Stress and temperature effects study on anhysteretic curves behavior of magnetostrictive materials
TAMAZIRT souad
Numerical Modeling of Electromagnetic Phenomena and Components Laboratory
MouloudMammeri University TiziOuzou, Algeria.
MOHELLEBI Hassane
Numerical Modeling of Electromagnetic Phenomena and Components Laboratory
MouloudMammeri University TiziOuzou, Algeria.
Abstract—The present work deals with the effect of stress and temperature on the effective magnetic field which occurs in magnetostrictive materials when it is subjected to external applied field. The magnetostriction is then computed for changes made on stress and temperature respectively. The anhysteretic curves representing the magnetization behavior with applied magnetic field are deduced. The results obtained are comparable with those provided in literature.
Keywords— Stress; Magnetostrictive materials; Magnetization, Temperature; Stress field; Anhysteretic curve;
I. INTRODUCTION
Magnetostriction materials are widely used as actuators and sensors because they provide a high displacement resolution with a high force and fast response. In this work we consider at first the effect of the stress on the magnetic field stress which depends on the variation of magnetostriction. The latter is consisting on the computation of the effective magnetic fields.
The anhysteretic curve could be then computed and plotted by introducing the field induced by the applied stress onto the magnetostrictive material. The developments realized are exploiting the thermodynamic approach which uses the Taylor series expansion of Gibbs free energy function [1]-[3], and considering magnetostriction models given in [1], when the magnetization changes by considering different levels of applied stress. At the second step, the effect of the temperature on the magnetic magnetization and then the anhysteretic magnetic curve is obtained and plotted. The effect of stress and temperature on the anhysteretic magnetization is then investigated and the anhysteretic curve is computed for the different values of applied stress and temperature.
II. MAGNETOSTRICTION DEFORMATION MODEL The model of magnetostrictive deformation used in this work permits to us with a magnetostrictive coefficient of approximation ( , ), depending on the stress σ and the magnetic magnetization M, to consider the following formulae [1]:
( , ) = ( ) + ( )
( )and ( )are the constraints, they define how the change looks like when the stress varies considering this variation is linear, the parameters ( )and ( )are written as following
[8]: ( ) = (0) + (0)
( ) = (0) + (0)
The parameters (0), (0), (0), (0) are constants values deduced from experimental data [1].
Figure1 represents the magnetostriction versus magnetization using the model defined by equation (1).The comparison with the experimental data is performed in case of high stress (σ=11.31 MPa) [1]-[2].It could note that the model gives results very shift to experimental one given in [8]..
Fig1.Magnetostriction versus Magnetization (σ=11.31 MPa)
III. EFFECT OF STRESS FIELD ON ANHYSTERETIC CURVE The developments realized are exploiting at first the thermodynamic approach which uses the Taylor series expansion of Gibbs free energy function. From the energy principle, one can wrote [3]:
A=1 2⁄ +-TS+3 2⁄ + Where A is the Helmholtz free energy density.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x 106 0
0.5 1 1.5 2 2.5
3x 10-3
Magnetisation (A/m)
Magnetostriction
Experimental data
Magnetostriction using S.Valadkhane model
(1)
(4) (2)(3)
Therefore, the stress dependence of the anhysteretic magnetization curve is defined as follows [2]:
M = M coth H a −
a H
Where is saturation magnetic magnetization and “a” is the domain density.
It then follows that the effective field and stress field could be expressed by [5]:
= + +
: effective magnetic field, H: is the applied magnetic field.
:Quantifies the field due to magnetic interactions between moments, the parameter quantifies the amount of domain interaction, : is the stress-magnetic field component.
The stress magnetic field due to the variation of magnetization induced by the applied stresses can be expressed by [5]:
=3 2
( , )
In the current study we have investigated the stress effect on the effective magnetic field by computing the stress magnetic field resulting from the effect of stress on the magnetostriction variation by computing the derivative of the Helmholtz free energy density A according to the applied stress represented by the parameter . This was performed by considering that A does not change with the temperature.
H = ∂A
From equation (8), the stress magnetic field∂σ depending on stress is simplified to have the form below:
=3 2
( , )
By considering magnetostriction model for Nickel given by equation (1), the values considered of parameters (0), (0), (0), (0)of Nickel are defined from [2] as follow:
TABLE I. Magnetostriction parameters of Nickel
Parameters
γ (0)
(m2A-2) γ (0)
(m2A2Pa-1) γ (0)
(m4A-4) γ (0) (m4A-4Pa-1) Values 8.51 10-24 1.31 10-24 1.22 10-35 -4.56 10-36 Figure.2 shows the anhysteretic curves of Nickel calculated using equation (5), by considering stress value σ = 0MPa, σ = 50MPa and σ = 100MPa.
Figure.3 represents the anhysteretic curves of nickel calculated using equation (5), by considering stress value σ =50MPa and experimental data given in [2].
Fig2.Nickel Anhysteretic curves under different values of stress (σ=0MPa, σ= 50MPa and σ= 100MPa).
Fig3. Nickel Anhysteretic curve using Valadkhan model (σ= 50 MPa and T=300K)
The anhysteretic curves behavior obtained when considering different value of stress are those predicted and given in literature. The result obtained by adding the effect of the variation of magnetostriction with stress in evaluation of stress magnetic field, are presented too in figure 3.
