1. Radially Polarized Piezoelectric Transducer
© 2013 COMSOL. All rights reserved.

2. Introduction
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

3. Select Model Wizard and 3D Space Dimension

4. Structural Mechanics > Piezoelectric Devices

5. Select a Stationary Study

6. Geometry – Create a disk

7. Definitions > Coordinate Systems > Base Vector System

8. You can find the same in the ribbon

9. Base Vector coordinate system
By default, the local coordinate system is oriented along the global rectangular coordinate system

10. 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)

11. 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.

12. 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

13. Unit vectors in cylindrical coordinate system
A unit vector can be expressed as:
where
The cylindrical and material coordinate systems can be related using the following unit vectors
This is the information we typed in as base vectors

14. Materials – PZT-5H

15. Change the coordinate system
Piezoelectric Devices (pzd) > Piezoelectric Material 1
Change the Coordinate system from Global coordinate system to Base Vector System 2 (sys2)

16. 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]

17. Structural boundary conditions
All other boundaries are free to deform
Restrict normal displacement of inner surface
Restrict vertical displacement of lower surface

18. Electrical boundary conditions
All other boundaries are at zero charge
Outer surfaces are at electrical ground (zero voltage)
Inner surfaces are at 100 V

19. Mesh and Compute
The Normal Swept mesh creates 72 hexahedral elements

20. 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

21. 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

22. 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

23. Transformation into local coordinate sytem
Turn 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

24. Strains in local coordinate system
Plot on a radial section

25. 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