Contents lists available at ScienceDirect

Ultrasonics journal homepage: www.elsevier.com/locate/ultras

3D modeling of circumferential SH guided waves in pipeline for axial cracking detection in ILI tools Shen Wang ⇑, Songling Huang, Wei Zhao, Zheng Wei State Key Lab. of Power System, Dept. of Electrical Engineering, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

Article history: Received 9 January 2014 Received in revised form 3 July 2014 Accepted 13 August 2014 Available online 1 September 2014 Keywords: Guided waves Circumferential SH waves FEM Reﬂection and transmission coefﬁcients Defect sizing

a b s t r a c t In this paper, SH (shear horizontal) guided waves propagating in the circumferential direction of pipeline are modeled in 3 dimensions, with the aim for axial cracking detection implementation in ILI (in-line inspection) tools in mind. A theoretical formulation is given ﬁrst, followed by an explanation about the 3D numerical modeling work. Displacement wave structures from the simulation and dispersion equation are compared to verify the effectiveness of the FEM package. Transverse slots along the axial direction are modeled to simulate axial cracking. Reﬂection and transmission coefﬁcients curves are obtained to provide insight in using circumferential SH guided waves for quantitative testing of axial pipeline cracking. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction A large number of pipelines exist in petroleum, chemical, electric power and many other industries. These pipelines, mostly buried underground, serve as critical transmission tools for liquid and gas products. Measures must be adopted to check the pipelines for threats like pipe wall defects and ensure pipeline safety, so as to avoid potential economic loss, environmental pollution and casualties. Various non-destructive testing methods based on different principles are suitable for in-line inspection of these pipelines, and the choice of one particular method depends on the operation condition, transmitted product and defect types. Because of the special underground working condition, the transmission pipelines are generally difﬁcult or very expensive to access from the outside, so large-scale inner inspectors propelled by the high pressure of the transferred liquid or gas products are suitable for the inspection of these pipelines. Volumetric defects like corrosion are mostly detected using the MFL (magnetic ﬂux leakage) method. Leading companies owning MFL inspectors include GE PII, ROSEN, etc. SCC (Stress corrosion cracking), often found in the axial direction of the pipeline, is the result of the combined action of stress and corrosion environment. SCC in pipe walls is difﬁcult to detect, compared with normal volumetric corrosion defects, because its opening is often too tight to introduce a sufﬁcient amount of ﬂux leakage to be detected by the magnetic sensors, so the traditional axial direction MFL method is not applicable and the circumferential ⇑ Corresponding author. http://dx.doi.org/10.1016/j.ultras.2014.08.018 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.

direction MFL method also faces serious difﬁculty. One promising pipeline SCC detection method is based on ultrasonic guided waves traveling along the circumferential direction of the cross section of the pipeline, which can hopefully interact with axial cracking strongly. Ultrasonic guided waves are special ultrasonic waves propagating along the extension direction of bounded elastic media called waveguides [1]. Unlike bulk waves in traditional ultrasonic NDE, guided waves have the unique characteristic called dispersion which describes the phenomenon that the propagation velocity of guided wave depends on material properties, the excitation frequency and the structure of the material under investigation, with only surface wave as an exception. Typical waveguides include plate, pipe, half space surface, bar with arbitrary cross-section [2–4], etc. According to the propagation direction, guided waves in pipe wall could be further classiﬁed into two types, i.e. longitudinal and circumferential guided waves. Gazis derived ﬁrst the complete solution of longitudinal guided waves in pipes [5,6], including longitudinal, torsional and ﬂexural modes, and this kind of guided waves is the main subject of research [7] and application because it’s suitable for implementation in a portable inspection device. Circumferential guided waves were initially applied empirically using a plate approximation model, until Liu derived the detailed formulation of circumferential Lamb waves [8]. Valle studied the crack sizing and locating method using circumferential guided waves [9]. Zhao derived the formulation for circumferential shear horizontal (written as CSH hereafter) guided waves [10] and later used EMAT (electromagnetic acoustic transducer)-generated

