# Study on Coercivity-Stress Relationship of X80 Steel under Biaxial Stress

Authors: Shulu Feng; Zhijiu Ai; Jian Liu; Jiayi He; Yukun Li; Qifeng Peng; Chengkun Li
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

Buried pipelines are prone to deformation under surface displacement, pressure occupation, and/or other actions, resulting in stress concentration. Conducting regular stress detection is an important approach to reducing pipeline accidents. This study performed experiments on measuring the coercivity stress of X80 steel, the most commonly used material for high-pressure large-diameter gas transmission pipelines. The coercivity of the specimen under different stress combinations was measured using the DC-driven coercive force measurement system, and the variation of coercive force with biaxial stress was also analyzed. The modified coercive force-stress expression of the generalized Hooke’s law was proposed; the model for the coercive force-stress relationship of X80 steel under the biaxial stress state was established based on this expression. The study indicates that the coercive force method is reliable for accurately measuring the stress of pipeline steel and evaluating the safety margin of in-service pipelines.


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## Body

## 1. Introduction

Large-diameter and high-pressure gas transmission has been an important development to meet the demands of long-distance large-capacity transmission of resources to improve efficiency and economic benefits of pipelines [1]. The X80 pipeline steel is the preferred steel grade for such transmission pipelines because of its strength, toughness, and weldability [2]. During long-term operations of an oil or gas pipeline, deformation is inevitable under the actions of surface displacement or other loads, resulting in additional stress concentration in some areas of the pipeline, eventually leading to pipeline damage. For in-service pipelines, destructive stress detection methods are unsafe and uneconomical. Therefore, it is important to conduct regular stress measurements on the risk sections of pipelines using nondestructive stress detection methods to evaluate the safety margin and prevent safety accidents. Present, more mature nondestructive stress detection methods such as the X-ray diffraction method and ultrasonic method require higher accuracy of the measured surfaces, so it is difficult to accurately evaluate the stress states of in-service pipelines. The coercivity method is a promising new high-efficiency nondestructive stress detection technology, which is an important means to evaluate the pipeline safety margin and prevent pipeline safety accidents (Ni et al., 2018; [3]).Herzer [4] found experimentally that the coercivity of a ferromagnetic material is inversely proportional to the grain size. Jiles and Atherton [5] derived the hysteretic magnetization curve by means of the mean field method. Then Jiles [6] studied the influence of mechanical stress on magnetization and proposed the JA model of force-magnetic coupling and the concept of the effective field. Novikov et al. [7] derived the analytical function of the coercivity along the tensile stress axis with the stress and conducted the experiments on low carbon steel. Novikov et al. [8] continued to carry out relevant theoretical research and established the general calculation curve of the average coefficient of all test low carbon steels. Huang et al. [9] explored the relationship between coercivity, external tensile stress, and residual stress using Q235 steel as the experimental material. Ivanov and Vashenko [10] studied the properties of coercivity of structural steel specimens under uniaxial static tension. Vengrinovich et al. [11] studied the difference in magnetic characteristic parameters under the biaxial stress state and concluded that the effect of the plane stress on the coercive force required further verification. Shen et al. [12] quantitatively evaluated the residual stress of a 35 CrMo steel cylinder before and after heat treatments using the metal magnetic memory technology, and their evaluation revealed the structural mechanical dependence of coercivity.To sum up, the coercive force method is a new research hotspot in the field of nondestructive testing [13, 14]. However, the previous studies on coercive force measurements mainly focus on the micromechanism of the coercive force change. So far, only a few studies have investigated the influence of one-way stress on the coercive force of materials, and the impacts of complex stresses on the measurement results have not been considered, preventing successfully applying the research to engineering practices [15, 16]. It is an effective way to ensure the safety of pipelines to adopt various advanced and targeted inspections to identify pipeline defects, judge the types of defects, and then carry out safety evaluation and maintenance [16]. To solve the above problems, this study conducted experiments on the relationship between coercivity and stress of X80 steel under different stress combinations. By measuring the coercivity values of specimens under different plane stress combinations and combining with the existing theories and experimental phenomena, the causes of the coercivity difference under different plane stress combinations were analyzed, and the coercivity-stress expression modified by the generalized Hooke law was established. According to the expression, a binary first-order model for coercive force stress under the biaxial stress state was proposed. The experimental data were fitted using the least square method, and the fitting accuracy was verified.


## 2. Theoretical Basis of Coercive Force-Stress Relation

### 2.1. Magnetic Domains and Magnetization Theory of Ferromagnetic Materials

When a ferromagnetic material is subjected to stress, the internal microstructure changes, the magnetic domain structure deflects, and the magnetic characteristic parameters vary significantly, which is called the force-magnetic coupling effect [17, 18]. Among them, the magnetic domain is a small area with spontaneous magnetization inside the ferromagnetic material, and the magnetic moment direction of each magnetic domain is different. The boundary between adjacent magnetic domains is called a domain wall, as shown in Figures 1 and 2.Figure 1
Direction of magnetic moment of ferromagnetic material in original state.Figure 2
Direction of magnetic moment of ferromagnetic material under tensile stress.The magnetization curve of the material is shown in Figure3. The horizontal axis represents the applied magnetic field strength; the vertical axis represents the induced magnetic field strength. The Om section is the initial magnetization curve, that is, the curve of the variation of the induced magnetic field with the applied magnetic field when the material is first magnetized in the initial state. When the external magnetic field is removed, the material cannot completely recover to the unmagnetized state due to the movement resistance of the domain wall, forming a hysteresis. The hysteresis causes the change of the magnetic induction intensity with the external magnetic field to lag behind the initial magnetization curve. For example, at the first demagnetization (mr segment), when the applied magnetic field strength is reduced to 0, the induced magnetic field strength of the material is not 0. The induced magnetic field strength represented by the intersection point Br of the mr segment and the y-axis is called remanence. When the applied magnetic field changes for one cycle, the magnetic induction intensity also changes to form a hysteresis loop (Bin Liu et al., 2015; [19]). From the hysteresis loop, the important parameters characterizing the hysteresis property of the material can be obtained [20]. When the material deforms due to the applied load, the magnetic domain structure inside the material varies, and the irreversible displacement of the domain wall occurs, resulting in changes in the material magnetic properties. The coercive force Hc represents the value of the applied magnetic field H when the magnetization M of the material is 0, and the coercive force changes accordingly with the magnetic domain structure; therefore, the stress state of the material can be evaluated by measuring the coercive force of the material ([21]; Leng et al. 2021).Figure 3
Schematic diagram of hysteresis loop.


### 2.2. Theory of Force-Magnetic Relation under One-Way Stress State

To study the effect of stress on magnetization, Jiles and Atherton established a force-magnetic coupling J-A model (Jiles and Atherton 1984; [5]) and proposed the concept of the effective field. The relationship between the magnetization intensity, the deformation, and elastic properties of materials in the model is(1)dMirrdσ=1ε2σMan−Mirr,ε=Eξ1/2,where Mirr is the irreversible magnetization intensity, Mrev is the reversible magnetization intensity, Man is the total magnetization intensity, ε is the stress size constant that can be determined by the test, and ξ is the unit volume energy coefficient.After differential transformation of equation (1), equation (2) is obtained:(2)dMirrMan−Mirr=1Eγξσdσ.Equation (2) demonstrates that a certain relationship exists between the magnetization and the stress. The magnetization M and the magnetic field strength H satisfy M=χH, where χ is the magnetic susceptibility. It can be considered that magnetic field strength H and stress are related; hence, the change law of stress can be obtained by exploring the magnetic field strength [22].Based on the force-magnetic coupling model, the relationship between coercive force and stress under the uniaxial stress was proposed by Novikov et al. [7]:(3)Hc∥=B1+A1e−β1σ+B1k1σ,B1=Hc0−A1,Hc⊥=B2−A2e−β2σ+B2k2σ,B2=Hc0+A2,where Hc∥ is the coercive force measured in the direction parallel to the applied stress axis, Hc⊥ is the coercive force measured in the direction perpendicular to the applied stress axis, Hc0 is the coercive force in the no-load state, A and B are the characterization constants of the crystal, and k1 and k2 are the parameters in the polycrystal proportional to the average of the magnetostriction constants λ100.When theσ value is large, the value of the exponential term is small, which can be ignored, and the coercive force is approximately linear with the stress.


