Keywords

1 Introduction

Aluminum alloys are widely used in the aerospace industry due to their good specific properties such as high-thermal conductivity and high strength-to-weight ratio [1]. However, they are prone to localized corrosion pits due to corrosive environments such as plain water and saltwater [2, 3]. Because the aerospace structure is affected by cyclic loads, it is very easy to generate stress concentrations at the location of corrosion pits. Therefore, after corrosive pit formation, the next steps under complicated fatigue loading condition include pit growth, transition from pitting to fatigue crack initiation and short crack growth in arbitrary-angles [4,5,6]. A US Air Force airframe teardown analysis showed that 80% of fatigue crack initiation features are corrosion pits [7]. Short cracks expand along the depth and surface of the structure, and then form typical surface cracks [8]. The entire structure may fail before the crack penetrates, which is concealed and hazardous. Therefore, it is necessary to inspect the process of pit-to-crack transition and the size and direction of fatigue crack growth for the safety assessment of the aerospace equipment.

The periodical non-destructive inspection of key parts is very important for the safety of equipment [9]. Nevertheless, conventional non-destructive testing methods to inspect structures regularly are generally expensive and labor-intensive. Moreover, it is not possible to repair and replace parts of the structure with cracks immediately, especially if the aircraft component is expensive. A new maintenance concept that is called structural health monitoring (SHM) technology has been proposed, which is to rationally allocate maintenance resources, and improve the safety and reliability of in-service aircraft structures [10]. Current SHM technologies mainly include GPS technology, optical fiber, strain gauges, accelerometers, acoustic emission, ultrasonic and electromagnetic. The GPS technology is used as an overall structure monitoring method, and it is difficult to monitor small cracks within large structures in time [11]. The optical fiber and strain gauges are commonly used techniques for monitoring structural stress and strain, but cannot monitor crack initiation and propagation [12, 13]. The accelerometers evaluate the structural state by measuring the mode of structural vibration, but the monitoring effect of nonlinear damage such as fatigue cracks needs to be improved [14]. The acoustic emission technology can detect dynamic cracks under external structural stress, but static cracks do not produce signals, and the application cost is high [15]. The ultrasound technology is not suitable for monitoring crack growth due to the need for couplants [16].

Electromagnetic monitoring methods are low-cost and easy-to-use based on electromagnetic induction and have good perspectives for use in the field of metallic structure health monitoring, which mainly include eddy current [17], alternating current potential drop [18] and metal magnetic memory (MMM) [19]. Li et al. measured the normal and tangential components of the stress-induced MMM signal by permanently installed magnetic sensor arrays [20]. MMM, that is susceptible to environmental interference and stress, is a weak magnetic signal which can easily cause misjudgment of crack monitoring results. Chaudhuri et al. proposed the alternating current potential drop method for weld toe fatigue crack initiation [21]. The alternating current potential drop requires alternating current to be applied to both ends of the structure, and complex array electrodes are placed on the surface of the structure for the crack monitoring. It destroys the anti-corrosion coating on the surface of the structures, and the monitoring results are greatly affected by human factors such as electrode installation. JENTEK Sensors Inc. has developed the Meandering Winding Magnetometer (MWM) sensor for monitoring of crack initiation and growth during fatigue tests and in service [22]. A rosette-like eddy current array sensor with high sensitivity was proposed for quantitatively monitoring hole-edge crack of aircraft structure [23, 24]. Sun et al. used one exciting coil covering the entire thickness and several sensing coils distributed along the axial length of the hole to quantitatively monitor a bolt hole crack in the radial and the axial directions [25]. The change-prone micro eddy current sensor was designed and fabricated with flexible printed circuit board (FPCB) technology to monitor fatigue cracks of a metal structure [26]. The annular flexible eddy current array (FECA) sensor was developed for quantitative monitoring of cracks in ferromagnetic steels under varying loads and temperatures [27]. Yusa et al. designed the arrayed uniform eddy current probe for crack monitoring and sizing of surface breaking cracks with the aid of a computational inversion technique [28]. However, the unidirectional eddy current generated by the above method is only sensitive to cracks perpendicular to the direction of the current. In addition, the traditional electromagnetic monitoring method cannot obtain the direction of crack propagation.

