Effect of Magnetic Excitation Intensity on Stress Recognition and Quantitative Evaluation in Ferromagnetic Pipelines

Abstract

Stress detection is an effective way to prevent pipeline failure, but stress recognition alone can hardly meet the safety and maintenance requirements of pipelines. Rather, improving the accuracy of stress detection and quantification has long been a top priority in the engineering sector. In the present study, stress detection models for pipelines were developed under varying magnetic excitation intensities, and the influence of a changing magnetic excitation field on stress recognition capacity was investigated. The variation law of the accuracy of stress detection under different excitation intensities was determined and validated experimentally. The results showed that at an excitation intensity of 2.5 of kA/m, the polarity of weak magnetic signals flipped when used to detect stress below 40 MPa, making the stress quantification difficult. The stress recognition capacity was the greatest under an excitation intensity of 7.5 kA/m for the stress below 40 MPa and the greatest under an excitation intensity of 5 kA/m for the stress of 40–160 MPa. Our research findings offer theoretical clues for choosing an appropriate excitation intensity for stress detection. The findings provide technical support for pipeline integrity assessment and risk warning, playing an important role in ensuring the safe operation of oil and gas transportation systems.

1. Introduction

Pipelines are one of the major pathways for oil and gas transport throughout the world and exhibit the advantages of large transport volume, freedom from restrictions caused by climate and ground surface factors, continuous operation ability, and low cost [1,2,3,4]. As a primary mode of energy transportation, pipelines are considered an integral part of the national economy and social security [5]. Failure is the most hazardous situation for newly built and in-service pipelines, and stress corrosion induced by stress concentration is one of the most important causes of pipeline failure [6,7]. Stress detection has become a heatedly discussed issue among pipeline inspectors in recent years due to its importance for emergency alerts and eliminating potential risks in advance [8,9,10]. Given the critical role of stress detection in pipeline integrity assessment, it is necessary to systematically compare the existing typical stress-measurement methods to clarify their applicability and limitations (see

Table 1. Comparison of typical stress-measurement methods and their applicability to in-service pipeline inspection.

Method	Principle/Quantity Measured	Advantages	Limitations	Applicability
Hole-drilling	Local strain relief → residual stress	Quantitative, standardized	Semi-destructive, low efficiency	Not suitable for long-distance pipelines
X-ray diffraction (XRD)	Lattice strain → near-surface stress	High accuracy	Shallow penetration, surface preparation required	Limited for field/on-pipe inspection
Neutron diffraction	Lattice strain → bulk stress	Deep penetration	Very expensive, non-portable	Not applicable to engineering practice
Ultrasonic acoustoelastic	Wave-velocity change → stress	Non-destructive, deeper probing	Sensitive to coupling/temperature; slow scanning	Useful for local welds
Magnetic Barkhausen noise	Barkhausen activity → near-surface stress & microstructure	High sensitivity; portable	Needs excitation; surface preparation; shallow depth	Not for long-distance pipeline inspection
MMMT	Stress-induced residual magnetic field	No excitation needed; early-damage sensitive	Strongly interfered; poor repeatability; weak quantification	Screening only; low confidence
WMD	Controlled low-field excitation → stress-induced magnetic response	Non-contact, coating-free, stable signals, suitable for long-range scanning	Calibration required; affected by lift-off/material	Highly suitable for in-service pipelines (used in this work)

). Studies have shown that the degree of material magnetization has a significant influence on the sensitivity and stability of stress-related magnetic signals [11]. Weak magnetic detection (WMD) emphasizes the existence of an optimal low-field intensity range, within which magnetic domain wall motion is more active, and stress sensitivity reaches its maximum [12,13]. Related research has mainly focused on the optimization of excitation parameters in magnetic flux leakage (MFL) and weak magnetic stress detection [14]. Some research teams have developed adjustable excitation systems to enhance adaptability under different steel grades and residual stress conditions and have proposed quantitative calibration models describing the relationship between excitation intensity and stress-induced magnetic signal variations [15]. In the earlier years of stress detection, Dubov, the Russian researcher, proposed metal magnetic memory testing (MMMT), which detects stress in ferromagnetic materials by analyzing changes in their magnetic properties caused by stress and the Earth’s magnetic field [16,17,18]. As MMMT is being more intensively applied and studied, some defects with this technique have become widely recognized. For example, poor anti-interference ability, low stability, and low reproducibility of magnetic signals in the presence of the Earth’s magnetic field. These problems have resulted in a low degree of confidence when using magnetic memory signals for stress recognition. It is hardly possible to reach a confidence interval above 90%, as otherwise expected from the MFL detection [19,20,21,22,23]. WMD has been proposed in recent years for stress recognition, offering improved performance [24,25,26,27]. A weak magnetic excitation is applied externally to avoid the influence of the Earth’s magnetic field on stress-induced magnetic signals. Since an external excitation field has a stable and adjustable intensity, the weak magnetic signals carrying stress information are typically more stable and reproducible, with a higher confidence interval. This finally increases the accuracy for recognizing stress concentration areas [28,29,30].

Higher requirements are usually placed on WMD for stress recognition, since a higher stress detection accuracy is of great importance for pipeline safety evaluation [31,32,33,34]. To improve the accuracy of WMD for stress recognition, we studied the variation law of the pipeline’s magnetic parameters under different excitation intensities and built analytical models for weak magnetic signals. The variation pattern of the characteristics of weak magnetic signals with stress under different excitation intensities was analyzed. Changes in the accuracy of stress detection under different excitation intensities were observed. The research findings can be used to inform stress detection in pipelines with varying diameters based on weak magnetic signals.

