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.