Liu Qingyou, Luo Xu, Zhu Haiyan, Liu Jianxun, Han Yiwei. Modified magnetomechancial model in the constant and low intensity magnetic field based on J–A theory. Chinese Physics B, 2017, 26(7): 077502
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Modified magnetomechancial model in the constant and low intensity magnetic field based on J–A theory
Liu Qingyou1, 2, Luo Xu1, †, Zhu Haiyan3, Liu Jianxun1, Han Yiwei1
College of Mechatronic Engineering of Southwest Petroleum University, Chengdu 610500, China
Key Laboratory of Fluid and Power Machinery (Ministry of Education), Xihua University, Chengdu 600300, China
College of Petroleum Engineering of Southwest Petroleum University, Chengdu 610500, China
† Corresponding author. E-mail: 402585133@qq.com
Abstract
Abstract
The existing magnetomechancial models cannot explain the different experimental phenomena when the ferromagnetic specimen is respectively subjected to tension and compression stress in the constant and low intensity magnetic field, especially in the compression case. To promote the development of magnetomechancial theory, the energy conservation equation, effective magnetic field equation, and anhysteretic magnetization equation of the original Jiles–Atherton (J–A) theory are elucidated and modified, an equation of the local equilibrium status is employed and the differential expression of the modified magnetomechancial model based on the modified J–A theory is established finally. The effect of stress and plastic deformation on the magnetic parameters is analyzed. An excellent agreement is achieved between the theoretic predictions by the present modified model and the previous experimental results. Comparing with the calculation results given by the existing models and experimental results, it is seen indeed that the modified magnetomechanical model can describe the different magnetization features during tension-release and compression-release processes much better, and is the only one which can accurately reflect the experimental observation that the magnetic induction intensity reverses to negative value with the increase of the compressive stress and applied field.
Because of the combined action of geomagnetic field (about 40 A/m), stress–strain, and change in material microstructure, a self-spontaneous leakage magnetic field will be formed on the surface of the stress concentration or plastic deformation region in the ferromagnetic material. Based on this effect, the Russian researcher Dubov first proposed the concept of metal magnetic memory (MMM) in 1994,[1] and put forward the MMM technique in the 1998 fiftieth International Conference on welding.[2] As a new nondestructive detection method, which can effectively detect the early damage and assess the development trend of fatigue damage, it has been widely studied by the European and Chinese scholars.[3–7]
Essentially, the MMM effect is a result of the magnetomechanical effect, which is the change of magnetization resulting from the application of stress in the constant and low intensity field. Because of its complexity, lots of theoretical and experimental studies have been carried out.[8–11] Among them, Jiles and Atherton assumed that the main effect on the magnetization of the ferromagnetic material caused by cycling the applied stress is an irreversible change in the prevailing magnetization towards the anhysteretic magnetization , which is named the approach law, and the bulk magnetization M is the sum of a reversible component and an irreversible component . Finally, they established the famous Jiles–Atherton (J–A) theory.[12–14] Then, based on the J–A theory, Sablik[11,15] and Jiles[16] took the elastic stress as an equivalent field and established the primal J–A–S model for the magnetomechanical effect, but the theoretical and experimental studies of them mainly focus on analyzing the magnetization due to a changed magnetic field under the constant stress or deformation, not the magnetization along with the changed stress in the constant magnetic field. In order to make the magnetomechanical model suitable for the magnetization along with the stress in the constant magnetic field, Sablik[17] employed two different relaxation constants and respectively for loading and releasing stress processes, and modified the approach law to improve the accuracy. Li et al.[18,19] applied the Rayleigh law to the stress, as well as a new linear stress-dependent term to improve the agreement near zero stress. During the stage of plastic deformation, Leng,[4] Wang,[5] Li,[6] Jiles,[20] Sablik,[21,22] Lo,[23]et al. analyzed the effect of plastic deformation and stress on the parameters, such as pinning constant k and coupling coefficient α, and derived different magnetomechanical models for plastic deformation.
