain using a FFT (Fast Fourier Transform). The parts of the spectrum corresponding respectively to the first and the third harmonic are then separately transformed back to the time domain using an IFFT (Inverse Fast Fourier Transform). The amplitudes of the
FIG. 4. Evolution of At3 relative to n for different incident amplitudes. The normal load varies from 0.5 to 4 MPa. Fundamental frequency fx ¼ 1 MHz and the friction coefficient l ¼ 0:3. 1712
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where s is the shear stress and u_ is the speed at point x. For x ¼ 0, this energy corresponds to the dissipation of energy by friction. Figure 5 shows the evolution of the dissipated energy against n for the same three configurations as described in Fig. 4. These curves are normalized by the incident energy defined by Eq. (12). Obviously, the bigger the input amplitude is, the bigger the dissipation of energy by the frictional interface is. However, the relative change remains constant as shown in the normalized data of Fig. 5. For each case, the peak value is around n ’ 0:33. This result suggests the existence of a link between the third harmonic A3x and the amount of energy dissipated by friction. However the exact relationship is unclear at this point. Finally, we look at the energy balance between the incident and the reflected wave. Figure 6 shows the evolution of the dissipated energy, the total reflected energy, and the reflected energy at the first, third, and fifth harmonics. As expected, the maximum dissipated energy corresponds with the minimum of reflected energy. Moreover we can verify that the model is conservative since
FIG. 5. Evolution of dissipated energy by friction for three input amplitudes. Blanloeuil et al.: Shear wave and frictional interface
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FIG. 6. Transfer of energy from the incident wave to the reflected wave and dissipation by friction.
the sum of the reflected and dissipated energy equals the incident energy. It is worth noting that the simple model of friction employed here is responsible for this complex behavior. In order to determine if this model is sufficient, experimental validation of the behavior is essential. To conclude, a normal stress r0 2 ½0:4; 4 MPa and an input amplitude A 2 ½1; 8 nm should provide a large enough parameter space to study the evolution of A3 with a friction coefficient of 0.3. This simple numerical model can provide quantitative evaluation as regards to third harmonic generation and dissipation of energy. An experimental study was carried out to validate this behavior, informed by the range of values determined here. III. EXPERIMENTAL INVESTIGATION
The experimental setup is illustrated schematically in Fig. 7. Two steel blocks are brought into contact and loaded
in compression. The force is applied using a threaded rod and nuts to ensure a constant load. This force is monitored through a load-cell situated between the screw thread and the samples. In practice the system is mounted horizontally to allow the application of loads lower than the block weight. A 25.4 mm (1 in.) broadband shear transducer of center frequency 1 MHz (V152, Panametrics, Waltham, MA) sends wave pulses into the system at location T1. A 12.7 mm (1/2 in.) transducer of center frequency 5 MHz (V155, Panametrics, Waltham, MA) is used as a receiver at location T2. A personal computer controls the process of excitation and reception. The input signal is generated by a wave generator (33120A, Agilent Tec., Santa Clara, CA) and the received signal is measured on an oscilloscope (LT224, Lecroy, Chestnut Ridge, NY) with both communicating with the computer over a GPIB interface. The excitation is a 15 cycle pulse which is amplified (GA-2500A, Ritec Inc., Warwick, RI) and then sent to the transducer. A longer signal than in the numerical modeling is used to ensure the best chance of measuring the generated nonlinearity. The two steel blocks used have dimensions of 46 46 55 mm3 and were machined to ensure parallel faces, with a dimensional tolerance estimated to be 0.4 mm. The contacting faces were then polished with P80 and P180 paper to produce an interface of known quality and with an average roughness of Ra ¼ 0:4 lm. The amplitude of the incident shear wave at the interface, effectively giving shear stress, is of prime importance when using the dimensionless parameter n defined in Eq. (11). This was measured by transmitting an input wave into one of the steel blocks. The response was then measured on the parallel face using a laser vibrometer (OFV 353, Polytec, Waldbronn, Germany). The measurement was made at an angle of 45 to ensure sensitivity to the largely in plane shear mode and the measured displacement corrected to give the true amplitude. This was repeated for both input and receiving transducers at the frequencies of interest, namely input, and third harmonic. This process ensures that the values that are measured in the later load experiments can be directly converted to the amplitude at the interface, without having to specifically correct for the effects of attenuation or beam spreading, increasing the confidence in the results. The same procedure was repeated to calibrate the input amplitude that the amplifier would produce when set using the amplifiers arbitrary amplitude scale. The resulting amplitudes are shown in Table I. Although the surfaces are polished, there still remain asperities that could interfere in the contact mechanism when the two blocks are first put into contact. In order to ensure the reproducibility of the measurements made, the contacting blocks are subjected to three loading cycles prior to ultrasonic measurements being made. The laboratory TABLE I. Input displacement A at the interface against arbitrary amplifier amplitude scale. The measure is done on a single block with a laser vibrometer measurement for each graduation of the amplifier scale.
