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2.1  Piezoelectric constitutive laws in 3D

Up to a certain level of electric field and strain, piezoelectric materials behave linearly. This tutorial is restricted to linear piezoelectricity, but the interested reader can refer to [] for more details on non-linear piezoelectricity.
Assuming a linear piezoelectric material and adopting the notations of the IEEE Standards on piezoelectricity [], the 3D constitutive equations are given by:

(2)

where E_i and D_i are the components of the electric field vector and the electric displacement vector, and T_i and S_i are the components of stress and strain vectors, defined according to:

(3)

Matrix notations are usually adopted leading to:

(4)

A widely used alternative and equivalent representation consists in writing the constitutive equations in the following form:

(5)

where the following relationships hold:

(6)

(7)

(8)

There are also two additional possibilities to write these constitutive equations, which are less commonly used but are given here for completeness:

(9)

(10)

The following relationships hold:

(11)

(12)

(13)

The piezoelectric coefficients are contained in the matrix [d] whose structure is specific to each type of piezoelectric material. The typical structure for a z-polarized PZT material is

(14)

Regular PZT ceramics are isotropic in the plane perpendicular to the poling direction (d₃₁=d₃₂, d₁₅=d₂₄), but piezoelectric composites can have orthotropic properties []. PVDF material does not exhibit piezoelectricity in the shear mode, so that the typical structure is:

(15)

PVDF can be either isotropic or orthotropic in the plane perpendicular to the poling direction, depending on the fabrication process (uni-axial or bi-axial). Table 2.1 gives typical piezoelectric coefficients for PZT ceramics and PVDF films. Note that these properties can vary significantly from the figures in the table, as there are many different material types. The permittivity is usually given with its relative value which is the ratio of the permittivity by the permittivity of vacuum (є₀=8.854  10⁻¹² F/m).

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Material properties	PZT	PVDF (bi-axial)
Piezoelectric properties
d33 (pC/N)	440	-25
d31 (pC/N)	-185	3
d32 (pC/N)	-185	3
Relative permittivity
єr	1800	12
Young's Modulus
Y1(GPa)	54	3
Y2(GPA)	54	3
Y3(GPA)	48	10
ρ (kg/m3)	7600	1800

Table 2.1: Typical piezoelectric properties of PZT ceramics and PVDF films

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