As for the approach law of the original J–A theory, the is a global equilibrium status, it only happens when the magnetization is on the major hysteresis loop or initial magnetization curve. However, in the constant and low intensity field, the magnetization is on the minor hysteresis loop, and the application of stress induces change in magnetization toward the local equilibrium status , so the approach law of cannot be directly applied to model the magnetomechanical effect in the constant and low intensity field.[24–26] In order to solve this problem, Xu[27] established an equation of and derived the J–A–F model at elastic deformation stage. Nevertheless, the calculated results of the J–A–F model are only in agreement with the experiment results during the tension-release process, and cannot reflect the experimental observation that the magnetic induction intensity reverses to a negative value with the increase of the compressive stress and the external field. Additionally, there are also some mistakes in the existing models, such as having confused with M.
The objective of this paper is to systematically identify and modify the inaccuracy of the energy conservation equation, effective magnetic field equation, and anhysteretic magnetization equation of the original J–A model and to develop a modified magnetomechanical model in the constant and low intensity magnetic field, which is suitable for both tension-release and compressive-release processes.
2. Modification of the J–A theory
The key parts of the J–A theory are the energy conservation equation, the effective magnetic field equation, and the anhysteretic magnetization equation. However, there are several queries about them by analyzing the classic papers introducing the J–A theory. So it is necessary to determine which expressions are reasonable.
2.1. The energy conservation equation
As is known, there are defects, irregularities of crystalline lattice defects, impurities, etc. within the ferromagnetic material, which hamper the movement of the domain wall, resulting in the hysteresis loss. According to the law of conservation of energy, the total magnetization energy is equal to the anhysteretic energy which would be in the ideal or unpinned case minus the hysteresis loss due to the pinning effect. By analyzing three different forms of energy conservation equation in the classic papers [11,13], and [28], it is concluded that it is inaccurate to replace with M to calculate the hysteresis loss in Ref. [11], and unreasonable to regard the irreversible magnetization energy as the total magnetization energy in Ref. [13]. Additionally, Jiles added without a clear interpretation to the hysteresis loss, and there is also a mistake replacing with in Ref. [28]. Consequently, the correct expression of the energy conservation equation should be
Differentiating Eq. (1) leads to
2.2. The effective magnetic field
The notion of effective magnetic field plays a crucial role in the J–A theory, and the effect of the applied stress is represented by an equivalent field in the model of effective magnetic field .[16] However, there is a discrepancy in the definition of the equivalent field among different authors: some of them claimed that the applied stress σ should be employed to calculate , such as Sablik in Ref. [11]; some of them assumed that the σ should be replaced with the residual stress , such as Leng in Ref. [4] and Sablik in Refs. [22] and [29]; and the others introduced the decomposition of into the magneto-elastic induced by elastic deformation and the magneto-plastic induced by plastic deformation, such as Wang in Ref. [5] and Li in Ref. [6]. Moreover, which one of and M should be applied to also remains a point of divergence.
Actually, depending on the situation, should be discriminated between and , where is the anhysteretic effective magnetic field for anhysteretic calculating, and is the hysteretic effective magnetic field for hysteretic calculating. So the could be respectively written as
2.3. The anhysteretic magnetization
For the description of the anhysteretic magnetization, Jiles and Sablik have applied the modified Langevin function[15–20]
Nevertheless, some researchers claimed that the anhysteretic magnetization should be given as an implicit of itself[21,22,30]
According to the physical definition of , it can be concluded that the curve of along with the changed magnetic field H or stress σ should be a single value curve without hysteretic loop. It can be easily noticed that the assumption of the definition given by Eq. (5) leads to the conclusion that even the anhysteretic curve itself reveals hysteresis with respect to the stress σ, which is inconsistent with the physical definition of the hysteresis magnetization, as shown in Fig. 1(a). On the contrary, it is obvious that the calculated result of Eq. (6) is reasonable, as shown in Fig. 1(b), so equation (6) is the proper one.
