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This paper proposes the equivalent magnetic charge (EMC) method, to analyze the magnetic field presented in ferromagnetic pipe. Distributions of the magnetic charge density on pipes’ surface are calculated, and the magnetic field on several measurement lines is predicted using numerical calculation with several different directions of magnetization. The correctness of the analysis results is then validated by the comparison between the results of the ANSOFT and measurement of the magnetic field inside several actual pipes. With the EMC method, it is of great interest to found that when the magnetization direction changes, the radial and axial magnetic field components inside the pipe will shift accordingly. Moreover, the variation law of magnetic field along the axial retains substantially unaltered, and the magnetic field distribution in the pipe is obviously symmetrical with a plurality of extreme points. As a result, the magnetic field distributes evenly if the pipe is long enough.

In order to solve the problem of the quantification of detection signals in the magnetic flux leakage (MFL) of defective in-service oil and gas pipelines, a non-uniform magnetic charge model was established based on magnetic effects. The distribution patterns of magnetic charges under different stresses were analyzed. The influences of the elastic load and plastic deformation on the characteristic values of MFL signals were quantitatively assessed. The experimental results showed that the magnetic charge density was large at the edges of the defect and small at the center, and approximately decreased linearly with increasing stress. The eigenvalues of the axial and radial components of the MFL signals were compared, and it was found that the eigenvalues of the radial component exhibited a larger decline rate and were more sensitive to stress. With the increase in the plastic deformation, the characteristic values of the MFL signals initially decreased and then increased, and there was an inflection point. The location of the inflection point was associated with the magnetostriction coefficient. Compared with the uniform magnetic charge model, the accuracy of the axial and radial components of the MFL signals in the elastic stage of the improved magnetic charge model rose by 17% and 16%, respectively. The accuracy of the axial and radial components of the MFL signals were elevated by 9.15% and 9%, respectively, in the plastic stage.

During the operation of long-distance oil and gas pipelines, the pipeline wall is very prone to stress concentration under the continuous action of complex loads. Weak magnetic internal detection can detect stress concentration during the service of the pipeline, which has significant advantages in the diagnosis of pipeline stress damage. However, the traditional weak magnetic detection forward model mostly adopts the assumption of uniform stress distribution, which is difficult to accurately characterize the actual non-uniform distribution state of the stress concentration zone, resulting in theoretical deviations in the magnetic field characterization. Therefore, this study applies the finite element simulation platform to construct a three-dimensional model of X70 pipeline containing prefabricated defects to obtain the stress distribution at the defects. And based on the magnetoelastic effect, the coupling relationship model between stress distribution and magnetization intensity is constructed, and the magnetic charge density is used as a bridge to combine with the magnetic charge theory to establish a kind of weak magnetic detection forward model under non-uniform stress field, through which the influence of defect depth on the weak magnetic signals is systematically investigated, which provides a theoretical basis for the pipeline stress assessment.

Weak magnetic detection technology can detect stress concentration areas in ferromagnetic materials. However, the stress non-uniform characteristics of pipeline welds lead to significant differences in stress distribution range and values between inner wall welds and outer wall welds. This discrepancy makes it crucial to further evaluate the impact of stress non-uniformity on magnetic signals. To study the magnetic signal characteristics under the influence of residual stress in weld seams, a magneto-mechanical analytical model was established based on the magnetic charge theory and the distribution characteristics of residual stress in the weld seam. The magneto-mechanical relationship and magnetic signal distribution characteristics at the inner and outer wall welds of the pipeline are calculated. Furthermore, the effects of different excitation intensities on the amplitude growth characteristics of magnetic signals are analyzed and compared. To verify the analysis model, weld detection experiments with different excitation intensities were designed. The results show that both the peak-to-valley values of the normal component and the peak values of the tangential component of the outer wall weld are lower than those of the inner wall weld. Conversely, the peak-to-valley width of the normal component and the peak width of the tangential component are greater than those of the inner wall weld. Additionally, the rate of increase in weak magnetic signal amplitude decreases in a first-order exponential relationship with increasing excitation intensity. The average decay rates of the normal and tangential component amplitude growth rates for the inner wall weld are 34.03% and 27.9%, respectively, while for the outer wall weld, they are 31.75% and 28.01%, respectively. This study contributes to the identification and quantitative assessment of weak magnetic signals in inner and outer wall welds.

