ULTRAS 4757
No. of Pages 10, Model 5G
21 February 2014 Ultrasonics xxx (2014) xxx–xxx 1
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras 5 6
Corrosion and erosion monitoring in plates and pipes using constant group velocity Lamb wave inspection
3 4 7
Q1
8 9 10
Q2
a
Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH 45221, USA Cincinnati NDE, Cincinnati, OH 45244, USA c ClampOn AS, 5162 Laksevaag, Bergen, Norway b
11 12
a r t i c l e
1 2 4 2 15 16 17 18 19 20 21
Peter B. Nagy a,b,⇑, Francesco Simonetti a,b, Geir Instanes b,c
i n f o
Article history: Available online xxxx
Q4
Keywords: Lamb waves EMATs Corrosion monitoring
a b s t r a c t Recent improvements in tomographic reconstruction techniques generated a renewed interest in shortrange ultrasonic guided wave inspection for real-time monitoring of internal corrosion and erosion in pipes and other plate-like structures. Emerging evidence suggests that in most cases the fundamental asymmetric A0 mode holds a distinct advantage over the earlier market leader fundamental symmetric S0 mode. Most existing A0 mode inspections operate at relatively low inspection frequencies where the mode is highly dispersive therefore very sensitive to variations in wall thickness. This paper examines the potential advantages of increasing the inspection frequency to the so-called constant group velocity (CGV) point where the group velocity remains essentially constant over a wide range of wall thickness variation, but the phase velocity is still dispersive enough to allow accurate wall thickness assessment from phase angle measurements. This paper shows that in the CGV region the crucial issue of temperature correction becomes especially simple, which is particularly beneficial when higher-order helical modes are also exploited for tomography. One disadvantage of working at such relatively high inspection frequency is that, as the slower A0 mode becomes faster and less dispersive, the competing faster S0 mode becomes slower and more dispersive. At higher inspection frequencies these modes cannot be separated any longer based on their vibration polarization only, which is mostly tangential for the S0 mode while mostly normal for the A0 at low frequencies, as the two modes become more similar as the frequency increases. Therefore, we propose a novel method for suppressing the unwanted S0 mode based on the Poisson effect of the material by optimizing the angle of inclination of the equivalent transduction force of the Electromagnetic Acoustic Transducers (EMATs) used for generation and detection purposes. Ó 2014 Elsevier B.V. All rights reserved.
23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
45 46
1. Introduction
47
Corrosion and erosion detection and monitoring are essential prognostic means of preserving material integrity and reducing the life-cycle cost of industrial infrastructure, ships, aircraft, ground vehicles, pipelines, oil installations, etc. Long-range guided wave inspection has the potential to extend ultrasonic corrosion measurements in pipes over very long distances [1–6]. Carefully selected extensional, flexural, or torsional ultrasonic guided waves in the pipe wall provide an attractive solution for long-range corrosion monitoring because they can be excited at one location on the pipe and will propagate along the pipe, returning echoes indicating the presence of corrosion or other pipe features. However, reflec-
48 49 50 51 52 53 54 55 56 57
Q3
⇑ Corresponding author at: Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH 45221, USA. Tel.: +1 513 556 3353; fax: +1 513 556 5038. E-mail address: [email protected] (P.B. Nagy).
tion measurements are rather sensitive to the presence of a distinct sharp transition between sections of different thickness. Transmission measurements in pitch–catch mode work better when no such localized transition exists and the wall thickness varies in a gradual manner. Dry-coupled piezoelectric transducer systems were shown to detect corrosion in chemical plant pipework using cylindrical Lamb waves in pulse-echo mode over distances approaching 50 m in steel pipes [7] and they can propagate through multiple bends [8]. It was also shown that low-frequency axisymmetric modes can propagate over long distances even in buried, water-filled iron pipes [9]. Most of such inspections are based on reflection measurements in pitch–catch mode [2,6,10,11]. Carefully selected extensional, flexural, or torsional ultrasonic guided waves in the pipe wall provide an attractive solution for long-range corrosion monitoring because they can be excited at one location on the pipe and will propagate along the pipe, returning echoes indicating the
http://dx.doi.org/10.1016/j.ultras.2014.01.017 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: P.B. Nagy et al., Corrosion and erosion monitoring in plates and pipes using constant group velocity Lamb wave inspection, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.01.017
58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74
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133 135 136 137 138 139 140
presence of corrosion or other pipe features. However, reflection measurements are rather sensitive to the presence of a distinct transition between sections of different thickness. Transmission measurements in pitch–catch mode work better when no such localized transition exists and the wall thickness varies in a gradual manner [12]. It was shown that ultrasonic guided wave attenuation measurements can be also exploited for the detection of wall loss due to corrosion [13]. Various wave modes can be used to best detect thinning of the pipe wall based on mode cutoff, group and phase velocity, transmission coefficient or attenuation measurements. For example, by carefully selecting the inspection frequency to match the range of wall thickness in the pipe, one can measure the group velocity of the S0 mode for corrosion monitoring [14]. Ultrasonic guided wave inspection methods can be also distinguished based on the generation and detection principles they rely on as well as the different physical principles of the transducers used. Conventional normal and angle beam transducers exhibit very different spatial and temporal frequency characteristics that can be analyzed using source influence theory [15]. Typically, inspection is based on a single carefully selected guided mode. However, in some cases, a multi-mode approach is adapted, e.g., by using a linear array comb transducer [16]. Guided waves generated by axisymmetric and non-axisymmetric surface loading have their distinct advantages and disadvantages [17]. Time-delay periodic ring arrays have been used to generate axisymmetric guided wave modes in hollow cylinders [18]. Most structural health monitoring (SHM) systems focus on crucial areas that are particularly susceptible for damage, e.g., erosion or corrosion. In such cases localized inspection strategies are preferable over long-range inspection that inevitably sacrifices detection sensitivity to maximize area coverage. Recently, Cawley et al. devised an optimal inspection strategy for designing a permanently installed corrosion/erosion monitoring (CEM) system [19]. When relatively small wall thickness loss is expected more or less uniformly distributed over the area of interest, a small number of spot sensors should be used. When the loss tends to be severe and concentrated at a few unpredictable locations, an averagingtype area monitoring system is preferable. The decision is harder when moderate loss is expected over a significant but unpredictable fraction of the surface [19]. Short-range ultrasonic guided wave tomography (GWT) is especially well suited to map the wall loss distributed over the targeted area from a limited number of transducer locations [20–24]. In a typical GWT configuration, a pair of transmitting and receiving ring arrays of ultrasonic transducers surrounds the area to be monitored. Different combinations of the array elements are used to transmit and receive guided wave signals to interrogate the area of interest from multiple directions. Each received signal carries information about the geometrical characteristics of the encountered defects, which is then decoded using appropriate reconstruction algorithms. Ultrasonic guided waves are particularly well suited for inspection of pipelines. In relatively thin-walled pipes, the guided waves can be approximated as Lamb modes propagating along helical paths that allow the same mode to arrive to the receiver at different times [25]. Fig. 1 shows a schematic diagram of the three lowest-order helical paths along and around a cylindrical pipe. The propagation length of the nth-order helical path is.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ‘n ¼ z2 þ ða þ pnDÞ2 ;
ð1Þ
where z and a are the axial and azimuthal distances between the transducers, respectively, D is the average diameter of the pipe, and n is the azimuthal order. Although higher-order helical modes are somewhat more affected by the circumferential curvature of the pipe, in thin-walled
n = +1 n=0
D
75
P.B. Nagy et al. / Ultrasonics xxx (2014) xxx–xxx
a
2
n = -1 z Fig. 1. A schematic view of the three lowest-order helical paths in a cylindrical pipe.
pipes their velocity is still almost the same as that of the zero-order mode that follows the most direct path between the transmitter and the receiver therefore all modes can be crudely approximated by the corresponding Lamb mode in a flat plate. The problem is complicated by the fact that a number of dispersive Lamb modes can propagate in each direction in a pipe of given material depending on its wall thickness and the inspection frequency range used by the monitoring system. Because of the highly dispersive nature of Lamb modes, fast modes following longer helical paths (n = 1, 2, 3, . . .) can actually beat slower modes following the shortest direct route (n = 0), therefore the observed vibration at the location of the receiver is much more complicated than one would assume based on direct Lamb wave propagation only. As an example, Fig. 2 shows the (a) phase and (b) group velocity dispersion curves, respectively, for Lamb waves in a steel plate. Both velocities were normalized to the shear velocity cs of the material. In these calculations the longitudinal and shear bulk velocities were assumed to be cd = 5900 m/s and cs = 3200 m/s. Of particular interest in the following will be the region surrounding the point where the group velocity of the A0 mode reaches its maximum, the so-called constant group velocity (CGV) point, which is around fd 1.4 MHz mm for steel. This region is indicated by an open circle in Fig. 2. Even without the added complexity of higher-order helical modes, separation of numerous dispersive Lamb modes presents a formidable problem and renders reliable inversion all but impossible. Therefore, most guided wave inspections are conducted at frequencies well below the cut-off frequency of the first-order asymmetric Lamb mode so that only the two fundamental, i.e., zeroth-order, Lamb modes are present. At low frequencies, the fundamental symmetric or S0 mode is a simple dilatational plate vibration with weak dispersion and mostly in-plane vibration while the fundamental asymmetric or A0 mode is a flexural plate vibration with strong dispersion and mostly out-of-plane vibration.Generally speaking, dispersion is good since it provides sensitivity to wall thickness variations, while out-of-plane displacement is bad since it provides strong coupling to the surrounding medium and results in strong attenuation through energy leakage. Consequently, neither of the fundamental modes is particularly useful below a certain minimum frequency, especially because the increasing wavelength limits the spatial resolution of any inspection scheme. As the frequency increases the differences between the S0 and A0 modes decrease and, in some respects, even reverse. The S0 mode becomes more dispersive while the A0 mode becomes less dispersive and the aspect ratio of their elliptically polarized surface displacement trajectories decreases. Reversal occurs roughly around the point when the clockwise rotation of the surface particle displacement produced p by ffiffiffi the S0 mode changes to counterclockwise rotation (cp ¼ 2cs ) that is around fd 2.4 MHz mm for steel. Above this frequency thickness product, the S0 mode exhibits lower group velocity than the A0 mode, but this happens outside the interest of frequency range where only the two fundamental modes exist.
