Keywords

1 Introduction

Alternating Current Field Measurement (ACFM) technology is a newly emerging electromagnetic non-destructive testing technology in recent years. It has many advantages such as lift-off insensitivity, non-contact detection, no need to clean structural attachments and coatings, and quantitative detection. It has been widely used in underwater structures, high-speed rail, nuclear power and non-destructive testing of surface defects of special equipment [1,2,3]. The alternating current magnetic field relies on a rectangular excitation coil to induce a uniform current on the surface of the conductive test block. The induced current passes through the crack vertically, gathers at both ends of the crack, causing the magnetic field (Bz) perpendicular to the direction of the test block to generate positive and negative peaks at both ends of the crack. Therefore, the distance between the positive and negative peaks of Bz reflects the crack length information; the induced current alternately generates peaks and valleys in the magnetic field (By) perpendicular to the direction of the crack at the end of the crack [4,5,6]; the induced current bypasses the bottom of the crack center, the current density at the center of the crack weakens, causing the magnetic field (Bx) along the direction of the crack to generate a valley [7], the valley reflects the crack depth information, as shown in Fig. 1.

Fig. 1
An illustration depicts the A C F M system. It includes five different directions. Counterclockwise current corresponds to the B Z wave peak, clockwise current corresponds to the B Z wave valley, sparse current corresponds to the B X wave valley, and dense current corresponds to the B X wave peak.

Principle of ACFM

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Under normal circumstances, the crack length can be obtained based on the distance between the peaks and valleys of the characteristic signal Bz, and the approximate depth of the crack can be obtained based on the change in the valley of the characteristic signal Bx. However, due to the large error between the crack length and the distance between the peaks and valleys of the characteristic signal Bz, and the impact of the crack length on the Bx valley, the traditional characteristic signal evaluation method has a large error, affecting the evaluation of the remaining life of the structure and maintenance decisions. W. D. Dover, A. M. Lewi, etc. established the classic ACFM theoretical model, and gave an analytical model of the electromagnetic field disturbance around the crack based on the two-dimensional plane assumption and uniform induced current [8, 9]. Hu Shuhui and others proposed a linear interpolation method to invert the crack length and depth dimensions [10]. A. L. Ribeiro and others gave a forward model of the crack and characteristic signal under uniform induced current [11]. G. L. Nicholson and others applied ACFM to the detection and evaluation of cluster rolling contact fatigue (RCF) cracks in rails [12, 13]. R. K. Amineh [14] and others proposed a crack depth evaluation method under different lift-off heights based on the model inversion algorithm. The research group and other scholars have proposed a crack size inversion algorithm based on neural network self-learning in previous studies [15,16,17,18]. The above research uses the characteristic signal Bx to analyze the crack depth, and uses the characteristic signal Bz to analyze the crack length, laying the foundation for the ACFM detection and evaluation of the crack. However, the above studies did not consider the interaction between the crack length and depth on the characteristic signals Bx and Bz, resulting in insufficient evaluation accuracy. In addition, self-learning methods such as neural networks require a large amount of sample data, which is difficult to apply and implement quickly on site.

In response to the above problems, this paper establishes a three-dimensional finite element simulation model of ACFM in a seawater environment, analyzes in detail the interaction rules of the changes in crack length and depth on characteristic signals, and proposes a two-step interpolation crack size evaluation algorithm and a segmented interpolation crack profile reconstruction algorithm. This provides an effective method for real-time detection and high-precision evaluation of cracks in underwater structures.

2 Marine Environment ACFM Simulation Model

A three-dimensional simulation model of ACFM in a seawater environment is established using the ANSYS finite element simulation software [19, 20]. As shown in Fig. 2, the simulation model mainly consists of an excitation coil, a U-shaped manganese-zinc ferrite magnetic core, a steel plate test block, and a crack defect. The excitation coil is wound on the manganese-zinc ferrite crossbeam, and an alternating current of 0.1 A with a frequency of 2000 Hz is loaded on the excitation coil. To realistically simulate the distribution of the electromagnetic field in the seawater environment, the inside of the simulation model probe is filled with air, and the outside is surrounded by the seawater environment. The characteristic parameters of the simulation model are shown in Table 1, and the structural parameters are shown in Table 2.