### 2.3. Derivation of Magnetomechanical Relationship under Plane Stress State Based on Hooker’s Law

The Jiles–Atherton magnetomechanical effect theory and the V. F. Novikov coercive force-stress relation expression only consider the uniaxial stress state of the material, which explains the linear change of the coercive force with the uniaxial stress but ignores the dependence of the principal stress components in the classical elastic theory. Since the stress-strain always satisfiesσ=Eε, the change of material coercivity is assumed to be directly related to the strain [23], and equation (3) is modified.When the value in equation (3) is large, the value of the exponential term is small and can be ignored. Therefore, the coercive force is approximately linear with the stress [10, 24]. The uniaxial tensile specimens of X80 steel were loaded step by step, and the coercive force during loading was measured to verify the coercive force-stress relationship. The experimental results are shown in Figure 4. It can be seen that the coercive force is linearly related to the strain when the strain value exceeds a certain threshold value. In the uniaxial tensile test, the relationship between stress and strain is also linear; thus, the coercive force and stress are linear when the stress is large enough.Figure 4
Coercive force-strain relationship under uniaxial stress state.In the existing experiments, the coercivity-stress relationship of X80 steel is linear. Since the working stress of X80 steel is generally greater than the maximum stress applied in the experiment, the coercivity-stress relationship of X80 steel can be considered linear under the state of unidirectional stress. In actual measurements, the coercivity in one direction is considered:(4)Hcx=a1σx+b1,Hcy=a2σy+b2,where a1, a2, b1, and b2 are the slopes and intercepts in the linear expression of X80 steel, respectively, which can be measured by experiments.The expression of stress-strain relationship under plane stress state is derived from the generalized Hooke’s law:(5)σx=E1−ν2εx+νεy,σy=E1−ν2εy+νεx,where E is the modulus of elasticity of the material and ν is Poisson’s ratio of the material.In the plane stress state, equation (4) is no longer applicable because according to equation (5), the corresponding vertical stress and strain need to be known to obtain the principal stress in one direction. Ignoring this condition may lead to the error of stress determination and even the failure of stress detection.When the stress changes in only one direction, the relationship between transverse strainΔεy and axial strain Δεx is(6)Δεy=−νΔεx.Equation (6) explains why Hcx changes opposite to Hcy when the stress increases in one direction.Hooker’s law relates the stress and strain in two vertical directions by the superposition principle, and then the coercive force linear to stress can also be superposed. According to this idea, equation (3) can be superposed to obtain(7)Hc=B1+B2+A1e−β1σ1−A2e−β2σ2+B1k1σ1+B2k2σ2.Equation (7) indicates that the coercive force under two-way stress state is the superposition of the coercive force under two-way stress states. When the stress is large, the exponential term can be ignored, and the coercive force can be increased by the stress in both directions; when the stress is small, the primary term can be ignored, and the influence of the stress in both directions on the coercive force also depends on the crystal constant [25]. In this experiment, the loading gradient is 30 MPa, the exponential term in the formula can be ignored, and the coercive force is considered linear under this loading condition. Equation (4) is superposed on the basis of generalized Hook’s law.(8)σx=E1−ν2εx+νεy=11−ν2Hcx−b1a1+νHcy−b2a2,σy=E1−ν2εy+νεx=11−ν2Hcy−b2a2+νHcx−b1a1.Equation (8) is the relation expression of coercive force stress under the biaxial stress state. This formula explains the difference between coercive force-stress relations under biaxial and uniaxial stress states and proves that coercive force is associated with stress in both directions. However, the accuracy of the formula needs further experimental verification. The experimental concept of the stress-coercivity relationship of X80 steel under the biaxial stress state is given as follows.


## 2.1. Magnetic Domains and Magnetization Theory of Ferromagnetic Materials

When a ferromagnetic material is subjected to stress, the internal microstructure changes, the magnetic domain structure deflects, and the magnetic characteristic parameters vary significantly, which is called the force-magnetic coupling effect [17, 18]. Among them, the magnetic domain is a small area with spontaneous magnetization inside the ferromagnetic material, and the magnetic moment direction of each magnetic domain is different. The boundary between adjacent magnetic domains is called a domain wall, as shown in Figures 1 and 2.Figure 1
Direction of magnetic moment of ferromagnetic material in original state.Figure 2
Direction of magnetic moment of ferromagnetic material under tensile stress.The magnetization curve of the material is shown in Figure3. The horizontal axis represents the applied magnetic field strength; the vertical axis represents the induced magnetic field strength. The Om section is the initial magnetization curve, that is, the curve of the variation of the induced magnetic field with the applied magnetic field when the material is first magnetized in the initial state. When the external magnetic field is removed, the material cannot completely recover to the unmagnetized state due to the movement resistance of the domain wall, forming a hysteresis. The hysteresis causes the change of the magnetic induction intensity with the external magnetic field to lag behind the initial magnetization curve. For example, at the first demagnetization (mr segment), when the applied magnetic field strength is reduced to 0, the induced magnetic field strength of the material is not 0. The induced magnetic field strength represented by the intersection point Br of the mr segment and the y-axis is called remanence. When the applied magnetic field changes for one cycle, the magnetic induction intensity also changes to form a hysteresis loop (Bin Liu et al., 2015; [19]). From the hysteresis loop, the important parameters characterizing the hysteresis property of the material can be obtained [20]. When the material deforms due to the applied load, the magnetic domain structure inside the material varies, and the irreversible displacement of the domain wall occurs, resulting in changes in the material magnetic properties. The coercive force Hc represents the value of the applied magnetic field H when the magnetization M of the material is 0, and the coercive force changes accordingly with the magnetic domain structure; therefore, the stress state of the material can be evaluated by measuring the coercive force of the material ([21]; Leng et al. 2021).Figure 3
Schematic diagram of hysteresis loop.


## 2.2. Theory of Force-Magnetic Relation under One-Way Stress State

To study the effect of stress on magnetization, Jiles and Atherton established a force-magnetic coupling J-A model (Jiles and Atherton 1984; [5]) and proposed the concept of the effective field. The relationship between the magnetization intensity, the deformation, and elastic properties of materials in the model is(1)dMirrdσ=1ε2σMan−Mirr,ε=Eξ1/2,where Mirr is the irreversible magnetization intensity, Mrev is the reversible magnetization intensity, Man is the total magnetization intensity, ε is the stress size constant that can be determined by the test, and ξ is the unit volume energy coefficient.After differential transformation of equation (1), equation (2) is obtained:(2)dMirrMan−Mirr=1Eγξσdσ.Equation (2) demonstrates that a certain relationship exists between the magnetization and the stress. The magnetization M and the magnetic field strength H satisfy M=χH, where χ is the magnetic susceptibility. It can be considered that magnetic field strength H and stress are related; hence, the change law of stress can be obtained by exploring the magnetic field strength [22].Based on the force-magnetic coupling model, the relationship between coercive force and stress under the uniaxial stress was proposed by Novikov et al. [7]:(3)Hc∥=B1+A1e−β1σ+B1k1σ,B1=Hc0−A1,Hc⊥=B2−A2e−β2σ+B2k2σ,B2=Hc0+A2,where Hc∥ is the coercive force measured in the direction parallel to the applied stress axis, Hc⊥ is the coercive force measured in the direction perpendicular to the applied stress axis, Hc0 is the coercive force in the no-load state, A and B are the characterization constants of the crystal, and k1 and k2 are the parameters in the polycrystal proportional to the average of the magnetostriction constants λ100.When theσ value is large, the value of the exponential term is small, which can be ignored, and the coercive force is approximately linear with the stress.


## 2.3. Derivation of Magnetomechanical Relationship under Plane Stress State Based on Hooker’s Law

The Jiles–Atherton magnetomechanical effect theory and the V. F. Novikov coercive force-stress relation expression only consider the uniaxial stress state of the material, which explains the linear change of the coercive force with the uniaxial stress but ignores the dependence of the principal stress components in the classical elastic theory. Since the stress-strain always satisfiesσ=Eε, the change of material coercivity is assumed to be directly related to the strain [23], and equation (3) is modified.When the value in equation (3) is large, the value of the exponential term is small and can be ignored. Therefore, the coercive force is approximately linear with the stress [10, 24]. The uniaxial tensile specimens of X80 steel were loaded step by step, and the coercive force during loading was measured to verify the coercive force-stress relationship. The experimental results are shown in Figure 4. It can be seen that the coercive force is linearly related to the strain when the strain value exceeds a certain threshold value. In the uniaxial tensile test, the relationship between stress and strain is also linear; thus, the coercive force and stress are linear when the stress is large enough.Figure 4
Coercive force-strain relationship under uniaxial stress state.In the existing experiments, the coercivity-stress relationship of X80 steel is linear. Since the working stress of X80 steel is generally greater than the maximum stress applied in the experiment, the coercivity-stress relationship of X80 steel can be considered linear under the state of unidirectional stress. In actual measurements, the coercivity in one direction is considered:(4)Hcx=a1σx+b1,Hcy=a2σy+b2,where a1, a2, b1, and b2 are the slopes and intercepts in the linear expression of X80 steel, respectively, which can be measured by experiments.The expression of stress-strain relationship under plane stress state is derived from the generalized Hooke’s law:(5)σx=E1−ν2εx+νεy,σy=E1−ν2εy+νεx,where E is the modulus of elasticity of the material and ν is Poisson’s ratio of the material.In the plane stress state, equation (4) is no longer applicable because according to equation (5), the corresponding vertical stress and strain need to be known to obtain the principal stress in one direction. Ignoring this condition may lead to the error of stress determination and even the failure of stress detection.When the stress changes in only one direction, the relationship between transverse strainΔεy and axial strain Δεx is(6)Δεy=−νΔεx.Equation (6) explains why Hcx changes opposite to Hcy when the stress increases in one direction.Hooker’s law relates the stress and strain in two vertical directions by the superposition principle, and then the coercive force linear to stress can also be superposed. According to this idea, equation (3) can be superposed to obtain(7)Hc=B1+B2+A1e−β1σ1−A2e−β2σ2+B1k1σ1+B2k2σ2.Equation (7) indicates that the coercive force under two-way stress state is the superposition of the coercive force under two-way stress states. When the stress is large, the exponential term can be ignored, and the coercive force can be increased by the stress in both directions; when the stress is small, the primary term can be ignored, and the influence of the stress in both directions on the coercive force also depends on the crystal constant [25]. In this experiment, the loading gradient is 30 MPa, the exponential term in the formula can be ignored, and the coercive force is considered linear under this loading condition. Equation (4) is superposed on the basis of generalized Hook’s law.(8)σx=E1−ν2εx+νεy=11−ν2Hcx−b1a1+νHcy−b2a2,σy=E1−ν2εy+νεx=11−ν2Hcy−b2a2+νHcx−b1a1.Equation (8) is the relation expression of coercive force stress under the biaxial stress state. This formula explains the difference between coercive force-stress relations under biaxial and uniaxial stress states and proves that coercive force is associated with stress in both directions. However, the accuracy of the formula needs further experimental verification. The experimental concept of the stress-coercivity relationship of X80 steel under the biaxial stress state is given as follows.