In order to realize the detection of cracks at different angles, the RACFM technique was proposed by different scholars. Hoshikawa et al. proposed a new ECT probe called the Hoshi Probe which utilizes a uniform direction rotating eddy current to reduce noises [29, 30]. Cracks of different angles are detected using the designed probe, but it does not quantitatively analyze the relationship between the angle and size of the crack and the amplitude and phase of the characteristic signal. Udpa et al. designed a rotating current probe with sensor arrays to detect cracks at different angles at a fastener site in layered structures [31, 32]. However, only the amplitude of the characteristic signal is analyzed, the angle of the crack is obtained through the C-scan image. All the above methods are non-destructive testing methods, but neither of them involves the fixed-point monitoring of cracks.

In this paper, a novel fatigue crack monitoring method is proposed based on the RACFM. The angle of crack is quantitative measured by the Bz (the magnetic field perpendicular to the specimen) phase difference method. Firstly, two alternating current signals with a phase difference of 90° and the same frequency are respectively loaded on the orthogonal excitation coils, and the Bz signal is extracted by a sensor. Secondly, the probe is placed in a position where the specimen has no defects, and the real and imaginary components of the Bz signal are recorded as the calibration signal. Then, the probe is placed at the corrosion pits, the real and imaginary component of the crack is recorded as the monitoring signal. The differential signal is obtained by subtracting the calibration signal from the monitoring signal. The real and imaginary parts of the differential signal are converted into the amplitude and phase. Thirdly, taking the 0° crack phase as the reference, the reference is subtracted from the Bz phase of the measured crack to obtain the crack angle measurement result. Finally, the crack angle measurement result is modified by the amplitude of the Bz signal. The method proposed in this paper provides a new idea for monitoring the fatigue crack direction of aluminum alloy materials in the aerospace industry. Compared with the traditional eddy current monitering technology, the testing system is simpler and the quantification accuracy of the crack angle measurement is improved.

The rest of the paper is organized as follows. The theoretical model of the RACFM is established to analyze the influence of the angle of the crack on the distorted electromagnetic field in Sect. 2. The influences of crack angles and sizes on the characteristic signal are simulated by the 3D FEM model in Sect. 3. The RACFM monitoring probe and testing system are developed and cracks of different angles, lengths and depths are monitored in Sect. 4. Finally, the conclusions and future work are outlined in Sect. 5.

2 Theoretical Model

The rotating uniform alternating current field can be induced using two orthogonal unidirectional coils with 90° phase shift alternating currents. The induced current Jx1 and Jx2 generated by the two excitation coils can be represent respectively as follows:

$$J_{{{\text{x1}}}} \left( t \right) = J_{0}^{{}} \sin \left( {\omega t + \alpha_{0} } \right)\vec{X}_{1}$$
(1)
$$J_{y1} \left( t \right) = J_{0}^{{}} \cos \left( {\omega t + \alpha_{0} } \right)\vec{Y}_{1}$$
(2)

where \(\vec{X}_{1}\) and \(\vec{Y}_{1}\) are unit vectors along the x and y axis respectively, J0 is the amplitude of induced current density, α0 is the phase of induced current, ω is the frequency of the induced current, and t is the time.

The directions of the two induced electric fields are orthogonal, their frequencies are equal and the phase difference is 90°, as shown in Fig. 1. When there are no defects, the induced electric field is uniform. When defects exist, the induced electric field is distorted.

Fig. 1
A 3 D diagram of a rectangular specimen with a thin cylindrical object at the center. It marks the crack and corrosion pits with arrows.

