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Higher harmonics induced in lamb wave due to partial debonding of piezoelectric wafer transducers
NDT&E International 63 (2014) 21–27
Contents lists available at ScienceDirect
NDT&E International journal homepage: www.elsevier.com/locate/ndteint
Higher harmonics induced in lamb wave due to partial debonding of piezoelectric wafer transducers Nitesh P. Yelve n, Mira Mitra, P.M. Mujumdar Department of Aerospace Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India
art ic l e i nf o
a b s t r a c t
Article history: Received 1 July 2013 Received in revised form 28 November 2013 Accepted 21 January 2014 Available online 29 January 2014
Piezoelectric wafer (PW) transducers used for Lamb wave actuation may get partially debonded from the host structure, because of their prolonged use, excessive voltage supply, or improper bonding onto the host structure. In this paper, higher harmonics induced in Lamb wave because of such debonding of the PW actuator are studied both experimentally and through finite element simulation. In experiments, an artificial partial debond is created while bonding the actuator patch onto a pristine aluminium plate. Lamb wave transduced by this actuator in the plate is picked up by a PW sensor which does not have any debonding. In FE simulation, Augmented Lagrangian algorithm is used to solve the contact problem at the breathing debond. Three higher harmonics are observed in the experiments and also in the FE simulation. To ensure that the generated higher harmonics correspond to Lamb wave, time–frequency analysis is carried out using Morlet wavelet transform, and the results are reported in the paper. Spectral damage index (SDI), obtained from spectral attributes of first four harmonics in experiments and simulation, is found to be decreasing with an increase in debonding area. This shows that actuator debonding introduces contact nonlinearity which induces higher harmonics in Lamb wave. Therefore, in damage detection using Lamb wave based nonlinear techniques, the higher harmonics produced may get influenced by the false higher harmonics produced by actuator debonding, leading to incorrect results, if bonding of the actuator is not taken care of properly. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Piezoelectric wafer transducer debonding Contact nonlinearity Lamb wave Higher harmonics Morlet wavelet transform Spectral damage index
1. Introduction In material characterization [1,2] or damage detection [3–6], using Lamb wave based nonlinear techniques, the presence of higher harmonics is used as a measure of evaluation of the material or damage state. Nonlinear elasticity spread in the material continuum and local contact nonlinearity are the two mechanisms, which give rise to higher harmonics in Lamb wave [7]. The stronger the nonlinearity, the larger is the energy transferred from the fundamental harmonic to the higher harmonics and therefore an increase in the number and amplitude of higher harmonics is observed. In such nonlinear techniques used for material characterization or damage detection, piezoelectric wafer (PW) transducer is a good candidate for actuating Lamb wave in the structure because of its portability, small size and cost effectiveness. These characteristics of PW transducer make it suitable for in situ applications. Higher harmonics generation, in the PW transduced Lamb wave, in an aluminium plate with a breathing crack is studied in Reference [6]. Three higher harmonics are observed in this study
n
Corresponding author. E-mail address: [email protected] (N.P. Yelve).
0963-8695/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ndteint.2014.01.004
and time–frequency analysis using Morlet wavelet transform showed that these higher harmonics do lie on the dispersion curves and hence correspond to Lamb wave. The PW transducer excites both symmetric and anti-symmetric modes of Lamb wave and therefore it is able to generate both odd and even higher harmonics. This is shown both analytically and experimentally in Reference [8], in a quasi-statically loaded aluminium plate, with the aid of macro-fiber composite (MFC) piezoelectric patches. An important issue with PW transducer is its debonding from the host structure because of prolonged use, excessive deformation caused by high voltage supply, or improper bonding. Contact nonlinearity may prevail in this case, leading to higher harmonics in the Lamb wave response. This may show illusory presence of defect in a pristine material. This is proved both experimentally and through finite element (FE) simulation in the present paper. In experiments, an artificial partial debond is created in a PW actuator while bonding it onto a thin aluminium plate. In the simulation, partial breathing debonding is modelled between patch and plate by assigning contact attributes to the contacting faces [6]. Augmented Lagrangian algorithm [9] is used to implement contacting conditions at the meeting faces. The wave generated by such a debonded PW actuator, in both experiments and simulation, is picked up by the perfectly bonded PW sensor. The signal shows the presence of three higher harmonics. To
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N.P. Yelve et al. / NDT&E International 63 (2014) 21–27
ensure that the generated higher harmonics belong to Lamb wave, time–frequency analysis of both experimental and simulation results is carried out using Morlet wavelet transform and the results of this analysis are reported in the paper. Further, a spectral damage index (SDI) [6] is extracted from spectral attributes of higher harmonics for various debonding sizes. Variation of this SDI with increasing debonding size is found to be decreasing in experimental as well as simulation studies. This indicates that PW actuator debonding is very much susceptible to induce higher harmonics in Lamb wave which may lead to show illusory presence of defect in a pristine material. Therefore good care should be taken while bonding the PW transducers, and bonding should be always checked using some appropriate techniques. Patches may even be replaced after their use over a certain duration.
