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Micro-crack detection with nonlinear wave modulation technique and its application to loaded cracks
NDT and E International 107 (2019) 102132
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Micro-crack detection with nonlinear wave modulation technique and its application to loaded cracks
T
Sang Eon Lee, Hyung Jin Lim, Suyeong Jin, Hoon Sohn, Jung-Wuk Hong∗ Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Republic of Korea
ARTICLE INFO
ABSTRACT
Keywords: Lamb wave Piezoelectric Nonlinear Noise Finite-element method
Nonlinear ultrasonic modulation, a promising fatigue-crack-detection technique, is investigated with various frequency combinations, and an algorithm that can comprehensively analyze the results by comparison with ambient noise is proposed. The magnitudes at the 1st sideband frequencies of the Fourier-transformed result and surrounding noises are measured. Experiments with aluminum specimens before and after fatigue-crack generation show that cracks are detected correctly in both pristine and damaged specimens. When the developed crack-detection method is employed while applying an external force, specimen defects are correctly detected, although the magnitudes of modulated waves decreased. The time and effort required for monitoring the structure's health can be reduced by using ambient noise as a reference.
1. Introduction Most failures of metal structures are caused by fatigue and corrosion, and the proportion of fatigue failure is particularly high [1]. Fatigue cracks occur when repeated loads are applied to a structure, even if the stress is lower than the yield stress of the material. Fatigue cracks easily initiate at weak points such as stress-concentration positions and welded areas. The monitoring of weak parts and detection of cracks at the initiation phase are important tasks to increase the lifetimes of structures because the growth speed of a crack gradually increases with time. Depending on the size, type, and location of fatigue cracks, a variety of nondestructive testing methods have been used for detection [2,3]. Among those methods, the nonlinear ultrasonic modulation technique is suitable for micro-crack detection because of its high sensitivity which is retained through the opening and closing of crack surfaces [4,5]. Nonlinear ultrasonic modulation techniques have been developed continuously since their introduction in US patents [6]. Zaitsev et al. conducted nonlinear wave modulation tests with intact and cracked specimens, and concluded that nonlinear ultrasonic modulation techniques can detect fine cracks [7]. Didenkulov et al. conducted nondestructive tests on concrete beams based on wave modulation [8]. Sutin et al. applied ultrasonic modulation techniques for crack detection and estimated crack locations for various specimens using the pulseecho method [9]. Duffour et al. tested mild steel beams containing cracks of different shapes and sizes with a vibro-modulation technique
∗
[10]. Solodov et al. experimentally confirmed harmonics and modulation phenomena due to crack clapping and wave mixing [11]. Klepka et al. conducted nonlinear wave modulation tests on cracked aluminum plates and observed the effect of low-frequency vibration excitation on the modulation intensity [12]. Sohn et al. found internal defects by observing spectral sidebands while performing wave modulation tests on complex fitting-lug specimens [13,14]. The nonlinear ultrasonic modulation method involves a post-signalprocessing step because cracks are detected based on the modulation of the signal in the frequency domain. Lim et al. suggested a crack-detection method through outlier analysis, in which the results obtained by using several tests with various excitation conditions are investigated [15]. For the analysis, the nonlinearity index (NI), which represents the degree of nonlinearity, was defined and calculated for each test. In specific, the index was determined from the magnitudes at the sideband and input excitation frequencies. When the calculated NIs were sorted in the ascending order, an outlier was observed only if the specimen is damaged. It is possible to identify the integrity of the specimen observing the existence of the outlier, but the accuracy of crack-detection methods based on outliers needs to be improved. Defects can also be detected by using two indexes, namely the skewness and median of the NI distribution [16]. In this method, the skewness and median of the NI values are calculated, and the safety of the structure is determined based on the sign of these two values. In the skewness-median method, the nonlinearity index is calculated by subtracting the threshold from the modulation amplitude, and the
Corresponding author. E-mail addresses: [email protected], [email protected] (J.-W. Hong).
https://doi.org/10.1016/j.ndteint.2019.102132 Received 18 December 2018; Received in revised form 11 June 2019; Accepted 21 June 2019 Available online 23 June 2019 0963-8695/ © 2019 Published by Elsevier Ltd.
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specimen is categorized as damaged if at least the skewness or the median is positive. Although the skewness-median method is more accurate than outlier-based methods, the method has a limitation since an additional experiment with a single frequency is required in order to calculate the threshold. Nonlinear wave modulation has been studied not only experimentally but also numerically. Kawashima et al. observed harmonics by performing a numerical analysis with a model generated by assuming that the crack surfaces are perfectly flat and parallel to each other [17]. Lim et al. performed a numerical analysis with a crack generated by removing one layer of elements, and they simulated crack detection using ultrasonic waves [18]. Liu et al. numerically analyzed the Lamb wave propagation by using lead zinc titanate (PZT) patches and confirmed that the Lamb wave was deformed as it passes through a defect located in the plate [19,20]. Nonlinear wave modulation phenomena have been explained by researchers as follows. Richardson et al. suggested that harmonics generation is caused by opening and closing of an unbonded interface [21]. Kawashima et al. proposed that the nonlinearity originates from the integration of the stress-strain relationship, which varies depending on the crack width and wave amplitude [17]. Sutin explained that waves are modulated by the partial decoupling of the high-frequency signal while the low-frequency signal opens the crack [9]. Solodov et al. explained the wave modulation phenomenon by introducing a mechanical diode model of the crack surfaces [11]. Lee et al. suggested that the harmonics and wave modulation are caused by the truncation of waves propagating through the crack surface [22]. Nonlinear ultrasonic wave modulation techniques have been studied experimentally, numerically, and theoretically. However, the detection of cracks that are subjected to external forces has not been studied. Fatigue cracks accompany dislocations near the crack tip during propagation [23]. Furthermore, plastic deformation occurs at the back side of a crack tip [24]. In a fatigue-crack surface, various types of striations occur [25]. Therefore, both crack surfaces cannot be perfectly matched even if they are compressed. Because of the irregularity of crack surfaces, modulated waves are observed if the crack breathing phenomenon is obstructed by an external force, although the magnitudes of modulated waves are very weak. On the other hand, noise and unexpected nonlinearities are frequently observed in field tests, and they reduce the signal-to-noise ratio. Therefore, it is necessary to improve the accuracy of crack detection using the nonlinear ultrasonic modulation technique by separating noise and weak modulated waves. In this study, we introduce algorithms for crack detection using nonlinear ultrasonic wave modulation and propose a new algorithm that can reduce the effort and time required for testing. The source of nonlinear wave modulation varies depending on the strain range, specimen material, etc. In this study, the clapping of the crack surface caused by excitation is considered as the main factor modulating the wave. The excitation amplitude was selected by considering the size of the specimen and position of the crack. Experiments are performed before and after generating fatigue cracks on aluminum specimens with various excitation frequency combinations. Next, the magnitudes of modulated waves generated under tensile and compressive force are observed, and crack detection is performed using the proposed algorithm. The occurrence of nonlinear waves and the magnitude change of the modulated waves due to an external force are numerically verified. It was possible to distinguish intact and damaged specimens by applying the proposed algorithm, and cracks could be detected even when a strong external force was applied to the specimen.
