Past exam of the materials science course of the University of Cambridge / 2018 / Paper 1 / 2

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.

For a conventional initially random, untextured piezoceramic, orientational averaging suppresses the macroscopic linear response. Electrical poling of ferroelectric domains is the usual way to generate a lasting net response. Under that conventional preparation assumption, a ferroelectric constituent is the expected route.

However, the word polycrystalline alone does not logically require ferroelectricity. A deliberately textured aggregate of nonswitchable piezoelectric grains can have a nonzero effective tensor if their orientations and polar directions do not cancel. Ferroelectricity means electrically switchable spontaneous polarization, whereas piezoelectricity means linear stress–charge coupling. Thus the unconditional statement that every polycrystalline piezoelectric must be ferroelectric is too strong.

No. A piezoelectric single crystal need not be ferroelectric. A noncentrosymmetric crystal can couple strain to polarization without having a spontaneous polarization that an electric field can switch. Quartz is a standard example.

A single crystal already has an oriented crystallographic tensor, so the random-grain cancellation issue does not arise. The relevant tests are crystal symmetry and the nonzero piezoelectric coefficients; ferroelectric switching is a separate property.

If removal restores the original random-domain magnetic state, the magnetostrictive strain and induced voltage return toward their original values. Elastic strain transfer then relaxes the stress on the central layer.

A ferromagnetic nickel layer can retain remanent magnetization and an altered domain population, so removing the external field does not guarantee removal of all magnetostriction. The electrical time response also depends on leakage and the input impedance of the voltmeter: an open-circuit strain-induced voltage can decay through a finite resistance even while some strain remains. The final observed voltage is therefore history- and circuit-dependent.

Reversing a saturating longitudinal field gives approximately the same strain and the same voltage polarity relative to the original unmagnetized reference. In the isotropic orientational model,It is unchanged under $\mathbf{m}\to -\mathbf{m}$. Both positive and negative longitudinal saturation therefore contract the negative-magnetostriction nickel in the same axial direction. The resulting stress on the piezoelectric layer has the same sign.

During field reversal there can be a hysteretic transient as domains reorganize, and finite circuit leakage can alter the measured waveform. Those effects do not turn the saturated strain law into a response odd in the field.

Assume that the transverse field is large enough to saturate through the thickness despite demagnetizing effects, and retain the thin-layer plane-stress and open-circuit assumptions from part(a). The two imposed in-plane strains now becomeTheir sum is $-\Lambda$, whereas it was $\Lambda /2$ for longitudinal saturation. The in-plane stress sum, and hence the induced open-circuit voltage, is multiplied by $-2$:Thus the magnitude is about $24\mathrm{mV}$ with the opposite polarity, under this model. The larger demagnetizing field for an out-of-plane direction means the same external field magnitude might not produce the same degree of saturation. Changes in thickness constraint or strain transfer would also modify the estimate.