Perturbations of uniform alternating current around corrosion pits

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As shown in Fig. 2, the induced currents are decomposed into the current component Jx2 perpendicular to the crack direction and the current component Jy2 parallel to the crack direction as follows:

$$J_{{{\text{x2}}}} \left( t \right) = (J_{0}^{{}} \sin \left( {\omega t + \alpha_{0} } \right)\cos (\theta ) + J_{0}^{{}} \cos \left( {\omega t + \alpha_{0} } \right)\sin (\theta ))\vec{X}_{2}$$
(3)
$$J_{{{\text{y2}}}} \left( t \right) = (J_{0}^{{}} \cos \left( {\omega t + \alpha_{0} } \right)\cos (\theta ) - J_{0}^{{}} \sin \left( {\omega t + \alpha_{0} } \right)\sin (\theta ))\vec{Y}_{2}$$
(4)
Fig. 2
A schematic diagram of a rectangular specimen with a corrosion pit and a diagonal thin cylindrical object of crack at the center at an angle of theta degrees. It has color gradient arrows with marked dimensions.

Schematic diagram of induced current decomposition

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where \(\vec{X}_{2}\) and \(\vec{Y}_{2}\) are unit vectors along the x2 and y2 axis respectively and θ is the crack angle.

According to the vector composition theorem, the total induced current density J(t) in the specimen can be regarded as the superposition of two orthogonal induced currents Jx2 and Jy2, and the total induced current amplitude AJ(t) and phase angle θJ(t) can be represent as follows [33]:

$${ }A_{J} \left( {\text{t}} \right) = \sqrt {J_{x2} \left( t \right)^{2} + J_{y2} \left( t \right)^{2} } = J_{0}$$
(5)
$${ }\theta_{J} \left( {\text{t}} \right) = \arctan \left( {\frac{{J_{x2} \left( t \right)}}{{J_{y2} \left( t \right)}}} \right) = \omega t + \alpha_{0} { + }\theta$$
(6)

When there is no defect, the magnitude of the induced electric field generated by the rotating alternating current field is a fixed value. The direction of the induced electric field rotates periodically with time, and the rotation period is equal to the excitation current period. Furthermore, the phase of the induced electric field and the angle of the crack show a linear relationship.

The induced electric field is regarded as a number of straight wires carrying current, and the wire currents Ix2 and Iy2 in two directions can be expressed as follow:

$$I_{{{\text{x2}}}} \left( t \right) = (I_{0}^{{}} \sin \left( {\omega t + \alpha_{0} } \right)\cos (\theta ) + I_{0}^{{}} \cos \left( {\omega t + \alpha_{0} } \right)\sin (\theta ))\vec{X}_{2}$$
(7)
$$I_{{{\text{y2}}}} \left( t \right) = (I_{0}^{{}} \cos \left( {\omega t + \alpha_{0} } \right)\cos (\theta ) - I_{0}^{{}} \sin \left( {\omega t + \alpha_{0} } \right)\sin (\theta ))\vec{Y}_{2}$$
(8)

where I0 is amplitude of the wire current.

At the no crack position, the magnetic induction intensity in the Z direction (Bz) is zero, since the magnetic field components of each long straight wire cancel each other out. When the crack exists, the current line at the tip of the crack forms a curved arc, so Bz is not zero.

The micro-current arc \(I{\text{d}}\tau\) on the current line near one tip of the crack is selected. The radius of the arc is r. According to the principle of electromagnetic field superposition, the integral of the magnetic field formed by the current micro-element along the current deflection path in the specimen must be along the Z direction. According to the Biot-Savart law, Bzx2 and Bzy2 can be expressed as [34]:

$$B{\text{z}}_{x2} (t){ = }\oint {\frac{{\mu_{0} I_{x2} {\text{(t)d}}\tau }}{{4\pi \left( {r^{2} { + }l^{{2}} } \right)}}}$$
(9)
$$Bz_{y2} (t){ = }\oint {\frac{{\mu_{0} I_{y2} {\text{(t)d}}\tau }}{{4\pi \left( {r^{2} { + }l^{{2}} } \right)}}}$$
(10)

where ÎĽ0 is vacuum permeability and l is lift-off.

