Magnetic Dipole Model and Experimental Analysis of Magnetic Signal Detection Based on Magnetic Memory

By assuming that the magnetic charge is evenly distributed in the defect groove, the linear magnetic dipole is integrated in the depth direction of the defect, and the numerical calculation is carried out by using MATLAB to establish a two-dimensional magnetic dipole theoretical analysis model of double-correlated defects, which is suitable for analyzing the influence of stress on magnetic signal in the magnetic memory detection of rectangular and V-shaped combined macroscopic crack defects. To explore the influence factors and change rules of magnetic memory signal. In order to verify the correctness of the model analysis results, tensile tests were carried out on Q235 steel plate with double associated defects with different morphological characteristics, and the Hp(y) value of the normal component of the leakage magnetic field on the surface of the member was measured. The results show that the model based on magnetic dipole theory can explain some experimental phenomena and rules in magnetic memory detection.


Introduction
Ferromagnetic materials have good mechanical properties, which are mainly used in industrial production [1].Residual stress and hidden damage generated in the production and manufacturing process will lead to its failure or even fracture under long-term fatigue damage.Serious or even catastrophic accidents will seriously threaten people's property and life safety [2][3], so it is particularly important to assess the damage of ferromagnetic specimens in the early stage [4].Metal magnetic memory detection technology was first proposed by Professor DOUBOV in 1994 [5][6].Based on the force-magnetic coupling, this technology can accurately determine the stress concentration area, so as to realize early identification of defects.When a ferromagnetic specimen with defects is subjected to external stress, stress concentration occurs near the defects, and the leakage magnetic field is generated near the defect under the action of the geomagnetic field.The existence of the defect can be determined by measuring the leakage magnetic field on the specimen surface.In recent years, a large number of scholars at home and abroad began to study the defect leakage magnetic field model: Xie Yangguang [7] established the magnetic field distribution in the solenoid based on the double magnetic dipole, and compared with the single magnetic dipole, the maximum positioning error decreased by 90.26%.Minhhuy Le [8] proposed a magnetic dipole model method for non-destructive evaluation to estimate the crack shape and volume, which is consistent with the experimental results.Shi Pengpeng [9] gave the variation of leakage magnetic field in rectangular and V-shaped notch space, and analyzed the influencing factors of magnetic memory signal in detail.
2 At present, there are few researches on the magnetic dipole model of double-correlation defects at home and abroad.Therefore, this paper establishes a two-dimensional magnetic dipole model of spatial leakage field of double-correlation defects composed of rectangular and V-shaped macroscopic cracks to analyze the influence of stress on magnetic memory signal, and discusses the factors affecting the spatial leakage field distribution in detail.

Force-magnetic Coupling Model under Weak Magnetic Field
In the weak magnetization state, the relationship model between stress, applied magnetic field and material magnetization M can be directly given by reference [9]: where,  0 is the vacuum permeability,  0 = 4π × 10 −7 ; H 0 is the ambient magnetic field; M s is the saturation magnetization of the material; is Weiss molecular field coupling parameter;  is Weisss molecular field coupling parameter； as stress;  is the proportional coefficient, which affects the degree of stress magnetization effect; σ s is the yield stress of the material, λ s is the saturation magnetostrictive strain of the material.

Magnetic Dipole Model of Defect-free Leakage Field
Under the influence of tensile stress and external magnetic field, the ferromagnetic material is approximately magnetized into a magnet, with one end of the S-level and one end of the N-pole, as shown in figure 1.There are equal amounts of dissimilar magnetic charges at both left and right ends of the specimen.Assuming that the magnetic charges are evenly distributed at both left and right ends of the specimen, the magnetic memory signal value of any point P in the regional space where the specimen is located can be directly given by reference [9]: (2) where: L is the half length of the specimen; H is the thickness of the specimen;  0 is the vacuum permeability; 0 is the magnetic charge surface density,  0 =  0 , M is the magnetization.x and y are the coordinates of any point P in space.

