Keyword

1 Introduction

Austenitic stainless steel possesses excellent corrosion resistance, high toughness, and machinability, making it widely used in various fields such as petrochemical equipment, marine equipment, and power generation equipment [1]. Generally, austenitic stainless steel serves in high-temperature, high-pressure, and corrosive environments, where various types of stress corrosion cracking (SCC), fatigue cracking, and intergranular corrosion are prone to occur on the surface of the structure, ultimately forming irregular crack defects [2, 3]. Irregular cracks can rapidly accumulate and propagate under external forces, leading to structural failure and posing a serious threat to structural safety and serviceability [4]. Therefore, conducting research on the visualization and reconstruction techniques of irregular cracks on the surface of austenitic stainless steel is of significant importance and engineering application value for timely grasping crack propagation morphology information and facilitating structural safety assessment and maintenance decision-making.

Austenitic stainless steel exhibits non-magnetic properties, weak electrical conductivity, and coarse grain size, which make conventional magnetic particle testing (MT) and magnetic flux leakage testing (MFL) techniques impractical. Ultrasonic testing (UT) is mainly used for internal defect detection and is not sensitive to surface cracking [5, 6]. Additionally, due to the coarse grain size and complex acoustic reflection signals within austenitic stainless steel, UT is not suitable for detecting surface irregular cracks [7]. Penetrant testing (PT) can be used for detecting small surface cracks and revealing the opening direction of surface cracks. However, PT requires thorough cleaning of the surface of austenitic stainless steel from adhering substances, oil contamination, and coatings, resulting in low operational efficiency and difficulties in on-site testing. Moreover, the penetrant agents can cause environmental pollution. Alternating current potential drop (ACPD) method can detect and monitor local structural crack detection by injecting current and using contact probes, but the probes need to penetrate the coating and directly contact the structure surface [8]. Eddy current testing (ET) is a widely used non-destructive testing technique for surface defect detection in structures. It relies on impedance analysis to detect and evaluate defects. However, ET is susceptible to lift-off disturbances and requires high surface smoothness of the structure [9, 10].

Alternating Current Field Measurement (ACFM) is an emerging electromagnetic non-destructive testing technique in recent years. It integrates the advantages of eddy current testing and alternating current potential drop method. By using an excitation coil to induce uniform current on the surface of a structure, ACFM can detect and evaluate surface cracks by measuring the distorted magnetic field above the defect when cracks occur, causing the current to accumulate at the crack endpoints and perturb the spatial magnetic field [11, 12]. The advantage of a uniform electric field makes the probe less susceptible to lift-off height and enables crack detection and evaluation without the need to remove attachments and coatings (up to a thickness of 10 mm). The ACFM mathematical model is precise and allows for accurate assessment of crack length and depth [13]. In addressing the issue of irregular crack detection and evaluation, Nicholson et al. applied ACFM technology to the detection of irregular rolling contact fatigue cracks on steel rails and proposed an evaluation method for inclined surface cracks [14]. Noroozi et al. utilized ACFM technology and fuzzy learning methods for arbitrary profile crack detection and reconstruction [15]. Ravan et al. developed a computational model for calculating the electromagnetic field perturbations around arbitrary profile cracks and solved for the analytical solution of the electromagnetic field inside the crack region using the finite difference method [16]. Pasadas et al. investigated the current perturbation patterns around irregular surface cracks on aluminum plate specimens under uniform current excitation and proposed a Tikhonov normalization surface crack visualization inversion method [17]. Li Yong et al. [18] achieved visualization and imaging display of buried depth defects in aluminum plates using pulse uniform eddy current technology and characteristic signal gradient fields. In previous research, our research group proposed a method for detecting arbitrarily oriented cracks using rotating alternating current electromagnetic fields, achieving high sensitivity detection of arbitrarily oriented cracks [19, 20]. While the aforementioned studies have laid the foundation for irregular crack detection and evaluation, there is limited research and reporting on the visualization and reconstruction methods of surface contour irregular cracks specifically in austenitic stainless steel.

In response to the issue of detecting and evaluating irregular cracks on the surface of austenitic stainless steel, this study proposes a method for visual reconstruction of such cracks based on the Alternating Current Field Measurement (ACFM) technique. By analyzing the electromagnetic field distortion patterns around irregular cracks through simulation models, the study introduces a technique for visualizing the surface contour of irregular cracks using the gradient field of the vertical magnetic field (Bz) image. Experimental tests are conducted to detect irregular cracks on austenitic stainless steel using ACFM, and the proposed visualization reconstruction method based on the Bz image gradient field is employed for precise imaging display and accurate assessment of the surface contour of irregular cracks.

