Magneto-thermo-mechanical characterization of giant magnetostrictive materials
Province-Ministry Joint Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability, Hebei University of Technology
International Center for Materials Physics, The Chinese Academy of Sciences
School of Science, Tianjin University of Commerce
摘 要：
The variations of magnetization and magnetostriction with temperature and stress were investigated through the analysis of the effective field, induced by temperature and stress.A nonlinear magnetostrictive model of giant magnetostrictive materials was proposed.The proposed model can be used to calculate the magnetostrictive characterization of giant magnetostrictive materials in different temperatures and under different stresses.The coupling effects of axial stress, magnetic field, and temperature on the magnetostriction of a Terfenol-D rod were numerically simulated as well as experimentally tested.Comparison between the calculating and experimental results shows that the proposed model can better describe the magneto-thermo-mechanical characteristics of Terfenol-D rod under different temperatures and compressive stress.Therefore, the proposed model possesses an important significance for the design of magnetostrictive devices.
收稿日期：7 August 2012
基金：supported by the National Natural Science Foundation of China (Nos. 50971056 and 51171057);the Youth Natural Science Foundation of Hebei Province (No. E2011202002);
Magneto-thermo-mechanical characterization of giant magnetostrictive materials
Li Wang Bo-Wen Wang Zhi-Hua Wang Ling Weng Wen-Mei Huang Yan Zhou
Province-Ministry Joint Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability, Hebei University of Technology
International Center for Materials Physics, The Chinese Academy of Sciences
School of Science, Tianjin University of Commerce
Abstract：
The variations of magnetization and magnetostriction with temperature and stress were investigated through the analysis of the effective field, induced by temperature and stress. A nonlinear magnetostrictive model of giant magnetostrictive materials was proposed. The proposed model can be used to calculate the magnetostrictive characterization of giant magnetostrictive materials in different temperatures and under different stresses. The coupling effects of axial stress, magnetic field, and temperature on the magnetostriction of a Terfenol-D rod were numerically simulated as well as experimentally tested. Comparison between the calculating and experimental results shows that the proposed model can better describe the magneto-thermo-mechanical characteristics of Terfenol-D rod under different temperatures and compressive stress. Therefore, the proposed model possesses an important significance for the design of magnetostrictive devices.
Keyword：
Nonlinear model; Giant magnetostrictive materials; Temperature; Stress;
Author： Bo-Wen Wang e-mail: bwwang@hebut.edu.cn;
Received： 7 August 2012
1 Introduction
There are continuously increasing interests in the mechanism of magneto-mechanical coupling and temperature-sensitive characteristic with wide applications in actuators and transducers of giant magnetostrictive materials[1–3].Moffett et al.[4]discussed the magneto-mechanical characteristic of a Terfenol-D rod and found that the magnetostriction and magnetization curves were greatly depended on the mechanical prestress and magnetic fields.Kendall and Piercy[5]studied the magnetization and magnetostriction and found that they decreased with the increase of temperature.However, it is difficult to model the characteristic of magnetostrictive materials since their nonlinearity is sensitive to stress and temperature[6].In order to model the nonlinear relations of magnetostrictive materials, Carman and Mitrovic[7]founded the standard square model, which can predict magnetostriction of a Terfenol-D rod in moderate magnetic fields, and is not suitable in high magnetic fields.A new nonlinear constitutive model was proposed, however, the difference between the calculation result and the experimental one was significant[8].Zheng and Sun[9]founded a magneto-thermo-mechanical coupling constitutive model by using thermodynamics theory, and it can be used to calculate the magnetostriction of giant magnetostrictive materials.In this paper, we tried to found a nonlinear magnetostrictive model according to the magnetism theory and to investigate the magnetostriction and magnetization of giant magnetostrictive materials, with considering the effect of temperature and stress.