Bz can be expressed as:

$$B{\text{z(t)}} = B{\text{z}}_{x2} (t) + Bz_{y2} (t)$$
(11)

As shown in Fig. 3, the induced current is much more significantly sensitive to vertical cracks than to parallel cracks, so \(B{\text{z}}_{x2} (t) \ll Bz_{y2} (t)\). Bz can be expressed as:

$$B{\text{z(t)}} = \oint {\frac{{\mu_{0} I_{0} {\text{cos(}}\omega t + \alpha_{0} { + }\theta ){\text{d}}\tau }}{{4\pi \left( {r^{2} { + }l^{{2}} } \right)}}}$$
(12)
Fig. 3
2 diagrams labeled A and B. A has the current lines with a downward flow and a rectangular bar at the center. B has the current lines with a rightward flow and a rectangular bar at the center.

Schematic diagram of current disturbance. a Current lines are perpendicular to the crack; b Current lines are parallel to the crack

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It can be seen from (12) that at the crack tip, the angle of crack does not affect the amplitude of the Bz signal, but it has a linear relationship with the phase. Finally, the establishment process of the theoretical model is shown in Fig. 4.

Fig. 4
A process flow diagram has the following flow, orthogonal excitation, induced current, current decomposition, and characteristic signal.

Process of the theoretical model

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3 Finite Element Analysis

3.1 Model Set Up

The 3D simulation model for the fatigue crack angle quantitative monitoring is built using the finite element software COMSOL, which is widely used to solve Maxwell's equations for modeling the electromagnetic field response due to its efficient computing performance and outstanding multi-field bidirectional direct coupling analysis capabilities [35, 36], as shown in Fig. 5. The model includes a specimen, coil-1, coil-2, crack, pick-up point and air. The lift-off of coil-1 is 0.8 mm and the lift-off of coil-2 is 0.5 mm. In order to ensure that the induced current is uniform in any direction, the two coils are loaded with different currents due to different lift-off heights. Coil-1 carries the alternating currents with a 0.31 A amplitude, 10 kHz frequency, and 0° phase. Coil-2 carries the alternating currents with a 0.3 A amplitude, 10 kHz frequency, and 90° phase. Two excitation coils are perpendicular to each other. The pick-up point is located at the coil center, whose lift-off distance is 1.5 mm. The thickness of the specimen is 10 mm, and the conductivity is 3.77 E7 S/m. The center of the excitation coils is at the tip of the crack. The mesh of the model adopts free tetrahedral. The grid size of specimen, coil-1, coil-2 is set to finer and air is set to fine in mesh module. A transient analysis is set for the model. The dimensions of the model and the characteristic parameters are shown in Table 1.

Fig. 5
A 3 D model of a rectangular slab labeled air has the rectangular specimen at the center with coils 1 and 2. It marks the crack and pick-up point at the center.

Fatigue crack angle quantitative monitoring FEM model

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Table 1 Parameters of the model
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To explore the influence of corrosion pits on characteristic signals, cracks with and without corrosion pits are established as shown in Fig. 6. The length of the crack is 5 mm, the width is 0.2 mm, and the depth is 2 mm. The size of the corrosion pit is generally small, and the corrosion pit is spherical with a radius of 0.5 mm. The Bz signals of cracks with and without corrosion pits are shown in Fig. 7. The phase difference is 1.09°, and the amplitude difference is 3.18%. It shows that corrosion pits have little effect on the amplitude and phase of the Bz signal, which is because the boundary of the corrosion pit is smoother than the tip of the crack, and the current is more likely to gather at the tip of the crack. So, the following simulations and experiments adopt a model without corrosion pits to simplify the study.

Fig. 6
2 F E M model diagrams of 2 blocks labeled without corrosion pit and with corrosion pit. Without corrosion pit has a rectangular thin bar, while with corrosion pit has a rectangular thin bar attached to a circular object on the left.

