Quantitative Detection for Fatigue Natural Crack in Aero-Aluminum Alloy Based on Pulsed Eddy Current Technique

: Aero-space aluminum alloys, as vital materials in aerospace engineering, find extensive application in various aerospace components. However, prolonged usage often leads to the emergence of fatigue natural cracks, posing significant safety risks. Therefore, research on accurate quantitative detection techniques for the cracks in aerospace-aluminum alloys is of vital importance. Firstly, based on the three-points bending experimental model, this paper prepared the fatigue natural crack specimen, and the depth of the natural crack is calibrated. Then, given the complexity of geometric characteristics inherent in natural cracks, the pulsed eddy current signal under the different natural crack depth is acquired and analyzed using an experimental study. Finally, to better exhibit the non-linearity between PEC signal and crack depth, a GA-based BPNN algorithm is proposed. The Latin Hypercube method is considered to optimize the population distribution in the genetic algorithm. The results indicate that the characterization accuracy reaches 2.19% for the natural crack.


Introduction
In engineering applications, cracks can lead to structural failure, posing severe safety hazards in industrial fields such as transportation, energy, aircraft, and aerospace.Due to the continuous changing loads on aircraft during work, fatigue cracks will occur [1,2], which is an important cause of safety accidents.Consequently, there is significant attentions paid to the initiation and propagation of cracks, leading to research efforts aimed at understanding the formation and expansion of cracks in engineering materials like aeroaluminum alloys [3][4][5][6].On the other hand, some of the attentions are involved in seeking the effective detection of cracks to ensure the safety operation of equipment [7][8][9].The ability to accurately identifying and assessing cracks is integral to maintaining the structural integrity and performance of engineering systems.
The detection of metal cracks heavily relies on non-destructive testing (NDT) techniques, which evaluate the integrity of materials and structures without causing any damage.Based on the different physical principles, a range of methods have been explored, including eddy current testing (ECT), magnetic particle testing (MPT), penetrant testing (PT), ultrasonic testing (UT), and radiographic testing (RT) [10].Compared to other detection methods, ECT offers several distinct advantages in crack detection, such as being low cost, having a rapid response and no coupling agent, and being sensitive to both surface cracks and subsurface cracks.Additionally, the pulsed eddy current technique (PECT), which has evolved from the ECT method, is widely employed in defect detection due to its rich information in the frequency domain [11][12][13].
In addition, artificial intelligence algorithms have also been more and more applied in non-destructive testing.Z.W. Wang proposed a genetic algorithm (GA) based backpropagation neural network (BPNN) to invert the defect size on the ferromagnetic material [14].L. Wang used an Artificial Neural Network (ANN) to show the parameters of the combined cracks through ECT signals and found some highly implicit nonlinear mapping relationships [15].R. Cormerais proposed the data enhancement method of principal component analysis (PCA) to solve the problem of insufficient data of the ANN algorithm used for ECT inversion [16].Neural networks have become an effective way for defect inversion with ECT.
But the existing investigation indicated that PECT is an appropriate way to quantitatively characterize the cracks in metallic specimens, such as aero-aluminum alloys.However, an issue is that numerous studies have primarily focused on regular artificial cracks as experimental specimens [17][18][19][20].Actually, the characteristics of natural cracks significantly differ from those of artificial cracks [21,22].
Therefore, the motivation of this paper is to promote a quantitative approach for the natural crack in aero-aluminum alloys based on PEC.Section 2 introduces the three-points bending experiment, and 7075 aero-aluminum alloy specimens with natural cracks are prepared based on this method.Moreover, the cracks are measured using the optical method.Then, in Section 3, the specimens with natural cracks are measured using PECT.The influence of the natural crack on the output of the PEC probe are analyzed according to the experiment results.In Section 4, an BPNN algorithm based on GA is proposed to improve the characterization accuracy for the natural cracks.In the proposed algorithm, the Latin Hypercube is explored to obtain the a more uniform population distribution in GA, and consequently to optimize the weights for BPNN.Finally, some conclusions are arrived at in Section 5.