detecting cracks that uses the modulation phenomenon of waves passing through a crack. The compressive stress of ultrasonic waves can be transmitted through cracks; however, the tensile stress is not completely transmitted. Therefore, wave truncation occurs, which is observed as a variation of frequency components in the frequency domain through a Fourier transform. In particular, the peaks occurring at integer multiples of the excitation frequency are called as harmonics [26], and the peaks observed at frequencies corresponding to the sum and difference of the two excitation frequencies are referred to as sideband peaks, which are caused by modulation. A sinusoidal incident wave u (t ) with two different frequencies and a truncated wave utr (t ) propagating through a crack are expressed, respectively, as follows: (1)
u (t ) = A cos(2 f t ) + Ah cos(2 fh t ),
u tr (t ) = {A cos(2 f t ) + Ah cos(2 fh t )
Wcr } ×
(t ),
(2)
where A and f are the amplitude and frequency of the low-frequency wave, respectively, and Ah and fh are the amplitude and frequency of the high-frequency wave, respectively. Wcr is the crack width, and (t ) is a rectangular function that is equal to unity during crack closing, and zero otherwise. The Fourier-transformed utr (t ) can be expressed as follows:
F {utr (t)} = +
A Cn { (f 2
(n + 1) f ) + (f
Ah Cn { (f 2
(fh + nf ) + (f
Wcr Cn
nf ,
f
where
(n
1) f )}
( fh + nf ) )
(3)
( )
is the Dirac-delta function, Cn = T sinc T , is the pulse tr tr width, and Ttr is the period. The second term on the right-hand side of Eq. (3) indicates sideband frequencies that are equal to the summation and difference of the two input frequencies. n
2. Theory
2.1.2. High-order nonlinearity for a rough crack surface Wave truncation is an intuitive and well-formulated description of nonlinear wave modulation, but it has limitations in that it cannot accurately reflect the shape of the actual crack. Therefore, researchers have suggested various models to describe mechanism occurring in a real crack. Kawashima et al. explained the source of the wave modulation as a nonlinear stress-strain relationship [15]. Each part of rough crack surfaces behaves as a mechanical diode with a bilinear stressstrain relation, but the shape of the bilinear curve depends on the crack width and incident-wave amplitude. The nonlinear stress-strain relation formed by superposing these bilinear curves induced the wave modulation. Some studies additionally introduced the material hysteresis phenomenon as the source of wave modulation [27,28]. With the hysteresis effect, nonlinear characteristics such as higher harmonics and the natural frequency shift can be explained. In addition to the superposed bilinear model and hysteresis effect, which are relatively simple and easy to be implement, more realistic and complex models have been proposed. The rough surface contact model based on the Hertzian contact theory can cover all physical phenomena, but it is extremely complex [5,29]. Recently, non-classical dissipation mechanisms such as thermal dissipation at crack edges and the Luxemburg-Gorky (LG) effect have been proposed [30,31]. Nonclassical dissipation phenomena are generally observed at smaller strain ranges, and the approach can explain phenomena that cannot be explained using the classical models [12,29].
2.1. Generation of modulated wave
2.2. Crack-detection algorithms
2.1.1. Wave truncation when passing through a planar crack The nonlinear ultrasonic modulation technique is a method of
2.2.1. Direct comparison before and after crack generation The nonlinear ultrasonic modulation technique uses the sideband 2
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Fig. 1. Fourier-transformed result obtained from the nonlinear wave modulation technique performed with two different sinusoidal waves. In (a) and (c), the signals were measured from an intact specimen, and in (b) and (d), the signals were measured from a damaged specimen. Locations of the modulation frequencies (MF) are annotated.
occurrence in the frequency domain after the Fourier transform of the measured signal as a criterion. The most basic method is to compare the data before and after crack generation in the specimen. It is possible to determine whether a crack has occurred by comparing the magnitude of the sideband measured in the initial intact state and that measured after some time has passed. Fig. 1 shows the results of a nonlinear wave modulation test with a 3-mm-thick aluminum plate specimen in the frequency domain obtained through Fourier transformation. Figs. 1(a) and (c) show the results obtained from the intact specimen, while Fig. 1(b) and (d) show the results obtained after a fatigue crack was formed. By comparing Fig. 1(a) and (b), the difference in the generation of peaks at the sideband frequency is observed. Clear peaks are observed only when the specimen contains defects. However, in Fig. 1(c) and (d), it is difficult to identify the integrity of the specimen using the sideband generation. Even when the specimen is in an intact condition, a small amount of wave modulation might occur because of an internal pores or material nonlinearity. In contrast, the magnitude of the modulated wave is insufficient when the movement of the crack surfaces is inactivated depending on the wavelength and plate geometry. Therefore, it is necessary to repeat the test with various conditions and integrate the data synthetically because finding defects in a single test does not guarantee accuracy.
Fig. 2. Range of interest in the frequency domain when low- and high-frequency waves are applied for excitation. By measuring in the gray hatched areas, one modulated wave and (n-1) noise values will be obtained on both sides of the high frequency.
The n2 data from the left sideband can be represented by a threedimensional plot, as shown in Fig. 3. The figure shows 11 test results with a high frequency of 181 kHz, and it contains 112 data. The surface plot represents the magnitude measured at each frequency of interest and is projected as a contour. In this contour grid, the sideband frequencies are located on the diagonal line. On the contour grid, the position with the largest value for one low frequency is marked with a red inverted triangle, and the sideband frequencies are indicated with blue diamonds. The colocation of the two shapes indicates the occurrence of wave modulation. Fig. 3(a) and (b) show the results measured from an intact specimen and damaged specimen, respectively. Because the modulation frequency lies on the diagonal line in the surface plot, values significantly larger than the surrounding noise are observed in Fig. 3(b). Although remarkable peaks are found on the diagonal line in Fig. 3(a), those are not as obvious as in Fig. 3(b). This section describes a crack-detection algorithm that determines the integrity of the structure based on two factors, namely the damage index (DI) and intactness index (II), which are calculated from the observation of modulated-wave generation. DI is a factor related to the occurrence of the modulated wave in each test. For each test, a
2.2.2. Comparison with ambient noise along two axes This section proposes an algorithm for safety diagnosis by comparing the modulated wave with ambient noises without any additional experiment. In this method, ambient noises around the sideband frequencies are used to determine the occurrence of wave modulation. If n low frequencies are used for a high frequency fhj in the experiment, each low frequency can be expressed as f i , where i is a natural number ranging from one to n . The magnitude of the frequency spectrum ranges from fhj ± f 1 to fhj ± f n when two incident waves f i and fhj are applied to the specimen, as shown in Fig. 2. The k th frequency from the high frequency is denoted as fhj ± f ik , and the sideband frequency is denoted as fhj ± f ii . The values measured at the remaining (n-1) frequencies are noise values that are not related to the modulated wave. For one high frequency, n low frequencies are used, and n data are measured for each frequency combination. Therefore, a total of 2n2 data are collected when the experiments are performed with n low frequencies because sideband frequencies are located on both side of high frequency. 3
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Fig. 3. Three-dimensional surface plot representing the magnitudes of the frequency spectra from a nonlinear ultrasonic modulation test measured in a range of interest. The frequencies of the sideband are located on the diagonal line, and the off-diagonal elements are noise. The tests were performed with (a) an intact specimen and (b) a cracked specimen.
