= Free strain and blocked magnetostrictive stress
{title2=$\sigma=E(\varepsilon-\varepsilon^{\rm mag})$}
An <eigenstrain> is the deformation a material would prefer locally without applied <stress>. In a one-dimensional <linear elasticity> model, let $E$ be <Young's modulus>, take tension as positive <stress>, and write $\sigma=E(\varepsilon-\varepsilon^{\rm mag})$. A free rod has $\sigma=0$ and follows $\varepsilon=\varepsilon^{\rm mag}$. A perfectly blocked rod has $\varepsilon=0$ and $\sigma=-E\varepsilon^{\rm mag}$. Thus positive preferred elongation produces compressive <stress> when blocked.
For stipulated $E=100\,\mathrm{GPa}$, cross-sectional area $A=1\,\mathrm{mm^2}$, and $\varepsilon^{\rm mag}=20\times10^{-6}$, blocking gives $\sigma=-2\,\mathrm{MPa}$ and axial <force> $F=\sigma A=-2\,\mathrm N$. This idealization ignores changes of <magnetization> caused by the load itself. A real <magnetostrictive actuator> therefore needs both magnetic and mechanical constitutive information: a free-length measurement alone does not determine its loaded motion.
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