Illustration of piezoelectricity in statics: patch example

SDT-piezo Contents Functions

2.4 Illustration of piezoelectricity in statics: patch example

Consider a thin piezoelectric patch of dimensions b x h x w. The poling direction, noted 3 in the IEEE Standards on piezoelectricity is perpendicular to the plane of the piezoelectric patch. Continuous electrodes are present on the top and bottom surfaces (z=0, z=h) so that the electric potential is constant on these surfaces and denoted by V₁ and V₂ respectively. We assume that a difference of potential is applied between the electrodes, resulting in an electric field parallel to the poling direction and equal to (Figure 2.5)

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Figure 2.5: A piezoelectric patch poled through the thickness with continuous electrodes on the top and bottom surfaces

We adopt the following expression for the constitutive equations:

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The patch is assumed to be unconstrained so that it can expand freely, leading to {T}=0, so that we have :

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We have taken into account the fact that the electric field is in the z-direction only. This shows that when applying a difference of potential across the thickness (in the poling direction), strains will be induced in the directions 1,2, and 3. The magnitude of these different strains is proportional to the d_3i coefficients of the piezoelectric material. For a ceramic PZT material, d₃₁=d₃₂<0, and d₃₃>0 and is generally between 2 and 3 times larger in magnitude than d₃₁ and d₃₂.
The second equation can be used in order to assess the amount of charge that is accumulated on both electrodes. We have :

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The only non-zero component of the D vector is D₃ given by :

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The charge accumulated on the electrode is given by :

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where {n} is the normal to the electrode. For the top electrode, this leads to :

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where A is the surface of the electrode. For the bottom electrode

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When (V₁−V₂) is positive, the electric field is in the direction of poling and the charge on the top electrode is negative, while the charge accumulated on the bottom electrode is positive (Figure 2.5). Note that this equation corresponds to the equation linking the charge to the difference of potential for a capacitor (q=C Δ V). The value of the capacitance is therefore :

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which corresponds to the capacitance of the free piezoelectric patch ({T}=0).
If we now consider the case where the piezoelectric patch is fully mechanically constrained ({S}=0 ), we have:

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leading to :

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The fact that the patch is not allowed to expand is responsible for the generation of internal stresses which are proportionnal to the e_3i coefficients. In this case, the capacitance is given by:

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which corresponds to the capacitance of the constrained piezoelectric patch ({S}=0). This illustrates the fact that the capacitance of a piezoelectric patch depends on the mechanical boundary conditions. This is not the case for other types of dielectric materials in which the piezoelectric effect is not present, and for which therefore the capacitance is independent on the mechanical strain or stress.