Nonlinear Magnetostrictive Transducer

Magnetostriction describes the change in dimensions of a material due to a change in its magnetization. This phenomenon is a manifestation of magnetoelastic coupling, which is exhibited by all magnetic materials to some extent. The effects related to magnetoelastic coupling are described by various names. The Joule effect describes the change in length due to a change in the magnetization state of the material. This magnetostrictive effect is used in transducers for applications in sonars, acoustic devices, active vibration control, position control, and fuel injection systems.

Magnetostriction has a quantum-mechanical origin. The magneto-mechanical coupling takes place at the atomic level due to spin-orbit coupling. From a system level, the material can be assumed to consist of a number of tiny ellipsoidal magnets which rotate due to the torque produced by the externally applied magnetic field. The rotation of these elemental magnets produces a dimensional change leading to free strain in the material. The strain (or magnetostriction) has a nonlinear dependence on the magnetic field and the mechanical stress in the material.

This tutorial demonstrates how to model the nonlinear response of a magnetostrictive material.

Model Definition

A typical magnetostrictive transducer shown in Figure 1 has a steel housing enclosing a drive coil. A magnetostrictive material is placed in the core that works as an actuator when a magnetic field is applied by passing a current through the drive coil.

Figure 1: Sectional view of a cylindrical transducer.

Due to the rotational symmetry of the geometry, the problem is solved as a 2D axisymmetric model, which leads to reduced computation time. The corresponding 2D axisymmetric geometry is shown in Figure 2.

Figure 2: 2D axisymmetric view of a magnetostrictive transducer surrounded by an air domain. The geometric dimensions are in millimeters.

It is assumed that the current in the coil is DC, and hence it can be solved as a stationary problem. The first study performed considers a constant current density of 106 A/m2 in the coil. A second study is set up where the current density in the drive coil is varied from 0 to 107 A/m2 using the parametric sweep feature in COMSOL. The solution from this parametric sweep is then used to generate the characteristic nonlinear magnetostriction (λ) vs. magnetic field (H) curve. The ramping of current density using the parametric sweep option is performed under the assumption that the current in the coil changes quasi-statically without producing any inductive effect.

Notes on the magnetic and magnetoelastic problems

An air domain is created around the transducer to realistically model the magnetic flux path. The boundaries of this air domain are magnetically insulated which ensures that flux does not diverge out of the modeling domain. An alternative technique of implementing this air domain in COMSOL Multiphysics involves the use of infinite elements. For more information on infinite elements, please refer to the AC/DC Module User’s Guide.

The drive coil is modeled as a homogenized current-carrying domain. Individual wires and their electrical conductivity are not resolved. It is assumed that the externally applied current density in the coil is known a priori. In a 2D axisymmetric model, the external current density is the total current through the coil divided by the longitudinal cross-section area (coil length times coil thickness). The coil can also be modeled alternately using the Multiturn Coil Domain feature available in the AC/DC Module. Please refer to the AC/DC Module User’s Guide for more details on using this alternative technique.

Traditionally, the magnetic flux density (also called the B-field) is obtained as a function of the applied magnetic field (the H-field). Such relationship is usually called a B-H curve. The steel housing used in this example is designed to create a closed magnetic flux path, thereby minimizing flux leakage. The nonlinear magnetic behavior of the steel housing is modeled by using a B-H curve to specify the magnetic constitutive relation in the material. The nonlinear B-H curve is obtained by choosing the material Soft Iron (Without Losses) from the AC/DC material library. Incorporation of a nonlinear B-H curve helps in modeling magnetic saturation effects at a sufficiently high magnetic field. Furthermore, you can examine the results of the model to find out specific locations in a material where magnetic saturation has taken place whereas other regions of that material have remained unsaturated.

The stress in the magnetostrictive material is modeled as

The material is assumed to be isotropic, so that the elasticity tensor CH can be represented in terms of two parameters, Young’s modulus and Poisson’s ratio.

The magnetostrictive strain is modeled as the following quadratic isotropic function of the magnetization field M:

where λs is the saturation magnetostriction, which is the maximum magnetostrictive strain reached at the saturation magnetization Ms. The tensor product of two vectors is defined as

Note that the magnetostrictive strain is represented by a deviatoric tensor. This is because the deformation can be related to the magnetic domain rotation associated with the magnetization of the material; such process should not change the material volume.

Nonlinear magnetization in the magnetostrictive material is found from the nonlinear relation

where L is the Langevin function

with χ0 being the magnetic susceptibility in the initial linear region, and the effective magnetic field in the material is given by

where Sdev = dev(Sel) is the deviatoric part of the elastic stress tensor. The second term in the above relation represents the mechanical stress contribution to the effective field, and thus to the material magnetization, which is called the Villari effect. Note that the term is also proportional to magnetization, which implies that some applied magnetic field is needed for the Villari effect to occur. Thus, pure mechanics load cannot produce any magnetization of the material.

In addition, the magnetization and magnetic field are related to each other and to the B-field by

The effective tangential piezomagnetic coupling coefficients can be computed as

where

is the tangential magnetic susceptibility. An important observation from the above formula is that the piezomagnetic coefficients should reach their maximum (or minimum) at certain strength of the applied bias field. This is because M is zero at zero applied field, while χ tends to zero at large applied field magnitudes because of saturation.

The piezomagnetic coupling tensor d is a third order tensor. Due to the symmetry, it can be conventionally represented by a 3-by-6 matrix dHT with only a few nonzero components.