2. Experimental work The experiments are performed on a pristine aluminium plate, with the objective of studying higher harmonics induced in Lamb wave as a result of debonding in PW transducer. The size of PW transducers used here is 10 mm 7 mm 0.5 mm and material type is SP-5H. Properties given by the manufacturer [10] for these transducers are mentioned in Table 1. The material of the plate used here is aluminium (grade 5052-H32) with modulus of elasticity 70.3 GPa, Poisson's ratio 0.33, and density 2680 kg/m3 [11] and its size is 1200 mm 1200 mm 1.6 mm. Cyanoacrylate adhesive is used for bonding PW transducer onto the plate. The sensor patch is bonded completely whereas actuator patch is partially bonded so as to introduce artificial debonding in it, as shown in Fig. 1. Three such actuator-sensor sets are experimented with different debondings of the actuators. In these three sets, the actuators are debonded over their entire width, and 10%, 20%, and 30% of their lengths at the centre as shown in Fig. 1. Therefore, the debonded area of three actuators is 10%, 20%, and 30% of total actuator area available for bonding. Cyanoacrylate adhesive is a fast solidifying adhesive, therefore its curing time is very less. Because of this, the chances of spreading the adhesive beyond the required bonding area of the PW transducer are very less. Also, the limiting ink lines made on the bonding side of the PW transducer further prohibits the spreading of the adhesive. The size of debonding in the transducers is checked by removing them after completing the experiments. The debonding is found to be in close limits of accuracy. Table 1 Properties of SP-5H piezo wafers [10]. Properties Piezoelectric coupling coefficients kp k33 Piezoelectric charge constants d33 10 12 C=N d31 10 12 C=N Piezoelectric voltage constants g 33 10 3 V m=N g 31 10 3 V m=N Relative dielectric constant kT3 Density ρ kg=m3 Elastic constants
Values
0.63 0.73 550 265 19 9
Fig. 1. Schematic of patch debonding.
The experimental setup as shown in Fig. 2 consists of arbitrary function generator (Make: Tektronix, Model: 3021B), high speed bipolar amplifier (Make: NF, Model: BA4825), digital storage oscilloscope (Make: Tektronix, Model: 1002B), and a computer. A 8.5 cycle sine wave tone burst signal windowed by Gaussian function is generated in MATLAB&. The Gaussian window of length N is defined by the equation [12]
ωðnÞ ¼ e½ ð1=2Þ½αðn=ððN 1Þ=2ÞÞ ; 2
where
N1 N1 rn r ; 2 2
SE11 10 12 m2 =N
21
SE33 10 12 m2 =N
15
ð2Þ
and α is inversely proportional to the standard deviation of Gaussian random variable. In the present case N ¼251 and α ¼ 2:5. This tone burst is then input to the function generator with a resolution of 40,000 points as shown in Fig. 3a, and set at frequency 170 kHz (Fig. 3b). Using the amplifier, this tone burst is magnified to 7100 V and given to the debonded PW actuator. The received Lamb wave at the PW sensor is shown by the oscilloscope and the corresponding data are sent to the computer for further signal processing. Fig. 4a–c shows the input signal and the output signals in time domain corresponding to the case of the perfectly bonded and the 30% debonded PW actuator. The received time domain data is cluttered and it is difficult to make any clear interpretation from it. Frequency domain analysis is useful to get the information about the nonlinearity induced in Lamb wave, in terms of higher harmonics. Here, the output time domain data is converted into the frequency domain using the discrete Fourier transform (DFT) given as [13] M1
XðkÞ ¼ ∑ x½qe½ j2π qk=M q¼0
3100 7500
ð1Þ
ð3Þ
where k ¼ 0 : M 1;
ð4Þ
and M is the number of data points, which is 2048 in the present case. For this purpose the Fast Fourier Transform (FFT) algorithm is
N.P. Yelve et al. / NDT&E International 63 (2014) 21–27
23
Fig. 2. Experimental set-up.
relatively lower excitation frequency of 170 kHz where no situation of identical group and phase velocities exists. Therefore it can be concluded that the higher harmonics observed are solely because of the contact nonlinearity introduced by the actuator debonding. Such higher harmonics are also observed in the case of 10% and 20% debonding. It is necessary to check whether the signal picked up at the PW sensor is corresponding to Lamb wave. For this purpose time– frequency analysis of the received signal is done and arrival times of frequencies are identified. Continuous Wavelet Transform (CWT) is used for this purpose as its resolution is better than short time Fourier transform [14]. The CWT of a time domain signal x(t) is given as [14,15] Z þ1 1 t b Tða; bÞ ¼ pffiffiffiffiffiffi dt; ð5Þ xðtÞψ a jaj 1 where a is the dilation parameter, b is the translation parameter and ψ ðtÞ is a wavelet basis function. In the present case Morlet basis function is used, according to which [15,16]
ψ ðtÞ ¼
1
eðiω0 tÞ e ðt
2
=2Þ
;
ð6Þ
π ð1=4Þ where ω0 is the central frequency parameter and is taken as 10 to
Fig. 3. Input signal used in experiments. (a) Input signal in time domain. (b) Frequency spectrum of input signal.
used in MATLAB& with sampling frequency of 5 MHz. Fig. 5a and b respectively shows the frequency spectrum of the signal received in the case of the perfectly bonded PW actuator and the 30% debonded actuator. From Fig. 5a it can be seen that the received wave response shows no higher harmonics in the case of the perfectly bonded actuator. However, in the case of the 30% debonded actuator, Fig. 5b shows the presence of three higher harmonics at 340 kHz, 510 kHz, and 680 kHz. This means that the observed higher harmonics are not generated because of instrumentation nonlinearity. Further, at certain high excitation frequencies, higher harmonics are seen in the wave response as a result of nonlinear elasticity spread in the material, when the condition of identical group and phase velocities exists for the harmonics [4]. The current experiments are carried out at a
satisfy the admissibility condition [15,16]. Morlet wavelet transform for no debonding case and 30% debonding case, with dispersion plots of A0 and S0 modes of Lamb wave, is shown in Fig. 6a and b respectively. MATLAB& is used for this purpose. In Fig. 6, arrival times of frequency components present in the response signal are compared with those of A0 and S0 modes of Lamb wave [17]. The Morlet wavelet plot in Fig. 6a shows frequency 170 kHz, and in Fig. 6b shows frequencies 170 kHz and 340 kHz, contained in the response signal. They all lie close to the dispersion curves. Thus, from these two figures it can be said that the no debonding case shows only A0 and S0 modes of Lamb wave at fundamental frequency in the response signal and no higher harmonics, whereas in the case of 30% debonding case, the frequency 170 kHz corresponds to A0 and S0 modes of Lamb wave, and the second harmonic 340 kHz corresponds to S0 mode of Lamb wave. Third and fourth harmonic frequencies are not visible in Fig. 6b as their amplitudes are feeble compared to that of the fundamental frequency. Similar time frequency plots are observed in the case of 10% and 20% debonding.
