Novel Phase Reversal Feature for Inspection of Cracks Using Multi-frequency Alternating Current Field Measurement Technique

Abstract

Aluminum and its alloys have been widely used in aerospace and other industrial fields. Aluminum and its alloy structures are prone to surface and subsurface cracks when they are used in harsh environments. In this paper, a novel phase reversal feature is found to classify and evaluate cracks using the multi-frequency alternating current field measurement (ACFM) technique. The theoretical model of the phase reversal feature is developed. The distorted electromagnetic field and response signals of surface and subsurface cracks are analyzed by the finite element method. The multi-frequency ACFM testing system is set up. The experiments are carried out to test surface and subsurface cracks. The results show that the response signals of the surface and subsurface cracks have distinct characteristic due to the phase reversal feature using the multi-frequency ACFM technique. The surface and subsurface cracks can be classified by the amplitude reversal phenomenon of the Bz signal caused by the novel phase reversal feature. The buried depth of the subsurface crack can be evaluated accurately by the reversal frequency component.

1 Introduction

Aluminum is the second largest metal material next only to the steel material [1]. Because of the advantages of the light weight, the high strength, the good corrosion resistance, and the high thermal conductivity, aluminum and its alloys have been widely used in the power generation, the transportation, the construction and the aerospace industries [2,3,4]. The aluminum and alloy structures usually surfer from the huge wind load, the thermal shock, and other extreme conditions. As a result, cracks initiate and propagate in the surface and subsurface of the aluminum structure, which threatens the safety of the structure [5, 6]. Hence, it is of prime importance to inspect and evaluate these cracks for the health assessment of the structure.

It is well known that the electromagnetic nondestructive testing (ENDT) technique such as the eddy current testing (ET) is an excellent method for the detection of surface cracks in the aluminum. Generally, the excitation signal of the conventional ET is a high frequency sinusoidal signal [7]. Due to the skin effect, the conventional ET cannot be used to detect subsurface defects. The pulsed excitation signal and the multi-frequency excitation signal are introduced by some scholars [8,9,10,11].

In the pulsed eddy current testing (PECT) field, a strong and transient square pulsed excitation signal is loaded on the excitation coil. Because of the strong energy and the rich spectrum information, the response signal contains more characterizations about the subsurface defect. Ali Sophian et al. [12] developed a new PCA-based feature extraction method for the PECT. The method can effectively classify defects, and its performance is better than the conventional method using the response peak characteristics. He et al. [13] employed the peak amplitude and zero-crossing time of response signal in time domain to detect and characterize defects. Giguere, Fan, Tian, et al. [14,15,16] found the lift-off point of intersection (LOI) in the PECT method. The LOI is regarded as the potential feature for the evaluation of the subsurface defect. However, due to the wide frequency band in the frequency domain, the energy wasting is a great and unavoidable malpractice. Especially, the energy of the high frequency component is limited, which reduces the detection sensitivity of surface defect. Besides, because of the critical time domain analytical method, there are many features and factors that affect the testing results using the PECT.

In the multi-frequency eddy current testing (MFECT) field, two and more sine signals are added together as a multi-frequency excitation signal. The multi-frequency response signal can achieve specific permeation depth and obtain accurate characteristic information about the defect. Bernieri et al. [17] proposed a combination of a multi-frequency excitation and an optimized support vector machine for regression (SVR) for the reliable estimation of the geometrical characteristics of a thin defect. Zhang et al. [18] proposed a new approach to measure the multi-layer conductive coatings’ thickness based on the MFECT. Xie et al. [19] presented a novel frequency-band-selecting pulsed eddy current testing (FSPECT) method for the detection of local wall thinning defects in a certain depth range. In their work, a specific excitation signal was designed to replace the pulsed signal, which improved the detection sensitivity of subsurface defects in a certain depth range significantly. Gao et al. [20] presented the spectrum method for the identification and classification of defects using the MFECT method. From the above, the multi-frequency excitation method has gained great progress in the ET field. However, there are still two critical issues should be addressed for the inspection of cracks in the aluminum material. Firstly, the ET method is sensitive to the lift-off effect. Thus, there are many interference signals when the probe variations above the roughened surface, such as the weld. Secondly, the MFECT usually is presented to detect the surface and subsurface cracks in the aluminum by the amplitude of the response signal. Both surface and subsurface cracks have response signals from the multi-frequency excitation. The classification and evaluation of surface and subsurface cracks is still a challenge only by the amplitude characteristic when the buried depth of subsurface cracks is shallow.