frequency that has the maximum value among n data obtained near the sideband frequency is found. If the magnitude at the sideband frequency is more significant than ambient noise, DI is increased by 1. By performing (n × m ) tests with combinations of n low frequencies and m high frequencies, it is possible to make 2 × (n × m) comparisons for both sidebands. In general, DI has a value close to 2 × (n × m) because a modulation occurs in most cases, as shown in Fig. 3(b). If DI is 2 × (n × m) , it means that wave modulation has occurred for all frequency combinations, which may indicate the existence of a defect in the specimen. If DI is less than 2 × (n × m) , the wave modulation might not appear with some frequency combinations. Modulation may not occur in damaged specimens depending on the frequency combination. Therefore, the integrity of the specimen cannot be ensured even when DI is less than 2 × (n × m) . Additional comparisons will proceed with II. II is determined based on the noise level measured at fhj ± f ik , which is greater than the magnitude at the sideband frequency fhj ± f ii , by comparing it with the magnitude at another sideband frequency fhj ± f kk that is obtained from a test with a different frequency combination. If the noise level at fhj ± f ik is larger, II will be increased by 1. The existence of a noise level larger than the magnitude at two different sideband frequencies implies that the modulation of waves does not occur actively, which suggests that the specimen does not have contain a crack. In contrast, if the specimen is damaged, the existence of a noise level greater than the magnitude of two different modulated wave signals is highly unlikely. Therefore, the specimen is considered damaged when II is zero although the DI is less than 2 × (n × m) . If DI is less than 2 × (n × m) but II is larger than a threshold T , the judgment is postponed. This means that the modulated wave does not occur in several tests, but the frequency of occurrence of modulation at the sideband frequency is not negligible. In this case, additional tests are required with a different frequency combination or stronger excitation. The threshold T can be determined based on the experimental conditions and environment. Fig. 4 shows a flowchart of the data post-processing algorithm described in this section. M (f ) is the magnitude of the frequency spectrum at frequency f . Fig. 4 shows the procedure to check whether the noise value measured at fhj ± f ik is an outlier during the process. Depending on the experiment environment, an external signal that is significantly larger than ambient noise is occasionally observed. It reduces the accuracy of the test, which is based on the occurrence of wave modulation. Therefore, a noise level greater than a threshold calculated under the assumption that (n 1) noise values in each test follow an exponential distribution is regarded as an outlier. A confidence interval of 99.99% was used for threshold calculation. The description above corresponds to the upper half of Fig. 4. The analysis was performed for both sidebands and ambient noise
separately, following which the process was repeated once more with the sum of magnitude at both sidebands. When using vibration, rather than wave propagation, by extending the duration, the magnitudes at both sideband frequencies show a difference because it is correlated with the vibration characteristics of the structure such as geometry and boundary conditions [32]. The magnitude at one sideband can be consistently small. When a wave in a specific frequency band has difficulties in propagating, the specimen is determined to have no damage with a test considering both sidebands separately. Therefore, the accuracy of the test can be improved by applying the algorithm once more after adding two magnitudes at both sidebands. 3. Experimental verification 3.1. Tests before and after crack generation without external load Experiments were conducted to verify the accuracy of the proposed algorithm before and after generating fatigue cracks in the specimen, and the algorithm was applied to the results. Fig. 5 shows the geometry and arrangement of the aluminum specimen and PZT patches used for the experiment. The specimen is made of 3-mm thick aluminum 6061 material, which has a density of 2700 kg/m3 and Young's modulus of 70 GPa. Two rectangular PZTs were attached to the surface of the specimen, and one circular PZT was attached to the opposite side of the crack for measurement. Both the rectangular and circular PZTs are made by APC International with APC 850 piezoelectric material because the characteristic of APC 850 material is suitable for both sensing and actuating. The piezoelectric properties of APC 850 are listed in Table 1. A notch of 5-mm length is made at the center of the specimen to concentrate the stress during the fatigue test. Sections of 40-mm length on each side of the specimen were fixed on an Instron 8801 dynamic test machine in the fatigue-crack generation phase and nonlinear wave modulation test phase under loads. A low frequency of 4 V and a high frequency of 2 V with a conversion rate of 1 MHz were used for excitation, and the response was measured with a 1-MHz sampling rate for 100 ms. The width of the crack is smaller than 20 µm , and the specimen is quite small. The excitation amplitudes were selected to induce crack clapping in consideration of the crack and specimen size. Excessively strong excitations are not suitable in this experiment because they prevent observation, which is based on the application of an external force on the specimen. The nonlinear ultrasonic modulation test was performed after the intact specimen was fixed on the test machine. After the test, a fatigue crack was generated using a sinusoidal tensile force, following which the wave modulation test was performed once more with the cracked specimen. The specimen is fixed to the machine to unify the boundary conditions across all experiments. The nonlinear wave modulation test was performed 33 times with combinations of 11 low frequencies and 3 4
NDT and E International 107 (2019) 102132
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Fig. 4. Flowchart of the algorithm for analyzing the nonlinear wave modulation test. The process is performed for each sideband and repeated for the summation of both sidebands. Table 1 Piezoelectric properties of APC 850 material. Property
Symbol
Value
Electromechanical coupling factor
kp
0.63
Piezoelectric charge constant (10−12 m/V)
Fig. 5. Geometrical information of the aluminum plate and the arrangement of the PZT patches used in the experiment.
Poisson ratio
Density (kg/m3 )
high frequencies. Experiments were conducted while increasing the low frequency from 30 kHz to 40 kHz and the high frequency from 181 kHz to 183 kHz by increment of 1 kHz. The excitation frequencies were selected carefully to avoid the case where sideband frequencies coincide with the harmonic frequencies of the low frequency. After testing in the intact state, a sinusoidal tensile load ranging from 0.7 kN to 7 kN was applied with 10-Hz frequency to generate a fatigue crack until the length of the crack reached 6 mm. After the fatigue-crack generation, the tests were repeated with the damaged specimen. Table 2 lists the results by analyzing the experiment results obtained before and after fatigue cracking using the proposed algorithm. Three
k33 k31 k15 d33 d31 d15
0.72 0.36 0.68 400 - 175 590 0.35 7700
specimens were diagnosed based on the obtained DI and II, and the values that were used to determine the specimen damage are indicated in bold characters. All three specimens were concluded to be intact owing to the failure to find evidence of damage to the specimen from the experiment before crack generation. Specimens 1 and 3 showed sidebands greater than the ambient noise level in all tests. In the case of specimen 2, the two sidebands showed values smaller than the ambient noise level. However, since II was zero, the specimen seems to be damaged. The damage states of all three specimens were accurately diagnosed by the proposed algorithm. 5
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Table 2 Analysis of the experiment results by applying the proposed algorithm. DIs and IIs are counted, and the integrity of the specimen is tested with these two indexes.
Specimen #1 Specimen #2 Specimen #3
Condition
Intactness index (II)
Damage index (DI)
Test result
Intact Damaged Intact Damaged Intact Damaged
30, 17 0 4, 5 0 7, 2 0
23, 12 66 31, 16 64 41, 25 66
Intact Crack Intact Crack Intact Crack
Table 3 Experiment results for two damaged specimens under external forces. A lowfrequency excitation of 4 V and high-frequency excitation of 1 V were used. Specimen #4
3.2. Crack detection under tensile and compressive force
Specimen #5
When the crack surfaces are exposed to external forces, the signalto-noise ratio is decreased because the occurrence of wave modulation is reduced, causing difficulties in crack detection. In this section, the test is performed while applying tensile and compressive forces, and crack detection is attempted by analyzing the results with the proposed algorithm. As in the previous test, the specimen was fixed on the Instron 8801 test machine, and a nonlinear ultrasonic modulation test was performed with an external force ranging from 10 kN to −10 kN in decrements of 2 kN. The tests are repeated 33 times for combinations of 11 low frequencies and 3 high frequencies. In a 3-mm-thick aluminum specimen, slight bending was observed under 10-kN compression, and buckling occurred at −11 kN. Fig. 6 shows the changes in NI when tensile and compressive forces are applied on two identical specimens, where MD and MS are the magnitudes of frequency spectra at the sum and difference of the two excitation frequencies, respectively. In this case, NI is calculated by adding all magnitudes of frequency spectra at sideband frequencies for the 33 excitation combinations. The x-axis represents the magnitude of the external force applied to the specimens, and the y-axis represents NI. Fig. 6(a) and (c) show the change of modulation-wave generation according to the compression. The magnitude of the modulated waves has a maximum value when the compressive force is between 2 kN and
Tensile force (kN) Intactness index (II) Damage index (DI) Test result
0 0
2 0
4 0
6 0
8 0
10 0
63 crack
65 crack
65 crack
64 crack
63 crack
62 crack
Compressive force (kN) Intactness index (II) Damage index (DI) Test result
–
2
4
6
8
10
–
0
0
0
0
0
– –
65 crack
63 crack
65 crack
65 crack
65 crack
Tensile force (kN) Intactness index (II) Damage index (DI) Test result
0 0
2 0
4 0
6 0
8 0
10 2,0
60 crack
61 crack
63 crack
63 crack
62 crack
53,41 crack
Compressive force (kN) Intactness index (II) Damage index (DI) Test result
–
2
4
6
8
10
–
0
0
0
1,0
0
– –
61 crack
63 crack
58 crack
54,31 crack
53 crack
4 kN because the interaction of crack surfaces becomes more active as the distance between crack surfaces decreases because of the compression. In Fig. 6(a), the modulated-wave generation slightly increases under compressions of 8 kN and 10 kN. During the test with specimen 1, bending was observed at compression greater than 8 kN, and it seems that the entire crack surface was not pressurized. This phenomenon will be verified by numerical analysis in the next section. Fig. 6(b) and (d) show the change of modulated-wave generation with the tensile force. The amount of nonlinear wave generation increased as the tensile force increased up to 2 kN, and it decreased with further increase in the tensile force. The magnitudes are decreased continuously beyond 2 kN because the plate did not warp under tension.