The main methodological contributions of this work can be summarized as follows:

• A stress-dependent magneto–mechanical coupling model is developed by introducing an equivalent stress-induced field into the excitation and hysteretic magnetization processes. The resulting M(H, σ) curves under different excitation intensities form the basis for analyzing stress-related weak-magnetic signals.
• An analytical weak-magnetic field model for stress-concentrated regions is established. By modeling the stressed zone as a cuboid with stress-dependent equivalent magnetic charge density, explicit tangential and normal field expressions are obtained, enabling analytical mapping from stress level to signal features.
• A quantitative accuracy-evaluation method is proposed by combining the signal resolution ϕ with the ambient magnetic field H₀. These indices allow systematic assessment of how excitation intensity influences stress recognition and help identify optimal excitation intervals.
• An engineering-oriented excitation selection procedure is formulated by integrating material characterization, excitation design and calibration. This provides practical guidance for applying weak-magnetic stress detection in pipeline integrity assessment.

Although the external excitation field in WMD is controllable and independent of the geomagnetic field, the measured weak-magnetic signals can still be distorted in practice due to environmental magnetic noise, excitation-field non-uniformity near bends or welds, magnetization history, residual stress, and probe lift-off variations. In engineering applications, these effects are commonly mitigated through differential multi-component sensor arrays, background-field calibration, shielding or compensation coils, and digital filtering techniques. This study, conducted under controlled laboratory conditions, primarily investigates how excitation intensity affects stress-related signal amplitude and detection accuracy. More comprehensive modeling and compensation of field distortion in complex pipeline environments will be addressed in future work.

2. Weak Magnetic Stress Detection Model

2.1. Force-Magnetic Coupling Model

The weak magnetic signals are influenced by the pipelines’ permeability and magnetization intensity. In other words, the magnetization curves of the pipelines under different excitation intensities determine the variation pattern of the detected signals. It is necessary to analyze the magnetization curves of the pipelines under different stresses to reveal the variation pattern of the weak magnetic signals used to detect varying stress levels [35,36]. According to Sablik et al. [37], the non-hysteresis magnetization intensity M_an in an ideal state is given by

$$M_{an}=M_s[\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}({\displaystyle \frac{H_e}{a}})-{\displaystyle \frac{a}{H_e}}]$$

(1)

where $M_s$ is the material’s saturated magnetization intensity, in the unit of A/m; $\alpha$ is the shape coefficient of the magnetization curve; $H_e$ is the equivalent magnetization field, in the unit of A/m, and its intensity is represented by

$$H_e=H+\alpha M+H_{\sigma }$$

(2)

where $H$ is the intensity of the external magnetic field, in the unit of A/m. During WMD, $H$ is the intensity-adjustable excitation field; $M$ is the material’s magnetization intensity under the action of $H$, in the unit of A/m; α is the magnetization coupling coefficient; $H_{\sigma }$ is the stress-induced equivalent magnetic field, in the unit of A/m, and its intensity is represented by

$$H_{\sigma }={\displaystyle \frac{3\sigma }{{\mu }_0}}[\left(r_{1\left(0\right)}+r_{1\left(0\right)}^{,}\sigma \right)M+2\left(r_{2\left(0\right)}+r_{2\left(0\right)}^{,}\sigma \right)M^3]$$

(3)

where $r_{1\left(0\right)}$, $r_{1\left(0\right)}^{,}$, $r_{2\left(0\right)}$, and $r_{2\left(0\right)}^{,}$ are all constants related to the pipeline material; μ₀ is vacuum permeability, in the unit of H/m; $\sigma$ is the stress acting on the pipeline, in the unit of Pa. The magnetization intensity is related to the non-hysteresis magnetization intensity in the following manner:

$$M=M_{an}-{\displaystyle \frac{k\delta }{{\mu }_0}}{\displaystyle \frac{\mathrm{d}{M}_{irr}}{\mathrm{d}{H}_e}}$$

(4)

where k is the material’s pinning coefficient, representing the hysteresis characteristic; $\delta$ is the orientation coefficient; M_irr is the irreversible magnetization intensity, in the unit of A/m, as given by

$$M_{irr}={\displaystyle \frac{1}{1-c}}\left(M-c{M}_{an}\right)$$

(5)

where c is the irreversible coefficient. After merging and correction, the relationship of the magnetization intensity M vs. the external magnetic field H is written as follow:

$${\displaystyle \frac{dM}{dH}}={\displaystyle \frac{-\frac{{\mu }_0}{k\delta }(M_{an}-M)-\frac{c}{1-c}\frac{d{M}_{an}}{dH}}{\frac{{\mu }_0}{k\delta }(M_{an}-M)\{\alpha +\frac{3\sigma }{{\mu }_0}[{(r}_{1\left(0\right)}+r_{1\left(0\right)}^{,}\sigma )+{6(r}_{2\left(0\right)}+r_{2\left(0\right)}^{,}\sigma )M^2]\}-\frac{1}{1-c}}}$$

(6)

The above equation depicts the magnetization curve of the pipeline incorporating the influence of stress. It reflects the force-magnetic coupling relationship. From the magnetic parameters of the material, we plotted the magnetization curves for varying stress levels. The parameter values are below: μ₀ = 4π × 10⁻⁷ N A⁻², M_s = 1.58 × 10⁶ A m⁻¹, $c=0.25$, $r_{1\left(0\right)}=7\times {10}^{-18}$ A⁻² m², $r_{1\left(0\right)}^{,}=-1\times {10}^{-25}$ A⁻² m² Pa⁻¹, $r_{2\left(0\right)}=-3.3\times {10}^{-30}$ A⁻⁴ m⁴, $r_{2\left(0\right)}^{,}=2.1\times {10}^{-38}$ A⁻⁴ m⁴ Pa⁻¹, $\delta$ = 1, k = 0.002, and a = 1000.

