This document provides instructions for creating a radially polarized piezoelectric material model in COMSOL using a cylindrical coordinate system. It describes defining a custom local cylindrical coordinate system oriented with x1 as the azimuthal direction, x2 as the axial direction, and x3 as the radial direction. With this system defined, radial polarization is modeled by applying a voltage across the piezoelectric disk. Stresses, strains, displacements and electric fields are then visualized in the local cylindrical coordinate system.

1. Radially Polarized
Piezoelectric Transducer

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
 The method of evaluating anisotropic material properties in both cylindrical and global
(rectangular) coordinate is shown

3. Physics, Geometry, and Mesh
V
Radial polarization of
piezo material
Voltage across
piezo disc
Roller condition on the inside and
bottom of the disc

4. Cylindrical
Coordinate System
 By default, the local coordinate
system is oriented along the
global rectangular 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)

5. Some Technical Notes
 Why do we not use the predefined Cylindrical Coordinate System?
The more automatic option for creating a cylindrical coordinate system 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.

6.  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)
How can we transform coordinates?
Relation between material
and cylindrical coordinates
𝑋 = 𝑟 cos 𝜙
𝑌 = 𝑟 sin 𝜙
Z = 𝑧

7. Unit Vectors in Cylindrical Coordinate System
 A unit vector can be expressed as:
 The cylindrical and material coordinate systems are related through the following unit
vectors:
 This is entered in the base vectors section
Z
Y
X
e
Z
Y
X
e
Z
Y
X
e
r
z
ˆ
0
ˆ
sin
ˆ
cos
ˆ
ˆ
1
ˆ
0
ˆ
0
ˆ
ˆ
0
ˆ
cos
ˆ
sin
ˆ















Z
c
Y
b
X
a
e ˆ
ˆ
ˆ
ˆ 

 where 1
2
2
2


 c
b
a

8. Change the Coordinate System

9.  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
Displacement, Electric Fields, and Electric Potential

10. 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

11. Stresses and Strains
 Stresses and Strains are available in the Local Coordinate System for post processing
 Stresses in the local coordinate system are named:
Normal components: solid.sl11, solid.sl22, solid.sl33
Shear components: solid.sl12, solid.sl13, solid.sl23
 Strains in the local coordinate system are named:
Normal components: solid.el11, solid.el22, solid.el33
Shear components: solid.el12, solid.el13, solid.el23
 For our example this notation can interpreted as:
Index 1 → φ direction
Index 2 → z direction
Index 3 → r direction

12. Strains in Local Coordinate System
Plot on a radial section

13. Summary
 The radial polarization was modeled by creating a custom cylindrical coordinate system
 The tutorial showes how to create plots to visualize the new coordinate system and
stresses and strains in this coordinate system
 It also showed how anisotropic material properties can be evaluated in both user-
defined coordinate system as well as COMSOL’s global coordinate system