Free strain and blocked magnetostrictive stress ($\sigma =E(\epsilon -{\epsilon }^{\mathrm{mag}})$)

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 $\sigma =E(\epsilon -{\epsilon }^{\mathrm{mag}})$. A free rod has $\sigma =0$ and follows $\epsilon ={\epsilon }^{\mathrm{mag}}$. A perfectly blocked rod has $\epsilon =0$ and $\sigma =-E{\epsilon }^{\mathrm{mag}}$. Thus positive preferred elongation produces compressive stress when blocked.

For stipulated $E=100\mathrm{GPa}$, cross-sectional area $A=1\mathrm{m}{\mathrm{m}}^2$, and ${\epsilon }^{\mathrm{mag}}=20\times 10^{-6}$, 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.