⇦Chinese Physics Letters, 2020, Vol. 37, No. 8, Article code 087502 A Nonlinear Theoretical Model of Magnetization and Magnetostriction for Ferromagnetic Materials under Applied Stress and Magnetic Fields Pengpeng Shi (时朋朋)1,2* Affiliations 1School of Civil Engineering & Institute of Mechanics and Technology, Xi'an University of Architecture and Technology, Xi'an 710055, China 2State Key Laboratory for Strength and Vibration of Mechanical Structures, Shaanxi Engineering Research Center of NDT and Structural Integrity Evaluation, School of Aerospace, Xi'an Jiaotong University, Xi'an 710049, China Received 18 April 2020; accepted 11 June 2020; published online 28 July 2020 Supported by the National Natural Science Foundation of China (Grant No. 11802225) and the Natural Science Basic Research Plan in the Shaanxi Province of China (Grant No. 2019JQ-261).
*Corresponding author. Email: shipengpeng@mail.xjtu.edu.cn Citation Text: Shi P P 2020 Chin. Phys. Lett. 37 087502    Abstract A thermodynamic and micro-statistical model is proposed to explain the magnetization and magnetostriction mechanisms for isotropic ferromagnetic materials. Here a nonlinear magnetostrictive expression enhances the characterization of the nonlinear magnetic-mechanical effect, and the Brillouin function makes it possible to describe the relationship between the equivalent field and magnetization for various types of materials. Through detailed comparisons with the recent models of Wu et al. [Appl. Phys. Lett. 115 (2019) 162406] and Daniel [Eur. Phys. J.: Appl. Phys. 83 (2018) 30904], it is confirmed that the proposed model can provide greater physical insight and a more accurate description of the complex magnetostriction and magnetization behaviors, especially the complex nonlinearity of stress effects. DOI:10.1088/0256-307X/37/8/087502 PACS:75.80.+q, 75.60.Ej, 75.50.Bb © 2020 Chinese Physics Society Article Text The basic theory of ferromagnetism suggests that length of a ferromagnetic material placed in an external magnetic field can change due to variation of its magnetization state. That is, ferromagnetic materials have magnetostrictive properties, which are also known as the Joule effect.[1] Applied stress can change orientations of magnetic domains inside a ferromagnetic material, which alters its magnetic properties. Thus, ferromagnetic materials have an inverse magnetostrictive effect, also known as the Villaiy effect.[2] The magnetostrictive and inverse magnetostrictive effects reflect a mutual conversion between the stress and magnetism in ferromagnetic materials, known as the magneto-mechanical coupling effect.[3] Research on these magneto-mechanical coupling effects provides guidance to understand the magnetization and magnetostrictive behaviors of ferromagnetic materials under external magnetic fields, as well as to design and develop several different intelligent structures and nondestructive testing technology.[4–6] The magneto-mechanical coupling behaviors of ferromagnetic materials have attracted strong scientific interest for a long time. In 1984, Jiles proposed a hysteresis model of ferromagnetic materials based on the principle of magnetic domain motion. Since anhysteresis magnetization in the early Jiles model was based on the classic Weiss magnetization model, the theoretical curves under different applied stresses are realized by adjusting the model parameters.[7] In 1987, Sablik et al. modified the early model by introducing a stress equivalent field, which considered the stress effect on the anhysteresis and hysteresis behavior under the same set of parameters.[8] Based on the differential equivalent stress field expression, Jiles et al. established the final form of the Jiles model with clearer physical meaning by suggesting the even-order polynomial expression between magnetostriction and magnetization in 1994.