The stress effect study for the Nickel showed that when stress effect is added into the effective field ,it provides better results by comparing to experimental data given in [2].
IV. TEMPERATURE EFFECT ON MAGNETOSTRICTION MODELS Thermal effects can be incorporated on the an-hysteretic curve by expressing microscopic hysteresis parameters as a function of temperature, specifically the spontaneous magnetization
" ", the domain coupling “α” and domain density “a”.
The temperature dependence of spontaneous magnetization , can be expressed (Curie Law) [10]:
( ) = ( − )
Where is the value of spontaneous magnetization, is the Curie temperature, and C is a constant to be identified according to the value.
The domain coupling, “α” which represents the strength of magnetic interaction between domains in the material, can be expressed as [10]:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
H ( relative value) Manhy ( relative value)
Experimental data
Manhy using H function of/M &/ (=50)
(5)
(6)
(7)
(8)
(10) (9)
= μ
μ is the permeability of vacuum , w is the Weiss coefficient.
The domain density ‘’a ‘’ is given by [10]:
=
is Boltzmann’s constant, T is temperature, and m is the magnetic moment of typical domain.
A numerical process enables us to have variation of parameters “a” and “ ”, figure 4 and figure5 show the behavior of the previous parameters according to the temperature variation.
One can note that when the values of temperature increase, the amplitude of the saturation magnetization Msand the domain density become lowers.
Fig4. Saturation magnetization as a function of temperature in Nickel material (Tc= 631K).
Fig5. Analytical curve of domain density as function of temperature The an-hysteretic curve is then computed and plotted for Nickel magnetic material (Figure 6) by considering different values of temperature300K, 400K, 500 K and 600 K.
In each case the effective field associates the computation of the magnetic stress field and the contribution of the stress onto the total effective magnetic field. For each temperature the magnetic magnetization at saturation is different and becomes less important when the temperature increase. The results
obtained are those known for physical behavior of magnetization according to the temperature variation.
V. CONCLUSION
A magnetostriction model is considered and the effect of the magnetization is presented and validated by adding the stress effect in the computation of the stress magnetic field.
Consequently this yields to a changes in the anhysteretic curve behavior and affect the accuracy of the magnetostriction modeling phenomenon. The effect of temperature on the anhysteretic curve is then investigated and presented by computing the anhysteretic magnetization in case of Nickel material. One can note that when the values of stress and temperature respectively increases, the amplitude of the anhysteretic magnetization becomes lower in both two cases.
These results are in good agreement with those provided in literature.
References
[1] S.Valadkhan, K.Morris, A.Shum, A New Load-Dependent Hysteresis Model For Magnetostrictive Materials, Smart Materials and Structures IOP Science, Vol 19, 10 pp., 2010.
[2] Li, Lu, "Stress effects on ferromagnetic materials: investigation of stainless steel and nickel " (2004). Retrospective Theses and Dissertations. Paper 1179.
[3] M. J. Sablik, D. C. Jiles, Coupled Magnetoelastic Theory of Magnetic and Magnetostrictive Hysteresis, IEEE Transaction on Magnetics 29, 2113- 2123, 1993
[4] SinaValadkhan , Nano Positioning Control Using Magnetostrictive Actuators, Mechanical Engineering Waterloo, Ontario, Canada 2007 [5] D. C. Jiles and D. L. Atherton. Theory of ferromagnetic hysteresis, J. Magn.
Magn. Mater. Vol 61, pp. 48-60. 1984.
[6] Slaughter JC, Dapino MJ, Smith RC, and Flatau AB. Modeling of a Terfenol-D ultrasonic transducer. Proc SPIE Smart Struct Mater;3985:366–77., 2000
[7] B. Bergmair, T.Huber, F. Bruckner, C. Vogler, M.Fuger, and D. Suess.
Fully coupled dynamic model of a magnetostrictive amorphous ribbon and its validation. Journal of Applied Physics .Vol 115, 023905. 2014.
[8] F.Hocini, H.Mohellebi, M.Féliachi, S.H. OuldOuali, The inverse Jiles- Atherton model for the magnetostrictive materials. Application to Terfenol-D, European Journal of Electrical Engineering, Vol. 15/2-3, 2012.
[9] A.Arrot and J. Noakes, “Approximate equation of state for nickel near its critical temperature,” phys. Rev. lett.,vol 19, no 1,pp 786-789, 1967.
[10] B.Nait-Kaci, « Modélisation de l’hystérésis magnétique tenant compte des contraintes thermiques », mémoire de magister, université Mouloud Mammeri, Tizi-Ouzou, 2001.
300 350 400 450 500 550 600 650
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Temperature (K) saturation Magnetization (Ms relative values)
300 350 400 450 500 550 600 650
0 0.2 0.4 0.6 0.8 1
Temperature (K)
Domain density "a" ( relative value)
Fig6. Temperature effect on Nickel Anhysteretic curve (σ= 50 MPa.) (11)
(12)
0 2 4 6 8 10 12 14
x 104 0
0.5 1 1.5 2 2.5 3 3.5x 105
H(A/m) Manhy for different values of temperature
Manhy using S.VALADKHAN model 300K
400K 500K 600K