Magnetic Dipole Model of Double Correlation Defects
Magnetic dipole leakage field analysis was carried out for double-correlated defects.As shown in figure 2 and figure 3, rectangular and V-shaped defects in double-correlated defects are located on both sides of the Y axis respectively, and the distance from the Y axis is both m, the width of the two defects is 2b, and the depth of the two defects is h.
According to the magnetic dipole theory, the left and right sides of the double-correlated defect specimens have the same amount of dissimilar magnetic charges, which induce new magnetic charges on the left side of the rectangular defect and on the right side of the V-shaped defect, and the new magnetic charges generate new dissimilar magnetic charges at the right end of the rectangular defect and the left end of the V-shaped defect.In this case, magnetic poles of equal size and opposite polarity are generated by both the rectangular defect and the V-shaped defect.Then the tangential and normal magnetic field strength components of any point P in the space can be expressed as: (5) where  1 ,  1 ,  2 and  2 are the magnetic field strength components generated by rectangular defects,  3 ,  3 ,  4 and  4 are the magnetic field strength components generated by V-shaped defects.From equations ( 4)-( 11), it can be obtained that the leakage magnetic field strength of double-correlated defect P point can be expressed as:  ) ) ) (13)

Analysis and Discussion of the Influence of Defect Characteristic Parameters on Magnetic Memory Signal
The magnetic leakage field generated by ferromagnetic test block in the air can be simplified to the superposition of multiple magnetic dipole fields, so the magnetic dipole model can be used for forward modelling [10].

Effect of Crack Width on Magnetic Signal
Figure 3 shows the variation curve of magnetic memory signal of double correlation defect when σ=120Mpa and b takes 0.3mm, 0.5mm, 0.7mm and 0.9mm.With the increase of b, the range of stress concentration of double correlation defects increases, and the peaks of tangential and normal components of magnetic memory signal also increase.It can be seen from the figure that the magnetic memory signal value of rectangular defects is greater than that of V-shaped notch, so it is easier to detect rectangular defects than V-shaped defects in actual detection.

Effect of Crack Depth on Magnetic Signal
Figure 5 shows that influence of crack depth on magnetic memory signal: when stress σ=120Mpa and h is 0.5mm, 1mm, 1.5mm and 2mm.When the defect width remains unchanged, the detection signal increases with the increase of h, and the increase of the magnetic memory signal gradually decreases.In the stretched state, the amplitude and variation of detection signal induced by rectangular defects with the same crack depth are larger than that of V-shaped defects.

Effect of Lift Height on Magnetic Signal
Figure 6 shows the stress σ=120Mpa.The lifting height y is 1mm, 2mm, 3mm, 4mm.y induced space leakage magnetic field intensity variation law.When y increases, the value of detection signal of double correlation defect gradually decreases.The signal of rectangular defect is larger than that of V-shaped defect.The higher the lift height, the weaker the nonlinear degree of magnetic signal.

Effect of Double Defect Spacing on Magnetic Signal
Figure 7 shows the stress σ=120Mpa, m is taken as 2mm, 3mm, 4mm and 5mm.With the increase of the distance, the peak value of the tangential and normal components of the magnetic signal increases slightly.Therefore, the smaller the distance between the two defects, the more difficult it is to detect them.

Effect of Stress on Magnetic Signal
Figure 8 shows the theoretical model analysis results of the influence of stress on the spatial leakage magnetic field strength of specimens when stress σ is 30MPa, 60MPa, 90MPa, 120MPa, 150MPa, 180MPa, 210MPa, h=4mm, b=3mm, m=3mm, y=1mm.The peak values of the tangential and normal components of the leakage magnetic field on the surface of the double correlation defect decrease with the increase of the applied load.As can be seen from the figure, the internal magnetization of the specimen increases with the increase of applied stress, resulting in the nonlinear degree of magnetic memory signal induced by double-correlation defects of the specimen decreases with the increase of applied stress.The value of the normal component Hp(y) of the leakage magnetic field on a rectangular defect surface is larger than that of the V-shaped notch, so it is easier to detect the rectangular defect than the V-shaped defect in practice.

Test Materials and Test Equipment
In order to verify the correctness of the analysis results of the forward modeling model, three steel plates with a length of 450mm, a thickness of 10mm and a width of 40mm were designed respectively for the change of defect width, depth and double-associated defect spacing.Each steel plate preset 3 pairs of double-associated defects.Technical parameters of steel plates are shown in table 1. Path 1, 2 and 3 of leakage magnetic field extraction are set 1mm above the steel plate respectively.The length of measurement path 1 is 400mm, the length of path 2 is 350mm, and the length of path 3 is 350mm.In order to ensure that the signals between each group of defects do not affect each other, the interval between each group of defects is set to be 100mm.At room temperature, the static load tensile test was carried out by WDW-E100D tensile testing machine, and the pre-processed Q235 steel plate was vertically placed on the tensile machine for static load tensile test.TSC-1M-4 metal magnetic memory detector was used to measure the Hp(y) value of Q235 steel surface magnetic memory signal along the measuring path.