Experiment Setup
In our research, a 7075 aluminum alloy is selected as the object, which is widely used in the aerospace field.It is of great practical significance and value to study it.To obtain the natural crack, a three-point bending test platform is built.A schematic of the three-point bending test is shown in Figure 1, where F represents the cyclic applied load and L denotes the distance between the two supports.In the test, the specimen intended for fatigue natural crack generation is first secured with two supports.Then, a load is applied at the midpoint between these two supports.After a repeated action of stress is received at the loading point, fatigue cracks will occur along the loading direction of the force.
Appl.Sci.2024, 14, 4326 2 of 13 In addition, artificial intelligence algorithms have also been more and more applied in non-destructive testing.Z.W. Wang proposed a genetic algorithm (GA) based backpropagation neural network (BPNN) to invert the defect size on the ferromagnetic material [14].L. Wang used an Artificial Neural Network (ANN) to show the parameters of the combined cracks through ECT signals and found some highly implicit nonlinear mapping relationships [15].R. Cormerais proposed the data enhancement method of principal component analysis (PCA) to solve the problem of insufficient data of the ANN algorithm used for ECT inversion [16].Neural networks have become an effective way for defect inversion with ECT.
But the existing investigation indicated that PECT is an appropriate way to quantitatively characterize the cracks in metallic specimens, such as aero-aluminum alloys.However, an issue is that numerous studies have primarily focused on regular artificial cracks as experimental specimens [17][18][19][20].Actually, the characteristics of natural cracks significantly differ from those of artificial cracks [21,22].
Therefore, the motivation of this paper is to promote a quantitative approach for the natural crack in aero-aluminum alloys based on PEC.Section 2 introduces the three-points bending experiment, and 7075 aero-aluminum alloy specimens with natural cracks are prepared based on this method.Moreover, the cracks are measured using the optical method.Then, in Section 3, the specimens with natural cracks are measured using PECT.The influence of the natural crack on the output of the PEC probe are analyzed according to the experiment results.In Section 4, an BPNN algorithm based on GA is proposed to improve the characterization accuracy for the natural cracks.In the proposed algorithm, the Latin Hypercube is explored to obtain the a more uniform population distribution in GA, and consequently to optimize the weights for BPNN.Finally, some conclusions are arrived at in Section 5.

Experiment Setup
In our research, a 7075 aluminum alloy is selected as the object, which is widely used in the aerospace field.It is of great practical significance and value to study it.To obtain the natural crack, a three-point bending test platform is built.A schematic of the threepoint bending test is shown in Figure 1, where F represents the cyclic applied load and L denotes the distance between the two supports.In the test, the specimen intended for fatigue natural crack generation is first secured with two supports.Then, a load is applied at the midpoint between these two supports.After a repeated action of stress is received at the loading point, fatigue cracks will occur along the loading direction of the force.In the experiment, we machined a 7075 aluminum alloy material into a cube shape; the shape of the specimens is shown in Figure 2. The size of the specimen is 300 mm × 60 mm × 10 mm.This design allows for the observation of fatigue natural crack propagation behavior at both side A and Side B of the specimen.To generate the natural crack easily, to make sure the crack locations have the consistency on different specimens after experiment, and to ensure that the location of natural cracks on different specimens is basically the same, all specimens were prefabricated with the same artificial cracks.The artificial In the experiment, we machined a 7075 aluminum alloy material into a cube shape; the shape of the specimens is shown in Figure 2. The size of the specimen is 300 mm × 60 mm × 10 mm.This design allows for the observation of fatigue natural crack propagation behavior at both side A and Side B of the specimen.To generate the natural crack easily, to make sure the crack locations have the consistency on different specimens after experiment, and to ensure that the location of natural cracks on different specimens is basically the same, all specimens were prefabricated with the same artificial cracks.The artificial prefabricated crack is a rectangular groove that penetrates side A and side B with a width of 0.2 mm and a depth of 5 mm.The location of the crack is shown in Figure 2. The artificial crack is fabricated with electric discharge machining (EDM).
Appl.Sci.2024, 14, 3 of 13 prefabricated crack is a rectangular groove that penetrates side A and side B with a width of 0.2 mm and a depth of 5 mm.The location of the crack is shown in Figure 2. The artificial crack is fabricated with electric discharge machining (EDM).To facilitate a comprehensive comparison of fatigue natural crack propagation under various cyclic loads, three-points bending tests are carried out under different loading conditions.The specific loading parameters are detailed in Table 1.The specimens with natural crack are generated on the electro-hydraulic servo fatigue experiment platform QBS-100, whose maximum static test force is ±100 kN.The equipment is manufactured by Qianbang Test Equipment Co., LTD., from Changchun, China.The cyclic loadings are vertically acted along the middle line of the upper surface of the specimens.Figure 3 is the three-point bending experimental platform.During loading, a fixed amplitude of S = 0.5 mm is used with a sinusoidal loading pattern.

