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2.2  Piezoelectric constitutive laws in plates

When thin piezoelectric transducers are used with plate structures, the common plane stress hypothesis (T₃=0) must be used together with an hypothesis for the electric field. When the ceramic is poled through the thickness, the hypothesis commonly adopted is that the electric field is zero in the plane of the transducer (E₁=E₂=0). The constitutive equations then reduce to:

(16)

where the superscript ^* denotes the properties under the "piezoelectric plates" hypothesis (T₃=E₁=E₂=0). These properties are related to the 3D properties with the following relationships:

(17)

(18)

(19)

(20)

(21)

(22)

The distinction is very important, as it is often not well understood and many errors can arise from the confusion between plate and 3D properties of piezoelectric materials. Note however that the d_ij, s_ij^E and є^T coefficients are equal for plate and 3D constitutive equations. It is therefore preferable to handle the material properties of piezoelectric materials in the form of (5).
Similarly to the 3D equations, the constitutive equations can be written in a matrix form, separating the mechanical and the electrical parts:

Using (7) in equations ((20),(21),(22)), one can further show that

(24)

and for the permittivity:

(25)

with

(26)

and

(27)

The values of e₃₁^*, e₃₂^* and є₃₃^S* can therefore be computed knowing the elastic matrix [c^E* ] and the values of d₃₁ and d₃₂ and є₃₃^T