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The inverse magnetostrictive effect (also known as magnetoelastic effect or Villarian effect) is the name given to the change of the magnetic susceptibility of a material when subjected to a mechanical stress.

Explanation

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

Qualitative explanation of magnetoelastic effect

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

According to Le Chatelier's principle:

(dλdH)σ=(dBdσ)H{displaystyle left({frac {dlambda }{dH}}right)_{sigma }=left({frac {dB}{dsigma }}right)_{H}}

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

Quantitative explanation of magnetoelastic effect

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

Eσ=32λsσsin2⁡(θ){displaystyle E_{sigma }={frac {3}{2}}lambda _{s}sigma sin ^{2}(theta )}

where λs{displaystyle lambda _{s}} is the magnetostrictive expansion at saturation, and θ{displaystyle theta } is the angle between the saturation magnetization and the stress's direction.
When λs{displaystyle lambda _{s}} and σ{displaystyle sigma } are both positive (like in iron under tension), the energy is minimum for θ{displaystyle 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 λ100{displaystyle lambda _{100}} and λ111{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 λ100{displaystyle lambda _{100}} and λ111{displaystyle lambda _{111}}.


• Under assumptions of isotropic magnetostriction (i.e. domain magnetization is the same in any crystallographic directions), then λ100=λ111=λ{displaystyle lambda _{100}=lambda _{111}=lambda } and the linear dependence between the elastic energy and the stress is conserved, Eσ=32λσ(α1γ1+α2γ2+α3γ3)2{displaystyle E_{sigma }={frac {3}{2}}lambda sigma (alpha _{1}gamma _{1}+alpha _{2}gamma _{2}+alpha _{3}gamma _{3})^{2}}. Here, α1{displaystyle alpha _{1}}, α2{displaystyle alpha _{2}} and α3{displaystyle alpha _{3}} are the direction cosines of the domain magnetization, and γ1{displaystyle gamma _{1}}, γ2{displaystyle gamma _{2}},γ3{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:

• in civil engineering.^[4]
  


• for monitoring of large diesel engines in locomotives.^[10]
  


• for monitoring of ball valves.^[10]
  


• for biomedical monitoring.^[11]
  

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

See also

• Magnetostriction


• Magnetocrystalline anisotropy