Fig. 6. Nonlinearity index obtained from the two specimens under external forces. (a) and (b) show the results for specimen #1, while (c) and (d) show the results for specimen #2. 6
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4.2. Numerical modeling including realistic crack model
Table 4 Experiment results for two damaged specimens under external forces. A lowfrequency excitation of 4 V and high-frequency excitation of 3 V were used. DIs were increased compared to the results in 〈Table 3〉. Specimen #4
Specimen #5
Tensile force (kN) Intactness index (II) Damage index (DI) Test result
0 0
2 –
4 –
6 0
8 –
10 0
64 crack
66 crack
66 crack
65 crack
66 crack
65 crack
Compressive force (kN) Intactness index (II) Damage index (DI) Test result
–
2
4
6
8
10
–
–
–
–
–
–
– –
66 crack
66 crack
66 crack
66 crack
66 crack
Tensile force (kN) Intactness index (II) Damage index (DI) Test result
0 0
2 0
4 –
6 –
8 0
10 1
62 crack
64 crack
66 crack
66 crack
65 crack
60, 33 crack
Compressive force (kN) Intactness index (II) Damage index (DI) Test result
–
2
4
6
8
10
–
0
0
0
0
0
– –
64 crack
65 crack
64 crack
63 crack
54 crack
The modulation phenomenon as the wave passes through the cracks and the magnitude change of the modulated wave caused by the external force are observed in the numerical simulation with finite-element analysis [34]. Numerical analysis is performed with the commercial finite-element analysis software LS-Dyna [35]. The shape of the numerical crack model becomes realistic as the number of elements used for mimicking the crack increases, but the calculation time increases dramatically as the number of elements increases. Therefore, the size of the elements near the crack and that of the others are set differently, and they are combined by use of the tied contact function. The size of the elements near the fatigue crack is set to 0.1 mm, and others have a size of 1 mm. The length of the crack was 6 mm, and the numerical crack model was discretized with 60 elements of 0.1 mm size and 61 nodes. Moreover, the 3-mm thick aluminum plate near the crack was modeled with thirty layers. Thus, 31 nodes are positioned along the thickness direction. To express the crack more realistically in the numerical simulation, a set of 1891 widths is sampled from the measured crack width dataset in Section 4.1 and reflected in the numerical model [33]. Therefore, the numerical aluminum plate model has a crack with an irregular shape. The crack width tends to decrease as it approaches the crack tip. In the experiment, sections of 40-mm length at each side of the specimen were fixed to the test machine, but in the numerical analysis, only a 150-mm-long section at the middle of the specimen was modeled to reduce the number of elements. The right side of the numerical plate is fixed, and the external force was adjusted through the displacement control of the left side of the plate, as shown in Fig. 7. Instead of attaching PZT patches, lateral sinusoidal waves of 10 nm in amplitude were incident on the nodes located at the position of rectangular PZTs. The lateral displacement was measured at the node at the center of the circular PZT attachment. Although the amplitudes of incident waves were 10 nm, a few µm movements were observed in the output signal owing to the reflection and interference effect of waves, and it is sufficient to induce the contact of crack surfaces. The excitation frequencies for numerical simulations were 41 kHz and 188 kHz, and the sideband frequencies were 147 kHz and 229 kHz. The wavelengths of the incident waves were approximately 151 mm and 33 mm for the low and high frequency, respectively. The analyses were conducted with
Table 3 summarizes the analysis of the experiment results performed with a low-frequency excitation of 4 V and high-frequency excitation of 1 V by using the proposed algorithm. We counted II and DI in each test under different external forces and diagnosed the specimen based on these indexes. Values that indicate specimen damage are expressed in bold characters. DI did not reach 66 owing to the weak excitation and the effect of the external force. However, II was mostly zero owing to the frequent wave modulation. In the test of specimen 5 under compression of 8 kN, 12 sidebands were not observed, and it was concluded that the damage had occurred in the specimen because the second II (II2) was 0. Table 4 lists the experiment results obtained when the high frequency was increased to 3 V. The DI value shows that modulation has occurred more actively compared to the results in Table 3. If the diagnosis is difficult owing to insufficient modulation, it is necessary to perform the test with a strong intensity. Fatigue cracks in two specimens could be found by using the algorithm in all cases with external forces. 4. Numerical simulations 4.1. Crack-width measurement Numerical analyses have been performed assuming that the crack plane is flat or that a crack has been modeled by deleting one element layer [17,18]. However, actual fatigue cracks do not have a flat and parallel shape owing to the plastic deformation and fragmentation. Therefore, to mimic the surface motion of fatigue cracks realistically, the width of the fatigue crack should be measured and applied to the finite-element model. The widths of the fatigue crack were measured from image-processed photomicrographs, and the finite-element model including a realistic fatigue crack was generated according to the method introduced by Lee et al. [33]. The image processing consists of three steps: skeletonization, pruning, and crack-width measurement. In this method, the crack widths were measured in the direction perpendicular to the centerline of the crack.
Fig. 7. Geometry of the numerical simulation using the finite-element model. The right side of the plate is fixed, and the left side is moved to pressurize the specimen. Waves are incident by controlling the displacement of the nodes located at the position of rectangular PZTs, and a signal is measured at the node located at the center of the circular PZT. 7
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Fig. 8. Numerical results of the nonlinear wave modulation technique in the frequency domain. The sidebands are marked with black arrows. The simulations are performed (a) without an external force, (b) with a compressive force of 10 kN, and (c) with a warped plate under a compressive force of 10 kN.
three different boundary conditions: without any external force, under a compression of 10 kN, and warping due to a compression of 10 kN.
Several methods for performing safety diagnosis with various input conditions and synthetically judging the integrity of a structure with the data have been suggested. In this paper, we present an algorithm that shows sufficient accuracy while reducing the cost of the experiment compared to existing methods. Nonlinear modulated wave tests were conducted before and after generating fatigue cracks on aluminum specimens. The excitation amplitudes were set to make crack clapping the main source of nonlinearity generation. If experiments are performed with a smaller strain range, additional reference data may be required for diagnosis. As a result, it was possible to determine the specimen damage accurately. In addition, tests were performed while applying an external force to the specimen containing fatigue cracks. Modulated wave generation was temporarily increased by weak tensile and compressive forces. However, as the force increased above a certain level, the interaction of cracks was disturbed, and the wave modulation decreased. In the case of compressive stress, the increment of the modulated wave was observed because of the warping of the thin plate specimen. This phenomenon was also observed in numerical analysis. Even with data that have a relatively small signal-to-noise ratio due to the external force, the crack could be detected by applying the proposed algorithm. The proposed nondestructive evaluation method with ultrasonic waves and post-processing algorithms shows a high accuracy and can reduce the effort and time required for testing. It is also possible to detect cracks under compression, which are known to be difficult to detect with conventional methods. Modulated waves could occur even if the signal-to-noise ratio is decreased because of the irregular shape of crack surfaces that is caused by the plastic deformation and fragmentation during crack propagation. Future studies may investigate the change of generation of the modulated wave according to the shape and direction of the crack under an external force.
4.3. Simulation results Fig. 8 contains the displacement measured in each numerical analysis in the frequency domain through the Fourier transform. Fig. 8(a) shows various frequency components such as harmonics and sidebands resulting from the active interaction of crack surfaces due to the absence of an external force. Numerical results also include numerical errors due to finite precision and truncation errors caused by approximate mathematical solutions. Various components were observed due to nonlinearity and numerical error, but sideband which means modulated wave was observed clearly. The 1st sideband marked with black arrows is clearly observed. However, in Fig. 8(b), a sideband distinct from surrounding noise is not observed. It can be seen that, under the strong compressive force, the generation of modulated waves decreases because the relative motion of crack surfaces is hindered by the compression. In Fig. 8(c), the nonlinear wave modulation test is performed with an initial displacement to the center of the plate in the normal direction of the plate to induce bending. Compared to Fig. 8(a), the size of the sideband reduced, but a clear peak is observed in contrast to Fig. 8(b). It is numerically confirmed that the generation of the modulated wave increased when bending occurred because of compression in the plate. 5. Summary and conclusions Micro-cracks that are difficult to detect with conventional linear methods can be successfully detected with the nonlinear wave modulation technique owing to its high sensitivity resulting from the use of crack-surface interaction. In the method, two distinct waves with different frequencies are incident on the structure, and safety diagnosis is performed based on the occurrence of a modulated wave that appears as a sideband in the frequency domain. However, owing to the presence of pores or material nonlinearity, weak modulated waves are observed even without cracks. Moreover, depending on the excitation frequency, wave modulation does not occur when the movement of a crack surface is insufficient. Therefore, it is necessary to repeat the test with various frequency combinations.