Fig. 1. (color online) Comparison between (a) Eq. (5) and (b) Eq. (6) for anhysteretic magnetization .
3. Modification of the magnetomechanical model
3.1. The local equilibrium status
The existing magnetomechanical models established by Sablik, Jiles, Leng et al. are all based on the global equilibrium status . From the microcosmic perspective, the global equilibrium status is the case of ideal crystal in which the domain walls move until they reach thermodynamic equilibrium without any pinning site obstruction. However, according to the results of the experiment, the application of stress in a constant and low intensity magnetic field induces change in magnetization toward a local equilibrium status , which should be a stabilized loop around the anhysteretic magnetization curve, as shown in Fig. 2. So can be taken as a middle status from M to . In contrast with status , must have some pinning sites in the crystal structure of the material, and the pinning coefficient k1 is employed to characterize the pinning loss from to . The smaller the k1 is, the less the amount of pinning sites that cannot be overcome is, the closer the is to , indicating that the hysteresis loss is less from to .
Fig. 2. (color online) The experimental hysteresis loops during different stress cycles in Figs. 4 and 5 of Ref. [31]: (a) cycles 500–505, (b) cycles 1600–1650.
where is the magnetization energy at the local equilibrium status, is the hysteretic loss due to the pinning effect from to , k1 is the pinning coefficient from to , which mathematically controls the amount of pinning sites that cannot be overcome by applying a stress in low magnetic field, is the inelastic energy loss, η is the coefficient with the dimensions of the energy per unit volume, and W is the elastic energy per volume. Differentiating Eq. (7) leads to
Because , , replacing with in Eq. (3) and solving Eqs. (3) and (8) simultaneously give
With the model parameters of Fig. 9 in Ref. [16], and by taking , σ = −100 MPa to 100 MPa, the variation with stress is shown in Fig. 3.
Fig. 3. (color online) The –σ curves with different k1 when H = 40 A/m.
It can be seen that the curve of –σ is a loop around the curve of –σ. With the increase of k1, the curve of –σ is far away from the curve of –σ, and the enclosed area increases, indicating a larger energy loss from to , which is nicely coincident with the theoretical predications above and the experimental results show that the magnetization caused by stress is moving towards a stabilized loop, as seen in Fig. 3.
3.2. The modification of magnetization M
According to the modified energy conservation equation in subSection 2.1, by replacing the with in Eq. (3), the energy conservation equation from M to is
where k2 is the pinning coefficient from M to . Differentiating and ordering Eq. (10) give
Because and ,
Differentiating Eq. (12) leads to
Differentiating Eq. (4) leads to
The combination of Eqs. (11), (13), and (14) leads to the differential magnetization susceptibility
Solving Eqs. (3), (4), (6), (9), and (15) simultaneously by using the standard variable-step Runge–Kutta method, the M changing with can be determined in the constant and low intensity field.
3.3. The interrelating effects of plastic deformation and stress on the model parameters
Figure 4 shows the stress against strain (.[21] It is seen that starting from zero applied stress, the deformation is increased linearly as the stress is increased. Once the yield stress is reached and the applied stress continues to increase, the deformation is increased nonlinearly from until some final deformation is reached. The deformation we shall call , the plastic deformation after yield. The difference is known as the strain hardening stress , which is the part of stress producing plastic deformation, resulting in the increase of dislocation density . According to the experimental result of the x-ray test, the relationship between and is[21]
where αk is a constant which is given as 0.76, G is the specimen shear modulus as given by , Y is the Yang’s modulus, v is the Poisson’s ratio, b is the appropriate Burgers vector magnitude for the specimen’s dislocations, and is the initial dislocation density prior to plastic deformation.
Fig. 4. (color online) Schematic diagram of tensile stress–strain curves for ferromagnetic materials. Applying a stress ( to a specimen, the specimen deforms elastically and linearly up to yield stress , and then deforms plastically to strain , and the plastic deformation , in which is the strain at the yield point. After it is relaxed, the strain returns elastically to a residual strain .
