The Behavior Laws of Piezoelectricity
The piezoelectric effect is a complex way to say that a pressure on a special material will generate some electric voltage.
The piezoelectric effect is a reversible process so if you apply a voltage at the 2 extremities of a piezo material, it will then deform mechanically.
For more details about what is piezoelectricity, read this article
The piezoelectric effect described previously can be translated into 2 important behavior laws which describe the combined effect of electrical and elastic mechanical behavior.
The linear electrical behavior of the material:
- D is the electric charge density displacement (electric displacement)
- ? is permittivity (free-body dielectric constant)
- E is electric field strength
The Hooke’s law for elastic materials:
- S is the strain
- s is the compliance under short-circuit conditions
- T is the stress
These relations may be combined into so-called coupled equations, of which the strain-charge form is:
Strain-charge Matrix relation for the tetragonal PZT C4V
For example, the strain-charge for a material of the 4mm (C4v) crystal class (such as a poled piezoelectric ceramic such as tetragonal PZT or BaTiO3) as well as the 6mm crystal class may also be written as (ANSI IEEE 176):
That’s the relations I will use to calculate practically the coefficients afterwards, so those matrices are very important.
The 4 Constitutive Piezoelectric equations
In practice, piezoelectric coefficients can be defined in four ways as follows:
And because of that, we can define the previous 2 equations in 4 different “formats” in function of the coefficients we want to use:
Following is a description of all matrix variables used in the piezoelectric constitutive equations: