Bulk Forces and Interface Forces in Assemblies of Magnetized Pieces of Matter

Abstract

When two magnets are stuck together, where do magnetic forces operate and which formulas should one apply to compute them? Such frequently asked questions do not find immediate answers in the literature on forces, mainly because the force field is obtained, by the Virtual Power Principle (VPP), as a (mathematical, vector-valued) distribution, not as a plain vector field, which would be more convenient for practical computation. We intend to show, in a few important cases of contact (between two linear materials with different permeabilities, between magnet and magnetizable metal, linear or not, between two hard magnets, etc.), how to represent this single distribution by two vector fields, one of them borne by the bulk of the matter, the other one localized at material interfaces where discontinuities of permeability, of magnetization, etc., do occur. Suitable extensions of the classical Maxwell tensor play an important role in the computation (by the so-called 'pillbox trick') of this interface vector field.

Essential Measurements for Finite Element Simulations of Magnetostrictive Materials

“…The first term is called the inhomogeneity term, the second one is the anisotropy term, and the third term is called the magnetostrictive term. According to [21], the magnetostrictive stress tensor σ M inside M is defined as…”

“…We defined the stress tensor for the elastic system in the standard way. According to [21], it is possible to define an augmented Maxwell stress tensor σ T , whose divergence yields all the magnetic force components given in (24). It follows that the equilibrium equation of linear elasticity can be written as:…”

“…It is our strategic choice not to define the elastic constitutive relation for a magnetoelastic system. The reason is that it is not obvious how the coupled constitutive relation models the effects of the inhomogeneity force component in (24) on the boundary ∂ M. This is particularly an issue, when one employs (21) to solve the elastic equations. Consequently, we define only the elastic constitutive relation σ () for elastic systems without magnetization.…”

“…In order to run finite element simulations for a magnetostrictive material M, we need to choose a discretization and a proper amount of test functions δd i and δa i in (6) and (21). Since we assume that the elastic constitutive relation σ () is measured a priori, all what is needed is to measure relation h(, b).…”

Magnetic forces in and on a magnet

“…Isotropy, requiring parallel B and H, would result from the invariance of magnetic energy density under rotations, which cannot hold, as a rule, at places where M = 0. (See [3] for an example of anisotropy due to the law B = µH when µ is not a scalar but a 3 × 3 matrix. Then H × B = 0. )…”