2 An Irregular Crack Simulation Model

2.1 Simulation Model

An ACFM simulation model of an irregular crack on austenitic stainless steel is established using the ANSYS finite element simulation software, as illustrated in Fig. 1. The simulation model primarily consists of an excitation coil, a U-shaped magnetic core, a test specimen, and the irregular crack. The excitation coil is wound around the crossbeam of the U-shaped magnetic core with 500 turns, and the irregular crack is located beneath the excitation coil.

Fig. 1
An illustration of an A C F M simulation model. It depicts an excitation coil surrounded by a magnetic core, placed on a test block. A crack is present at the bottom.

ACFM simulation model of austenitic stainless steel irregular crack

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The simulation model and experimental specimen utilize austenitic stainless steel 316L for the test. The U-shaped magnetic core is made of manganese-zinc ferrite, while the excitation coil is composed of copper wire. The remaining medium is air. The material parameters for the simulation model are provided in Table 1. The irregular crack consists of four segments, each with the same length of 20 mm and a depth of 3 mm. The crack angles are oriented at 0°, 30°, 60°, and 90° relative to the scanning direction.

Table 1 Parameters for the simulation model
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2.2 Simulation Analysis of Electromagnetic Field

A sinusoidal excitation signal with a frequency of 2 kHz and an amplitude of 5 Vpp is applied to the excitation coil. The excitation coil induces a uniform current field on the surface of the austenitic stainless steel. The surface induction current vector map of the test specimen is extracted, as shown in Fig. 2. In the defect-free region, the induction current appears uniform. However, due to the presence of the irregular crack, the induction current accumulates at the endpoints and on both sides of the irregular crack.

Fig. 2
A graphical representation of a crack surrounded by arrows in different directions for the induced current. An index strip with different magnetic field intensities is given below.

Law of perturbation around irregular cracks by induction current

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In the defect-free region, the induction current is uniformly distributed, resulting in a vertical magnetic field Bz of 0. However, in the presence of an irregular crack, the accumulation of current causes spatial magnetic field distortion. The currents with different rotational directions lead to peak or trough values of Bz at the crack endpoints. The Bz image is extracted at a position 2 mm from the surface of the test specimen, as shown in Fig. 3. Bz exhibits peaks and troughs at different positions along the crack, and the distorted peak and trough positions align with the locations where the current accumulates at the endpoints of the irregular crack.

Fig. 3
A heatmap graph of a magnetic field plots Y direction versus X direction in millimeters. An index strip with different magnetic field intensities is given on the right.

Vertical magnetic field Bz image

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The simulation results indicate that the ACFM induction current can accumulate at the endpoints and on both sides of the irregular crack. This accumulation of current causes distortion in the vertical magnetic field Bz. The Bz image exhibits positive and negative peaks at the locations where the current accumulates, reflecting the surface topography information of the irregular crack.

3 Visualization Reconstruction Method

3.1 Gradient Field Algorithm

The gradient field reflects both the magnitude and direction of a scalar field. It can be obtained by taking the gradient of the scalar field. The gradient field is defined as follows:

$${\varvec{grad}}\;u\left( {x,y,z} \right) = \left( {\frac{\partial u}{{\partial x}},\frac{\partial u}{{\partial y}},\frac{\partial u}{{\partial z}}} \right) = \nabla u\left( {x,y,z} \right)$$
(1)

In the formula, \(u\left( {x,y,z} \right)\) is the quantity field, \({\varvec{grad}}\;u\left( {x,y,z} \right)\) or \(\nabla u\left( {x,y,z} \right)\) is referred to as the gradient field of the quantity field \(u\left( {x,y,z} \right)\).

The positions of positive and negative peaks in the vertical magnetic field Bz reflect the surface topography information of the irregular crack. By calculating the gradient field of the Bz image, we can obtain information about the distorted peak positions and their orientations. As shown in Fig. 4, the gradient fields in the X-direction (probe scanning direction) and Y-direction (direction of the induction current, perpendicular to the scanning direction) are calculated from the Bz image. These gradient field images represent the feature signals in both directions (the gradient images are dimensionless and only represent relative changes in magnitude).

Fig. 4
2 heatmap graphs A and B plot Y direction versus X direction in millimeters. A plots the X direction and Y plots the Y direction gradient fields. An index strip with different magnetic field intensities is given on the right.

Bz image gradient field signal

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In Fig. 4a, the gradient field image in the X-direction of the Bz image represents the surface contour of the irregular crack. In Fig. 4b, the gradient field in the Y-direction of the Bz image reflects the contours on both sides of the irregular crack. The gradient field image in the X-direction of the Bz image is more suitable for reconstructing the surface contour of the irregular crack. Therefore, this paper proposes a visualization reconstruction method for the surface contour of irregular cracks using the gradient field of the vertical magnetic field Bz image. The specific steps of the method are illustrated in Fig. 5.

