Magnetomechanics
The theory for The Magnetomechanics, Solid Interface, The Magnetomechanics, No Currents, Solid Interface, The Magnetic–Elastic Interaction in Rotating Machinery Interface and The Magnetic–Rigid-Body Interaction in Rotating Machinery Interface is given in this section.
The Electromagnetic Stress Tensor
Within a vacuum or other medium, forces between magnetized bodies can be computed on the assumption that a fictitious state of stress exists within the field. Historically, the consistency of this approach led 19th-century physicists to postulate the existence of the ether, a ubiquitous medium through which electromagnetic forces propagate. While these ideas have been superseded by the development of particle physics, the use of an electromagnetic stress tensor (also known as the Maxwell stress tensor) remains an accurate and convenient technique to compute electromagnetic forces.
Maxwell stresses exist in all materials, including air and even free space. However, the force magnitude is rather small, and usually it can only cause significant deformations at small dimension.
The electromagnetic stress tensor in a vacuum (in the absence of electric fields) is given by (Ref. 151):
(3-174)
Where H is the magnetic field, I is the identity tensor, μ0 is the magnetic permeability of free space, and ⊗ is the outer product of two vectors
Within an isotropic linear magnetic solid under small deformations, the magnetic flux density vector is related to the magnetic field as (Ref. 152):
where the magnetic susceptibility χ can be a function of the mechanical strain in the material
where χ0 is the magnetic susceptibility of the material without deformation, and T is the small-strain tensor given by
The corresponding electromagnetic tress tensor takes the following form:
(3-175)
where the relative magnetic permeability is introduced as μ =  1+ χ. It can be written equivalently as
where the two terms represent the contributions from the underlying free space and the material magnetization, respectively.
The Magnetomechanics, Solid and Magnetic Forces, Rotating Machinery coupling features apply the material and underlying free space contributions to the stress as body load. In weak form: −σEM:test(T).
The electromagnetic stress tensor can be used to compute the forces acting on a solid body.
The balance of forces at the surface of the solid (material 1) in vacuum or air (material 2) implies:
where the total stress tensors in media 1 and 2 are, respectively:
where σmech is a mechanical component of the total stress in the material, and p2 is the pressure in the surrounding air (or other medium). Using Equation 3-174:
where the magnetic field H is computed in material 2.
The Magnetomechanics, Solid; Magnetomechanics, Boundary; and Magnetic Forces, Rotating Machinery coupling features apply the above defined traction as a boundary load on the boundaries which are external for the coupling feature selection but internal for the corresponding magnetic fields interface.
COMSOL Multiphysics does not explicitly include the ambient pressure definition on the coupling features. However, it is possible to add an additional surface force to the corresponding Solid Mechanics interface if the pressure is known or computed by another physics interface.
Large Deformations
For finite deformations, the expressions for the electromagnetic stress and material magnetization can be derived using the following thermodynamic potential called magnetic enthalpy:
where the subscript m indicates that the vector components must be taken on the material frame, and the right Cauchy–Green deformation tensor is
with
and J = det(F). The mechanical energy function Ws(C) depends on the solid model used.
The total second Piola–Kirchhoff stress tensor is given by
and the magnetic flux density vector can be calculated as
About Different Stress Forms
There exists a long-lasting controversy in scientific literature about the definition of electromagnetic forces acting on solids. An extensive review can be found in Ref. 143. Many classic textbooks (for example, Ref. 151 and Ref. 152) operate with the so-called Minkowski magnetic stress tensor that is usually written as
The corresponding electromagnetic body force can be written as
which sometimes is referred to as the Korteweg–Helmholtz magnetic force. For homogeneous materials without deformation, one has ∇χ = 0. Hence, in the absence of electric currents (J = 0), the body force becomes zero. Thus, the whole magnetomechanical load on the solid is due to the Maxwell stress jump at the boundaries between domains with different material properties.
There exists two most often used alternatives. The first one is called the Einstein–Laub stress tensor:
This form is widely accepted in modern magnetoelasticity and material science, see Ref. 144. The corresponding body force can be written as
which is also called the Kelvin magnetic force. Note that it is nonzero as soon as there are magnetic field variation and magnetization within the material
The other alternative is the Chu stress tensor:
and the corresponding magnetic body force is given by
so that variations of both the magnetic susceptibility and magnetic field can contribute.
Magnetomechanics, Boundary and Magnetic Forces, Rotating Machinery coupling features always operate with Minkowski stress tensor form.
For Magnetomechanics, Solid coupling feature, the choice of stress tensor is available only when one of the coupled interfaces is Magnetic Fields, No Currents.