Calibration Depth of Fatigue Natural Cracks
The experiments in Section 2.1 produced four specimens.One of the specimens is illustrated in Figure 4a.One visible natural crack can be found.To calibrate the depth of fatigue natural cracks, high-definition image acquisition equipment was employed, which is shown in Figure 4b.To facilitate a comprehensive comparison of fatigue natural crack propagation under various cyclic loads, three-points bending tests are carried out under different loading conditions.The specific loading parameters are detailed in Table 1.The specimens with natural crack are generated on the electro-hydraulic servo fatigue experiment platform QBS-100, whose maximum static test force is ±100 kN.The equipment is manufactured by Qianbang Test Equipment Co., Ltd., from Changchun, China.The cyclic loadings are vertically acted along the middle line of the upper surface of the specimens.Figure 3 is the three-point bending experimental platform.During loading, a fixed amplitude of S = 0.5 mm is used with a sinusoidal loading pattern.
Appl.Sci.2024, 14, 3 of 13 prefabricated crack is a rectangular groove that penetrates side A and side B with a width of 0.2 mm and a depth of 5 mm.The location of the crack is shown in Figure 2. The artificial crack is fabricated with electric discharge machining (EDM).To facilitate a comprehensive comparison of fatigue natural crack propagation under various cyclic loads, three-points bending tests are carried out under different loading conditions.The specific loading parameters are detailed in Table 1.The specimens with natural crack are generated on the electro-hydraulic servo fatigue experiment platform QBS-100, whose maximum static test force is ±100 kN.The equipment is manufactured by Qianbang Test Equipment Co., LTD., from Changchun, China.The cyclic loadings are vertically acted along the middle line of the upper surface of the specimens.Figure 3 is the three-point bending experimental platform.During loading, a fixed amplitude of S = 0.5 mm is used with a sinusoidal loading pattern.

Calibration Depth of Fatigue Natural Cracks
The experiments in Section 2.1 produced four specimens.One of the specimens is illustrated in Figure 4a.One visible natural crack can be found.To calibrate the depth of fatigue natural cracks, high-definition image acquisition equipment was employed, which is shown in Figure 4b.

Calibration Depth of Fatigue Natural Cracks
The experiments in Section 2.1 produced four specimens.One of the specimens is illustrated in Figure 4a.One visible natural crack can be found.To calibrate the depth of fatigue natural cracks, high-definition image acquisition equipment was employed, which is shown in Figure 4b.
After capturing images, the GetData2.12software tool is utilized to measure and annotate the crack depths.The results are shown in Table 2 (where A/B denotes the two side of the fatigue natural crack specimen, as that shown in Figure 2).The average depth in the table represents the mean of the crack depths measured on side A and side B. After capturing images, the GetData2.12software tool is utilized to measure and annotate the crack depths.The results are shown in Table 2 (where A/B denotes the two side of the fatigue natural crack specimen, as that shown in Figure 2).The average depth in the table represents the mean of the crack depths measured on side A and side B. To analyze the relationship between the number of cycles and crack depth for different specimens, the data from the table are organized as shown in Figure 5.As shown in Figure 5, it is evident that the depth of the cracks increases with the number of cycles.By comparing the crack depths on both side A and side B of the same specimen, it can be seen that there is a discrepancy in the crack depths on both sides for all specimens.This indicates that natural cracks, unlike artificial ones, do not have uniformly regular walls and depths.There are many factors that lead to this phenomenon.The initiation and propagation of fatigue natural cracks are mainly influenced by the microstructure characteristics of the material itself.In the study of material mechanics, an assumption of continuity, an assumption of homogeneity, and an assumption of isotropy are usually followed, but the actual situation of materials is affected by these factors.For example, the   To analyze the relationship between the number of cycles and crack depth for different specimens, the data from the table are organized as shown in Figure 5.As shown in Figure 5, it is evident that the depth of the cracks increases with the number of cycles.By comparing the crack depths on both side A and side B of the same specimen, it can be seen that there is a discrepancy in the crack depths on both sides for all specimens.After capturing images, the GetData2.12software tool is utilized to measure and annotate the crack depths.The results are shown in Table 2 (where A/B denotes the two side of the fatigue natural crack specimen, as that shown in Figure 2).The average depth in the table represents the mean of the crack depths measured on side A and side B. To analyze the relationship between the number of cycles and crack depth for different specimens, the data from the table are organized as shown in Figure 5.As shown in Figure 5, it is evident that the depth of the cracks increases with the number of cycles.By comparing the crack depths on both side A and side B of the same specimen, it can be seen that there is a discrepancy in the crack depths on both sides for all specimens.This indicates that natural cracks, unlike artificial ones, do not have uniformly regular walls and depths.There are many factors that lead to this phenomenon.The initiation and propagation of fatigue natural cracks are mainly influenced by the microstructure characteristics of the material itself.In the study of material mechanics, an assumption of continuity, an assumption of homogeneity, and an assumption of isotropy are usually followed, but the actual situation of materials is affected by these factors.For example, the  This indicates that natural cracks, unlike artificial ones, do not have uniformly regular walls and depths.There are many factors that lead to this phenomenon.The initiation and propagation of fatigue natural cracks are mainly influenced by the microstructure characteristics of the material itself.In the study of material mechanics, an assumption of continuity, an assumption of homogeneity, and an assumption of isotropy are usually followed, but the actual situation of materials is affected by these factors.For example, the heterogeneities of the microstructure, such as inclusions and grain boundaries, will affect the initiation and propagation of cracks [23].In addition, the stress direction cannot be guaranteed to be completely consistent during the stress cyclic loading process.All these factors make it so that natural fractures have complex geometric distribution compared with artificial fractures, especially in fracture depth and direction, which poses great challenges for the quantitative detection of natural cracks.