326

S. Wang et al. / Ultrasonics 56 (2015) 325–331

CSH waves for the in-line nondestructive inspection of mechanical dents [11,12]. Luo used 2D BEM and the normal mode expansion technique to study the circumferential SH guided wave-based defect sizing for SCC [13,14], and compared the new circumferential model of pipeline with the plate model to study the validity of the plate approximation [15]. These studies enable a better understanding of the behavior of circumferential guided waves. In this paper, SH guided waves propagating in the circumferential direction of pipeline (CSH waves) are modeled in 3 dimensions, with the aim for axial cracking detection implementation in ILI tools in mind. A theoretical formulation is given ﬁrst, followed by an explanation about the 3D numerical modeling work. Displacement wave structures from the simulation and dispersion equation are compared to verify the effectiveness of the FEM package. Transverse slots along the axial direction are modeled to simulate axial cracking. Reﬂection and transmission coefﬁcients curves are obtained to provide insight in using circumferential guided waves for quantitative testing of axial pipeline cracking.

The formulation of circumferential SH guided waves [10] starts from Navier equation,

@2u @t 2

ð1Þ

in which k and l are Lamé constants, q is density, u is displacement vector. For CSH guided waves described by the coordinate system in Fig. 1, only the displacement component along the z axis/uz is nonzero, and uz is independent of variable z,

ur ¼ uh ¼ 0;

uz – 0;

@uz ¼0 @z

l

! ¼q

"

x

2 # kR U ðr Þ ¼ 0 r

2

CT

ð5Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ in which C T ¼ l=q is shear wave velocity. Eq. (5) is recognized as a Bessel equation, whose solution is Bessel function,

xr

CT

þ BY M

xr

CT

ð6Þ

in which M ¼ kR is the order of the Bessel functions of the ﬁrst and second kinds, J and Y. A and B are arbitrary unknown variables. The expression of the stress component srz is,

srz ¼ l

@ur @uz þ @z @r

Substituting (2) and (4), the previous expression is simpliﬁed as,

srz ¼ lU 0 ðrÞeiðkRhxtÞ The traction free boundary condition

ð2Þ

Navier Eq. (1) could then be simpliﬁed to the following expression,

1 @uz @ 2 uz 1 @ 2 uz þ 2 þ 2 r @r r @h2 @r

1 U ðr Þ þ U 0 ðr Þ þ r 00

U ðr Þ ¼ AJ M

2. Theoretical background of CSH guided waves

lr2 u þ ðk þ lÞrr u ¼ q

plate SH guided waves, expression like eiðkxxtÞ is used, in which x is the distance or position in the propagation direction. For CSH guided waves, Rh assumes the similar role, but its expression needs further consideration because of the existence of pipe curvature. R ¼ b was used in [10], (possibly) considering that the ultrasonic transducers used to generate circumferential SH guided waves are often placed at the outer pipe wall surface. In this paper, R ¼ ða þ bÞ=2 is proposed instead to represent the propagation distance (together with h). That is, the average value of outer and inner radii is used here. Substituting the above expression of uz to (3) we can get,

srz jr¼a;b ¼ 0 is applied to obtain,

@ 2 uz @t 2

ð3Þ

U 0 ðr Þr¼a;b ¼ 0

uz is only the function of r; h and t, and assumed to have the following form,

Substituting (6) we get a set of linear homogeneous equations with A and B as the unknown variables,

uz ðr; h; t Þ ¼ U ðr ÞeiðkRhxtÞ

8h i h i > < Ma J M xC a Cx J Mþ1 xC a A þ Ma Y M xC a Cx Y Mþ1 xC a B ¼ 0 T T T T T T h i h i > : Mb J M xC b Cx J Mþ1 xC b A þ Mb Y M xC b Cx Y Mþ1 xC b B ¼ 0 T T T T T T

ð4Þ

which states uz has a distribution along the radial direction. k is wave number deﬁned as k ¼ x=C P , in which C P is the phase velocity. R is the radius of the path of propagation for CSH guided waves. For

r uθ

ur uz

θ

O z

b a

Fig. 1. Cylindrical coordinate system used in this work.

the determinant must be zero to obtain non-trivial solutions,

d11

d12

d21

d22

¼0

in which dmn corresponds to the coefﬁcients in the previous set of homogeneous equations. This is just the dispersion equation from which dispersion curves could be calculated numerically. Fig. 2 shows the phase velocity and group velocity dispersion curves for CSH guided waves, propagating in a pipe with inner radius a = 49 mm, and outer radius b = 50 mm. The guided wave modes are numbered as ‘CSHn’, n = 0, 1, 2, . . .. The CSH0 mode is almost non-dispersive, while other modes are all dispersive. These dispersion curves seem just like the dispersion curves of SH guided waves propagating in a plate, which indicates the close relationship between these two kinds of guided waves. The material properties for the steel pipeline used in the calculation of dispersion curves are as follows, Young’s modulus E is 207 109 Pa, Poisson’s ratio v is 0.3, and the mass density q is 7800 kg/m3. These material properties will be used throughout this work.

S. Wang et al. / Ultrasonics 56 (2015) 325–331

To save some computation time, only the lower half section of pipeline is modeled, with the axial length of this section equal to 0.1 m. As used in the calculation of dispersion curves, the inner radius of the pipeline is a = 49 mm, and the outer radius is b = 50 mm, thus the ratio between them is g ¼ a=b ¼ 98%. Obviously this is a reduced model of the large diameter pipeline used for liquid and gas transmission. P1 and P2 are points to record the simulated waveforms, and they are at the inner surface of the pipeline model, and on the same half circle dashed line on the z = 0.05 m plane of the model. On the same z = 0.05 plane, P1 is at ðr ¼ a; h ¼ 315 Þ, and P2 is at ðr ¼ a; h ¼ 225 Þ. The type of the ﬁnite elements is C3D8R, a kind of three-dimensional solid element. When there is not any defect in the pipe, the cross section of the lower half pipe model is one single region. When there exists an axial defect in the pipe, the cross section of the pipe is divided into a defect region and two regions without any defects, and ﬁner ﬁnite elements will be used in the defect region to capture the behavior of guided waves near the simulated defect. The circumferential width in degrees of the defect region is 15°. The sizes of the elements must be selected carefully to obtain relatively accurate simulation results. The size of the element along the circumferential direction (also the propagation direction) leC must be selected to ensure that there exists a sufﬁcient number of elements in one wave length k, calculated from the phase velocity and frequency of the selected working point, of the chosen guided wave mode, which means that, if

10000

(a) 9000

CP (m/s)

8000

CSH2

7000 6000

CSH

1

5000 4000

CSH0 3000

0

1

2

3

4 x 10

Frequency (Hz)

6

3500

(b) CSH

0

2500

k ¼N leC

CSH

G

C (m/s)

3000

1

2000

1500

CSH 1000

2

0

1

2

3

Frequency (Hz)

4 x 10

6

Fig. 2. Dispersion curves of CSH waves. (a) Phase velocity dispersion curves, (b) group velocity dispersion curves.

3. Numerical modeling of CSH guided waves The FEM package of Abaqus is used in this study to simulate the behavior of circumferential SH guided waves. The numerical model is shown in Fig. 3.

Fig. 3. Numerical model used in the FEM simulation.

327

ð7Þ

then N should be at least 10 for a good spatial resolution, and the value of 20 is recommended [16]. Using more elements means longer computation times and a higher consumption of CPU resources and hard drive space, so a moderate number of elements should be applied while ensuring that sufﬁcient accuracy is achieved. When there’s no defect, N = 18 is used. When there exists a defect in the pipe, N = 10 is used for the regions without any defects and N = 15 is used for the defect region in this work. The selection of the size of the element along the radial direction (also the thickness direction) leR should ensure that the number of nodes in this direction is big enough to describe the wave structure (the distribution of displacement and stress components along the thickness direction) of the guided wave mode, and characterize the shape of the simulated defects. For the model without any defect, there exist 20 elements in the thickness direction, while for the model with axial defects, 60 elements are used. Finally, from various tests of simulation, the size of the element along the axial direction leA should also be small enough to capture accurately the beam spreading of the selected CSH mode in the axial direction, and leA ¼ 2 leC is used in the simulation, which proves sufﬁcient in most cases. In order to generate pure CSH modes, displacement boundary conditions are applied to one end of the lower half pipeline model (‘excitation zone’ in Fig. 3) according to the displacement wave structure of the desired mode. Even though the displacement wave structure could be described with an analytical expression for CSH guided waves as in (6), it could not be input into Abaqus directly because it contains Bessel functions, which are not supported by the math module in the Python programming environment in the Abaqus software. The displacement wave structure is thus obtained numerically from the dispersion equation and ﬁtted to a 3rd order polynomial, so the coefﬁcients of this polynomial could be used to establish the necessary expression to be input into the Abaqus FEM package. A 5.5 cycle sine wave modulated with a Hamming window is used as the time waveform in this work. The axial length of the excitation zone is 0.02 m. Except the displacement boundary condition on the excitation zone, there’re

