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# 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.

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:

# W. G. Cady: The Father of Modern Piezoelectricity

American physicist and electrical engineer Dr. Walter Guyton Cady (1874–1974) was, during his lifetime, described as the “Father of Modern Piezoelectricity”.

Cady became a professor of physics at Wesleyan University (Connecticut) in 1902, but his work with piezoelectricity did not begin until 1917. Cady’s interest in submarine dete…

## How to simulate Piezoelectricity?

Piezoelectric devices can be simulated in 1D, 2D or 3D using the Finite Element Method and the appropriate multiphysic solvers. Research have demonstrated that the best method to obtain accurate solution of such system is to use Time Response Dynamic Analysis FEA Simulation and Nonlinear Explicit/Implicit Coupled Algorithms. OnScale is the only software on the market truly efficient to perform this kind of advanced analysis.

METHOD
Finite Element Method is the best way to simulate transducers
The finite element method reduces the electromechanical partial differential equations (PDEs) over the model domain to a system of ordinary differential equations (ODEs) in time that can be solved.
EQUATIONS
What is the system of ordinary differential equations (ODEs) which is solved?
The electromechanical finite element equations are derived from the piezoelectricity constitutive relations and the equations of mechanical and electrical equilibrium
DYNAMIC ANALYSIS
Frequency domain or Time domain?
Frequency Domain poses the assumption of harmonic behavior, whereas Time-domain solutions assume general temporal evolution of the system, requiring step-by-step integration of the equations.
INTEGRATION
Implicit or an Explicit integration Scheme?
OnScale uses a mixed explicit/implicit algorithm which performs a direct time integration of the 2D or 3D electromechanical equations. This special method increases the speed at which such 2D or 3D piezoelectric can be calculated while maintaining a very good level of results accuracy.