It can be observed from Figure 1a that as the stress acting on the pipeline increased, the material’s magnetization intensity decreased under the same excitation intensity. Under a lower stress level, the material’s magnetization intensity did not vary significantly with stress. However, a higher stress level caused the magnetization intensity to differentiate more dramatically. As the excitation intensity increased, the degree of differentiation between the magnetization intensities under the varying stress levels first increased and then decreased until barely any differentiation was observed at all. From the variation pattern in the magnetization curves, we calculated the characteristics of the weak magnetic signals in stress concentration areas for varying stress levels.

Compared with conventional MMMT or MFL-based models that either work solely under the geomagnetic field or focus on defect-related leakage flux, the present magneto–mechanical framework explicitly incorporates the combined effects of controllable excitation intensity and applied stress on the magnetization curve of pipeline steel, which enables a targeted optimization of weak-magnetic stress-detection conditions.

2.2. Weak Magnetic Model for Stress Detection

Figure 1b illustrates the physical modeling and experimental schematic of pipeline stress detection. The right section depicts the weak-magnetic detection mechanism. A local stress-concentrated region on the pipe wall is assumed and simplified as a rectangular cuboid element with a length l, width w, and depth d. An external excitation coil applies a uniform magnetic field along the pipe’s axial (X) direction, and the magnetization vector is primarily oriented along this axis. Due to the stress concentration, the magnetic permeability and susceptibility of the material in this region decrease, resulting in a spatial gradient of magnetization intensity. Magnetic discontinuity occurs at both axial ends of the stress-concentrated region, leading to the accumulation of equivalent magnetic charges (indicated by the blue “+” and “−” symbols). These magnetic charges generate an additional stray magnetic field, which is captured by a high-sensitivity magnetic sensor array installed inside the detection module (shown in red), thereby producing a measurable weak-magnetic stress signal. The inset above illustrates the overall structure of the detection device, including the axially arranged excitation coils, Hall sensors, and mechanical support components. The photograph on the right corresponds to the actual stress-concentrated area on the pipeline’s outer surface (marked by a red dashed circle), which matches the defect location in the schematic model.

The weak magnetic signals captured by the probe can be calculated analytically. The analytical formula for tangential and normal signals are provided below:

$$H_x={\displaystyle \frac{\rho }{4\pi {\mu }_0}}{\int }_{-w/2}^{w/2}{\int }_{-d}^0{\displaystyle \frac{x+l/2}{\left|r\right|}}-{\displaystyle \frac{x-l/2}{\left|r\right|}}d{y}_{0}d{z}_0$$

(7)

$$H_y={\displaystyle \frac{\rho }{4\pi {\mu }_0}}{\int }_{-w/2}^{w/2}{\int }_{-d}^0{\displaystyle \frac{y-y_0}{\left|r\right|}}-{\displaystyle \frac{y-y_0}{\left|r\right|}}d{y}_{0}d{z}_0$$

(8)

where $H_x$ is the axial weak magnetic signal in the pipeline wall, in the unit of A/m; $H_y$ is the axial weak magnetic signal in the pipeline wall, in the unit of A/m; $\rho$ is the magnetic charge density on the axial cross-section of the pipeline wall, in the unit of T; $\left|r\right|$ is the distance from the magnetic charges in the pipeline wall to the probe, in the unit of m, as given by

$$\left|r\right|={[{\left(x-l/2\right)}^2+{\left(y-y_0\right)}^2+{\left(z-z_0\right)}^2]}^{{\displaystyle \frac{3}{2}}}$$

(9)

where $y_0$ and $z_0$ are the circumferential and radial coordinates of each micro-element magnetic charge on the defective cross-section, respectively. According to the definition of the magnetic charge density, the equivalent magnetic charge density over the cross-section of the stress concentration area is given by

$$\rho ={\mu }_0(M_1-M_2)$$

(10)

where $M_1$ is the magnetization intensity at sites without stress concentration in the pipeline wall; $M_2$ is the material’s magnetization intensity in the stress concentration area. Thus, for stress detection, the magnetization intensity of the stress concentration area under the varying stress levels was estimated based on the relationship between the magnetization intensity and excitation intensity in Figure 1. So far, we have obtained the equivalent magnetic charge density over the cross-section, from which the characteristics of the weak magnetic signals in the stress concentration area under the varying stress levels were estimated.

3. Influence of Excitation Intensity on the Weak Magnetic Signals for Stress Detection

3.1. Selection of the Weak Excitation Intensity

As the stress applied to the pipeline increases, the magnetization intensity of the material under the same excitation field gradually decreases. With increasing excitation intensity, the distinction in magnetization between different stress levels first increases and then gradually diminishes until it becomes nearly indistinguishable. Therefore, when conducting stress detection, the selected external magnetic excitation intensity should be optimized to avoid excessive excitation—which could weaken the stress-dependent magnetic response—as well as insufficient excitation, which could lead to interference similar to that observed in MMMT. In this study, Q235 steel was chosen as the test material, and its magnetic hysteresis loop was measured to characterize its stress-related magnetic behavior.

According to the quasi-static magnetic characterization results, the coercivity of Q235 steel was determined to be 1154 A/m. To ensure that the material reached a stable and repeatable magnetic state during magnetization, an excitation intensity range of 2.5 kA/m to 10 kA/m was selected as the effective analysis interval. The minimum excitation intensity was set to more than twice the coercivity value, based on the theoretical consideration that when the applied magnetic field significantly exceeds the material’s coercivity, it can effectively overcome the domain wall pinning effect. This enables the magnetization process to transition from an irreversible Barkhausen-jump behavior to a continuous and controllable magnetization stage. In practical testing, even if the specimen initially resides at a reverse remanent point on the hysteresis loop due to magnetic history effects, the application of a forward excitation field not less than twice the coercivity rapidly drives the magnetization along the initial magnetization curve, crossing the coercive point and entering the main magnetization region. This design effectively eliminates the uncertainty of the initial magnetic state, ensuring consistency and comparability of magnetization responses under different stress conditions, and thereby providing a reliable experimental basis for stress-related magnetic detection. In addition, selecting the excitation intensity to be higher than twice the coercivity not only ensures a stable and repeatable magnetization state, but also reduces the variability of the initial magnetic condition caused by magnetization history, which is one of the important sources of field distortion in weak magnetic measurements.

3.2. Weak Magnetic Signals for Stress Detection at Different Excitation Intensities

To systematically investigate how stress level influences weak-magnetic testing signals, a standardized stress-concentration zone was defined on the pipe wall with fixed geometry: 10 mm in width and 2 mm in depth. The stress in non-concentration regions of the wall was taken as zero. Different stress values within the concentration zone lead to different magnetization intensities; the resulting magnetization mismatch across regions causes magnetic charges to accumulate on the two cross-sections flanking the zone along the magnetization direction. The magnetic charge density can be computed using Equation (10). Magnetic charge density directly reflects the extent of stress-induced changes in the material’s magnetic properties and is the key physical quantity linking the stress state to the spatial magnetic-flux-leakage field. By systematically varying the value of $M_2$ in the stress-concentration zone, one can calculate the equivalent magnetic charge density for different stress states, thereby providing the theoretical input for subsequent quantitative evaluation of weak-magnetic signals. This modeling framework establishes the physical basis for elucidating the mapping between stress level and weak-magnetic signal amplitude.

3.2.1. Weak Magnetic Signals at the Excitation Intensity of 2.5 kA/m and Varying Stress Levels

Under an excitation intensity of 2.5 kA/m, the probe was passed through the centerline along the width direction, at 1 mm above the stress concentration area. The weak magnetic signals in the stress concentration area for varying stress levels are shown in Figure 2.

Figure 2 shows the spatial distribution of weak-magnetic signals in the defect region under various stress levels (20–160 MPa), along with the corresponding radial and axial components. The 3D maps on the left depict the spatial field distribution, while the line scans on the right illustrate the magnetic response across the defect center. Both components exhibit a clear increase in amplitude with rising stress, indicating that external loading enhances magnetization and intensifies field distortion near the defect.

For the radial component (Figure 2a), the waveform displays a typical peak–valley shape with strong symmetry and high stress sensitivity. At 20 MPa, the amplitude is weak and close to the baseline; when the stress increases to 160 MPa, the amplitude rises by nearly an order of magnitude and the extrema become sharper. This trend suggests that the radial field is dominated by the polarization of equivalent magnetic charges within the stress concentration zone, closely associated with local magnetic permeability and hysteresis characteristics.

The axial component (Figure 2b) exhibits a valley–peak pattern with larger peak width, reflecting the superposition of flux leakage from both sides of the defect. Its overall amplitude increases with stress, and the waveform shifts upward due to enhanced dipole strength along the magnetization direction. When the stress exceeds 80 MPa, the growth of amplitude tends to saturate, indicating that the local magnetic domains approach saturation and the effect of additional stress becomes negligible.

Under a low excitation intensity of 2.5 kA/m, the weak-magnetic signals behave differently. At 20 MPa, the tangential signal reverses its peak polarity, and the normal component also shows opposite extrema compared with higher stress levels (40–160 MPa). This inversion occurs because, under weak excitation, the magnetization of the low-stress specimen exceeds that of the unstressed state, leading to a polarity reversal of equivalent magnetic charges on both sides of the stress concentration zone. As the stress increases above 40 MPa, the magnetization becomes weaker than that of the unstressed material, the signal polarity returns to normal, and the amplitude continues to rise.

3.2.2. Weak Magnetic Signals at the Excitation Intensity of 5 kA/m for Varying Stress Levels

Using the same method as above, we estimated the characteristics of weak magnetic signals from the pipelines at the excitation intensities of 5 kA/m, 7.5 kA/m, and 10 kA/m, respectively. The weak magnetic signals in the stress concentration area for varying stress levels under the excitation intensity of 5 kA/m are shown in Figure 3.

Figure 3 illustrates the spatial distribution of weak-magnetic signals in the defect region and their variation along the sampling path under different stress levels (20–160 MPa). The overall distribution pattern is similar to that shown in Figure 2, and only the distinctive features and differences are highlighted here. As the applied stress increases, the amplitudes of both the radial and axial components continuously rise, consistent with the results of Figure 2, further confirming that external stress enhances the magnetization of the material and intensifies magnetic field distortion around the defect. For the radial component (Figure 3a), the waveform exhibits a typical peak-valley structure, with the amplitude increasing nearly linearly as stress grows, while the positions of the extrema remain stable. This indicates that stress mainly affects the field intensity rather than its spatial distribution. The axial component (Figure 3b) shows a valley–peak pattern with smaller amplitude but smoother variation, reflecting the enhanced superposition of MFL on both sides of the defect. Compared with Figure 2, the excitation level here lies within a stable magnetization region, where no polarity reversal occurs and the signal variation shows a steady, monotonic trend. At higher stress levels (above 80 MPa), the growth rate of the radial amplitude slightly decreases, and the axial response tends toward saturation, implying that part of the magnetic domains have approached magnetic saturation and that the influence of additional stress on local flux density gradually weakens. The signal characteristics at the varying stress levels are shown in

Table 2. Signal characteristics under the excitation intensity of 5 kA/m for varying stress levels (A/m).