Detection Natural Cracks by Using PECT 3.1. PECT Platform
To facilitate the detection of cracks within specimens, a PECT platform was constructed.This platform comprises an excitation signal source, an excitation coil and detection probe, an amplification circuit, the specimen with fatigue natural cracks, and a signal acquisition and processing circuit.Illustrations of the PECT platform and the schematic diagram of the platform are shown in Figure 6, respectively.
heterogeneities of the microstructure, such as inclusions and grain boundaries, will affect the initiation and propagation of cracks [23].In addition, the stress direction cannot be guaranteed to be completely consistent during the stress cyclic loading process.All these factors make it so that natural fractures have complex geometric distribution compared with artificial fractures, especially in fracture depth and direction, which poses great challenges for the quantitative detection of natural cracks.

PECT Platform
To facilitate the detection of cracks within specimens, a PECT platform was constructed.This platform comprises an excitation signal source, an excitation coil and detection probe, an amplification circuit, the specimen with fatigue natural cracks, and a signal acquisition and processing circuit.Illustrations of the PECT platform and the schematic diagram of the platform are shown in Figure 6, respectively.As is shown in Figure 6, the detection probes are pivotal components.In this investigation, the probe is composed of an excitation coil and a hall magnetic sensor.The main geometric and electric parameters of the excitation coil are detailed in Table 3.The corresponding probe structure diagram and actual probe are presented in Figure 7. Additionally, a hall sensor UGN3503 is encapsulated at the bottom of the coil to detect the magnetic field strength.The output voltage is acquired using Altec's PCI8532 data acquisition card.

Experiment Testing
To investigate the influence of natural cracks on the detection signal and to more accurately evaluate the depth of the natural crack using pulsed eddy current detection, measurements were carried out at three detection points along the fatigue natural crack in Y direction on the specimen, as illustrated in Figure 8.As is shown in Figure 6, the detection probes are pivotal components.In this investigation, the probe is composed of an excitation coil and a hall magnetic sensor.The main geometric and electric parameters of the excitation coil are detailed in Table 3.The corresponding probe structure diagram and actual probe are presented in Figure 7. Additionally, a hall sensor UGN3503 is encapsulated at the bottom of the coil to detect the magnetic field strength.The output voltage is acquired using Altec's PCI8532 data acquisition card.heterogeneities of the microstructure, such as inclusions and grain boundaries, will affect the initiation and propagation of cracks [23].In addition, the stress direction cannot be guaranteed to be completely consistent during the stress cyclic loading process.All these factors make it so that natural fractures have complex geometric distribution compared with artificial fractures, especially in fracture depth and direction, which poses great challenges for the quantitative detection of natural cracks.

PECT Platform
To facilitate the detection of cracks within specimens, a PECT platform was constructed.This platform comprises an excitation signal source, an excitation coil and detection probe, an amplification circuit, the specimen with fatigue natural cracks, and a signal acquisition and processing circuit.Illustrations of the PECT platform and the schematic diagram of the platform are shown in Figure 6, respectively.As is shown in Figure 6, the detection probes are pivotal components.In this investigation, the probe is composed of an excitation coil and a hall magnetic sensor.The main geometric and electric parameters of the excitation coil are detailed in Table 3.The corresponding probe structure diagram and actual probe are presented in Figure 7. Additionally, a hall sensor UGN3503 is encapsulated at the bottom of the coil to detect the magnetic field strength.The output voltage is acquired using Altec's PCI8532 data acquisition card.

Experiment Testing
To investigate the influence of natural cracks on the detection signal and to more accurately evaluate the depth of the natural crack using pulsed eddy current detection, measurements were carried out at three detection points along the fatigue natural crack in Y direction on the specimen, as illustrated in Figure 8.

