The relationship between the fatigue damage and the group speed of guided waves in the steel wire

Abstract

Failure of the cables can cause a bridge to collapse. Fatigue damage of steel wire is one of the causes of cable failure. In this paper, we study the relationship between the fatigue damage and the group speed of guided waves in the steel wire. The relationship between cyclic loading times and group speed of steel wire is obtained by applying tension to steel wire. The results show that the group speed of guided waves increases linearly before the 6.02 million cyclic-loading times, increases exponentially from 6.02 to 6.97 million times. The curve of the relationship between the group speed of the guided waves and the number of cyclic loading times can be fitted to an exponential function, and this curve can be used as a calibration curve to evaluate the fatigue damage of steel wire.

Keywords

Group speed guided waves fatigue damage steel wire

1. Introduction

The cable is the main force-bearing component of cable-stayed bridge. And the bridge cables will experience different degrees of fatigue damage due to the increase in the number of years of service. As the main load bearing unit of bridge cables, the fatigue damage assessment of steel wire is necessary for bridge detection. Fatigue damage is microscopic defect, which brings great challenges to testing. At present, the main methods of cable detection are acoustic emission, magnetic detection and guided wave detection [1–4]. Xu [4] studied the effect of fatigue damage on the notch frequency of the pre-stressing strand under tension and the notch frequency can also be employed to evaluate the fatigue damage of strands after 1.85 million cyclic-loading times. Li [5] used acoustic emission technology to detect the evolution process of fatigue damage of a cable-stayed cable in a bridge in China, and analyzed the causes of damage formation. Sun [6,7] conducted acoustic emission experiments on high-strength steel wires and cables, and obtained the time-frequency characteristics and energy attenuation laws of acoustic emission waves propagating on the steel wires. Zhou [8] studied the variation law of circumferential leakage flux during the fatigue damage of steel wire ropes and it was found that the axial magnetic permeability of the linear reciprocating section of the steel wire rope gradually increased and the distribution tends to be consistent, while the magnetic permeability of the steel wire in the curved section of the pulley gradually decreases and the distribution points to unevenness. Zhou [9] used the coupling efficiency of magnetostrictive guided waves to distinguish the fatigue damage of strands. Zhou [10] proposed a method of using nonlinear coefficients as characteristic parameters to detect fatigue damage of components.

Recently, there are few methods for detecting fatigue damage of steel wires, and there is a lack of characteristic parameters that can be used to evaluate cable fatigue damage on site. Magnetostrictive guided wave detection technology is widely used in the detection of steel wire, steel strand and cable because it has the characteristics of non-contact, no coupling and high energy conversion efficiency [11,12]. Therefore, the relationship between the fatigue damage and the group speed of guided waves in the steel wire is studied in this paper. The experiment obtained the relationship between the group speed of the guided waves and the number of cyclic loadings by conducting guided wave detection on the cyclically loaded steel wire. The group speed increases almost linearly with the increase of cyclic loading times, and the trend is basically the same under different tensions. The results show that the group speed can be used to detect the fatigue damage of steel wire.

2. Theory background

Under the action of cyclic load, the steel wire components often cause fatigue damage, causing dislocation microstructure changes and crack initiation, which leads to degradation of material properties. The propagation velocity of guided wave in the structure is only related to the properties of the material, and the longitudinal wave velocity and the transverse wave velocity are as shown in Eq. (1) and Eq. (2), respectively [13]. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle C_{L}=\sqrt{{\displaystyle \frac{E}{{\rho}(1-v^{2})}}}\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle C_{T}=\sqrt{{\displaystyle \frac{E}{{\rho}(1+v)}}}\end{eqnarray}$ (2) where 𝜌 denotes material density, v denotes Poisson’s ratio, and E denotes elastic modulus.

The modulus of elasticity is one of the mechanical properties of a material. The damage evolution of the material is characterized by measuring the change of the effective elastic modulus of the damaged material, and the damage model is shown in Eq. (3) [14]. $\begin{eqnarray}\displaystyle D=1-\frac{\tilde{E}}{E} & & \displaystyle\end{eqnarray}$ (3) where D denotes damage variable, $\tilde{E}$ denotes modulus of deformation.