FEM models of cracks with and without corrosion pits

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Fig. 7
A double-line graph of B z versus time. It plots the overlapping without corrosion pit and with corrosion pit lines as concave down, increasing curves with a dip at 0.5 on the x-axis. The value is estimated.

Simulation results of cracks with and without corrosion pits

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As shown in Fig. 8, the induced currents at different moments are extracted. It can be seen from the figure that the angle of the induced current change between two adjacent times is about 45°, and the induced current rotates counterclockwise. So, the structure of the orthogonal excitation coils can induce a periodic rotating uniform alternating current on the surface of the specimen, and the induced current will be disturbed when it encounters a crack.

Fig. 8
8 diagrams labeled A to H of current flows are represented using arrows. They have a leftward flow, a diagonal upward flow, an upward flow, a diagonal rightward flow, a rightward flow, a diagonal downward flow, a downward flow, and a diagonal leftward flow of current from A to H, respectively.

Results of induced current at different transient times. a T/8; b 2 T/8; c 3 T/8; d 4 T/8; e 5 T/8; f 6 T/8; g 7 T/8; h 8 T/8

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3.2 Characteristic Signal Analysis of Cracks with Different Angles

To explore the correlations between the crack angle and the Bz signal, the crack (length = 20 mm, width = 0.2 mm, depth = 2 mm) is established, and the cracks with different angles (0°, 60°, 120°, 180°, 240°, 300°) are simulated. The Bz signals at the tip of crack are extracted, as shown in Fig. 9a. It can be seen from the figure that the Bz signals of cracks at different angles have the same amplitude and different phases. Hence, the Bz phases of the cracks at different angles are extracted, as shown in Fig. 9b. It shows that the phase of the Bz signal has a linear relationship with the angle of the crack. This is because the orthogonal excitation coils generate a rotating uniform alternating current in the specimen and the direction of the initial distorted current has a linear relationship with the angle of the crack.

Fig. 9
3 graphs labeled A to C. A has a multi-line graph of B z versus time that plots several color gradient sinusoidal waves. B has a line graph of phase angle versus angle that plots an upward slope. C has a polar diagram of degrees that plots a circular shape.

Characteristic signals of cracks with different angles. a Bz signals of cracks with different angles; b Relationships between the crack angle and the phase of the Bz signal; c Crack angle measurement results

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Taking the Bz phase of the 0° crack as the reference, the reference is subtracted from the Bz phases of the cracks at all angles, and crack angle measurement results are obtained, as shown in Fig. 9c. It shows that, in the polar diagram, the distance from the signal point to the origin is the amplitude of the Bz signal, and the phase of the signal point is the measured angle of the crack.

3.3 Characteristic Signal Analysis of Cracks with Different Lengths and Depths

In order to explore the correlations between the size of the crack and the Bz signal, multiple angles (0°, 60°, 120°, 180°, 240°, 300°) of cracks with different lengths (2 mm, 4 mm, 6 mm, 8 mm, 10 mm) and different depths (1 mm, 2 mm, 3 mm, 4 mm, 5 mm) are simulated to determine the angle quantification algorithm for cracks of different sizes. Taking the 0° crack phase (length = 20 mm, width = 0.2 mm, depth = 2 mm) as the reference, according to the method in Sect. 3.3, the polar diagrams of different length and depth cracks are obtained, as shown in Fig. 10a and Fig. 10b. The amplitude of the Bz signal is mainly affected by the size of the crack. The longer the crack length or the deeper the crack depth, the greater the amplitude of the Bz signal. This is because the increase in the size of the crack leads to an increase in the perturbation current density. The phase of the Bz signal is mainly affected by the angle of the crack. For the same crack, the phase difference of the Bz signal is equal to the angle difference of that crack taken in any two directions.

Fig. 10
2 polar diagrams of degrees labeled A and B plot various symbols for 2, 4, 6, 8, and 10 millimeters, and 2, 3, 4, 5, and 6 millimeters, respectively.