no other boundary conditions applied in the model, and from the resulting recorded waveforms, this conﬁguration of boundary conditions is acceptable since there’re no unwanted oscillations. A Python script using the Abaqus Scripting Interface is written to ease the modeling process. It’s also helpful for the parametric study to be discussed in Section 4. Different means can be used to verify the capability of the FEM package. The easiest one is to check the propagation velocity of the guided wave packet in a ﬂawless pipe model and compare that with the group velocity of the selected working point on the dispersion curves. The results of this test for CSH0 mode at 1 MHz are summarized in Table 1. D is the propagating distance, Dt is the time difference, and v is the calculated velocity of wave packet. P1 and P2 are the points as described in Fig. 3. On the z = 0.05 plane, P3 is at ðr ¼ ða þ bÞ=2; h ¼ 315 Þ, P5 is at ðr ¼ b; h ¼ 315 Þ, P2 is at ðr ¼ ða þ bÞ=2; h ¼ 225 Þ, and P4 is at ðr ¼ b; h ¼ 225 Þ. The theoretical group velocity obtained from the dispersion curves is C G ¼ 3194:66 m=s. The errors with respect to C G are also given. It could be seen that the velocities from the simulations are very close to the theoretical group velocity from the dispersion curves. Besides this simple veriﬁcation of propagating velocities, the authors also calculate the displacement wave structures of the CSH0 mode and the CSH1 mode propagating in a ﬂawless pipeline from the simulations, and compare those with the theoretical wave structures obtained from the dispersion equation of the CSH guided waves. For this purpose, 11 spatially equally spaced points at the mid-way of propagation are selected along the thickness or radial direction of the pipe wall, which is free of defects. The waveforms of the axial displacement component (uz ) are recorded for all of these 11 points, and the absolute values of the amplitudes of the direct arrivals in these waveforms are solved, normalized by their maximum, and plotted against the radial position of the points to form the simulated displacement wave structure, as shown in Figs. 4(a) and 5(a). The ‘’ markers indicate the 11 calculated (radial position, normalized axial displacement) pairs. The corresponding theoretical displacement wave structures are in Figs. 4(b) and 5(b), as solved from CSH wave formulation and dispersion curves. The simulated and theoretical displacement wave structures are very similar, which means the applied displacement wave structures at the excitation zone are preserved, and proves the generation of the desired CSH0 mode at f = 1 MHz and C P ¼ 3194:83 m=s, and CSH1 mode at f = 2.5 MHz and C P ¼ 4154:34 m=s. The phase velocities are solved from the dispersion curves in Fig. 2. The frequencies of the wave modes are used to specify the operating points in the dispersion curves, and also as the central frequencies of the applied burst signals in the simulations. For the study of the interactions of CSH waves with axial cracking, three types of axial slots are modeled in this work, and they are outer pipe surface slots, inner pipe surface slots, and internal symmetrical slots, as shown in Fig. 6. The cross sectional areas of these slots are all rectangular in cylindrical coordinates, and ’symmetric’ means that the slot is symmetric with respect to the middle ‘curved surface’ of the pipe wall. These axial slots are assumed to be inﬁnitely long along the axial direction. W c stands for the circumferential width of the opening of the axial slot in degrees/°, and Dr is the percentage radial depth of the slot.

Table 1 Propagating velocities of wave packets from the simulations for CSH0 mode at 1 MHz.