Fig. 2
A 3-D schematic diagram of A C F M in the seawater environment. It includes a crack, a U-shaped core, an excitation coil, seawater, a steel plate, and an air domain.

3D FEM model of ACFM in seawater environment

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Table 1 Parameters of simulation model
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Table 2 Dimensions of simulation model
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The current vector diagram on the surface of the steel plate test block in the simulation model is extracted, as shown in Fig. 3a. It can be seen that the excitation coil induces a uniform current on the surface of the test block, and the current passes through the crack vertically, gathering at both ends and deflecting in opposite directions. The current density on the surface of the steel plate test block is extracted, as shown in Fig. 3b. The current density gathers at the end of the crack to form a maximum peak, and the induced current bypasses from the bottom of the crack to form a valley in the center of the crack, causing Bx to appear a valley at the center of the crack, as shown in Fig. 4a. Due to the different deflection directions, the disturbance current causes Bz to present positive and negative peaks at the position of the crack end, as shown in Fig. 4b. The change rules of the characteristic signals Bx and Bz in the simulation model in the seawater environment are consistent with the ACFM principle.

Fig. 3
2 graphs. a. A current vector diagram with a color bar at the bottom ranges from 0 to 355333. b. A 3-D graph of current density versus X and Y directions. The irregular plane has two peaks around 1.6 amperes per meter square at negative 15 and 15 millimeters.

Distribution of induced current in the surface of specimen. a Current vector diagram. b Current density plot

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Fig. 4
Two 3-D spectral graphs. a. It plots B X versus X and Y directions. It has 2 peaks at 11 tesla at negative 11 and 10 millimeters. b. It plots B Z versus X and Y directions. It has a peak at 4 tesla at negative 10 millimeters and a dip at negative 10 tesla at 15 millimeters. Values are estimated.

Distorted magnetic field. a Bx. b Bz

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The characteristic signal Bx is related to the crack depth because the current bypasses in the direction of the crack depth, and the peaks and valleys of the characteristic signal Bz are located at the two ends of the crack, so the distance between the peaks and valleys reflects the crack length. However, the changes in crack length and depth have an interactive effect on the characteristic signals Bz and Bx, and a single change in length or depth is not the only factor affecting the characteristic signals. In order to determine the influence of crack size changes on the characteristic signals Bx and Bz, the characteristic signals of different size cracks are analyzed with the help of simulation.

3 Two-Step Interpolation Algorithm

Four groups of simulation models with different sizes of cracks were established, all with a crack width of 0.5 m. The first group of simulation models are cracks of the same length (10 mm) with different depths (1, 2, 3, 4, 5, 6 mm). The second group is cracks of the same length (20 mm) with different depths (1, 2, 3, 4, 5, 6 mm). The third group is cracks of the same length (30 mm) with different depths (1, 2, 3, 4, 5, 6 mm). The fourth group is cracks of the same length (40 mm) with different depths (1, 2, 3, 4, 5, 6 mm). The fifth group is cracks of the same length (50 mm) with different depths (1, 2, 3, 4, 5, 6 mm). The distance between the positive and negative peaks of the characteristic signal Bz is defined as PBz, and the PBz change graph of cracks with different depths is obtained, as shown in Fig. 5a. The change rule of PBz of the characteristic signal with different depths is basically consistent with the crack length, indicating that PBz is basically not affected by the crack depth.

Fig. 5
2 multiline graphs. a. The P B z versus length graph plots the 6 curve for 1, 2, 3, 4, 5, and 6 millimeters. exhibiting a linear upward trend. b. The S B x versus depth graph plots 5 curves for 10, 20, 30, 40, and 50 millimeters. All curves are increasing.