[9,10] Over the next two decades, researchers have further promoted the Jiles model and the influence of plasticity, temperature, non-coaxial loads and other complex factors on the magneto-mechanical coupling behavior.[11–14] By considering the nonlinear stress effects in great depth, Zheng and Liu proposed a nonlinear magneto-mechanical constitutive model in 2005 based on a thermodynamic framework.[15] Because the model is easy to apply in practice and the meaning of its parameters is clear, the Zheng–Liu model has been widely used to analyze the magneto-mechanical phenomenon of ferromagnetic materials. In 2007, Zheng et al. established the magneto-thermo-elastic modified model by considering the temperature effect.[16,17] Further, by considering the approach law based on elastic energy, Shi et al. established a magneto-mechanical model for the inverse magnetostrictive effect of ferromagnetic materials under a constant weak magnetic field,[18] and further analyzed the complicated magneto-thermo-elastoplastic coupling phenomenon.[19] In addition, researchers have conducted theoretical research on the magneto-mechanical coupling phenomenon of ferromagnetic materials based on other methods. Daniel proposed a theoretical model based on the magnetoelastic energy of ferromagnetic materials.[20] Wu et al. developed a micro-statistical model calculating the magnetization and magnetostriction of isotropic ferromagnetic materials.[21] Compared with the previous experimental results,[22] the model proposed by Wu et al.[21] can better describe the relation between the magnetostriction and magnetization than the model proposed by Daniel,[20] especially for the experimental results under a compressive stress. Li et al. analyzed magnetization dynamics of nanocomposite magnet based on the micromagnetism simulation.[23] In this letter, a theoretical model is proposed for the magnetization and magnetostriction of ferromagnetic materials based on thermodynamic principles and the magnetic moment statistical method. The existing experimental results are compared with the classic Jiles model,[10] the two newer models,[20,21] and the proposed model. It is confirmed that the proposed model provides greater physical insight and gives a more accurate depiction of complex magnetostriction and magnetization behaviors. A magneto-mechanical coupling constitutive relationship for ferromagnetic materials is established based on thermodynamic theory. The total differential of the elastic Gibbs free energy $G_{\rm e}$ is found, regardless of the temperature change, and we obtain the following thermodynamic relationship[24] $$ \varepsilon =-\frac{\partial G_{\rm e} }{\partial \sigma },~~~\mu_{0} H=\frac{\partial G_{\rm e} }{\partial M},~~ \tag {1} $$ where $\varepsilon$ is the strain of the material, $H$ is the environmental magnetic field, $\mu _0$ is the vacuum permeability, $\sigma$ is the stress, and $M$ is the magnetization. Based on the previous experimental results[25] and considering that the magnetostriction of a material is an even function of the magnetization $M$ related to the stress $\sigma$, an expression for the magnetostriction is[18,26] $$ \lambda =\left[ {1-g(\sigma)} \right]\frac{\lambda_{\rm s} M^{2}}{M_{\rm ws}^{2} }-\frac{\theta \lambda_{\rm s} \left({M^{4}-M_{0}^{4} (\sigma)} \right)}{M_{\rm ws}^{4} },~~ \tag {2} $$ $$ g(\sigma)=\left\{ {{ \begin{array}{*{20}c} {\tanh \left({\beta \sigma /\sigma_{\rm s} } \right), ~~\sigma \geqslant 0,} \hfill \\ {\tanh \left({2\beta \sigma /\sigma_{\rm s} } \right)/2, ~~\sigma < 0.} \hfill \\ \end{array} }} \right.