Inverse magnetostrictive effect

The inverse magnetostrictive effect, magnetoelastic effect or Villari effect is the change of the magnetic susceptibility of a material when subjected to a mechanical stress.
Explanation.
The magnetostriction formula_1 characterizes the shape change of a ferromagnetic material during magnetization, whereas the inverse magnetostrictive effect characterizes the change of sample magnetization formula_2(for given magnetizing field strength formula_3) when mechanical stresses formula_4 are applied to the sample.
Qualitative explanation of magnetoelastic effect.
Under a given uni-axial mechanical stress formula_4, the flux density formula_6 for a given magnetizing field strength formula_3 may increase or decrease. The way in which a material responds to stresses depends on its saturation magnetostriction formula_8. For this analysis, compressive stresses formula_4 are considered as negative, whereas tensile stresses are positive.
According to Le Chatelier's principle:
formula_10
This means, that when the product formula_11 is positive, the flux density formula_6 increases under stress. On the other hand, when the product formula_11 is negative, the flux density formula_6 decreases under stress. This effect was confirmed experimentally.
Quantitative explanation of magnetoelastic effect.
In the case of a single stress formula_4 acting upon a single magnetic domain, the magnetic strain energy density formula_16 can be expressed as:
formula_17
where formula_8 is the magnetostrictive expansion at saturation, and formula_19 is the angle between the saturation magnetization and the stress's direction. 
When formula_8 and formula_4 are both positive (like in iron under tension), the energy is minimum for formula_19 = 0, i.e. when tension is aligned with the saturation magnetization. Consequently, the magnetization is increased by tension.
Magnetoelastic effect in a single crystal.
In fact, magnetostriction is more complex and depends on the direction of the crystal axes. In iron, the [100] axes are the directions of easy magnetization, while there is little magnetization along the [111] directions (unless the magnetization becomes close to the saturation magnetization, leading to the change of the domain orientation from [111] to [100]). This magnetic anisotropy pushed authors to define two independent longitudinal magnetostrictions formula_23 and formula_24.
Method of testing the magnetoelastic properties of soft magnetic materials.
Method suitable for effective testing of magnetoelastic effect in magnetic materials should fulfill the following requirements:
Following testing methods were developed:
Applications of magnetoelastic effect.
Magnetoelastic effect can be used in development of force sensors. This effect was used for sensors:
Magnetoelastic effect have to be also considered as a side effect of accidental application of mechanical stresses to the magnetic core of inductive component, e.g. fluxgates.