Please cite this article in press as: P.B. Nagy et al., Corrosion and erosion monitoring in plates and pipes using constant group velocity Lamb wave inspection, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.01.017
141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193
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No. of Pages 10, Model 5G
21 February 2014 3
Normalized Group Velocity [mm/μs]
Normalized Phase Velocity [mm/μs]
P.B. Nagy et al. / Ultrasonics xxx (2014) xxx–xxx
2.5 2 1.5 1 symmetric modes
0.5
asymmetric modes CGVpoint
0 0
2
4
6
8
10
Frequency ×Thickness [MHz mm]
(a)
2.5 symmetric modes asymmetric modes CGVpoint
2 1.5 1 0.5 0 0
2
4
6
8
10
Frequency ×Thickness [MHz mm]
(b)
Fig. 2. Normalized (a) phase and (b) group velocities versus frequency thickness in steel.
195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224
In a recent study, Huthwaite et al. presented a detailed investigation of the advantages and disadvantages of the S0 and A0 modes for guided wave tomography and compared their performance using both numerical and experimental data [26]. As expected, the sensitivity of the A0 mode to thickness variations was shown to be superior to that of the S0 mode, while the leaky attenuation caused by liquid loading was found much higher for the A0 mode than for the S0. The A0 mode was shown to be less sensitive to the presence of various surface coatings than the S0 mode. Finally, the authors found that both modes could achieve similar levels of spatial resolution in thickness mapping. The final decision between the two fundamental modes depends on the issue of transduction. Since the S0 mode is faster in the frequency region of interest for corrosion/erosion monitoring, it will always arrive first along the direct route, so any other modes that will arrive later can be removed easily by gating. In contrast, when using the A0 mode the transducers must be carefully optimized to suppress the excitation of S0 waves. When relying on later arriving helical modes in pipe inspection, the speed advantage of the S0 mode is less significant, so if the S0 mode is used, A0 mode generation and detection must be suppressed [26]. In order to illustrate the necessity of single mode generation and reception, Fig. 3 shows the cascade plot composed of the received signals of 16 receivers distributed along the circumference of a D = 226 mm diameter steel pipe of d = 7.4 mm wall thickness. The transmitter was z = 600 mm away from the plane of the receiver array positioned at the same circumferential position as Receiver #1. The transmitted signal was a 4-cycle windowed toneburst of f = 180 kHz carrier frequency, corresponding to fd 1.33 MHz mm. The first arrival at each receiver is the direct (n = 0) S0 mode followed by the direct A0 mode at Receiver #1
Channels
194
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0
100
200
300
400
500
600
700
Time [µs] Fig. 3. Cascade plot of the received signals detected by 16 transducers distributed along the circumference of a D = 226 mm diameter steel pipe.
and by the S0 mode along the shortest helical path (n = 1 for a positive offset value of a) for most other receivers. The complexity of the wave patterns exhibited by the later parts of these signals demonstrates that it is essential to suppress one of the fundamental modes in order to facilitate unequivocal identification of the direct path from the higher-order helical paths. Additional requirements in short-range GWT include the need to minimize ultrasonic scattering from neighboring transducers and to maximize the temperature stability of not only the transducers themselves but also the coupling between the transducers and the structure to be monitored. These requirements heavily favor noncontacting Electromagnetic Acoustic Transducers (EMATs) over conventional contact piezoelectric transducers, especially since, in SHM with permanently deployed transducers, the inherently lower transduction sensitivity of EMATs can be fairly easily compensated by more excessive signal averaging. Later in this paper we will demonstrate that in the frequency range of interest the S0 mode transduction sensitivity of EMATs can be fairly easily suppressed without affecting the A0 sensitivity by controlling the geometrical features of the transducer while the opposite is not feasible. For this reason, in the following we will focus solely on ultrasonic guided wave monitoring of wall thickness variations using the A0 mode.
225
2. Constant group velocity mode
248
The fundamental asymmetric A0 mode is particularly well suited for wall thickness monitoring in the frequency range around the previously defined CGV point (see Fig. 1). CGV inspection exploits the wide plateau region where the group velocity is essentially independent of wall thickness variations, therefore the time of arrival of the signals to be monitored does not change [27]. Additional advantages of conducting wall thickness monitoring in the CGV region include relatively low susceptibility to attenuation due to absorbing coatings as well as to leakage into the surrounding fluid in the case of fluid-filled and/or immersed pipes. The good sensitivity of A0 guided wave inspection in the CGV regime is entirely due to the significant phase dispersion exhibited by this mode in this region. Fig. 4 shows the fundamental symmetric and asymmetric arrivals after z = 400 mm propagation in steel plates of different thickness. The leading S0 mode exhibits weak group and phase velocity dispersion. In comparison, the lagging A0 mode exhibits very strong phase velocity dispersion, but not perceivable group velocity dispersion. In order to determine the sensitivity of guided wave inspection based on phase measurement to wall thickness variation, let us assume that the fundamental flexural mode signal h(t) is gated by a
249
Please cite this article in press as: P.B. Nagy et al., Corrosion and erosion monitoring in plates and pipes using constant group velocity Lamb wave inspection, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.01.017
226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247
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303
sp ¼
Amplitude [a.u.]