Xie [15] discussed the physical concept of the elastic modulus of damaged materials in the elastic modulus method, pointed out the limitations and applicable conditions of the elastic modulus method, and the problems in describing the damage behaviour of elastoplastic materials. The elastic modulus method requires special processing of the test piece and accurate measurement of the strain of the material during loading.

It can be seen from the above analysis that the fatigue damage of the steel wire causes a change in the elastic modulus, which in turn causes a change in the guided wave propagation speed. We can detect the fatigue damage of the steel wire by measuring the propagation velocity of the guided wave.

3. Experimental setup

In the experiment, a hot-dip galvanized steel wire with 7 mm diameter 1770 MPa ultimate tensile strength (UTS) and 38.5 mm² cross-sectional area was selected. In the middle of the specimen is an arc transition with 20 mm length and 6.7 mm diameter as shown in Fig. 1. The arc transition is made of a hand-held electric grinder, the photo of the steel wire arc transition is shown in Fig. 1(a) and the dimension diagram of wire arc transition section is shown in Fig. 1(b). The wire was cyclically loaded using a fatigue testing machine and the upper limit stress was 30.67 kN (about 45% UTS). The cyclic stress amplitude was 410 MPa. Fatigue testing machine cyclic loading mode is sine wave, and the loading frequency was set to 5 Hz.

Fig. 1.

The photo of the steel wire arc transition and the dimension diagram of wire arc transition section.

Fig. 2.

The figure of experimental sensor layout and guided wave propagation.

The experimental sensor layout is shown in Fig. 2. The transmitter consists of an exciting coil and a permanent magnet magnetizer, and the receiver comprises a receiving coil and a permanent magnet magnetizer [16,17]. The distance between the transmitter and the receiver is 400 mm. The distance between the transmitter and the left end of the wire is 450 mm and the clamping part of the fatigue testing machine is 200 mm at the left end of the wire. The arrangement of the receiver and the transmitter are symmetric about the arc transition The excitation signal was a 70 kHz sine wave burst with three cycles and the signal amplitude was 400 V. The sampling frequency was set to 5 MHz. In this experiment, after the steel wire is cyclically loaded, the data is collected by the magnetostrictive guided wave instrument under four different tensile stresses. The photo of the experimental setup to collect the data from the steel wire in the fatigue testing machine is shown in Fig. 3. When conducting the guided wave detection on the steel wire under different cycle times, it is necessary to consider the influence of the tension on the wire on the guided wave signal. Therefore, when the steel wire is cyclically loaded for a certain number of guided waves, the data should be collected under different tensile stresses. The guided wave detection was performed on steel wires with different cyclic loading times under four tensile forces of 6.81 kN (about 10% UTS), 13.63 kN (about 20% UTS), 20.44 kN (about 30% UTS) and 27.26 kN (about 45% UTS), respectively.

4. Results and discussions

The steel wire was subjected to cyclic loading from 0 to 6.97 million times and the interval is 0.2 million times The propagation path of the guided wave in the steel wire is shown in Fig. 2. The guided wave propagates from the excitation coil to both ends of the steel wire When the guided wave propagates from the excitation coil along the right side of the steel wire to the receiving coil, there is a wave called the first passing wave. The position of the excitation coil is recorded as position A. The guided wave travels from the excitation coil to the position A at a distance of 300 mm. Similarly, when the guided wave propagates from the excitation coil along the right side of the steel wire to the clamping position of the fatigue test machine, a reflection waves is generated. It should be noted that due to the clamping of the fatigue test machine, the guided wave is not transmitted to the end of the wire. Then, the reflected wave continues to propagate along the right side of the wire. When the guided wave propagates to the receiving coil, there is a second wave, and the guided wave travel distance is 800 mm. The guided wave propagates to the right clamping position of the wire and then propagates along the left side of the wire to the receiving coil. This wave is the third wave with a propagation distance of 1300 mm and is recorded as position B. Then the guided wave propagation distance between the position A and the position B is 1000 mm. Figure 4 shows the time-domain signal collected by the guided wave instrument with 0 cyclic loading times and 6.81 kN tension. The initial pulse, the first passing wave and the end reflection waves are shown in Fig. 4. According to the position A and the position B in Fig. 2, the corresponding position is taken in Fig. 4. From Fig. 4, the propagation time of the guided wave from the excitation coil to the position A and the position B is 0.0826 ms and 0.2674 ms, respectively. Then the guided wave propagation time between the position A and the position B is 0.1848 ms. Therefore, the group speed of guided waves can be calculated to be 5411 m/s by the propagation time and the propagation distance between the the position A and the position B.

