CN107423506B - Method for calculating maximum external magnetization intensity of magnetostrictive material - Google Patents

Abstract

The invention provides a method for calculating the maximum external magnetization of a magnetostrictive material, including: establishing a nonlinear free energy equation with a single magnetic domain in the magnetostrictive material as a research object; The preset value of the magnetic field is brought into the nonlinear free energy equation to solve; the influence of the applied magnetic field and stress on the magnetic domain deflection is analyzed according to the solution result; the optimal value of the applied magnetic field and stress is obtained; magnetization. Compared with the prior art, the present invention solves the extremum value of the nonlinear free energy equation of the single domain particle in the material, studies the law of the angle deflection of the magnetic domain in the magnetostrictive material, and then describes the magnetization and magneto-mechanical coupling characteristics of the material, and obtains The optimal value of the external magnetic field and stress can be used to calculate the magnetization generated by the magnetostrictive material. This calculation method has high accuracy and can more accurately provide important parameters for the magnetostrictive material device.

Description

A method for calculating the maximum magnetization of magnetostrictive materials

technical field

The invention relates to the technical field of magnetostrictive materials, in particular to a method for calculating the maximum external magnetization of magnetostrictive materials.

Background technique

The establishment and improvement of the magnetic energy coupling model of the magnetostrictive material machine has always been a hot and difficult research topic by scholars and experts at home and abroad, and it is also the premise that directly affects and restricts the application of magnetostrictive materials in broadband and high-performance devices. When the actuator based on magnetostrictive material works, it will be subjected to the double load coupling action of the excitation magnetic field and the mechanical stress. The material constitutive relationship is established as a mathematical model that is not lower than the two coupled fields, and the interaction between the mechanical and magnetic coupling fields The superposition greatly increases the difficulty of establishing the material theoretical model. In engineering applications, the approximate linear piezo-magnetic relationship is generally used to simplify the material model and establish the coupling relationship of materials. The currently established linear constitutive relationship of materials is used to analyze and describe the internal transformation relationship between magnetism and machine in the material. It can only be used for engineering applications with low precision. The established model can only reflect the material in a specific linear region. The coupling relationship cannot be used to describe the nonlinear relationship of giant magnetostrictive materials.

Since the establishment of the magnetostrictive material model in the prior art is too simple, it is difficult to accurately value important parameters in the magnetostrictive material device, and the magnetostrictive material cannot obtain the best application performance.

SUMMARY OF THE INVENTION

In view of this, the purpose of the present invention is to provide a method for calculating the maximum external magnetization of the magnetostrictive material. The method provided by the present invention can more accurately calculate the externally generated maximum magnetization of the magnetostrictive material.

The invention provides a method for calculating the maximum external magnetization of magnetostrictive materials, including:

The nonlinear free energy equation is established by taking the single magnetic domain in the magnetostrictive material as the research object;

Substitute the preset value of the applied stress and the preset value of the applied magnetic field into the nonlinear free energy equation to solve;

According to the solution results, the influence of the applied magnetic field and stress on the magnetic domain deflection is analyzed;

According to the influence law of the applied magnetic field and stress on the magnetic domain deflection, the optimal values of the applied magnetic field and stress are obtained;

The maximum external magnetization of the magnetostrictive material is calculated according to the optimal value of the applied magnetic field and stress.

Preferably, the nonlinear free energy equation is:

E=E _k +E _σ +E _H formula 1;

In formula 1, E is the free energy;

E _k is the magnetocrystalline anisotropy energy;

E _σ is the stress anisotropy;

E _H is the magnetization energy.

Preferably, the formula 1 is expressed in the X coordinate system as:

In formula 2, K ₁ and K ₂ are the magnetocrystalline anisotropy constants,

α ₁ is the direction cosine of the magnetization M to the x-axis, α ₂ is the direction cosine of the magnetization M to the y-axis, and α ₃ is the direction cosine of the magnetization M to the z-axis;

β1 is the direction cosine _of the applied magnetic field H and the stress σ to the x-axis, _β2 is the direction cosine _of the applied magnetic field H and the stress σ to the y-axis, and β3 is the applied magnetic field H and the stress σ The direction cosine of the z-axis;

λ ₁₀₀ is the magnetostriction coefficient of the magnetostrictive material in the <100> direction, and λ ₁₁₁ is the magnetostriction coefficient of the magnetostrictive material in the <111>direction;

H is the strength of the applied magnetic field, σ is the applied stress;

μ ₀ is the vacuum permeability, M _s is the saturation magnetization;

The X coordinate system takes the [100] crystal axis direction as the x axis, the [010] crystal axis direction as the y axis, and the [001] crystal axis direction as the z axis.

Preferably, the directions of the applied magnetic field and the stress are both in the direction of the [110] crystallographic axis.