The alternating current field measurement (ACFM) is an emerging ENDT method for the detection and evaluation of cracks [21,22,23,24]. Due to the advantages of the non-contact testing, the high tolerance to lift-off and the quantitative evaluation ability, it has been widely used in the petrochemical industry [25,26,27,28,29], the electric power industry and the rail transportation industry. As the same with the conventional ET, the ACFM probe is excited by a sinusoidal signal. Thus, the ACFM is usually used to inspect surface cracks. In our previous work, a double frequency circumferential current field testing (CCFT) method was presented for the detection and evaluation of both inner and outer cracks in the aluminum tube [30]. The amplitude feature is used to classify and evaluate the surface and subsurface cracks. However, the amplitude of response signal is affected by the length, the depth or the buried depth of the crack. Thus, the evaluation result is obtained at the certain conditions, such as the same length crack, the large buried depth. Thus, new features should be find to classify and evaluate surface and subsurface cracks in the aluminum using the multi-frequency ACFM.

In this paper, a novel phase reversal feature is found to classify and evaluate surface and subsurface cracks based on the multi-frequency ACFM. The phase reversal feature is a inherent feature of the multi-frequency excitation ACFM, which is not affected by the dimensionality of the subsurface crack. Thus the surface and subsurface cracks can be classified and the buried depth can be evaluated accurately. The rest of the paper is organized as follows. In Sect. 2, the physical principle and the 3D finite element method (FEM) model of the multi-frequency ACFM are developed to analyze the phase reversal feature. In Sect. 3, the multi-frequency ACFM system is developed. In Sect. 4, the surface and subsurface cracks are detected. The surface and subsurface cracks are classified using the amplitude reversal phenomenon of the Bz signal. The buried depth of the subsurface crack is evaluated by the reversal frequency. In Sect. 5, the conclusion and further work are drawn.

2 Methodology

• A. Physics Principle

In the classical ACFM model, an induced uniform current field is excited on the aluminum specimen when an alternating excitation signal is loaded on the excitation coil, as shown in Fig. 1a. Due to the skin effect, the induced currents mainly gather in the thin surface of the specimen. When a surface crack is presented, the surface current field will be disturbed. It makes the magnetic field distorted. Thus, the surface crack can be detected by measuring the distorted magnetic field.

Fig. 1

Theoretical model. a ACFM model. b Transformer circuit model

Generally, we only focus on the surface thin induced current. In fact, the induced current field penetrates inside the aluminum specimen. The induced currents at different depths inside the aluminum specimen can be given by Eq. (1). It can be seen that the amplitude of the induced current field decreases exponentially and the phase of the induced current field lags linearly with the increase of depth.

$$J_{Z} = J_{0} e^{{ - \frac{z}{\delta }\left( {1 + j} \right)}}$$

(1)

where J_z is the induced current at the depth Z, J₀ is the current in the surface of the specimen. When the current density in a certain depth of the specimen decays to 1/e of that on the surface of the specimen, the certain depth is called skin depth, as given in Eq. (2).

$$\delta = \left( {{1 \mathord{\left/ {\vphantom {1 {\pi \rho fu_{r} u_{0} }}} \right. \kern-0pt} {\pi \rho fu_{r} u_{0} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}$$

(2)

where $\delta$ is the skin depth, f is the excitation frequency, u_r is the relative magnetic permeability, u₀ is the vacuum permeability, $\rho$ is the electric conductivity of the specimen.

