 Solution

The voltage is a mechanically mediated effect: nickel magnetostriction strains the bonded central layer, and the direct piezoelectric effect converts stress into electrical displacement. An estimate of $|d_{31}|$ requires a strain-transfer and electrical-boundary assumption.

Let $x$ run along the applied field and $z$ through the layers. Assume saturated, approximately volume-preserving isotropic nickel magnetostriction, perfect bonding, mechanically dominant nickel faces and a high-impedance voltage measurement. Relative to the random-domain state,Treat the thin central layer as in plane stress, with these imposed in-plane strains, negligible converse-effect backreaction and isotropic elastic constants. ThensoFor a transversely polarized layer,The open-circuit increment has $\Delta D_3\simeq 0$, and $|E_3|=|V|/t=800\mathrm{V}{\mathrm{m}}^{-1}$. HenceThe observed voltage magnitude does not specify the sign without electrode polarity and poling direction. A one-dimensional approximation retaining only axial stress $E\Lambda$ instead gives $4.78\mathrm{pC}{\mathrm{N}}^{-1}$. That is a different mechanical model. Nickel stiffness and thickness, interface compliance, central-layer tensor coefficients and electrical loading are not specified, so the experiment does not determine a unique exact intrinsic coefficient without these assumptions.