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2 0 1 2 C OP i e z o c e r am i c Tub e

Introduction

This example involves a static 2D axisymmetric analysis of a piezoelectric actuator using the Piezoelectric Devices physics interface. It models a radially polarized piezoelectric tube, as described by S. Peelamedu and co-authors (Ref. 1). An application area where radially polarized tubes are employed is in nozzles for fluid control in inkjet printers.

Model Definition

G E O M E T R Y

The tube has a height of 0.62 mm and an inner and outer radius of 0.38 mm and 0.62 mm, respectively. It is represented in an axisymmetric geometry by a single off-axis rectangle, as shown in Figure 1.

Figure 1: The axisymmetric geometry.M S O L 1 | P I E Z O C E R A M I C TU B E

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B O U N D A R Y C O N D I T I O N SThe model studies two cases, distinguished by different boundary conditions. Case 1 represents the direct piezoelectric effect, and Case 2 represents the inverse piezoelectric effect.

Case 1Direct Piezoelectric Effect:

Structural mechanics boundary conditionconstrain the bottom surface from moving axially (in the z direction), but also add an internal fluid pressure of 0.1 MPa.

Electrostatics boundary conditionground the inner and outer surfaces.

Case 2Inverse Piezoelectric Effect:

Structural mechanics boundary conditionconstrain the bottom surface from moving axially (in the z-direction).

Electrostatics boundary conditionapply a 1 V potential difference between the tubes inner and outer surfaces.

M A T E R I A L O R I E N T A T I O N

COMSOLs material library data is entered in a form which assumes that the crystal polarization is aligned with the global co-ordinate z axis. For the radially polarized case treated in this model, the orientation must be rotated so that the material polarization direction is aligned with the r direction (radially polarized). To do so, specify the co-ordinate system in the Piezoelectric Material feature. By selecting the co-ordinate system as the predefined zx-plane system, you rotate the material so that its z direction is aligned with the r direction of the model, and the materials x direction is aligned with the models z direction.

The piezoceramic material in this example (PZT-5H) is a transversely isotropic material, which is a special class of orthotropic materials. Such a material has the same properties in one plane (isotropic behavior) and different properties in the direction normal to this plane. Thus you can use either the zx-plane material orientation or the zy-plane material orientation; both give the same solution.

Results and Discussion

The image in Figure 2 shows the radial displacement due to the applied pressure in case 1. Figure 3 shows the corresponding induced electric potential. Both the potential and radial displacement are shown along a cut line 300 m above the base of the tube in Figure 4 and Figure 7.O C E R A M I C TU B E 2 0 1 2 C O M S O L

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2 0 1 2 C OFigure 2: Deformed shape and radial displacement due to an internal pressure of 0.1 MPa (case 1 the direct piezoelectric effect).

Figure 3: Induced electric potential within the deformed tube due to an internal pressure of 0.1 MPa (case 1 the direct piezoelectric effect).M S O L 3 | P I E Z O C E R A M I C TU B E

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4 | P I E ZFigure 4: Radial displacement as a function of r-co-ordinate at a height of 300 m above the base of the tube. The results are for case 1the direct piezoelectric effect.

Figure 5: Electric potential as a function of r-co-ordinate at a height of 300 m above the base of the tube. The results are for case 1the direct piezoelectric effect.O C E R A M I C TU B E 2 0 1 2 C O M S O L

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2 0 1 2 C OFigure 6: Deformed shape and radial displacement of the piezoceramic-tube actuator due to the radial electric field (Case 2Inverse Piezoelectric Effect).

Figure 7: Electric potential applied to the tube to induce the displacements shown in Figure 6 (Case 2Inverse Piezoelectric Effect).M S O L 5 | P I E Z O C E R A M I C TU B E

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6 | P I E ZFigure 8: Radial displacement as a function of r-co-ordinate at a height of 300 m above the base of the tube. The results are for case 2the inverse piezoelectric effect.

Figure 9: Electric potential as a function of r-co-ordinate at a height of 300 m above the base of the tube. The results are for case 2the inverse piezoelectric effect.O C E R A M I C TU B E 2 0 1 2 C O M S O L

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2 0 1 2 C OFigure 6 shows the radial displacement resulting from the applied potential shown in Figure 7. Once again the potential and radial displacement are shown along a cut line 300 m above the base of the tube in Figure 8 and Figure 9.

These results show good agreement with those from S. Peelamedu (Ref. 1).

Reference

1. S. M. Peelamedu, C. B. Kosaraju, R. V. Dukkipati and N. G. Naganathan, Numerical Approach for Axisymmetric Piezoceramic Geometries towards Fluid Control Applications, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 214, no. 2, pp. 87-97, 2000.

