B(Stress) magneto-mechanical model: dependency between B(H) constitutive law and mechanical stress


The magnetic behavior of a material is known to be sensitive to the application of mechanical stress. This may lead to significant effects that modify the performance of electromagnetic devices. For instance, the iron losses in a mechanically stressed magnetic circuit may increase up to 40%. The mechanical stress may be applied in different ways: during the construction of a magnetic device (punching, laser cut, assembly...) or during its operation (magnetostriction, centrifugal forces due to high-speed rotations, etc.).

Flux can model this dependency between the B(H) constitutive relation of an electrical steel and mechanical stress through its B(Stress) analytical multiscale model. This model allows considering the influence of an equivalent mechanical stress on the B(H) magnetic behavior of the material. More specifically, this coupled property allows Flux to account for plastic or elastic deformation effects that modify the B(H) relationship that would be verified in an unconstrained sample of the material. The modified B(H) curves are computed automatically by Flux and used in the FEM computations.

It is important to note that the descriptions of both the B(H) and B(Stress) properties of a material are mandatory to account for magneto-mechanical stress effects. Figure 1 shows the computed B(H) curves that result from different values of mechanical stress for an electrical steel sheet (FeSi alloy):

Figure 1. Example of modified B(H) curves of a FeSi steel sheet that are automatically computed by Flux, in accordance with the coupled B(Stress) property for different mechanical constraint values.

In this model, the influence of mechanical stress on the shape of the B(H) curve is governed by the saturation magnetostriction constant λ. In Figure 1, the unconstrained material may be described by the Isotropic analytic saturation + knee adjustment law with the following parameters:

  • Initial relative permeability: 10000;
  • Saturation magnetization: 1.9 T;
  • Knee adjusting coefficient: 0.02.
On the other hand, the coupled B(Stress) property of this material is characterized by a saturation magnetostriction constant λ equal to 9e-6.
Parameter λ may be determined with the help of measurements (i.e., by applying a controlled stress on a magnetic sample) and depends directly on the type of alloy composing the steel sheet. A list containing the most frequent alloys and metals and the corresponding values of their saturation magnetostriction constants is provided in Table 1 below.
Table 1. Typical saturation magnetostriction constant values for metals and alloys frequently used in electric machinery.
Alloy / metal Saturation magnetostriction constant
FeSi 9e-6
FeNi 27e-6
FeCo 70e-6
Fe -7.0e-6
Ni -33e-6
Co -62e-6
The following topics are covered in the next sections:
  • How to create a B(Stress) magneto-mechanical property coupled to a B(H) magnetic property;
  • Specificities for the solving process;
  • Application example.

Creating a material with a magneto-mechanical model and assigning it to a region

The magneto-mechanical model is available only in Flux 2D for Transient Magnetic and Magneto-static applications and may be assigned to a material either during its creation or during its modification.

More specifically, the user must set both the B(H) property of the material to the appropriate subtype (i.e., isotropic analytic saturation + knee adjustment) and the B(Stress) property.

The procedure is detailed below:

  • First, create or edit an existing material in a Flux project. Two methods are available to perform these actions:
    1. Using the Physics menu, by selecting the option Material and then New or Edit;
    2. Through the Flux Data Tree located on the left of the main project view, i.e., in its Physics section, by double-clicking on Material to create a new one or by double-clicking on an existing material to edit it.
    Depending on the case, Flux will display either the New Material or the Edit Material dialog box.
  • Enable the Magnetic Property option that is available in the B(H) tab.
  • Then, choose the following model:
    • Isotropic analytic saturation + knee adjustment and fill the properties of the model: the Initial relative permeability, the Saturation magnetization and the Knee adjusting coefficient.
  • Enable the Magneto-mechanical property option that is available in the B(Stress) tab and then choose the following model:
    • Analytic multiscale model and fill the value of the Saturation magnetostriction constant λ.
  • Finally, the previously created material must be assigned to a Laminated magnetic non conducting region for a proper consideration of its magneto-mechanical behavior during solving process and post-processing operations (e.g. computation of iron losses). In practice, while creating or editing the chosen laminated magnetic non-conducting region, the user must enable the Mechanical stress dependence option in the main tab of the region dialog box, and then choose between one of the three following constraint types (and thus models):
    1. Uniform over the whole region or
    2. Exponential decay towards region center or
    3. Defined by a spatial variable distribution.

    In any case, the user must then fill the Equivalent uniaxial stress (MPa) field with the value or the parametric expression of the intensity of the mechanical constraint in MPa. For the first case (Uniform over the whole region), this value impacts the B(H) law of all nodes of the region.

    The second model (Exponential decay towards region center) applies this value to all boundary nodes of the region (except symmetry and periodicity lines), and the internal nodes are constrained by a lower stress (in absolute value) determined by means of an exponential decay equation. To define such a model, the user is also required to fill the Distance-to-boundary defining the stress decay rate (mm) field, which is the inverse value of the exponential decay constant (in other words, the value to fill plays the same role as the "time constant" of RL or RC circuits).

    The third model (Defined by a spatial variable distribution) uses an expression or formula (which may include a spatial quantity with values varying with position) to define the stress values on nodes of the region.

