Inverse magnetostrictive 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]

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.

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.