2 Nonlinear model
A simple series relation between the magnetostriction k (M, r, Dt) and magnetization M is given as[10]
where Dt is the difference between the actual temperature and reference one, r is prestress.The coefficients ci?r?are relative with stress and can be determined by solving a simultaneous equation group.For implementation purposes, the series is truncated after i=2, then
According to Boltzmann statistics, the magnetization M can be quantified by[11]
Since the stress and temperature have an influence on the effective field, the total effective field, including the stress contribution and temperature effect, can be represented as[10, 12]
The temperature dependence of saturation magnetization can be obtained according to ferromagnetics theory[13]:
The parameter a can be expressed as Eq. (9) [14]:
where N is the volumetric domain density, kBis the Boltzmann constant, T is actual temperature (the unit of T is Kelvin temperature scale) .Equation (9) can describe the variation of the parameter a with the temperature and saturation magnetization.
Substitution of Eqs. (3) , (4) , (7) – (9) into Eq. (2) yields the total magnetostriction, including the stress and temperature influence, as follows:
Equation (10) gives the mathematical relation among the magnetostriction, magnetic field, magnetization, stress, and temperature for ferromagnetic materials.The magnetostriction as the function of the magnetic field, stress, and temperature can be calculated by using Eq. (10) .Also, the Eq. (10) can be used to determine the relation between the magnetostriction and magnetization and to investigate of the magnetization process of ferromagnetic materials.Clearly, Eq. (10) can be simplified as the form of quadratic moment domain rotation model and can be used conveniently in engineering fields.
3 Numerical calculation
The numerical calculation of Terfenol-D rod under different temperatures and prestresses is made according to the proposed model.The magnetization dependences on applied magnetic fields of the Terfenol-D rod under different temperatures are shown in Fig.1.The magnetization increases with an increasing magnetic field for a given temperature and decreases with an increasing temperature for a given magnetic field.The decreasing value of the magnetization is due to the magnetic-order disturbed with the increase of the thermal disturbance energy.With the increase of temperature, the influence of thermal disturbance on magnetic-order is more obvious, which is shown as magnetization decreases with an increasing temperature.
Fig.1 Magnetization curves for Terfenol-D rod under different temperatures
The variations of the magnetostriction with the magnetic field strength at different temperatures are shown in Fig.2.The magnetostriction decreases with the increase of the temperature and it means that the increase of temperature can induce the thermal disturbance.The experimental result of magnetostriction, measured by standard strain gage, is also shown in Fig.2.In comparing the result of calculation with that of experiment in Fig.2, we can find that the calculation result of magnetostriction is in a good agreement with experimental result at 20°C.With the increase of the temperature, the calculation result of magnetostriction is reasonable and almost consistent with experimental result.It means that the Eq. (10) can be used to calculate the magnetostriction as the function of the magnetic field, stress, and temperature.However, small deviation exists between the calculation result and experimental result when the temperature is larger than 80°C.In fact, the parameter a in the Eq. (9) is related with the volumetric domain density N, and here we suppose that the domain density N is constant with the increase of the temperature for simplifying.It seems that the small deviation comes from the variation of the domain density N with the increase of the temperature.
Fig.2 Comparison of calculation curves (solid lines) with experi-mental data (dot lines)
Fig.3 Magnetic field dependence of room temperature magneto-striction under different prestress
4 Conclusion
A nonlinear model to determine the relation of magnetothermo-mechanical characteristics for Terfenol-D is founded.The calculation result of magnetostriction is in a good agreement with experimental result at 20°C.With the increase of the temperature, the calculation result of magnetostriction is reasonable and almost consistent with experimental result.The calculation result shows that the rotation of magnetic domains becomes difficult with increasing the prestress when the field is less than 150 kA.m-1, and can easily move when the field is larger than 250 kA.m-1.The model can be used to calculate the magnetostriction as the function of the magnetic field, stress, and temperature.
参考文献
[10] Jiles DC.Theory of the magnetomechanical effect.J Phys D.1995;28 (8) :1537.
[13] Feng D.Superconductivity and Magnetic in Metal Physics, vol.4.Beijing:Science Press;1998.