Preferably, the polar coordinate transformation is performed on Equation 2, so that the direction of the magnetization M in the X coordinate system is

Preferably, the formula 2 is expressed in the Y coordinate system as:

z轴的方向为

In the Y coordinate system, the direction of the x-axis is [001], and the direction of the y-axis is

The direction of the z-axis is

Preferably, the method for solving the nonlinear free energy equation is to use MATLAB software to solve.

Preferably, the preset value of the applied stress is 0-12 MPa.

Preferably, the preset value of the external magnetic field is 0-120,000 A/m.

Preferably, the optimal value of the external magnetic field is 45000-55000 A/m.

Preferably, the optimum value of the applied stress is -4.5 to -5.5 MPa.

Compared with the prior art, the present invention takes a single domain particle in the magnetostrictive material as the research object of the magnetic domain deflection model, and studies the giant magnetostrictive material by solving the extreme value of the nonlinear free energy equation of the single domain particle in the material. The law of the angle deflection of the internal magnetic domain, and then describe the magnetization and magneto-mechanical coupling characteristics of the material, obtain the best values of the external magnetic field and stress, and then calculate the external magnetization of the magnetostrictive material. The method provided by the present invention calculates accurately , can more accurately provide important parameters for the magnetostrictive material device, so that the magnetostrictive material shows the best magnetization performance in use.

Description of drawings

In order to explain the embodiments of the present invention or the technical solutions in the prior art more clearly, the following briefly introduces the accompanying drawings that need to be used in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only It is an embodiment of the present invention. For those of ordinary skill in the art, other drawings can also be obtained according to the provided drawings without creative efforts.

1 is a schematic diagram of an X-system coordinate system in an embodiment of the present invention;

Fig. 2 is the calculation result of MATLAB when the applied magnetic field and stress are 0 in the embodiment of the present invention;

Fig. 3 is the calculation result of MATLAB when the applied stress is 0 and the applied magnetic field is 10000A/m in the embodiment of the present invention;

Fig. 4 is the calculation result of MATLAB when the applied stress is 0 and the applied magnetic field is 50000A/m in the embodiment of the present invention;

Fig. 5 is the calculation result of MATLAB when the applied stress is 0 and the applied magnetic field is 90000A/m in the embodiment of the present invention;

Fig. 6 is the calculation result of MATLAB when the applied magnetic field is 0 and the applied stress is -1MPa in the embodiment of the present invention;

Fig. 7 is the calculation result of MATLAB when the applied magnetic field is 0 and the applied stress is -1MPa in the embodiment of the present invention;

Fig. 8 is the calculation result of MATLAB when the applied magnetic field is 0 and the applied stress is -5MPa in the embodiment of the present invention;

Fig. 9 is the calculation result of MATLAB when the applied magnetic field is 0 and the applied stress is -5MPa in the embodiment of the present invention;

Fig. 10 is the calculation result of MATLAB when the applied magnetic field is 0 and the applied stress is -10MPa in the embodiment of the present invention;

Fig. 11 is the calculation result of MATLAB when the applied magnetic field is 0 and the applied stress is -10MPa in the embodiment of the present invention;

Fig. 12 is the calculation result of MATLAB when the applied magnetic field is 50000A/m and the applied stress is -5MPa in the embodiment of the present invention.

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, but not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

The invention provides a method for calculating the maximum external magnetization of magnetostrictive materials, including:

The nonlinear free energy equation is established by taking the single magnetic domain in the magnetostrictive material as the research object;

Substitute the preset value of the applied stress and the preset value of the applied magnetic field into the nonlinear free energy equation to solve;

According to the solution results, the influence of the applied magnetic field and stress on the magnetic domain deflection is analyzed;

According to the influence law of the applied magnetic field and stress on the magnetic domain deflection, the optimal values of the applied magnetic field and stress are obtained;

The maximum external magnetization of the magnetostrictive material is calculated according to the optimal value of the applied magnetic field and stress.

In the present invention, under the action of an external magnetic field H and stress σ, the generated magnetocrystalline anisotropy energy E _k , stress anisotropy energy E _σ and magnetization energy E _H all affect the deflection of a single magnetic domain in the magnetostrictive material , thereby affecting the free energy of a single magnetic domain; the present invention preferably establishes a nonlinear free energy equation according to the magnetocrystalline anisotropy energy, stress anisotropy energy and magnetization energy:

E=E _k +E _σ +E _H formula 1;

In formula 1, E is the free energy;

E _k is the magnetocrystalline anisotropy energy;

E _σ is the stress anisotropy;

E _H is the magnetization energy.

