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**Radially Polarized Piezoelectric Transducer**

© 2013 COMSOL. All rights reserved.

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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

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**Select Model Wizard and 3D Space Dimension**

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**Structural Mechanics > Piezoelectric Devices**

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**Select a Stationary Study**

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**Geometry – Create a disk**

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**Definitions > Coordinate Systems > Base Vector System**

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**You can find the same in the ribbon**

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**Base Vector coordinate system**

By default, the local coordinate system is oriented along the global rectangular coordinate system

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**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)

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**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.

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**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

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**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

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Materials – PZT-5H

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**Change the coordinate system**

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

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**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]

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**Structural boundary conditions**

All other boundaries are free to deform Restrict normal displacement of inner surface Restrict vertical displacement of lower surface

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**Electrical boundary conditions**

All other boundaries are at zero charge Outer surfaces are at electrical ground (zero voltage) Inner surfaces are at 100 V

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Mesh and Compute The Normal Swept mesh creates 72 hexahedral elements

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**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

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**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

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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

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**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

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**Strains in local coordinate system**

Plot on a radial section

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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

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