# Inverse magnetostrictive effect

The inverse magnetostrictive effect, magnetoelastic effect or Villari effect, after its discoverer Emilio Villari, is the change of the magnetic susceptibility of a material when subjected to a mechanical stress.

## Explanation

The magnetostriction $\lambda$ characterizes the shape change of a ferromagnetic material during magnetization, whereas the inverse magnetostrictive effect characterizes the change of sample magnetization $M$(for given magnetizing field strength $H$) when mechanical stresses $\sigma$ are applied to the sample.

### Qualitative explanation of magnetoelastic effect

Under a given uni-axial mechanical stress $\sigma$, the flux density $B$ for a given magnetizing field strength $H$ may increase or decrease. The way in which a material responds to stresses depends on its saturation magnetostriction $\lambda _{s}$. For this analysis, compressive stresses $\sigma$ are considered as negative, whereas tensile stresses are positive. According to Le Chatelier's principle:
$\left({\frac {d\lambda }{dH}}\right)_{\sigma }=\left({\frac {dB}{d\sigma }}\right)_{H}$
This means, that when the product $\sigma \lambda _{s}$ is positive, the flux density $B$ increases under stress. On the other hand, when the product $\sigma \lambda _{s}$ is negative, the flux density $B$ decreases under stress. This effect was confirmed experimentally.

### Quantitative explanation of magnetoelastic effect

In the case of a single stress $\sigma$ acting upon a single magnetic domain, the magnetic strain energy density $E_{\sigma }$ can be expressed as:
$E_{\sigma }={\frac {3}{2}}\lambda _{s}\sigma \sin ^{2}(\theta )$
where $\lambda _{s}$ is the magnetostrictive expansion at saturation, and $\theta$ is the angle between the saturation magnetization and the stress's direction. When $\lambda _{s}$ and $\sigma$ are both positive (like in iron under tension), the energy is minimum for $\theta$ = 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 $\lambda _{100}$ and $\lambda _{111}$.
* In cubic materials, the magnetostriction along any axis can be defined by a known linear combination of these two constants. For instance, the elongation along [110] is a linear combination of $\lambda _{100}$ and $\lambda _{111}$.
* Under assumptions of isotropic magnetostriction (i.e. domain magnetization is the same in any crystallographic directions), then $\lambda _{100}=\lambda _{111}=\lambda$ and the linear dependence between the elastic energy and the stress is conserved, $E_{\sigma }={\frac {3}{2}}\lambda \sigma (\alpha _{1}\gamma _{1}+\alpha _{2}\gamma _{2}+\alpha _{3}\gamma _{3})^{2}$. Here, $\alpha _{1}$, $\alpha _{2}$ and $\alpha _{3}$ are the direction cosines of the domain magnetization, and $\gamma _{1}$, $\gamma _{2}$,$\gamma _{3}$ those of the bond directions, towards the crystallographic directions.

## Method of testing the magnetoelastic properties of magnetic materials

Method suitable for effective testing of magnetoelastic effect in magnetic materials should fulfill the following requirements:
* magnetic circuit of the tested sample should be closed. Open magnetic circuit causes demagnetization, which reduces magnetoelastic effect and complicates its analysis.
* distribution of stresses should be uniform. Value and direction of stresses should be known.
* there should be the possibility of making the magnetizing and sensing windings on the sample - necessary to measure magnetic hysteresis loop under mechanical stresses.
Following testing methods were developed:
* tensile stresses applied to the strip of magnetic material in the shape of a ribbon. Disadvantage: open magnetic circuit of the tested sample.
* tensile or compressive stresses applied to the frame-shaped sample. Disadvantage: only bulk materials may be tested. No stresses in the joints of sample columns.
* compressive stresses applied to the ring core in the sideways direction. Disadvantage: non-uniform stresses distribution in the core .
* tensile or compressive stresses applied axially to the ring sample. Disadvantage: stresses are perpendicular to the magnetizing field.

## Applications of magnetoelastic effect

Magnetoelastic effect can be used in development of force sensors. This effect was used for sensors:
* in civil engineering.
* for monitoring of large diesel engines in locomotives.
* for monitoring of ball valves.
* for biomedical monitoring.
Inverse magnetoelastic effects have to be also considered as a side effect of accidental or intentional application of mechanical stresses to the magnetic core of inductive component, e.g. fluxgates or generator/motor stators when installed with interference fits.