~~ \tag {3} $$ Thus, $\lambda \left({\sigma,M} \right)=\varepsilon \left({\sigma,M} \right)-\varepsilon (\sigma)$, indicating that the magnetostrictive strain is equal to the total strain minus the pure elastic strain component. The first term on the right-hand side of Eq. (2) indicates that the magnetostrictive strain in soft ferromagnetic materials increases due to the magnetic domain wall shifting process. The second term indicates that the magnetostrictive strain decreases due to magnetic domain rotations after the magnetization exceeds the saturation value. Here $M_{\rm ws}$ is the saturation wall shift magnetization of the material in an unstressed state, $\sigma_{\rm s}$ is the stress related to the yield of the material, $\lambda_{\rm s}$ is the maximum magnetostriction of the material in an unstressed state, $\beta$ is a material parameter related to the growth rate of pure elastic strain, $M_{0} (\sigma)$ represents the stress-dependent saturation wall shift magnetization; $\theta$ is dimensionless step function, $\theta$ is zero when $M < M_0$, and $\theta$ is a constant value when $M \geq M_0$. The constant is positive, which is related to the value of the magnetostriction reduction caused by the magnetic domain rotation. The above expression describes the relationship between the magnetostriction, magnetization, and applied stress. The magnetization can be expressed as a function of effective field as $$ M=f(H_{\rm eff}).~~ \tag {4} $$ The effective field of a ferromagnetic material under the combined action of external forces and a magnetic field can be expressed as the sum of the environmental magnetic field and the stress equivalent field. That is, $$ H_{\rm eff} =H+H_{\sigma }.~~ \tag {5} $$ Equations (1) and (2) are used to calculate the stress equivalent field caused by the inverse magnetostrictive effect as $$\begin{alignat}{1} \mu_{0} H_{\sigma } ={}&\frac{\partial }{\partial M}\int {\lambda d\sigma }=\frac{2\lambda_{\rm s} M}{M_{\rm ws}^{2} }\int_0^\sigma {\Big({1-\tanh \frac{\sigma }{\sigma_{\rm s} }} \Big)d\sigma }\\ & -\frac{4\theta \lambda_{\rm s} \sigma }{M_{\rm ws}^{4} }({M^{3}-M_{0}^{3} (\sigma)}).~~ \tag {6} \end{alignat} $$ Finally, the equivalent stress field of a material under the combined action of the stress and magnetic field can be obtained as $$\begin{alignat}{1} H_{\rm eff} ={}& H+\frac{2\lambda_{\rm s} M}{\mu_{0} M_{\rm ws}^{2} }\left({\sigma -G(\sigma)} \right)\\ &-\frac{4\theta \lambda_{\rm s} \sigma }{\mu_{0} M_{\rm ws}^{4} }\left({M^{3}-M_{0}^{3} (\sigma)} \right),~~ \tag {7} \end{alignat} $$ $$ G(\sigma)=\left\{ {{ \begin{array}{*{20}c} {\sigma_{\rm s} \ln \cosh \left({\beta \sigma /\sigma_{\rm s} } \right)/\beta,~~ \sigma \geqslant 0,} \hfill \\ {\sigma_{\rm s} \ln \cosh \left({2\beta \sigma /\sigma_{\rm s} } \right)/4\beta, ~~\sigma < 0.} \hfill \\ \end{array} }} \right.~~ \tag {8} $$ In the Jiles model, the relationship of the magnetostriction with applied stress and magnetization satisfies the expression $\lambda =\gamma_{1} (\sigma)M^{2}+\gamma_{2} (\sigma)M^{4}$. It is worth noting that the second-order and fourth-order coefficients of magnetization in the Jiles model are all linearly related to stress, which makes the model impossible to accurately describe the complex nonlinear magnetostriction variation of ferromagnetic material with applied stress and magnetic fields. The proposed nonlinear magnetostrictive expression enhances the quantitative advantage of the model in describing the nonlinear magnetic-mechanical coupling effect, which is confirmed in the subsequent analysis. The expression of the function $f$ in $M=f(H_{\rm eff})$ is further obtained based on the magnetic moment statistical method.