3. Finite element simulation In order to study higher harmonics induced in Lamb wave due to PW transducer debonding through finite element (FE) simulation, the most decisive issue is modelling the debonding between the plate and the PW actuator patch which can closely approximate
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N.P. Yelve et al. / NDT&E International 63 (2014) 21–27
Fig. 5. Experimental frequency domain results. (a) No debonding case. (b) 30% debonding case.
Fig. 4. Experimental time domain data. (a) Input tone burst. (b) Output in the case of no debonding. (c) Output in the case of 30% debonding.
the physical contact between the plate and the patch faces. ANSYS© is used here for FE simulation. The aluminium plate with PW actuator and sensor for transmitting and receiving Lamb wave respectively, is modelled in the thickness plane with plane strain approximation. This approximation saves computational time without affecting the trend of results significantly [18]. Properties assigned to the plate and the transducers are same as that mentioned in the previous section. PLANE82 (8-node quadrilateral) elements are used for plate and PLANE223 (8-node quadrilateral coupled-field) elements are used for PW patches. Selection of element size is crucial in FE simulation of wave propagation. The size of the elements must be sufficiently small for there to be at least 10 nodes per wavelength of the propagating wave [19]. The excitation frequency used in the study is 170 kHz and its fourth harmonic, 680 kHz is expected to be generated as a result of
contact nonlinearity at patch debond. Therefore the mesh size should be small enough to be able to capture 680 kHz. Only A0 and S0 modes of Lamb wave are involved in the analysis and A0 mode has the smaller wavelength at 680 kHz which is 3.075 mm. Accordingly, the distance between two nodes should not be more than 0.3 mm and as the elements used in the simulation have midside nodes, the element size should not exceed 0.6 mm. In the simulation study, a fine discretized mapped mesh is used, with the element size of 0.5 mm which is less than the threshold mesh size 0.6 mm required to capture the fourth harmonic of 170 kHz. The nodes on the free and the bonded faces of PW transducers are coupled to form master nodes on the respective faces. Therefore, in the case of actuator the voltage can be applied at the master node on the free face, and in the case of sensor, the voltage can be recorded from the master node on the free face. The PW transducer grounding is done by applying zero voltage at the master nodes on the bonded faces. Newmark algorithm is used here for time integration in FE analysis with α ¼0.25 and δ ¼0.5 where α and δ are the Newmark parameters. For these values of α and δ the Newmark algorithm is unconditionally stable [19]. In this case, the time step size to be used is based on the trade-off between the desired accuracy and the computational efforts. The accuracy in the integration can be stated in terms of period elongation and amplitude decay [19]. In the FE simulation of Lamb wave propagation, both period elongation and amplitude decay affect the group velocity. Therefore a convergence study is carried out to decide the appropriate time step for solution so that the group velocities of A0 and S0 Lamb wave modes obtained through simulation are in good agreement with those obtained analytically while requiring optimal simulation efforts both in terms of time of simulation and space required
N.P. Yelve et al. / NDT&E International 63 (2014) 21–27
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Fig. 6. Morlet wavelet transform of the experimental output data. (a) No debonding case. (b) 30% debonding case.
Fig. 8. Simulation time domain data. (a) Input tone burst. (b) Output in the case of no debonding. (c) Output in the case of 30% debonding.
Fig. 7. Deformed state of PW actuator in FE showing patch debonding.
in the computer. Accordingly, the time step of 0:2 μs is used for the simulation. CONTA172 (3-node surface-to-surface contact) and TARGE169 (target segment) elements are assigned to the faces of the PW
patch and the plate respectively over the debonded area to model the contact nonlinearity. Fig. 7 shows the deformed state of the PW actuator in FE, which clearly shows the patch debonding. Augmented Lagrangian (AL) contact algorithm under frictionless condition is used here to solve the contact problem at the debonding. It is an iterative series of penalty updates to find the contact tractions. It offers advantages such as more penetration control, better conditioning of governing equations, and satisfaction of constraints with finite penalties [9]. In this case the contact pressure R is defined by [20] ( R¼
0
if up 40;
K p up þ λi þ 1
if up r0;
ð7Þ
26
where
N.P. Yelve et al. / NDT&E International 63 (2014) 21–27
(
λi þ 1 ¼
λi þ K p up if jup j 4 ɛ; λi if jup j o ɛ;
ð8Þ
and Kp is the normal contact stiffness, up is the contact gap size, λi is the Lagrange multiplier at ith iteration, ɛ is the compatibility tolerance. The Lagrange multiplier component λi is computed locally i.e. for each element and iteratively. In AL treatment of frictionless contact of debonding faces, the admissible deformation of prospective points of contact satisfies [9] hðxÞ r 0;
ð9Þ
t S ¼ rðxÞ PS Z0;
ð10Þ
t S ðxÞhðxÞ ¼ 0;
ð11Þ
t R ¼ PS þ t S r ¼ 0;
ð12Þ
where h is a scalar-valued gauge function, P is the first Piola– Kirchoff stress tensor, r is the outward normal in the current configuration, S is the outward normal in reference configuration. Eq. (9) represents the condition of impenetrability, Eq. (10) represents the restriction that the normal component of surface traction be compressive, and Eq. (11) is a condition ensuring that tS may only be nonzero when hðxÞ ¼ 0. Eqs. (9)–(11) are therefore recognized as the Kuhn–Tucker conditions. Eq. (12) asserts that no friction is present. Parameters required for solving this frictionless contact problem in FE using AL contact algorithm are normal contact stiffness and maximum allowable penetration. Normal contact stiffness is based on Young's modulus and the size of the elements underlying
Fig. 9. Simulation frequency domain results. (a) No debonding case. (b) 30% debonding case.