The phase change of the induced current field can be modelled as a transformer circuit [31], as shown in Fig. 1b. The excitation coil is regarded as the input side of the transformer and the specimen is regarded as the output of the transformer. The electromagnetic coupling equivalent circuit follows the Kirchhoff's law, as given in Eqs. (3)–(5).

$$V_{1} = I_{1} \left( {R_{0} + j\omega L_{0} } \right) + I_{e} \left( {j\omega M} \right)$$

(3)

$$0 = I_{e} \left( {R_{e} + j\omega L_{e} } \right) + I_{1} \left( {j\omega M} \right)$$

(4)

$$M = k\sqrt {L_{0} L_{e} }$$

(5)

where k is the coefficient of coupling between the two inductors. M is the mutual inductance in the circuits.

I_e is given in Eq. (6) by rearranging Eq. (4).

$$I_{e} = - I_{1} \frac{j\omega M}{{{\varvec{R}}_{{\varvec{e}}} + \omega^{2} L_{e} }} = - I_{1} \frac{\omega M}{{{\varvec{R}}_{{\varvec{e}}}^{2} + \omega^{2} L_{e}^{2} }}\left( {\omega L_{e} + j{\varvec{R}}_{{\varvec{e}}} } \right)$$

(6)

For plane wave excitation, the resistive and reactive components of the induced current impedance are equal in magnitude (R_e = ωL_e = X_e) [2]. So that I_e is calculated in Eq. (7).

$$I_{e} = - I_{1} \omega L_{0} \frac{{k^{2} }}{2}\left( {1 + j} \right) \equiv I_{1} \omega L_{0} \frac{{k^{2} }}{\sqrt 2 }e^{{ - \frac{3\pi }{4}j}}$$

(7)

Therefore, when the phase of the excitation current is 0°, the phase of the induced current on the surface of the aluminum specimen is −135°. In addition, the phase of the induced current field lags linearly with the increase of penetration depth. The phase suddenly reverses from −180° to 180° at a specific penetration depth, which is called a phase reversal feature in this paper. The phase reversal feature represents a change of the induced current direction. When the thickness of the aluminum specimen is infinite, the direction of the induced current changes periodically. In practice, the plate thickness is limited. The induced currents inside the specimen are more severely attenuated and the disturbance around cracks is weaker. Thus, only one time phase change of the induced current field is considered in this paper. So the induced currents in the specimen can be divided into the forward phase current and the reverse phase current.

In this paper, the phase of the excitation signal is set 0°. The reverse phase induced current field is in the surface of the aluminum specimen. The forward phase induced current field is under the reverse phase induced current field in the depth direction of the specimen, as shown in Fig. 1a. The phase reversal feature can take place with variational depths as the excitation frequency is different.

For surface cracks, when the crack is shallow, only the reverse phase induced current field is disturbed, as shown in Fig. 2a. The induced currents turn around at the tips of the surface crack in one direction all the time. Thus, the space magnetic field distorted in the vertical direction (Called Bz) is always in one direction because the reverse phase induced current field deflects in the same direction. When the crack is deep, the reverse phase and forward phase induced current fields are disturbed simultaneously, as shown in Fig. 2b. The two induced current fields turn around at the tips of the surface crack at the same time and the deflection directions are opposite. However, the density of the reverse phase induced current field is much greater than that of the forward phase current field. Thus, the direction of the Bz signal is still determined by the deflection direction of the reverse phase induced current field. In a conclusion, the direction of the distorted magnetic field is always the same direction regardless of the depth of surface cracks.