In the present invention, formula 1 is preferably represented by an X coordinate system, wherein the X coordinate system takes the [100] crystallographic axis direction as the x-axis, the [010] crystallographic axis direction as the y-axis, and the [001] crystallographic axis direction as the y-axis. for the z-axis. The present invention has no special restrictions on the directions of the applied magnetic field and stress, and those skilled in the art can select the applied directions of the applied magnetic field and stress according to actual conditions. In the present invention, a rod-shaped magnetostrictive material is preferably selected, and both the external magnetic field and the stress are loaded in the axial direction of the rod-shaped magnetostrictive material, that is, the crystal axis [110] direction.

In the present invention, formula 1 in the X coordinate system can be expressed as:

In formula 2, K ₁ and K ₂ are the magnetocrystalline anisotropy constants,

α ₁ is the direction cosine of the magnetization M to the x-axis, α ₂ is the direction cosine of the magnetization M to the y-axis, and α ₃ is the direction cosine of the magnetization M to the z-axis;

β1 is the direction cosine _of the applied magnetic field H and the stress σ to the x-axis, _β2 is the direction cosine _of the applied magnetic field H and the stress σ to the y-axis, and β3 is the applied magnetic field H and the stress σ The direction cosine of the z-axis;

λ ₁₀₀ is the magnetostriction coefficient of the magnetostrictive material in the <100> direction, and λ ₁₁₁ is the magnetostriction coefficient of the magnetostrictive material in the <111>direction;

H is the strength of the applied magnetic field, σ is the applied stress;

μ ₀ is the vacuum permeability, and M _s is the saturation magnetization.

In the present invention, the first and second terms in Equation 2 are the representation of the magnetocrystalline anisotropy energy E _k , the third and fourth terms are the representation of the stress anisotropy energy E _σ , and the fifth term is the magnetization Representation of energy E _H.

In the present invention, the process of solving the non-free energy equation shown in Equation 2 is complicated and the amount of calculation is large. The present invention preferably adopts the method of polar coordinate transformation to transform Equation 2 to simplify the solution of the nonlinear free energy equation.

In the present invention, the method for polar coordinate transformation is preferably:

则α₁、α₂和α₃变换为：Let the direction of the magnetization M in the X coordinate system be

Then α ₁ , α ₂ and α ₃ are transformed into:

外加应力的方向为

由于本发明优选以X坐标系中轴向[110]方向为外加磁场和应力的方向，经过极坐标变换后，外加磁场和应力的方向为：Let the direction of the applied magnetic field in the X coordinate system be

The direction of the applied stress is

Because the present invention preferably takes the axial [110] direction in the X coordinate system as the direction of the applied magnetic field and stress, after the polar coordinate transformation, the direction of the applied magnetic field and stress is:

Similarly, β ₁ , β ₂ and β ₃ are transformed into:

z轴的方向为

Continue to transform the X coordinate system to obtain a Y coordinate system, the Y coordinate system takes the [110] direction as the z axis, and establishes the x and y axes with the plane [110] perpendicular to the [110] direction; the Y coordinate system The direction of the x-axis is [001], and the direction of the y-axis is

The direction of the z-axis is

In the Y coordinate system, β ₁ , β ₂ and β ₃ , α ₁ , α ₂ and α ₃ continue to be transformed into:

Putting Equation 5 and Equation 6 into Equation 2, the nonlinear free energy equation after coordinate transformation is obtained:

In the present invention, the nonlinear free energy equation is preferably solved by MATLAB software, and the influence law of the applied magnetic field and stress on the magnetic domain deflection is analyzed according to the solution result. The present invention can use MATLAB software to solve the nonlinear free energy equation shown in formula 7. , the specific solution process is as follows:

Select the values of K ₁ , K ₂ , _Ms and λ ₁₀₀ parameters in formula 7;

的数值。Program Equation 7 to obtain E, θ and

value of .

In the present invention, each parameter in the nonlinear free energy equation shown in formula 7 is preferably as follows:

In the present invention, formula 7 is preferably calculated by programming as follows:

Clear;clc;close;

k1=-60000; % anisotropy constant k1

k2=-200000; % anisotropy constant k2

A0=50; % magnetostriction coefficient [100] direction

A1=1640; % magnetostriction coefficient [111] direction

b1=1/2^0.5; % stress and magnetic field direction cosine

b2=1/2^0.5;

b3 = 0;

o=0; % stress load

H=0; % magnetic field load

x = 0:1:180; % angle 1 division

y=0:1:360; % angle 2 division

[p0,p1]=meshgrid(x,y); % define the cell grid

z=cos(p0/180*pi); % direction cosine calculation

y=sin(p1/180*pi).*sin(p0/180*pi);

x=sin(p0/180*pi).*cos(p1/180*pi);

B=[0 0 1; sqrt(2)/2-sqrt(2)/2 0; sqrt(2)/2sqrt(2)/2 0]; % coordinate transformation criterion definition

B1=B(:,1);

B2 = B(:, 2);

B3=B(:,3);

B1 = B1';

B2 = B2';