9 mm 9.5 mm 10 mm 10.5 mm 11 mm
k
x
ð7Þ
;
and group slowness
sg ¼
306
307
@k @sp ¼ sp þ x @x @x
ð8Þ
are both related to the wave number k and the angular frequency x, the true phase angle can be re-written from Eq. (6) as follows
U ¼ x2 z 80
90
100
110
120
130
140
150
160
170
180
@sp : @x
ð9Þ
It is advantageous to introduce a normalized frequency
Time [µs] Fig. 4. Fundamental symmetric and asymmetric arrivals after z = 400 mm propagation in steel walls of different thickness.
n¼
136.8 µs 136.1 µs 135.3 µs 134.6 µs 133.9 µs
. Phase Angle [degree]
-90 -135
ð10Þ
cs
-270
FðnÞ ¼ n2
0.05
0.10
0.15
0.20
Frequency [MHz] Fig. 5. Phase spectra of the fundamental flexural mode at different gate positions (tw) in a steel plate of wall thickness d = 10 mm over propagation distance z = 400 mm.
271 272
273
275 276 277
278 280 281 282 283 284 285
286 288 289 290 291 292
293 295 296 297 298
299 301 302
ð12Þ
rectangular window of tw center delay and Tw total length. The magnitude H(x) and phase u(x) spectra of the gated signal can be obtained from the complex Fourier transform as follows
jHðx; t w Þjeiuðx;tw Þ ¼
Z
t w þT2w
tw T2w
hðtÞeixðttw Þ dt;
ð2Þ
where t and x denote time and angular frequency. The phase spectrum can be calculated from
uðx; tw Þ ¼ x tw sp ðxÞz ;
ð3Þ
where sp denotes the phase slowness and z is the propagation distance. As it is illustrated in Fig. 5, the phase spectrum is a function of the gate position even if no truncation of the signal occurs. Within close vicinity of the center frequency of inspection x0, this relationship is more or less linear
uðtw ; xÞ u0 þ u1 ðx x0 Þ;
ð4Þ
where u0 and u1 are linear regression coefficients. It is advantageous to introduce the so-called ‘‘true’’ phase angle U as the interception point of the local linear regression with the ordinate axis (x = 0)
u ¼ u0 u1 x0 ;
ð5Þ
which is independent of the gate position [28]. Assuming that the center of the gate position in Eq. (3) is equal to the group time of arrival at the center frequency
U ¼ x sg ðxÞ sp ðxÞ z; where sg is the group slowness. Since the phase slowness
ð6Þ
@PðnÞ : @n
ð13Þ
2
!
@sg 1 @P @ P ¼ 2 þn 2 : @n @n cs @n
320 322
326 327 328
329 331 332 333 334 335 336 337 338 339
340
342
!
@F @P @2P ¼ n 2 þn 2 : @n @n @n
343 344
345
ð15Þ 347
It can be easily seen from Eqs. (14) and (15) that they have the same roots, which proves the above mentioned coincidence. Therefore, in a fairly wide range around the CGV point, the true phase angle can be approximated using the peak of the normalized sensitivity function as follows
0.80 CGV point
0.70
0.60
0.50 1
318
ð14Þ
From Eq. (13), the derivative of the normalized sensitivity function is
0
314
324
Fig. 6 shows the normalized sensitivity function versus the normalized frequency for steel. Interestingly, the sensitivity function peaks at the same value of the normalized frequency as the group velocity, i.e., at the CGV point of n0 2.85, where its value is F0 0.73 radian or 42°. The coincidence of the peak group velocity and peak sensitivity locations is of great significance from the point of view of thermal stability [30]. From Eq. (8), the derivative of the group slowness is
Normalized Sensitivity, F [rad]
-360 0.00
312
323
where F is a normalized sensitivity function that is dependent only on Poisson’s ratio m through the normalized phase slowness P [29]
-315
270
ð11Þ
z U ¼ FðnÞ ; d
-225
311
319
Then, the true phase angle can be written as
-180
310
316
xd
PðnÞ ¼ sp cs :
-45
309
315
and normalized phase slowness 0
305
2
3
4
5
Normalized Frequency, ξ Fig. 6. Normalized sensitivity function versus the normalized frequency for steel.
Please cite this article in press as: P.B. Nagy et al., Corrosion and erosion monitoring in plates and pipes using constant group velocity Lamb wave inspection, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.01.017
348 349 350 351 352
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40
absolute phase true phase
Phase Angle [deg]
30 20 10 0 -10 -20 -30 -40 0
10
20
30
40
50
60
70
80
90
100
Temperature [°C] Fig. 7. Measured absolute and true phase angle variations as functions of temperature in a steel pipe (D = 200 mm, d = 12.5 mm, z = 330 mm and f = 120 kHz).
