The correlation of guided wave propagation characteristics with structural prestress is of paramount importance to the structural health monitoring of gas pipelines. A variable section structure and inhomogeneous prestress are common conditions in the pipeline. However, most of the existing guided wave finite element models focus on the structure size and stress distribution under two-dimensional conditions, and it is difficult to analyze the three-dimensional structure with non-uniform stress and variable cross section. In this paper, an acoustoelastic theory combined with a semi-analytical finite element based on the three-dimensional mapping method is proposed to investigate guided wave propagation. It provides a generalized tool to study guided waves in waveguides with a variable cross section under inhomogeneous prestress. Then it is applied to two cases, a hollow cylinder with a variable cross section subjected to axial force and radial force, to demonstrate the capability of the method. Dispersive solutions are obtained in terms of the three-dimensional dispersion surface and the change in phase velocity in a variable cross section. The results show that there is a propagation mode, which is insensitive to the change in the section but sensitive to the change in prestress. The effectiveness of the proposed method is verified by comparing with the experimental results. This study provides a good application prospect for the structural design and performance analysis of variable cross section waveguides.
I. INTRODUCTION
On land, almost all natural gas and the vast majority of oil in the world are transmitted through pipelines. Therefore, the safety of oil and gas pipelines has become a focus of attention.1,2 Due to the harsh working conditions of pipelines, it is easily subjected to failure because of poor prestressed states, such as residual stress3 and cyclic stress.4 Structural health monitoring (SHM)5,6 is one of the important means to maintain the pipeline state and predict pipeline life. Furthermore, guided waves are widely used in prestress detection of pipelines because of their reliable response, large detection range, and long propagation distance.
It is not easy to study the effect of cross section on guided waves. Scholars have been studying this for a long time. Theoretical methods are both effective and convenient when solving problems with simple geometry, such as flat and cylindrical structures. However, when the object problem has a complex cross section, the propagation of guided waves will become complicated, and complexities exist in multiple modes. Compared with the analytical method, the numerical method has more advantages in complex cross section and boundary conditions.7 The time-domain finite element (TFE) method8 has been used to analyze the propagation, scattering, and transduction. Because the TFE method is generally numerically expensive and time-consuming, the semi-analytical method, including the wave finite element (WFE) method,9 the scaled boundary finite element (SBFE) method,10 and the semi-analytical finite element (SAFE) method,11,12,13 is developed. The SAFE method is widely used in various applications due to its unique advantages in exploring the scattering property and the wave propagation behavior.
Considering the performance and function of the structure, a variable section structure often appears in practical engineering. It is also very important to study the influence of a waveguide structure with a variable cross section. Pagneux and Maurel14 studied the propagation of Lamb waves in an elastic waveguide with varying thickness. On this basis, other scholars extended this work to guided wave propagation in waveguides with a variable curvature and cross section.15 El-Kettani et al.16 studied the phase velocity changing pattern and wavenumber propagation of an elastic plate with a gradually thinned small-angle wedge-shape by using the FEM method. Then, they assumed that the thickness of the plate changed along the Gaussian curve and thus studied the mode changing law between the symmetric and asymmetric mode in an aluminum plate.17 Chen et al.18 proposed an analysis method for the characteristics of guided wave propagation characteristics in a straight switch rail with a turnout that has a variable section. The research results indicated that the scattering characteristics of steel rails are relatively stable along the longitudinal direction and have little variation over time.
There is a prerequisite for conducting guided wave structure health monitoring to obtain the relationship between the stress state and guided wave signal. For the purpose of analyzing the guided wave propagation law under the influence of prestress, many researchers have carried out a lot of studies on axial stress.19,20,21 Loveday22,23 analyzed the influence of the guided wave propagation law in an arbitrary cross section structure under axial load by using the SAFE method. Michaels24 and Mohabuth25 studied how the high-order constants affect the Lamb wave propagation and then developed the acoustoelastic theory of Lamb wave propagation and created a model in prestressed plates. The results show consistency between numerical and experimental results at high frequencies. Ma et al.26 combined the theory of acoustic elasticity with the SAFE method to study the effect of axial stress on guided waves in T-shaped structural components and demonstrated through experiments that this method can be used to monitor the axial stress. In addition to axial stress, Zuo27 studied the variation in guided wave propagation under shear stress. The conclusion is that shearing deformation can rotate the waveguide significantly. In other words, it is necessary to study the influence of stress distribution on waveguide propagation characteristics.