Fig. 3.

The photo of the experimental setup to collect the data from the steel wire in the fatigue testing machine.

Fig. 4.

The time-domain signal with the cyclic loading of 0 times and the tension of 6.81 kN.

Fig. 5.

The time-domain signal with the cyclic loading of 2.12 million times and the tension of 6.81 kN.

Figure 5 shows the time-domain signal collected by the guided wave instrument with 2.12 million cyclic loading times and 6.81 kN tension. According to the position A and the position B in Fig. 2, the corresponding position is taken in Fig. 5. From Fig. 5, the propagation time of the guided wave from the excitation coil to the position A and the position B is 0.0836 ms and 0.2664 ms, respectively. The guided wave propagation time between the position A and the position B is 0.1828 ms, and the group speed of guided waves is 5470 m/s. It can be seen from the group speed of the guided waves loaded in 0 cycles and the group speed of the guided waves of 2.12 million times that the group speed of the guided waves increases as the number of cycles of wire loading increases.

Fig. 6.

The relation curves of the group speed and the cyclic loading times under different tension.

Fig. 7.

Fitting curves of the relationship between the group speed of the guided waves and the number of cyclic loadings under the tension of 6.81 kN.

The relation curves between the cyclic loading times and the group speed under different tensile forces are shown in Fig. 6. According to Fig. 6, the group speed of guided waves increases linearly before the 6.02 million cyclic-loading times, increases exponentially from 6.02 to 6.97 million times. In order to better evaluate the fatigue damage of steel wire, the curve of the relationship between the velocity of the guided wave group and the number of cyclic loadings under the 6.81 kN tensile force is exponentially fitted. The fitting curve is shown in Fig. 7 and the curve equation is shown in Eq. (4). The fitting curve of the relationship between the group speed of the guided waves and the number of cyclic loading times can be used as a calibration curve to evaluate the fatigue damage of steel wire. $\begin{eqnarray}\displaystyle y=5.428\ast 10^{-14}\ast e^{4.889\ast x}+5410\ast e^{0.0049\ast x} & & \displaystyle\end{eqnarray}$ (4) where the unit of y is m/s, the unit of x is one million times.

Equation (4) shows the relationship between the number of cyclic loadings and the guided wave group speed when the tensile stress is 6.81 kN. Considering the effect of tensile stress on the velocity of the guided wave group according to Fig. 6, the relationship between the velocity of the guided wave group and the number of cyclic loading under different tensile stresses is shown in Eq. (5). $\begin{eqnarray}\displaystyle y=5.428\ast 10^{-14}\ast e^{4.889\ast x}+(-0.02689\ast F^{2}-1.798\ast F+5424)\ast e^{0.0049\ast x} & & \displaystyle\end{eqnarray}$ (5) where the unit of y is kN.

5. Conclusions

The relationship between the group speed of the guided waves and the number of cyclic loadings is studied in this paper The results show that the group speed of the guided waves would changes under different cyclic loading times. The group speed of guided waves increases linearly before the 6.02 million cyclic-loading times, increases exponentially from 6.02 to 6.97 million times. The entire curve can be fitted to a monotonically increasing exponential function. The fitting curve of the relationship between the group speed of the guided waves and the number of cyclic loading times can be used as a calibration curve to evaluate the fatigue damage of steel wire And the group speed can be employed to evaluate the fatigue damage of steel wire.

Footnotes

Acknowledgements

This work was supported by the National Key Research and Development Program of China (Grant No. 2019YFB1310403) and the National Natural Science Foundation of China (Grant No.51575213).

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