Magnetic Permeability

A material's magnetic permeability:

must be defined for Eddy Current Analysis and Magnetostatic Analysis;
can be specified directly for linear magnetic behavior or through one or more B–H curves for nonlinear magnetic behavior;
can be isotropic, orthotropic, or (in the case of linear behavior) fully anisotropic;
can be specified as a function of temperature and/or field variables;
can be specified as a function of frequency in a time-harmonic eddy current procedure; and
can be combined with permanent magnetization.

This page discusses:

Linear Magnetic Behavior
Nonlinear Magnetic Behavior
Permanent Magnetization
Elements

Linear Magnetic Behavior

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

Directional Dependence of Magnetic Permeability

Isotropic, orthotropic, or fully anisotropic magnetic permeability can be defined. For non-isotropic magnetic permeability a local orientation for the material directions must be specified (Orientations).

Isotropic Magnetic Permeability

For isotropic magnetic permeability only one value of magnetic permeability is needed at each temperature and field variable value. Isotropic magnetic permeability is the default.

Orthotropic Magnetic Permeability

For orthotropic magnetic permeability three values of magnetic permeability ( $μ_{11}$ , $μ_{22}$ , $μ_{33}$ ) are needed at each temperature and field variable value.

Anisotropic Magnetic Permeability

For fully anisotropic magnetic permeability six values ( $μ_{11}$ , $μ_{12}$ , $μ_{22}$ , $μ_{13}$ , $μ_{23}$ , $μ_{33}$ ) are needed at each temperature and field variable value.

Frequency-Dependent Magnetic Permeability

Magnetic permeability can be defined as a function of frequency in a time-harmonic eddy current analysis.

Nonlinear Magnetic Behavior

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.

Directional Dependence of Nonlinear Magnetic Behavior

Isotropic, orthotropic, or transversely isotropic nonlinear magnetic behavior can be defined. For non-isotropic nonlinear magnetic behavior a local orientation for the material directions must be specified (Orientations).

Isotropic Nonlinear Magnetic Behavior

For isotropic nonlinear magnetic response only one B–H curve is needed at each temperature and field variable value. Isotropic magnetic permeability is the default. Abaqus assumes that the nonlinear magnetic behavior is governed by

B = B (| H |) (\frac{H}{| H |})

Orthotropic Nonlinear Magnetic Behavior

For orthotropic nonlinear magnetic response three B–H curves (one curve to define the behavior in each of the local directions 1, 2, and 3) are needed at each temperature and field variable value. Abaqus assumes that the nonlinear magnetic behavior in the local material directions is governed by

B = d i a g (B_{1} (| H |), B_{2} (| H |, B_{3} (| H |)) (\frac{H}{| H |}),

where $d i a g ()$ refers to a diagonal matrix.

Transversely isotropic nonlinear magnetic behavior is a special case of orthotropic behavior, in which the behavior in any two directions is the same and is different from that in the third direction.

Permanent Magnetization

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.

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

H = μ^{- 1} B - H_{c}

for linear isotropic, orthotropic, or anisotropic magnetic behavior and

H = \hat{H} (| B |) (\frac{B}{| B |}) - H_{c}

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

Elements

Magnetic material behavior is active only in electromagnetic elements (see Choosing the Appropriate Element for an Analysis Type).