[27] In the absence of an external magnetic field, the magnetic moments at each atom are distributed chaotically under thermal equilibrium, giving a total magnetic moment of zero. When an external magnetic field is considered, the magnetic moment of each atom tends to point along the direction of the magnetic field, while the exact angle deviations $\vartheta$ from the magnetic field are different. The energy of each atomic magnetic moment under the action of a magnetic field is $-\mu_{0} H_{\rm eff} \cos \vartheta$. According to quantum theory, the momentum moment of each atom is spatially quantized in an external magnetic field. Thus, the component of the momentum moment along the magnetic field is $m_{J} \hbar$, and the magnetic moment component is $m_{J} g\mu_{\rm B}$, where $m_{J}$ is the magnetic quantum number given as $m_{J} = -J,- (J-1),\ldots,0,1,2,\ldots,+J$. Therefore, the partition function and magnetic energy of a single magnetic moment can be expressed as $$ Z_{m} =\sum\limits_{m_{J} =-J}^{+J} \exp \Big(\frac{m m_J}{k_{\rm B} T}H_{\rm eff} \Big),~~ \tag {9} $$ $$ G=-k_{\rm B} T\ln {[Z_{m}] } ^{N}=-k_{\rm B} TN\ln {[Z_{m}] } ,~~ \tag {10} $$ where $m=g_{\rm J} \mu _{\rm B}$ with $g_{\rm J}$ and $\mu _{\rm B}$ being the Lande spectroscopic $g$ factor and Bohr magneton, respectively; $k_{\rm B}$ is Boltzmann's constant, $T$ is the temperature, $J$ is total quantum number, and $N$ is the number of atoms. The magnetization $M$ can be expressed as $$ M=-\frac{1}{\Delta V}\Big({\frac{\partial G}{\partial H}} \Big)=\frac{Nk_{\rm B} T}{\Delta V}\frac{1}{Z_{m} }\frac{dZ_{m} }{dH}.~~ \tag {11} $$ Substituting Eq. (10) into Eq. (11) yields $$ M=M_{\rm s} B_{J} ({kH_{\rm eff} }),~~ \tag {12} $$ where $k=(J_{\rm m})/({k_{\rm B} T}),~M_{\rm s} =NJ_{\rm m}/\Delta V$, and $B_{J} (x)$ is the Brillouin function. $J_{\rm m}$ corresponds to the maximum magnetic moment value displayed by the external energy of the atom. In this study, the Brillouin function describing the relationship between the equivalent field and magnetization is proposed. The existence of the parameter $J$ in the nonlinear Brillouin function makes it possible to accurately describe the relationship between the equivalent field and magnetization of various types of materials.[28] As $J\to \infty$, $$ M=M_{\rm s} \Big(\coth x-\frac{1}{x}\Big).~~ \tag {13} $$ This shows that when the magnetic quantum number is sufficiently high, the magnetization curve approaches the results obtained from the classic Langevin model for isotropic materials (3D case). In addition, when $J = 1/2$, we have $B_{1/2} (x)=\tanh x$, and this expression is suitable to describe the anhysteretic magnetization for an axially anisotropic material which is magnetized along the easy axis (1D case). Finally, under the combined effects of stress and a magnetic field, the theoretical model of magnetization and magnetostriction for ferromagnetic materials based on micro-statistics and thermodynamic theory can be expressed as $$ M=M_{\rm s} B_{J} ({kH_{\rm eff} }),~~ \tag {14} $$ $$ \lambda =\varepsilon -\lambda_{\rm s} g(\sigma)=[ {1-g(\sigma)} ]\frac{\lambda_{\rm s} M^{2}}{M_{\rm ws}^{2} }-\frac{\theta \lambda_{\rm s} ({M^{4}-M_{0}^{4} (\sigma)})}{M_{\rm ws}^{4} },~~ \tag {15} $$ $$ H_{\rm eff} =H+\frac{2\lambda_{\rm s} M}{\mu_{0} M_{\rm ws}^{2} }({\sigma -G(\sigma)})-\frac{4\theta \lambda_{\rm s} \sigma }{\mu_{0} M_{\rm ws}^{4} }({M^{3}-M_{0}^{3} (\sigma)}).~~ \tag {16} $$ Equations (14) and (15) give analytic expressions for the theoretical models of magnetization and magnetostriction in ferromagnetic materials. This theory is mainly applicable to ferromagnetic materials with magnetic domain rotations, such as low-carbon steel. For magnetostrictive materials that do not undergo magnetic domain rotations during magnetization, a simplified theoretical model suitable for magnetostrictive materials is obtained by setting $M_{0} (\sigma)=M_{\rm s}$. To facilitate comparisons between the proposed model and the previous models, the theoretical formulas for the classic Jiles