Radially Polarized Piezoelectric Transducer

© 2013 COMSOL. All rights reserved.

• This tutorial provides a step-by-step instruction on how to create a piezoelectric material that is radially polarized in a cylindrical coordinate system

• This model can be created using any of the Acoustics Module, MEMS Module or Structural Mechanics Module

• The method of visualizing stress and strain in the cylindrical coordinate system is shown

Introduction

Select Model Wizard and 3D Space Dimension

Structural Mechanics > Piezoelectric Devices

Select a Stationary Study

Geometry – Create a disk

Definitions > Coordinate Systems > Base Vector System

You can find the same in the ribbon

Base Vector coordinate system• By default, the local coordinate system is oriented along the

global rectangular coordinate system

Cylindrical coordinate system

• In order to model radial polarization of the piezo disk, we need to define a cylindrical (local) coordinate system

• The cylindrical coordinate directions will correspond to the local coordinates in the following manner

Local axis Cylindrical coordinates

x1 φ (Azimuthal)

x2 z (Axial)

x3 r (Radial)

Related Technical Notes

• Why do we not use COMSOL’s predefined Cylindrical Coordinate System?– COMSOL has a more automatic option for creating a cylindrical

coordinate system but that option fixes the relation between the local axes and the axes of the cylindrical coordinate system using the following relation: x1 → r, x2 → φ, x3 → z which is not what we want

• Why do we use upper case X and Y instead of lower case x and y to define the base vectors?– The coordinate system will be used to transform material properties.

The material properties are defined in the Material Coordinate System (X,Y,Z) and not the Spatial Coordinate System (x,y,z). Hence the Base Vector Coordinate System needs to be defined in terms of the material coordinates. This is important especially when the material is expected to deform significantly and exhibit geometric nonlinearity.

How can we transform coordinates?

• In order to create a new local coordinate system (cylindrical), we need to define the unit vectors of the cylindrical coordinates in terms of the material coordinates (X,Y,Z)

• For that purpose we will use the relation between the material and cylindrical coordinates (r,φ,z)

Relation between material and cylindrical coordinates

cosrX sinrY

zZ

Unit vectors in cylindrical coordinate system

ZYXe

ZYXe

ZYXe

r

z

ˆ0ˆsinˆcosˆ

ˆ1ˆ0ˆ0ˆ

ˆ0ˆcosˆsinˆ

A unit vector can be expressed as:

ZcYbXae ˆˆˆˆ where 1222 cba

The cylindrical and material coordinate systems can be related using the following unit vectors

This is the information we typed in as base vectors

Materials – PZT-5H

Change the coordinate system

• Piezoelectric Devices (pzd) > Piezoelectric Material 1

• Change the Coordinate system from Global coordinate system to Base Vector System 2 (sys2)

What happens to the material properties?

• d33 denotes the polarization along the local z-direction

• By default this would correspond to the material’s z-direction

• In the newly defined cylindrical coordinate system this would correspond to the radial direction

d33 = 5.93e-10[C/N]

Structural boundary conditions

Restrict vertical displacement of lower surface

Restrict normal displacement of inner surface

All other boundaries are free to deform

Electrical boundary conditions

Outer surfaces are at electrical ground (zero voltage)

Inner surfaces are at 100 V

All other boundaries are at zero charge

Mesh and Compute

The Normal Swept mesh creates 72 hexahedral elements

Displacement, Electric Fields and Electric Potential

The radial displacement produced by a radial electric field (black cones) shows that the piezo disk is radially polarized

Voltage distribution in the piezo disk

Cylindrical coordinate system

The blue arrows pointing radially within the disk indicates that the third axis (x3) of the Base Vector System is aligned with the radial direction

Stresses and Strains

• Stresses and Strains are available in the Local Coordinate System for postprocessing

• Stresses in the local coordinate system are named:– Normal components: pzd.sl11, pzd.sl22, pzd.sl33– Shear components: pzd.sl12, pzd.sl13, pzd.sl23

• Strains in the local coordinate system are named:– Normal components: pzd.el11, pzd.el22, pzd.el33– Shear components: pzd.el12, pzd.el13, pzd.el23

• For our example this notation can interpreted as:– Index 1 → φ direction– Index 2 → z direction– Index 3 → r direction

Transformation into local coordinate sytemTurn on the Equation View to see how the components of the coordinate transformation tensor sys2.Tij (i,j = 1,2,3) influence the stress and strain computation

Strains in local coordinate system

Plot on a radial section

Summary

• This tutorial showed how to setup a static analysis on a radially polarized piezoelectric disk

• The radial polarization was modeled by creating a custom cylindrical coordinate system

• The tutorial showed how to create plots to visualize the new coordinate system and stresses and strains in this coordinate system