P1 to P2 P3 to P4 P5 to P6

D (m)

Dt (s)

v (m/s)

Error (%)

0.0770 0.0778 0.0785

2.4432 105 2.4432 105 2.4433 105

3150.34 3182.44 3214.54

1.39 0.38 0.62

Normalized amplitude of displacement

S. Wang et al. / Ultrasonics 56 (2015) 325–331

1

0.8

0.6

0.4

0.2

(a) 0 0.049

0.0492

0.0494

0.0496

0.0498

0.05

0.0498

0.05

Radial position (m)

Normalized amplitude of displacement

328

1

0.8

0.6

0.4

0.2

(b) 0 0.049

0.0492

0.0494

0.0496

Radial position (m) Fig. 4. CSH0 mode displacement wave structure, f = 1 MHz. (a) From simulation, (b) from CSH wave formulation and dispersion curves.

Fig. 7 is the contour plot showing the interaction of CSH0 mode at 1 MHz with an outer surface axial slot. The plotted variable is the displacement component in the axial direction. W c is 0.5°, and Dr is 50%. Reﬂection and transmission of CSH0 mode guided wave could be seen clearly from this ﬁgure. 4. Parametric sweeping of reﬂection and transmission coefﬁcients A parametric sweeping process is applied next to quantitatively describe the interaction of circumferential SH guided waves with axial cracking of different sizes. CSH0 mode at 1 MHz is used for this parametric sweeping study. Since the defect model used is essentially a 2D one because the simulated cracking is supposed to be inﬁnitely long along the axial direction, only W c and Dr are considered in this work. When W c is ﬁxed, the variations of reﬂection coefﬁcients (R coefs) and transmission coefﬁcients (T coefs) with the increasing Dr are calculated. R coef is deﬁned as the ratio of amplitudes of the reﬂected wave and the direct arrival wave at the point P1 in Fig. 3, and T coef is deﬁned as the ratio of amplitudes of the transmitted wave at the point P2 and the direct arrival wave at the point P1, that is,

RCoef ¼

AP1 reflected ; AP1 direct

T Coef ¼

AP2 transmitted AP1 direct

ð8Þ

329

Normalized amplitude of displacement

S. Wang et al. / Ultrasonics 56 (2015) 325–331

(a) 1

0.8

0.6

0.4

0.2

0 0.049

0.0492

0.0494

0.0496

0.0498

0.05

Normalized amplitude of displacement

Radial position (m)

1

(b) Fig. 7. Contour plot of the axial displacement component showing the interaction of the CSH0 mode with an outer surface axial slot, with W c ¼ 0:5 and Dr ¼ 50%.

0.8

0.6

0.4

0.2

0 0.049

0.0492

0.0494

0.0496

0.0498

0.05

Radial position (m) Fig. 5. CSH1 mode displacement wave structure, f = 2.5 MHz. (a) From simulation, (b) from CSH wave formulation and dispersion curves.

plane wave without beam spreading or any other factor to induce energy attenuation, this normalized energy should always be equal to 1, while the CSH waves in the 3D model of this work are not plane waves, so beam spreading in the transverse direction (that is, axial direction for CSH waves) exists, which will induce serious amplitude and energy attenuation. Other factors inﬂuencing the energy include viscosity parameters used in the FEM model, and the dispersion of the guided wave mode, even though CSH0 mode is almost non-dispersive. For the above reasons at least, normalized energy less than 1 will be observed. Fig. 8 shows variations of R coefs, T coefs and normalized energy with increasing Dr , with W c ¼ 0:5 , for outer surface axial slots. Fig. 9 shows the curves for inner surface axial slots, and Fig. 10 shows the curves for internal symmetrical axial slots. The coefﬁcients and normalized energy are solved at 11 slot depths of 0%, 10%, 20%, . . . , 90%, 100%, as indicated by the ‘+’, ‘’ and ‘}’ markers. The depth of 0% means a ﬂawless model, and the depth of 100% just corresponds to a further reduced pipe model occupying the range of h ¼ ð270 þ W c =2Þ 360 in the cylindrical coordinate system.