Test Process
The initial magnetic field value of the specimen was measured before the tensile test.The steel plate is placed in a north-south direction between the upper and lower clamping ends of the tensile testing machine, and the lifting height was kept 1mm.The initial magnetic field value was measured along the measurement paths 1, 2 and 3, respectively.
In the tensile test, the steel plate is loaded step by step at the same rate.The three kinds of steel plates prefabricated with different defect characteristic parameters are loaded to 30MPa, 60MPa, 90MPa, 120MPa, 150MPa, 180MPa and 210MPa respectively.After the predetermined load is reached, Three kinds of Q235 steel plates were measured along paths 1, 2 and 3 according to the method of measuring the initial magnetic field value, and the value of Hp(y) of the normal component of the leakage magnetic field was collected.

Test Results and Analysis
In figure 11, the crack half-width is 2mm, 3mm and 4mm from left to right, the crack depth is 4mm, and the double defect distance is 6mm.It is found that the normal component Hp(y) of Q235 steel plate crosses zero and its symbol changes.With the increase of load, the magnetic memory distribution curve rotates counterclockwise.When the load is constant, the nonlinear degree of the normal component of the leakage magnetic field signal becomes more obvious with the increase of the crack half-width, and the defects are easier to be detected.The change rule of the normal component in the test results is consistent with that in figure 4. Figure 10 shows the effect of crack depth on magnetic leakage signal, h from left to right is 2mm, 4mm, 6mm, crack half-width is 3mm, double defect distance is 6mm.  Figure 11 is the schematic diagram of steel plate fracture.During the test, when the load is applied to 79KN (197.5MPa), the steel plate breaks, and the fracture location is at the rectangular gap in the double-correlated defect at the crack depth of 6mm.Compared with the normal component of Magnetic leakage signal in the double-correlated defect compared in figure 5 and figure 8, the stress concentration of the rectangular defect is larger.It's easier to detect consistent conclusions.Figure 12 is the stressstrain curve of Q235 steel plate with different crack depth under tensile state.Figure 13 shows the influence of double correlation defect spacing on magnetic memory signal.The defect spacing from left to right is 4mm, 6mm, 8mm, depth is 4mm, and half-width is 3mm.With the increase of load, the magnetic memory distribution curve rotates counterclockwise, and the degree of nonlinearity increases, making it easier to detect defects.The test results are consistent with figure 7.

Conclusion
(1) By assuming that the magnetic charge is uniformly distributed on the groove surface, a twodimensional magnetic dipole theoretical analysis model of double-correlated defects is established, and the numerical calculation is carried out by MATLAB programming, and the analytical solution of the magnetic dipole is given.
(2) When the load is constant, with the increase of crack half-width, crack depth and defect spacing of double-correlated defects in Q235 steel plate, the degree of nonlinearity of defect induced magnetic signal increases, the stress concentration becomes more obvious, and the defects are more easily detected.The experimental phenomenon is basically consistent with the model results, which indicates that the magnetic dipole theoretical analysis model proposed in this paper can explain some experimental phenomena of magnetic memory.
(3) The defect characteristic parameters of Q235 steel plate are certain.With the increase of load, the peak value of normal component decreases, the nonlinearity of the normal component of magnetic signal at the double correlation gap weakens, and the defects are not easy to be detected.The experimental results are roughly the same as those of the magnetic dipole theoretical analysis model.
(4) The magnetic dipole theoretical analysis model established in this paper can initially realize the theoretical analysis of some basic test phenomena and variation rules of double-correlation defects, but for the quantitative analysis of test signals in magnetic memory detection, the calculation of magnetic charge density and a more comprehensive force-magnetic coupling model need to be taken into account.

Figure 4 .
Figure 4. Influence of crack width on magnetic signal of double correlation defect.

Figure 5 .
Figure 5. Influence of crack depth on magnetic signal of double correlation defect.

Figure 6 .
Figure 6.The influence of disjunction height on magnetic signal.

Figure 7 .
Figure 7. Influence of double-association defect m value on magnetic memory signal.

Figure 8 .
Figure 8.Effect of stress on magnetic memory signal.

Figure 9 .
Figure 9. Influence of crack half-width on normal component of magnetic leakage signal.

Figure 10 .
Figure 10.Influence of crack depth on the normal component of magnetic leakage signal.

Figure 13 .
Figure 13.Effect of defect spacing on normal component Hp(y) of magnetic leakage signal.

Table 1 .
Technical parameters of Q235 steel.