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:
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.
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.
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 c is the elasticity tensor.
where Hs and
H are the magnetic fields in the deformed (spatial) and undeformed configuration, respectively.
The Maxwell stress SM is related to the magnetization of the free space occupied by the deformed body, and it is computed as
The piezomagnetic strain tensor is introduced as , where d is the piezomagnetic coupling tensor. The piezomagnetic deformation gradient is modeled as
where μr is the relative magnetic permeability.
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.
where . Note that the magnetostrictive strain is represented by a deviatoric tensor, thus tr(εme) = 0. This is because the deformation is related to the magnetic domain rotation, and such process should not change the material volume.
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.
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.
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
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
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 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.
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.
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.