Magnetostriction and Piezomagnetism
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.
The inverse effect accounts for the change in magnetization due to mechanical stress in the material. This effect is also known Villari effect. This effect is mostly useful in sensors.
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.
Piezomagnetism
The magnetostriction has a nonlinear dependence on the magnetic field and the mechanical stress in the material. However, the effect can be modeled using linear coupled constitutive equations if the response of the material consists of small deviations around an operating point (bias point). This type of coupling is reffed to as Piezomagnetic Effect.
It is possible to express the relation between the stress S, strain ε, magnetic field H, and magnetic flux density B in either a stress-magnetization form or strain-magnetization form:
Stress-Magnetization
Strain-Magnetization
where μ0 is the magnetic permeability of free space; cH and sH are the stiffness and compliance matrices measured at constant magnetic field, respectively; and μrS and μrT are the relative magnetic permeabilities measured at constant strain and constant stress, respectively. The matrices dHT and eHS are called piezomagnetic coupling matrices.
In COMSOL Multiphysics, both constitutive forms can be used; simply select one, and the software makes all necessary transformations. The following equations transform strain-magnetization material data to stress-magnetization data:
One can rewrite the system of constitutive relations in the following equivalent form:
where the magnetostrictive strain is introduced as
and the material magnetization due to the applied field is given by
Hence, the stress in material is proportional to the elastic strain
and it contributes to the material magnetization.
All the necessary material data inputs are placed within the Piezomagnetic Material node under the Solid Mechanics interface, which are added automatically when adding a predefined Piezomagnetism multiphysics interface. Such a node can be also added manually under any Solid Mechanics interface similar to all other material model features. The Piezomagnetic Material uses Voigt notation for the anisotropic material data. More details about the data ordering can be found in the Orthotropic and Anisotropic Materials section.
For a crystalline material with tetragonal symmetry, the strain-magnetization form of the constitutive relations is the following:
The following material data corresponds to Terfenol-D at 100 kA/m bias and 30 MPa prestress (Ref. 6):
Multiplicative Formulation for Piezomagnetism
The total deformation gradient is computed from the structural displacement field as
and the right Cauchy–Green is defined as .
The decomposition between elastic and inelastic deformation is made using a multiplicative decomposition of the deformation gradient
where the inelastic deformation tensor depends on the inelastic process, such as piezomagnetic effect and thermal expansion .
The elastic right Cauchy–Green deformation tensor is then computed from Fel
and the elastic Green–Lagrange strain tensor is computed as:
Note that .
The strain tensor C, the magnetic field H, and the temperature T are used as the state variables. The free energy density in the undeformed configuration is defined as
where and .
For structurally linear material,
where c is the elasticity tensor.
The free space contribution is computed as
where Hs and H are the magnetic fields in the deformed (spatial) and undeformed configuration, respectively.
The second Piola–Kirchhoff stress is computed as
The Maxwell stress SM is related to the magnetization of the free space occupied by the deformed body, and it is computed as
The inelastic deformation gradient is further decomposed into a thermal and piezomagnetic parts as
The piezomagnetic strain tensor is introduced as , where d is the piezomagnetic coupling tensor. The piezomagnetic deformation gradient is modeled as
Introduce
which are the strain and stress tensors in the intermediate configuration, where the thermal expansion part have been removed. These quantities are independent of the magnetic field H.
Then,
and
where
The stress is computed then as
The magnetic flux density (in the undeformed configuration) is computed as
where the material magnetization is given by
For a magnetically linear material, it is computed as
where μr is the relative magnetic permeability.
Nonlinear Magnetostriction
A commonly accepted micromagnetic description of the magnetostriction is as follows (Ref. 2):
All domains have magnetization of the same magnitude |M| = Ms, but the magnetization can have different orientations characterized by the corresponding direction vector m = M/Ms for each domain. The applied magnetic field changes the domain orientation.
In this section, the term domain refers to a small part of magnetic material. This is typical for micromagnetics literature, and it should not be mistaken with the concept of domain as part of the model geometry, the latter is often used in COMSOL Multiphysics documentation.
For a single crystal with cubic symmetry, the magnetostrictive strain tensor can be written as the following quadratic form using tensor notation:
(3-153)
where . Note that the magnetostrictive strain is represented by a deviatoric tensor, thus trme) = 0. This is because the deformation is related to the magnetic domain rotation, and such process should not change the material volume.