	20 MPa	40 MPa	60 MPa	80 MPa	160 MPa
Tangential amplitude	1159	3999	9614	30,506	93,026
Normal peak amplitude	2254	7777	18,697	59,327	180,914

.

3.2.3. Weak Magnetic Signals at the Excitation Intensity of 7.5 kA/m for Varying Stress Levels

The weak magnetic signals in the stress concentration area for varying stress levels under the excitation intensity of 7.5 kA/m are shown in Figure 4.

Under Figure 4 illustrates the spatial distribution of weak-magnetic signals in the defect region and their variation along the sampling path under different stress levels (20–160 MPa). Similar to Figure 2 and Figure 3, both radial and axial components are presented to examine the effect of stress variation on weak-magnetic field responses. Overall, as the applied stress increases, the amplitudes of both components rise markedly, and their evolution trends are consistent. This indicates that higher stress enhances the material’s magnetization and intensifies the local magnetic field distortion around the defect. The radial component (Figure 4a) exhibits a typical peak–valley waveform with pronounced positive–negative symmetry. At low stress (20 MPa), the signal amplitude is weak and approaches the baseline, whereas at 160 MPa the peak–valley difference increases significantly—nearly by an order of magnitude. This trend demonstrates the high sensitivity of the radial magnetic field component to stress variation, primarily arising from magnetic charge polarization within the stress concentration zone and the enhancement of magnetization response due to stress-induced changes in magnetic permeability. In the medium-to-high stress range (80–160 MPa), the growth rate of the peak amplitude slightly decreases, suggesting that local magnetic domains gradually approach saturation and that the influence of additional stress tends to stabilize. The axial component (Figure 4b) also shows a valley–peak profile, though with smaller amplitude and smoother variation. As stress increases, the overall waveform shifts upward, indicating a synchronous enhancement of magnetic dipole strength along the magnetization direction. Compared with the radial component, the axial signal demonstrates higher spatial stability, reflecting the balanced superposition of MFL on both sides of the defect. Owing to its broader peak width and stable extrema spacing, the axial component is better suited for defect boundary identification and signal-shape compensation.

The signal characteristics at the varying stress levels are shown in

Table 3. Signal characteristics under the excitation intensity of 7.5 kA/m for varying stress levels (A/m).

	20 MPa	40 MPa	60 MPa	80 MPa	160 MPa
Tangential amplitude	1614	3854	6777	18,079	63,741
Normal peak amplitude	3138	7495	13,180	35,159	123,962

.

Comparison between the results in Table 1 and Table 2 would show that under the stress of 40 MPa, the peak amplitudes of the tangential and normal magnetic signals at the excitation intensity of 7.5 kA/m were larger than those under the same stress but an excitation intensity of 5 kA/m. Above the stress of 40 MPa, the peak amplitudes of the tangential and normal magnetic signals at the excitation intensity of 7.5 kA/m were smaller than those under the same stress but an excitation intensity of 5 kA/m. The above results indicated that the weak magnetic signals in the pipelines were more conspicuous at 7.5 kA/m under a lower stress level, while the signals were more easily differentiated at 5 kA/m under higher stress levels (40–160 MPa).

3.2.4. Weak Magnetic Signals at the Excitation Intensity of 10 kA/m for Varying Stress Levels

To further assess the differentiation ability of the weak magnetic signals under different excitation intensities and varying stress levels, we obtained the weak magnetic signals in the stress concentration are under varying stress levels for the excitation intensity of 10 kA/m, as shown in Figure 5.

Figure 5 illustrates the spatial distribution of weak-magnetic signals in the defect region and their variation along the sampling path under different stress levels (20–160 MPa). The three-dimensional magnetic field distributions are shown on the left, while the line-scan profiles across the defect center are presented on the right. Overall, both the radial and axial components exhibit a continuous increase in amplitude with rising stress. Compared with the results obtained under lower excitation intensities, the higher excitation strength used here places the material in a stable magnetization region, resulting in stronger signal amplitudes. This indicates that external stress promotes magnetic domain realignment and enhances magnetization intensity, thereby intensifying the magnetic field distortion around the defect. For the radial component (Figure 5a), the waveform exhibits a typical peak–valley pattern with clear positive–negative symmetry. At a low stress of 20 MPa, the amplitude is small and close to the baseline, while at 160 MPa, the peak-to-valley difference increases significantly, and the extrema slightly shift toward the defect center. This behavior demonstrates the high sensitivity of the radial magnetic field component to stress variation, which primarily arises from the polarization of magnetic charges in the stress concentration zone and the stress-induced changes in magnetic permeability. In the medium-to-high stress range (80–160 MPa), the rate of amplitude growth slightly decreases, suggesting that local magnetic domains gradually approach saturation and the influence of additional stress tends to stabilize. The axial component (Figure 5b) shows a valley–peak waveform with smaller amplitude and smoother variation. As the stress increases, the overall waveform shifts upward, indicating a synchronous enhancement of magnetic dipole strength along the magnetization direction. Compared with the radial component, the axial signal exhibits a more uniform spatial distribution, reflecting the balanced superposition of MFL on both sides of the defect. Due to its broader peak width and stable extrema spacing, the axial component demonstrates higher stability and is particularly suitable for defect boundary identification and signal-shape compensation. Moreover, under an excitation intensity of 10 kA/m, no signal polarity reversal occurs in the low-stress state, indicating that the applied magnetic field is sufficient to prevent magnetization inversion in the weak-field region. With increasing stress, both the tangential signal amplitude and the peak-to-peak value of the normal component rise continuously, showing a one-to-one correspondence between stress magnitude and signal characteristics. This result confirms that, under high excitation, the mapping relationship between stress and MFL response becomes more stable, and the signal variation exhibits excellent monotonicity and reproducibility. The signal characteristics at the varying stress levels are shown in

Table 4. Characteristic values of different stress signals at 10 kA/m excitation strength(A/m).