Experiment Testing
To investigate the influence of natural cracks on the detection signal and to more accurately evaluate the depth of the natural crack using pulsed eddy current detection, measurements were carried out at three detection points along the fatigue natural crack in Y direction on the specimen, as illustrated in Figure 8.
The excitation coil is a square wave signal, with a duty cycle of 50%, frequency of 2 kHz, and an exciting voltage of 12 V.After the test, the output voltage signal V out of the hall sensor in the probe can be obtained.
The difference signal obtained by calculating the difference between the signal of the non-defective specimen and that of the defective specimen can effectively reflect the information of the defect.In our team's previous research, the feasibility of this scheme has been verified, and the impact of different excitation signals, different lifting heights, and other variables on the detection results has been studied [13].The excitation coil is a square wave signal, with a duty cycle of 50%, frequency of 2 kHz, and an exciting voltage of 12 V.After the test, the output voltage signal Vout of the hall sensor in the probe can be obtained.
The difference signal obtained by calculating the difference between the signal of the non-defective specimen and that of the defective specimen can effectively reflect the information of the defect.In our team's previous research, the feasibility of this scheme has been verified, and the impact of different excitation signals, different lifting heights, and other variables on the detection results has been studied [13].
This method is also used in this paper.The output signal obtained during the detection of samples without crack is defined as the reference signal Vref, and the differential signal ∆V is obtained by the difference between Vout and Vref.The maximum and minimum values of ∆V are the changes in the position of the rising edge and falling edge of the output signal, and the absolute values of the maximum and minimum values of ∆V are marked as ∆VR and ∆VF, respectively, and their values are shown in Table 4.The relation between the crack depths and the peak values are displayed in Figure 9.It was observed that the voltage values at three detection points generally increased with the depth of natural cracks.However, due to non-uniformity in the depth of fatigue natural cracks along the Y axis in Figure 8, the values at three detection points for the same specimen are differed.The most significant variance was observed in Specimen 1, while Specimen 3 exhibited the least variance.This further illustrates that the depth distribution of natural cracks along the Y direction is more complex than that of artificial cracks [21,22].This method is also used in this paper.The output signal obtained during the detection of samples without crack is defined as the reference signal V ref , and the differential signal ∆V is obtained by the difference between V out and V ref .The maximum and minimum values of ∆V are the changes in the position of the rising edge and falling edge of the output signal, and the absolute values of the maximum and minimum values of ∆V are marked as ∆V R and ∆V F , respectively, and their values are shown in Table 4.The relation between the crack depths and the peak values are displayed in Figure 9.It was observed that the voltage values at three detection points generally increased with the depth of natural cracks.However, due to non-uniformity in the depth of fatigue natural cracks along the Y axis in Figure 8, the values at three detection points for the same specimen are differed.The most significant variance was observed in Specimen 1, while Specimen 3 exhibited the least variance.This further illustrates that the depth distribution of natural cracks along the Y direction is more complex than that of artificial cracks [21,22].The excitation coil is a square wave signal, with a duty cycle of 50%, frequency of 2 kHz, and an exciting voltage of 12 V.After the test, the output voltage signal Vout of the hall sensor in the probe can be obtained.
The difference signal obtained by calculating the difference between the signal of the non-defective specimen and that of the defective specimen can effectively reflect the information of the defect.In our team's previous research, the feasibility of this scheme has been verified, and the impact of different excitation signals, different lifting heights, and other variables on the detection results has been studied [13].
This method is also used in this paper.The output signal obtained during the detection of samples without crack is defined as the reference signal Vref, and the differential signal ∆V is obtained by the difference between Vout and Vref.The maximum and minimum values of ∆V are the changes in the position of the rising edge and falling edge of the output signal, and the absolute values of the maximum and minimum values of ∆V are marked as ∆VR and ∆VF, respectively, and their values are shown in Table 4.The relation between the crack depths and the peak values are displayed in Figure 9.It was observed that the voltage values at three detection points generally increased with the depth of natural cracks.However, due to non-uniformity in the depth of fatigue natural cracks along the Y axis in Figure 8, the values at three detection points for the same specimen are differed.The most significant variance was observed in Specimen 1, while Specimen 3 exhibited the least variance.This further illustrates that the depth distribution of natural cracks along the Y direction is more complex than that of artificial cracks [21,22].Based on this consideration, the average voltage V average of three detection points are taken as output in the inversion process.The difference signal between V average and V ref is shown in Figure 10.The corresponding maximum and minimum values are labeled ∆V RA and ∆V FA in Table 4, respectively.