## 3. Biaxial Tensile Test of X80 Steel

### 3.1. Coercive Force Measurement Principle

The methods of measuring coercivity include the open-circuit method and closed-circuit method. The closed-circuit measuring method is generally used to measure the coercivity of the ring core; the open-circuit measuring method is usually used to measure the coercivity of the strip specimen [26]. Compared with the open magnetic circuit method, the closed magnetic circuit method has negligible or no air gap, so the magnetic resistance in the magnetic circuit is smaller. In contrast, the open magnetic circuit method has an air gap, and the magnetic resistance of the magnetic circuit is relatively larger. The closed magnetic circuit measurement method is used in this study. Figure 5 shows its basic principle. The excitation coil and induction coil of certain turns are wrapped around the magnetic core. The magnetic core contacts the surface of the tested part to connect the excitation and induction circuits. Thus, the magnetic core and the tested part form a closed magnetic circuit. When an alternating signal passes through the excitation coil, a changing magnetic field is generated in the closed magnetic circuit. According to Faraday’s law of electromagnetic induction, when the magnetic flux and field intensity change, the induced current appears in the induction coil. The electrical signals from both excitation and induction ends are connected to the display system; hence, when the signal at the excitation end changes for one cycle, the complete hysteresis loop of the sample can be displayed.Figure 5
Schematic diagram of coercive force measurement by the yoke method.


### 3.2. Introduction to Experimental Equipment and Flow

#### 3.2.1. DC Drive Coercive Force Measurement System

Independently developed high-precision DC-driven coercive force measurement equipment that measures the hysteresis loop and the magnetic parameters on the loop by electromagnetic scanning was adopted for the coercive force measurements, as shown in Figure6.Figure 6
DC drive coercive force measurement system.The measurement system can measure the following parameters of a soft magnetic material: coercive forceHC, residual magnetism Br, saturation magnetic induction Bs, initial permeability μi, and maximum permeability μm. The shape of the tested sample is mainly annular; however, after being connected to other auxiliary measuring devices such as a solenoid or magnetometer, the samples of various shapes such as a rod shape or strip shape can also be possibly tested.During the measurement, the U-shaped detection probe matched with the system is placed on the surface of the tested pipeline to form a closed magnetic circuit, as shown in Figure7. The magnetic loop equation of the closed magnetic circuit can be written as(9)N1I=HL,Figure 7
Schematic diagram of closed magnetic circuit.whereI is the intensity of the excitation current, N1 is the number of turns of the excitation coil, H is the intensity of the excitation magnetic field, and L is the average length of the measured magnetic circuit.The excitation power module generates a DC gradual signal with a period of 8 seconds so that the magnetic field in the closed magnetic circuit can change periodically, resulting in the change of the magnetic flux in the induction coil. According to the electromagnetic induction law, the corresponding induced electromotive force is generated in the coil. The induced electromotive force signal in the induction coil is transmitted to the data acquisition module through a filtering process, and the processed data are sent to the display module. The display module describes the current-excitation magnetic field diagram and hysteresis loop diagram of the tested part, and the coercive force value is calculated according to the hysteresis loop, as shown in Figures8 and 9, respectively.Figure 8
Waveform of excitation current-excitation magnetic field.Figure 9
Measurement results of hysteresis loop.


#### 3.2.2. Biaxial Tensile Test Piece

The coercive force of X80 steel under the plane stress state is mainly considered in the design experiment. The biaxial tensile test piece made of self-designed X80 pipeline steel was used in the experiment. Figure10(a) shows the shape of the test piece. Width of biaxial tensile test piece is 500 mm, and thickness is 10 mm. After a series of comparisons, it is difficult to obtain a large enough stress value if the specimen is too thick, and the strain linearity is poor if the specimen is too thin. Finally, a 10 mm thick steel plate is selected to make the specimen. The test piece is 500 mm wide to reserve enough flat area for measurement after being clamped by the tensile testing machine. Figure 10(b) shows the stress distribution of the test piece after the symmetrical tensile force is applied by ANSYS software. The type of element used in the model is a steel plane element with a length of 0.01, and the symmetric constraint is used to apply tensile force in the loading hole of the steel plate.Figure 10
Stress distribution of specimen under loading. (a) Biaxial tensile test piece. (b) Stress distribution of test piece under loading.
(a)(b)The result of the finite element simulation indicates that the center area of the test piece can maintain a uniform stress state during the biaxial tensile process, which is a suitable measurement area to avoid the measurement error caused by uneven stress.In view of the stress concentration around the loading hole, it is difficult to accurately calculate the working stress of the measured area based on the external tensile force. Two groups of biaxial strain gauges are mounted on the back center of the measured area of the biaxial tensile test piece. The average of the strain gauge readings from the two groups is taken as the actual strain, and the stress value of the measurement area is obtained by calculating the strain and the elastic modulus of the test piece.


#### 3.2.3. Biaxial Tensile Tester

Figure11 shows the biaxial tensile testing machine used for loading. The biaxial tensile platform consists of a biaxial tensile device and compression device. During the measurement, the hydraulic pump exerts the tensile force on the biaxial tensile device. The biaxial tensile test platform can realize unidirectional and biaxial tensile. The biaxial tensile test piece is fastened on the tensile testing machine by 16 bolts; the dovetail groove and slider on the tensile testing machine exert force on the test piece through the bolts.Figure 11
Biaxial tensile tester. (a) Biaxial stretching device. (b) Pressurization device.
(a)(b)


#### 3.2.4. Experimental Process

In the experiment, the coercive force measurement direction aligns with the line connecting the two ends of the probe core, as shown in Figure12, and the connecting lines also coincide with the loading directions. The two mutually perpendicular loading directions are the X direction and Y direction, as marked on the test piece. During the measurement, the tensile force was applied to the specimen by stepwise loading, the load in the X direction was kept unchanged, and the coercive force changes in X and Y directions were recorded by stepwise loading in the Y direction. Then, the load in the Y direction was unchanged, and the load in the X direction was gradually applied. The process was repeated with the stress incremental of 30 MPa in both directions.Figure 12
Schematic diagram of coercive force measurement direction.


## 3.1. Coercive Force Measurement Principle

The methods of measuring coercivity include the open-circuit method and closed-circuit method. The closed-circuit measuring method is generally used to measure the coercivity of the ring core; the open-circuit measuring method is usually used to measure the coercivity of the strip specimen [26]. Compared with the open magnetic circuit method, the closed magnetic circuit method has negligible or no air gap, so the magnetic resistance in the magnetic circuit is smaller. In contrast, the open magnetic circuit method has an air gap, and the magnetic resistance of the magnetic circuit is relatively larger. The closed magnetic circuit measurement method is used in this study. Figure 5 shows its basic principle. The excitation coil and induction coil of certain turns are wrapped around the magnetic core. The magnetic core contacts the surface of the tested part to connect the excitation and induction circuits. Thus, the magnetic core and the tested part form a closed magnetic circuit. When an alternating signal passes through the excitation coil, a changing magnetic field is generated in the closed magnetic circuit. According to Faraday’s law of electromagnetic induction, when the magnetic flux and field intensity change, the induced current appears in the induction coil. The electrical signals from both excitation and induction ends are connected to the display system; hence, when the signal at the excitation end changes for one cycle, the complete hysteresis loop of the sample can be displayed.Figure 5
Schematic diagram of coercive force measurement by the yoke method.