Fig. 5
A flowchart of reconstruction of an image gradient field. The X direction of the gradient field is calculated and extrema is determined. The data is normalized and converted to a grayscale image at the end.

Visual reconstruction method of Bz image gradient field

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Step 1: Calculate the X-direction gradient field of the vertical magnetic field Bz. Define the probe scanning direction as the X-direction and calculate the X-direction gradient field (GXBz) from the feature signal Bz.

Step 2: Determine the extrema. Determine the sign of the gradient field GXBz (PGXBz). If it is positive, proceed to the next step. If it is negative, multiply it by (−1).

Step 3: Remove negative background values. Check if GXBz is greater than 0. Keep the positive values and discard the negative values.

Step 4: Normalize the data. Normalize GXBz to the range of 0–1, obtaining the normalized signal representing the surface of the crack. Plot it as a color map.

Step 5: Convert to a grayscale image. Convert the obtained normalized signal color map to a grayscale image, resulting in a visualization of the surface contour of the crack.

3.2 Simulation Results Visualization Refactoring

Using the algorithm described above, the second and third steps are applied to further process the X-direction gradient field of the vertical magnetic field Bz image in Fig. 4a. This results in a gradient field signal with the background field removed, as shown in Fig. 6a. The gradient field signal with the background field removed clearly depicts the visual contour of the crack surface. Next, in step four, the signal in Fig. 6a is normalized to the range of 0–1, resulting in a normalized signal color map shown in Fig. 6b. Finally, step five is applied to convert the normalized color map in Fig. 6b into a grayscale image, resulting in the visualization reconstruction of the irregular crack in the austenitic stainless steel, as shown in Fig. 6c. The visualization reconstruction result in Fig. 6c closely matches the contour of the irregular crack, indicating that the gradient field method using the vertical magnetic field Bz image can effectively visualize the surface contour of irregular cracks.

Fig. 6
3 heatmap graphs from A to C plot Y direction versus X direction in millimeters. A, B, and C remove the background field signals. An index strip with different values is given next to A.

Visual reconstruction simulation results of irregular crack in austenitic stainless steel l

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4 Experimental Verification

4.1 Test System Construction

To verify the feasibility of the visualization and reconstruction method for the vertical magnetic field gradient (Bz) of irregular cracks in austenitic stainless steel, an ACFM experimental test system was constructed. The irregular cracks on the surface of the austenitic stainless steel specimen were visualized and reconstructed. As shown in Fig. 7, the ACFM test system mainly consists of a probe, a signal box, a control cabinet, and a three-axis platform. The probe includes an excitation coil, a U-shaped magnetic core, a magnetic field sensor, and an amplification and filtering circuit. The signal box generates a 2 kHz sinusoidal excitation signal with an amplitude of 5 Vpp, which is amplified and loaded onto the excitation coil inside the probe. The excitation coil induces a uniform current field on the surface of the specimen. When cracks are present, the current disturbance causes spatial magnetic field distortion. The magnetic field sensor (tunneling magnetoresistance magnetic field sensor) inside the probe picks up the distorted magnetic field signal. After initial processing by the amplification and filtering module inside the probe, the signal is further amplified by the processing module inside the enclosure. Finally, the magnetic field sensor detects the analog signal, which is then converted to a digital signal by the signal acquisition module and sent to the computer. The computer program performs digital filtering, phase-locked amplification, and averaging on the signal to obtain the characteristic signal Bz [21, 22]. The PLC inside the control cabinet controls the three-axis platform to move in a grid pattern along the surface of the specimen. The platform drives the probe to extract the vertical magnetic field Bz above the specimen using a step-by-step method, and finally, the Bz image of the scanning area is plotted.

Fig. 7
A diagram A, an illustration B, and a photograph C. A depicts an A C F M test system. B depicts a 3-D probe structure with an excitation coil and a magnetic field sensor. C depicts an A C F M test system with a computer.

ACFM test system

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4.2 Experimental Test

As shown in Fig. 8b, a 316L austenitic stainless steel specimen was used for the experiment. The specimen surface was processed using electrical discharge machining to create irregular cracks consisting of four sections. The crack length was 30 mm, and the crack angles with respect to the scanning direction were 0°, 30°, 60°, and 90°, as shown in Fig. 8. The crack width was 0.5 mm, and the crack depth was 3.0 mm.

Fig. 8
A photograph of a test block with a crack. An illustration of crack angles is given on top of the block from 0 to 90 degrees.