Characteristic signal of cracks with different lengths and depths. a Polar diagram of cracks with different lengths; b Polar diagram of cracks with different depths

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However, the angle measurement results of cracks with the same angle and different sizes have certain deviations by the above method. It can be seen from Fig. 10a, b that the amplitude of the Bz signal and the crack angle measurement result show a certain correlation. When the amplitude of the Bz signal increases, the result of the angle measurement will be smaller, so the amplitude of the Bz signal can be used to modify the measured angle of the crack.

4 Experimental Setup and Result

4.1 Probe and System Setup

The RACFM monitoring probe is built, as shown in Fig. 11. It includes two planar excitation coils and a magnetic sensor. The length and width of the excitation coils are 57 mm and 59 mm, respectively. The two coils are designed to be perpendicular to induce a periodic rotating uniform alternating current. Each planar excitation coil is composed of two symmetrically distributed coils, and the winding directions of the coils are clockwise and counterclockwise respectively. The number of turns of each coil is 32, the width of the wire is 0.2 mm, and the pitch of the wire is 0.15 mm. The magnetic field sensor is located in the center above the excitation coils, which is the commercial TMR packages type 2301 from Duowei in China.

Fig. 11
2 images labeled A and B. A has the probe schematic with 2 coils, a T M R chip, and the surface in different color gradients. B has a close-up photograph of the probe.

Design of probe. a Schematic diagram of probe; b Picture of probe

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The RACFM testing system is developed, as shown in Fig. 12, which includes a signal generator, a power amplifier, a specimen, a low pass filter module, a signal amplifier module, a lock-in amplifier module, a computer and a capture card. Two sinusoidal signals with 90° phase shift are generated by a signal generator and amplified by a power amplifier. Two amplified signals are loaded on two orthogonal excitation coils respectively. The periodic rotating electromagnetic field is generated by the excitation coils. The TMR sensor picks up the induced magnetic field and converts it into a voltage signal. The voltage signal is filtered and amplified. The DC output of the lock-in amplifier is acquired by the DAQ card. These signals can be processed and analyzed by an intelligent software developed in LabVIEW and MATLAB [37].

Fig. 12
A block diagram connects the signal processing and display from a laptop and the data acquisition device to blocks of lock-in amplifier, signal amplification, low pass filter, signal generator, power amplifier, and specimen with probe and crack.

Block diagram of experimental system

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4.2 Crack Length Monitoring

The first specimen tested in this paper is an aluminum plate with five 3 mm deep and 0.2 mm wide cracks. Artificial cracks of different length are machined into the plate by electrical discharge machining (EDM) method. The lengths of the cracks are 9 mm, 7 mm, 5 mm, 3 mm and 2 mm, as shown in Fig. 13.

Fig. 13
2 diagrams. The top one has a block with cracks 1 to 5 marked 9, 7, 5, 3, and 2, respectively. The bottom one has a shaded thin rectangular bar with cracks marked 9, 7, 5, 3, and 2 from left to right.

Specimen with different length cracks

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The established testing system is used for monitoring the cracks of different lengths. The excitation frequency is 10 kHz, and the voltage is 3 V. Firstly, the probe is placed in a position where the specimen has no defects, and the real and imaginary components of the Bz signal are recorded as the calibration signal. Secondly. the probe is placed at the tip of the crack, and the probe is rotated counterclockwise from 0°. At the same time, the real and imaginary components of the cracks of different length are recorded as the monitoring signal. The differential signal is obtained by subtracting the calibration signal from the monitoring signal. Thirdly, the real and imaginary parts of the differential signal are converted into the amplitude and phase. Finally, taking the 0° crack phase (length = 9 mm, width = 0.2 mm, depth = 3 mm) as the reference, according to the method in Sect. 3.3, the angle measurement results of different length cracks is obtained, as shown in Fig. 14. It can be seen from the figure that when the probe rotates about the centre of the circle relative to the crack tip once, the phase and amplitude of the Bz signal also behave in the same manner, and the phase angle changes by 360°, which proves that the phase of the Bz signal can be used to measure the angle of the crack. What’s more, when the crack length is long, the graph drawn by the phase and amplitude is close to a circle, and when the crack is short, the graph drawn by the phase and amplitude is elliptical. This is because the induced current in the specimen is not ideally uniform in any direction. When the crack length is short, the distance between the two tips of the crack is closer, the perturbation electromagnetic field generated at the other tip will affect the amplitude of the response signal resulting in large changes in the Bz amplitude of cracks at different angles.