Fig. 2

Cracks in different current fields. a Surface crack in the reverse phase current field. b Surface crack in the reverse and forward phase current fields. c Subsurface crack in the forward phase current field. d Subsurface crack in the reverse and forward phase current fields

For subsurface cracks, when the crack is only in the forward phase induced current field [32], as shown in Fig. 2c. The induced currents turn around at the tips of the subsurface crack in one direction and the direction of the Bz signal is always in one direction. However, when the subsurface crack is in the reverse phase and forward phase induced current fields, as shown in Fig. 2d, the turned direction of the reverse phase current filed is opposite to that of the forward phase current filed. What's more, the reverse phase current filed is stronger. As a result, the direction of the distorted magnetic field changes to the opposite direction due to the deeper buried depth of the surface cracks (Called amplitude reversal phenomenon). Thus, the phase reversal feature can be used to classify the surface crack and the subsurface crack when the multi-frequency excitation ACFM is carried out. When the phase reversal feature appears near the buried depth, there will be no obvious peak or trough (Called transitional signal) in the Bz signal. The buried depth can be obtained by the transitional signal using the multi-frequency response signal.

• B. FEM Modeling and Analyzing

To verify the theoretical model proposed above, a 3D finite element method (FEM) model of ACFM was set up by the COMSOL software, as shown in Fig. 3. The simulation model consisted of a specimen, a U-shape core, a coil, and an air domain. The excitation coil (500 turns) was wound on the U-shape Mn–Zn ferrite yoke. The excitation signal was loaded on the excitation coil. The amplitude of the excitation current was 100 mA and the frequency was 1 kHz. The specimen was an aluminum plate, whose conductivity was 3.774 × 10⁷ S/m and relative permeability was 1. The lift-off value of the probe was 1 mm.

Fig. 3

FEM model

The induced uniform current field at different depths in the aluminum were extracted, as shown in Fig. 4a. It is worth noting that the direction of the induced current field turns to the opposite direction at the depth of 3 mm. This is because the phase of induced current field changes from −180° to 180° from depth 2 mm to 3 mm. The crossing 0° phase point is called phase reverse point (PRP) at this specific depth. The current density attenuates exponentially and the phase of the induced current lags linearly with the increasing depth of the aluminum specimen, as shown in Fig. 4b. This phenomenon is consistent with the theoretical model proposed above.

Fig. 4

Induced current field at different depths. a Directions of induced currents. b Induced current density and phase

As mentioned in the theorical model, the depth of the phase reverse feature is affected by the excitation frequency. The phase of the induced current field at different depths was obtained with different excitation frequencies from 200 to 1000 Hz, as shown in Fig. 5a. The original phase of the induced current field is around the −135° with different excitation frequencies. The phase goes down sharply as the excitation frequency increases. All the phases reverse at a specific depth with different excitation frequencies. The depth of the PRP drops with the increasing of the excitation frequency, as shown in Fig. 5b. For a lower excitation frequency, the PRP is in a deeper depth in the aluminum specimen. For example, when the excitation frequency is 300 Hz, the depth of the PRP is 4.89 mm. It means that the direction of the induced current field is in the reverse direction when the penetration depth is less than 4.89 mm. However, the direction of the induced current field is in the forward direction when the penetration depth is greater than 4.89 mm.

Fig. 5

Phase of induced current with different excitation frequencies. a Phase of induced current. b Depths of PRP

• C. Characteristic Signal Analysis

To analyze the magnetic field response signals of cracks, a surface crack and a subsurface crack were set up in the simulation model. The size of the surface crack was 30 mm (Length) × 0.5 mm (Width) × 2 mm (Depth). The buried depth of the subsurface crack was 3 mm, and the length and width were the same as the surface crack. The 200 Hz and 1 kHz were selected as the frequencies of the excitation signal. According to the simulated results, the depths of the PRP were 2.19 mm (1 kHz excitation) and 6.6 mm (200 Hz excitation), respectively.