Signal characteristic with crack size. a PBz with cracks of different depths. b SBx with cracks of different lengths

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In order to compare the distortion amplitude of the characteristic signals of cracks at different depths, and to eliminate the influence of linear parameters such as current and coil turns on the simulation or experimental results, the sensitivity of the characteristic signal Bx is defined as SBx.

$$SBx = \left( {Bx_{0} - Bx_{\min } } \right)Bx_{0} = 1 - Bx_{\min } /Bx_{0}$$
(1)

The sensitivity SBx of the maximum value position distortion of the characteristic signal of cracks of different lengths is obtained, as shown in Fig. 5b. The characteristic signal Bx is mainly the secondary magnetic field distortion caused by the current disturbance in the center of the crack, and the Bx disturbance situation is more obviously affected by the depth of the crack, and the Bx sensitivity of the characteristic signal of cracks of different depths differs greatly. At the same time, the length of the crack also affects the disturbance of the current in the depth direction. The current in the center of the shorter crack is affected by the length of the crack, causing the maximum sensitivity change of Bx. When the crack extends to a certain extent, the induced current in the center of the crack reaches a minimum and is not affected by the length of the crack, and the Bx sensitivity of the characteristic signal is also not affected by the length of the crack. As can be seen from Fig. 5b, when the crack length is greater than 30 mm, the Bx sensitivity of the characteristic signal of cracks of different lengths basically remains consistent. When the crack length is less than 30 mm, there is a large difference in the sensitivity of the characteristic signal of cracks of different lengths.

In summary, the peak-to-valley distance of the characteristic signal Bz is affected by the crack length and is not affected by the change in crack depth; the sensitivity of the characteristic signal Bx is affected by the crack depth and is also affected by the crack length. Based on the above rules, in order to achieve high-precision evaluation of cracks, a two-step interpolation algorithm based on characteristic signals Bx and Bz is proposed, with the following specific steps:

  1. (1)

    Use the characteristic signal Bz to obtain the peak-to-valley distance PBz, and use PBz to obtain the crack length L.

  2. (2)

    When the crack length L ≥ 30 mm, the sensitivity SBx and the crack depth interpolation formula are obtained by simulation or experimental fitting, and the crack depth is evaluated by the interpolation method measured by the experiment SBx. When the crack length L < 30 mm, the sensitivity SBx and the crack depth interpolation formula of a specific length (such as 10, 20 mm, etc.) crack are obtained by simulation or experimental fitting, and the crack depth is evaluated by selecting the approximate length fitting formula measured by the experiment SBx. The two-step interpolation method can quickly obtain crack length and depth information, does not require a large amount of sample data, and is conducive to the online real-time evaluation of crack size.

The crack length, depth and characteristic signal rules obtained from the simulation model in this article, and the formulas (2)–(4) obtained from polynomial interpolation, where formula (2) is the first step to obtain the crack length L, formula (3) is the second step to obtain the crack depth D30 greater than 30 mm in length, and formula (4) is the second step to obtain the crack depth D20 of 20 mm in length.

$$L = 1.02 \times P_{{{\text{Bz}}}} + 1.02$$
(2)
$$D_{30} = - 0.2106 \times S_{Bx}^{3} + 2.207 \times S_{Bx}^{2} - 5.171 \times S_{Bx} + 4.491$$
(3)
$$D_{20} = 0.07196 \times S_{Bx}^{3} - 0.4911 \times S_{Bx}^{2} + 2.296 \times S_{Bx} - 1.908$$
(4)

4 Establishment of the Experimental System

The underwater ACFM detection system consists of an underwater probe, a hull, and an above-water computer. The probe is connected to the hull through a water-sealed joint, and the hull is connected to the above-water computer through an optical fiber, as shown in Fig. 6a. The internal lithium battery of the hull powers the entire system. The excitation module generates a sinusoidal excitation signal with a frequency of 2000 Hz and an amplitude of 10 V and loads it into the excitation coil in the probe. The excitation coil induces a uniform current field on the surface of the test piece. When a defect is present, the induced current is disturbed, causing spatial magnetic field distortion. The magnetic field sensor inside the probe measures the distorted magnetic field and transmits it to the amplification module inside the hull through the primary amplification filter circuit [21, 22]. After the signal is amplified, it is transmitted to the acquisition card, transmitted to the processor after A/D conversion, and transmitted to the surface through the optical fiber. The optical fiber receiver on the surface converts the optical signal into an electrical signal and transmits it to the computer. The internal program of the computer analyzes and evaluates the crack size in real time. The developed underwater ACFM system is shown in Fig. 6b.