## 3.2. Introduction to Experimental Equipment and Flow

### 3.2.1. DC Drive Coercive Force Measurement System

Independently developed high-precision DC-driven coercive force measurement equipment that measures the hysteresis loop and the magnetic parameters on the loop by electromagnetic scanning was adopted for the coercive force measurements, as shown in Figure6.Figure 6
DC drive coercive force measurement system.The measurement system can measure the following parameters of a soft magnetic material: coercive forceHC, residual magnetism Br, saturation magnetic induction Bs, initial permeability μi, and maximum permeability μm. The shape of the tested sample is mainly annular; however, after being connected to other auxiliary measuring devices such as a solenoid or magnetometer, the samples of various shapes such as a rod shape or strip shape can also be possibly tested.During the measurement, the U-shaped detection probe matched with the system is placed on the surface of the tested pipeline to form a closed magnetic circuit, as shown in Figure7. The magnetic loop equation of the closed magnetic circuit can be written as(9)N1I=HL,Figure 7
Schematic diagram of closed magnetic circuit.whereI is the intensity of the excitation current, N1 is the number of turns of the excitation coil, H is the intensity of the excitation magnetic field, and L is the average length of the measured magnetic circuit.The excitation power module generates a DC gradual signal with a period of 8 seconds so that the magnetic field in the closed magnetic circuit can change periodically, resulting in the change of the magnetic flux in the induction coil. According to the electromagnetic induction law, the corresponding induced electromotive force is generated in the coil. The induced electromotive force signal in the induction coil is transmitted to the data acquisition module through a filtering process, and the processed data are sent to the display module. The display module describes the current-excitation magnetic field diagram and hysteresis loop diagram of the tested part, and the coercive force value is calculated according to the hysteresis loop, as shown in Figures8 and 9, respectively.Figure 8
Waveform of excitation current-excitation magnetic field.Figure 9
Measurement results of hysteresis loop.


### 3.2.2. Biaxial Tensile Test Piece

The coercive force of X80 steel under the plane stress state is mainly considered in the design experiment. The biaxial tensile test piece made of self-designed X80 pipeline steel was used in the experiment. Figure10(a) shows the shape of the test piece. Width of biaxial tensile test piece is 500 mm, and thickness is 10 mm. After a series of comparisons, it is difficult to obtain a large enough stress value if the specimen is too thick, and the strain linearity is poor if the specimen is too thin. Finally, a 10 mm thick steel plate is selected to make the specimen. The test piece is 500 mm wide to reserve enough flat area for measurement after being clamped by the tensile testing machine. Figure 10(b) shows the stress distribution of the test piece after the symmetrical tensile force is applied by ANSYS software. The type of element used in the model is a steel plane element with a length of 0.01, and the symmetric constraint is used to apply tensile force in the loading hole of the steel plate.Figure 10
Stress distribution of specimen under loading. (a) Biaxial tensile test piece. (b) Stress distribution of test piece under loading.
(a)(b)The result of the finite element simulation indicates that the center area of the test piece can maintain a uniform stress state during the biaxial tensile process, which is a suitable measurement area to avoid the measurement error caused by uneven stress.In view of the stress concentration around the loading hole, it is difficult to accurately calculate the working stress of the measured area based on the external tensile force. Two groups of biaxial strain gauges are mounted on the back center of the measured area of the biaxial tensile test piece. The average of the strain gauge readings from the two groups is taken as the actual strain, and the stress value of the measurement area is obtained by calculating the strain and the elastic modulus of the test piece.


### 3.2.3. Biaxial Tensile Tester

Figure11 shows the biaxial tensile testing machine used for loading. The biaxial tensile platform consists of a biaxial tensile device and compression device. During the measurement, the hydraulic pump exerts the tensile force on the biaxial tensile device. The biaxial tensile test platform can realize unidirectional and biaxial tensile. The biaxial tensile test piece is fastened on the tensile testing machine by 16 bolts; the dovetail groove and slider on the tensile testing machine exert force on the test piece through the bolts.Figure 11
Biaxial tensile tester. (a) Biaxial stretching device. (b) Pressurization device.
(a)(b)


### 3.2.4. Experimental Process

In the experiment, the coercive force measurement direction aligns with the line connecting the two ends of the probe core, as shown in Figure12, and the connecting lines also coincide with the loading directions. The two mutually perpendicular loading directions are the X direction and Y direction, as marked on the test piece. During the measurement, the tensile force was applied to the specimen by stepwise loading, the load in the X direction was kept unchanged, and the coercive force changes in X and Y directions were recorded by stepwise loading in the Y direction. Then, the load in the Y direction was unchanged, and the load in the X direction was gradually applied. The process was repeated with the stress incremental of 30 MPa in both directions.Figure 12
Schematic diagram of coercive force measurement direction.


## 3.2.1. DC Drive Coercive Force Measurement System

Independently developed high-precision DC-driven coercive force measurement equipment that measures the hysteresis loop and the magnetic parameters on the loop by electromagnetic scanning was adopted for the coercive force measurements, as shown in Figure6.Figure 6
DC drive coercive force measurement system.The measurement system can measure the following parameters of a soft magnetic material: coercive forceHC, residual magnetism Br, saturation magnetic induction Bs, initial permeability μi, and maximum permeability μm. The shape of the tested sample is mainly annular; however, after being connected to other auxiliary measuring devices such as a solenoid or magnetometer, the samples of various shapes such as a rod shape or strip shape can also be possibly tested.During the measurement, the U-shaped detection probe matched with the system is placed on the surface of the tested pipeline to form a closed magnetic circuit, as shown in Figure7. The magnetic loop equation of the closed magnetic circuit can be written as(9)N1I=HL,Figure 7
Schematic diagram of closed magnetic circuit.whereI is the intensity of the excitation current, N1 is the number of turns of the excitation coil, H is the intensity of the excitation magnetic field, and L is the average length of the measured magnetic circuit.The excitation power module generates a DC gradual signal with a period of 8 seconds so that the magnetic field in the closed magnetic circuit can change periodically, resulting in the change of the magnetic flux in the induction coil. According to the electromagnetic induction law, the corresponding induced electromotive force is generated in the coil. The induced electromotive force signal in the induction coil is transmitted to the data acquisition module through a filtering process, and the processed data are sent to the display module. The display module describes the current-excitation magnetic field diagram and hysteresis loop diagram of the tested part, and the coercive force value is calculated according to the hysteresis loop, as shown in Figures8 and 9, respectively.Figure 8
Waveform of excitation current-excitation magnetic field.Figure 9
Measurement results of hysteresis loop.


## 3.2.2. Biaxial Tensile Test Piece

The coercive force of X80 steel under the plane stress state is mainly considered in the design experiment. The biaxial tensile test piece made of self-designed X80 pipeline steel was used in the experiment. Figure10(a) shows the shape of the test piece. Width of biaxial tensile test piece is 500 mm, and thickness is 10 mm. After a series of comparisons, it is difficult to obtain a large enough stress value if the specimen is too thick, and the strain linearity is poor if the specimen is too thin. Finally, a 10 mm thick steel plate is selected to make the specimen. The test piece is 500 mm wide to reserve enough flat area for measurement after being clamped by the tensile testing machine. Figure 10(b) shows the stress distribution of the test piece after the symmetrical tensile force is applied by ANSYS software. The type of element used in the model is a steel plane element with a length of 0.01, and the symmetric constraint is used to apply tensile force in the loading hole of the steel plate.Figure 10
Stress distribution of specimen under loading. (a) Biaxial tensile test piece. (b) Stress distribution of test piece under loading.
(a)(b)The result of the finite element simulation indicates that the center area of the test piece can maintain a uniform stress state during the biaxial tensile process, which is a suitable measurement area to avoid the measurement error caused by uneven stress.In view of the stress concentration around the loading hole, it is difficult to accurately calculate the working stress of the measured area based on the external tensile force. Two groups of biaxial strain gauges are mounted on the back center of the measured area of the biaxial tensile test piece. The average of the strain gauge readings from the two groups is taken as the actual strain, and the stress value of the measurement area is obtained by calculating the strain and the elastic modulus of the test piece.


## 3.2.3. Biaxial Tensile Tester

Figure11 shows the biaxial tensile testing machine used for loading. The biaxial tensile platform consists of a biaxial tensile device and compression device. During the measurement, the hydraulic pump exerts the tensile force on the biaxial tensile device. The biaxial tensile test platform can realize unidirectional and biaxial tensile. The biaxial tensile test piece is fastened on the tensile testing machine by 16 bolts; the dovetail groove and slider on the tensile testing machine exert force on the test piece through the bolts.Figure 11
Biaxial tensile tester. (a) Biaxial stretching device. (b) Pressurization device.
(a)(b)


## 3.2.4. Experimental Process

In the experiment, the coercive force measurement direction aligns with the line connecting the two ends of the probe core, as shown in Figure12, and the connecting lines also coincide with the loading directions. The two mutually perpendicular loading directions are the X direction and Y direction, as marked on the test piece. During the measurement, the tensile force was applied to the specimen by stepwise loading, the load in the X direction was kept unchanged, and the coercive force changes in X and Y directions were recorded by stepwise loading in the Y direction. Then, the load in the Y direction was unchanged, and the load in the X direction was gradually applied. The process was repeated with the stress incremental of 30 MPa in both directions.Figure 12
Schematic diagram of coercive force measurement direction.