the contact elements. Value of maximum allowable penetration considered for the analysis is 0.1 times the thickness of underlying elements because at its large value the AL method works like the Penalty method [20] which is not desirable here. Gaussian input signal of 8.5 cycles, 7100 V having central frequency 170 kHz is given to the debonded PW actuator. The generated wave is picked up at the PW sensor which does not have any debonding. The input and output time domain data in the case of the perfectly bonded PW actuator and 30% debonded PW actuator are shown in Fig. 8. Frequency domain plot of the signal received at the sensor for the case of no debonding is shown in Fig. 9a. It does not show any higher harmonic. Whereas, the Fourier transform of the signal received in the case of 30% debonded PW actuator is shown in Fig. 9b. It clearly shows the presence of three higher harmonics at 340 kHz, 510 kHz, and 680 kHz similar to that obtained in experiments. Such higher harmonics are also observed in the case of 10% and 20% debonding. From Figs. 5 and 9 it can be seen that amplitudes of higher harmonics observed in the experiments and in the FE simulation respectively vary to some extent. This is because, FE simulation is based on the plane strain assumption [18]. Morlet wavelet transform is used to plot the arrival times of the frequency components contained in the output signal, in the case
Fig. 10. Morlet wavelet transform of the simulation output data. (a) No debonding case. (b) 30% debonding case.
N.P. Yelve et al. / NDT&E International 63 (2014) 21–27
27
response of a pristine aluminium plate. Time–frequency analysis using Morlet wavelet transform shows that the generated higher harmonics are also Lamb wave. These results emphasize that observed higher harmonics are induced in Lamb wave because of contact nonlinearity at the breathing debond. These higher harmonics are considerably sensitive to the patch debonding, which is confirmed by the decreasing trend of spectral damage index with the increasing debond area. This is a serious issue as it may show illusory presence of defect in a pristine material. Also, this may influence the higher harmonics occurring because of a defect in the material and hence may affect the accuracy and reliability of the results. Therefore it is important to bond the PW transducers very carefully onto the structure and also to check the bond quality often by using some appropriate techniques. Solution could also be to replace the patches after their use over a certain span of time. Fig. 11. Variation of SDI with debond area.
of completely bonded, and partially debonded PW actuators. These arrival times are then compared with those of A0 and S0 modes of Lamb wave. The results are shown in Fig. 10 for completely bonded and 30% debonded cases and they conform well with those obtained in experiments. Similar results are observed in the case of 10% and 20% debonding.
Acknowledgements The authors gratefully acknowledge the financial support by the Aeronautics Research and Development Board, Government of India for this research work (Project no. 1642).
References 4. Estimation of spectral damage index In the present work, to estimate the dependency of higher harmonics generation on the extent of PW transducer debonding, the spectral damage index (SDI) [6] proposed earlier by the authors of this paper is extracted from the spectral attributes of higher harmonics and its variation with the debond size is studied. For this purpose, first four harmonics are considered. These four harmonics are mapped between one and four and their amplitudes are normalized with respect to the fundamental harmonic observed in the case of the completely bonded PW actuator. Peaks of these four harmonics are joined by a series of lines in frequency domain plot, which is known as spectral envelop [6]. As debond area increases the contact nonlinearity increases which eventually causes energy transfer from fundamental harmonic to higher harmonics. This is evident from the fact that as debond area increases amplitudes of higher harmonics increase with simultaneous decrease of fundamental harmonic amplitude, as observed in the present work. This changes the angle θ between the bilinear fit [6] of spectral envelop and the change in the angle depicts the extent of debonding. The SDI is defined as tan θ [6]. This index is obtained for 10%, 20%, and 30% debonding in PW actuator in both experiment and simulation as shown in Fig. 11. It can be seen from this figure that as debond area increases, the SDI decreases which clearly depicts the dependence of higher harmonics and in turn contact nonlinearity on PW actuator debonding. 5. Conclusion In the present experimental and simulation studies, an artificially created partial debonding of the PW actuator is found to be showing presence of three higher harmonics in Lamb wave
[1] Deng M, Yang J. Characterization of elastic anisotropy of a solid plate using nonlinear Lamb wave approach. J Sound Vib 2007;308:201–11. [2] Bermes C, Kim J, Qu J, Jacobs L. Nonlinear Lamb waves for the detection of material nonlinearity. Mech Syst Signal Process 2008;22:638–46. [3] Deng M, Pei J. Assessment of accumulated fatigue damage in solid plates using nonlinear Lamb wave approach. Appl Phys Lett 2007;90 121902 1–3. [4] Pruell C, Kim J, Qu J, Jacobs L. Evaluation of plasticity driven material damage using Lamb waves. Appl Phys Lett 2007;91 231911 1–3. [5] Pruell C, Kim J, Qu J, Jacobs L. Evaluation of fatigue damage using nonlinear guided waves. Smart Mater Struct 2009;18 035003 1–7. [6] Yelve N, Mitra M, Mujumdar P. Spectral damage index for estimation of breathing crack depth in an aluminum plate using nonlinear Lamb wave. Struct Control Health Monit, http://dx.doi.org/10.1002/stc.1604, in press. [7] Jhang K. Nonlinear ultrasonic techniques for nondestructive assessment of micro damage in material: a review. Int J Precis Eng Manuf 2009;10 (1):123–35. [8] Srivastava A, DiScalea F. On the existence of antisymmetric or symmetric Lamb waves at nonlinear higher harmonics. J Sound Vib 2009;323:932–43. [9] Simo J, Laursen T. An augmented Lagrangian treatment of contact problems involving friction. Comput Struct 1992;42(1):97–116. [10] 〈http://www.sparklerceramics.com/piezoelectricproperties.html〉. [11] ASM Handbook. Properties and selection: nonferrous alloys and special purpose materials, vol. 2. Materials Park, OH: ASM International; 1992. [12] Smith J. Spectral audio signal processing. California: W3K Publishing; 2011. [13] Yang W, Cao W, Chung T, Morris J. Applied numerical methods using MATLAB&. New Jersey: John Wiley & Sons, Inc. Publication; 2005. [14] Goswami JC, Chan AK. Fundamentals of wavelets: theory, algorithms, and applications. New Jersey: John Wiley & Sons; 2011. [15] Addison P. The illustrated wavelet transform handbook: introductory theory and applications in science, engineering, medicine and finance. London: IOP Publishing Ltd; 2002. [16] Sinha S, Routh P, Anno P, Castagna J. Spectral decomposition of seismic data with continuous-wavelet transform. Geophysics 2005;70(6):19–25. [17] Su Z, Ye L, Lu Y. Guided Lamb waves for identification of damage in composite structures: a review. J Sound Vib 2006;295:753–60. [18] Poddar B, Bijudas CR, Mitra M, Mujumdar PM. Damage detection in a wovenfabric composite laminate using time-reversed Lamb wave. Struct Health Monit 2012;11:602–12. [19] Bathe D. Finite element procedures in engineering analysis. New Jersey: Prentice-Hall Inc; 1982. [20] ANSYS 14.0 2012 Theory Reference.