For simple forces such as axial force and shear force and a simple waveguide cross section, the above-mentioned research is capable and effective. However, for the combination of simple stress and a complex waveguide cross section, structural stress often presents complex distribution, that is, non-uniform stress field. More attention has been paid to the study of guided wave propagation characteristics in non-uniform prestressed fields. Lematre et al.28 explored the propagation characteristics of guided wave propagation subjected to a prestress gradient based on the sublayer method. The research results found that the horizontal mode and Lamb mode are sensitive to prestress. Akbarov and Bagirov29 explored the propagation law of axisymmetric longitudinal waves in a hollow cylinder under non-uniform initial stress. Analysis of the results showed that the inhomogeneous initial stresses act on the character of the dispersion curves. However, most of the existing studies focus on two-dimensional space, and the study of guided wave propagation under inhomogeneous stress distribution in three-dimensional space is limited.
Through the analysis of the above-mentioned research, the current research on guided waves is either for the structure with a variable cross section without external force or for the structure with a constant cross section under external force. Meanwhile, due to the complexity in the variable cross section structure, the modes of guided waves are necessarily complex. Therefore, how to select a suitable mode for stress detection is a problem worth exploring. It is necessary to research a three dimensional distribution surface mode that is insensitive to the change in the cross section but sensitive to stress changes.
This paper proposes an acoustoelastic theory combined with a semi-analytical finite element based on the three-dimensional mapping (AE-SAFE-3D) method. Moreover, we attempt to study the propagation law of guided waves in a variable hollow cylinder under external force. First, in Sec. II, a semi-analytical finite element model based on the AE-SAFE-3D method is established by using affine transformation to map the complex three-dimensional stress distribution into two-dimensional space. Next, Sec. III provides two practical cases about a hollow cylinder with a variable cross section: One is to study the guided wave propagation under axial force, and the other case is to study the effect of radial force on the guided wave propagation. Section IV outlines the feasibility of the proposed model by comparing with previous studies. Finally, Sec. V provides some concluding remarks.
II. PRINCIPLES AND METHODS
A. Acoustoelasticity model for wave propagation
B. Formulation of the AE-SAFE-3D method
Acoustoelastic guided waves are promising for prestress measurements in waveguides,27 and the SAFE method has been praised as an effective method for modal study in waveguides. In addition, this method adopts the assumption that harmonic guided waves propagate along the axis, greatly reducing computational costs.31 Hence, it is very important to consider the three-dimensional stress state in the two-dimensional cross section.
It is assumed that the cross section of the analysis object undergoes a continuous and differentiable slow change. As shown in Fig. 1, according to the idea of differentiation, the local characteristics of the analyzed object can be equivalent to the structure of a constant cross section. The structure with a variable cross section is divided into n − 1(n ≥ 1) segments. The spacing should be as dense as possible to reflect the continuous axial variation in the dispersion characteristics.
III. APPLICATIONS OF THE AE-SAFE-3D METHOD
In this section, the part where the sectional size of the gas pipeline changes is analyzed. The AE-SAFE-3D method is applied to two examples of a hollow cylinder with a variable cross section. For gas pipelines under axial force and radial force, the propagation characteristics of different guided wave modes with stress and space geometry are analyzed.
A. Guided wave in a hollow cylinder subjected to axial force
The dispersion characteristics of waveguides subjected to axial force have been widely studied. In this section, for a hollow cylinder with a variable cross section subjected to axial force, the propagation characteristics of guided waves under stress are studied.
1. Model description
The geometry of the gas pipeline is shown in Fig. 2. The hollow cylinder consists of three parts, such as a constant cross section at both ends and a variable cross section in the middle. The length of each part is set to L = 200 mm, and the cross section of each part is modeled as a ring shape with a wall thickness of t = 3 mm. The outer diameters of both ends are set to d1 = 30 mm and d2 = 50 mm, respectively. In addition, the boundary conditions are set so that one end is fixed and the other end is applied with an axial force ranging from 0 to 80 kN in increments of 20 kN. The material used is steel,34 and the material constants are listed in Table I.
The material constants of a hollow cylinder.
