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Theory: How to Calculate Piezoelectric Material Properties from a Material Datasheet

Obtaining the correct piezoelectric material properties from a manufacturer datasheet and transforming those properties into the correct format for simulation can be a hassle.

Let’s take the example of the datasheet below. We will show you how to convert the soft ceramic material "C-6" present on this datasheet into a format which can be inserted into the material database of OnScale.

 

Piezoelectric Datasheet

Before we explain how to do that, first we need to look at some of the equations defining piezoelectricity to understand more closely what those material coefficients mean.

What is the Piezoelectric Effect?

 

There’s no shame in refreshing our memory before going into the equations, as you will see, it will be extremely useful. 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.

piezo-gif-animation_v1

Equations of Piezoelectricity

 

The piezoelectric effect that we previously described 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:

 

Linear electrical behavior
  • D is the electric charge density displacement (electric displacement)
  • ε is permittivity (free-body dielectric constant)
  • E is electric field strength

The Hooke’s law for elastic materials:

Hookes Law
  • S is strain
  • s is compliance under short-circuit conditions
  • T is stress

These relations may be combined into so-called coupled equations, of which the strain-charge form is:

Coupled Equations

Of course, we need to think about those equations in terms of their matrix equivalent.

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):

Strain Charge for material

These are the relations we will use to practically calculate 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:

Piezoelectric Equations

And because of that, we can define the previous 2 equations in 4 different "formats" in function of the coefficients we want to use:

 

Coefficients

 

Following is a description of all matrix variables used in the piezoelectric constitutive equations:

Symbol Object Type Size Units Meaning
piezoelectric constitutive equations T
vector
6 x 1
piezoelectric constitutive equations
stress components (e.g. s1)
piezoelectric constitutive equations S
vector
6 x 1
piezoelectric constitutive equations
strain components (e.g. e3)
piezoelectric constitutive equations E
vector
3 x 1
piezoelectric constitutive equations
electric field components
piezoelectric constitutive equations D
vector
3 x 1
piezoelectric constitutive equations
electric charge density displacement components
piezoelectric constitutive equations s
matrix
6 x 6
piezoelectric constitutive equations
compliance coefficients
piezoelectric constitutive equations c
matrix
6 x 6
piezoelectric constitutive equations
stiffness coefficients
piezoelectric constitutive equations e
matrix
3 x 3
piezoelectric constitutive equations
electric permittivity
piezoelectric constitutive equations d
matrix
3 x 6
piezoelectric constitutive equations
piezoelectric coupling coefficients for Strain-Charge form

 

What to do with all that?

In a second, more practical approach, Part 2 will show you:

1- Which coefficients OnScale need

2- A practical algorithm in Python to calculate all those coefficients

3- How to input them in OnScale as a new custom piezoelectric material

 

Ready for Part 2 of this blog tutorial? We start by answering what parameters are needed in OnScale to define a Piezoelectric material. Click here to start reading!

 

Get Started With OnScale Today

 

Cyprien Rusu, Director of Engineering for Asia at OnScale
Cyprien Rusu, Director of Engineering for Asia at OnScale
Cyprien Rusu is our Director of Engineering for Asia at OnScale. He has a extensive background in FEM, Technical Marketing, Sales and Support. Cyprien recieved his MS in Civil Engineering from Tsinghua University. At OnScale he is a trusted advisor for our client base in Asia, while creating and providing OnScale training.

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