B3 = B3';

B11=B1(:,1);

B12=B1(:,2);

B13=B1(:,3);

B21=B2(:,1);

B22=B2(:,2);

B23=B2(:,3);

B31=B3(:,1);

B32=B3(:,2);

B33=B3(:,3);

x1=B11*x+B12*y+B13*z;

y1=B21*x+B22*y+B23*z;

z1=B31*x+B32*y+B33*z;

a1=x1; % complete coordinate transformation

a2=y1;

a3=z1;

E=k1*(a1.^2.*a2.^2+a1.^2.*a3.^2+a3.^2.*a2.^2)+k2.*a1.^2.*a2. ^2.*a3.^2-1.5*A0*o*(a1.^2.*b1.^2+a2.^2.*b2.^2+a3.^2.*b3.^2)- 3*A1*o*(a1.*b1.*a2.*b2+a1.*b1.*a3.*b3+a3.*b3.*a2.*b2)-0.961*H*(a1.*b1 +a2.*b2+a3.*b3); % Calculated free energy value

%mesh(p0,p1,E) % draw a 3D graph

contour(p0,p1,E,50) % draw an equipotential curve, that is, a two-dimensional projection map

xlabel('p0'), ylabel('p1') % set the label

的数值。In the present invention, the above programming calculation program is the calculation of formula 7 when the preset value of the applied magnetic field and stress is 0. When it is necessary to calculate the applied magnetic field and stress of other preset values, it is only necessary to use o and o in the above program. H is set to the corresponding applied stress value and the preset value of the applied magnetic field, and E, θ and E, θ and

value of .

In the present invention, the method for analyzing the influence law of external magnetic field and stress on magnetic domain deflection is preferably:

When the preset value of the applied stress is 0, and the preset value of the applied magnetic field is in the range of 0-120000A/m, the nonlinear free energy equation is calculated, and the influence law of the applied magnetic field on the magnetic domain deflection is analyzed;

The nonlinear free energy equation is calculated within the range of the preset value of the applied magnetic field is 0, and the preset value of the applied stress is 0～-12Mpa (the negative value of the applied stress indicates the applied compressive stress), and the influence of the applied stress on the magnetic domain deflection is analyzed. law.

In the present invention, the influence rule of the external magnetic field on the magnetic domain deflection is:

When the applied magnetic field and stress are both 0, the free energy appears at 8 minimum values. When the applied magnetic field increases to 10000A/m, the free energy minimum value rotates and gradually approaches the direction of the applied magnetic field; when the applied magnetic field increases When it reaches 50000A/m, 6 of the 8 free energy minima disappear, and the remaining 2 free energy minima gradually approach the direction of the applied magnetic field; when the applied magnetic field exceeds 50000A/m to 120000A/m, still There are only 2 free energy minima.

In the present invention, the influence rule of applied stress on magnetic domain deflection is:

When the applied stress increases from -1MPa to -5MPa, the effect of stress changes the magnetic field energy of the magnetic domain transition at the two free energy minima. The increase of stress is beneficial to reduce the magnetic domain at the two free energy minima. Continue to increase the stress so that the stress exceeds -5MPa to reach -12MPa, and increasing the stress causes the magnetic domain to deflect in the direction perpendicular to the magnetization.

In the present invention, the method for obtaining the best value of the applied magnetic field and stress is preferably:

When the magnetostrictive material is in the external magnetic field, the volume of the magnetic domain whose spontaneous magnetization direction forms a small angle with the direction of the external magnetic field expands with the increase of the external magnetic field, so that the magnetization direction of the magnetic domain is further turned to the direction of the external magnetic field; other spontaneous magnetization directions and The volume of the magnetic domain with a large angle in the direction of the external magnetic field is gradually reduced. At this time, the magnetostrictive material exhibits macroscopic magnetism to the outside; the external magnetic field continues to increase until all the magnetic domains are aligned along the external magnetic field and reach saturation. The moments are aligned, and the magnetostrictive material produces the maximum magnetization to the outside.

According to the influence of the applied magnetic field on the magnetic domain deflection law, as the applied magnetic field increases, the magnetic domain gradually deflects in the direction of the applied magnetic field. When the applied magnetic field is low (<50000A/m), the free energy minimum value is too much, With the increase of the applied magnetic field, the free energy minimum value gradually decreases, but when the applied magnetic field is too large (>50000A/m), the free energy minimum value decreases slowly and basically keeps 2 (theoretically increasing continuously The magnetic field can finally obtain a minimum value of free energy, but the required external magnetic field is very large); considering the influence of the external magnetic field on the deflection of the magnetic domain and the energy consumption of the magnetic field in the actual application process, the optimal value of the external magnetic field is preferred. It is 45000A/m - 55000A/m, More preferably, it is 50000A/m.

