Abstract
The aim of the investigation presented here was to understand how the viscosity parameters of an adhesive layer affect group velocity and attenuation of the double-layer adhered pipe. Various parameter combinations (attenuation of longitudinal wave and shear wave, pαL and qαT; thickness, d; and density, nρ) were utilized in order to generate different uncured degrees of the adhesive layer. In the frequency range 0∼500 kHz, the group velocity dispersion curves and attenuation dispersion curves were obtained from these models. Then, the group velocity and attenuation of the two commonly used modes, and , were compared and analyzed. The results have shown that it is important to remark that little effect on group velocity was caused, and significant linear increases of attenuation occur with increase in q, d, and n. However, variable had little effect on attenuation; more modes emerged when d increased or n decreased, causing difficulties on mode identification and signal processing. The numerical results provided a useful way to evaluate bonding quality by measuring the group velocity and attenuation in the pipelines.
1. Introduction
Glass fiber reinforced pipes (GFRP) are widely applied to aerospace, construction, chemical equipment, medical devices, sports equipment, and other fields because of many advantages, such as high-pressure resistance, corrosion preventive, excellent flexibility, convenient installation, long service life, and so on. The adhesive layer can transmit stress, block crack, and absorb and scatter energy. A poorly cured interface can degrade the viscoelastic strength, which may cause fatigue cracks, and a debonding interface can reduce the mechanical properties, which may cause brittle failure. The existence of uncured defects will inevitably affect the safety and reliability of pipelines in use. Therefore, it is of great significance to carry out bonding quality inspection of composite pipes online and in service.
Conventional ultrasonic nondestructive testing (NDT) methods become difficult and inefficient to evaluate bonding quality of pipelines [1–7]. Guided wave is widely used in nondestructive inspection because of its capacity of traveling long distances without substantial attenuation [8–12]. However, guided wave signals are difficult to analyze due to the multimode and dispersion characteristics. Based on material properties of the adhesive layer, the dispersion behavior can be described by obtaining the group velocity curve and attenuation curve with the different adhesive quality [13–15].
To date, several studies have been investigated by many scholars to inspect bonded structures and have proved that guided wave detection is a very effective method. Matt [16] performed semianalytical finite element (SAFE) analyses for CFRP plate-to-spar joints in unmanned aerial vehicles and provided substantial insight into the guided wave behavior within pristine and damaged joints; the SAFE method was adopted to the dispersive properties of the guided wave across a pipe elbow and in materials with viscoelastic properties (Shorter [17]; Lhémery et al. [18]; Yan et al. [19]); Scalea et al. [20] studied the propagation of guided waves in adhesively bonded lap shear joints. The lowest-order, antisymmetric A0 strength of transmission was studied for three different bond states in aluminum joints. Hong [21] combined Hamilton’s principle and the semianalytical finite element method to study phase velocity dispersion curves of 16 layer adhesively bonded composites. The result showed phase velocity changed slightly when the state of the adhesive layer changed from properly bond to poorly bond. Siryabe et al. [22] excited the Lamb wave by an air-coupled transducer in the aluminum substrate and analyzed the time-frequency relationship of S0 under different bonding conditions. Castaings [23] found that SH0 was sensitive to the change of properties of the interfacial adhesive, which can analyze the bonding quality by quantifying the shear properties at the interface. Kharrat et al. [24] selected the torsional wave to detect the defects of pipeline. Rojas et al. [25] combined dispersion curve with short-time wavelet entropy of Lamb mode to detect flat bottom hole defects in the center of the plates.
In summary, most of the literature on guided waves focused on the propagation of elastic materials with little or no damping, not considering the effect of material absorption on guided wave attenuation, and research objects are more focused on the plate structure and lap joints. Therefore, the dispersion characteristics of the bonded composite pipes are studied in this paper. Based on the fact that the defects of bonding structures are mainly related to acoustic properties, thickness, and density of the adhesive layer, this paper mainly studied the effect of these variations on dispersion characteristics of guided waves.
2. Basic Theory of Guided Wave in Pipes
In a cylindrical structure with a global cylindrical coordinate system (r, θ, z), as shown in Figure 1, guided waves can propagate in the axial or circumferential direction. SAFE has been widely adopted for the computation of guided wave dispersive features in waveguides. Navier’s governing guided wave equation can be expressed as the following equation [8, 26]:where are the volume integral, ; is the second derivative of displacement u with respect to time t, ; T represents the matrix transpose; r is the density; is the strain; and is the stress.
Assuming that guided waves propagate in the z direction, the displacement can be represented by the following equation:where is the nodal displacement vector of the element and is the shape function with respect to the thickness r. For a two-node element, is a 3 × 6 matrix, as follows:where
is the natural coordinate in the r direction. The strain-displacement relations in cylindrical coordinates is as follows:in which
According to Hooke’s law, the stress-strain relation is where C is the stiffness matrix, real in elastic materials, or complex in viscoelastic materials. The elastic constants of all the layers in a composite pipe must be expressed in the global cylindrical coordinate system . For anisotropic composite materials, this can be achieved through the rotation of the stiffness matrix in the rectangular coordinate system (x, y, z) of each lamina. Substituting the displacement equation (2), strain equation (4), and stress equation (6) into the governing equation (1), eigenvalues for wavenumber k can be solved at each frequency ω and the corresponding eigenvector contains the wave structure information.