The strain in any direction given by the directional cosines βi can be written as
Using Equation 3-153, one gets
(3-154)
When both magnetization and measurement direction are parallel to the same crystal direction [100], one has m1 = β1 = 1 and all other components are zero, so that
In a similar way for the [111] direction, one has for all components and λ = λ111.
If the strain is measured in [100] direction, while all the magnetization vectors are aligned perpendicular to it, one has only the following two nonzero components: m2 = β1 = 1 and consequently:
In many applications, such alignment of the domains is achieved by applying a compressive prestress. Thus, the maximum usable magnetostriction is achieved via a 90-degree rotation of the domains
For an isotropic material, λ100 = λ111 = λs, and Equation 3-153 becomes
For a polycrystalline material without preferred orientation, the following approximation can be used (Ref. 1):
In COMSOL Multiphysics, this description of the magnetostriction is modeled using the following equation for the magnetostrictive strain:
(3-155)
The strain field is deviatoric, and Equation 3-155 exhibits the same properties as Equation 3-153 at saturation, that is, when |M| → Ms. Equation 3-154 is replaced by
Note that the strain vanishes when |M| → 0, which makes the model applicable in the whole range from full demagnetization to saturation.
For isotropic materials, the magnetostrictive strain is modeled as the following quadratic isotropic form of the magnetization field using tensor notation (Ref. 3):
(3-156)
If the stress and strain tensors are represented by 6-component vectors using Voigt notation, the stress in the magnetostrictive material is modeled as
where cH is a 6-by-6 stiffness matrix. For isotropic materials, the matrix can be represented in terms of two parameters, for example, using the Young’s modulus and Poisson’s ratio. Cubic materials possess only three independent components: c11, c12, and c44.
One can derive a linear response around a given bias state characterized by a premagnetization vector M0. Thus,
where M1 is a perturbation.
Using Equation 3-156,one finds
If one assumes a unidirectional state, for example
it will further simplify into
Magnetization
The magnetization in the magnetostrictive material is found from the following nonlinear implicit relation (Ref. 4 and Ref. 5):
(3-157)
where L is the Langevin function
with χ0 being the magnetic susceptibility in the initial linear region.
Other possible choices of the L function are a hyperbolic tangent, which is sometimes referred to as the Ising model
and a linear function
The latter option will make it possible to find an explicit expression for the magnetization. However, such model does not have a proper saturation behavior, and thus it should be used only in the operating range far from saturation. Both the Langevin function and hyperbolic tangent models requires the magnetization vector components to be treated as extra dependent variables.
For cubic crystals, the effective field in the material is given by
(3-158)
where H is the applied magnetic field. The second term in Equation 3-158 represents the mechanical stress contribution to the effective magnetic field, and thus to the material magnetization, which is called the Villari effect. The deviatoric stress tensor is related to the strain as
For isotropic materials, the effective magnetic field expression simplifies into
(3-159)
In addition, the magnetization and magnetic field are related to each other and to the magnetic flux density (also called the B-field) by
COMSOL Multiphysics solves for the magnetic vector potential A whose curl yields the vector B-field. The H-field is then obtained as a function of the B-field and magnetization
The effective tangent piezomagnetic coupling coefficients can be computed as
For an isotropic material, the derivative can be evaluated to give
where
is the tangent magnetic susceptibility. The corresponding expression in case of cubic material is more complicated, but it has a similar structure involving products of the magnetization and tangent susceptibility components. An important observation from the above formulas is that the piezomagnetic coefficients should reach their maximum (or minimum) at certain strength of the applied bias field if the saturation effect is taken into account. This is because M is zero at zero applied field, while χm 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 few nonzero components.
Hysteresis modeling
The Jiles–Atherton hysteresis model for magnetostrictive materials is available COMSOL Multiphysics. The model assumes that the total magnetization can be represented as a sum of hysteretic and anhysteretic parts, the latter one is given by Equation 3-157, thus
Compared to Equation 3-158 and Equation 3-159, the effective magnetic field Heff gets one more term αM, where α is the interdomain coupling parameter.
The change in the total magnetization caused by the change on the effective magnetic field is represented as
(3-160)
where cr is the reversibility parameter, and kp is the pining loss parameter.
Equation 3-160 can be solved using either a time-dependent analysis or a stationary parametric sweep.