The relationship between crack depth and detection voltage is shown in Figure 11, ∆V RA and ∆V FA are represented by "o" and "x", respectively.The horizontal coordinates of the four points, from left to right, correspond to different crack depths of specimens 1, 2, 3, and 4. The depth of crack is observed using the optical method described in Section 2.2.The average depth in Table 2 is used as the characterization of the depth of each specimen.
Based on this consideration, the average voltage Vaverage of three detection points are taken as output in the inversion process.The difference signal between Vaverage and Vref is shown in Figure 10.The corresponding maximum and minimum values are labeled ∆VRA and ∆VFA in Table 4, respectively.The relationship between crack depth and detection voltage is shown in Figure 11, ∆VRA and ∆VFA are represented by "o" and "x", respectively.The horizontal coordinates of the four points, from left to right, correspond to different crack depths of specimens 1, 2, 3, and 4. The depth of crack is observed using the optical method described in Section 2.2.The average depth in Table 2 is used as the characterization of the depth of each specimen.By performing linear regression on both the average peak voltage values of the rising edge and the falling edge, the linearity mapping relationship between the natural depth and the rising edge or the falling edge can be summarized as Equation ( 1), where d is the crack depth.
41.83 110.6 79. 14 20.56 Table 5 lists the inversion results.The actual crack depth is the result that is observed using the optical method, which can be thought of as the true depth of the crack.Corresponding inversion errors for specimens 1#, 2#, 3#, and 4# are based on the mapping model shown in Equation (1).
According to Table 5, the error range for the four specimens lies between 0.50% and 11.77% generally.Especially when utilizing the falling edge inversion model, the relative errors for the four specimens are in the range of 0.5%~11.77%,while when utilizing the rising edge inversion model, the inversion error is in the range of 1.11%~2.34%for four   The relationship between crack depth and detection voltage is shown in Figure 11, ∆VRA and ∆VFA are represented by "o" and "x", respectively.The horizontal coordinates of the four points, from left to right, correspond to different crack depths of specimens 1, 2, 3, and 4. The depth of crack is observed using the optical method described in Section 2.2.The average depth in Table 2 is used as the characterization of the depth of each specimen.By performing linear regression on both the average peak voltage values of the rising edge and the falling edge, the linearity mapping relationship between the natural depth and the rising edge or the falling edge can be summarized as Equation (1), where d is the crack depth.
41.83 110.6 79.14 20.56 Table 5 lists the inversion results.The actual crack depth is the result that is observed using the optical method, which can be thought of as the true depth of the crack.Corresponding inversion errors for specimens 1#, 2#, 3#, and 4# are based on the mapping model shown in Equation ( 1).
According to Table 5, the error range for the four specimens lies between 0.50% and 11.77% generally.Especially when utilizing the falling edge inversion model, the relative errors for the four specimens are in the range of 0.5%~11.77%,while when utilizing the rising edge inversion model, the inversion error is in the range of 1.11%~2.34%for four  By performing linear regression on both the average peak voltage values of the rising edge and the falling edge, the linearity mapping relationship between the natural depth and the rising edge or the falling edge can be summarized as Equation (1), where d is the crack depth.
∆V RA = 41.83d+ 110.6 Table 5 lists the inversion results.The actual crack depth is the result that is observed using the optical method, which can be thought of as the true depth of the crack.Corresponding inversion errors for specimens 1#, 2#, 3#, and 4# are based on the mapping model shown in Equation ( 1).
According to Table 5, the error range for the four specimens lies between 0.50% and 11.77% generally.Especially when utilizing the falling edge inversion model, the relative errors for the four specimens are in the range of 0.5~11.77%,while when utilizing the rising edge inversion model, the inversion error is in the range of 1.11~2.34%for four specimens.For specimen 1#, 2#, and #3, the measured accuracy is higher according the rising edge inversion model; and for specimen 4#, the reverse case is shown.Therefore, it is necessary to seek a generalized approach for the characterization all specimens with natural crack depth which has satisfactory accuracy.

