= Quadratic magnetostriction and frequency doubling
{title2=$\varepsilon_{zz}=\kappa H^2$}
As an original stipulated low-field model, take the axial <magnetic field intensity> $H$ in $\mathrm{A/m}$ and prescribe
$$
q(H)=\kappa H^2,\qquad
\varepsilon_{zz}=q(H),\qquad
\varepsilon_{xx}=\varepsilon_{yy}=-\frac{q(H)}2,
\qquad \kappa=5\times10^{-14}\,\mathrm{m^2/A^2}.
$$
All <strain> components are dimensionless, and their <trace> vanishes. This is an even, quasistatic, unsaturated toy law with a fixed measuring axis. It is distinct from the fixed-magnitude directional model in <saturated isotropic magnetostrictive strain>. In particular, one must not replace local unit <magnetization> by normalized net <magnetization> in a many-domain specimen and assume this law follows.
For $H(t)=H_0\sin(2\pi ft)$, $H_0=20\,\mathrm{kA/m}$, and $f=50\,\mathrm{Hz}$, a <trigonometric identity> gives
$$
q(t)=\frac{\kappa H_0^2}{2}[1-\cos(4\pi ft)]
=10^{-5}[1-\cos(2\pi(100\,\mathrm{Hz})t)].
$$
The axial <strain> ranges from zero to $20\times10^{-6}$, with mean $10\times10^{-6}$ and a $100\,\mathrm{Hz}$ component of amplitude $10\times10^{-6}$. The <cosine> coefficient is negative with the chosen time origin. For $L=100\,\mathrm{mm}$ the length ranges from zero to $2\,\mathrm{\mu m}$ relative to $H=0$. There is no $50\,\mathrm{Hz}$ term in this unbiased model.
\Image[/magnetostriction-strain-reversal-and-frequency-model.png]
{title=Stipulated angular, reversal, and frequency responses of magnetostrictive strain}
{description=Original calculations, not experimental data: the angular panel uses a saturated isotropic model with positive saturation magnetostriction; the other panels use a separately stipulated quadratic field law. Microstrain denotes a relative length change of one part per million.}
A static bias $H_b$ changes the conclusion. Expanding $\kappa[H_b+H_0\sin(2\pi ft)]^2$ produces a fundamental-frequency term $2\kappa H_bH_0\sin(2\pi ft)$ as well as the doubled-frequency term. This is an incremental <linear response> about a state with a chosen magnetic axis; it does not contradict <isotropic linear magnetostriction vanishes> at an unbiased isotropic reference. Real <hysteresis>, <thermal expansion> from <Joule heating>, and mechanical <resonance> can add further components or change their phases. Frequency alone cannot identify the cause of motion: <Joule heating> also contains an even-in-<electric current> contribution.
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