Contents lists available at ScienceDirect
NDT&E International journal homepage: www.elsevier.com/locate/ndteint
Higher harmonics induced in lamb wave due to partial debonding of piezoelectric wafer transducers Nitesh P. Yelve n, Mira Mitra, P.M. Mujumdar Department of Aerospace Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India
art ic l e i nf o
a b s t r a c t
Article history: Received 1 July 2013 Received in revised form 28 November 2013 Accepted 21 January 2014 Available online 29 January 2014
Piezoelectric wafer (PW) transducers used for Lamb wave actuation may get partially debonded from the host structure, because of their prolonged use, excessive voltage supply, or improper bonding onto the host structure. In this paper, higher harmonics induced in Lamb wave because of such debonding of the PW actuator are studied both experimentally and through finite element simulation. In experiments, an artificial partial debond is created while bonding the actuator patch onto a pristine aluminium plate. Lamb wave transduced by this actuator in the plate is picked up by a PW sensor which does not have any debonding. In FE simulation, Augmented Lagrangian algorithm is used to solve the contact problem at the breathing debond. Three higher harmonics are observed in the experiments and also in the FE simulation. To ensure that the generated higher harmonics correspond to Lamb wave, time–frequency analysis is carried out using Morlet wavelet transform, and the results are reported in the paper. Spectral damage index (SDI), obtained from spectral attributes of first four harmonics in experiments and simulation, is found to be decreasing with an increase in debonding area. This shows that actuator debonding introduces contact nonlinearity which induces higher harmonics in Lamb wave. Therefore, in damage detection using Lamb wave based nonlinear techniques, the higher harmonics produced may get influenced by the false higher harmonics produced by actuator debonding, leading to incorrect results, if bonding of the actuator is not taken care of properly. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Piezoelectric wafer transducer debonding Contact nonlinearity Lamb wave Higher harmonics Morlet wavelet transform Spectral damage index
1. Introduction In material characterization [1,2] or damage detection [3–6], using Lamb wave based nonlinear techniques, the presence of higher harmonics is used as a measure of evaluation of the material or damage state. Nonlinear elasticity spread in the material continuum and local contact nonlinearity are the two mechanisms, which give rise to higher harmonics in Lamb wave [7]. The stronger the nonlinearity, the larger is the energy transferred from the fundamental harmonic to the higher harmonics and therefore an increase in the number and amplitude of higher harmonics is observed. In such nonlinear techniques used for material characterization or damage detection, piezoelectric wafer (PW) transducer is a good candidate for actuating Lamb wave in the structure because of its portability, small size and cost effectiveness. These characteristics of PW transducer make it suitable for in situ applications. Higher harmonics generation, in the PW transduced Lamb wave, in an aluminium plate with a breathing crack is studied in Reference [6]. Three higher harmonics are observed in this study
n
Corresponding author. E-mail address: [email protected] (N.P. Yelve).
0963-8695/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ndteint.2014.01.004
and time–frequency analysis using Morlet wavelet transform showed that these higher harmonics do lie on the dispersion curves and hence correspond to Lamb wave. The PW transducer excites both symmetric and anti-symmetric modes of Lamb wave and therefore it is able to generate both odd and even higher harmonics. This is shown both analytically and experimentally in Reference [8], in a quasi-statically loaded aluminium plate, with the aid of macro-fiber composite (MFC) piezoelectric patches. An important issue with PW transducer is its debonding from the host structure because of prolonged use, excessive deformation caused by high voltage supply, or improper bonding. Contact nonlinearity may prevail in this case, leading to higher harmonics in the Lamb wave response. This may show illusory presence of defect in a pristine material. This is proved both experimentally and through finite element (FE) simulation in the present paper. In experiments, an artificial partial debond is created in a PW actuator while bonding it onto a thin aluminium plate. In the simulation, partial breathing debonding is modelled between patch and plate by assigning contact attributes to the contacting faces [6]. Augmented Lagrangian algorithm [9] is used to implement contacting conditions at the meeting faces. The wave generated by such a debonded PW actuator, in both experiments and simulation, is picked up by the perfectly bonded PW sensor. The signal shows the presence of three higher harmonics. To
22
N.P. Yelve et al. / NDT&E International 63 (2014) 21–27
ensure that the generated higher harmonics belong to Lamb wave, time–frequency analysis of both experimental and simulation results is carried out using Morlet wavelet transform and the results of this analysis are reported in the paper. Further, a spectral damage index (SDI) [6] is extracted from spectral attributes of higher harmonics for various debonding sizes. Variation of this SDI with increasing debonding size is found to be decreasing in experimental as well as simulation studies. This indicates that PW actuator debonding is very much susceptible to induce higher harmonics in Lamb wave which may lead to show illusory presence of defect in a pristine material. Therefore good care should be taken while bonding the PW transducers, and bonding should be always checked using some appropriate techniques. Patches may even be replaced after their use over a certain duration.