According to the influence of the above applied stress on the magnetic domain deflection law, with the increase of the applied stress (<-5MPa), the applied stress is beneficial to reduce the magnetic field energy required for the magnetic field transition; but with the further increase of the applied stress (>-5MPa) ), the direction of the magnetic domain is deflected in the direction perpendicular to the magnetization, and the various stress properties will hinder the deflection and transition of the magnetic domain, increase the magnetic field required for saturation magnetization, and make the magnetization of the material more difficult. It can be seen that the use of a smaller applied stress is beneficial to the magnetization of the material, and the optimal value of the applied stress is preferably -4.5 to -5.5 MPa, more preferably -5 MPa.

In the present invention, the calculation method of the maximum magnetization generated by the magnetostrictive material is:

Substitute the optimal values of the applied magnetic field and stress into the nonlinear free energy equation for calculation;

值；According to the calculation results, θ _n and θ n at the minimum value of free energy are obtained

value;

According to the formula:

M=M _S cos θ _n Equation 9

where M _s is the saturation magnetization;

Calculate the maximum magnetization produced by the magnetostrictive material to the outside.

Example

Taking the giant magnetostrictive rod of Terfenol-D material composition as an example to calculate the maximum magnetization generated by the magnetostrictive material to the outside, the length of the giant magnetostrictive rod is 40mm and the diameter is 5mm.

The free energy equation is established by taking a single magnetic domain in the magnetostrictive material as the research object:

E=E _k +E _σ +E _H Formula 1

In formula 1, E is the free energy;

E _k is the magnetocrystalline anisotropy energy;

E _σ is the stress anisotropy;

E _H is the magnetization energy.

Equation 1 is represented in the X coordinate system. The X coordinate system is shown in Figure 1, with the [100] crystal axis direction as the x axis, the [010] crystal axis direction as the y axis, and the [001] crystal axis direction as the The z-axis, the direction of the applied magnetic field and stress is the [110] crystallographic axis, and M is the direction of the magnetization:

In formula 2, K ₁ and K ₂ are the magnetocrystalline anisotropy constants,

α ₁ is the direction cosine of the magnetization M to the x-axis, α ₂ is the direction cosine of the magnetization M to the y-axis, and α ₃ is the direction cosine of the magnetization M to the z-axis;

β1 is the direction cosine _of the applied magnetic field H and the stress σ to the x-axis, _β2 is the direction cosine _of the applied magnetic field H and the stress σ to the y-axis, and β3 is the applied magnetic field H and the stress σ The direction cosine of the z-axis;

λ ₁₀₀ is the magnetostriction coefficient of the magnetostrictive material in the <100> direction, and λ ₁₁₁ is the magnetostriction coefficient of the magnetostrictive material in the <111>direction;

H is the strength of the applied magnetic field, σ is the applied stress;

μ ₀ is the vacuum permeability, and M _s is the saturation magnetization.

式2中α₁、α₂和α₃变换为：Perform polar coordinate transformation on Equation 2, let the direction of magnetization M in the X coordinate system be

In formula 2, α ₁ , α ₂ and α ₃ are transformed into:

The directions of the applied magnetic field and stress in the X coordinate system are:

In formula 2, β ₁ , β ₂ and β ₃ are transformed into:

z轴的方向为

式2中，β₁、β₂和β₃，α₁、α₂和α₃继续变换为：Continue to transform the X coordinate system, take the [110] direction as the z-axis, and establish the x-axis and y-axis with the plane [110] perpendicular to the [110] direction, and obtain the Y coordinate system. In the Y coordinate system, the direction of the x-axis is [001], and the direction of the y-axis is

The direction of the z-axis is

In formula 2, β ₁ , β ₂ and β ₃ , α ₁ , α ₂ and α ₃ continue to be transformed into:

Substituting Equation 5 and Equation 6 into Equation 2, we get:

Use MATLAB software to calculate Equation 7:

The value of each parameter in formula 7 is:

MATLAB, the programming procedure is as follows:

Clear;clc;close;

k1=-60000; % anisotropy constant k1

k2=-200000; % anisotropy constant k2

A0=50; % magnetostriction coefficient [100] direction

A1=1640; % magnetostriction coefficient [111] direction

b1=1/2^0.5; % stress and magnetic field direction cosine

b2=1/2^0.5;

b3 = 0;

o=0; % stress load

H=0; % magnetic field load

x = 0:1:180; % angle 1 division

y=0:1:360; % angle 2 division

[p0,p1]=meshgrid(x,y); % define the cell grid

z=cos(p0/180*pi); % direction cosine calculation

y=sin(p1/180*pi).*sin(p0/180*pi);

x=sin(p0/180*pi).*cos(p1/180*pi);

B=[0 0 1; sqrt(2)/2-sqrt(2)/2 0; sqrt(2)/2sqrt(2)/2 0]; % coordinate transformation criterion definition

