Size effects and magnetoelastic couplings: a link between Hall–Petch behaviour and coercive field in soft ferromagnetic metals

ABSTRACT

Size effects regarding Hall–Petch (HP) relation are studied in this work for cobalt, nickel and Fe–3wt.%Si (FeSi), from polycrystalline to multicrystalline states. The materials show a breakdown in HP plot for thickness (t) to grain size (d) ratio less than a critical value. This appears in the beginning of plasticity for cobalt and FeSi whereas a plastic strain threshold must be overcome for nickel. Measurements of the coercive field on strained samples are able to depict such modification for low t/d ratio. Values of the coercive field in the polycrystalline domain allow an estimation of the magnetocrystalline anisotropy energy, related to the grain volume fraction concerned by reversal mechanisms for magnetic domains. Multicrystalline samples of cobalt and FeSi becomes magnetically softer at the yield stress. This is linked to a delay of the maximum intergranular stress towards higher strains for FeSi. For cobalt, non-linear elasticity and strong basal texture modify the magnetoelastic effects in coarse grain samples. For nickel, size effect on the coercive field appears after a few per cent of plastic strain as for HP relationship. A mean internal stress can be captured by magnetic measurements on polycrystals, related to the intragranular part of the kinematic stress. The softening of the magnetic properties for strained nickel multicrystals is due to a competition between the apparition of dislocation cells, which increases the coercive field by mechanisms of magnetic domain wall pinning, and surface softening of multicrystals, which tends to decrease the value of H_c.

KEYWORDS:

• Grain size
• magnetic properties
• deformation mechanisms
• size effects
• internal stresses
• coercive field

Acknowledgements

Authors would like to thank Prof. Ivan Guillot from ICMPE, CNRS, Paris France, for the technical support and fruitful discussions related to the TEM observations on FeSi Alloys.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 A ${\chi }_i^2$ statistical test was systematically performed on the least square regression considering one or two different stages for HP plots. An error bar of 5% was associated with each stress measurement in agreement with the experimental scattering analysis performed in previous works [6].

2 As for HP relationships plotted in Figure 4, a ${\chi }_i^2$ statistical test was also systematically performed on the least square regression considering one or two different stages for coercive field plots. An error bar of 20 A/m was associated with each coercive field measurement.

3 Strictly speaking, K = |K₁|/5 concerns an average value of the anisotropy energy at the macroscale level. The corresponding experimental value of K obtained via Equation (9) is a value at the scale of the domain wall, in link with its intrinsic surface energy through Equation (8). Following Chen [31], K = K₁ in this case within the order of magnitude. K falls in the range K₁/8–K₁, the former value being representative of 180°walls and the latter one representative of less favourable oriented walls as secondary closure walls encountered in the neighbourhood of grain boundaries.

4 More generally, internal stress levels are represented by a kinematic hardening tensor $\overline{\overline{X}}$ which is a part of the stress tensor $\overline{\overline{\sigma }}$ [82]. Considering the case of a tensile test: $\overline{\overline{\sigma }}=\left[\begin{array}{ccc}\sigma & 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$ the corresponding plastic strain tensor is diagonal and deviatoric (plasticity without variation of volume): $\overline{\overline{{\epsilon }_p}}=\left[\begin{array}{ccc}{\epsilon }_p& 0& 0\\ 0& -{\epsilon }_p/2& 0\\ 0& 0& -{\epsilon }_p/2\end{array}\right]$. The kinematic hardening being collinear with the plastic strain tensor, $\overline{\overline{X}}$ takes the general form: $\overline{\overline{X}}=\left[\begin{array}{ccc}X& 0& 0\\ 0& -X/2& 0\\ 0& 0& -X/2\end{array}\right]$.