For the surface crack, the reverse phase current field deflects in the clockwise direction at one end of the surface crack with the excitation frequency of 200 Hz, as shown in Fig. 6a. Meanwhile, the reverse phase current field deflects in the anticlockwise direction at another end of the surface crack. According to the Ampere's Law, the Bz shows a trough at one tip of the surface crack and a peak at another tip of the surface crack, as shown in Fig. 6c. Because the PRP of the 200 Hz is 6.6 mm, the surface crack is not affected by the phase reverse feature. When the excitation frequency is 1000 Hz (PRP is 2.19 mm), although the surface crack is located in the reverse phase and forward phase current fields at the same time, the reverse phase current field maily deflects in the same way around the surface crack, as shown in Fig. 6b. Thus, the Bz also shows a trough at one tip of the surface crack and a trough at another tip of the surface crack, as shown in Fig. 6c. Due to the different current density, the peaks of the Bz with 1000 Hz excitation frequency are stronger than that of the 200 Hz excitation frequency.

Fig. 6

Disturbed current field and distorted magnetic field of surface crack. a 200 Hz excitation frequency. b 1000 Hz excitation frequency. c Distorted magnetic field signal Bz

For the subsurface crack (Buried depth 3 mm), the reverse and forward phase induced currents are disturbed at the same time when the excitation frequency is 200 Hz, as shown in Fig. 7a. When the depth is less than the PRP (6.6 mm), the deflection direction of the induced current is in the clockwise direction at one end of the subsurface crack (reverse phase induced current field). However, When the depth is more than the PRP (6.6 mm), the deflection direction of the induced current is in the anticlockwise direction at the same end of the subsurface crack (forward phase induced current field). Because the density of the reverse phase current is greater than that of the forward phase current, the Bz still shows a trough at one end and a peak at another end of the subsurface crack, as shown in Fig. 7c. However, when the excitation frequency is 1000 Hz, there is only the forward phase current field which is disturbed by the subsurface crack, as shown in Fig. 7b. This is because the buried depth of the subsurface crack is under the depth of the PRP (2.19 mm). Thus, the deflection direction of the induced current is in the anticlockwise direction at one end of the subsurface crack. As a result, the Bz shows a peak at one end and a trough at another end of the subsurface crack, which is opposite to the Bz signal excited by the 200 Hz sine signal, as shown in Fig. 7c.

Fig. 7

Disturbed current field and distorted magnetic field of subsurface crack. a 200 Hz excitation frequency. b 1000 Hz excitation frequency. c Distorted magnetic field signal Bz

When the excitation frequency is 500 Hz, the depth of the PRP is 3.4 mm. The edge of the subsurface crack is near the PRP area. The phase of the induced current changes from the reverse direction to the forward direction, which shows stray state, as shown in Fig. 8a. The stray currents cannot turn regularly. As a result, there is no obvious peak or trough in the Bz signal (Called transitional signal), as shown in Fig. 8b. In a conclusion, there are always peak and trough in the Bz signal for the surface crack with different frequency excitation signals. The peak and trough of the Bz signal can turn to the opposite direction for the subsurface crack with different frequency excitation signals. Especially, the transitional signal of the Bz is excited by a specific excitation frequency because of the stray current. Thus, the surface and subsurface cracks can be classified by the amplitude reversal phenomenon of the peak and trough of the Bz with multi-frequency excitation method. Because the transitional signal is caused by the stray current near the top side of the surface crack, the buried depth of the subsurface can be evaluated using the specific excitation frequency.

Fig. 8

Disturbed current field and distorted magnetic field of subsurface crack. a Stray current field. b Transitional signal

3 Multi-frequency ACFM Testing System

• A. Multi-frequency Excitation Signal Synthesis

To verify the theoretical and simulated results, the multi-frequency excitation signal was synthesized. It has been proved that the 1 kHz was the optimal excitation frequency to detect surface cracks in the aluminum specimen [30]. Therefore, 1 kHz was set as the highest frequency component of the multi-frequency excitation signal. To get a penetration depth of 5 mm and above, 200 Hz was set as the minimum excitation frequency in this paper. The sinusoidal signals were added together with 200, 300, 400, 500, 600, 700, 800, 900, and 1000 Hz to generate the multi-frequency excitation signal. The amplitude of the sinusoidal signals was 1 V and the phase was 0°. The multi-frequency excitation signal was generated by LABVIEW software and output through an acquisition card (NI, USB6351) that was provided analog signal output function. The output multi-frequency excitation signal is shown in Fig. 9.