B1=B(:,1);

B2 = B(:, 2);

B3=B(:,3);

B1 = B1';

B2 = B2';

B3 = B3';

B11=B1(:,1);

B12=B1(:,2);

B13=B1(:,3);

B21=B2(:,1);

B22=B2(:,2);

B23=B2(:,3);

B31=B3(:,1);

B32=B3(:,2);

B33=B3(:,3);

x1=B11*x+B12*y+B13*z;

y1=B21*x+B22*y+B23*z;

z1=B31*x+B32*y+B33*z;

a1=x1; % complete coordinate transformation

a2=y1;

a3=z1;

E=k1*(a1.^2.*a2.^2+a1.^2.*a3.^2+a3.^2.*a2.^2)+k2.*a1.^2.*a2. ^2.*a3.^2-1.5*A0*o*(a1.^2.*b1.^2+a2.^2.*b2.^2+a3.^2.*b3.^2)- 3*A1*o*(a1.*b1.*a2.*b2+a1.*b1.*a3.*b3+a3.*b3.*a2.*b2)-0.961*H*(a1.*b1 +a2.*b2+a3.*b3); % Calculated free energy value

%mesh(p0,p1,E) % draw a 3D graph

contour(p0,p1,E,50) % draw an equipotential curve, that is, a two-dimensional projection map

xlabel('p0'), ylabel('p1') % set the label

According to the above programming procedure, when o=0, change the H value, increase the H value from 0 to 120000A/m, and perform multiple calculations for each increase of 1000A/m to obtain multiple calculation results.

According to the above programming procedure, when H=0, change the value of o, increase o from 0 to -12MPa, and perform multiple calculations with each increase of 0.1MPa to obtain multiple calculation results.

According to the above calculation results, the influence of the applied magnetic field and stress on the magnetic domain deflection is analyzed:

Figure 2 shows the free energy calculation results (two-dimensional projection diagram) obtained when the applied magnetic field and stress are 0. It can be seen from Figure 2 that there are 8 free energy minima in the figure, and the corresponding X-system coordinates are:

[111],

[111],

When the stress is 0, an external magnetic field is applied, and when the magnetic field increases to H=10000A/m, the 8 free energy minima rotate with the increase of the magnetic field, and gradually approach the direction of the external magnetic field, as shown in Figure 3, Figure 3 is A 2D projection plot of the output of a MATLAB calculation.

方向的磁畴产生跃迁效应，极小值

彻底消失，随着磁场不断增大极小值

也渐渐淡化，如图4所示，图4为MATLAB计算输出的二维投影图。When the magnetic field reaches H=50000A/m,

The direction of the magnetic domain produces a transition effect, the minimum value

disappears completely, and the minimum value as the magnetic field continues to increase

It also gradually fades, as shown in Figure 4, which is a two-dimensional projection map of the output calculated by MATLAB.

方向的磁畴产生跃迁效应，极小值

消失，极小值

[111]逐渐向外加磁场方向靠近，如图5所示，图5为MATLAB计算输出的二维投影图。When the magnetic field reaches H=90000A/m,

The direction of the magnetic domain produces a transition effect, the minimum value

disappear, minimum

[111] gradually approached the direction of the applied magnetic field, as shown in Fig. 5, which is a two-dimensional projection map of the output calculated by MATLAB.

方向的极小值稍微高于

方向，这是由于应力作用下

方向的部分发生偏转，跃迁至

方向。应力的作用将改变磁畴跃迁所需的磁场能，应力的增加有利于减少

方向的磁畴跃迁所需磁场，如图6和图7所示，图6为MATLAB计算输出的三维图，图7为图6中三维图的某一截面图。When the magnetic field is 0, external stress is applied, and when the stress increases to -1MPa,

The minimum value of the direction is slightly higher than

direction, which is due to the stress

The part of the direction is deflected and jumps to

direction. The effect of stress will change the magnetic field energy required for the magnetic domain transition, and the increase of stress is beneficial to decrease

The magnetic field required for the magnetic domain transition in the direction is shown in Figure 6 and Figure 7, Figure 6 is a three-dimensional map calculated by MATLAB, and Figure 7 is a cross-sectional view of the three-dimensional map in Figure 6.

方向的极小值明显高于

方向，

方向多数磁畴已发生偏转，跃迁至

方向。对于

方向的磁畴，压应力将成为其磁弹性过程中磁畴跃迁的阻力，从而增大磁畴跃迁所需磁场，如图8和图9所示，图8为MALAB输出的三维图，图9为图8中三维图的某一截面图。When the stress reaches -5MPa,

The minimum value of the direction is significantly higher than

direction,

The direction of the majority of the magnetic domains has been deflected, transitioning to

direction. for

The compressive stress will become the resistance of the magnetic domain transition in the magneto-elastic process, thereby increasing the magnetic field required for the magnetic domain transition. It is a cross-sectional view of the three-dimensional diagram in FIG. 8 .