ρ . | λ . | μ . | l . | m . | n . |
---|---|---|---|---|---|
7850 kg/m3 | 150 GPa | 75 GPa | −300 GPa | −620 GPa | −720 GPa |
ρ . | λ . | μ . | l . | m . | n . |
---|---|---|---|---|---|
7850 kg/m3 | 150 GPa | 75 GPa | −300 GPa | −620 GPa | −720 GPa |
The model is developed in a two-step solution. The first step is to calculate the three-dimensional stress, and the second step is to calculate the dispersion characteristics based on the stress results using the AE-SAFE-3D method. The solved frequencies of the AE-SAFE-3D method are set from 0.1 to 25 kHz with increments of 0.2 kHz.
2. Results
Figure 3 shows the three-dimensional stress distribution under axial force. It is obvious that the state of stress does not vary with the distance at the front and the back. In the constant section, the stress hardly varies with the distance. However, in the variable cross section, the stress reaches the maximum when the section size is the largest, and the stress reaches the minimum when the section size is the smallest.
Figure 4 presents the dispersion curves for wave propagation in a typical cross section. Figure 4(a) shows the relationship between phase velocity and frequency at the large cross section, and Fig. 4(b) shows the relationship between phase velocity and frequency at the small cross section. It can be found that the number of modes at large cross sections is more than that at small cross sections in the frequency range below 25 kHz. The reason is that the product of frequency and diameter determines the number of modes, which has the same result as that in the previous literature.11 Therefore, the number of modes decreases with the decrease in the cross section diameter in the same frequency range.
Dispersion curves for wave propagation in a typical cross section under axial force from 0 to 80 kN: (a) x = 200 cross section; (b) x = 400 cross section.
Dispersion curves for wave propagation in a typical cross section under axial force from 0 to 80 kN: (a) x = 200 cross section; (b) x = 400 cross section.
Figure 4 shows the change in phase velocity of the L(0,1) mode and T(0,1) mode in a typical cross section under axial force from 0 to 80 kN. Most scholars35,36 found that low frequency wave modes are sensitive to local damage; hence, the L(0,1) mode and T(0,1) mode are widely used in guided wave stress detection because of their simpler dispersion characteristics. It can be seen that the change in phase velocity in a large cross section is smaller than that in a small cross section under the same axial force. This is because compared to a small cross section, the stress in a large cross section is relatively small. Therefore, when guided waves propagate from a large cross section to a small cross section along a hollow cylinder under the same axial force, the change in phase velocity increases with the propagation distance.
The L(0,1) mode and T(0,1) mode are sensitive to the stress state. Figure 5(a) shows the three-dimensional dispersion surface of the L(0,1) mode under an unstressed state. It can be found that the dispersion characteristics of guided wave propagation change with the increase in the distance. When the frequency is the same, the propagation speed of the L(0,1) mode increases with the increase in the distance. In addition, compared with the low frequency condition, the rate of change of propagation velocity under the high frequency condition has a great change. Therefore, when the L(0,1) mode is used for guided wave detection of a gas pipeline with a variable cross section, the change in propagation velocity cannot be ignored.
Three-dimensional dispersion surface under an unstressed state: (a) L(0,1) mode; (b) T(0,1) mode.
Three-dimensional dispersion surface under an unstressed state: (a) L(0,1) mode; (b) T(0,1) mode.
Figure 5(b) shows the three-dimensional dispersion surface of the T(0,1) mode under an unstressed state. It can be seen that the phase velocity of the T(0,1) mode hardly changes with the propagation distance. The T(0,1) mode is not only insensitive to structural size but also sensitive to the stress state. Therefore, the T (0,1) mode is preferred for the detection of waveguides with a variable cross section.
B. Guided wave in a hollow cylinder under radial force
On the basis of selecting the T(0,1) mode in the first case, the second case is to study the dispersion characteristics of the T(0,1) mode in a hollow cylinder under radial force.
1. Model description
The geometry of the model is also shown in Fig. 2, and the material constants are the same parameters as the first case. In this case, the boundary conditions are set so that the two ends are fixed and an axial force from 0 to 8 kN is applied in the middle with an increment of 2 kN. The solution form of the model is the same as that of the first case, which adopts a two-step solution. In the first step, the three-dimensional stress distribution under the radial force is calculated. In the second step, the mapping between the three-dimensional stress and the three-dimensional geometry is completed by the AE-SAFE-3D method to realize the calculation of the dispersion characteristics.