	20 MPa	40 MPa	60 MPa	80 MPa	160 MPa
Tangential amplitude	1492	2906	4661	11,002	37,398
Normal peak amplitude	2901	5652	9064	21,396	72,730

.

Comparison between the results in Table 2 and Table 3 would show that under the stress of 40 MPa, the peak amplitudes of the tangential and normal magnetic signals at the excitation intensity of 7.5 kA/m were larger than those under the same stress but at an excitation intensity of 10 kA/m. The characteristic values of the weak magnetic signals at a lower stress level (below 40 MPa) and an excitation intensity of 7.5 A/m were higher than those at either 5 kA/m or 10 kA/m. This indicated that the excitation intensity of 7.5 kA/m had the highest differentiation ability for stress below 40 MPa. But above the stress of 40 MPa, the peak amplitudes of the tangential and normal signals under the excitation intensity of 10 kA/m were smaller than those under the same stress level and an excitation intensity of 7.5 kA/m. These results indicated that the differentiation ability of the weak magnetic signals was lower at 10 kA/m for higher stress levels. That is, as the excitation intensity increased from 5 kA/m to 10 kA/m, the differentiation ability of weak magnetic signals for higher stress levels was attenuated. The differentiation ability was the best under higher stress levels at an excitation intensity of 5 kA/m.

3.3. Accuracy Analysis of Stress Detection

After analyzing the influence of the excitation intensity on the differentiation ability for stresses, we further studied the influence of the excitation intensity on the accuracy of stress detection. The latter is not only influenced by the characteristic intensity of the weak magnetic signals, but also by the ambient magnetic field. Even when the stress-induced magnetic signals have large characteristic values, the accuracy of stress detection would be compromised in the presence of an intense ambient magnetic field, which further affects the accuracy of stress quantification. Given the above, the accuracy analysis for stress detection will inform the design of the excitation intensity for real-world stress detection.

When it comes to stress signal detection in pipelines, the probe captures the stress-induced magnetic signals along with the ambient magnetic fields where it passes through. This explains the variability of the baseline values of weak magnetic signals under different excitation intensities. In this study, the resolution $\phi$ of the stress-induced magnetic signals is defined as the ratio of the characteristic value variation in the weak magnetic signals to that of the stress intensity. Hence, the resolution $\phi$₁ of the tangential signals and the resolution of $\phi$₂ of the normal signals are, respectively, given by

$${\phi }_1=∆H_1/∆\sigma , {\phi }_2=∆H_2/∆\sigma$$

(11)

where $∆\sigma$ is the variation gradient of stress; $∆H_1$ is the variation gradient of the tangential signal amplitude in the stress concentration area; $∆H_2$ is the variation gradient of the normal signal amplitude in the stress concentration area. The higher the resolution of the stress-induced magnetic signals, the more significant the variation in the weak magnetic signals per megapascal of stress would be.

The accuracy $\omega$ of stress detection is defined as the ratio of the resolution of stress-induced magnetic signals to the intensity H₀ of the ambient magnetic field. Hence, the accuracy $\omega$₁ of stress detection using tangential signals and the accuracy $\omega$₂ using normal signals are, respectively, given by

$${\omega }_1={\phi }_1/H_0, {\omega }_2={\phi }_2/H_0$$

(12)

The higher the accuracy, the better the differentiation ability for stresses and hence the higher accuracy of stress quantification using the weak magnetic signals would be. As defined in Equation (12), $\phi$ represents the resolution of the weak-magnetic signal, i.e., the variation in the characteristic signal value per unit change in stress. This term reflects how sensitively the magnetic signal responds to stress. $H_0$ denotes the magnitude of the ambient/background magnetic field experienced by the probe. Therefore, the ratio $\mathit{\omega }=\phi /H_0$ directly measures how much stress-related signal change can be distinguished above the background field. A larger $\mathit{\omega }$ implies that, for a given change in stress, the relative change in the measured signal with respect to the background field is larger, which corresponds to higher detection accuracy in practical engineering applications.

Under the excitation intensity of 2.5 kA/m, 5 kA/m, 7.5 kA/m, and 10 kA/m, the detection accuracy using tangential and normal signals for varying stress levels are shown in

Table 5. Accuracy of weak magnetic signal detection under different excitation intensities and varying stress levels.

		20 MPa	40 MPa	60 MPa	80 MPa	160 MPa
2.5 kA/m	Tangential	0.032	0.066	0.168	0.623	0.265
	Normal	0.062	0.128	0.326	1.212	0.515
5 kA/m	Tangential	0.012	0.028	0.056	0.209	0.156
	Normal	0.023	0.055	0.109	0.406	0.304
7.5 kA/m	Tangential	0.011	0.015	0.019	0.075	0.076
	Normal	0.021	0.029	0.038	0.147	0.148
10 kA/m	Tangential	0.007	0.007	0.009	0.032	0.033
	Normal	0.015	0.014	0.017	0.062	0.064

.

It can be found that, except for larger errors at 2.5 kA/m and 20 MPa, the stress detection accuracy decreased as the excitation intensity increased. The accuracy using normal signals was higher than using tangential signals, which means it is easier to detect and quantify stresses using tangential signals. Based on the data from Table 4, we plotted the variation curves of the detection accuracy vs. excitation intensity for varying stress levels, as shown in Figure 6.