## 4. Study on Coercive Force-Stress Relationship of X80 Steel under Biaxial Stress

### 4.1. Analysis of Experimental Data

The coercivity of two X80 steel specimens produced in the same batch was measured under the same experimental conditions to achieve the control effect. Considering the stress in bothX and Y directions and the coercive force as dependent variables, the collected experimental data were depicted by a three-dimensional graph X with MATLAB software. Figures 13 and 14 show the trends of the Y direction coercive force versus the biaxial stress. It can be seen that the coercive force changes obviously with the biaxial stress. The coercive force perpendicular to the loading direction increases with the stress, and the coercive force parallel to the loading direction decreases with the increase of the stress, which agrees with the coercive force-stress equation (8) extended by Hooke’s law. However, as shown in the image, the coercive force levels of the two specimens under the same stress state are at different intervals. The coercive force in both directions of specimen 1 is larger, which may be due to the nonuniformity of steel. Therefore, the specific quantitative relationship of a single specimen needs to be analyzed according to the specific experimental data.Figure 13
Three-dimensional diagram of test data of test piece 1. (a) Relation between coercive force and stress inX direction. (b) Relationship between coercive force and stress in Y direction.
(a)(b)Figure 14
Three-dimensional diagram of test data of test piece 2. (a)X direction coercive force. (b) Coercive force in Y direction.
(a)(b)Analyzing the coercive force-stress relationship under the plane stress state started with the cross section of the image to mainly consider the influence of single direction stress. The scheme of this paper is to select the relation ofHc−σy at σy = 0.120 and 240 MPa as an example to draw a plot and control the variable σx, while observing the influence of σy on the coercive force in both directions to study how the two stresses perpendicular to each other simultaneously affect the coercive force.


#### 4.1.1.Hc−σy Relationship of Test Piece 1 under Different σx Conditions

TheHcx−σy and Hcy−σy relationships of test piece 1 at different σx conditions are shown in Figures 15(a) and 15(b). The figures demonstrate that the coercive force perpendicular to the loading direction (Y direction) increases with the stress in the loading direction (X direction), while the coercive force parallel to the loading direction decreases. The stress of the second direction (X direction) affects the initial coercive force and then affects the overall level of the coercive force. The initial value of coercive force parallel to the direction of tensile force decreases, while the initial value of coercive force perpendicular to the direction of tensile force increases, which is inconsistent with the theory of the force-magnetic relation under the uniaxial stress state. The influence of two uniaxial stresses on coercive force cannot be simply superposed into the influence of bidirectional stress on coercive force. Therefore, it is necessary to explore the influence of bidirectional stress on coercive force from another perspective.A least-squares fit of theHc−σy curve for each X direction stress was performed to investigate how the stress level affects the coercivity. The above experimental data were subjected to a linear regression using the least-squares method. Table 1 provides the fit results, and Figure 15 shows the fit function image. The fitting results show that the slopes and intercepts of Hc−σy fitting line are different under different σx, indicating that the stress in the second direction greatly impacts the coercive force-stress relationship. Although Hc−σy is at different levels under different σx, the coercivity curves in both directions show the trend of translation with the change of σx. Specifically, σx is increased by 120 MPa, the initial value of coercive force in X direction is decreased by about 50∼100 Hc/A·m−1, and the initial value of coercive force in Y direction is increased by about 50 Hc/A·m−1. This shows that the variation of coercivity under the influence of single stress σy is consistent with the derivation of V. F. Novikov, but when the stresses in both directions change, the one-way force-magnetic coupling relationship is no longer accurate. Therefore, further study of how the plane stress affects the coercive force together is necessary.Table 1
Hc−σy fitting formula under different σx of test piece 1.
σx (MPa)Hcx−σyHcy−σyσx = 0Hcx = 0.53σy + 1608.6Hcy = −0.67σy + 1624.1σx = 120Hcx = 0.74σy + 1495.8Hcy = −0.77σy + 1680.8σx = 240Hcx = 0.57σy + 1469.1Hcy = −0.73σy + 1721.6Figure 15
Stress-coercivity relationship of specimen 1 under threeσx. (a) Hcx−σy diagram of test piece 1 under different σx. (b) Hcy−σy diagram of test piece 1 under different σx.
(a)(b)


#### 4.1.2.Hc−σy Relationship of Test Piece 2 under Different σx Conditions

TheHc−σy relationship of test piece 2 under different σx is shown in Figure 16. Similarly, a linear fitting was performed on the experimental data using the least-squares method, and the fitting results are given in Table 2. It can be seen from the figure that, under the same experimental steps, the change of coercive force with stress on test piece 2 is similar to the change on test piece 1, but the slopes of Hcx−σy and Hcy−σy relational graph of test piece 2 under different σx are lower than those of test piece 1. Moreover, the initial coercive force of test piece 2 is small under three fixed loads in the X direction. Therefore, the coercive force of test piece 2 is less sensitive to stress than that of test piece 1. However, since both test pieces of X80 pipeline steel were produced by the same manufacturer in the same batch, it implies that the steel unevenness causes the different coercive force levels in the 2 test pieces under the same stress. Therefore, it is essential to determine the coercive force sensitivity coefficient of different pipe sections in practical application for more accurate stress estimation results.Figure 16
Stress-coercivity relationship of test piece 2 under threeσx. (a) Hcx−σy diagram of test piece 2 under different σx. (b) Hcy−σy diagram of test piece 2 under different σx.
(a)(b)Table 2
Hc−σy fitting formula under different σx of test piece 2.
σx (MPa)Hcx−σyHcy−σyσx = 0Hcx = 0.36σy + 1072.4Hcy = 0.45σy + 1082.7σx = 120Hcx = 0.50σy + 997.1Hcy = 0.52σy + 1120.5σx = 240Hcx = 0.39σy + 979.4Hcy = 0.49σy + 1147.7


### 4.2. Binary Fitting of Coercive Force-Stress Relation

#### 4.2.1. Fitting Method

Based on the coercivity-stress formula extended by Hooke’s law and the experimental data image, it is assumed that the coercivity of X80 steel is only affected by the stresses inX and Y directions, and this effect is linear. From equation (8), equation (10) is(10)Hcx=kxσx+hxσy+Cx,Hcy=kyσx+hyσy+Cy,where Hcx and Hcy are the coercive force values in X and Y directions, respectively; σx and σy are the stresses in X and Y directions, respectively; kx, ky, hx, hy, Cx, and Cy are constants related to material properties, which are calculated according to actual measurement results.It can be seen from equation (8) that the coercive force stress is in binary one-order relation under a biaxial stress state, and the coercive force is affected by stress in two directions. The experimental results were fitted in MATLAB, and the degree of coincidence between one-time fitting, multiple-time fitting, and actual results were compared. Finally, the binary one-time model was selected to fit the experimental results, and the fitting effect is shown in Figures 17 and 18.Figure 17
One-time binary fitting of test data of test piece 1. (a) Coercive force fitting of test piece 1 inX direction Hcx=−0.4936σx+0.7491σy+1564.0. (b) Coercive force fitting of test piece 1 in Y direction Hcy=0.3916σx−0.7503σy+1630.0.
(a)(b)Figure 18
Binary one-time fitting of test piece experimental data. (a) Coercive force fitting of test piece 2 inX direction Hcx=−0.4944σx+0.3291σy+1043. (b) Coercive force fitting of test piece 2 in Y direction Hcy=0.2611σx−0.5002σy+1087.
(a)(b)As shown in the fitting effect diagram, using the binary primary model, the difference between the actual coercive force and the fitted value is small; thus, it can be considered that the coercive force and the plane stress approximately satisfy the binary primary relationship, and the experimental results validate the rationality of equations (5) and (6).Comparing the coercivity fitting formulas of Specimen 1 and Specimen 2, the coefficients corresponding to each stress direction and the intercepts of the two sets of formulas are not exactly the same; however, both sets of formulas show the same regularity. According to the previous derivation, the coefficient difference between the two sets of formulas is mainly due to the different Poisson’s ratios of the two specimens, caused by the inhomogeneity of materials.


