·Table of Contents ·Computer Processing and Simulation | Wave propagation Analysis by Finite Element method for Flaw Sizing of Circular PipesShan Lin, Koichiro Kawashima, Toshihiro ItoNagoya Institute of Technology, JAPAN, Hisashi Nagamizo Mitsubishi Chemical Co., JAPAN Contact |
(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 h_{12} as equation (3) is assumed . The component h_{44} 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.
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/m^{3}) | 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 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.
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, Dt_{1} denotes the time difference between the reflected signal at the wedge/pipe interface and the signal reflected from the slot. On the contrary,Dt_{2} denotes the time difference between the transmitted signal and the signal reflected from the slot. We can estimate the position of the slot with Dt_{1} or Dt_{2}, namely, the slot location is given by
(8) |
(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) | Dt_{1}(µs) | Dt_{2}(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. |
(10) |
where A_{d} is the maximum amplitude of the transmitted waves through the slot, A_{0} 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. |
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