It can be found that above 20 MPa, the detection accuracy for varying stress levels decreased as the excitation intensity increased. Therefore, the excitation intensity should be above 5 kA/m to ensure the one-to-one correspondence between the stress-induced magnetic signals and the stress values. Within the range from 5 kA/m to 8 kA/m, the accuracy of stress detection first increased and then decreased as the stress increased. The accuracy was the highest at about 80 MPa. From 8 kA/m to 10 kA/m, the accuracy of stress detection was similar under the varying stress levels. As analyzed above, a higher excitation intensity (8 kA/m–10 kA/m) is preferable for stress detection within a broader range of stresses (above 80 MPa). This will help reduce errors caused by stress fluctuations and achieve a filtering effect. For a narrower range of stresses, a lower excitation intensity (5 kA/m) can be chosen for stress detection in pipelines, which will increase the stress detection rate and resolution and quantify stresses.

From a practical viewpoint, the above analysis implies that, once the optimal excitation range has been determined for a given material and stress range, it should be combined with strict control of the sensor lift-off, baseline calibration and environmental compensation in field inspections. In this way, the influence of magnetic-field distortion can be constrained within an acceptable level, and the quantitative relationship between stress and weak magnetic signals established in this study can be reliably applied to pipeline integrity assessment.

3.4. Engineering Procedure for Determining the Optimal Excitation Intensity

In practical engineering applications, the excitation intensity cannot be taken as a fixed empirical value; instead, it should be determined according to the pipeline material, the target stress range, and the ambient magnetic environment. Based on the above magneto-mechanical analysis and accuracy evaluation, an engineering-oriented selection procedure can be summarized as follows:

• Define detection requirements. Specify the pipeline steel grade, wall thickness and diameter, expected stress range to be monitored (early-warning low-stress range or medium-to-high stress range), and the background magnetic field level.
• Obtain magnetic properties. Measure the quasi-static hysteresis loop of the pipeline steel (or use data from material characterization) to determine the coercive field and the approximate near-saturation field.
• Set a candidate excitation interval. Select a preliminary excitation interval in which the applied magnetic field is typically 2–3 times the coercive field to overcome magnetic history effects, while remaining below the strong-saturation region so that the stress-dependent magnetization maintains sufficient sensitivity.
• Evaluate stress–signal mapping. Within this interval, use the proposed magneto-mechanical coupling model to calculate the weak-magnetic signal characteristics and then derive the resolution and accuracy indices according to Equations (11) and (12). This step provides a quantitative mapping between stress and weak-magnetic response under different excitation intensities.
• Select the optimal intensity. For the target stress range, choose the excitation intensity that yields the highest accuracy, ensures a one-to-one and monotonic relationship between stress and signal (no polarity reversal), and satisfies constraints on coil power, thermal load, and structural integration. For the Q235 steel considered in this work, this procedure leads to an excitation of about 7.5 kA/m for low-stress detection (<40 MPa) and about 5 kA/m for quantitative evaluation in the 40–160 MPa range.
• Verify and adjust. Finally, verify the selected excitation intensity through tensile tests on representative specimens and adjust slightly according to the on-site noise level and detection requirements.

A flowchart of this engineering procedure for determining the excitation intensity is provided in Figure 7, which is expected to facilitate practical use of the proposed method by engineering readers.

4. Tests and Result Analysis

4.1. Tests on the Fluence of Excitation Intensity on Weak Magnetic Signals for Stress Detection

The influence of excitation intensity on the weak magnetic signals for stress detection was validated by tensile tests in the pipelines. The pipeline specimen and the tensile machine (Jinan Dongfang Testing Instrument Co., Ltd., Jinan, China) are shown in Figure 8. The tensile specimens were machined from steel plates used for engineering pipelines. The specimen was 800 mm long, 60 mm wide, and 16 mm thick. There was a crack, measuring 0.5 mm wide, 2 mm deep, and 60 mm long, prefabricated in the middle of the specimen. Each specimen contained a prefabricated axial surface crack so that a localized stress-concentration zone was formed at the crack tip, analogous to the stress concentration that develops in defective pipeline sections. Controlled by the host machine, the tensile machine applied tensile forces of varying intensity on the specimen. Stress concentration areas would appear at the bottom of the crack under the tensile forces. The stress magnitudes in these areas were estimated by ANSYS2025R2 simulation under varying tensile forces. An excitation coil was placed outside the specimen, and a weak magnetic detector probe was immobilized in the center of the coil. The probe was moved up and down on the specimen surface without the crack to collect weak magnetic signals from the stress concentration area. The weak magnetic signals were then uploaded to the host machine via the signal collection system for real-time display and storage. The excitation coil was connected to the 220 V alternating current (AC) power supply through the bridge circuit and the transformer, which converted AC to direct current (DC) for excitation of the coil. The current size was adjusted by the transformer, which further changed the excitation intensity on the specimen.

The magnetic field at the coil center was measured using the magnetometer. The magnetic field intensity was 10 kA/m under the excitation current of 2.5 A. An excitation current of 1.25 A, 2 A, and 2.5 A was imposed to induce an excitation magnetic field with an intensity of 5 kA/m, 8 kA/m, and 10 kA/m, respectively. Under different magnetic field intensities, a tensile force of 10 kN, 20 kN, 40 kN, and 80 kN was imposed on the specimen, resulting in an axial stress of 20 MPa, 40 MPa, 80 MPa, and 160 MPa at the defects, respectively. The signal characteristics were recorded at the defects at different excitation intensities for varying stress levels.

4.2. Test Data and Analysis

Figure 9 presents, under an 80 kN tensile load, the weak-magnetic signal characteristics in the defect region measured at different excitation intensities (corresponding to currents of 2.5 A, 2.0 A, and 1.25 A). Panel (a) shows the tangential component waveform, and panel (b) shows the normal component waveform. The plots clearly indicate that, with increasing excitation intensity, the amplitudes of both components increase markedly.

Figure 10 shows, under a tensile load of 40 kN, the weak-magnetic signal characteristics in the defect region measured at different excitation intensities (corresponding to currents of 2.5 A, 2.0 A, and 1.25 A).