Fig. 6. Three types of axial slots modeled in this work, cross sectional view.

in which A represents amplitudes of the recorded wave packets. The defective pipe model is used for all the calculations of the amplitudes of direct arrivals, reﬂections and transmissions. Based on the reﬂection and transmission coefﬁcients, a normalized energy could also be deﬁned as,

ENormalized ¼ R2Coef þ T 2Coef

R, T coefs and normalized energy

0.7

Uz R coefs Uz T coefs

0.6

Normalized energy 0.5 0.4 0.3 0.2 0.1 0

ð9Þ

This normalized energy represents the ratio of the total energy of the waves after the propagation in the waveguide and the interaction with the defect, and the input energy. For an ideal situation of

0

20

40

60

80

100

Axial outer slot depth (%) Fig. 8. Variations of R coefs, T coefs and normalized energy with increasing Dr , with W c ¼ 0:5 , for outer surface axial slots.

330

S. Wang et al. / Ultrasonics 56 (2015) 325–331

5. Conclusion

0.7

R, T coefs and normalized energy

U R coefs z

Uz T coefs

0.6

Normalized energy 0.5 0.4 0.3 0.2 0.1 0 0

20

40

60

80

100

Axial inner slot depth (%) Fig. 9. Variations of R coefs, T coefs and normalized energy with increasing Dr , with W c ¼ 0:5 , for inner surface axial slots.

0.7

SH guided waves propagating in the circumferential direction of pipeline are modeled in 3 dimensions in this work, with the aim for axial cracking detection implementation in ILI tools. A theoretical formulation of CSH guided waves is given ﬁrst, followed by an explanation about the 3D numerical modeling work. For both CSH0 and CSH1 modes, displacement wave structures from the simulation and the dispersion equation are compared to verify the effectiveness of the used FEM package. Three types of transverse slots along the axial direction are modeled to simulate axial cracking of pipelines. Reﬂection and transmission coefﬁcients curves are obtained to provide insight in using circumferential guided waves for quantitative testing of axial pipeline cracking. From the results of simulations, it could be concluded that the R and T coefﬁcients vary monotonously with increasing slot depth for the case of CSH0 mode at 1 MHz. Because the dispersion, beam spreading of the CSH waves in the transverse direction and other factors will generally inﬂuence the calculated R and T coefﬁcients, an amplitude compensation process would be necessary. Experiments in a large diameter pipeline are reserved for a future study. Acknowledgments

U R coefs

R, T coefs and normalized energy

z

Uz T coefs

0.6

This work is ﬁnancially supported by the National Natural Science Foundation of China (Grant No. 51107058), the Major Programs of the National High Technology Research and Development Program of China (863 Program) (Grant No. 2011AA090301), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20110002120031) and Tsinghua University Initiative Scientiﬁc Research Program (Grant Nos. 20111080983 and 20131089198).

Normalized energy 0.5 0.4 0.3 0.2

References

0.1 0 0

20

40

60

80

100

Axial internal symmetrical slot depth (%) Fig. 10. Variations of R coefs, T coefs and normalized energy with increasing Dr , with W c ¼ 0:5 , for internal symmetrical axial slots.

From these calculated curves of R and T coefﬁcients, we could see that they generally vary monotonously with increasing slot depth, and the curves for the outer surface slots, the inner surface slots and the internal symmetric slots are almost the same, for the speciﬁed pipe geometry and CSH0 mode at 1 MHz. The normalized energy is always between 0.3 and 0.4, as opposed to the ideal situation of being equal to 1. As discussed previously, like SH guided waves propagating in a plate generated by a normal transducer with a limited size in the transverse direction, the amplitudes of the recorded waveforms will be altered by the dispersion of the guided mode and the beam spreading. Viscosity parameters deﬁned in the FEM model will also inﬂuence the amplitudes. These factors will have impact on the calculation of R, T coefﬁcients and hence the normalized energy, which means these coefs and energy will depend on the propagation distance of the selected wave mode. Generally an amplitude compensation process is necessary in practical applications. An approach based on reﬂection and transmission coefﬁcients surfaces [17] proposed by the authors previously could be considered for cracking sizing, since for the special problem discussed here, only two unknowns (W c and Dr ) exist.