353 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375
z U F 0 : d
ð16Þ
This approximation illustrates not only the easiness of inverting the measured true phase angle into the average wall thickness, but it also assures the thermal stability of the measurement without external temperature compensation. As an example, Fig. 7 shows measured absolute and true phase angles versus temperature in a steel pipe (D = 200 mm, d = 12.5 mm, z = 330 mm and f = 120 kHz). Over a 90 °C temperature range the total variation of the true phase angle is less than ±5°, far less than the ±30° variation of the absolute phase, which is due to the temperature dependence of the shear wave velocity in steel. Since the effect of temperature on the Poisson coefficient is negligible, the sensitivity function F(n) is temperature independent to a very good approximation. Therefore, a change in temperature only causes a shift of the non-dimensional parameter n due to its dependence on cs, which however does not result in a significant change in the value of F due to the plateau region around the CGV point. Therefore, in the vicinity of the CGV point the true phase angle is effectively independent of temperature. For small wall thickness variations Dd
No. of Pages 10, Model 5G
21 February 2014 Ultrasonics xxx (2014) xxx–xxx 1
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras 5 6
Corrosion and erosion monitoring in plates and pipes using constant group velocity Lamb wave inspection
3 4 7
Q1
8 9 10
Q2
a
Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH 45221, USA Cincinnati NDE, Cincinnati, OH 45244, USA c ClampOn AS, 5162 Laksevaag, Bergen, Norway b
11 12
a r t i c l e
1 2 4 2 15 16 17 18 19 20 21
Peter B. Nagy a,b,⇑, Francesco Simonetti a,b, Geir Instanes b,c
i n f o
Article history: Available online xxxx
Q4
Keywords: Lamb waves EMATs Corrosion monitoring
a b s t r a c t Recent improvements in tomographic reconstruction techniques generated a renewed interest in shortrange ultrasonic guided wave inspection for real-time monitoring of internal corrosion and erosion in pipes and other plate-like structures. Emerging evidence suggests that in most cases the fundamental asymmetric A0 mode holds a distinct advantage over the earlier market leader fundamental symmetric S0 mode. Most existing A0 mode inspections operate at relatively low inspection frequencies where the mode is highly dispersive therefore very sensitive to variations in wall thickness. This paper examines the potential advantages of increasing the inspection frequency to the so-called constant group velocity (CGV) point where the group velocity remains essentially constant over a wide range of wall thickness variation, but the phase velocity is still dispersive enough to allow accurate wall thickness assessment from phase angle measurements. This paper shows that in the CGV region the crucial issue of temperature correction becomes especially simple, which is particularly beneficial when higher-order helical modes are also exploited for tomography. One disadvantage of working at such relatively high inspection frequency is that, as the slower A0 mode becomes faster and less dispersive, the competing faster S0 mode becomes slower and more dispersive. At higher inspection frequencies these modes cannot be separated any longer based on their vibration polarization only, which is mostly tangential for the S0 mode while mostly normal for the A0 at low frequencies, as the two modes become more similar as the frequency increases. Therefore, we propose a novel method for suppressing the unwanted S0 mode based on the Poisson effect of the material by optimizing the angle of inclination of the equivalent transduction force of the Electromagnetic Acoustic Transducers (EMATs) used for generation and detection purposes. Ó 2014 Elsevier B.V. All rights reserved.
23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
45 46
1. Introduction
47
Corrosion and erosion detection and monitoring are essential prognostic means of preserving material integrity and reducing the life-cycle cost of industrial infrastructure, ships, aircraft, ground vehicles, pipelines, oil installations, etc. Long-range guided wave inspection has the potential to extend ultrasonic corrosion measurements in pipes over very long distances [1–6]. Carefully selected extensional, flexural, or torsional ultrasonic guided waves in the pipe wall provide an attractive solution for long-range corrosion monitoring because they can be excited at one location on the pipe and will propagate along the pipe, returning echoes indicating the presence of corrosion or other pipe features. However, reflec-
48 49 50 51 52 53 54 55 56 57
Q3
⇑ Corresponding author at: Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH 45221, USA. Tel.: +1 513 556 3353; fax: +1 513 556 5038. E-mail address: [email protected] (P.B. Nagy).
tion measurements are rather sensitive to the presence of a distinct sharp transition between sections of different thickness. Transmission measurements in pitch–catch mode work better when no such localized transition exists and the wall thickness varies in a gradual manner. Dry-coupled piezoelectric transducer systems were shown to detect corrosion in chemical plant pipework using cylindrical Lamb waves in pulse-echo mode over distances approaching 50 m in steel pipes [7] and they can propagate through multiple bends [8]. It was also shown that low-frequency axisymmetric modes can propagate over long distances even in buried, water-filled iron pipes [9]. Most of such inspections are based on reflection measurements in pitch–catch mode [2,6,10,11]. Carefully selected extensional, flexural, or torsional ultrasonic guided waves in the pipe wall provide an attractive solution for long-range corrosion monitoring because they can be excited at one location on the pipe and will propagate along the pipe, returning echoes indicating the
http://dx.doi.org/10.1016/j.ultras.2014.01.017 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: P.B. Nagy et al., Corrosion and erosion monitoring in plates and pipes using constant group velocity Lamb wave inspection, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.01.017
58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74
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133 135 136 137 138 139 140