Figure 11 shows, under a tensile load of 20 kN, the weak-magnetic signal characteristics in the defect region measured at different excitation intensities (corresponding to currents of 2.5 A, 2.0 A, and 1.25 A).

Figure 12 shows, under a tensile load of 10 kN, the weak-magnetic signal characteristics in the defect region measured at different excitation intensities (corresponding to currents of 2.5 A, 2.0 A, and 1.25 A).

Our tests showed that all defect-induced magnetic signals could be detected under different excitation intensities and that the normal signals had a better differentiation ability than the tangential signals. Under the tensile force of 80 kN, the characteristics of defect-induced magnetic signals at the excitation current of 1.25 A were more conspicuous than those at 2 A, indicating that the weak magnetic signals are more suitable for stress detection at a higher stress level.

Under the tensile force of 10 kN to 40 kN, the characteristics of defect-induced magnetic signals at the excitation current of 2.5 A were more conspicuous than those at 2 A. Therefore, increasing the excitation intensity is more appropriate for stress detection under a smaller stress level.

The test results agreed with the theoretical analyses. The tensile tests under different excitation intensities validated the theories for conceptualizing stress detection using weak magnetic signals. Specifically, the numerical model in Section 3 assumes an idealized rectangular stress-concentration region with uniform magnetization and a simplified magnetic charge distribution, whereas in the experiments the actual stress and magnetization distributions near the crack tip are non-uniform and affected by the specimen geometry, machining tolerances, and the finite size of the excitation coil. Moreover, the experimental probe has a finite lift-off distance and is influenced by environmental magnetic noise, which further distorts the tangential component and reduces its signal-to-noise ratio. As a result, the simulated tangential signal exhibits a clearer peak–valley structure than the measured curve, while the normal component shows better agreement with the simulation.

Practical Considerations

These experimental results further indicate that, when appropriate shielding and baseline calibration are adopted, the influence of external field distortion can be effectively controlled, and the optimized excitation intensity obtained in this work remains applicable to engineering stress detection.

4.3. Experimental Verification of Stress Identification and Quantitative Evaluation Accuracy

To experimentally verify the stress identification performance and quantitative evaluation accuracy of the proposed method, the weak-magnetic signals measured in Section 4.2 were further processed. For each excitation intensity and each tensile load level, the peak-to-peak value of the normal component and the amplitude of the tangential component in the defect region were extracted as characteristic quantities. For each excitation intensity, a calibration relationship between the applied stress and the measured weak-magnetic characteristics was established by least-squares fitting. Figure 13 shows the experimental scatter plots and fitted curves of stress versus normal peak-to-peak value under the three excitation intensities. It can be observed that, within the practical stress range of 20–160 MPa, the experimental data exhibit a clear monotonic trend and an approximate one-to-one correspondence between the stress level and the weak-magnetic signal amplitude. In particular, an excitation intensity of 7.5 kA/m provides the largest slope and thus the highest sensitivity in the low-stress range (20–40 MPa), whereas 5 kA/m yields a higher slope and better differentiation in the medium-to-high stress range (40–160 MPa). These observations are consistent with the theoretical analysis in Section 3.3. Based on the calibration curves, the stresses were quantitatively inverted from the measured weak-magnetic characteristics, and the results were compared with the reference stresses obtained from the tensile machine and finite-element analysis.

5. Conclusions

This study focuses on the application of WMD technology for quantitative stress identification in pipelines under varying excitation magnetic field strengths. A systematic analytical framework was established and validated through multiple sets of experiments. The main conclusions and innovations are as follows:

• Theoretical Advancement in the Excitation–Stress Relationship

A stress-dependent magneto–mechanical coupling model and a three-dimensional analytical model of weak-magnetic fields in a stress-concentrated zone are established for ferromagnetic pipelines. These models clarify how the external excitation field, hysteretic magnetization and stress-induced equivalent field jointly determine the weak-magnetic response and provide an analytical mapping from stress level to tangential and normal signal components.

• Enhanced Engineering Applicability and Detection Reliability

Based on the above models, a quantitative evaluation method for stress-detection performance is proposed. By defining the resolution ϕ of stress-induced signals and the accuracy index ω that incorporates the background field intensity, the influence of excitation intensity on stress recognition is systematically analyzed. The results show that 7.5 kA/m is the optimal excitation for low stress below 40 MPa, whereas 5 kA/m provides higher accuracy for medium–high stress in the range of 40–160 MPa.

• Expansion of Theoretical and Methodological Boundaries

The numerical analyses are verified by tensile experiments on Q235 pipeline specimens with a prefabricated crack under different excitation intensities. The experimental results confirm the polarity reversal of weak-magnetic signals at low excitation (2.5 kA/m) and validate the optimal excitation intervals obtained from the proposed accuracy-evaluation method. On this basis, an engineering procedure for selecting excitation intensity is suggested, which enhances the quantitative accuracy and practical applicability of weak-magnetic stress detection in pipeline integrity management. Moreover, calibration experiments under different excitation intensities confirm that the proposed weak-magnetic features enable a monotonic and nearly one-to-one mapping between stress level and signal amplitude within the practical stress range, and the inverted stresses agree well with the reference values obtained from tensile tests, which experimentally verifies the quantitative evaluation accuracy of the proposed method.

In conclusion, this study proposes a systematic method for selecting optimal excitation conditions for different stress ranges, significantly improving the quantitative accuracy and engineering applicability of weak magnetic stress detection. The outcomes not only enrich the fundamental understanding of weak magnetism in ferromagnetic pipelines but also provide effective technical tools and theoretical support for the safe operation and integrity management of oil and gas pipeline systems.