2. Experimental work The experiments are performed on a pristine aluminium plate, with the objective of studying higher harmonics induced in Lamb wave as a result of debonding in PW transducer. The size of PW transducers used here is 10 mm 7 mm 0.5 mm and material type is SP-5H. Properties given by the manufacturer [10] for these transducers are mentioned in Table 1. The material of the plate used here is aluminium (grade 5052-H32) with modulus of elasticity 70.3 GPa, Poisson's ratio 0.33, and density 2680 kg/m3 [11] and its size is 1200 mm 1200 mm 1.6 mm. Cyanoacrylate adhesive is used for bonding PW transducer onto the plate. The sensor patch is bonded completely whereas actuator patch is partially bonded so as to introduce artificial debonding in it, as shown in Fig. 1. Three such actuator-sensor sets are experimented with different debondings of the actuators. In these three sets, the actuators are debonded over their entire width, and 10%, 20%, and 30% of their lengths at the centre as shown in Fig. 1. Therefore, the debonded area of three actuators is 10%, 20%, and 30% of total actuator area available for bonding. Cyanoacrylate adhesive is a fast solidifying adhesive, therefore its curing time is very less. Because of this, the chances of spreading the adhesive beyond the required bonding area of the PW transducer are very less. Also, the limiting ink lines made on the bonding side of the PW transducer further prohibits the spreading of the adhesive. The size of debonding in the transducers is checked by removing them after completing the experiments. The debonding is found to be in close limits of accuracy. Table 1 Properties of SP-5H piezo wafers [10]. Properties Piezoelectric coupling coefficients kp k33 Piezoelectric charge constants d33 10 12 C=N d31 10 12 C=N Piezoelectric voltage constants g 33 10 3 V m=N g 31 10 3 V m=N Relative dielectric constant kT3 Density ρ kg=m3 Elastic constants
Values
0.63 0.73 550 265 19 9
Fig. 1. Schematic of patch debonding.
The experimental setup as shown in Fig. 2 consists of arbitrary function generator (Make: Tektronix, Model: 3021B), high speed bipolar amplifier (Make: NF, Model: BA4825), digital storage oscilloscope (Make: Tektronix, Model: 1002B), and a computer. A 8.5 cycle sine wave tone burst signal windowed by Gaussian function is generated in MATLAB&. The Gaussian window of length N is defined by the equation [12]
ωðnÞ ¼ e½ ð1=2Þ½αðn=ððN 1Þ=2ÞÞ ; 2
where
N1 N1 rn r ; 2 2
SE11 10 12 m2 =N
21
SE33 10 12 m2 =N
15
ð2Þ
and α is inversely proportional to the standard deviation of Gaussian random variable. In the present case N ¼251 and α ¼ 2:5. This tone burst is then input to the function generator with a resolution of 40,000 points as shown in Fig. 3a, and set at frequency 170 kHz (Fig. 3b). Using the amplifier, this tone burst is magnified to 7100 V and given to the debonded PW actuator. The received Lamb wave at the PW sensor is shown by the oscilloscope and the corresponding data are sent to the computer for further signal processing. Fig. 4a–c shows the input signal and the output signals in time domain corresponding to the case of the perfectly bonded and the 30% debonded PW actuator. The received time domain data is cluttered and it is difficult to make any clear interpretation from it. Frequency domain analysis is useful to get the information about the nonlinearity induced in Lamb wave, in terms of higher harmonics. Here, the output time domain data is converted into the frequency domain using the discrete Fourier transform (DFT) given as [13] M1
XðkÞ ¼ ∑ x½qe½ j2π qk=M q¼0
3100 7500
ð1Þ
ð3Þ
where k ¼ 0 : M 1;
ð4Þ
and M is the number of data points, which is 2048 in the present case. For this purpose the Fast Fourier Transform (FFT) algorithm is
N.P. Yelve et al. / NDT&E International 63 (2014) 21–27
23
Fig. 2. Experimental set-up.
relatively lower excitation frequency of 170 kHz where no situation of identical group and phase velocities exists. Therefore it can be concluded that the higher harmonics observed are solely because of the contact nonlinearity introduced by the actuator debonding. Such higher harmonics are also observed in the case of 10% and 20% debonding. It is necessary to check whether the signal picked up at the PW sensor is corresponding to Lamb wave. For this purpose time– frequency analysis of the received signal is done and arrival times of frequencies are identified. Continuous Wavelet Transform (CWT) is used for this purpose as its resolution is better than short time Fourier transform [14]. The CWT of a time domain signal x(t) is given as [14,15] Z þ1 1 t b Tða; bÞ ¼ pffiffiffiffiffiffi dt; ð5Þ xðtÞψ a jaj 1 where a is the dilation parameter, b is the translation parameter and ψ ðtÞ is a wavelet basis function. In the present case Morlet basis function is used, according to which [15,16]
ψ ðtÞ ¼
1
eðiω0 tÞ e ðt
2
=2Þ
;
ð6Þ
π ð1=4Þ where ω0 is the central frequency parameter and is taken as 10 to
Fig. 3. Input signal used in experiments. (a) Input signal in time domain. (b) Frequency spectrum of input signal.