    Note: A positive stress value corresponds to a tension and a negative value to compression (i.e., the typical case for punched laminated sheets employed in electrical machines). Moreover, this stress is said to be equivalent to an uniaxial stress in the sense that the change in the B(H) property in the region corresponds to the change that would take place in a sample of the material subjected to uniaxial stress. The region remains isotropic, i.e., the modified B(H) property is used for all directions in the computations.
    Note: The first approach (Uniform over the whole region) is useful when the user decides to split the original region in sub-regions, which may include parts that are not affected by mechanical stress and others that are impacted by a mechanical constraint.
    Note: The second approach (Exponential decay towards region center) may be used as a straightforward way to model a magnetic region that is subjected to a mechanical constraint only in a narrow band lying along its boundaries (e.g. effects of punching operations). Since it does not require splitting the original region into sub-regions, this approach makes the geometrical and the mesh descriptions easier. As a rule of thumb for meshing, it is advised to have in this area a maximum element size equal to the Distance-to-boundary defining the stress decay rate (mm) value. Remark that in this second approach, all the nodes of the region are subjected to a mechanical stress (and consequently to a modified B(H) property). Its value is the highest on boundaries and decays exponentially as the nodes are located inside the region. In such a manner, 95% of the mechanical stress is applied to the narrow band lying along the region boundaries and whose width is three times the distance value filled in the Distance-to-boundary defining the stress decay rate (mm) field. A comparison between the first two approaches is shown in Figure 2 below where the obtained stress distributions (isovalues) are reported.
    Figure 2. Isovalues of the mechanical stress applied on a stator tooth through two of the available approaches. In the first (a), the stator is split into two regions: a region (lying along the boundaries) damaged by punching where a uniform stress value is applied and an intact region (in the center). In the second (b), the mechanical constraint decays exponentially from the boundaries towards the center.

    Note: With the third approach (Defined by a spatial variable distribution), the user may define a spatial distribution for the stress value which can be non-uniform over the region. The spatial stress distribution can be defined by means of classical Flux quantities and parameters as well as by import of external values obtained from simulations performed with mechanical solvers (e.g., Altair OptiStruct). Further information on how to import such a spatial quantity may be found in the following chapter: Importing mechanical stress distributions into Flux 2D.

Specificities for the solving process

Flux solving process is customized in case the project to solve requires the use of this magneto-mechanical model. In particular:
  • the default options for the nonlinear Newton-Raphson solver are set in a way to ensure - in most cases - the best trade-off between accuracy and rapidity. For more details, please refer to Adjusting parameters in the Newton-Raphson method.
  • the Knee adjusting coefficient defined in the B(H) property of the material is not considered for regions where the Mechanical stress dependence option is activated. This means that in case of Equivalent uniaxial stress (MPa) value set to zero, the results may be slightly different when compared to the configuration where the Mechanical stress dependence option is not activated. This choice allows a better representation of the phenomena occurring in case of high values of mechanical compression (e.g., -150 MPa).

Application example

Let's consider a Permanent Magnet Synchronous Machine (PMSM) modeled with Flux 2D and use it as an example to investigate the effect of punching on iron losses computations. Let's also consider two of the approaches described in the previous section for the description of the required Laminated magnetic non-conducting region when applied to this specific machine:
  • Approach based on the constraint Uniform over the whole region: the stator of the PMSM is split in two separate regions. The first is a narrow band corresponding to the edges of the stator and has been damaged by the punching process. The second represents the innermost parts of the stator that have not been damaged through punching.
  • Approach based on the Exponential decay towards region center of the constraint: the stator is represented by a single region. The compressive mechanical stress is set to its maximum value on the boundary nodes and it exponentially decays as the nodes are located inside the region. This approach assumes that the width of the damaged zone along the boundaries is small when compared to the stator dimensions.
Figure 3. The geometry of the stator and the adopted meshes for each approach. The mesh used together with the approach based on uniform stress values applied to two regions is shown in (a). The mesh used together with the approach based on the application of a decaying mechanical stress towards the center of the unique region used is displayed in (b).

Several computations were performed with the compressive mechanical stress going from 0 to -210 MPa and with both approaches: in the case of uniform stress a damaged zone width of 0.25 mm has been used, while in the case of exponential decaying stress, the decay rate has been fixed at 0.08 mm, so that 95% of the mechanical stress is applied on the three-times (0.24 mm) width external band. The total Bertotti iron losses were evaluated while in post-processing, after resolution of a Transient Magnetic scenario. A comparison of the total iron losses obtained for each value of mechanical stress is summarized in Figure 4 below:
Figure 4. The iron losses dissipated in the stator as a function of the mechanical stress. The blue line corresponds to the approach based on a stator split in two parts: a boundary region with an active magneto-mechanical property and an inner region without the mechanical stress dependency. The orange line, on the other hand, corresponds to the approach in which the stress decays exponentially from the boundaries towards the center of a single region.

The magnetic flux density distribution may be visualized through an isovalue plot in the stator, as shown in the Figure 5 below. In this figure we can clearly see the impact of the punching process and of the mechanical stress on the stator teeth boundaries. The intensity of the magnetic flux density in a narrow region along the perimeter of a tooth is visibly lower when compared to its innermost parts.

Figure 5. Distribution of the magnetic flux density in the bulk of the stator. Results obtained with approach based on the application of mechanical stress of -60 MPa decaying from the boundaries towards the region center.

The weakening of the magnetic flux density along the boundaries is a consequence of the localized degradation of the magnetic permeability resulting from punching during the fabrication of the stator, as shown in Figure 1.