model[10] and that proposed by Daniel in 2018[20] are given. Jiles proposed an empirical model to describe the relationships between magnetization and magnetostriction in ferromagnetic materials, which can be expressed as[10] $$ M=M_{\rm s} \left[ {\coth \left({H_{\rm eff} /a} \right)-a/H_{\rm eff} } \right],~~ \tag {17} $$ $$ \lambda =\gamma_{1} (\sigma)M^{2}+\gamma_{2} (\sigma)M^{4},~~ \tag {18} $$ $$ H_{\rm eff} =H+\alpha M+\frac{3\sigma }{\mu_{0} }\left({\gamma_{1} (\sigma)M+2\gamma_{2} (\sigma)M^{3}} \right),~~ \tag {19} $$ where $M_{\rm s}$ is the saturation magnetization, $H_{\rm eff}$ is the effective magnetic field, $\lambda$ is the magnetostriction, $H$ is the environmental magnetic field, $\gamma_{1}$ and $\gamma_{2}$ are the stress-dependent material coefficients, $\alpha$ is the self-coupling energy coefficient, $\sigma$ is the stress, $\mu_{0}$ is the vacuum permeability, and $a$ is a material parameter that controls the degree of magnetization. Daniel[20] proposed a theoretical model based on the magnetoelastic energy of ferromagnetic materials. Under a uniaxial tensile stress, the analytical model is expressed as $$ M=M_{\rm s} \Big({\frac{A_{x} \sinh \left({\kappa H} \right)}{A_{x} \cosh \left({\kappa H} \right)+2}} \Big),~~ \tag {20} $$ $$ \lambda =\lambda_{\rm s} \Big({1-\frac{3}{A_{x} \cosh \left({\kappa H} \right)+2}} \Big),~~ \tag {21} $$ where $A_{x} =e^{\beta \sigma}$, $\beta$ is the material parameter, $\lambda_{\rm s}$ is the saturation magnetostrictive coefficient, $\kappa ={3\mu_{0}^{0} } / {M_{\rm s}}$, and $\mu_{0}^{0}$ is the initial anhysteretic magnetic permeability of the material in an unstressed state.
cpl-37-8-087502-fig1.png
Fig. 1. Strain vs magnetization under different stresses for an iron-cobalt alloy. Solid lines: proposed model. Points: experimental data.[22] Dashed lines: Daniel's model (2018).[20]
Figure 1 compares the predicted results of the proposed model with those of Daniel[20] and the related experimental results.[22] The material parameters[22] used in our theoretical model are taken as $M_{\rm s} =1.9\times 10^{6}$ A/m, $\lambda_{\rm s} =83$ ppm, $k=1/1500$ m/A, $\sigma_{\rm s} =300$ MPa, $\beta =8$, $\theta =0$. The magnetostriction of a ferromagnetic material is symmetric with respect to the magnetization $M=0$. When $\sigma =100$ MPa, the magnetostriction of the material remains nearly unchanged. Both Daniel's model and the proposed model can closely predict the experimental results in this case. However, when $\sigma =0$ MPa and $\sigma =100$ MPa, there is a gap between Daniel's model and the experiments. Compared with Daniel's model,[20] the magnetostrictive curve predicted using the proposed model is in better agreement with the experimental results and can accurately reflect the magnetostrictive behavior of ferromagnetic materials with magnetization. Figure 2 depicts the change of magnetostrictive strain for an iron-cobalt alloy under the applied stress and magnetic field, where the comparison between the experimental results[20] and the prediction results of the model proposed here and the model of Wu et al.[21] is given. The model of Wu et al. predicts that the magnetostrictive strain decreases monotonically with increasing stress,[21] which is inconsistent with the nonlinear stress effect depicted by the experiment.[20] The material parameters[20] used in our theoretical model are taken as $M_{\rm s} =2\times 10^{6}$ A/m, $\lambda_{\rm s} =70$ ppm, $k=1/350$ m/A, $\sigma_{\rm s} =245$ MPa, $\theta =0$, $\beta =9$ when $\sigma \geqslant 0$, and $\beta =3$ when $\sigma < 0$. As shown in Fig. 2, the prediction results of the model proposed in this study are more accurately consistent with the experimental values.[20] This further confirms the effectiveness and advantages of this proposed model.