Acknowledgments This research was supported by a grant (17CTAP-C133220-01) from the Infrastructure and Transportation Technology Promotion Research Program funded by Ministry of Land, Infrastructure and Transport of the Korean government, and was also supported by the National Research Foundation of Korea grant funded by the Korean government (MSIT) (No.2018R1A2A1A05019453). 8
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Micro-crack detection with nonlinear wave modulation technique and its application to loaded cracks
T
Sang Eon Lee, Hyung Jin Lim, Suyeong Jin, Hoon Sohn, Jung-Wuk Hong∗ Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Republic of Korea
ARTICLE INFO
ABSTRACT
Keywords: Lamb wave Piezoelectric Nonlinear Noise Finite-element method
Nonlinear ultrasonic modulation, a promising fatigue-crack-detection technique, is investigated with various frequency combinations, and an algorithm that can comprehensively analyze the results by comparison with ambient noise is proposed. The magnitudes at the 1st sideband frequencies of the Fourier-transformed result and surrounding noises are measured. Experiments with aluminum specimens before and after fatigue-crack generation show that cracks are detected correctly in both pristine and damaged specimens. When the developed crack-detection method is employed while applying an external force, specimen defects are correctly detected, although the magnitudes of modulated waves decreased. The time and effort required for monitoring the structure's health can be reduced by using ambient noise as a reference.
1. Introduction Most failures of metal structures are caused by fatigue and corrosion, and the proportion of fatigue failure is particularly high [1]. Fatigue cracks occur when repeated loads are applied to a structure, even if the stress is lower than the yield stress of the material. Fatigue cracks easily initiate at weak points such as stress-concentration positions and welded areas. The monitoring of weak parts and detection of cracks at the initiation phase are important tasks to increase the lifetimes of structures because the growth speed of a crack gradually increases with time. Depending on the size, type, and location of fatigue cracks, a variety of nondestructive testing methods have been used for detection [2,3]. Among those methods, the nonlinear ultrasonic modulation technique is suitable for micro-crack detection because of its high sensitivity which is retained through the opening and closing of crack surfaces [4,5]. Nonlinear ultrasonic modulation techniques have been developed continuously since their introduction in US patents [6]. Zaitsev et al. conducted nonlinear wave modulation tests with intact and cracked specimens, and concluded that nonlinear ultrasonic modulation techniques can detect fine cracks [7]. Didenkulov et al. conducted nondestructive tests on concrete beams based on wave modulation [8]. Sutin et al. applied ultrasonic modulation techniques for crack detection and estimated crack locations for various specimens using the pulseecho method [9]. Duffour et al. tested mild steel beams containing cracks of different shapes and sizes with a vibro-modulation technique
∗
[10]. Solodov et al. experimentally confirmed harmonics and modulation phenomena due to crack clapping and wave mixing [11]. Klepka et al. conducted nonlinear wave modulation tests on cracked aluminum plates and observed the effect of low-frequency vibration excitation on the modulation intensity [12]. Sohn et al. found internal defects by observing spectral sidebands while performing wave modulation tests on complex fitting-lug specimens [13,14]. The nonlinear ultrasonic modulation method involves a post-signalprocessing step because cracks are detected based on the modulation of the signal in the frequency domain. Lim et al. suggested a crack-detection method through outlier analysis, in which the results obtained by using several tests with various excitation conditions are investigated [15]. For the analysis, the nonlinearity index (NI), which represents the degree of nonlinearity, was defined and calculated for each test. In specific, the index was determined from the magnitudes at the sideband and input excitation frequencies. When the calculated NIs were sorted in the ascending order, an outlier was observed only if the specimen is damaged. It is possible to identify the integrity of the specimen observing the existence of the outlier, but the accuracy of crack-detection methods based on outliers needs to be improved. Defects can also be detected by using two indexes, namely the skewness and median of the NI distribution [16]. In this method, the skewness and median of the NI values are calculated, and the safety of the structure is determined based on the sign of these two values. In the skewness-median method, the nonlinearity index is calculated by subtracting the threshold from the modulation amplitude, and the
Corresponding author. E-mail addresses: [email protected], [email protected] (J.-W. Hong).
https://doi.org/10.1016/j.ndteint.2019.102132 Received 18 December 2018; Received in revised form 11 June 2019; Accepted 21 June 2019 Available online 23 June 2019 0963-8695/ © 2019 Published by Elsevier Ltd.
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specimen is categorized as damaged if at least the skewness or the median is positive. Although the skewness-median method is more accurate than outlier-based methods, the method has a limitation since an additional experiment with a single frequency is required in order to calculate the threshold. Nonlinear wave modulation has been studied not only experimentally but also numerically. Kawashima et al. observed harmonics by performing a numerical analysis with a model generated by assuming that the crack surfaces are perfectly flat and parallel to each other [17]. Lim et al. performed a numerical analysis with a crack generated by removing one layer of elements, and they simulated crack detection using ultrasonic waves [18]. Liu et al. numerically analyzed the Lamb wave propagation by using lead zinc titanate (PZT) patches and confirmed that the Lamb wave was deformed as it passes through a defect located in the plate [19,20]. Nonlinear wave modulation phenomena have been explained by researchers as follows. Richardson et al. suggested that harmonics generation is caused by opening and closing of an unbonded interface [21]. Kawashima et al. proposed that the nonlinearity originates from the integration of the stress-strain relationship, which varies depending on the crack width and wave amplitude [17]. Sutin explained that waves are modulated by the partial decoupling of the high-frequency signal while the low-frequency signal opens the crack [9]. Solodov et al. explained the wave modulation phenomenon by introducing a mechanical diode model of the crack surfaces [11]. Lee et al. suggested that the harmonics and wave modulation are caused by the truncation of waves propagating through the crack surface [22]. Nonlinear ultrasonic wave modulation techniques have been studied experimentally, numerically, and theoretically. However, the detection of cracks that are subjected to external forces has not been studied. Fatigue cracks accompany dislocations near the crack tip during propagation [23]. Furthermore, plastic deformation occurs at the back side of a crack tip [24]. In a fatigue-crack surface, various types of striations occur [25]. Therefore, both crack surfaces cannot be perfectly matched even if they are compressed. Because of the irregularity of crack surfaces, modulated waves are observed if the crack breathing phenomenon is obstructed by an external force, although the magnitudes of modulated waves are very weak. On the other hand, noise and unexpected nonlinearities are frequently observed in field tests, and they reduce the signal-to-noise ratio. Therefore, it is necessary to improve the accuracy of crack detection using the nonlinear ultrasonic modulation technique by separating noise and weak modulated waves. In this study, we introduce algorithms for crack detection using nonlinear ultrasonic wave modulation and propose a new algorithm that can reduce the effort and time required for testing. The source of nonlinear wave modulation varies depending on the strain range, specimen material, etc. In this study, the clapping of the crack surface caused by excitation is considered as the main factor modulating the wave. The excitation amplitude was selected by considering the size of the specimen and position of the crack. Experiments are performed before and after generating fatigue cracks on aluminum specimens with various excitation frequency combinations. Next, the magnitudes of modulated waves generated under tensile and compressive force are observed, and crack detection is performed using the proposed algorithm. The occurrence of nonlinear waves and the magnitude change of the modulated waves due to an external force are numerically verified. It was possible to distinguish intact and damaged specimens by applying the proposed algorithm, and cracks could be detected even when a strong external force was applied to the specimen.