#### 4.2.2. Error Analysis

Select two groups of data segments with large fitting error in the figure, that is,σx=0 and σx=180 coercive force fitting results, and verify them with the sectional verification method to calculate relative error and absolute error. The calculation results are given in Tables 3 and 4.Table 3
Error analysis of fitting results of test piece 1.
Serial no.σx (MPa)σy (MPa)Hcx (A/m)Hcx′ (A/m)ERRxa (A/m)ERRxr (%)Hcy (A/m)Hc'y (A/m)ERRya (A/m)ERRyr (%)1001618.657156454.6573.381634.42416304.4242.8220301629.6351586.47343.1622.651607.311607.4910.18150.1230601638.6121608.94629.6661.811590.1951584.9825.2133.3240901653.1851631.41921.7661.321563.1251562.4730.65150.41501201669.7581653.89215.8660.951516.0541539.96423.9115.22601501678.8251676.3652.460.151510.4541517.4557.00154.46701801687.8911698.838−10.947−0.651494.8531494.9460.0930.06802101715.3991721.311−5.912−0.341491.1161472.43718.67911.89902401764.3071743.78420.5231.161477.3791449.92827.45117.481018001497.1071475.15221.9551.471694.9091700.4885.5793.1011180301495.8111497.625−1.814−0.121675.6551677.9792.3241.2912180601494.5141520.098−25.584−1.711656.4011655.470.9310.5213180901520.8071542.571−21.764−1.431639.1011632.9616.13953.41141801201547.11565.044−17.944−1.161621.81610.45211.3486.30151801501609.251587.51721.7331.351589.31587.9431.3570.75161801801671.41609.9961.413.671556.81565.4348.6344.79171802101671.30251632.46338.83952.321535.5351542.9257.394.10181802401671.2051654.93616.2690.971514.271520.4166.1463.41Table 4
Error analysis of fitting results of test piece 2.
Serial no.σx (MPa)σy (MPa)Hcx (A/m)Hcx′ (A/m)ERRxa (A/m)ERRxr (%)Hcy (A/m)Hc'y (A/m)ERRya (A/m)ERRyr (%)1001079.1104336.13.351089.61087.52.10.1920301086.41052.933.53.081071.51072.4−0.9−0.0830601092.41062.729.72.721060.11057.42.70.2540901102.11072.629.52.681042.11042.5−0.4−0.04501201113.21082.530.72.761010.71027.4−16.7−1.65601501119.21092.426.82.3910071012.8−5.8−0.58701801125.31102.223.12.05996.6997.6−1.0−0.10802101143.61112.131.52.75994.1982.711.41.15902401176.2112254.24.61984.9967.217.71.80101800998.11132−133.9−13.421129.91134.5−4.6−0.411118030997.21141.9−144.7−14.511117.11119.6−2.5−0.221218060996.31151.7−155.4−15.601104.31104.8−0.5−0.0513180901013.91161.6−147.7−14.571092.71089.13.60.33141801201031.41171.5−140.1−13.581081.21074.56.70.62151801501072.81181.4−108.6−10.121059.510590.50.05161801801114.31191.2−76.9−6.901037.91044.5−6.6−0.64171802101114.21201.1−86.9−7.801023.71029.9−6.2−0.61181802401114.11211−96.9−8.701009.51014.6−5.1−0.51Absolute deviationERRa and relative deviation ERRr are calculated by(11)ERRxa=Hcx−Hcx′,ERRya=Hcy−Hcy′,ERRxr=Hcx−Hcx′Hcxmax−Hcxmin×100%,ERRyr=Hcy−Hcy′Hcymax−Hcymin×100%.According to Tables3 and 4, six out of the 72 fitting results are significantly different from the measured results, and the relative error of the other binary primary fitting is generally within 15%. The relative error of the well-fitting data segment is generally less than 5%, which further proves the correctness of equations (8) and (10). At the same time, it is noted that the fitting result of Hcx has larger relative error. It is speculated that the relative error is caused by the large residual stress inside the X direction after the test piece was made; the measurement error of the equipment itself and the position deviation of the probe during the measurement also account for part of the reasons.After the coefficients of equation (10) are determined, Hcx and Hcy are obtained by measurement, while σx and σy are obtained by solving the two-element primary equation.


## 4.1. Analysis of Experimental Data

The coercivity of two X80 steel specimens produced in the same batch was measured under the same experimental conditions to achieve the control effect. Considering the stress in bothX and Y directions and the coercive force as dependent variables, the collected experimental data were depicted by a three-dimensional graph X with MATLAB software. Figures 13 and 14 show the trends of the Y direction coercive force versus the biaxial stress. It can be seen that the coercive force changes obviously with the biaxial stress. The coercive force perpendicular to the loading direction increases with the stress, and the coercive force parallel to the loading direction decreases with the increase of the stress, which agrees with the coercive force-stress equation (8) extended by Hooke’s law. However, as shown in the image, the coercive force levels of the two specimens under the same stress state are at different intervals. The coercive force in both directions of specimen 1 is larger, which may be due to the nonuniformity of steel. Therefore, the specific quantitative relationship of a single specimen needs to be analyzed according to the specific experimental data.Figure 13
Three-dimensional diagram of test data of test piece 1. (a) Relation between coercive force and stress inX direction. (b) Relationship between coercive force and stress in Y direction.
(a)(b)Figure 14
Three-dimensional diagram of test data of test piece 2. (a)X direction coercive force. (b) Coercive force in Y direction.
(a)(b)Analyzing the coercive force-stress relationship under the plane stress state started with the cross section of the image to mainly consider the influence of single direction stress. The scheme of this paper is to select the relation ofHc−σy at σy = 0.120 and 240 MPa as an example to draw a plot and control the variable σx, while observing the influence of σy on the coercive force in both directions to study how the two stresses perpendicular to each other simultaneously affect the coercive force.


### 4.1.1.Hc−σy Relationship of Test Piece 1 under Different σx Conditions

TheHcx−σy and Hcy−σy relationships of test piece 1 at different σx conditions are shown in Figures 15(a) and 15(b). The figures demonstrate that the coercive force perpendicular to the loading direction (Y direction) increases with the stress in the loading direction (X direction), while the coercive force parallel to the loading direction decreases. The stress of the second direction (X direction) affects the initial coercive force and then affects the overall level of the coercive force. The initial value of coercive force parallel to the direction of tensile force decreases, while the initial value of coercive force perpendicular to the direction of tensile force increases, which is inconsistent with the theory of the force-magnetic relation under the uniaxial stress state. The influence of two uniaxial stresses on coercive force cannot be simply superposed into the influence of bidirectional stress on coercive force. Therefore, it is necessary to explore the influence of bidirectional stress on coercive force from another perspective.A least-squares fit of theHc−σy curve for each X direction stress was performed to investigate how the stress level affects the coercivity. The above experimental data were subjected to a linear regression using the least-squares method. Table 1 provides the fit results, and Figure 15 shows the fit function image. The fitting results show that the slopes and intercepts of Hc−σy fitting line are different under different σx, indicating that the stress in the second direction greatly impacts the coercive force-stress relationship. Although Hc−σy is at different levels under different σx, the coercivity curves in both directions show the trend of translation with the change of σx. Specifically, σx is increased by 120 MPa, the initial value of coercive force in X direction is decreased by about 50∼100 Hc/A·m−1, and the initial value of coercive force in Y direction is increased by about 50 Hc/A·m−1. This shows that the variation of coercivity under the influence of single stress σy is consistent with the derivation of V. F. Novikov, but when the stresses in both directions change, the one-way force-magnetic coupling relationship is no longer accurate. Therefore, further study of how the plane stress affects the coercive force together is necessary.Table 1
Hc−σy fitting formula under different σx of test piece 1.
σx (MPa)Hcx−σyHcy−σyσx = 0Hcx = 0.53σy + 1608.6Hcy = −0.67σy + 1624.1σx = 120Hcx = 0.74σy + 1495.8Hcy = −0.77σy + 1680.8σx = 240Hcx = 0.57σy + 1469.1Hcy = −0.73σy + 1721.6Figure 15
Stress-coercivity relationship of specimen 1 under threeσx. (a) Hcx−σy diagram of test piece 1 under different σx. (b) Hcy−σy diagram of test piece 1 under different σx.
(a)(b)


### 4.1.2.Hc−σy Relationship of Test Piece 2 under Different σx Conditions

TheHc−σy relationship of test piece 2 under different σx is shown in Figure 16. Similarly, a linear fitting was performed on the experimental data using the least-squares method, and the fitting results are given in Table 2. It can be seen from the figure that, under the same experimental steps, the change of coercive force with stress on test piece 2 is similar to the change on test piece 1, but the slopes of Hcx−σy and Hcy−σy relational graph of test piece 2 under different σx are lower than those of test piece 1. Moreover, the initial coercive force of test piece 2 is small under three fixed loads in the X direction. Therefore, the coercive force of test piece 2 is less sensitive to stress than that of test piece 1. However, since both test pieces of X80 pipeline steel were produced by the same manufacturer in the same batch, it implies that the steel unevenness causes the different coercive force levels in the 2 test pieces under the same stress. Therefore, it is essential to determine the coercive force sensitivity coefficient of different pipe sections in practical application for more accurate stress estimation results.Figure 16
Stress-coercivity relationship of test piece 2 under threeσx. (a) Hcx−σy diagram of test piece 2 under different σx. (b) Hcy−σy diagram of test piece 2 under different σx.
(a)(b)Table 2
Hc−σy fitting formula under different σx of test piece 2.
σx (MPa)Hcx−σyHcy−σyσx = 0Hcx = 0.36σy + 1072.4Hcy = 0.45σy + 1082.7σx = 120Hcx = 0.50σy + 997.1Hcy = 0.52σy + 1120.5σx = 240Hcx = 0.39σy + 979.4Hcy = 0.49σy + 1147.7


