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Wave propagation Analysis by Finite Element method for Flaw Sizing of Circular Pipes Shan Lin, Koichiro Kawashima, Toshihiro Ito Nagoya Institute of Technology, JAPAN, Hisashi Nagamizo Mitsubishi Chemical Co., JAPAN Contact

ABSTRACT

Circumferential wave propagation along a pipe wall has been analyzed with an explicit finite element method to detect axial flaws from remote positions. An angle-beamed transducer is located opposite to a slot and different position to the slot. In the analysis, the plastic wedges of the angle-beamed transducer are taken into consideration to simulate the real pitch-catch measurement. The result of the simulation shows that the ultrasonic beam path is deviated from that predicted by the ray theory. The wavefront is propagated macroscopically along the circumference, however, the energy is carried locally along the radial direction. At the inner and outer surfaces of the pipe, the mode-converted waves are excited, thus the wave train becomes longer after multiple reflection. Two equations are proposed to estimate the position of the slot. The relation between the slot depth and received signal has been confirmed by experiments.

INTRODUCTION

Nondestructive detection of corrosion defects in pipelines is one of the major problems in the oil, chemical and other industries. In particular, there are lots of defects in the supported part of these pipes. In this case, evaluation of the defects without lifting or removing the pipe is required. One solution is to detect such defects from a remote position. Lamb waves technique is a candidate because the waves are propagated for long distance[1-4]. The waves, however, are propagated with different group velocities corresponding to various modes. The identification of these modes is not easy. If shear wave excited by an angle-beamed transducer is propagated along the circumference of pipes, the defects are detected at a remote position. For thick-walled pipes, a measurement scheme using shear waves has been used, where the ray theory is applied to predicate and explain the results of the measurment[5].
The ray theory assumes that ultrasonic beam is propagated and is reflected at interfaces under the same principle of the geometrical optics, where each ray is independent from other rays. Due to diffraction, the diameter of ultrasonic beams diverges during propagation. For a thin-walled pipe, the ultrasonic wave excited by an angle-beamed transducer is reflected and mode-converted frequently at the outer and the inner surfaces, which result in an explosive increase of the beam number. Therefore, the application of the ray theory is difficult.
Finite element method is a powerful approach for analyzing wave propagation. In this paper, the circumferential wave propagation along a pipe wall is performed with an explicit finite element method. To simulate the real pitch-catch measurement configuration, the plastic wedges of angle-beamed transducer of large attenuation are also taken into consideration. The formula to estimate the position of the slot is proposed and the relation between the depth of slots and the amplitude of received signals is presented. The simulated results are in good agreement with the measured.

ANALYZED METHOD

Uniform deformation is assumed in the axial direction of pipes, namely, plane strain condition. Analyzed model is composed of a pipe and an angle-beamed transducer shown in Fig.1. To simplify the simulation, the acoustic couplant used in the real measurement is neglected. Thus, the stresses and displacements at the interface between the pipe and the transducer are considered to be continuous.
The attenuation of the ultrasonic waves in the wedges is considered, where the relation between stresses and strains is expressed by

(1)

where s and e are the stress and strain, D and h are the tensors of elastic stiffness and viscous coefficient, respectively[6]. The wedges is considered to be isotropic and viscoelastic, so equation (1) is rewritten as

(2)

where E and n are Young's module and Poisson's ratio.
In high frequency, the relationship between the viscous coefficient and attenuation coefficient is given by

(3)

where ais attenuation coefficient of longitudinal wave and w is angular frequency. The same relation for h₁₂ as equation (3) is assumed . The component h₄₄ is calculated by

(4)

With viscous damping being considered, the discrete equations of motion are given by

(5)

where the suffix e stands for an element, {u} and {F} are the vectors of displacement and force, respectively. [M], [C] and [K] are the matrixes of mass, viscous damping and stiffness. They are given by

	(6a)
	(6b)
	(6c)

where r is mass density, [N]_e is the matrix of shape function and [B]_e is the matrix relating the strains with nodal displacements in an element.