Fig. 14
A polar diagram of degrees plots 9, 7, 5, 3, and 2 millimeter lines as circles along with the 0 degrees points from the outermost to the innermost layer, respectively.

Testing results of different length cracks

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In order to analyze the relationship between the amplitude of the Bz signal and the length of the crack, the amplitudes of the Bz signals of the different length cracks with an angle of 0° are extracted, as shown in Fig. 15. It can be seen from the figure that as the length of the crack expands, the amplitude of the Bz signal also increases, and the quadratic function is used fit the relationship between the crack length and the Bz amplitude, as shown in Eq. (13).

$${\text{A = }}0.01L^{2} + 0.12L - 0.10$$
(13)
Fig. 15
A line graph of B z amplitude versus crack length. It plots a line with an increasing trend.

Relationship between crack length and Bz amplitude

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where L is the length of crack and A is amplitude of Bz signal.

4.3 Crack Depth Monitoring

The second specimen tested in this paper is an aluminum plate with five 16 mm long and 0.5 mm wide cracks. The thickness of the plate is 9 mm. The depths of the cracks are 9 mm, 7 mm, 5 mm,3 mm and 1 mm, as shown in Fig. 16.

Fig. 16
2 diagrams. The top one has a block with cracks 6 to 10 marked 16, respectively. The bottom one has a shaded thin rectangular bar with cracks marked 9, 7, 5, 3, and 1 from left to right.

Specimen with different depth cracks

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According to the above experimental steps, the cracks of different depth are measured, and the amplitudes and phases are obtained as shown in Fig. 17. The phase angle of the Bz changes with the rotation of the probe, and the amplitude and phase of the Bz form a complete circle. Compared with cracks of different lengths, the depth of the cracks has no obvious effect on the Bz amplitude of different angles. But relative to No.5, polar diagrams with different depth cracks are more elliptical, because the width of the crack becomes larger, and the tip effect of the rectangular crack has an impact on the amplitude of each angle.

Fig. 17
A polar diagram of degrees plots 9, 7, 5, 3, and 1 millimeter lines as circles along with the 0 degrees points from the outermost to the innermost layer, respectively.

Testing results of different depth cracks

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In order to analyze the relationship between the amplitude of the Bz signal and the depth of the crack, the amplitudes of the Bz signals of the different depth cracks with an angle of 0° are extracted, as shown in Fig. 18. It can be seen from the figure that as the depth of the crack expands, the amplitude of the Bz signal also increases, and the quadratic function is used fit the relationship between crack depth and Bz amplitude, as shown in Eq. (14).

$$A{ = - 0}{\text{.04}}D^{2} + 1.16D{ + 0}{\text{.05}}$$
(14)
Fig. 18
A scatterplot and line graph of B z amplitude versus crack depth. It plots a line with an increasing trend with cross marks on it.

Relationship between crack depth and Bz amplitude

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where D is the depth of crack and A is amplitude of the Bz signal.

4.4 Modification of the Measured Angle of the Crack

According to Sect. 3.3, it can be seen that the crack angle measurement result has a certain deviation, which is also verified by the experimental results. As shown in Figs. 14 and 17, the curves of cracks of different sizes do not all start from 0° in the polar diagram. So further modification is required to obtain a more accurate angle measurement result. The angle measurement result of different size cracks with an angle of 0° are shown in Table 2. It can be seen from the table that the maximum error of the crack angle is 13.77°. This is because the angle measurement result is also affected by the crack size, which leads to inaccuracy in the measured angle of the crack.