2. Results
Figure 6 shows the three-dimensional stress distribution under a radial force of 8 kN. It can be seen that the stress at the end of the fixed constraint and the stress at the position subjected to the radial force are larger than other positions. The difference from the hollow cylinder under axial force is that the stress state of this case at the constant cross section changes with the distance.
Figure 7 shows the changes in phase velocity for the T(0,1) mode at different positions. It can be found that the change in phase velocity is large in a place with high stress. In other words, there is a great change in the phase velocity at the end of the fixed constraint and at the position subjected to the radial force, which is consistent with the trend of stress distribution. Therefore, the T(0,1) mode has the characteristics of being sensitive to the variation in stress states.
Changes in phase velocity for the T(0,1) mode at different positions: (a) 400 mm < × < 600 mm; (b) 200 mm < × < 400 mm; (c) 0 mm < × < 200 mm.
Changes in phase velocity for the T(0,1) mode at different positions: (a) 400 mm < × < 600 mm; (b) 200 mm < × < 400 mm; (c) 0 mm < × < 200 mm.
IV. MODEL VERIFICATION
The structure of a constant cross section is a common example of analyzing guided wave propagation. Therefore, the validation case of wave guides with a regular cross section is chosen to calculate using the AE-SAFE-3D method, and the results are compared with the experimental results.24 According to Ref. 24, the initial stress (σ = 60 MPa) is considered. The material is the same as the 6061-T6 aluminum used in Ref. 24, and the material constants used are listed in Table II.
The material constants for 6061-T6 aluminum.
ρ . | λ . | μ . | l . | m . | n . |
---|---|---|---|---|---|
2704 kg/m3 | 54.308 GPa | 27.174 GPa | −281.5 GPa | −339.0 GPa | −416 GPa |
ρ . | λ . | μ . | l . | m . | n . |
---|---|---|---|---|---|
2704 kg/m3 | 54.308 GPa | 27.174 GPa | −281.5 GPa | −339.0 GPa | −416 GPa |
Figure 8 shows a comparison of the experimental results and the SAFE numerical method. It can be seen that the results of the AE-SAFE-3D method are closer to the experimental results, which verifies the correctness of the AE-SAFE-3D method. It can be also seen that the results of the SAFE method are quite different from the experimental results and the stress states have little influence on the results. The reason is that the SAFE method ignores high-order elastic constants on guided wave propagation. Therefore, the inclusion of high-order elastic constants has a significant effect on the stress detection of guided waves.
Comparison of experimental results24 and SAFE numerical method:23 (a) S0 mode at 1.59 MHz mm; (b) A1 mode at 2.54 MHz mm; (c) S1 mode at 3.81 MHz mm.
V. CONCLUSIONS
In this paper, the AE-SAFE-3D method has been developed for waveguides with a variable cross section under inhomogeneous prestress. The governing equations are derived by combined the acoustoelastic theory with the SAFE method based on three-dimensional mapping. From this work, the following conclusions are obtained:
The number of modes decreases with the decrease in the cross section diameter in the same frequency range. In addition, when guided waves propagate from a large cross section to a small cross section along a hollow cylinder under the same axial force, the change in phase velocity increases with the propagation distance.
Compared with the L(0,1) mode, the T(0,1) mode is not only insensitive to structural size but also sensitive to the stress state. The T(0,1) mode has the characteristics of being sensitive to the variation in the stress states in the waveguides with a variable cross section. The T (0,1) mode is preferred for the detection of waveguides with a variable cross section.
The effectiveness of the proposed method is verified by comparison with experimental results. It confirms that the inclusion of high-order elastic constants has a significant effect on the stress detection of guided waves.
Although the object of this paper is a hollow cylinder, the method can be easily extended to other variable cross section waveguides. This method provides a good application prospect for the structural design and stress measurement of variable cross section waveguides.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China, through Grant No. 52005081.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Yunlong Wang: Conceptualization (equal); Writing – original draft (equal). Xiaokai Mu: Writing – review & editing (equal). Bo Yuan: Formal analysis (equal). Qingchao Sun: Project administration (equal). Wei Sun: Supervision (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.