方向曲线过于下垂，极小值太小，已经对

方向磁畴偏移产生阻力，不利于取得最大磁化强度，如图10和图11所示，图10为MATLAB计算输出的三维图，图11为图10中三维图的某一截面图。When the stress reaches -10MPa,

The direction curve is too sagging, the minimum value is too small, and the

The directional magnetic domain shift produces resistance, which is not conducive to obtaining the maximum magnetization. As shown in Figure 10 and Figure 11, Figure 10 is a 3D graph calculated by MATLAB, and Figure 11 is a cross-sectional view of the 3D graph in Figure 10.

The effect of external magnetic field and stress on the magnetic domain deflection is as follows:

处的自由能凹陷程度越大，也就是说增大应力使得磁畴向与外加磁场垂直的方向偏转，将增大磁致伸缩材料中

方向的磁畴的体积分数，磁化饱和所需磁场增大，这主要是因为应力各向异性能将阻碍磁畴的偏转和跃迁，使材料的磁化更加困难，从能量转换的角度来看，应力各向异性能不利于材料磁化的进行。The 8 free energy minima gradually approach the direction of the external magnetic field with the increase of the external magnetic field, that is, the magnetic domain gradually deflects in the direction of the magnetization, and the external magnetic field continues to increase until all the magnetic domains are aligned along the external magnetic field and reach saturation. The unit magnetic moments in each magnetic domain have been arranged neatly, and the magnetostrictive material generates the maximum magnetization to the outside;

The greater the degree of free energy depression at the location, that is to say, increasing the stress makes the magnetic domain deflect in the direction perpendicular to the applied magnetic field, which will increase the amount of magnetostrictive material in the magnetostrictive material.

The volume fraction of the magnetic domain in the direction of the direction, the magnetic field required for magnetization saturation increases, mainly because the stress anisotropy will hinder the deflection and transition of the magnetic domain, making the magnetization of the material more difficult. From the perspective of energy conversion, the stress Anisotropic energy is not conducive to the progress of material magnetization.

Get the maximum value of the applied magnetic field and stress:

With the increase of the applied magnetic field, the magnetic domain is deflected in the direction of the outward magnetic field, which is beneficial to obtain the maximum magnetization of the magnetostrictive material. However, when the applied magnetic field exceeds 50000A/m, the magnetic domain deflection basically reaches saturation, even if it increases further When an external magnetic field is applied, it is difficult to reduce the free energy at the minimum value; when the applied stress is -1 to -5MPa, the magnetic domain is deflected in the direction of the applied magnetic field with the increase of the applied stress, which is beneficial to the magnetostrictive material to generate the maximum magnetization to the outside. , when the applied stress exceeds -5MPa, the magnetic domain will be deflected in the direction perpendicular to the applied magnetic field, preventing the magnetostrictive material from generating the maximum magnetization to the outside.

Therefore, the optimal value of the applied magnetic field is 50000A/m, and the maximum value of the applied stress is -5MPa.

Calculate the maximum external magnetization of the magnetostrictive material:

Set H in the above programming program to 50000, and o to -5 to calculate;

The calculated two-dimensional projection map is shown in Figure 12. It can be seen from Figure 12 that the coordinates of the two free energy minima are:

Calculate the maximum magnetization generated by the magnetostrictive material according to the formula:

M=M _S cos θ _n formula 8;

Since there are two values of θ _n , Equation 8 is rewritten as:

In formula 9, Ms is 0.765A/m, θ ₁ =θ ₂ =18;

The calculated maximum magnetization M = 0.73 A/m.

Experiment to verify the calculation results:

Use a magnetic flux density measuring instrument to test the magnetic flux density B of the above Terfenol-D bar, and then according to the formula:

B=μ ₀ ·M formula 10;

Calculate the magnetization M, in formula 10, μ ₀ is the magnetic permeability of the Terfenol-D material.

When the applied magnetic field is 50000A/m, which should be -5MPa (compressive stress), the test result is that the magnetization is 0.7473A/m; the experimental result is close to the calculation result obtained by the theoretical calculation method provided by the present invention. The calculation method is more accurate.