NUMERICAL PROCEDURE

1.ANALYZED MODEL
The analyzed model composed of a pipe and an angle-beamed transducer is shown in Fig.1 with dimensions. The transducer has the transmitting and receiving piezoelectricity elements. The material properties of the pipe and the transducer are given in Table 1. The pipe has an axial slot as artificial flaw in the outer or inner surface, whose dimension is given in Table 2. The incident angle of the transmitting vibrator is 45 degrees, so the refraction angle for shear wave is 71.2 degrees. The diameter of the vibrator is 4mm..

Fig 1: Simulation model.

Depth (mm)
Inside	Outside
1.14	1.0
1.60	1.5
2.10	2.0
Table 2: Dimension of slot (width=3mm).

	Pipe	Wedge
Material	Steel	Polyimide
Young's modulus(GPa)	206	4.39
Poisson's ratio	0.28	0.38
Density(kg/m3)	7700	1410
Longitudinal wave velocity(m/s)	5850	2410
Table 1: Material properties of pipe and wedge materials.

Large attenuation coefficient is given to the part B of the transducer as shown in Fig.1 to eliminate the waves multiply reflected at the sides of the transducer. In this paper, the attenuation coefficient of longitudinal wave in the part B is greater than by 50 times that in the wedges. The attenuation in the steel pipe is ignored .

2.NUMERICAL CONDITIONS


Fig 2: Incident wave.

For the central node of the transmitting vibrator, a time-dependent force whose central frequency is 5MHz shown in Fig.2 was given. The forces having the same waveform shown in Fig.2 but variable amplitude according to Gaussian distribution are given to other nodes.

For satisfying the stability conditions, the integration time step Dt and mesh size h are restricted by

(7)

where l is the longitudinal wave length and T is the propagating time through the distance h[7]. In this paper,
h 0.03mm and Dt=2ns.

NUMERICAL RESULTS AND DISCUSSIONS

1. WAVE PROPAGATION IN A DEFECT-FREE PIPE
The distributions of the displacement component in x direction at 5.1 and 15 ms are shown in Figs.3 and 4, where R_in and R_on stand for the nth reflected waves at the inner and outer surface, respectively. The angle a or b denotes the longitudinal or transverse wavefront. From these figures, these leading wavefronts are parallel to the radius and they are propagated along the circumference. This is different from the predication by the ray theory. The reflected wave with different orientation is formed at the inner and outer surfaces every reflection as shown in Fig.4. Figures 3 and 4 show the complicated, time-dependent amplitude distribution of the shear waves over radial direction. The ray theory, however, predicts rather simple shift of the maximum amplitude of shear waves over the cross section.

Fig 3: Distribution of x-direction displacement (t=5.1µs).

Fig 4: Distribution of x-direction displacement (t=15.0µs).

As shown in Fig.5, the ray theory assumes the various beam paths of which the refraction angle is dependent on the position on the interface between the pipe and the transducer. The point where the central beam path is reflected on the outer surface is denoted by point 1 through which a section A is introduced. The magnitude of displacement vector over the section A for different time is shown in Fig.6. The arrival time of the transverse wavefront at point 1 is 8.73ms according to the ray theory. Different from the predication by the ray theory, the maximum displacement appears approximately at the position of one forth of the wall thickness from the outer surface at 9.0ms and 9.1ms. At 9.15 or 9.17<ms, the position of the maximum displacement moves toward the inner surface.

Fig 5: Propagation path according to ray theory.

Fig 6: Amplitude distribution over cross section A (Amplitude=).

2. RECEIVED SIGNALS FROM THE OUTER SLOTS AT VARIOUS POSITION
The signals received from an outer slot at various positions are shown in Fig.7. The position of slot is denoted by the angle q as shown in Fig.1. The width and depth of the slot are 0.5 and 1.5mm, respectively. When q=180°, the reflected and transmitted waves are received nearly at the same time, therefore, the identification of the slot is difficult. To judge whether a slot exists or not, one method is to compare the amplitude of the received signals with that for defect-free case. The other is to move slightly the transducer along circumference . If the reflected waves are observed, a slot exists. Depending on the slot position, the waveforms of the reflected and transmitted waves also change in the arrival time and amplitude.