[1] J.L. Rose, A baseline and vision of ultrasonic guided wave inspection potential, J. Press. Ves. Technol. Trans. ASME 124 (3) (2002) 273–282. [2] L. Gavric, Computation of propagative waves in free rail using a ﬁnite element technique, J. Sound Vib. 185 (3) (1995) 531. [3] P. Wilcox, M. Evans, O. Diligent, A. Lowe, A. Cawley, Dispersion and excitability of guided acoustic waves in isotropic beams with arbitrary cross section, in: D. Thompson, D. Chimenti (Eds.), Review of Progress in Quantitative Nondestructive Evaluation, vols. 21A & B, Vol. 615 of AIP Conference Proceedings, 2002, pp. 203–210 (28th Annual Conference on Quantitative Nondestructive Evaluation, Brunswick, ME, July 29–August 03, 2001). [4] T. Hayashi, W. Song, J. Rose, Guided wave dispersion curves for a bar with an arbitrary cross-section, a rod and rail example, Ultrasonics 41 (3) (2003) 175– 183. [5] D. Gazis, 3-dimensional investigation of the propagation of waves in hollow circular cylinder. 1. Analytical foundation, J. Acoust. Soc. Am. 31 (5) (1959) 568–573. [6] D. Gazis, 3-dimensional investigation of the propagation of waves in hollow circular cylinders. 2. Numerical results, J. Acoust. Soc. Am. 31 (5) (1959) 573– 578. [7] M. Lowe, D. Alleyne, P. Cawley, Defect detection in pipes using guided waves, Ultrasonics 36(1–5) (1998) 147–154 (17th Ultrasonics International Conference (UI 97), Delft, Netherlands, July 02–04, 1997). [8] G. Liu, J. Qu, Guided circumferential waves in a circular annulus, J. Appl. Mech. Trans. ASME 65 (2) (1998) 424–430. [9] C. Valle, M. Niethammer, J. Qu, L. Jacobs, Crack characterization using guided circumferential waves, J. Acoust. Soc. Am. 110 (3I) (2001) 1282–1290. [10] X. Zhao, J.L. Rose, Guided circumferential shear horizontal waves in an isotropic hollow cylinder, J. Acoust. Soc. Am. 115 (5 I) (2004) 1912–1916. [11] X. Zhao, V. Varma, G. Mei, B. Ayhan, C. Kwan, In-line nondestructive inspection of mechanical dents on pipelines with guided shear horizontal wave electromagnetic acoustic transducers, J. Press. Ves. Technol.-Trans. ASME 127 (3) (2005) 304–309. [12] X. Zhao, V. Varma, G. Mei, H. Chen, In-line nondestructive inspection and classiﬁcation of mechanical dents in a pipeline with sh wave EMATs, in: D. Thompson, D. Chimenti (Eds.), Review of Progress in Quantitative Nondestructive Evaluation, vols. 26A and 26B, Vol. 894 of AIP Conference Proceedings, 2007, pp. 144–151 (33rd Annual Review of Progress in Quantitative Nondestructive Evaluation, Portland, OR, July 30–August 04, 2006).

S. Wang et al. / Ultrasonics 56 (2015) 325–331 [13] W. Luo, J. Rose, H. Kwun, A two dimensional model for crack sizing in pipes, in: D. Thompson, D. Chimenti (Eds.), Review of Progress in Quantitative Nondestructive Evaluation, vols. 23A and 23B, Vol. 23 of Review of Progress in Quantitative Nondestructive Evaluation, 2004, pp. 187–192 (30th Annual Review of Progress in Quantitative Nondestructive Evaluation, Green Bay, WI, July 27–August 01, 2003). [14] W. Luo, J.L. Rose, H. Kwun, Circumferential shear horizontal wave axial-crack sizing in pipes, Res. Nondestruct. Eval. 15 (4) (2004) 149–171.

331

[15] W. Luo, X. Zhao, J.L. Rose, A guided wave plate experiment for a pipe, J. Press. Ves. Technol., Trans. ASME 127 (3) (2005) 345–350. [16] I. Bartoli, F. diScalea, M. Fateh, E. Viola, Modeling guided wave propagation with application to the long-range defect detection in railroad tracks, NDT & E Int. 38 (5) (2005) 325–334. [17] S. Wang, S. Huang, W. Zhao, X. Wang, Approach to lamb wave lateral crack quantiﬁcation in elastic plate based on reﬂection and transmission coefﬁcients surfaces, Res. Nondestruct. Eval. 21 (4) (2010) 213–223.