presence of corrosion or other pipe features. However, reflection measurements are rather sensitive to the presence of a distinct transition between sections of different thickness. Transmission measurements in pitch–catch mode work better when no such localized transition exists and the wall thickness varies in a gradual manner [12]. It was shown that ultrasonic guided wave attenuation measurements can be also exploited for the detection of wall loss due to corrosion [13]. Various wave modes can be used to best detect thinning of the pipe wall based on mode cutoff, group and phase velocity, transmission coefficient or attenuation measurements. For example, by carefully selecting the inspection frequency to match the range of wall thickness in the pipe, one can measure the group velocity of the S0 mode for corrosion monitoring [14]. Ultrasonic guided wave inspection methods can be also distinguished based on the generation and detection principles they rely on as well as the different physical principles of the transducers used. Conventional normal and angle beam transducers exhibit very different spatial and temporal frequency characteristics that can be analyzed using source influence theory [15]. Typically, inspection is based on a single carefully selected guided mode. However, in some cases, a multi-mode approach is adapted, e.g., by using a linear array comb transducer [16]. Guided waves generated by axisymmetric and non-axisymmetric surface loading have their distinct advantages and disadvantages [17]. Time-delay periodic ring arrays have been used to generate axisymmetric guided wave modes in hollow cylinders [18]. Most structural health monitoring (SHM) systems focus on crucial areas that are particularly susceptible for damage, e.g., erosion or corrosion. In such cases localized inspection strategies are preferable over long-range inspection that inevitably sacrifices detection sensitivity to maximize area coverage. Recently, Cawley et al. devised an optimal inspection strategy for designing a permanently installed corrosion/erosion monitoring (CEM) system [19]. When relatively small wall thickness loss is expected more or less uniformly distributed over the area of interest, a small number of spot sensors should be used. When the loss tends to be severe and concentrated at a few unpredictable locations, an averagingtype area monitoring system is preferable. The decision is harder when moderate loss is expected over a significant but unpredictable fraction of the surface [19]. Short-range ultrasonic guided wave tomography (GWT) is especially well suited to map the wall loss distributed over the targeted area from a limited number of transducer locations [20–24]. In a typical GWT configuration, a pair of transmitting and receiving ring arrays of ultrasonic transducers surrounds the area to be monitored. Different combinations of the array elements are used to transmit and receive guided wave signals to interrogate the area of interest from multiple directions. Each received signal carries information about the geometrical characteristics of the encountered defects, which is then decoded using appropriate reconstruction algorithms. Ultrasonic guided waves are particularly well suited for inspection of pipelines. In relatively thin-walled pipes, the guided waves can be approximated as Lamb modes propagating along helical paths that allow the same mode to arrive to the receiver at different times [25]. Fig. 1 shows a schematic diagram of the three lowest-order helical paths along and around a cylindrical pipe. The propagation length of the nth-order helical path is.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ‘n ¼ z2 þ ða þ pnDÞ2 ;
ð1Þ
where z and a are the axial and azimuthal distances between the transducers, respectively, D is the average diameter of the pipe, and n is the azimuthal order. Although higher-order helical modes are somewhat more affected by the circumferential curvature of the pipe, in thin-walled
n = +1 n=0
D
75
P.B. Nagy et al. / Ultrasonics xxx (2014) xxx–xxx
a
2
n = -1 z Fig. 1. A schematic view of the three lowest-order helical paths in a cylindrical pipe.
pipes their velocity is still almost the same as that of the zero-order mode that follows the most direct path between the transmitter and the receiver therefore all modes can be crudely approximated by the corresponding Lamb mode in a flat plate. The problem is complicated by the fact that a number of dispersive Lamb modes can propagate in each direction in a pipe of given material depending on its wall thickness and the inspection frequency range used by the monitoring system. Because of the highly dispersive nature of Lamb modes, fast modes following longer helical paths (n = 1, 2, 3, . . .) can actually beat slower modes following the shortest direct route (n = 0), therefore the observed vibration at the location of the receiver is much more complicated than one would assume based on direct Lamb wave propagation only. As an example, Fig. 2 shows the (a) phase and (b) group velocity dispersion curves, respectively, for Lamb waves in a steel plate. Both velocities were normalized to the shear velocity cs of the material. In these calculations the longitudinal and shear bulk velocities were assumed to be cd = 5900 m/s and cs = 3200 m/s. Of particular interest in the following will be the region surrounding the point where the group velocity of the A0 mode reaches its maximum, the so-called constant group velocity (CGV) point, which is around fd 1.4 MHz mm for steel. This region is indicated by an open circle in Fig. 2. Even without the added complexity of higher-order helical modes, separation of numerous dispersive Lamb modes presents a formidable problem and renders reliable inversion all but impossible. Therefore, most guided wave inspections are conducted at frequencies well below the cut-off frequency of the first-order asymmetric Lamb mode so that only the two fundamental, i.e., zeroth-order, Lamb modes are present. At low frequencies, the fundamental symmetric or S0 mode is a simple dilatational plate vibration with weak dispersion and mostly in-plane vibration while the fundamental asymmetric or A0 mode is a flexural plate vibration with strong dispersion and mostly out-of-plane vibration.Generally speaking, dispersion is good since it provides sensitivity to wall thickness variations, while out-of-plane displacement is bad since it provides strong coupling to the surrounding medium and results in strong attenuation through energy leakage. Consequently, neither of the fundamental modes is particularly useful below a certain minimum frequency, especially because the increasing wavelength limits the spatial resolution of any inspection scheme. As the frequency increases the differences between the S0 and A0 modes decrease and, in some respects, even reverse. The S0 mode becomes more dispersive while the A0 mode becomes less dispersive and the aspect ratio of their elliptically polarized surface displacement trajectories decreases. Reversal occurs roughly around the point when the clockwise rotation of the surface particle displacement produced p by ffiffiffi the S0 mode changes to counterclockwise rotation (cp ¼ 2cs ) that is around fd 2.4 MHz mm for steel. Above this frequency thickness product, the S0 mode exhibits lower group velocity than the A0 mode, but this happens outside the interest of frequency range where only the two fundamental modes exist.