Table 2 Angle measurement results of different size cracks
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The amplitudes of the Bz signal and the angle measurement results are extracted as shown in Fig. 19. It shows that when the amplitude of the Bz signal becomes larger, the angle measurement result becomes smaller, which is consistent with the simulation results. Therefore, the angle measurement result needs to be modified by the amplitude of the Bz signal. The quadratic function is used fit the relationship between the amplitude and the angle measurement result, as shown in Eq. (15).

$$P{ = 0}{\text{.73}}A^{2} - 8.85A{ + 12}{\text{.68}}$$
(15)
Fig. 19
A scatterplot and line graph of the crack measured angle versus amplitude. It plots a concave up, decreasing curve with cross marks above, on, and below it.

Relationship between phase and amplitude

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where P is the angle measurement result. and A is the amplitude of the Bz signal.

According to Eq. (15), the angle measurement results are modified using the amplitudes of the Bz signal, and the modified measured angles of the crack are shown in Table 3. And the error of crack angle measurement before and after the modification is compared, as shown in Fig. 20. It shows that the measurement error of the crack angle is significantly reduced after the modification, and the maximum error is reduced from 13.81° to 3.11°. It shows that this method can realize quantitative monitoring of the angle by the corrosion pit.

Table 3 Modified measured angles of the crack
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Fig. 20
A scatterplot of measured angles error versus crack numbers. It plots the angle measurement error cross symbols and modified measure angles error circle symbols with an increasing trend.

Comparison of crack angle measurement

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5 Conclusions and Further Work

In this work, a novel fatigue crack angle quantitative monitoring method is presented based on the RACFM, which measures the angle of the crack through the corrosion pit. The main research results are shown as follows:

  • The theoretical model of the RACFM is established to analyze the response principle of the induced uniform electromagnetic field on the plate and the disturbance of the induced current by different crack angles. It indicates that, at the crack tip, the angle of crack does not affect the amplitude of the Bz signal, and it has a linear relationship with the phase.

  • The 3D simulation model of the fatigue crack angle quantitative monitoring is built. When the sinusoidal signals with initial phases of 0° and 90° are loaded on the orthogonal excitation coils, a periodic rotating uniform current field is induced in the surface of the specimen. The center of the excitation coil is placed at the tip of the crack. The relationship between the angle, length, depth of the crack and the amplitude, phase of the Bz signal is analyzed. For the Bz signals of cracks with the same angle and different sizes, the larger the Bz amplitude, the smaller the crack angle measurement result.

  • The RACFM monitoring probe, which includes two planar excitation coils and a TMR sensor, and testing system are developed. Cracks with different angles, lengths and depths are measured. The amplitude and phase of the Bz differential signal are obtained and used to draw polar diagrams. The results show that the phase of the Bz signal can be used to quantify the angle of the crack. The amplitude of the Bz signal and the length and depth of the crack are in a quadratic function respectively, and the amplitude of the Bz signal can be used to modify the crack angle measurement result.

The experimental results suggest that the proposed method has the potential to quantitatively monitor crack propagation angle. In addition, the traditional eddy current monitoring technology only extracts the amplitude of the characteristic signal, and the accuracy of the crack angle monitoring depends on the density of the sensor. The proposed the Bz phase difference method utilizes the amplitude and phase of the detection signal, and the crack angle can be measured at one end of the crack, which greatly improves the quantification accuracy of the crack angle measurement. And the use of orthogonal excitation can eliminate the influence of crack angle on sensitivity. It should be noted that there are some limitations of the proposed method. Most aerospace equipment is a curved structure, and the influence of curvature on the induction signal is not considered in this paper. However, the development of flexible PCB technology provides an effective method to solve this problem. The stability and reliability of the detection system need to be improved if it is to be implemented in real case. Further work will focus on the monitoring of bolt hole cracks in multilayer structures.