Fig. 6
2 parts. a. A workflow of the underwater A C F M system. It begins with a computer, followed by fiber optic receivers, a capsule including the fiber optic, voltage regulator module, power amplifier, amplification module, incentive module, followed by probe-excitation coil, primary amplification, and others. b. A photo of an underwater A C F M system.

Underwater ACFM system. a System block diagram. b System photo

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5 Crack Evaluation Test

The probe and hull are placed in a water tank filled with seawater medium. The probe is driven by a mechanical arm to scan the crack area of the test block at a uniform speed of 40 mm/s, as shown in Fig. 7a. Two test blocks are set up in this test, both of which are made of Q235 material. Test block 1 has rectangular groove cracks of the same depth (4 mm), with a crack opening of 0.5 mm and crack lengths of 20, 40, and 45 mm respectively, as shown in Fig. 7b. Test block 2 has cracks of different cross-sectional shapes, with a crack opening of 0.5 mm. Crack 1# is a semi-elliptical crack with a length of 20 mm and a maximum defect depth of 4 mm; Crack 2# has a semi-elliptical defect with a length of 30 mm and a maximum defect depth of 5 mm; Crack 3# has a complex shape with a surface opening length of 40 mm and a maximum defect depth of 4 mm, as shown in Fig. 7c.

Fig. 7
3 parts. a. A photo of the underwater A C F M system testing setup. b. A photo of a test specimen block with cracks of different lengths at 45, 40, and 20 millimeters with a depth of 4 millimeters and a width of 0.5 millimeters. The cracks are circled. c. An illustration of three cracks with different complex shapes.

Underwater ACFM testing system. a Test photos. b Test block 1. c Specimen block 2 sections

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5.1 Crack Size Evaluation

The probe is driven by the mechanical arm to uniformly detect Test Block 1. The detection results of the sensitivity SBx of the characteristic signal Bx are shown in Fig. 8a, and the detection results of the characteristic signal Bz are shown in Fig. 8b.

Fig. 8
2 line graphs. a. It plots S B X versus X direction. The plot begins at 0% and has several rises and falls with an interval of approximately 30 millimeters. b. It plots B Z versus X direction. The plot begins at 0 on the y axis and then has a series of dips and peaks.

Testing results of cracks with same length. a SBx. b Bz

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From Bz, the peak-to-valley distances PBz can be obtained as 18.0, 37.5, and 42.0 mm, respectively. Using formula (2) for interpolation, the crack length sizes are obtained as 19.4, 39.3, and 43.9 mm. For cracks with lengths of 39.3 and 43.9 mm, the crack depths are calculated using formula (3) to be 3.7 and 3.8 mm, respectively, with errors of 7.5 and 5.0%, respectively, achieving high evaluation accuracy.

For cracks with a length less than 30 mm, the approximate formula (4) is used to evaluate the crack with a length of 19.4 mm, and the estimated crack depth is 3.8 mm, with an error of 7.5%. If formula (3) is used directly to estimate the crack depth, the crack depth is obtained as 7.2 mm, which is much larger than the actual size of the crack. It can be seen that the two-step interpolation algorithm can improve the evaluation accuracy of the crack depth for cracks with a length less than 30 mm. The results of the evaluation of the length and depth of the crack using the two-step interpolation algorithm are shown in Table 3. In summary, the two-step interpolation algorithm can achieve high-precision evaluation of crack length and depth, with a maximum depth error of 7.5% and a maximum length error of 3.0%.

Table 3 Evaluation results of cracks
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5.2 Crack Profile Evaluation

The probe uniformly detects Test Block 2, and the sensitivity SBx of the characteristic signal Bx is obtained by formula (1), and the part greater than the background magnetic field (the signal above the profile is removed), as shown in Fig. 9a.

Fig. 9
3 line graphs. a. It plots S B X versus X direction. The curve is almost flat, with three dip points. b. The B Z versus X direction graph plots a fluctuating curve. c. The depth versus S B X graph plots a curve that begins around (0, 0) and then rises gradually.