Please cite this article in press as: P.B. Nagy et al., Corrosion and erosion monitoring in plates and pipes using constant group velocity Lamb wave inspection, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.01.017
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Normalized Group Velocity [mm/μs]
Normalized Phase Velocity [mm/μs]
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2.5 2 1.5 1 symmetric modes
0.5
asymmetric modes CGVpoint
0 0
2
4
6
8
10
Frequency ×Thickness [MHz mm]
(a)
2.5 symmetric modes asymmetric modes CGVpoint
2 1.5 1 0.5 0 0
2
4
6
8
10
Frequency ×Thickness [MHz mm]
(b)
Fig. 2. Normalized (a) phase and (b) group velocities versus frequency thickness in steel.
195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224
In a recent study, Huthwaite et al. presented a detailed investigation of the advantages and disadvantages of the S0 and A0 modes for guided wave tomography and compared their performance using both numerical and experimental data [26]. As expected, the sensitivity of the A0 mode to thickness variations was shown to be superior to that of the S0 mode, while the leaky attenuation caused by liquid loading was found much higher for the A0 mode than for the S0. The A0 mode was shown to be less sensitive to the presence of various surface coatings than the S0 mode. Finally, the authors found that both modes could achieve similar levels of spatial resolution in thickness mapping. The final decision between the two fundamental modes depends on the issue of transduction. Since the S0 mode is faster in the frequency region of interest for corrosion/erosion monitoring, it will always arrive first along the direct route, so any other modes that will arrive later can be removed easily by gating. In contrast, when using the A0 mode the transducers must be carefully optimized to suppress the excitation of S0 waves. When relying on later arriving helical modes in pipe inspection, the speed advantage of the S0 mode is less significant, so if the S0 mode is used, A0 mode generation and detection must be suppressed [26]. In order to illustrate the necessity of single mode generation and reception, Fig. 3 shows the cascade plot composed of the received signals of 16 receivers distributed along the circumference of a D = 226 mm diameter steel pipe of d = 7.4 mm wall thickness. The transmitter was z = 600 mm away from the plane of the receiver array positioned at the same circumferential position as Receiver #1. The transmitted signal was a 4-cycle windowed toneburst of f = 180 kHz carrier frequency, corresponding to fd 1.33 MHz mm. The first arrival at each receiver is the direct (n = 0) S0 mode followed by the direct A0 mode at Receiver #1
Channels
194
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0
100
200
300
400
500
600
700
Time [µs] Fig. 3. Cascade plot of the received signals detected by 16 transducers distributed along the circumference of a D = 226 mm diameter steel pipe.
and by the S0 mode along the shortest helical path (n = 1 for a positive offset value of a) for most other receivers. The complexity of the wave patterns exhibited by the later parts of these signals demonstrates that it is essential to suppress one of the fundamental modes in order to facilitate unequivocal identification of the direct path from the higher-order helical paths. Additional requirements in short-range GWT include the need to minimize ultrasonic scattering from neighboring transducers and to maximize the temperature stability of not only the transducers themselves but also the coupling between the transducers and the structure to be monitored. These requirements heavily favor noncontacting Electromagnetic Acoustic Transducers (EMATs) over conventional contact piezoelectric transducers, especially since, in SHM with permanently deployed transducers, the inherently lower transduction sensitivity of EMATs can be fairly easily compensated by more excessive signal averaging. Later in this paper we will demonstrate that in the frequency range of interest the S0 mode transduction sensitivity of EMATs can be fairly easily suppressed without affecting the A0 sensitivity by controlling the geometrical features of the transducer while the opposite is not feasible. For this reason, in the following we will focus solely on ultrasonic guided wave monitoring of wall thickness variations using the A0 mode.
225
2. Constant group velocity mode
248
The fundamental asymmetric A0 mode is particularly well suited for wall thickness monitoring in the frequency range around the previously defined CGV point (see Fig. 1). CGV inspection exploits the wide plateau region where the group velocity is essentially independent of wall thickness variations, therefore the time of arrival of the signals to be monitored does not change [27]. Additional advantages of conducting wall thickness monitoring in the CGV region include relatively low susceptibility to attenuation due to absorbing coatings as well as to leakage into the surrounding fluid in the case of fluid-filled and/or immersed pipes. The good sensitivity of A0 guided wave inspection in the CGV regime is entirely due to the significant phase dispersion exhibited by this mode in this region. Fig. 4 shows the fundamental symmetric and asymmetric arrivals after z = 400 mm propagation in steel plates of different thickness. The leading S0 mode exhibits weak group and phase velocity dispersion. In comparison, the lagging A0 mode exhibits very strong phase velocity dispersion, but not perceivable group velocity dispersion. In order to determine the sensitivity of guided wave inspection based on phase measurement to wall thickness variation, let us assume that the fundamental flexural mode signal h(t) is gated by a
249
Please cite this article in press as: P.B. Nagy et al., Corrosion and erosion monitoring in plates and pipes using constant group velocity Lamb wave inspection, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.01.017
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303
sp ¼
Amplitude [a.u.]