With the change of stress and plastic deformation, the pinning coefficient k2, scaling constant a and coupling coefficient α would be changed as follows:[6,21,23]
where is the pinning coefficient related to dislocation, and will increase with the increase of plastic deformation, . , m is a constant, is the saturation magnetostriction constant, indicates that the pinning effect will decrease with the increase of stress. a0 is the initial scaling constant. d is the grain size. q1, q2, q3 are the material constants.
4. The result of modification
Consider the model parameters of Fig. 9 in paper [16], and take , , m = 0.2, , , , , , , , , , , c = 0.1, , , , , , , , , , . The magnetic induction calculated by the modified model is shown in Fig. 5(e) when the external field is 26.4 A/m, 80 A/m, and 132 A/m.
Fig. 5. (color online) Comparison of the different results given by the modified model, the original models, and the experiment. (a) The testing results of No. 2 specimen in paper [32]. (b) The results of Jiles’ model in Fig. 9 of paper [16]. (c) The results of Li’s model in paper [6]. (d) The results of Xu’s model in paper [27]. (e) The result of the modified model. (f) The result of the modified model after optimizing parameters.
Comparing with the results calculated by the existing models and the experimental results of No. 2 specimen in paper [32], we obtain the following results.
(i) According to the experimental results, the variation of during the tension-release process is much more significant than that during the compression-release process, and the residual after the tension-release process is much stronger than that after the compression-release process. The models established by Li and Xu and the modified model can describe the significant difference effectively, while Jiles’ model cannot.
(ii) During the tension-release process, with the increase of the stress, the magnetic intensity increases at first, then drops much slower for tension than for compression; the larger the external magnetic field H is, the more obvious the phenomenon is. It is at about 60 MPa that it reaches the maximum during tension. As shown in Fig. 5(c), it is at about 50 MPa where Li’s model shifts into a reversal and the decaying rate of Li’s model is much higher than that of the experimental result. In contrast, the results calculated by the Jiles, Xu, and the modified models can reflect this feature accurately.
(iii) During the compression-release process, the firstly increases and then decreases along with the increase of the stress. When H = 26.4 A/m, the experimental values are all positive; while when H = 80 A/m and 132 A/m, parts of the testing data are lower than zero, and with the applied field H increasing, the experimental decreases to negative. However, the theoretical predictions of magnetic intensity by Jiles’, Li’s, and Xu’s models are all positive, and the magnetic intensity increases with the increase of the applied field H, which is contrary to the experimental phenomenon. In comparison, it can be seen that the theoretic predictions by the modified model are in considerably good agreement with the test data, particularly in the compressive situation, and the modified model is the only one which can describe the experimental phenomenon correctly.
(iv) Additionally, because the calculated result is very sensitive to the parameters,[24] the slope of the –σ curve calculated by the modified model is much smaller in ascending and greater in descending than that of the experimental results when taking the same parameters of Fig. 9 in paper [16]. The problem can be solved by optimizing the related parameters. For example, the genetic algorithm (GA) is employed to optimize , , , and , and the objective function is defined as , where Xi is the calculated by the presented modified model, xi is the experimental , and i denotes the corresponding data point number. When , , , , the MSE reaches its minimum, indicating that the calculated error between the calculated and the experimental results is least, and the theoretic results are in better agreement with the experimental results, as shown in Fig. 5(f).
Based on the analysis above, it is concluded that the modified model can describe the experimental observation of the magnetomechanical effect in the constant and low intensity field much better than the existing models, and the modification is reasonable.
5. Conclusion
In this study, the simultaneous equations of the modified magnetomechanical model in the constant and low intensity field are established by modifying the J–A theory and employing the equation of . Comparing with the results calculated by the existing models, the modified magnetomechanical model can describe the different magnetization features in tension-release and compression-release processes much better, and is the only one which can reflect the experimental observation that the magnetic induction intensity reverses to negative value with the increase of the external field and compressive stress effectively. Nevertheless, in order to make the modified model more accurate and effective, future work shall be aimed at optimizing the related parameters.