Fig. 9

Multi-frequency excitation signal. a Time domain signal. b Frequency domain signal

• B. Probe and Testing system

As shown in Fig. 10a, a U-shape magnetic core, an excitation coil, a tunnel magneto resistance (TMR) sensor, and a signal processing circuit were packaged in the ACFM probe. The excitation coil (copper wire whose diameter was 0.15 mm) was wound around the U-shape core with 500 turns. The TMR sensor (Type: TMR2301, made by MULTI DIMENSION, China) was placed at the bottom of the probe, which was used to measure the Bz signal. The signal processing circuit was used to amplify the Bz signal and filter the interference noise.

Fig. 10

Probe and testing system. a Detection probe. b Testing system

As shown in Fig. 10b, the multi-frequency ACFM testing system included a probe, an acquisition card, a power amplifier, a DC power, and a personal computer (PC). The signal acquisition card was controlled by the PC to output the multi-frequency excitation signal. And then the excitation signal was amplified by the power amplifier. The amplified excitation signal was loaded on the excitation coil of the ACFM probe. The uniform current field was excited into the aluminum plate by the probe. When a crack was presented, the induced current field would be disturbed. The disturbed current field made the space vertical magnetic field distorted (Bz) the above the crack. The Bz signal was picked up by the TMR sensor. Then the Bz signal was sent to the acquisition card. In the end, the Bz signal was processed and displayed by software in the computer.

4 Experiment

• A. Classification of cracks

The specimen was an aluminum plate with a thickness of 10 mm, as shown in Fig. 11. There were five cracks (No. 1–5) with the same length (30 mm) and width (0.5 mm). The five cracks could be considered as surface cracks or subsurface cracks (turned to another side). The depths of the surface cracks were 5 mm (No. 5), 6 mm (No. 4), 7 mm (No. 3), 8 mm (No. 2), and 9 mm (No. 1) respectively. The buried depths of the subsurface cracks were 1 mm (No. 1), 2 mm (No. 2), 3 mm (No. 3), 4 mm (No. 4), and 5 mm (No. 5) respectively.

Fig. 11

Dimensions of the specimen and cracks

The probe was set on the aluminum plate to measure the response signal of the Bz signal (one end of a surface crack), as shown in Fig. 12. There were 9 frequency components in the Bz signal. The amplitude of each frequency component in the frequency domain was set as the Bz amplitude at this location.

Fig. 12

Bz signal. a Bz time domain signal. b Bz frequency signal

Firstly, the surface cracks were tested using the multi-frequency ACFM testing system. The peak value of each frequency component was extracted to obtain the Bz signals of the five surface cracks, as shown in Fig. 13. The Bz signals show negative peaks and positive peaks at the two ends of the surface crack. For a specific depth crack (for example 5 mm depth), the peak value of Bz increases as the excitation frequency goes down, as shown in Fig. 13a. This is because the amplitudes of different frequency components are different, as shown in Fig. 12b. For different depth surface cracks, all the Bz signals show peaks and troughs in the same direction. It indicates that the Bz signals of the surface cracks do not have the amplitude reversal phenomenon. This is mainly because the Bz signals are influenced by the deflection direction of the reversal current. Although the forward phase current field is also disturbed around the bottom of the surface cracks, it only affects the magnitude of the Bz signal due to the weak current density.