Group velocities describe the propagating speeds of guided wave packets, which can be calculated from the wavenumber-frequency relation:
If the adhesive material exhibits viscous properties, the viscoelastic matrix C depends on Lame constants and , considering attenuation coefficients and . The density and Lame constants and can determine the longitudinal wave velocity and shear wave velocity , as shown in the following equation:where , . Complex longitudinal wave velocity and complex shear wave velocity are used to solve the Lamb wave problem. A complex wavenumber can be calculated, in which the real part describes the Lamb propagation and the imaginary part represents the Lamb attenuation. The Lamb attenuation caused by the viscosity of the medium is determined by the imaginary part of the complex wavenumber at different frequencies, .
3. Experimental Works
In the experiment, three layer (GFRP-adhesive-GFRP) pipes (thickness: 0.45 mm/0.1 mm/0.45 mm, inner R: 20 mm) were created. The material parameters of GFRP and adhesive are shown in Tables 1 and 2, respectively.
This paper presents a comprehensive study about the effects of adhesive parameters on the dispersion property of guided waves in composite pipes. Group velocity curves and attenuation curves were obtained using a numerical analysis method in MATLAB language. It is suitable for structures with anisotropic materials and arbitrary shapes, considering the effect of acoustic impedance attenuation.
In view of the large attenuation property of composite materials, the frequency exciting guided wave is usually low. Therefore, the frequency range of this paper was determined to be 0∼500 kHz. The viscoelastic parameters of the adhesive were utilized in order to generate some different degree of uncured adhesive layers. For the study, group velocity curves and attenuation curves were obtained from these models (shown in Table 3), where symbols and represent the attenuation coefficient of the longitudinal wave and the shear wave, d indicates the thickness, and n is the density coefficient.
Figure 2 shows the group velocity curves about the nonattenuation guided waves with complete solidification. The result showed that there are two longitudinal modes and , one torsional mode , and three flexural modes , , and . From Figure 2, spreads the fastest and the velocity of and has little change with a frequency above 50 kHz. The normalized wave shapes of these modes in the radial direction with frequency 200 kHz are shown in Figure 3. The result showed that axial and radial displacements are dominant for , tangential displacement is dominant for , and three displacements exist for , respectively.
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4. Results and Discussion
Next, the effects of the adhesive parameters on the dispersion characteristics of guided waves were studied, especially, focused on two typical modes and . “CL200” represents the group velocity of at 200 kHz, “AT500” means the attenuation of at 500 kHz, etc.
4.1. Effect of Body Wave Attenuation of Adhesive Layer on Dispersion Curve
4.1.1. p = q = 0.5, 1, 2, and 4
Figures 4(a) and 4(b) present the group velocity curves of and when p = q. The curves show that the viscosity of the adhesive layer has little effect (<0.3%) on the group velocity of guided waves. The group velocity range of is 2698∼2702 m/s and of is 1497∼1501 m/s. The attenuation of the viscoelastic material will lead to the imaginary part of the wavenumber. The real part of a complex wavenumber describes the Lamb propagation, and the imaginary part represents the Lamb attenuation. Theoretically, the introduction of body wave attenuation does not affect the group velocity, which is consistent with the experimental results.
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Therefore, we will only consider the influence of p, q on the attenuation of guided waves. Figures 4(c) and 4(d) display the attenuation curves of and when p = q. As shown in Figure 4(c), the attenuation values of are the lowest at frequency 28 kHz and then gradually increase with the increase in frequency; Figure 4(d) shows that there is generally a linear relationship between the attenuation of and frequency. In addition, at the same frequency, the greater the value of p = q, the higher the attenuation of and . It is seen that the maximum attenuation value of is and of is , when p = q = 4. When p = q is 0.5, 1, 2, and 4, AL200,AL500,AT200, and AT500 have 6.77, 679, 659, and 6.60 times growth, respectively (see case 1–case 4).
4.1.2. p = 0.5, 1, 2, and 4 and q = 1
From Figures 5(a) and 5(b), we can see that while attenuation of gradually increases with the increase in , the attenuation value of remains unchanged. At frequency 500 kHz, the maximum attenuation values of are (see Figure 5(a)) and of are 40 (see Figure 5(b)), when p = 4 and q = 1. When q = 1 and p is 0.5, 1, 2, and 4, AL200,AL500, AT200, and AT500 have 0.77, 0.36, 0, and 0 times growth, respectively (see case 5–case 8).