In General Relativity, addressing coupling to a non-linear electromagnetic field, together with a negative cosmological constant, we obtain the general static spherical symmetric black hole solution with magnetic charges, which is asymptotic to anti-de Sitter (AdS) space-times. In particular, for a degenerate case the solution becomes a Hayward–AdS black hole, which is regular everywhere in the full space-time. The existence of such a regular black hole solution preserves the weak energy condition, while the strong energy condition is violated. We then derive the first law and the Smarr formula of the black hole solution. We further discuss its thermodynamic properties and study the critical phenomena in the extended phase space where the cosmological constant is treated as a thermodynamic variable as well as the parameter associated with the non-linear electrodynamics. We obtain many interesting results such as: the Maxwell equal area law in the P - V (or S - T ) diagram is violated and consequently the critical point ( T ∗ , P ∗ ) of the first order small–large black hole transition does not coincide with the inflection point ( T c , P c ) of the isotherms; the Clapeyron equation describing the coexistence curve of the Van der Waals (vdW) fluid is no longer valid; the heat capacity at constant pressure is finite at the critical point; the various exponents near the critical point are also different from those of the vdW fluid.

Pipeline magnetic flux leakage (MFL) internal detection technology is the most widely used and effective method in the field of long-distance oil and gas pipeline online detection. With the improvement of data quantization precision, the influence of stress on MFL signal has been paid more and more attention. In this paper, the relationship between stress and saturation magnetization is introduced based on J-A theory. The analytical model of MFL detection signal for pipeline composite defects is established. The MFL signal characteristics of composite defects are quantitatively calculated. The effect of stress on MFL signal is studied. The theoretical analysis is verified by experimental data and excavation results. The researches show that the saturation magnetization of ferromagnets decreases exponentially with the increase of stress in strong magnetic field. The MFL signal of composite defect is weaker than that of volumetric defects of the same dimension. The axial amplitude and radial peak-to-peak value of MFL signal decrease with the increase of stress around the defect. The axial amplitude and radial peak-to-peak value of MFL signal increase non-linearly with the increase of width and depth of defects. When using MFL signal to judge the defect depth, it is necessary to make clear whether there is stress concentration phenomenon around the defect because the stress will lead to underestimation of the defect depth.

Quantitative online detection of microcracks in long-distance oil and gas pipelines is an international problem, and the effective detection method is still lacking. In this paper, a mathematical model of non-uniform distribution of crack magnetic charges is established based on the stress distribution laws of pipeline cracks under internal pressure. The weak magnetic signal characteristics of pipeline cracks with different sizes are analyzed. The internal pressure increasing factor of weak magnetic signals are extracted to analyze the corresponding relationship between crack size and weak magnetic signals. The experimental study of the X70 pipeline is carried out. The results show that the axial component of the weak magnetic signal at the crack has a maximum value near the tip, and a minimum value appears in the middle of the crack. The internal pressure increasing factor is introduced to quantify the weak magnetic signal, the crack is in a safe state (not expanding) when the internal pressure increasing factor is positive, the weak magnetic signal has a linear relationship with the crack size. However, the crack is in a dangerous state when the internal pressure increasing factor is negative, and the pipeline crack will expand as the internal pressure increases.

Axial flux motors have attracted significant attention in recent years due to their advantages such as shorter axial length and high torque density. However, the optimization of axial flux motors is an extremely time-consuming process. To reduce the computational time required for motor optimization, this study employed a magnetic charge model to establish a coreless axial flux motor model and analyzed the advantages of this approach. This method is applicable to coreless axial flux motor optimizations with surface-mounted rotors and concentrated windings. Parameter optimization was subsequently performed based on the theoretical model. In terms of seeking optimal solutions, the torque obtained through the magnetic charge method (MCM) reached 99.67% of the finite element method (FEM) results. Finally, a prototype was fabricated, and a test platform was constructed based on the optimization results. The experimental torque showed a 4% deviation from simulations, validating the accuracy of the optimization.

We present a numerical study of magnetic ordering in spin ice on kagome, a two-dimensional lattice of corner-sharing triangles. The magnet has six ground states and the ordering occurs in two stages, as one might expect for a six-state clock model. In spin ice with short-range interactions up to second neighbours, there is an intermediate critical phase separated from the paramagnetic and ordered phases by Kosterlitz-Thouless (KT) transitions. In dipolar spin ice, the intermediate phase has long-range order of staggered magnetic charges. The high- and low-temperature phase transitions are of the Ising and 3-state Potts universality classes, respectively. Freeze-out of defects in the charge order produce a very large spin correlation length in the intermediate phase. As a result of that, the lower-temperature transition appears to be of the KT type.

Non-destructive tests of gantry cranes by means of the residual magnetic field (RMF) method were carried out for a duration of 7 years. Distributions of the residual magnetic field tangential and the normal components of their gradients were determined. A database of magnetograms was created. The results show that the gradients of tangential components can be used to identify and localize stress concentration zones in gantry crane beams. Special attention was given to the unsymmetrical distribution of the tangential component gradient on the surface of the crane beam No. 5 (which was the most loaded one). The anomaly was the effect of a slight torsional deflection of the beam as it was loaded. Numerical simulations with the finite element method (FEM) were used to explain this phenomenon. The displacement boundary conditions introduced into the simulations were established experimentally. Validation was carried out using the X-ray diffraction method, which confirmed the location of strain concentration zones (SCZs) identified by means of RMF testing.