Fig. 13

Bz signals of the surface cracks. a 5 mm deep crack. b 6 mm deep crack. c 7 mm deep crack. d 8 mm deep crack. e 9 mm deep crack

Secondly, the subsurface cracks were tested by turning the aluminum plate to another side. The testing results of the subsurface cracks are shown in Fig. 14. It can be seen that the peak and trough of Bz signal changed to the opposite direction for each subsurface crack. It indicates that the Bz signals of the subsurface cracks show the amplitude reversal phenomenon. For the lower frequency component (for example 200 Hz in Fig. 14c), the Bz shows a trough and then a peak. For the higher frequency component (for example 1000 Hz in Fig. 14c), the Bz shows a peak and then a trough. This is because the 200 Hz excitation frequency has a deeper depth of the PRP (6.6 mm), which is larger than the buried depth (3 mm) of the subsurface crack. Although the reverse and forward phase induced currents are disturbed at the same time, the density of the reverse phase current is greater than that of the forward phase current. Thus, the peak and trough directions of the Bz signals keep the same all the time for the 1 mm to 5 mm buried depth subsurface crack when the excitation frequency is 200 Hz. The 1000 Hz excitation frequency has a shallower depth of the PRP (2.19 mm), which is less than the buried depth (3 mm). Only the forward current turns around the subsurface crack. The peak and trough of the Bz excited by 1000 Hz is opposite to that of the 200 Hz excitation signal.

Fig. 14

Bz signals of the subsurface cracks. a 1 mm buried depth crack. b 2 mm buried depth crack. c 3 mm buried depth crack. d 4 mm buried depth crack. e 5 mm buried depth crack

All the peaks of the Bz goes down with the increases of the buried depth. This is because the attenuation of the current density in the depth direction. And it should be noted that the transitional signals always exist in all buried depth cracks. The transitional signals are caused by the reversal frequency component near the buried depth of the subsurface crack. Thus, the reversal frequency component of the transitional signal can be used to evaluated the buried depth of the subsurface crack.

In conclusion, the Bz signal has one peak and one trough all the time whether it is a surface crack or a subsurface crack. However, the Bz signal shows different peaks and troughs for the subsurface crack with different excitation frequency components. With the increase of the excitation frequency, the amplitude reversal phenomenon will occur in the Bz signals. The results consistent with the previous theorical and simulated results. Thus, the surface and subsurface cracks can be classified by the amplitude reversal phenomenon of the Bz signal caused by the novel phase reversal feature using the multi-frequency ACFM technique.

• B. Evaluation of subsurface cracks

Many scholars have proposed many methods to evaluate the depth of surface cracks. However, the buried depth of subsurface cracks is still a big challenge. Because the conventional amplitude characteristic is confused by the dimensionality and the buried depth of subsurface cracks, the buried depth cannot be evaluated well and truly. As mentioned above, there is transitional signal caused by the reversal frequency component near the buried depth of a subsurface crack. So the reversal frequency component can be used to evaluate the buried depth. Because the reversal frequency component is only related to the buried depth, it is not affected by the dimensionality of subsurface cracks. To find the reversal frequencies of the different buried depth subsurface cracks, the Bz signals of the five subsurface cracks were further processed using Eq. (8). All the frequency components were ploted in figures, as shown in Fig. 15.

$$Bz_{p} = Bz/Bz_{\max }$$

(8)

Fig. 15

Different frequency component signals of the subsurface cracks. a 1 mm buried depth crack. b 2 mm buried depth crack. c 3 mm buried depth crack. d 4 mm buried depth crack. e 5 mm buried depth crack

where Bz_p is the normalized Bz signal, Bz is unprocessed magnetic response field signal, Bz_max is the maximum value of the Bz signal.

As shown in Fig. 15, the distance between the peak and trough of the Bz signal indicates the crack length. The peak and trough of the Bz signals reverse at different reversal frequency components for the different buried depths cracks. The forward and reverse currents are mixed together to present the stray current state. Because the stray current caused by the reversal frequency component is a current area, it turns and gathers around the buried depth. The peak and trough of the Bz signals do not reverse at the same time as the frequency increases. Thus, the reversal frequency component is a transition zone, as shown in Fig. 15a. The two boundaries of the transition zone are the frequency of the peak reverse and the frequency of the trough reverse. To get a consistent and accurate reversal frequency component, the intermediate frequency component in the reversal frequency area was selected as the reversal frequency component to evaluate the buried depth. The reversal frequency components of the 5 subsurface cracks were marked with the dotted lines, as shown in Fig. 15.