detecting cracks that uses the modulation phenomenon of waves passing through a crack. The compressive stress of ultrasonic waves can be transmitted through cracks; however, the tensile stress is not completely transmitted. Therefore, wave truncation occurs, which is observed as a variation of frequency components in the frequency domain through a Fourier transform. In particular, the peaks occurring at integer multiples of the excitation frequency are called as harmonics [26], and the peaks observed at frequencies corresponding to the sum and difference of the two excitation frequencies are referred to as sideband peaks, which are caused by modulation. A sinusoidal incident wave u (t ) with two different frequencies and a truncated wave utr (t ) propagating through a crack are expressed, respectively, as follows: (1)
u (t ) = A cos(2 f t ) + Ah cos(2 fh t ),
u tr (t ) = {A cos(2 f t ) + Ah cos(2 fh t )
Wcr } ×
(t ),
(2)
where A and f are the amplitude and frequency of the low-frequency wave, respectively, and Ah and fh are the amplitude and frequency of the high-frequency wave, respectively. Wcr is the crack width, and (t ) is a rectangular function that is equal to unity during crack closing, and zero otherwise. The Fourier-transformed utr (t ) can be expressed as follows:
F {utr (t)} = +
A Cn { (f 2
(n + 1) f ) + (f
Ah Cn { (f 2
(fh + nf ) + (f
Wcr Cn
nf ,
f
where
(n
1) f )}
( fh + nf ) )
(3)
( )
is the Dirac-delta function, Cn = T sinc T , is the pulse tr tr width, and Ttr is the period. The second term on the right-hand side of Eq. (3) indicates sideband frequencies that are equal to the summation and difference of the two input frequencies. n
2. Theory
2.1.2. High-order nonlinearity for a rough crack surface Wave truncation is an intuitive and well-formulated description of nonlinear wave modulation, but it has limitations in that it cannot accurately reflect the shape of the actual crack. Therefore, researchers have suggested various models to describe mechanism occurring in a real crack. Kawashima et al. explained the source of the wave modulation as a nonlinear stress-strain relationship [15]. Each part of rough crack surfaces behaves as a mechanical diode with a bilinear stressstrain relation, but the shape of the bilinear curve depends on the crack width and incident-wave amplitude. The nonlinear stress-strain relation formed by superposing these bilinear curves induced the wave modulation. Some studies additionally introduced the material hysteresis phenomenon as the source of wave modulation [27,28]. With the hysteresis effect, nonlinear characteristics such as higher harmonics and the natural frequency shift can be explained. In addition to the superposed bilinear model and hysteresis effect, which are relatively simple and easy to be implement, more realistic and complex models have been proposed. The rough surface contact model based on the Hertzian contact theory can cover all physical phenomena, but it is extremely complex [5,29]. Recently, non-classical dissipation mechanisms such as thermal dissipation at crack edges and the Luxemburg-Gorky (LG) effect have been proposed [30,31]. Nonclassical dissipation phenomena are generally observed at smaller strain ranges, and the approach can explain phenomena that cannot be explained using the classical models [12,29].
2.1. Generation of modulated wave
2.2. Crack-detection algorithms
2.1.1. Wave truncation when passing through a planar crack The nonlinear ultrasonic modulation technique is a method of
2.2.1. Direct comparison before and after crack generation The nonlinear ultrasonic modulation technique uses the sideband 2
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Fig. 1. Fourier-transformed result obtained from the nonlinear wave modulation technique performed with two different sinusoidal waves. In (a) and (c), the signals were measured from an intact specimen, and in (b) and (d), the signals were measured from a damaged specimen. Locations of the modulation frequencies (MF) are annotated.
occurrence in the frequency domain after the Fourier transform of the measured signal as a criterion. The most basic method is to compare the data before and after crack generation in the specimen. It is possible to determine whether a crack has occurred by comparing the magnitude of the sideband measured in the initial intact state and that measured after some time has passed. Fig. 1 shows the results of a nonlinear wave modulation test with a 3-mm-thick aluminum plate specimen in the frequency domain obtained through Fourier transformation. Figs. 1(a) and (c) show the results obtained from the intact specimen, while Fig. 1(b) and (d) show the results obtained after a fatigue crack was formed. By comparing Fig. 1(a) and (b), the difference in the generation of peaks at the sideband frequency is observed. Clear peaks are observed only when the specimen contains defects. However, in Fig. 1(c) and (d), it is difficult to identify the integrity of the specimen using the sideband generation. Even when the specimen is in an intact condition, a small amount of wave modulation might occur because of an internal pores or material nonlinearity. In contrast, the magnitude of the modulated wave is insufficient when the movement of the crack surfaces is inactivated depending on the wavelength and plate geometry. Therefore, it is necessary to repeat the test with various conditions and integrate the data synthetically because finding defects in a single test does not guarantee accuracy.
Fig. 2. Range of interest in the frequency domain when low- and high-frequency waves are applied for excitation. By measuring in the gray hatched areas, one modulated wave and (n-1) noise values will be obtained on both sides of the high frequency.
The n2 data from the left sideband can be represented by a threedimensional plot, as shown in Fig. 3. The figure shows 11 test results with a high frequency of 181 kHz, and it contains 112 data. The surface plot represents the magnitude measured at each frequency of interest and is projected as a contour. In this contour grid, the sideband frequencies are located on the diagonal line. On the contour grid, the position with the largest value for one low frequency is marked with a red inverted triangle, and the sideband frequencies are indicated with blue diamonds. The colocation of the two shapes indicates the occurrence of wave modulation. Fig. 3(a) and (b) show the results measured from an intact specimen and damaged specimen, respectively. Because the modulation frequency lies on the diagonal line in the surface plot, values significantly larger than the surrounding noise are observed in Fig. 3(b). Although remarkable peaks are found on the diagonal line in Fig. 3(a), those are not as obvious as in Fig. 3(b). This section describes a crack-detection algorithm that determines the integrity of the structure based on two factors, namely the damage index (DI) and intactness index (II), which are calculated from the observation of modulated-wave generation. DI is a factor related to the occurrence of the modulated wave in each test. For each test, a
2.2.2. Comparison with ambient noise along two axes This section proposes an algorithm for safety diagnosis by comparing the modulated wave with ambient noises without any additional experiment. In this method, ambient noises around the sideband frequencies are used to determine the occurrence of wave modulation. If n low frequencies are used for a high frequency fhj in the experiment, each low frequency can be expressed as f i , where i is a natural number ranging from one to n . The magnitude of the frequency spectrum ranges from fhj ± f 1 to fhj ± f n when two incident waves f i and fhj are applied to the specimen, as shown in Fig. 2. The k th frequency from the high frequency is denoted as fhj ± f ik , and the sideband frequency is denoted as fhj ± f ii . The values measured at the remaining (n-1) frequencies are noise values that are not related to the modulated wave. For one high frequency, n low frequencies are used, and n data are measured for each frequency combination. Therefore, a total of 2n2 data are collected when the experiments are performed with n low frequencies because sideband frequencies are located on both side of high frequency. 3
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Fig. 3. Three-dimensional surface plot representing the magnitudes of the frequency spectra from a nonlinear ultrasonic modulation test measured in a range of interest. The frequencies of the sideband are located on the diagonal line, and the off-diagonal elements are noise. The tests were performed with (a) an intact specimen and (b) a cracked specimen.