cpl-37-8-087502-fig2.png
Fig. 2. Longitudinal magnetostriction strain vs applied stress under uniaxial stress for an iron-cobalt alloy: (a) theoretical analysis from the proposed model, (b) theoretical analysis from the model of Wu et al.[21]
cpl-37-8-087502-fig3.png
Fig. 3. Magnetization vs applied magnetic field under uniaxial stress for an iron–cobalt alloy.
Figure 3 shows the magnetization of an iron-cobalt alloy with the stress state and magnetic field based on the proposed model. The model parameters used here are the same as those in Fig. 2. Figure 3 shows that under the same stress, the magnetization of the iron-cobalt alloy gradually increases with the increase of magnetic field and tends to saturation. As the stress value increases, the material becomes easier to magnetize due to the enhancement effect of the stress equivalent field. For the TbDyFe giant magnetostrictive material, Fig. 4 analyzes the change of magnetostrictive strain with the applied stress and magnetic field. Figures 4(a) and 4(b) show the prediction results based on the model proposed in this study and the model of Wu et al.,[21] respectively. The material parameters[29] used in our theoretical model are taken as $M_{\rm s} =2\times 10^{6}$ A/m, $\lambda_{\rm s} =950$ ppm, $k=1/38000$ m/A, $\sigma_{\rm s} =245$ MPa, $\theta =0$, and $\beta =20$. Figure 4(b) shows that in the predicted results of the model of Wu et al.,[21] the magnetostriction increases monotonously with the increasing stress, which is obviously inconsistent with the non-monotonic law of stress effect reflected in the experimental results in Fig. 1 of Ref. [29]. Compared with the model of Wu et al.,[21] the model proposed here can basically reflect the nonlinear law of stress effect, and can more accurately reflect that the saturation magnetization decreases with increasing stress as revealed by experiments in Ref. [29].
cpl-37-8-087502-fig4.png
Fig. 4. Magnetostriction strain vs applied magnetic field under uniaxial stress for a TbDyFe magnetostrictive alloy: (a) theoretical analysis from the proposed model, (b) theoretical analysis from the model of Wu et al.[21]
cpl-37-8-087502-fig5.png
Fig. 5. Magnetostriction of materials under different stresses vs magnetization for carbon steel.
Figure 5 shows the changes in the magnetostriction of ferromagnetic materials with magnetization. The material parameters[25] used in the present theoretical model are taken as $M_{\rm s} =2\times 10^{6}$ A/m, $M_{\rm ws} =0.98\times 10^{6}$ A/m, $\lambda_{\rm s} =4$ ppm, $k = 1/1000$ m/A, $\sigma_{\rm s} = 300$ MPa, $\beta =2.8$, $\theta =3/4$. It is seen that the magnetostriction of ferromagnetic materials first increases before decreasing for a growing magnetization, and the maximum magnetostrictive strain gradually increases with a greater compressive stress. As shown in Fig. 2, the model from Jiles in 1995[10] and the proposed model both reflect the change trends of the magnetostrictive strain for ferromagnetic materials under different magnetizations and applied stresses, and the proposed model is more consistent with the experimental data.[25] However, for the latest model reported by Wu et al.,[21] the magnetostriction is monotonic with the increased magnetization, which is different from the experimental results for carbon steel. Figure 6 compares the predicted magnetization of a material with an applied magnetic field using the proposed model with the associated experimental results.[10] The material parameters[10] used in our theoretical model are taken as $M_{\rm s} =1.38\times 10^{6}$ A/m, $M_{\rm ws} =1.16\times 10^{6}$ A/m, $\lambda_{\rm s} =4.17$ ppm, $k=1/1600$ m/A, $\sigma_{\rm s} =245$ MPa, $\beta =1$, $\theta =3/4$. It is seen that the magnetization of the ferromagnetic material gradually increases with the external magnetic field and tends to converge to a saturation state. When the material is subjected to a compressive stress, it is more difficult to magnetize the material under a low magnetic field ($H < 10$ kA/m). The magnetization of a compressed material under the same magnetic field is lower than the value when it is not prestressed. Under a tensile stress, the magnetization of the material under a low magnetic field ($H < 10$ kA/m) coincides with the magnetization curve without the prestress. The predictions from the proposed model can reflect the effects of stress on the magnetization behavior of ferromagnetic materials.
