Magnetic permeability

Linear magnetic behavior is defined by direct specification of magnetic permeability.

Nonlinear magnetic behavior is characterized by magnetic permeability that depends on the strength of the magnetic field. The nonlinear magnetic material model in Abaqus is suitable for ideally soft magnetic materials without any hysteresis effects (see Figure 1) characterized by a monotonically increasing response in B–H space, where B and H refer to the strengths of the magnetic flux density vector and the magnetic field vector, respectively. Nonlinear magnetic behavior is defined through direct specification of one or more B–H curves that provide B as a function of H and, optionally, temperature and/or predefined field variables, in one or more directions. Nonlinear magnetic behavior can be isotropic, orthotropic, or transversely isotropic (which is a special case of the more general orthotropic behavior). More than one B–H curve is needed to define the nonlinear magnetic behavior if it is not isotropic. For each curve, the slope between the last pair of data points is assumed to be the permeability of the free space. Abaqus issues an error message if the slope between any two adjacent data points is less than 0.01 times the permeability of the free space, as computed based on the last two data points for that curve.

Ferromagnetic materials can be magnetized by placing them in a magnetic field, which is typically created by applying currents in a system of coil windings surrounding the material being magnetized. These materials can be classified into soft and hard magnetic materials (see Figure 1). Soft magnetic materials lose their magnetization after removal of the applied currents (see Nonlinear magnetic behavior). Hard magnetic materials retain their magnetization permanently after removal of the applied currents. The leftover magnetization in a permanent magnet is called remanence, denoted by $B_r$ in Figure 2. This magnetization can be removed by applying currents in the opposite direction; the strength of the opposing magnetic fields that remove magnetization entirely is called coercivity, denoted by $H_c$ in Figure 2.

Figure 1. Response of hard and soft magnetic materials.

Figure 2. Remanence and coercivity in permanent magnets.

Permanent magnetization in Abaqus is suitable for hard magnetic materials when the magnets are operating around the point of remanence. This behavior captures the response of magnetization or demagnetization around the point of remanence, as shown by the darker descending line of the hysteresis loop in Figure 2. The underlying magnetic permeability can be linear or nonlinear. In either case, permanent magnetization is defined by its coercivity such that

for linear isotropic, orthotropic, or anisotropic magnetic behavior and

for nonlinear isotropic $\mathrm{B}$-$\widehat{\mathrm{H}}$ response.

MAGNETIC PERMEABILITY
PERMANENT MAGNETIZATION
direction of magnetization in the global system
magnitude of coercivity

MAGNETIC PERMEABILITY, NONLINEAR
NONLINEAR BH
input $\mathrm{B}$-$\widehat{\mathrm{H}}$ by shifting the response to the right by $H_c$
PERMANENT MAGNETIZATION
direction of magnetization in the global system
magnitude of coercivity