Model Library path: MEMS_Module/Piezoelectric_Devices/piezoceramic_tube

Modeling Instructions

M O D E L W I Z A R D

1 Go to the Model Wizard window.

2 Click the 2D axisymmetric button.

3 Click Next.

4 In the Add physics tree, select Structural Mechanics>Piezoelectric Devices (pzd).

5 Click Add Selected.

6 Click Next.

7 Find the Studies subsection. In the tree, select Preset Studies>Stationary.

8 Click Finish.

G E O M E T R Y 1

Create the tube by adding an off-axis rectangle in the axisymmetric geometry.

Rectangle 11 In the Model Builder window, under Model 1 right-click Geometry 1 and choose

Rectangle.

2 In the Rectangle settings window, locate the Size section.M S O L 7 | P I E Z O C E R A M I C TU B E

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8 | P I E Z3 In the Width edit field, type 240 [um].

4 In the Height edit field, type 620 [um].

5 Locate the Position section. In the r edit field, type 380 [um].

6 Click the Build All button.

Add a PZT 5H to the model.

M A T E R I A L S

1 In the Model Builder window, under Model 1 right-click Materials and choose Open Material Browser.

2 In the Material Browser window, locate the Materials section.

3 In the tree, select Piezoelectric>Lead Zirconate Titanate (PZT-5H).

4 Right-click and choose Add Material to Model from the menu.

Lead Zirconate Titanate (PZT-5H)Initially set up the physics settings to model the direct piezoelectric effect.

P I E Z O E L E C T R I C D E V I C E S

Piezoelectric Material 11 In the Model Builder window, under Model 1>Piezoelectric Devices click Piezoelectric

Material 1.

2 In the Piezoelectric Material settings window, locate the Coordinate System Selection section.

3 From the Coordinate system list, choose Material ZX-Plane System.

By selecting the material orientation as the zx-plane, you rotate the material so that its z direction is aligned with the r direction of the model, and the material's x direction is aligned with the model's z direction.

Ground both the inner and outer surfaces of the cylinder.

Ground 11 In the Model Builder window, right-click Piezoelectric Devices and choose the

boundary condition Electrical>Ground.

2 Select Boundaries 1 and 4 only.

Add a pressure follower load to the inner surface of the cylinder.O C E R A M I C TU B E 2 0 1 2 C O M S O L

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2 0 1 2 C OBoundary Load 11 In the Model Builder window, right-click Piezoelectric Devices and choose the

boundary condition Structural>Boundary Load.

2 Select Boundary 1 only.

3 In the Boundary Load settings window, locate the Force section.

4 From the Load type list, choose Pressure.

5 In the p edit field, type 0.1[MPa].

Constrain the lower surface of the tube with a roller boundary condition.

Roller 11 In the Model Builder window, right-click Piezoelectric Devices and choose the

boundary condition Structural>Roller.

2 Select Boundary 2 only.

M E S H 1

Create a mapped mesh.

Mapped 11 In the Model Builder window, under Model 1 right-click Mesh 1 and choose Mapped.

2 In the Mapped settings window, click Build All.

S T U D Y 1

In the Model Builder window, right-click Study 1 and choose Compute.

R E S U L T S

The default plot groups show the displacement of the tube and the induced electric potential. Adapt these for comparison with Ref. 1.

Displacement (pzd)1 In the Model Builder window, under Results right-click Displacement (pzd) and choose

Rename.

2 Go to the Rename 2D Plot Group dialog box and type Radial Displacement (pzd) in the New name edit field.

3 Click OK.

Radial Displacement (pzd)1 In the Model Builder window, expand the Results>Radial Displacement (pzd) node,

then click Surface 1.M S O L 9 | P I E Z O C E R A M I C TU B E

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10 | P I E2 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Piezoelectric Devices>Displacement field (Material)>Displacement field, R component (u).

3 Click the Plot button.

Potential, 3D (pzd)Create a cross section through the geometry to use for line plots of the electric potential and displacement.

Data Sets1 In the Model Builder window, under Results right-click Data Sets and choose Cut Line

2D.

2 In the Cut Line 2D settings window, locate the Line Data section.

3 In row Point 1, set R to 380[um] and Z to 300[um].

4 In row Point 2, set R to 620 [um]and Z to 300[um].

Visualize the cross section line.

5 Click the Plot button.

Add line plots of the radial displacement and the potential along the cross section.

1D Plot Group 31 In the Model Builder window, right-click Results and choose 1D Plot Group.

2 In the 1D Plot Group settings window, locate the Data section.

3 From the Data set list, choose Cut Line 2D 1.

4 Right-click Results>1D Plot Group 3 and choose Line Graph.

5 In the Line Graph settings window, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Piezoelectric Devices>Displacement field (Material)>Displacement field, R component (u).