The reversal frequency components were picked up, as shown in Fig. 16. The reversal frequency component decreases linearly with the increasing of the buried depth. It is consistent with the phase linear transmission of the induced current field, as shown in Fig. 4b. Thus, the buried depth can be evaluated by the reversal frequency component using the linear relationship function.

Fig. 16

Reversal frequency components of different buried depth cracks

Three reversal frequency components of the subsurface cracks (1, 3 and 5 mm buried depth) were used to fit the linear relationship function. The relationship function between the buried depth and the reversal frequency component is shown in Eq. (9).

$$bd_{e} = - 0.01775 \times f_{r} + 10.54$$

(9)

$$E_{bd} = {{\left| {bd_{e} - bd} \right|} \mathord{\left/ {\vphantom {{\left| {bd_{e} - bd} \right|} {bd}}} \right. \kern-0pt} {bd}}$$

(10)

where bd_e is the measured buried depth, $f_{r}$ is the reversal frequency component. E_bd is the relative evaluation error, bd is the actual buried depth.

The remaining two subsurface cracks (No. 2 and No. 4) can be evaluated using Eq. (9). The evaluated results are 2.02 mm and 4.0612 mm respectively. The relative evaluation error of the buried depth is defined in Eq. (10). The buried depth evaluated results and the relative evaluation errors are shown in Table 1. The relative evaluation errors of No. 2 and No. 4 cracks are 1.0 and 1.53% respectively. The buried depth of the subsurface crack can be evaluated accurately by the reversal frequency component.

Table 1 Results of buried depth evaluation

In order to verify the generalisability of the proposed classification and evaluation method, a subsurface crack in another aluminium plate was detected using the developed system. The length of the subsurface crack was 14 mm, the width was 0.2 mm, and the buried depth was 4 mm. First of all, the probe was pushed to detect the subsurface crack. The detection result is shown in Fig. 17. The Bz signals of different frequencies show peaks and troughs. And as the frequency increases, the positions of the peak and trough exchange. It suggests that the Bz signals of the subsurface crack show the amplitude reversal phenomenon. It means that the proposed classification method is effective. Secondly, to evaluate the subsurface crack, the Bz signal was further processed using Eq. (8). The processed signal is shown in Fig. 17b and the reversal frequency of the subsurface crack is 353 Hz, which is marked in the figure. Finally, the subsurface crack was evaluated using Eq. (9), and the error was calculated using Eq. (10). The evaluated buried depth is 4.274 mm and the relative evaluation error is 6.85%. It shows that the novel phase reversal feature proposed in this paper is also effective and calculate for other specimens.

Fig. 17

Detection result of the subsurface crack. a Bz signals. b Bz_p signals

5 Conclusion

In this paper, the novel phase reversal feature is found to classify and evaluate cracks in the aluminum based on the multi-frequency ACFM technique. The physical principle of the phase reversal feature is developed. The distorted electromagnetic fields around the surface and subsurface cracks with different excitation frequencies are analyzed by the FEM model. The multi-frequency ACFM testing system is set up to test the surface and subsurface cracks. The results show that the peak and trough of the Bz signal caused by the subsurface crack can reverse with different excitation frequency component due to the phase reversal of the induced current field. The peak and trough of the Bz signal caused by the surface crack do not reverse. Thus, the surface and subsurface cracks can be classified by the amplitude reversal phenomenon of the Bz signal caused by the phase reversal feature. As the transitional signals of the Bz always exist in different buried depth caused by the stray current, the reversal frequency component is selected to evaluate the buried depth of the subsurface crack. The buried depth and the reversal frequency component have a good linear relationship. The buried depth of the subsurface crack can be evaluated accurately. Further work will focus on the evaluation of the buried depth with different lift-off distances and the evaluation of other complex subsurface defects.