## 4.1.1.Hc−σy Relationship of Test Piece 1 under Different σx Conditions

TheHcx−σy and Hcy−σy relationships of test piece 1 at different σx conditions are shown in Figures 15(a) and 15(b). The figures demonstrate that the coercive force perpendicular to the loading direction (Y direction) increases with the stress in the loading direction (X direction), while the coercive force parallel to the loading direction decreases. The stress of the second direction (X direction) affects the initial coercive force and then affects the overall level of the coercive force. The initial value of coercive force parallel to the direction of tensile force decreases, while the initial value of coercive force perpendicular to the direction of tensile force increases, which is inconsistent with the theory of the force-magnetic relation under the uniaxial stress state. The influence of two uniaxial stresses on coercive force cannot be simply superposed into the influence of bidirectional stress on coercive force. Therefore, it is necessary to explore the influence of bidirectional stress on coercive force from another perspective.A least-squares fit of theHc−σy curve for each X direction stress was performed to investigate how the stress level affects the coercivity. The above experimental data were subjected to a linear regression using the least-squares method. Table 1 provides the fit results, and Figure 15 shows the fit function image. The fitting results show that the slopes and intercepts of Hc−σy fitting line are different under different σx, indicating that the stress in the second direction greatly impacts the coercive force-stress relationship. Although Hc−σy is at different levels under different σx, the coercivity curves in both directions show the trend of translation with the change of σx. Specifically, σx is increased by 120 MPa, the initial value of coercive force in X direction is decreased by about 50∼100 Hc/A·m−1, and the initial value of coercive force in Y direction is increased by about 50 Hc/A·m−1. This shows that the variation of coercivity under the influence of single stress σy is consistent with the derivation of V. F. Novikov, but when the stresses in both directions change, the one-way force-magnetic coupling relationship is no longer accurate. Therefore, further study of how the plane stress affects the coercive force together is necessary.Table 1
Hc−σy fitting formula under different σx of test piece 1.
σx (MPa)Hcx−σyHcy−σyσx = 0Hcx = 0.53σy + 1608.6Hcy = −0.67σy + 1624.1σx = 120Hcx = 0.74σy + 1495.8Hcy = −0.77σy + 1680.8σx = 240Hcx = 0.57σy + 1469.1Hcy = −0.73σy + 1721.6Figure 15
Stress-coercivity relationship of specimen 1 under threeσx. (a) Hcx−σy diagram of test piece 1 under different σx. (b) Hcy−σy diagram of test piece 1 under different σx.
(a)(b)


## 4.1.2.Hc−σy Relationship of Test Piece 2 under Different σx Conditions

TheHc−σy relationship of test piece 2 under different σx is shown in Figure 16. Similarly, a linear fitting was performed on the experimental data using the least-squares method, and the fitting results are given in Table 2. It can be seen from the figure that, under the same experimental steps, the change of coercive force with stress on test piece 2 is similar to the change on test piece 1, but the slopes of Hcx−σy and Hcy−σy relational graph of test piece 2 under different σx are lower than those of test piece 1. Moreover, the initial coercive force of test piece 2 is small under three fixed loads in the X direction. Therefore, the coercive force of test piece 2 is less sensitive to stress than that of test piece 1. However, since both test pieces of X80 pipeline steel were produced by the same manufacturer in the same batch, it implies that the steel unevenness causes the different coercive force levels in the 2 test pieces under the same stress. Therefore, it is essential to determine the coercive force sensitivity coefficient of different pipe sections in practical application for more accurate stress estimation results.Figure 16
Stress-coercivity relationship of test piece 2 under threeσx. (a) Hcx−σy diagram of test piece 2 under different σx. (b) Hcy−σy diagram of test piece 2 under different σx.
(a)(b)Table 2
Hc−σy fitting formula under different σx of test piece 2.
σx (MPa)Hcx−σyHcy−σyσx = 0Hcx = 0.36σy + 1072.4Hcy = 0.45σy + 1082.7σx = 120Hcx = 0.50σy + 997.1Hcy = 0.52σy + 1120.5σx = 240Hcx = 0.39σy + 979.4Hcy = 0.49σy + 1147.7


## 4.2. Binary Fitting of Coercive Force-Stress Relation

### 4.2.1. Fitting Method

Based on the coercivity-stress formula extended by Hooke’s law and the experimental data image, it is assumed that the coercivity of X80 steel is only affected by the stresses inX and Y directions, and this effect is linear. From equation (8), equation (10) is(10)Hcx=kxσx+hxσy+Cx,Hcy=kyσx+hyσy+Cy,where Hcx and Hcy are the coercive force values in X and Y directions, respectively; σx and σy are the stresses in X and Y directions, respectively; kx, ky, hx, hy, Cx, and Cy are constants related to material properties, which are calculated according to actual measurement results.It can be seen from equation (8) that the coercive force stress is in binary one-order relation under a biaxial stress state, and the coercive force is affected by stress in two directions. The experimental results were fitted in MATLAB, and the degree of coincidence between one-time fitting, multiple-time fitting, and actual results were compared. Finally, the binary one-time model was selected to fit the experimental results, and the fitting effect is shown in Figures 17 and 18.Figure 17
One-time binary fitting of test data of test piece 1. (a) Coercive force fitting of test piece 1 inX direction Hcx=−0.4936σx+0.7491σy+1564.0. (b) Coercive force fitting of test piece 1 in Y direction Hcy=0.3916σx−0.7503σy+1630.0.
(a)(b)Figure 18
Binary one-time fitting of test piece experimental data. (a) Coercive force fitting of test piece 2 inX direction Hcx=−0.4944σx+0.3291σy+1043. (b) Coercive force fitting of test piece 2 in Y direction Hcy=0.2611σx−0.5002σy+1087.
(a)(b)As shown in the fitting effect diagram, using the binary primary model, the difference between the actual coercive force and the fitted value is small; thus, it can be considered that the coercive force and the plane stress approximately satisfy the binary primary relationship, and the experimental results validate the rationality of equations (5) and (6).Comparing the coercivity fitting formulas of Specimen 1 and Specimen 2, the coefficients corresponding to each stress direction and the intercepts of the two sets of formulas are not exactly the same; however, both sets of formulas show the same regularity. According to the previous derivation, the coefficient difference between the two sets of formulas is mainly due to the different Poisson’s ratios of the two specimens, caused by the inhomogeneity of materials.


### 4.2.2. Error Analysis

Select two groups of data segments with large fitting error in the figure, that is,σx=0 and σx=180 coercive force fitting results, and verify them with the sectional verification method to calculate relative error and absolute error. The calculation results are given in Tables 3 and 4.Table 3
Error analysis of fitting results of test piece 1.
Serial no.σx (MPa)σy (MPa)Hcx (A/m)Hcx′ (A/m)ERRxa (A/m)ERRxr (%)Hcy (A/m)Hc'y (A/m)ERRya (A/m)ERRyr (%)1001618.657156454.6573.381634.42416304.4242.8220301629.6351586.47343.1622.651607.311607.4910.18150.1230601638.6121608.94629.6661.811590.1951584.9825.2133.3240901653.1851631.41921.7661.321563.1251562.4730.65150.41501201669.7581653.89215.8660.951516.0541539.96423.9115.22601501678.8251676.3652.460.151510.4541517.4557.00154.46701801687.8911698.838−10.947−0.651494.8531494.9460.0930.06802101715.3991721.311−5.912−0.341491.1161472.43718.67911.89902401764.3071743.78420.5231.161477.3791449.92827.45117.481018001497.1071475.15221.9551.471694.9091700.4885.5793.1011180301495.8111497.625−1.814−0.121675.6551677.9792.3241.2912180601494.5141520.098−25.584−1.711656.4011655.470.9310.5213180901520.8071542.571−21.764−1.431639.1011632.9616.13953.41141801201547.11565.044−17.944−1.161621.81610.45211.3486.30151801501609.251587.51721.7331.351589.31587.9431.3570.75161801801671.41609.9961.413.671556.81565.4348.6344.79171802101671.30251632.46338.83952.321535.5351542.9257.394.10181802401671.2051654.93616.2690.971514.271520.4166.1463.41Table 4
Error analysis of fitting results of test piece 2.
Serial no.σx (MPa)σy (MPa)Hcx (A/m)Hcx′ (A/m)ERRxa (A/m)ERRxr (%)Hcy (A/m)Hc'y (A/m)ERRya (A/m)ERRyr (%)1001079.1104336.13.351089.61087.52.10.1920301086.41052.933.53.081071.51072.4−0.9−0.0830601092.41062.729.72.721060.11057.42.70.2540901102.11072.629.52.681042.11042.5−0.4−0.04501201113.21082.530.72.761010.71027.4−16.7−1.65601501119.21092.426.82.3910071012.8−5.8−0.58701801125.31102.223.12.05996.6997.6−1.0−0.10802101143.61112.131.52.75994.1982.711.41.15902401176.2112254.24.61984.9967.217.71.80101800998.11132−133.9−13.421129.91134.5−4.6−0.411118030997.21141.9−144.7−14.511117.11119.6−2.5−0.221218060996.31151.7−155.4−15.601104.31104.8−0.5−0.0513180901013.91161.6−147.7−14.571092.71089.13.60.33141801201031.41171.5−140.1−13.581081.21074.56.70.62151801501072.81181.4−108.6−10.121059.510590.50.05161801801114.31191.2−76.9−6.901037.91044.5−6.6−0.64171802101114.21201.1−86.9−7.801023.71029.9−6.2−0.61181802401114.11211−96.9−8.701009.51014.6−5.1−0.51Absolute deviationERRa and relative deviation ERRr are calculated by(11)ERRxa=Hcx−Hcx′,ERRya=Hcy−Hcy′,ERRxr=Hcx−Hcx′Hcxmax−Hcxmin×100%,ERRyr=Hcy−Hcy′Hcymax−Hcymin×100%.According to Tables3 and 4, six out of the 72 fitting results are significantly different from the measured results, and the relative error of the other binary primary fitting is generally within 15%. The relative error of the well-fitting data segment is generally less than 5%, which further proves the correctness of equations (8) and (10). At the same time, it is noted that the fitting result of Hcx has larger relative error. It is speculated that the relative error is caused by the large residual stress inside the X direction after the test piece was made; the measurement error of the equipment itself and the position deviation of the probe during the measurement also account for part of the reasons.After the coefficients of equation (10) are determined, Hcx and Hcy are obtained by measurement, while σx and σy are obtained by solving the two-element primary equation.