Natural Crack Recognition Algorithm Based on Neural Networks
Given the nonlinear relationship between fatigue natural crack depth and the peak values of the detection signal, employing artificial intelligence algorithms to construct a nonlinear mapping relationship model could further improve detection accuracy due to its excellent capability in expression the nonlinearity issues [14].
In this paper, a BPNN is constructed, which is shown in Figure 12.The input layer consists of two neurons, corresponding to ∆V RA and ∆V FA , respectively.The output layer has one neuron, which represents the depth of the fatigue natural crack.The model includes two hidden layers, with the number of neurons in these layers determined according to Equation (2).

Natural Crack Recognition Algorithm Based on Neural Networks
Given the nonlinear relationship between fatigue natural crack depth and the peak values of the detection signal, employing artificial intelligence algorithms to construct a nonlinear mapping relationship model could further improve detection accuracy due to its excellent capability in expression the nonlinearity issues [14].
In this paper, a BPNN is constructed, which is shown in Figure 12.The input layer consists of two neurons, corresponding to ∆VRA and ∆VFA, respectively.The output layer has one neuron, which represents the depth of the fatigue natural crack.The model includes two hidden layers, with the number of neurons in these layers determined according to Equation (2).
( ) In the equation, n2 represents the number of neurons in the hidden layers, n1 denotes the number of neurons in the input layer, and m is the number of layers in the output layer; and a is a constant, with a value range of 1 to 10.
Following this formula, given that the BPNN structure for crack prediction has two neurons in the input layer and one neuron in the output layer, and setting a = 8, the number of neurons in the hidden layers is calculated to be 10.In the equation, n 2 represents the number of neurons in the hidden layers, n 1 denotes the number of neurons in the input layer, and m is the number of layers in the output layer; and a is a constant, with a value range of 1 to 10.
Following this formula, given that the BPNN structure for crack prediction has two neurons in the input layer and one neuron in the output layer, and setting a = 8, the number of neurons in the hidden layers is calculated to be 10.
To train the neural network effectively, a substantial amount of sample data are required.However, for natural cracks, the actual measured data are limited and insufficient to meet the data volume requirements for neural network training.In a previous study by the other researchers from our team, cracks were detected based on PEC technology, and simulation results were used to expand the data.On this basis, a neural network algorithm was used to establish the transformation relationship between detection signal and crack size, and high inversion accuracy was obtained [14].This shows that the method of using the simulation results to expand the data is feasible.
This strategy is also used in this article.Some data are obtained through a simulation calculation.The big difference between the natural crack and the artificial crack is the wall of the artificial is smooth, while the natural crack is unsmooth.In a numerical simulation, the unsmooth wall of the crack is simulated by a special region with the specific conductivity.In this paper, a simulation is to obtain more samples to deal with the limitation of the natural cracks by using the three-point bending test.Therefore, the details of the numerical simulation is not described in this paper.The corresponding simulation data are shown in Table 6.When using the BPNN algorithm to predict the depth of fatigue natural cracks, an improved genetic algorithm is employed to optimize the network weights to address the issue of the network tending to converge to local optima.The underlying concept of optimization is to treat the weights as individuals in the genetic algorithm, where the fitness value of an individual corresponds to the prediction error of the BP neural network.Through operations such as selection, crossover, and mutation, superior weights are generated.The optimization process for the BP neural network includes encoding and decoding of weights, remodeling of the fitness function, and the determination of the selection, crossover, and mutation operators.The process of optimizing BP neural network weights based on the GA is illustrated in Figure 13.GA population initialization typically relies on random generation, which leads to certain levels of non-uniformity.This unevenness can slow down the search speed of the GA, leading to prolonged convergence times and significantly impacting the algorithm's efficiency.This paper adopted Optimal Latin Hypercube Sampling design method to generate a more uniform population distribution.
Figure 14 illustrates the Latin Hypercube sampling of ten samples in a two-dimensional space, where the process initially determines in which combined interval each sample falls before specifying the sample's exact location within that interval.An enhanced random evolutionary algorithm optimizes the specific positioning of sample points to achieve optimal Latin Hypercube sampling.Figure 14a  GA population initialization typically relies on random generation, which leads to certain levels of non-uniformity.This unevenness can slow down the search speed of the GA, leading to prolonged convergence times and significantly impacting the algorithm's efficiency.This paper adopted Optimal Latin Hypercube Sampling design method to generate a more uniform population distribution.
Figure 14 illustrates the Latin Hypercube sampling of ten samples in a two-dimensional space, where the process initially determines in which combined interval each sample falls before specifying the sample's exact location within that interval.An enhanced random evolutionary algorithm optimizes the specific positioning of sample points to achieve optimal Latin Hypercube sampling.Figure 14a predominantly shows sample distribution in the lower-left and upper-right corners, with no points in the upper-left and lower-right corners.However, the ten sample points of the optimized Latin Hypercube sampling are uniformly distributed across the two-dimensional sample space after optimization, as shown as in Figure 14b.
GA, leading to prolonged convergence times and significantly impacting the algorithm's efficiency.This paper adopted Optimal Latin Hypercube Sampling design method to generate a more uniform population distribution.
Figure 14 illustrates the Latin Hypercube sampling of ten samples in a twodimensional space, where the process initially determines in which combined interval each sample falls before specifying the sample's exact location within that interval.An enhanced random evolutionary algorithm optimizes the specific positioning of sample points to achieve optimal Latin Hypercube sampling.Figure 14a predominantly shows sample distribution in the lower-left and upper-right corners, with no points in the upperleft and lower-right corners.However, the ten sample points of the optimized Latin Hypercube sampling are uniformly distributed across the two-dimensional sample space after optimization, as shown as in Figure 14b.In GA, the two most crucial parameters that require repeated adjustment are the crossover probability Pc and mutation probability Pm, which directly influence the optimal weight values in the neural network prediction model.This paper adopts the concept of orthogonal experiments, dividing the crossover probability into three levels: 0.4, 0.65, and 0.9, and the mutation probability is similarly divided into three levels: 0.001, 0.01, and 0.1, resulting in nine combinations.The prediction results are shown in Table 7.In GA, the two most crucial parameters that require repeated adjustment are the crossover probability P c and mutation probability P m , which directly influence the optimal weight values in the neural network prediction model.This paper adopts the concept of orthogonal experiments, dividing the crossover probability into three levels: 0.4, 0.65, and 0.9, and the mutation probability is similarly divided into three levels: 0.001, 0.01, and 0.1, resulting in nine combinations.The prediction results are shown in Table 7.
By cross-comparing the results in Table 7, it can be seen that the minimum relative errors are 3.03%, 4.21%, and 3.03%, respectively, when the baseline is at different values.And it can be found that P m is set to 0.65 and P c to 0.01 in all of these combinations.Under this condition, the BPNN achieves the highest prediction accuracy.Consequently, we adopt this set of weights as the neural network's weights to finalize the predictive model and proceed to validate it using a set of validation samples.The validation results are presented in Table 8.
Compared with the results in Table 4, in which the average relative error is 3.73%, the relative error for natural cracks is 2.19% by using the optimized neural network algorithm shown in Table 8.The result indicated that the proposed GA-based BPNN can great improve the characterization accuracy for the natural crack.

