# 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 ${\displaystyle \lambda }$  characterizes the shape change of a ferromagnetic material during magnetization, whereas the inverse magnetostrictive effect characterizes the change of sample magnetization ${\displaystyle M}$ (for given magnetizing field strength ${\displaystyle H}$ ) when mechanical stresses ${\displaystyle \sigma }$  are applied to the sample.[1]

### Qualitative explanation of magnetoelastic effect

Under a given uni-axial mechanical stress ${\displaystyle \sigma }$ , the flux density ${\displaystyle B}$  for a given magnetizing field strength ${\displaystyle H}$  may increase or decrease. The way in which a material responds to stresses depends on its saturation magnetostriction ${\displaystyle \lambda _{s}}$ . For this analysis, compressive stresses ${\displaystyle \sigma }$  are considered as negative, whereas tensile stresses are positive.
According to Le Chatelier's principle:

${\displaystyle \left({\frac {d\lambda }{dH}}\right)_{\sigma }=\left({\frac {dB}{d\sigma }}\right)_{H}}$

This means, that when the product ${\displaystyle \sigma \lambda _{s}}$  is positive, the flux density ${\displaystyle B}$  increases under stress. On the other hand, when the product ${\displaystyle \sigma \lambda _{s}}$  is negative, the flux density ${\displaystyle B}$  decreases under stress. This effect was confirmed experimentally.[2]

### Quantitative explanation of magnetoelastic effect

In the case of a single stress ${\displaystyle \sigma }$  acting upon a single magnetic domain, the magnetic strain energy density ${\displaystyle E_{\sigma }}$  can be expressed as:[1]

${\displaystyle E_{\sigma }={\frac {3}{2}}\lambda _{s}\sigma \sin ^{2}(\theta )}$

where ${\displaystyle \lambda _{s}}$  is the magnetostrictive expansion at saturation, and ${\displaystyle \theta }$  is the angle between the saturation magnetization and the stress's direction. When ${\displaystyle \lambda _{s}}$  and ${\displaystyle \sigma }$  are both positive (like in iron under tension), the energy is minimum for ${\displaystyle \theta }$  = 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 ${\displaystyle \lambda _{100}}$  and ${\displaystyle \lambda _{111}}$ .

• 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 ${\displaystyle \lambda _{100}}$  and ${\displaystyle \lambda _{111}}$ .
• Under assumptions of isotropic magnetostriction (i.e. domain magnetization is the same in any crystallographic directions), then ${\displaystyle \lambda _{100}=\lambda _{111}=\lambda }$  and the linear dependence between the elastic energy and the stress is conserved, ${\displaystyle E_{\sigma }={\frac {3}{2}}\lambda \sigma (\alpha _{1}\gamma _{1}+\alpha _{2}\gamma _{2}+\alpha _{3}\gamma _{3})^{2}}$ . Here, ${\displaystyle \alpha _{1}}$ , ${\displaystyle \alpha _{2}}$  and ${\displaystyle \alpha _{3}}$  are the direction cosines of the domain magnetization, and ${\displaystyle \gamma _{1}}$ , ${\displaystyle \gamma _{2}}$ ,${\displaystyle \gamma _{3}}$  those of the bond directions, towards the crystallographic directions.

## 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:[3]

• 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.[4] Disadvantage: open magnetic circuit of the tested sample.
• tensile or compressive stresses applied to the frame-shaped sample.[5] 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.[6] Disadvantage: non-uniform stresses distribution in the core .
• tensile or compressive stresses applied axially to the ring sample.[7] Disadvantage: stresses are perpendicular to the magnetizing field.

## Applications of magnetoelastic effect

Magnetoelastic effect can be used in development of force sensors.[8][9] 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.[12]

## References

1. ^ a b Bozorth, R. (1951). Ferromagnetism. Van Nostrand.
2. ^ Salach, J.; Szewczyk, R.; Bienkowski, A.; Frydrych, P. (2010). "Methodology of testing the magnetoelastic characteristics of ring-shaped cores under uniform compressive and tensile stresses" (PDF). Journal of Electrical Engineering. 61 (7): 93.
3. ^ Bienkowski, A.; Kolano, R.; Szewczyk, R (2003). "New method of characterization of magnetoelastic properties of amorphous ring cores". Journal of Magnetism and Magnetic Materials. 254: 67. Bibcode:2003JMMM..254...67B. doi:10.1016/S0304-8853(02)00755-2.
4. ^ a b Bydzovsky, J.; Kollar, M.; Svec, P.; et al. (2001). "Magnetoelastic properties of CoFeCrSiB amorphous ribbons - a possibility of their application" (PDF). Journal of Electrical Engineering. 52: 205.
5. ^ Bienkowski, A.; Rozniatowski, K.; Szewczyk, R (2003). "Effects of stress and its dependence on microstructure in Mn-Zn ferrite for power applications". Journal of Magnetism and Magnetic Materials. 254: 547. Bibcode:2003JMMM..254..547B. doi:10.1016/S0304-8853(02)00861-2.
6. ^ Mohri, K.; Korekoda, S. (1978). "New force transducers using amorphous ribbon cores". IEEE Transactions on Magnetics. 14: 1071. Bibcode:1978ITM....14.1071M. doi:10.1109/TMAG.1978.1059990.
7. ^ Szewczyk, R.; Bienkowski, A.; Salach, J.; et al. (2003). "The influence of microstructure on compressive stress characteristics of the FINEMET-type nanocrystalline sensors" (PDF). Journal of Optoelectronics and Advanced Materials. 5: 705.
8. ^ Bienkowski, A.; Szewczyk, R. (2004). "The possibility of utilizing the high permeability magnetic materials in construction of magnetoelastic stress and force sensors". Sensors and Actuators A - Physical. Elsevier. 113: 270. doi:10.1016/j.sna.2004.01.010.
9. ^ Bienkowski, A.; Szewczyk, R. (2004). "New possibility of utilizing amorphous ring cores as stress sensor". Physica Status Solidi A. 189: 787. Bibcode:2002PSSAR.189..787B. doi:10.1002/1521-396X(200202)189:3<787::AID-PSSA787>3.0.CO;2-G.
10. ^ a b Bienkowski, A.; Szewczyk, R.; Salach, J. (2010). "Industrial Application of Magnetoelastic Force and Torque Sensors" (PDF). Acta Physica Polonica A. 118: 1008.
11. ^ Meydan, T.; Oduncu, H. (1997). "Enhancement of magnetostrictive properties of amorphous ribbons for a biomedical application". Sensors and Actuators A - Physical. Elsevier. 59: 192. doi:10.1016/S0924-4247(97)80172-0.
12. ^ Szewczyk, R.; Bienkowski, A. (2004). "Stress dependence of sensitivity of fluxgate sensor". Sensors and Actuators A - Physical. Elsevier. 110 (1–3): 232. doi:10.1016/j.sna.2003.10.029.