frequency that has the maximum value among n data obtained near the sideband frequency is found. If the magnitude at the sideband frequency is more significant than ambient noise, DI is increased by 1. By performing (n × m ) tests with combinations of n low frequencies and m high frequencies, it is possible to make 2 × (n × m) comparisons for both sidebands. In general, DI has a value close to 2 × (n × m) because a modulation occurs in most cases, as shown in Fig. 3(b). If DI is 2 × (n × m) , it means that wave modulation has occurred for all frequency combinations, which may indicate the existence of a defect in the specimen. If DI is less than 2 × (n × m) , the wave modulation might not appear with some frequency combinations. Modulation may not occur in damaged specimens depending on the frequency combination. Therefore, the integrity of the specimen cannot be ensured even when DI is less than 2 × (n × m) . Additional comparisons will proceed with II. II is determined based on the noise level measured at fhj ± f ik , which is greater than the magnitude at the sideband frequency fhj ± f ii , by comparing it with the magnitude at another sideband frequency fhj ± f kk that is obtained from a test with a different frequency combination. If the noise level at fhj ± f ik is larger, II will be increased by 1. The existence of a noise level larger than the magnitude at two different sideband frequencies implies that the modulation of waves does not occur actively, which suggests that the specimen does not have contain a crack. In contrast, if the specimen is damaged, the existence of a noise level greater than the magnitude of two different modulated wave signals is highly unlikely. Therefore, the specimen is considered damaged when II is zero although the DI is less than 2 × (n × m) . If DI is less than 2 × (n × m) but II is larger than a threshold T , the judgment is postponed. This means that the modulated wave does not occur in several tests, but the frequency of occurrence of modulation at the sideband frequency is not negligible. In this case, additional tests are required with a different frequency combination or stronger excitation. The threshold T can be determined based on the experimental conditions and environment. Fig. 4 shows a flowchart of the data post-processing algorithm described in this section. M (f ) is the magnitude of the frequency spectrum at frequency f . Fig. 4 shows the procedure to check whether the noise value measured at fhj ± f ik is an outlier during the process. Depending on the experiment environment, an external signal that is significantly larger than ambient noise is occasionally observed. It reduces the accuracy of the test, which is based on the occurrence of wave modulation. Therefore, a noise level greater than a threshold calculated under the assumption that (n 1) noise values in each test follow an exponential distribution is regarded as an outlier. A confidence interval of 99.99% was used for threshold calculation. The description above corresponds to the upper half of Fig. 4. The analysis was performed for both sidebands and ambient noise
separately, following which the process was repeated once more with the sum of magnitude at both sidebands. When using vibration, rather than wave propagation, by extending the duration, the magnitudes at both sideband frequencies show a difference because it is correlated with the vibration characteristics of the structure such as geometry and boundary conditions [32]. The magnitude at one sideband can be consistently small. When a wave in a specific frequency band has difficulties in propagating, the specimen is determined to have no damage with a test considering both sidebands separately. Therefore, the accuracy of the test can be improved by applying the algorithm once more after adding two magnitudes at both sidebands. 3. Experimental verification 3.1. Tests before and after crack generation without external load Experiments were conducted to verify the accuracy of the proposed algorithm before and after generating fatigue cracks in the specimen, and the algorithm was applied to the results. Fig. 5 shows the geometry and arrangement of the aluminum specimen and PZT patches used for the experiment. The specimen is made of 3-mm thick aluminum 6061 material, which has a density of 2700 kg/m3 and Young's modulus of 70 GPa. Two rectangular PZTs were attached to the surface of the specimen, and one circular PZT was attached to the opposite side of the crack for measurement. Both the rectangular and circular PZTs are made by APC International with APC 850 piezoelectric material because the characteristic of APC 850 material is suitable for both sensing and actuating. The piezoelectric properties of APC 850 are listed in Table 1. A notch of 5-mm length is made at the center of the specimen to concentrate the stress during the fatigue test. Sections of 40-mm length on each side of the specimen were fixed on an Instron 8801 dynamic test machine in the fatigue-crack generation phase and nonlinear wave modulation test phase under loads. A low frequency of 4 V and a high frequency of 2 V with a conversion rate of 1 MHz were used for excitation, and the response was measured with a 1-MHz sampling rate for 100 ms. The width of the crack is smaller than 20 µm , and the specimen is quite small. The excitation amplitudes were selected to induce crack clapping in consideration of the crack and specimen size. Excessively strong excitations are not suitable in this experiment because they prevent observation, which is based on the application of an external force on the specimen. The nonlinear ultrasonic modulation test was performed after the intact specimen was fixed on the test machine. After the test, a fatigue crack was generated using a sinusoidal tensile force, following which the wave modulation test was performed once more with the cracked specimen. The specimen is fixed to the machine to unify the boundary conditions across all experiments. The nonlinear wave modulation test was performed 33 times with combinations of 11 low frequencies and 3 4
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Fig. 4. Flowchart of the algorithm for analyzing the nonlinear wave modulation test. The process is performed for each sideband and repeated for the summation of both sidebands. Table 1 Piezoelectric properties of APC 850 material. Property
Symbol
Value
Electromechanical coupling factor
kp
0.63
Piezoelectric charge constant (10−12 m/V)
Fig. 5. Geometrical information of the aluminum plate and the arrangement of the PZT patches used in the experiment.
Poisson ratio
Density (kg/m3 )
high frequencies. Experiments were conducted while increasing the low frequency from 30 kHz to 40 kHz and the high frequency from 181 kHz to 183 kHz by increment of 1 kHz. The excitation frequencies were selected carefully to avoid the case where sideband frequencies coincide with the harmonic frequencies of the low frequency. After testing in the intact state, a sinusoidal tensile load ranging from 0.7 kN to 7 kN was applied with 10-Hz frequency to generate a fatigue crack until the length of the crack reached 6 mm. After the fatigue-crack generation, the tests were repeated with the damaged specimen. Table 2 lists the results by analyzing the experiment results obtained before and after fatigue cracking using the proposed algorithm. Three
k33 k31 k15 d33 d31 d15
0.72 0.36 0.68 400 - 175 590 0.35 7700
specimens were diagnosed based on the obtained DI and II, and the values that were used to determine the specimen damage are indicated in bold characters. All three specimens were concluded to be intact owing to the failure to find evidence of damage to the specimen from the experiment before crack generation. Specimens 1 and 3 showed sidebands greater than the ambient noise level in all tests. In the case of specimen 2, the two sidebands showed values smaller than the ambient noise level. However, since II was zero, the specimen seems to be damaged. The damage states of all three specimens were accurately diagnosed by the proposed algorithm. 5
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Table 2 Analysis of the experiment results by applying the proposed algorithm. DIs and IIs are counted, and the integrity of the specimen is tested with these two indexes.
Specimen #1 Specimen #2 Specimen #3
Condition
Intactness index (II)
Damage index (DI)
Test result
Intact Damaged Intact Damaged Intact Damaged
30, 17 0 4, 5 0 7, 2 0
23, 12 66 31, 16 64 41, 25 66
Intact Crack Intact Crack Intact Crack
Table 3 Experiment results for two damaged specimens under external forces. A lowfrequency excitation of 4 V and high-frequency excitation of 1 V were used. Specimen #4
3.2. Crack detection under tensile and compressive force
Specimen #5
When the crack surfaces are exposed to external forces, the signalto-noise ratio is decreased because the occurrence of wave modulation is reduced, causing difficulties in crack detection. In this section, the test is performed while applying tensile and compressive forces, and crack detection is attempted by analyzing the results with the proposed algorithm. As in the previous test, the specimen was fixed on the Instron 8801 test machine, and a nonlinear ultrasonic modulation test was performed with an external force ranging from 10 kN to −10 kN in decrements of 2 kN. The tests are repeated 33 times for combinations of 11 low frequencies and 3 high frequencies. In a 3-mm-thick aluminum specimen, slight bending was observed under 10-kN compression, and buckling occurred at −11 kN. Fig. 6 shows the changes in NI when tensile and compressive forces are applied on two identical specimens, where MD and MS are the magnitudes of frequency spectra at the sum and difference of the two excitation frequencies, respectively. In this case, NI is calculated by adding all magnitudes of frequency spectra at sideband frequencies for the 33 excitation combinations. The x-axis represents the magnitude of the external force applied to the specimens, and the y-axis represents NI. Fig. 6(a) and (c) show the change of modulation-wave generation according to the compression. The magnitude of the modulated waves has a maximum value when the compressive force is between 2 kN and
Tensile force (kN) Intactness index (II) Damage index (DI) Test result
0 0
2 0
4 0
6 0
8 0
10 0
63 crack
65 crack
65 crack
64 crack
63 crack
62 crack
Compressive force (kN) Intactness index (II) Damage index (DI) Test result
–
2
4
6
8
10
–
0
0
0
0
0
– –
65 crack
63 crack
65 crack
65 crack
65 crack
Tensile force (kN) Intactness index (II) Damage index (DI) Test result
0 0
2 0
4 0
6 0
8 0
10 2,0
60 crack
61 crack
63 crack
63 crack
62 crack
53,41 crack
Compressive force (kN) Intactness index (II) Damage index (DI) Test result
–
2
4
6
8
10
–
0
0
0
1,0
0
– –
61 crack
63 crack
58 crack
54,31 crack
53 crack
4 kN because the interaction of crack surfaces becomes more active as the distance between crack surfaces decreases because of the compression. In Fig. 6(a), the modulated-wave generation slightly increases under compressions of 8 kN and 10 kN. During the test with specimen 1, bending was observed at compression greater than 8 kN, and it seems that the entire crack surface was not pressurized. This phenomenon will be verified by numerical analysis in the next section. Fig. 6(b) and (d) show the change of modulated-wave generation with the tensile force. The amount of nonlinear wave generation increased as the tensile force increased up to 2 kN, and it decreased with further increase in the tensile force. The magnitudes are decreased continuously beyond 2 kN because the plate did not warp under tension.