FIG. 7. (Color online) Experimental setup for bulk transverse wave transmission through a plane interface. J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
Ritec A (nm) Ritec A (nm)
3 0.12 6.5 2.10
3.5 0.14 7 3.39
4 0.17 7.5 5.17
4.5 0.25 8 7.09
5 0.47 8.5 9.02
5.5 0.72 9 10.9
6 1.23
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environment is constant on the scale of the measurement time so a high degree of signal averaging was performed (1000 averages) in order to maximize the signal to random noise level. The evolution of the third harmonic as a function of the n parameter was then measured, paying special care to ensure a good range of measurements was made between 0 and 1 where the nonlinearity is expected. This was done using the parameters generated from the earlier numerical model. The experimental procedure consists first in increasing the normal load while the input amplitude is at its highest level; that is 10.9 nm at the interface. For this amplitude, the load was increased from 0.6 to 9.5 kN in 24 steps, which results in a normal stress r0 2[0.3;4.5] MPa, assuming plane strain. The peak load of 9.5 kN force is then maintained and the input amplitude is decreased from 10.9 to 0.12 nm in the 13 steps shown in Table I. In doing so, 36 increasing values of n have been obtained. For each of them, the amplitudes of the first and third harmonic have been evaluated, leading to 36 independent values. For the record, the definition of n is n ¼ lr0 =j2rInc j. Figure 8 shows an example of a received time signal with three different wave arrivals clearly visible. The first wave is a longitudinal wave. The second one corresponds to a longitudinal wave converted from the incident shear wave at the interface. The third is the shear wave that is the primary interest of this paper. Each measure is processed as described in Sec. II to extract the third harmonic. The amplitudes obtained outside of the temporal window give the noise level associated with each harmonic. Finally, these data are converted from Voltage (V) to displacement (m) using the calibration values obtained as described above. In order to use the nondimensional parameter defined in Eq. (11) the static friction coefficient at the interface must be measured. This was done by measuring at what angle relative to the horizontal the two blocks began to slip and resolving the forces to determine the friction coefficient. Though primitive this was repeated ten times and found to give the same slip angle within 1 . The resulting coefficient of friction used was 0.3. For each measurement of load and amplitude at interface, the n value is computed. Normal stress is defined as the ratio of the applied force over the surface of contact.
FIG. 8. Received signal for a normal load of 6.5 kN and displacement at the interface of 10.9 nm. 1714
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Incident shear stress of the wave is computed using the calibrated incident amplitude, and the measured coefficient of friction was used. Repeat measurements were made at the high and low end of the load and interface displacement amplitude with a range of 5%. A value that will be seen in Sec. IV is sufficient to ensure that the measured results are clear of the noise floor. The degree of nonlinearity input to the system increases with amplitude as expected. To prevent this effect masking the changes in nonlinearity at the interface, the input nonlinearity must be removed. Therefore, for the data corresponding to high values of n ðn > 2:5Þ where the system is not exhibiting nonlinear contact behavior, we determine a scale and factor a that minimizes the gap between A1 =Amax 1 for the measured data. In this case, a is such that, A3 =Amax 3 A3 A3 A1 ¼ ¼ a : (13) Amax Amax Amax 3 3 1 n>2:5 n>2:5 n>2:5 |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} measured
system
Thus, the final normalized value of the third harmonic due to contact nonlinearity is A3 A1 corrected ¼ a max Amax (14) A3 3 : Amax A1 3 Since the nonlinearity generated at the interface is of a different origin to that produced by the amplifier this will not be suppressed. For each starting input amplitude, this analysis is applied to the data. When the amplitude is small (0.47 nm, for example), there is no difference between A3 and A1 as shown in Fig. 9. Therefore, the corrected A3 is zero. The nonlinearity is not active at the interface since the input amplitude is not large enough to trigger sliding. The same treatment is applied to the data that start at the highest input amplitude, 10.9 nm. For high values of n, the trend of A3 is exactly the same as A1 , as shown in Figure 10. However, for low n between 0 and 1, a difference exists. Without removing the effects of the input nonlinearity, the true change in the region of interest is impossible to see.