used in MATLAB& with sampling frequency of 5 MHz. Fig. 5a and b respectively shows the frequency spectrum of the signal received in the case of the perfectly bonded PW actuator and the 30% debonded actuator. From Fig. 5a it can be seen that the received wave response shows no higher harmonics in the case of the perfectly bonded actuator. However, in the case of the 30% debonded actuator, Fig. 5b shows the presence of three higher harmonics at 340 kHz, 510 kHz, and 680 kHz. This means that the observed higher harmonics are not generated because of instrumentation nonlinearity. Further, at certain high excitation frequencies, higher harmonics are seen in the wave response as a result of nonlinear elasticity spread in the material, when the condition of identical group and phase velocities exists for the harmonics [4]. The current experiments are carried out at a
satisfy the admissibility condition [15,16]. Morlet wavelet transform for no debonding case and 30% debonding case, with dispersion plots of A0 and S0 modes of Lamb wave, is shown in Fig. 6a and b respectively. MATLAB& is used for this purpose. In Fig. 6, arrival times of frequency components present in the response signal are compared with those of A0 and S0 modes of Lamb wave [17]. The Morlet wavelet plot in Fig. 6a shows frequency 170 kHz, and in Fig. 6b shows frequencies 170 kHz and 340 kHz, contained in the response signal. They all lie close to the dispersion curves. Thus, from these two figures it can be said that the no debonding case shows only A0 and S0 modes of Lamb wave at fundamental frequency in the response signal and no higher harmonics, whereas in the case of 30% debonding case, the frequency 170 kHz corresponds to A0 and S0 modes of Lamb wave, and the second harmonic 340 kHz corresponds to S0 mode of Lamb wave. Third and fourth harmonic frequencies are not visible in Fig. 6b as their amplitudes are feeble compared to that of the fundamental frequency. Similar time frequency plots are observed in the case of 10% and 20% debonding.
3. Finite element simulation In order to study higher harmonics induced in Lamb wave due to PW transducer debonding through finite element (FE) simulation, the most decisive issue is modelling the debonding between the plate and the PW actuator patch which can closely approximate
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N.P. Yelve et al. / NDT&E International 63 (2014) 21–27
Fig. 5. Experimental frequency domain results. (a) No debonding case. (b) 30% debonding case.
Fig. 4. Experimental time domain data. (a) Input tone burst. (b) Output in the case of no debonding. (c) Output in the case of 30% debonding.
the physical contact between the plate and the patch faces. ANSYS© is used here for FE simulation. The aluminium plate with PW actuator and sensor for transmitting and receiving Lamb wave respectively, is modelled in the thickness plane with plane strain approximation. This approximation saves computational time without affecting the trend of results significantly [18]. Properties assigned to the plate and the transducers are same as that mentioned in the previous section. PLANE82 (8-node quadrilateral) elements are used for plate and PLANE223 (8-node quadrilateral coupled-field) elements are used for PW patches. Selection of element size is crucial in FE simulation of wave propagation. The size of the elements must be sufficiently small for there to be at least 10 nodes per wavelength of the propagating wave [19]. The excitation frequency used in the study is 170 kHz and its fourth harmonic, 680 kHz is expected to be generated as a result of
contact nonlinearity at patch debond. Therefore the mesh size should be small enough to be able to capture 680 kHz. Only A0 and S0 modes of Lamb wave are involved in the analysis and A0 mode has the smaller wavelength at 680 kHz which is 3.075 mm. Accordingly, the distance between two nodes should not be more than 0.3 mm and as the elements used in the simulation have midside nodes, the element size should not exceed 0.6 mm. In the simulation study, a fine discretized mapped mesh is used, with the element size of 0.5 mm which is less than the threshold mesh size 0.6 mm required to capture the fourth harmonic of 170 kHz. The nodes on the free and the bonded faces of PW transducers are coupled to form master nodes on the respective faces. Therefore, in the case of actuator the voltage can be applied at the master node on the free face, and in the case of sensor, the voltage can be recorded from the master node on the free face. The PW transducer grounding is done by applying zero voltage at the master nodes on the bonded faces. Newmark algorithm is used here for time integration in FE analysis with α ¼0.25 and δ ¼0.5 where α and δ are the Newmark parameters. For these values of α and δ the Newmark algorithm is unconditionally stable [19]. In this case, the time step size to be used is based on the trade-off between the desired accuracy and the computational efforts. The accuracy in the integration can be stated in terms of period elongation and amplitude decay [19]. In the FE simulation of Lamb wave propagation, both period elongation and amplitude decay affect the group velocity. Therefore a convergence study is carried out to decide the appropriate time step for solution so that the group velocities of A0 and S0 Lamb wave modes obtained through simulation are in good agreement with those obtained analytically while requiring optimal simulation efforts both in terms of time of simulation and space required
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Fig. 6. Morlet wavelet transform of the experimental output data. (a) No debonding case. (b) 30% debonding case.
Fig. 8. Simulation time domain data. (a) Input tone burst. (b) Output in the case of no debonding. (c) Output in the case of 30% debonding.