When the applied magnetic field is 60000A/m, which should be -5MPa (compressive stress), the test result is that the magnetization is 0.7478A/m (the magnetic field is increased by 10000A/m, the magnetization is only increased by 0.0004A/m, and the energy consumption is relatively large. ); when the applied magnetic field is 40000A/m, the test result is -5MPa (compressive stress), the magnetization is 0.6758A/m; when the applied magnetic field is 50000A/m, it should be -4MPa (compressive stress) The test result is 0.7398A/m; the test result is 0.7337A/m when the external magnetic field is 50000A/m, which should be -6MPa (compressive stress); when the external magnetic field is 60000A/m, it should be -6MPa (compressive stress). ), the test result is 0.7351A/m; when the external magnetic field is 40000A/m, the test result should be -4MPa (compressive stress), the test result is 0.6930A/m; when the external magnetic field is 60000A/m, it should be -4MPa ( compressive stress), the test result is 0.7038A/m; when the applied magnetic field is 40000A/m, which should be -6MPa (compressive stress), the test result is 0.6576A/m; it can be seen that the magnetostrictive The stretchable material does obtain the maximum magnetization at the optimal value of the external magnetic field and stress, and the method provided by the present invention has high accuracy.

It can be seen from the above embodiments that the present invention provides a method for calculating the maximum external magnetization of a magnetostrictive material, including: establishing a nonlinear free energy equation with a single magnetic domain in the magnetostrictive material as the research object; The preset value of the stress is substituted into the nonlinear free energy equation to solve; the influence of the applied magnetic field and stress on the magnetic domain deflection is analyzed according to the solution result; the optimal value of the applied magnetic field and stress is obtained; magnetization. Compared with the prior art, the present invention solves the extremum value of the nonlinear free energy equation of the single domain particle in the material, studies the law of the angle deflection of the magnetic domain in the magnetostrictive material, and then describes the magnetization and magneto-mechanical coupling characteristics of the material, and obtains The optimal value of the external magnetic field and stress can be used to calculate the magnetization generated by the magnetostrictive material. This calculation method has high accuracy and can more accurately provide important parameters for the magnetostrictive material device.

Claims (6)

1. A method of calculating the maximum externally generated magnetization of a magnetostrictive material, comprising:

establishing a nonlinear free energy equation by taking a single magnetic domain in a magnetostrictive material as a research object;

substituting a preset value of the external stress and a preset value of the external magnetic field into the nonlinear free energy equation to solve;

analyzing the influence rule of the external magnetic field and the stress on the magnetic domain deflection according to the solving result;

obtaining the optimal values of the external magnetic field and the stress according to the influence rule of the external magnetic field and the stress on the magnetic domain deflection;

calculating the maximum magnetization intensity of the magnetostrictive material generated outwards according to the optimal values of the external magnetic field and the stress;

the nonlinear free energy equation is:

E＝E_k+E_σ+E_Hformula 1;

in formula 1, E is free energy;

E_kvarious properties of magnetic crystal;

E_σthe stress is different in performance;

E_His magnetization energy;

the formula 1 is expressed in an X coordinate system as:

in the formula 2, K₁And K₂Is a constant of magnetocrystalline anisotropy, and,

α₁is the direction cosine of the magnetization M to the x-axis, alpha₂Is the direction cosine of the magnetization M to the y-axis, α₃Is the direction cosine of the magnetization M to the z-axis;

β₁for the direction cosine of the applied magnetic field H and stress sigma to the x-axis, beta₂For the direction cosine of the applied magnetic field H and stress sigma to the y-axis, beta₃Cosine of the direction of the applied magnetic field H and stress sigma to the z-axis;

λ₁₀₀is a magnetostrictive material in<100>Coefficient of magnetostriction in the direction, λ₁₁₁Is a magnetostrictive material in<111>The magnetostriction coefficient of the direction;

h is the external magnetic field intensity, and sigma is the external stress;

μ₀for vacuum permeability, M_sIs the saturation magnetization;

the X coordinate system takes the [100] crystal axis direction as an X axis, takes the [010] crystal axis direction as a y axis and takes the [001] crystal axis direction as a z axis;

the formula 2 is subjected to polar coordinate transformation, and the direction of the magnetization M in the X coordinate system is

The formula 2 is expressed in a Y coordinate system as:

the direction of the x axis in the Y coordinate system is [001]]The y-axis being in the direction

The z-axis is in the direction of

The external magnetic field is applied toSubstituting the optimal value of the force into the nonlinear free energy equation for calculation; obtaining theta at the minimum value of the free energy according to the calculation result_nAnd

a value;

the maximum magnetization of the magnetostrictive material to the outside is calculated according to equation 9:

M＝M_Scosθ_nformula 9;

wherein M is_sThe saturation magnetization.

2. The calculation method according to claim 1, wherein the nonlinear free energy equation is solved by using MATLAB software.

3. The calculation method according to claim 1, wherein the preset value of the applied stress is between 0 and-12 MPa.

4. The calculation method according to claim 1, wherein the predetermined value of the applied magnetic field is 0 to 120000A/m.

5. The calculation method according to claim 1, wherein the optimum value of the externally applied magnetic field is 45000-55000A/m.

6. The calculation method according to claim 1, wherein the optimum value of the applied stress is-4.5 to-5.5 MPa.

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