Austenitic stainless steel with irregular crack test block

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The probe, driven by the three-axis platform, performed grid scanning of the irregular crack area. The probe was lifted to a height of 2 mm, and the scanning area was 100 × 100 mm2 with a scanning step size of 0.5 mm. After completing the scanning, the Bz image was obtained, as shown in Fig. 9. The vertical magnetic field Bz exhibited positive and negative peak values in the area of the irregular cracks, with strong peak signals observed at the endpoints and both sides of the cracks.

Fig. 9
A heatmap graph plots Y direction versus X direction in millimeters. It depicts the magnetic field results with regions of high and low intensities. An index strip with different values is given to the right.

Experimental results of vertical magnetic field Bz

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The processing of Fig. 9 using the gradient field visualization and reconstruction method for the Bz image is performed in step one. This step involves obtaining the X-direction gradient field signal image of the Bz image. The positions of magnetic field distortion peaks are connected by lines, reflecting the contour direction of the irregular crack surface, as shown in Fig. 10a. In step two, step three, and step four of the visualization and reconstruction method, further processing is applied to Fig. 10a, resulting in a normalized gradient field image with the background removed, as shown in Fig. 10b. Step five is then applied to Fig. 10b, resulting in a visual reconstruction of the surface contour of the irregular crack, as shown in Fig. 10c. The visual reconstruction image reflects the positions of the crack endpoints and surface contours. Due to the current concentration effect at the crack endpoints, the vertical magnetic field Bz is stronger at the endpoint positions. The calculation results of the Bz image gradient field reflect the positions of extreme signals. Since the extreme signals are stronger and can overshadow the image of the intermediate crack region, they provide particularly prominent information about the positions of the crack endpoints.

Fig. 10
3 heatmap graphs from A to C plot Y direction versus X direction in millimeters. A plots a gradient field in X direction. B depicts a normalized image. C depicts a reconstructed result. An index strip with different values is given next to A and B.

Experimental results of visual reconstruction of irregular cracks in austenitic stainless steel

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4.3 Reconstruction Accuracy Evaluation

Based on the peak positions in the visualized image of the surface contour of the irregular crack in Fig. 10c, the irregular crack can be divided into four sections. The coordinates of the endpoints of each crack section are as follows: (13.5, 16.0), (42.0, 17.0), (75.0, 29.0), (89.0, 56.0), (86.5, 87.0). Using these endpoint coordinates, a plot of the crack endpoints and their directions is created, resulting in an accurate assessment of the irregular crack on the surface of the austenitic stainless steel, as shown in Fig. 11.

Fig. 11
A line graph plots Y direction versus X direction in millimeters from 0 to 100. It plots the crack evaluation results from lower to higher values of X and Y.

Evaluation results of irregular cracks

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Based on the coordinates of the crack endpoint positions, the length and angle of each crack section can be calculated. The calculated lengths and angles for each section are as follows: Length = 28.5 mm, Angle = 2.0°. Length = 35.1 mm, Angle = 20.0°. Length = 30.4 mm, Angle = 62.6°. Length = 31.1 mm, Angle = 85.4°. By comparing these values with the true dimensions and angles of the irregular cracks on the surface of the austenitic stainless steel specimen, we can see that the maximum reconstruction error in the length of the irregular cracks is 5.1 mm, and the maximum error in the angle is 10°. This indicates a relatively high level of assessment accuracy in the visualization and reconstruction of the irregular crack lengths and angles.

5 Conclusions

  1. (1)

    This study simulated the electromagnetic field distortion around irregular cracks in austenitic stainless steel. The results showed that the ACFM-induced currents can accumulate at the endpoints and sides of irregular cracks, causing distortion in the vertical magnetic field Bz. The Bz exhibits peak and valley values at the positions of the crack endpoints, reflecting the information of the irregular crack endpoints and contours.

  2. (2)

    The gradient field visualization and reconstruction method using the vertical magnetic field Bz image was applied to reconstruct the ACFM simulation results of irregular cracks in austenitic stainless steel. The results showed that the X-direction gradient field image of the Bz image can present the surface contour of the irregular crack. By removing the background, normalizing the image, and converting it to grayscale, a clear visualized image of the surface contour of the irregular crack can be obtained.

  3. (3)

    An ACFM testing system was constructed to conduct experiments on the detection of irregular cracks in austenitic stainless steel, and the feasibility of the gradient field visualization and reconstruction method using the vertical magnetic field Bz image was validated. The results showed that the X-direction gradient field image of the Bz image can present a visualized image of the surface contour of the irregular crack, exhibiting extreme values at the crack endpoint positions. Through evaluation of the visualized results, the maximum error in crack length was 5.1 mm, and the maximum error in angle was 10°, validating the feasibility of the method. The research results of this study have strong guiding significance and engineering application value for the detection and evaluation of irregular cracks on the surface of austenitic stainless steel.