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4.1.3. p = 1 and q = 0.5, 1, 2, and 4
The attenuation curves of and are presented in Figures 6(a) and 6(b), illustrating that attenuation of and increases with the increase in q. It is seen that the maximum attenuation value of is about and of is about , when p = 1 and q = 4. When p = 1 and q is 0.5, 1, and 2, AL200, AL500, AT200, and AT500 have 4.57, 5.60, 6.59, and 6.60 times growth, respectively (see case 9–case 12).
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The results obtained from the preliminary analysis (see Figures 4–6) are set out to understand that body wave attenuation has little effect on the group velocity dispersion, but causes a significant difference in the attenuation dispersion. What stands out in these figures is that shear wave attenuation has a dominance effect on attenuation dispersion characteristics of guided waves, contrasting the influence of longitudinal wave attenuation. In addition, guided waves in higher frequency will have higher attenuation.
4.2. Effects of Thickness of Adhesive Layer on the Dispersion Curve
The group velocity curves of and with different adhesive thickness are shown in Figures 7(a) and 7(b), respectively. From the curves, it is clear that increasing thickness does result in a slight drop on the group velocity. When d is 0.05, 0.1, 0.2, and 0.4 (mm), CL200, CL500, CT200, and CT500 have 5.80%, 18.1%, 4.23%, and 4.70% decrease, respectively. (see case 13–case 16). Figures 7(c) and 7(d) show the plot of attenuation of and , providing that the attenuation value has a linear relationship with thickness. The maximum attenuation value of is and of is , when d = 0.4 mm. Besides, when thickness is 0.4 mm, there is one more generated in the pipe. When d is 0.05, 0.1, 0.2, and 0.4, AL200,AL500, AT200, and AT500 have 6.62, 11.64, 6.15, and 6.66 times growth, respectively.
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4.3. Effects of Density of Adhesive Layer on the Dispersion Curve
In the effect study of density on dispersion characteristics, four different density values were set up. As the density decreases, the number of guided modes will increase. Figures 8(a) and 8(b) present the effect of density on group velocity. The results provide that group velocity values are tended to one value, except group velocity of which decreases sharply from 2617 m/s at 405 kHz to 854 m/s at 410 kHz when n = 0.1. When n is 0.1, 0.2, 0.5, and 1, CL200, CL500, CT200, and CT500 have 1.50% decrease, 135.42% increase, 1.32% decrease, and 1.38% decrease, respectively (see case 17–case 20). In addition, CL500 has a sharp decline, when n is 0.1.
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From Figures 8(c) and 8(d), we can see that the attenuation value of is less than and of is less than , except the attenuation of up to at 410 kHz when n = 0.1. AL200, AL500, AT200, and AT500 have 7.60, 0.95, 8.68, and 8.75 times growth, respectively (see case 17–case 20).
5. Discussion
Figure 9 displays the comparison result of group velocity and attenuation of and modes in different adhesive quality. From Figure 9(a), we can see no significant change of group velocity. This result is consistent with the theoretical analysis in the second section: group velocity depends on the real part of the wavenumber; however, body wave attenuation will only affect the imaginary part of the wavenumber. From the data in Figure 9(b), it is apparent that body wave attenuation of the adhesive layer greatly affects guided wave attenuation. Generally, the guided wave attenuation is mainly determined by the shear wave attenuation, compared with the longitudinal wave. Figure 10 shows the number of guided wave modes propagating in the waveguide under various models. The attenuation of the bonding layer will not affect the number of guided wave modes, but the increase in thickness and density will lead to the increase in the number, which will increase the difficulty of guided wave mode separation and signal processing.
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6. Conclusion
The propagation of guided waves in composite structures with a viscoelastic adhesive layer is a difficult topic, which has been rarely reported. In this investigation, the aim was to assess how the viscoelastic parameters, such as attenuation coefficients of body wave, thickness, and density of adhesive affect the dispersion characteristics of the GFRP/adhesive/GFRP pipes. Results show that guided wave attenuation increases with the increase in body wave attenuation of the adhesive layer, but group velocity does not change significantly. Body wave attenuation only affects the imaginary part of the wavenumber, but the group velocity is determined by the real part of the wavenumber; guided waves in higher frequency will have higher attenuation; body wave attenuation does not result in the change of guided wave modes’ number; the increase in the thickness and density of the adhesive layer will lead to an increase in guided wave attenuation and modes number. Therefore, the attenuation value of the guided wave mode can reflect the material properties of the adhesive layer and the adhesive quality. The attenuation study of guided waves propagating in composite pipes contributes a theoretical basis to evaluate the bonding quality of the adhesive layer. On the other hand, the significance of analyzing the attenuation dispersion characteristics is to ensure that the attenuation value at the detection frequency is the lowest in practical engineering application, so as to ensure the sensitivity, reliability, and accuracy of detection.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
There are no conflicts of interest regarding the publication of this article.
Acknowledgments
Thanks are due to Prof. Han and Dr. Qin for their academic supervision and personal support. This work has been supported by the Emergency Management Project of the Natural Science Foundation of China (grant no. 61842103), the Youth Science Foundation (grant no. 11604304), and the Natural Science Foundation of Shanxi, China (grant no. 201801D121156).