Fig. 6. Nonlinearity index obtained from the two specimens under external forces. (a) and (b) show the results for specimen #1, while (c) and (d) show the results for specimen #2. 6
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4.2. Numerical modeling including realistic crack model
Table 4 Experiment results for two damaged specimens under external forces. A lowfrequency excitation of 4 V and high-frequency excitation of 3 V were used. DIs were increased compared to the results in 〈Table 3〉. Specimen #4
Specimen #5
Tensile force (kN) Intactness index (II) Damage index (DI) Test result
0 0
2 –
4 –
6 0
8 –
10 0
64 crack
66 crack
66 crack
65 crack
66 crack
65 crack
Compressive force (kN) Intactness index (II) Damage index (DI) Test result
–
2
4
6
8
10
–
–
–
–
–
–
– –
66 crack
66 crack
66 crack
66 crack
66 crack
Tensile force (kN) Intactness index (II) Damage index (DI) Test result
0 0
2 0
4 –
6 –
8 0
10 1
62 crack
64 crack
66 crack
66 crack
65 crack
60, 33 crack
Compressive force (kN) Intactness index (II) Damage index (DI) Test result
–
2
4
6
8
10
–
0
0
0
0
0
– –
64 crack
65 crack
64 crack
63 crack
54 crack
The modulation phenomenon as the wave passes through the cracks and the magnitude change of the modulated wave caused by the external force are observed in the numerical simulation with finite-element analysis [34]. Numerical analysis is performed with the commercial finite-element analysis software LS-Dyna [35]. The shape of the numerical crack model becomes realistic as the number of elements used for mimicking the crack increases, but the calculation time increases dramatically as the number of elements increases. Therefore, the size of the elements near the crack and that of the others are set differently, and they are combined by use of the tied contact function. The size of the elements near the fatigue crack is set to 0.1 mm, and others have a size of 1 mm. The length of the crack was 6 mm, and the numerical crack model was discretized with 60 elements of 0.1 mm size and 61 nodes. Moreover, the 3-mm thick aluminum plate near the crack was modeled with thirty layers. Thus, 31 nodes are positioned along the thickness direction. To express the crack more realistically in the numerical simulation, a set of 1891 widths is sampled from the measured crack width dataset in Section 4.1 and reflected in the numerical model [33]. Therefore, the numerical aluminum plate model has a crack with an irregular shape. The crack width tends to decrease as it approaches the crack tip. In the experiment, sections of 40-mm length at each side of the specimen were fixed to the test machine, but in the numerical analysis, only a 150-mm-long section at the middle of the specimen was modeled to reduce the number of elements. The right side of the numerical plate is fixed, and the external force was adjusted through the displacement control of the left side of the plate, as shown in Fig. 7. Instead of attaching PZT patches, lateral sinusoidal waves of 10 nm in amplitude were incident on the nodes located at the position of rectangular PZTs. The lateral displacement was measured at the node at the center of the circular PZT attachment. Although the amplitudes of incident waves were 10 nm, a few µm movements were observed in the output signal owing to the reflection and interference effect of waves, and it is sufficient to induce the contact of crack surfaces. The excitation frequencies for numerical simulations were 41 kHz and 188 kHz, and the sideband frequencies were 147 kHz and 229 kHz. The wavelengths of the incident waves were approximately 151 mm and 33 mm for the low and high frequency, respectively. The analyses were conducted with
Table 3 summarizes the analysis of the experiment results performed with a low-frequency excitation of 4 V and high-frequency excitation of 1 V by using the proposed algorithm. We counted II and DI in each test under different external forces and diagnosed the specimen based on these indexes. Values that indicate specimen damage are expressed in bold characters. DI did not reach 66 owing to the weak excitation and the effect of the external force. However, II was mostly zero owing to the frequent wave modulation. In the test of specimen 5 under compression of 8 kN, 12 sidebands were not observed, and it was concluded that the damage had occurred in the specimen because the second II (II2) was 0. Table 4 lists the experiment results obtained when the high frequency was increased to 3 V. The DI value shows that modulation has occurred more actively compared to the results in Table 3. If the diagnosis is difficult owing to insufficient modulation, it is necessary to perform the test with a strong intensity. Fatigue cracks in two specimens could be found by using the algorithm in all cases with external forces. 4. Numerical simulations 4.1. Crack-width measurement Numerical analyses have been performed assuming that the crack plane is flat or that a crack has been modeled by deleting one element layer [17,18]. However, actual fatigue cracks do not have a flat and parallel shape owing to the plastic deformation and fragmentation. Therefore, to mimic the surface motion of fatigue cracks realistically, the width of the fatigue crack should be measured and applied to the finite-element model. The widths of the fatigue crack were measured from image-processed photomicrographs, and the finite-element model including a realistic fatigue crack was generated according to the method introduced by Lee et al. [33]. The image processing consists of three steps: skeletonization, pruning, and crack-width measurement. In this method, the crack widths were measured in the direction perpendicular to the centerline of the crack.
Fig. 7. Geometry of the numerical simulation using the finite-element model. The right side of the plate is fixed, and the left side is moved to pressurize the specimen. Waves are incident by controlling the displacement of the nodes located at the position of rectangular PZTs, and a signal is measured at the node located at the center of the circular PZT. 7
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Fig. 8. Numerical results of the nonlinear wave modulation technique in the frequency domain. The sidebands are marked with black arrows. The simulations are performed (a) without an external force, (b) with a compressive force of 10 kN, and (c) with a warped plate under a compressive force of 10 kN.
three different boundary conditions: without any external force, under a compression of 10 kN, and warping due to a compression of 10 kN.
Several methods for performing safety diagnosis with various input conditions and synthetically judging the integrity of a structure with the data have been suggested. In this paper, we present an algorithm that shows sufficient accuracy while reducing the cost of the experiment compared to existing methods. Nonlinear modulated wave tests were conducted before and after generating fatigue cracks on aluminum specimens. The excitation amplitudes were set to make crack clapping the main source of nonlinearity generation. If experiments are performed with a smaller strain range, additional reference data may be required for diagnosis. As a result, it was possible to determine the specimen damage accurately. In addition, tests were performed while applying an external force to the specimen containing fatigue cracks. Modulated wave generation was temporarily increased by weak tensile and compressive forces. However, as the force increased above a certain level, the interaction of cracks was disturbed, and the wave modulation decreased. In the case of compressive stress, the increment of the modulated wave was observed because of the warping of the thin plate specimen. This phenomenon was also observed in numerical analysis. Even with data that have a relatively small signal-to-noise ratio due to the external force, the crack could be detected by applying the proposed algorithm. The proposed nondestructive evaluation method with ultrasonic waves and post-processing algorithms shows a high accuracy and can reduce the effort and time required for testing. It is also possible to detect cracks under compression, which are known to be difficult to detect with conventional methods. Modulated waves could occur even if the signal-to-noise ratio is decreased because of the irregular shape of crack surfaces that is caused by the plastic deformation and fragmentation during crack propagation. Future studies may investigate the change of generation of the modulated wave according to the shape and direction of the crack under an external force.
4.3. Simulation results Fig. 8 contains the displacement measured in each numerical analysis in the frequency domain through the Fourier transform. Fig. 8(a) shows various frequency components such as harmonics and sidebands resulting from the active interaction of crack surfaces due to the absence of an external force. Numerical results also include numerical errors due to finite precision and truncation errors caused by approximate mathematical solutions. Various components were observed due to nonlinearity and numerical error, but sideband which means modulated wave was observed clearly. The 1st sideband marked with black arrows is clearly observed. However, in Fig. 8(b), a sideband distinct from surrounding noise is not observed. It can be seen that, under the strong compressive force, the generation of modulated waves decreases because the relative motion of crack surfaces is hindered by the compression. In Fig. 8(c), the nonlinear wave modulation test is performed with an initial displacement to the center of the plate in the normal direction of the plate to induce bending. Compared to Fig. 8(a), the size of the sideband reduced, but a clear peak is observed in contrast to Fig. 8(b). It is numerically confirmed that the generation of the modulated wave increased when bending occurred because of compression in the plate. 5. Summary and conclusions Micro-cracks that are difficult to detect with conventional linear methods can be successfully detected with the nonlinear wave modulation technique owing to its high sensitivity resulting from the use of crack-surface interaction. In the method, two distinct waves with different frequencies are incident on the structure, and safety diagnosis is performed based on the occurrence of a modulated wave that appears as a sideband in the frequency domain. However, owing to the presence of pores or material nonlinearity, weak modulated waves are observed even without cracks. Moreover, depending on the excitation frequency, wave modulation does not occur when the movement of a crack surface is insufficient. Therefore, it is necessary to repeat the test with various frequency combinations.
Acknowledgments This research was supported by a grant (17CTAP-C133220-01) from the Infrastructure and Transportation Technology Promotion Research Program funded by Ministry of Land, Infrastructure and Transport of the Korean government, and was also supported by the National Research Foundation of Korea grant funded by the Korean government (MSIT) (No.2018R1A2A1A05019453). 8
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