Fig. 7. Deformed state of PW actuator in FE showing patch debonding.
in the computer. Accordingly, the time step of 0:2 μs is used for the simulation. CONTA172 (3-node surface-to-surface contact) and TARGE169 (target segment) elements are assigned to the faces of the PW
patch and the plate respectively over the debonded area to model the contact nonlinearity. Fig. 7 shows the deformed state of the PW actuator in FE, which clearly shows the patch debonding. Augmented Lagrangian (AL) contact algorithm under frictionless condition is used here to solve the contact problem at the debonding. It is an iterative series of penalty updates to find the contact tractions. It offers advantages such as more penetration control, better conditioning of governing equations, and satisfaction of constraints with finite penalties [9]. In this case the contact pressure R is defined by [20] ( R¼
0
if up 40;
K p up þ λi þ 1
if up r0;
ð7Þ
26
where
N.P. Yelve et al. / NDT&E International 63 (2014) 21–27
(
λi þ 1 ¼
λi þ K p up if jup j 4 ɛ; λi if jup j o ɛ;
ð8Þ
and Kp is the normal contact stiffness, up is the contact gap size, λi is the Lagrange multiplier at ith iteration, ɛ is the compatibility tolerance. The Lagrange multiplier component λi is computed locally i.e. for each element and iteratively. In AL treatment of frictionless contact of debonding faces, the admissible deformation of prospective points of contact satisfies [9] hðxÞ r 0;
ð9Þ
t S ¼ rðxÞ PS Z0;
ð10Þ
t S ðxÞhðxÞ ¼ 0;
ð11Þ
t R ¼ PS þ t S r ¼ 0;
ð12Þ
where h is a scalar-valued gauge function, P is the first Piola– Kirchoff stress tensor, r is the outward normal in the current configuration, S is the outward normal in reference configuration. Eq. (9) represents the condition of impenetrability, Eq. (10) represents the restriction that the normal component of surface traction be compressive, and Eq. (11) is a condition ensuring that tS may only be nonzero when hðxÞ ¼ 0. Eqs. (9)–(11) are therefore recognized as the Kuhn–Tucker conditions. Eq. (12) asserts that no friction is present. Parameters required for solving this frictionless contact problem in FE using AL contact algorithm are normal contact stiffness and maximum allowable penetration. Normal contact stiffness is based on Young's modulus and the size of the elements underlying
Fig. 9. Simulation frequency domain results. (a) No debonding case. (b) 30% debonding case.
the contact elements. Value of maximum allowable penetration considered for the analysis is 0.1 times the thickness of underlying elements because at its large value the AL method works like the Penalty method [20] which is not desirable here. Gaussian input signal of 8.5 cycles, 7100 V having central frequency 170 kHz is given to the debonded PW actuator. The generated wave is picked up at the PW sensor which does not have any debonding. The input and output time domain data in the case of the perfectly bonded PW actuator and 30% debonded PW actuator are shown in Fig. 8. Frequency domain plot of the signal received at the sensor for the case of no debonding is shown in Fig. 9a. It does not show any higher harmonic. Whereas, the Fourier transform of the signal received in the case of 30% debonded PW actuator is shown in Fig. 9b. It clearly shows the presence of three higher harmonics at 340 kHz, 510 kHz, and 680 kHz similar to that obtained in experiments. Such higher harmonics are also observed in the case of 10% and 20% debonding. From Figs. 5 and 9 it can be seen that amplitudes of higher harmonics observed in the experiments and in the FE simulation respectively vary to some extent. This is because, FE simulation is based on the plane strain assumption [18]. Morlet wavelet transform is used to plot the arrival times of the frequency components contained in the output signal, in the case
Fig. 10. Morlet wavelet transform of the simulation output data. (a) No debonding case. (b) 30% debonding case.
N.P. Yelve et al. / NDT&E International 63 (2014) 21–27
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response of a pristine aluminium plate. Time–frequency analysis using Morlet wavelet transform shows that the generated higher harmonics are also Lamb wave. These results emphasize that observed higher harmonics are induced in Lamb wave because of contact nonlinearity at the breathing debond. These higher harmonics are considerably sensitive to the patch debonding, which is confirmed by the decreasing trend of spectral damage index with the increasing debond area. This is a serious issue as it may show illusory presence of defect in a pristine material. Also, this may influence the higher harmonics occurring because of a defect in the material and hence may affect the accuracy and reliability of the results. Therefore it is important to bond the PW transducers very carefully onto the structure and also to check the bond quality often by using some appropriate techniques. Solution could also be to replace the patches after their use over a certain span of time. Fig. 11. Variation of SDI with debond area.
of completely bonded, and partially debonded PW actuators. These arrival times are then compared with those of A0 and S0 modes of Lamb wave. The results are shown in Fig. 10 for completely bonded and 30% debonded cases and they conform well with those obtained in experiments. Similar results are observed in the case of 10% and 20% debonding.
Acknowledgements The authors gratefully acknowledge the financial support by the Aeronautics Research and Development Board, Government of India for this research work (Project no. 1642).
References 4. Estimation of spectral damage index In the present work, to estimate the dependency of higher harmonics generation on the extent of PW transducer debonding, the spectral damage index (SDI) [6] proposed earlier by the authors of this paper is extracted from the spectral attributes of higher harmonics and its variation with the debond size is studied. For this purpose, first four harmonics are considered. These four harmonics are mapped between one and four and their amplitudes are normalized with respect to the fundamental harmonic observed in the case of the completely bonded PW actuator. Peaks of these four harmonics are joined by a series of lines in frequency domain plot, which is known as spectral envelop [6]. As debond area increases the contact nonlinearity increases which eventually causes energy transfer from fundamental harmonic to higher harmonics. This is evident from the fact that as debond area increases amplitudes of higher harmonics increase with simultaneous decrease of fundamental harmonic amplitude, as observed in the present work. This changes the angle θ between the bilinear fit [6] of spectral envelop and the change in the angle depicts the extent of debonding. The SDI is defined as tan θ [6]. This index is obtained for 10%, 20%, and 30% debonding in PW actuator in both experiment and simulation as shown in Fig. 11. It can be seen from this figure that as debond area increases, the SDI decreases which clearly depicts the dependence of higher harmonics and in turn contact nonlinearity on PW actuator debonding. 5. Conclusion In the present experimental and simulation studies, an artificially created partial debonding of the PW actuator is found to be showing presence of three higher harmonics in Lamb wave
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