= 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. \b[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,
$$
\varepsilon_x=\Lambda=-4\times10^{-5},\qquad
\varepsilon_y=-\Lambda/2=2\times10^{-5}.
$$
Treat the thin central layer as in <plane stress>, with these imposed in-plane strains, negligible converse-effect backreaction and isotropic elastic constants. Then
$$
\sigma_x=\frac{E}{1-\nu^2}(\varepsilon_x+\nu\varepsilon_y),
\quad
\sigma_y=\frac{E}{1-\nu^2}(\varepsilon_y+\nu\varepsilon_x),
$$
so
$$
\sigma_x+\sigma_y=\frac{E(\varepsilon_x+\varepsilon_y)}{1-\nu}
=\frac{E\Lambda}{2(1-\nu)}=-1.80\,\mathrm{MPa}.
$$
For a transversely polarized layer,
$$
D_3=d_{31}(\sigma_x+\sigma_y)+\epsilon_0\kappa E_3.
$$
The open-circuit increment has $\Delta D_3\simeq0$, and $|E_3|=|V|/t=800\,\mathrm{V\,m^{-1}}$. Hence
$$
\boxed{|d_{31}|\simeq
\frac{\epsilon_0\kappa|V|}{t|\sigma_x+\sigma_y|}
=6.69\times10^{-12}\,\mathrm{C\,N^{-1}}
=6.69\,\mathrm{pC\,N^{-1}}.}
$$
The 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 $4.78\,\mathrm{pC\,N^{-1}}$. That is a different mechanical model. <Nickel> stiffness and thickness, interface compliance, central-layer tensor coefficients and electrical loading are not specified, so \b[the experiment does not determine a unique exact intrinsic coefficient without these assumptions].
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