# The constitutive equation of Piezoelectric materials ?

• ThangMMM
In summary: Additionally, the numbers following PZT- represent the composition of the material, with PZT-4 being made of 40% lead titanate and 60% lead zirconate, and PZT-5 being made of 50% lead titanate and 50% lead zirconate. In summary, piezoelectric materials, such as PZT, have unique electro-mechanical properties described by a constitutive equation and compliance matrix, with specific compositions that determine their behavior.
ThangMMM
I'm studying about piezoelectric material. The electro - mechanical property of this material can be described by the following constitutive equation:

Sij is the compliance matrix, and dmi is the piezoelectric constant for the materials

The matrix form of the equation:

Applying this equation for PZT (lead zirconate titanate) materials with tetragonal perovskite structure, some of these above Sij and dmi can be equal, or zero. The equation becomes:

I'm concerning about why S31 = S32 = S13 = S23 and S66 = 2(S11 - S12) ??

When I'm reading some documents about PZT, they usually classify PZT into PZT-4, PZT-5, etc (like this: http://www.efunda.com/Materials/piezo/material_data/matdata_index.cfm ). What does these number after it means ?

thank you very much\..

ThangMMM said:
I'm studying about piezoelectric material. The electro - mechanical property of this material can be described by the following constitutive equation:

View attachment 125487

Sij is the compliance matrix, and dmi is the piezoelectric constant for the materials

The matrix form of the equation:

View attachment 125488

Applying this equation for PZT (lead zirconate titanate) materials with tetragonal perovskite structure, some of these above Sij and dmi can be equal, or zero. The equation becomes:

View attachment 125489

I'm concerning about why S31 = S32 = S13 = S23 and S66 = 2(S11 - S12) ??
These conditions must mean that PZT has anisotropic material symmetries such that it is transversely isotropic.

## 1. What is the constitutive equation of piezoelectric materials?

The constitutive equation of piezoelectric materials is a mathematical relationship that describes how these materials respond to applied electric fields and mechanical stresses. It relates the strain or deformation of the material to the applied electric field, as well as how the material generates an electric field when subjected to mechanical stress.

## 2. How does the constitutive equation of piezoelectric materials differ from that of other materials?

Piezoelectric materials have a unique property called the piezoelectric effect, which means they can convert mechanical energy into electrical energy and vice versa. This is why their constitutive equation includes both mechanical and electrical variables, while the equations for other materials typically only involve one or the other.

## 3. What is the significance of the constitutive equation in understanding the behavior of piezoelectric materials?

The constitutive equation is essential in understanding the behavior of piezoelectric materials because it allows us to predict how they will respond to different types of forces and electric fields. By manipulating the variables in the equation, we can determine the electrical and mechanical properties of the material, such as its piezoelectric coefficient and Young's modulus.

## 4. Are there different versions of the constitutive equation for different types of piezoelectric materials?

Yes, there are different versions of the constitutive equation for different types of piezoelectric materials. This is because the material's crystal structure and composition can affect its piezoelectric properties and, therefore, the specific form of the constitutive equation.

## 5. Can the constitutive equation of piezoelectric materials be used to design and optimize devices?

Yes, the constitutive equation can be used in the design and optimization of devices that use piezoelectric materials. By understanding how the material will behave under different conditions, engineers can use the equation to determine the best material and dimensions for their specific application, such as in sensors, actuators, and energy harvesters.

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