Testing results of crack with different profiles. a SBx. b Bz. c Depth and SBx relationship

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In order to achieve crack profile reconstruction and evaluation, this paper proposes a crack profile reconstruction algorithm based on the segmentation interpolation of the characteristic signal Bx. First, use the peak-to-valley distance PBz of the characteristic signal Bz in Fig. 9b and formula (2) to revise the crack length, and obtain the lengths of the three cracks as 19.4, 29.6, and 38.8 mm. Secondly, using the segmentation interpolation method, Crack 2# is used as the calibration crack, which is divided into 15 equal parts in the length direction to form 15 crack depth points. At the same time, the sensitivity SBx of the characteristic signal of Crack 2# is set to 15 equal parts in the crack area to form the sensitivity of 15 position points. Then, the corresponding relationship between the depth of any position of the crack and the sensitivity SBx can be obtained, as shown in Fig. 9c. The relationship between crack depth D and sensitivity SBx can be expressed by the polynomial interpolation formula as follows:

$$D = 0.8762 \times S_{Bx}^{3} - 3.634 \times S_{Bx}^{2} + 6.174 \times S_{Bx} + 0.04038$$
(5)

Finally, the sensitivity SBx of Crack 1# is divided into 10 equal parts along the length direction to obtain the sensitivity of 10 position points. The sensitivity SBx of Crack 3# is divided into 20 equal parts along the length direction to obtain the sensitivity of 20 position points. Using formula (5), the crack depth corresponding to the sensitivity of each position point can be obtained, and the crack profile contour can be reconstructed, as shown in Fig. 10.

Fig. 10
4 parts. A and C are line graphs of depth versus X directions for the estimated profile and the true profile. The curves fall at first, almost flat in the middle, and then rise at the end. B and D are the illustrations of the crack profiles of plots A and C, respectively.

Reconstruction results of crack profile. a 1# Crack reconstruction results. b 1# Crack profile visualization morphology. c 2# Crack reconstruction results. d 2# Crack profile visualization morphology

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The vertical coordinate 0 position in Fig. 10a and c represents the upper surface of the test block, and the negative value represents the depth below the surface of the test block. Figure 10b and d show the visualized shape of the crack profile, which matches the defect profile height in Test Block 2. The actual profile area of Crack 1# is 60.76 mm2, and the area of the closed area that can be evaluated by the curve is 57.28 mm2, with an error of 5.7%; the actual profile area of Crack 3# is 118.94 mm2, and the evaluation result is 116.13 mm2, with an error of 2.4%, achieving high reconstruction accuracy. The above shows that the segmentation interpolation algorithm based on the characteristic signal Bx can achieve high-precision reconstruction and visual display of the crack profile.

6 Conclusion

This paper establishes a three-dimensional finite element simulation model of ACFM in a seawater environment, analyzes in detail the influence of crack length and depth on characteristic signals, proposes a two-step interpolation crack size evaluation algorithm and a segmentation interpolation crack profile reconstruction algorithm, builds an underwater ACFM testing system, and carries out crack detection and evaluation experiments. The main conclusions are as follows:

  1. (1)

    The distance between the peaks and valleys of the characteristic signal Bz is related to the crack length and is not affected by the crack depth; the depth of the characteristic signal Bx valley is mainly related to the crack depth, but is also affected by the crack length.

  2. (2)

    When the crack is larger than 30 mm, the crack length does not affect the depth of the characteristic signal Bx valley; when the crack is less than 30 mm, the crack length has a greater impact on the characteristic signal Bx valley.

  3. (3)

    The two-step interpolation crack size evaluation algorithm based on the characteristic signals Bx and Bz can achieve high-precision evaluation of crack length and depth, with a maximum error of 3.0% for length evaluation and a maximum error of 7.5% for depth evaluation.

  4. (4)

    The segmentation interpolation profile reconstruction algorithm based on the characteristic signal Bx can achieve high-precision reconstruction and visual display of the crack profile, with a maximum reconstruction error of 5.7%.