## 4.2.1. Fitting Method

Based on the coercivity-stress formula extended by Hooke’s law and the experimental data image, it is assumed that the coercivity of X80 steel is only affected by the stresses inX and Y directions, and this effect is linear. From equation (8), equation (10) is(10)Hcx=kxσx+hxσy+Cx,Hcy=kyσx+hyσy+Cy,where Hcx and Hcy are the coercive force values in X and Y directions, respectively; σx and σy are the stresses in X and Y directions, respectively; kx, ky, hx, hy, Cx, and Cy are constants related to material properties, which are calculated according to actual measurement results.It can be seen from equation (8) that the coercive force stress is in binary one-order relation under a biaxial stress state, and the coercive force is affected by stress in two directions. The experimental results were fitted in MATLAB, and the degree of coincidence between one-time fitting, multiple-time fitting, and actual results were compared. Finally, the binary one-time model was selected to fit the experimental results, and the fitting effect is shown in Figures 17 and 18.Figure 17
One-time binary fitting of test data of test piece 1. (a) Coercive force fitting of test piece 1 inX direction Hcx=−0.4936σx+0.7491σy+1564.0. (b) Coercive force fitting of test piece 1 in Y direction Hcy=0.3916σx−0.7503σy+1630.0.
(a)(b)Figure 18
Binary one-time fitting of test piece experimental data. (a) Coercive force fitting of test piece 2 inX direction Hcx=−0.4944σx+0.3291σy+1043. (b) Coercive force fitting of test piece 2 in Y direction Hcy=0.2611σx−0.5002σy+1087.
(a)(b)As shown in the fitting effect diagram, using the binary primary model, the difference between the actual coercive force and the fitted value is small; thus, it can be considered that the coercive force and the plane stress approximately satisfy the binary primary relationship, and the experimental results validate the rationality of equations (5) and (6).Comparing the coercivity fitting formulas of Specimen 1 and Specimen 2, the coefficients corresponding to each stress direction and the intercepts of the two sets of formulas are not exactly the same; however, both sets of formulas show the same regularity. According to the previous derivation, the coefficient difference between the two sets of formulas is mainly due to the different Poisson’s ratios of the two specimens, caused by the inhomogeneity of materials.


## 4.2.2. Error Analysis

Select two groups of data segments with large fitting error in the figure, that is,σx=0 and σx=180 coercive force fitting results, and verify them with the sectional verification method to calculate relative error and absolute error. The calculation results are given in Tables 3 and 4.Table 3
Error analysis of fitting results of test piece 1.
Serial no.σx (MPa)σy (MPa)Hcx (A/m)Hcx′ (A/m)ERRxa (A/m)ERRxr (%)Hcy (A/m)Hc'y (A/m)ERRya (A/m)ERRyr (%)1001618.657156454.6573.381634.42416304.4242.8220301629.6351586.47343.1622.651607.311607.4910.18150.1230601638.6121608.94629.6661.811590.1951584.9825.2133.3240901653.1851631.41921.7661.321563.1251562.4730.65150.41501201669.7581653.89215.8660.951516.0541539.96423.9115.22601501678.8251676.3652.460.151510.4541517.4557.00154.46701801687.8911698.838−10.947−0.651494.8531494.9460.0930.06802101715.3991721.311−5.912−0.341491.1161472.43718.67911.89902401764.3071743.78420.5231.161477.3791449.92827.45117.481018001497.1071475.15221.9551.471694.9091700.4885.5793.1011180301495.8111497.625−1.814−0.121675.6551677.9792.3241.2912180601494.5141520.098−25.584−1.711656.4011655.470.9310.5213180901520.8071542.571−21.764−1.431639.1011632.9616.13953.41141801201547.11565.044−17.944−1.161621.81610.45211.3486.30151801501609.251587.51721.7331.351589.31587.9431.3570.75161801801671.41609.9961.413.671556.81565.4348.6344.79171802101671.30251632.46338.83952.321535.5351542.9257.394.10181802401671.2051654.93616.2690.971514.271520.4166.1463.41Table 4
Error analysis of fitting results of test piece 2.
Serial no.σx (MPa)σy (MPa)Hcx (A/m)Hcx′ (A/m)ERRxa (A/m)ERRxr (%)Hcy (A/m)Hc'y (A/m)ERRya (A/m)ERRyr (%)1001079.1104336.13.351089.61087.52.10.1920301086.41052.933.53.081071.51072.4−0.9−0.0830601092.41062.729.72.721060.11057.42.70.2540901102.11072.629.52.681042.11042.5−0.4−0.04501201113.21082.530.72.761010.71027.4−16.7−1.65601501119.21092.426.82.3910071012.8−5.8−0.58701801125.31102.223.12.05996.6997.6−1.0−0.10802101143.61112.131.52.75994.1982.711.41.15902401176.2112254.24.61984.9967.217.71.80101800998.11132−133.9−13.421129.91134.5−4.6−0.411118030997.21141.9−144.7−14.511117.11119.6−2.5−0.221218060996.31151.7−155.4−15.601104.31104.8−0.5−0.0513180901013.91161.6−147.7−14.571092.71089.13.60.33141801201031.41171.5−140.1−13.581081.21074.56.70.62151801501072.81181.4−108.6−10.121059.510590.50.05161801801114.31191.2−76.9−6.901037.91044.5−6.6−0.64171802101114.21201.1−86.9−7.801023.71029.9−6.2−0.61181802401114.11211−96.9−8.701009.51014.6−5.1−0.51Absolute deviationERRa and relative deviation ERRr are calculated by(11)ERRxa=Hcx−Hcx′,ERRya=Hcy−Hcy′,ERRxr=Hcx−Hcx′Hcxmax−Hcxmin×100%,ERRyr=Hcy−Hcy′Hcymax−Hcymin×100%.According to Tables3 and 4, six out of the 72 fitting results are significantly different from the measured results, and the relative error of the other binary primary fitting is generally within 15%. The relative error of the well-fitting data segment is generally less than 5%, which further proves the correctness of equations (8) and (10). At the same time, it is noted that the fitting result of Hcx has larger relative error. It is speculated that the relative error is caused by the large residual stress inside the X direction after the test piece was made; the measurement error of the equipment itself and the position deviation of the probe during the measurement also account for part of the reasons.After the coefficients of equation (10) are determined, Hcx and Hcy are obtained by measurement, while σx and σy are obtained by solving the two-element primary equation.


## 5. Conclusions and Prospects

Based on the generalized Hooke’s law, the coercivity-stress relation in the uniaxial stress state is extended to the plane stress state. The coercive force-stress measurement experiment of X80 steel under the biaxial stress state was conducted using the self-built biaxial tensile test device and DC-driven coercive force measuring device. The experimental law is summarized based on the derivation formula. The influence of biaxial stress on the coercive force is linear, and the influence coefficient is related to the material Poisson’s ratio. In the direction of stress increase, the cocoercive force increases, and the vertical coercive force decreases.The coercivity-stress relationship model of X80 steel under the biaxial stress state was established in this paper based on the experimental data and the derivation. The fitting results of the binary-first-order model agree well with the experimental data, which verifies the correctness of the derivation. It is also shown that the coercive force method can accurately measure the stress level of X80 pipeline steel and is a reliable method to evaluate the safety margin of in-service pipelines.Many factors such as temperature, material inhomogeneity, and residual stress impact the coercive force measurement. Combined with other mature measurement methods, the causes of errors were further analyzed, the accuracy of the coercive force measurement equipment was improved, the influence of random errors during the measurement was eliminated, and the model accuracy was further improved. Since the data with large error occasionally appears in the measurement, more experiments need to be done in the future.



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*Source: 2510505-2022-06-29.xml*