# Inverse magnetostrictive effect

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The inverse magnetostrictive effect (also known as magnetoelastic effect or Villari effect) is the name given to 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.[1]

### Qualitative explanation of magnetoelastic effect

Under given uniaxial mechanical stresses $\sigma$, value of flux density $B$ for given magnetizing field strength $H$ may increase or decrease. The way in which material responds to stresses depends on its saturation magnetostriction $\lambda_s$. For the analysis, compressive stresses $\sigma$ are considered as negative, whereas tensile stresses are positive.
Accordingly to Le Chatelier's principle:

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

It means, that when result of the product $\sigma \lambda_s$ is positive, value of flux density $B$ increases under stresses. On the other hand, when the result of the product $\sigma \lambda_s$ is negative, value of flux density $B$ decreases under stresses. This effect was confirmed experimentally.[2]

### 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:[1]

$E_\sigma = \frac{3}{2} \lambda_s \sigma \sin^2(\theta)$

where $\lambda_s$ is the magnetostrictive expansion at saturation, and $\theta$ is the angle between the saturation magnetization and the stresses 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 the 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 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. 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 possitility 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. 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. 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 - Applied Research 189: 787. 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.