Fig 7: Received signals for different slot position

As show in Fig.7, Dt₁ denotes the time difference between the reflected signal at the wedge/pipe interface and the signal reflected from the slot. On the contrary,Dt₂ denotes the time difference between the transmitted signal and the signal reflected from the slot. We can estimate the position of the slot with Dt₁ or Dt₂, namely, the slot location is given by

(8)

or

(9)

where R is the outer radius of the pipe , c_t is the shear wave velocity and q is expressed in radian. The width of the slot can be neglected if it is narrow. The estimated position of the slot by equation (8) or (9) is shown in Table 3. The estimated value of q agrees within 1%.

q(degree) (measured)	q(radian) (measured)	Dt1(µs)	Dt2(µs)	q(radian) (estimated by (9 ))	q(radian) (estimated by (8 ))
37.6	0.656	12.20	46.15	0.654	0.657
45.0	0.785	14.53	43.67	0.782	0.790
75.2	1.312	24.21	33.93	1.303	1.315
180	3.141	58.00	0.0	3.122	3.141
Table 3: Comparison of the estimated position of the slot with measured

For an arbitrary configuration of the transducer and the slot, we can estimate the slot depth and width with the reflected and transmitted signals, however, this will be reported in a next paper. In this paper, the relation between slot depth and the transmitted wave amplitude is given in the next session for only when q=180° and the width of the slot is 3mm.

3.RELATION BETWEEN THE SLOT DEPTH AND RECEIVED SIGNAL AMPLITUDE


Fig 8: Relation between amplitude and slot depth.

To verify the results of the simulation, we made the experiment corresponding to the silmulation. The angle-beamed transducer was excited by a pulser/receiver. The transmitted waves were digitized and recorded in a computer. Figure 8 shows the relation between the slot depth and the received signal amplitude, where A and r are defined by

(10)

where A_d is the maximum amplitude of the transmitted waves through the slot, A₀ is the maximum amplitude of the transmitted waves for defect-free pipe, D is the depth of the slots and T is the thickness of the pipe.

The simulation is in good agreement with the experiment. When the slot locates on the outer surface the amplitude of received signals drops linearly with an increase in the slot depth. When the slot is on the inner surface, the amplitude of received signals has no obvious change for smaller r than 0.4. Then the amplitude drops with an increase in the slot depth.
For the defect-free pipe, the distribution of the displacement amplitude over the section S shown in Fig.1 at 28.2ms is shown in Fig.9. The time is the arrival time of the shear wave to the section S. Figure 9 shows that the amplitude near the inner surface is much smaller than that near the outer one. Therefore, the amplitude of the received signals from a shallow inner slot is not dependent on the slot depth.

Fig 9: Amplitude distribution over section S.

CONCLUSIONS

For remote detection of defects that exist in inner or/and outer surface of thin-walled pipes, circumferential shear wave propagation excited by an angle-beamed transducer was analyzed with FEM. The results of the simulation were compared with the experiment. Main results are summarized as follows:
1. The energy of ultrasonic waves isn't propagated along the path of the central beam predicated by the ray theory, but it is propagated mostly along the circumference.
2. The location of the slots is estimated with the arrival time of the reflected waves and the transmitted ones.
3. The slot depth is estimated with the received signal amplitude with a simple relation it the slot in on the outer surface.

REFERENCES

1. I.A.Viktorov, Rayleigh and Lamb Waves, Plenum press, (1967).
2. M.G. Silk et al, The propagation in metal tubing of ultrasonic wave modes equivalent to Lamb waves, Ultrasonics, Vol.17, p.11, (1979).
3. D.N.Alleyne et al: Long Range Propagation of Lamb Wave in Chemical Plant Pipe work, Material Evaluation, 55, p.504, (1997).
4. M.J.S.Lowe et al: Defect Detection in Pipes Using Guide Waves, Ultrasonics, 36, p.147, (1998)
5. J.Krautkramer et al, Ultrasonic testing of materials, 4th Edition, Springer, p.410, (1990)
6. B.A.Auld: Acoustic Fields and Waves in Solids, John Wiley Sons Inc., (1973).
7. K.Kawashima et al: Finite Element Simulation of Leaky Surface Wave Propagation Excited by a Line- Focused Transducer, Review of Progress in Quantitative Nondestructive Evaluation, Vol.17,p.995, (1998).

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