The Electromagnetic Stress Tensor
Within a vacuum or other medium, forces between charged 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 While these ideas have been superseded by the development of special relativity, the use of an electromagnetic stress tensor (also known as the Maxwell stress tensor) remains an accurate and convenient technique to compute electromagnetic forces. The electromagnetic stress tensor in a vacuum (in the absence of magnetic fields) is given by (Ref. 1):
(5-2)
Where E is the electric field, I is the identity tensor, and ε0 is the permittivity of free space.
Within an isotropic dielectric, in the absence of magnetic fields, the electromagnetic tress tensor takes the following form (Ref. 1):
(5-3)
Where D is the electric displacement and a1 and a2 are the electrostrictive material properties, which must be determined by experiment. Physically, the parameter a2 expresses the change in permittivity that results from elongation of the solid parallel to the field, whilst a1 corresponds to the change in permittivity produced by strains perpendicular to the field (Ref. 1). Note that for the case of an isotropic dielectric liquid (in which the shear stresses are necessarily zero) a1=a2=ρ(∂ε/∂ρ)T where ε is the permittivity of the fluid (Ref. 2—note that this reference uses alternative definitions of a1 and a2. Using a1’ and a2’ for the constants in Ref. 2, a1’=a1+a2 and a2’=a2.). By default a1=a2=0, which is equivalent to the assumption of no electrostriction.
Using the notation defined in Ref. 3, for a linear isotropic material, the coefficients a1 and a2 can be related to the electrostrictive coefficients Q11 and Q12 by the following equations:
Where ν is the material’s Poisson’s ratio, EYM is its Young’s modulus, and χ is its electric susceptibility.
The electromagnetic stress tensor can be used to compute the forces acting on a dielectric body. From Equation 5-1 the balance of forces at the surface of a dielectric body (material 1) in vacuum or air (material 2) implies:
where T1 and T2 are the total stress tensors in media 1 and 2, respectively. To compute the balance of forces the following total stress tensors are used:
Where σm is a mechanical component of the total stress, required to balance the electrostatic forces on the body and p is the pressure due to air (if present).
COMSOL Multiphysics does not explicitly include the pressure, p, in the Electromechanics interface. However, it is possible to add an additional surface force to the physics interface if the pressure is known or computed by another physics interface.
Because T2 is a fictitious stress, its effect need only be accounted for on the surfaces of mechanical bodies. The normal component of the stress tensor can then be applied as a surface force. Using Equation 5-2:
(5-4)
where n1 is the surface normal, a 1-by-3 vector, pointing out from the mechanical body. This force is applied by the Electromechanical Interface feature.
The Electromechanical Interface feature does not explicitly include the pressure term, p, in Equation 5-4. However, it is possible to add an additional surface force to the physics interface if the pressure is computed explicitly, for example by means of a Laminar Flow interface added to the model.
In the mechanical region the total stress tensor takes the form:
(5-5)
Within the Linear Elastic Dielectric feature, the electromechanical part of the stress tensor is applied as a prestress in the solid mechanics equations. Since the prestress is applied in the material frame, its expression in the equation view is complicated by the transformation that is applied to transform between frames.
Because COMSOL Multiphysics applies TEM,S as a prestress in the Electromechanics interface, the total stress still contains a fictitious component due to the “vacuum stress” used to apply the electromagnetic forces.
This stress is balanced by surface forces acting on the body, so the computed displacements and strains are accurate. However, the stress shown in postprocessing contains an offset due to the presence of the fictitious vacuum stress. This is apparent when, for example, modeling a solid beam with zero electrical susceptibility in an electric field. While the strains produced are zero, a uniform nonzero stress, equal to the vacuum stress, is present in the beam.
When Linear Elastic Material is used for a solid domain, it is assumed that the electric field is zero within the material and the term TEM,S is consequently zero. In this case the only electric forces acting on the solid are those on its surfaces, given by Equation 5-4 and applied through the Electromechanical Interface feature.
For a general dielectric material the total stress tensor takes the form:
where in σ(E, D) the dependence of the mechanical stress on the electric field and displacement has not been written out explicitly. COMSOL Multiphysics only includes the functionality to apply the correct electromechanical forces to an isotropic material.