The material properties used to describe the magnetostrictive material are shown in Table 1.

Table 1: Material properties of the magnetostrictive material.
Material property	Value	Description
E	60·109 Pa	Young’s modulus
ν	0.45	Poisson’s ratio
ρ	7870 kg/m3	Density
σ	5.96·106 S/m	Electric conductivity
εr	1	Relative permittivity
λs	2·10-4	Saturation magnetostriction
Ms	1.5·106 A/m	Saturation magnetization

The lower end of the magnetostrictive rod is modeled as fixed, while the upper one can be mechanically loaded in order to study the Villari effect.

Coupling the Magnetic and Structural Problems

The implementation is straightforward as you make use of a predefined multiphysics coupling interface available in COMSOL called Nonlinear Magnetostriction.

Selecting this interface in the Model Wizard adds and interfaces together with the corresponding multiphysics coupling feature, .

Most of the settings you need to configure the coupling are found in the window for either the coupling feature or the feature added under the interface.

Results and Discussion

The results obtained from the first study, where a constant external current density of 106 A/m2 is applied to the coil. Figure 3 shows the von Mises stress in the magnetostrictive material as a surface plot. This plot indicates that the stress due to magnetostriction is uniformly zero everywhere except the region near the bottom surface of the rod due to the fixed constraint boundary condition that was applied to this end of the rod. This is because the free strain due to magnetostriction should not produce any stress unless the material is mechanically constrained. Figure 4 shows that the corresponding strain field caused by the magnetostriction is also fairly uniform in the material except at the fixed end of the rod.

Figure 3: Surface plot of the von Mises stress and a scaled deformation plot of the displacement.

Figure 4: Surface plot of the axial strain component.

Figure 5 shows the magnetic flux concentration in the magnetostrictive core due to the closed magnetic path provided by the steel housing. The magnetic flux density in the rod is mostly uniform. Fringe effects can be seen at both ends of the rod where majority of the magnetic flux is forced to curl into the steel housing.

Figure 5: Surface plot of the norm of the magnetic flux density and a normalized arrow plot of its r and z-components showing the closed flux path in the model.

Figure 6 shows an interesting postprocessing feature in COMSOL Multiphysics. The solution obtained from the 2D axisymmetric model has been revolved by 225 degrees for 3D visualization of the solution. On solving a 2D axisymmetry model, COMSOL Multiphysics automatically creates a 3D solution dataset by revolving the solution, which is then plotted as a 3D plot.

Figure 6: A 225 degree sectional view in 3D of the norm of the magnetic flux density in the magnetostrictive rod, steel housing and in the region within the housing. The solution in the outer air domain has been suppressed to get a better view. The normalized arrow plot shows the direction of the magnetic flux density.

Figure 7 shows the magnetostriction curve of the material obtained from the parametric study that simulated a quasi-static ramping up of the current density in the coil for three different values of the mechanical load. The corresponding B-H curve is shown in Figure 8. Because the magnetic field is oriented mostly along the axial direction, only the Z-components of the corresponding vectors are plotted. Note the significantly nonlinear behavior in the region where the magnetic field Hz varies between 5 to 20 kA/m.

Figure 7: Magnetostriction versus magnetic field (at a point on the magnetostrictive material.

Figure 8: Magnetic flux density versus magnetic field at a point on the magnetostrictive material.

Finally, Figure 9 shows the components of the tangential piezomagnetic coupling matrix in case of no mechanical loading.

Figure 9: Tangential piezomagnetic coupling coefficients at a point on the magnetostrictive material.

Reference

1. S. Chikazumi, Physics of Ferromagnetism, Oxford University Press, New York, 1997.

Structural_Mechanics_Module/Magnetomechanics/nonlinear_magnetostriction

Modeling Instructions

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and choose .

See the figure below for the objects that need to be selected in the next step.

Select the objects , , and only.

In the window for , locate the section.

Clear the check box.

Point 1 (pt1)

In the toolbar, click

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Click the

button in the toolbar.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
J0	1e6[A/m^2]	1E6 A/m²	Current density
F0	0[MPa]	0 Pa	Mechanical load

Solid Mechanics (solid)

In the window, under click .

In the window for , locate the section.

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Select Domain 3 only.

The solid mechanics equations will be solved only in the magnetostrictive material.

Magnetic Fields (mf)

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Ampère’s Law 2

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From the list, choose .

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Ampère’s Law, Nonlinear Magnetostrictive 1

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Select Domain 3 only.

External Current Density 1

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Specify the Je vector as


0	r
J0	phi
0	z

Add Material

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Materials

Air (mat1)

Select Domains 1 and 4–6 only.

Soft Iron (Without Losses) (mat2)

In the window, click .

Select Domain 2 only.

Magnetostrictive

In the window, right-click and choose .

In the window for , type Magnetostrictive in the text field.

Select Domain 3 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	60e9	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.45	1	Young’s modulus and Poisson’s ratio
Density	rho	7870	kg/m³	Basic
Initial magnetic susceptibility	chi0_iso ; chi0ii = chi0_iso, chi0ij = 0	200	1	Magnetostrictive
Interdomain coupling	alphaJA_iso ; alphaJAii = alphaJA_iso, alphaJAij = 0	0	1	Jiles-Atherton model parameters
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	5.96e6	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic
Saturation magnetization	Ms	1.5e6	A/m	Magnetostrictive
Saturation magnetostriction	lambdas	200[ppm]	1	Magnetostrictive