FIG. 9. Correction of A3 by subtraction of A1 trend for an initial interface amplitude of 0.47 nm. The amplitude of the incident wave is not large enough to trigger sliding. Blanloeuil et al.: Shear wave and frictional interface
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FIG. 10. Correction of A3 by subtraction of A1 trend for an initial interface amplitude of 10.9 nm.
After subtraction of the linear trend there is still a clear increase in the harmonic value between 0 n 1, indicating that the third harmonic is greater than the inherent system nonlinearity (electronics and material), and is therefore due to the nonlinear effect at the contacting interface. The corrected A3 defined by Eq. (14) for a large interface amplitude (10.9 nm) is shown in Fig. 11. Included as a solid line is a comparison with the numerical result described earlier. In this example the parameters used in the numerical model are the same as those measured experimentally for the real system. The numerical results show a good qualitative match with the experimental results. The nonlinearity is clearly evident for n 2 ½0; 1 and the maximum of the third harmonic amplitude is observed at n ¼ 0.4, both for the experimental and numerical results. IV. DISCUSSION AND CONCLUSION
This study presents a numerical and experimental analysis of the propagation of a shear wave through a frictional interface with the express goal of using the generated nonlinearity to characterize the interface. A simple FD model is developed that describes the case of a classical Coulomb friction law and
a plane wave. This model gives the third harmonic behavior in the dimensionless parameter n space, with a maximum at n ¼ 0.4. This response corresponds to a specific signature of sliding with friction resulting in contact nonlinearity. Third harmonic behavior shows an explicit dependence on frictioninduced dissipated energy. The results are in accordance with previous studies16,17 and are used to determine suitable parameters for the experimental investigation. An experimental procedure to validate these numerical results was developed. Good agreement between the experimental and numerical results is demonstrated leading to a twofold conclusion. First, classical Coulomb’s law included in the model provides a sufficient description of friction during sliding at the contact interface. Therefore, this law can be used in more complex FE models like solids, including cracks. Second, an estimation of the friction coefficient can be made by fitting numerical curves to the experimental data through adjusting the numerical parameter n. Finally, when an acoustic wave with oblique incidence interacts with a crack, the sliding can be activated even with longitudinal waves. This work provides an experimental justification for the use of a Coulomb law to describe friction. The specific behavior of the third harmonic due to frictional effects may be useful to detect and identify closed-cracks or characterize interfaces exhibiting sliding with friction. Although potentially offering a unique way to measure the contact nonlinearity and the coefficient of friction the approach does present significant difficulties. Very high input amplitudes are necessary to activate the sliding mechanism and without these, no contact nonlinearity could be measured. This was made more complex in this specific case where the testing frame was unable to support much higher loads. The primary advantage of this approach relative to say conventional generation of second harmonics at a clapping interface is in the ability to remove the effects of input nonlinearity; that is, a measurement of nonlinearity can essentially be made at the same time as the interface nonlinearity. This is not possible with a conventional second harmonic generation approach. In comparison to the second harmonic approach the method presented here does not have the same sensitivity when a crack is open; however, when closed (by a load say) it may have greater sensitivity. This is, however, complicated by the potentially high coefficients of friction in such a scenario preventing the interface nonlinearity activating. Further work is necessary to determine the usefulness of this novel approach in these cases. ACKNOWLEDGMENTS
The authors would like to thank Christine Biateau and Philippe Malerne for their technical support and valuable comments.
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FIG. 11. Evolution of the third harmonic with increasing n measured experimentally and calculated numerically. The amplitude of the numerical result is fitted to the experimental data. J. Acoust. Soc. Am., Vol. 135, No. 4, April 2014
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Blanloeuil et al.: Shear wave and frictional interface
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