cpl-37-8-087502-fig6.png
Fig. 6. Magnetization of materials under different stresses vs an applied magnetic field for low-carbon steel.
In summary, the principles of thermodynamics and statistical methods of magnetic moments are used to develop a theoretical model to describe the magnetization and magnetostrictive effects of ferromagnetic materials. Compared with the existing models, the proposed model gives greater physical insight, and the material parameters can be determined through the experimental results. Compared with the results from the existing models, it is concluded that the proposed model more accurately describes the relationship between magnetostriction and magnetization. The proposed model qualitatively reflects the correspondence between the magnetostriction of a material under different prestresses and an applied magnetic field, and it can describe the effects of stress on the magnetization behavior of materials, especially the complex nonlinearity of stress effects. The proposed theoretical model has great prospects for applications of ferromagnetic materials in electronic devices and intelligent control. References Ueber die Aenderungen des magnetischen Moments, welche der Zug und das Hindurchleiten eines galvanischen Stroms in einem Stabe von Stahl oder Eisen hervorbringenProgress of converse magnetoelectric coupling effect in multiferroic heterostructuresOverview of Researches on the Nondestructive Testing Method of Metal Magnetic Memory: Status and ChallengesA magnetomechanical model for the magnetic memory methodTheory of the magnetisation process in ferromagnets and its application to the magnetomechanical effectModel for the effect of tensile and compressive stress on ferromagnetic hysteresisRecent developments in modeling of the stress derivative of magnetization in ferromagnetic materialsTheory of the magnetomechanical effectModeling plastic deformation effect on the hysteresis loops of ferromagnetic materials based on modified Jiles-Atherton modelMagneto-elastoplastic coupling model of ferromagnetic material with plastic deformation under applied stress and magnetic fieldsModeling the Temperature Dependence of Hysteresis Based on Jiles–Atherton TheoryA model for hysteretic magnetic properties under the application of noncoaxial stress and fieldA nonlinear constitutive model for Terfenol-D rodsA one-dimension coupled hysteresis model for giant magnetostrictive materialsA nonlinear magneto-thermo-elastic coupled hysteretic constitutive model for magnetostrictive alloysA general nonlinear magnetomechanical model for ferromagnetic materials under a constant weak magnetic fieldThermo-magneto-elastoplastic coupling model of metal magnetic memory testing method for ferromagnetic materialsAn analytical model for the magnetostriction strain of ferromagnetic materials subjected to multiaxial stressA micro-statistical constructive model for magnetization and magnetostriction under applied stress and magnetic fieldsMeasurement and Analytical Modeling of the $\Delta E$Effect in a Bulk Iron-Cobalt AlloyMicromagnetism simulation on effects of soft phase size on Nd 2 Fe 14 B/ α –Fe nanocomposite magnet with soft phase imbedded in hard phaseSmart Material SystemsEquilibrium conditions for internal stresses in non-euclidean continua and stress spacesA general theoretical model of magnetostrictive constitutive relationships for soft ferromagnetic material rodsGeneralized form of anhysteretic magnetization function for Jiles–Atherton theory of hysteresisThe stress dependence of magnetostriction hysteresis in TbDyFe [110] oriented crystal
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