9 mm 9.5 mm 10 mm 10.5 mm 11 mm
k
x
ð7Þ
;
and group slowness
sg ¼
306
307
@k @sp ¼ sp þ x @x @x
ð8Þ
are both related to the wave number k and the angular frequency x, the true phase angle can be re-written from Eq. (6) as follows
U ¼ x2 z 80
90
100
110
120
130
140
150
160
170
180
@sp : @x
ð9Þ
It is advantageous to introduce a normalized frequency
Time [µs] Fig. 4. Fundamental symmetric and asymmetric arrivals after z = 400 mm propagation in steel walls of different thickness.
n¼
136.8 µs 136.1 µs 135.3 µs 134.6 µs 133.9 µs
. Phase Angle [degree]
-90 -135
ð10Þ
cs
-270
FðnÞ ¼ n2
0.05
0.10
0.15
0.20
Frequency [MHz] Fig. 5. Phase spectra of the fundamental flexural mode at different gate positions (tw) in a steel plate of wall thickness d = 10 mm over propagation distance z = 400 mm.
271 272
273
275 276 277
278 280 281 282 283 284 285
286 288 289 290 291 292
293 295 296 297 298
299 301 302
ð12Þ
rectangular window of tw center delay and Tw total length. The magnitude H(x) and phase u(x) spectra of the gated signal can be obtained from the complex Fourier transform as follows
jHðx; t w Þjeiuðx;tw Þ ¼
Z
t w þT2w
tw T2w
hðtÞeixðttw Þ dt;
ð2Þ
where t and x denote time and angular frequency. The phase spectrum can be calculated from
uðx; tw Þ ¼ x tw sp ðxÞz ;
ð3Þ
where sp denotes the phase slowness and z is the propagation distance. As it is illustrated in Fig. 5, the phase spectrum is a function of the gate position even if no truncation of the signal occurs. Within close vicinity of the center frequency of inspection x0, this relationship is more or less linear
uðtw ; xÞ u0 þ u1 ðx x0 Þ;
ð4Þ
where u0 and u1 are linear regression coefficients. It is advantageous to introduce the so-called ‘‘true’’ phase angle U as the interception point of the local linear regression with the ordinate axis (x = 0)
u ¼ u0 u1 x0 ;
ð5Þ
which is independent of the gate position [28]. Assuming that the center of the gate position in Eq. (3) is equal to the group time of arrival at the center frequency
U ¼ x sg ðxÞ sp ðxÞ z; where sg is the group slowness. Since the phase slowness
ð6Þ
@PðnÞ : @n
ð13Þ
2
!
@sg 1 @P @ P ¼ 2 þn 2 : @n @n cs @n
320 322
326 327 328
329 331 332 333 334 335 336 337 338 339
340
342
!
@F @P @2P ¼ n 2 þn 2 : @n @n @n
343 344
345
ð15Þ 347
It can be easily seen from Eqs. (14) and (15) that they have the same roots, which proves the above mentioned coincidence. Therefore, in a fairly wide range around the CGV point, the true phase angle can be approximated using the peak of the normalized sensitivity function as follows
0.80 CGV point
0.70
0.60
0.50 1
318
ð14Þ
From Eq. (13), the derivative of the normalized sensitivity function is
0
314
324
Fig. 6 shows the normalized sensitivity function versus the normalized frequency for steel. Interestingly, the sensitivity function peaks at the same value of the normalized frequency as the group velocity, i.e., at the CGV point of n0 2.85, where its value is F0 0.73 radian or 42°. The coincidence of the peak group velocity and peak sensitivity locations is of great significance from the point of view of thermal stability [30]. From Eq. (8), the derivative of the group slowness is
Normalized Sensitivity, F [rad]
-360 0.00
312
323
where F is a normalized sensitivity function that is dependent only on Poisson’s ratio m through the normalized phase slowness P [29]
-315
270
ð11Þ
z U ¼ FðnÞ ; d
-225
311
319
Then, the true phase angle can be written as
-180
310
316
xd
PðnÞ ¼ sp cs :
-45
309
315
and normalized phase slowness 0
305
2
3
4
5
Normalized Frequency, ξ Fig. 6. Normalized sensitivity function versus the normalized frequency for steel.
Please cite this article in press as: P.B. Nagy et al., Corrosion and erosion monitoring in plates and pipes using constant group velocity Lamb wave inspection, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.01.017
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absolute phase true phase
Phase Angle [deg]
30 20 10 0 -10 -20 -30 -40 0
10
20
30
40
50
60
70
80
90
100
Temperature [°C] Fig. 7. Measured absolute and true phase angle variations as functions of temperature in a steel pipe (D = 200 mm, d = 12.5 mm, z = 330 mm and f = 120 kHz).
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z U F 0 : d
ð16Þ
This approximation illustrates not only the easiness of inverting the measured true phase angle into the average wall thickness, but it also assures the thermal stability of the measurement without external temperature compensation. As an example, Fig. 7 shows measured absolute and true phase angles versus temperature in a steel pipe (D = 200 mm, d = 12.5 mm, z = 330 mm and f = 120 kHz). Over a 90 °C temperature range the total variation of the true phase angle is less than ±5°, far less than the ±30° variation of the absolute phase, which is due to the temperature dependence of the shear wave velocity in steel. Since the effect of temperature on the Poisson coefficient is negligible, the sensitivity function F(n) is temperature independent to a very good approximation. Therefore, a change in temperature only causes a shift of the non-dimensional parameter n due to its dependence on cs, which however does not result in a significant change in the value of F due to the plateau region around the CGV point. Therefore, in the vicinity of the CGV point the true phase angle is effectively independent of temperature. For small wall thickness variations Dd