Conclusions
In order to quantitatively detect the depth of the natural cracks in the metallic material, especially in the areo-aluminum alloy, the natural cracks are generated by the three-point bending test under the cyclic applied loads, and then the depths of the natural cracks are calibrated.Based on the specimens with the natural fatigue crack, PECT platform are constructed, and a series of experiments are carried out to investigate the time domain signal above the cracks with different depth and different detection points.To improve the quantitative characterization accuracy, a GA-based BPNN algorithm is proposed in which the Latin Hypercube is explored to obtain the more uniform population distribution in GA and, consequently, to optimize the weight for BPNN.The investigation indicated that: (1) Compared with the artificial crack, the natural cracks has the complex geometric distribution, especially in crack depth and the direction.(2) PEC can sense the weak change of the natural crack depth even if there is a prefabricated opening.(3) The proposed GA-based BPNN algorithm can improve the characterization accuracy of the natural crack with the relative error as 2.19%.
In this paper, the number of specimens with the natural cracks are limited.Therefore, in future, more samples for the natural cracks are acquired or generated by the intelligence algorithm, such as Principal Component Analysis [16] or Generative Adversarial Network methods for data enhancement [24], to improve the characterization accuracy further.

Figure 1 .
Figure 1.Schematic diagram of the three-point bending test.

Figure 1 .
Figure 1.Schematic diagram of the three-point bending test.

Figure 5 .
Figure 5.The correlation between crack depth and the number of cycles.

Figure 5 .
Figure 5.The correlation between crack depth and the number of cycles.

Figure 5 .
Figure 5.The correlation between crack depth and the number of cycles.

Figure 8 .
Figure 8. Top view of the detection probe and specimen position.

Figure 8 .
Figure 8. Top view of the detection probe and specimen position.

Figure 8 .
Figure 8. Top view of the detection probe and specimen position.

Figure 9 .
Figure 9. Relationship between ∆V R /∆V F and crack depth.

Figure 10 .
Figure 10.The difference signal between Vaverage and Vref.

Figure 11 .
Figure 11.Relationship between average voltage and crack depth.

Figure 10 .
Figure 10.The difference signal between V average and V ref .

Figure 10 .
Figure 10.The difference signal between Vaverage and Vref.

Figure 11 .
Figure 11.Relationship between average voltage and crack depth.

Figure 11 .
Figure 11.Relationship between average voltage and crack depth.

Figure 13 .
Figure 13.Neural network weight optimization based on GA.

Figure 13 .
Figure 13.Neural network weight optimization based on GA.

Figure 14 .
Figure 14.Comparison of initial population probability distribution and optimal Latin Hypercube distribution.(a) Before; (b) after.

Figure 14 .
Figure 14.Comparison of initial population probability distribution and optimal Latin Hypercube distribution.(a) Before; (b) after.

Author Contributions:
Conceptualization, Y.Y. and C.S.; Formal analysis, C.S. and H.L.; Investigation, C.S. and H.L.; Methodology, Y.Y. and C.S.; Software, F.W. and D.L.; Validation, C.S. and H.L.; Writingoriginal draft, C.S. and F.W.; Writing-review and editing, all authors.All authors have read and agreed to the published version of the manuscript.Funding: This work was supported by the National Nature Science Foundation of China [grant numbers 52275522 and 61960206010], the Sichuan Science and Technology Program [grant number 2022NSFSC0323], and the Open Fund of Key Laboratory of Traction Power [grant number TPL2206].

Table 1 .
Comparison of fatigue test conditions under different cyclic loads.

Table 1 .
Comparison of fatigue test conditions under different cyclic loads.

Table 1 .
Comparison of fatigue test conditions under different cyclic loads.

Table 2 .
The crack depth of specimen that measured by optical method.

Table 2 .
The crack depth of specimen that measured by optical method.

Table 2 .
The crack depth of specimen that measured by optical method.

Table 3 .
Main parameters of the coil.

Table 3 .
Main parameters of the coil.

Table 3 .
Main parameters of the coil.

Table 4 .
Peak value of rising edge and falling edge.

Table 4 .
Peak value of rising edge and falling edge.

Table 4 .
Peak value of rising edge and falling edge.

Table 5 .
Analysis of fitting curve errors for each specimen.

Table 6 .
Fatigue natural crack simulation data for the neural network.

Table 8 .
Validation of fatigue natural crack prediction after weight optimization.