6 Locate the y-Axis Data section. From the Unit list, choose nm.

7 Locate the x-Axis Data section. From the Parameter list, choose Expression.

8 In the Expression edit field, type r.

9 From the Unit list, choose mm.

10 Click the Plot button.

11 In the Model Builder window, right-click 1D Plot Group 3 and choose Rename.

12 Go to the Rename 1D Plot Group dialog box and type Radial Displacement (direct effect) in the New name edit field.Z O C E R A M I C TU B E 2 0 1 2 C O M S O L

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2 0 1 2 C O13 Click OK.

Radial Displacement (direct effect) 11 Right-click 1D Plot Group 3 and choose Duplicate.

2 In the Model Builder window, under Results right-click Radial Displacement (direct effect) 1 and choose Rename.

3 Go to the Rename 1D Plot Group dialog box and type Voltage (direct effect) in the New name edit field.

4 Click OK.

Voltage (direct effect)1 In the Model Builder window, expand the Results>Voltage (direct effect) node, then

click Line Graph 1.

2 In the Line Graph settings window, locate the y-Axis Data section.

3 In the Expression edit field, type abs(V).

4 Click the Plot button.

Now change the physics settings to model the inverse piezoelectric effect. First add an electric potential feature on the outer boundary. This will over-ride the existing ground feature.

P I E Z O E L E C T R I C D E V I C E S

Electric Potential 11 In the Model Builder window, right-click Model 1>Piezoelectric Devices and choose

the boundary condition Electrical>Electric Potential.

2 Select Boundary 4 only.

3 In the Electric Potential settings window, locate the Electric Potential section.

4 In the V0 edit field, type 1.

Next disable the boundary load.

Ground 1In the Model Builder window, under Model 1>Piezoelectric Devices right-click Boundary Load 1 and choose Disable.

Finally add a new study to compute the results for the inverse effect separately.

R O O T

In the Model Builder window, right-click the root node and choose Add Study.M S O L 11 | P I E Z O C E R A M I C TU B E

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M O D E L W I Z A R D1 Go to the Model Wizard window.

2 Find the Studies subsection. In the tree, select Preset Studies>Stationary.

3 Click Finish.

S T U D Y 2

In the Model Builder window, right-click Study 2 and choose Compute.

R E S U L T S

Plot the radial displacement and potential as before.

Displacement (pzd)1 In the Model Builder window, under Results right-click Displacement (pzd) and choose

Rename.

2 Go to the Rename 2D Plot Group dialog box and type Radial Displacement, inverse (pzd) in the New name edit field.

3 Click OK.

Radial Displacement, inverse (pzd)1 In the Model Builder window, expand the Results>Radial Displacement, inverse (pzd)

node, then click Surface 1.

2 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Piezoelectric Devices>Displacement field (Material)>Displacement field, R component (u).

3 Click the Plot button.

Potential, 3D (pzd) 11 In the Model Builder window, under Results right-click Potential, 3D (pzd) 1 and

choose Rename.

2 Go to the Rename 3D Plot Group dialog box and type Potential, 3D, inverse (pzd) in the New name edit field.

3 Click OK.

Create a second cut line for the new solution.

Data Sets1 In the Model Builder window, expand the Results>Data Sets node.

2 Right-click Cut Line 2D 1 and choose Duplicate.

3 In the Cut Line 2D settings window, locate the Data section.Z O C E R A M I C TU B E 2 0 1 2 C O M S O L

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2 0 1 2 C O4 From the Data set list, choose Solution 2.

Reproduce the two line plots for the inverse effect.

Radial Displacement (direct effect) 11 In the Model Builder window, under Results right-click Radial Displacement (direct

effect) and choose Duplicate.

2 In the 1D Plot Group settings window, locate the Data section.

3 From the Data set list, choose Cut Line 2D 2.

4 Right-click Results>Radial Displacement (direct effect) 1 and choose Rename.

5 Go to the Rename 1D Plot Group dialog box and type Radial Displacement (inverse effect) in the New name edit field.

6 Click OK.

7 Click the Plot button.

Voltage (direct effect) 11 In the Model Builder window, under Results right-click Voltage (direct effect) and

choose Duplicate.

2 In the 1D Plot Group settings window, locate the Data section.

3 From the Data set list, choose Cut Line 2D 2.

4 Right-click Results>Voltage (direct effect) 1 and choose Rename.

5 Go to the Rename 1D Plot Group dialog box and type Voltage (inverse effect) in the New name edit field.

6 Click OK.

7 Click the Plot button.M S O L 13 | P I E Z O C E R A M I C TU B E

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Piezoceramic TubeIntroductionModel DefinitionResults and DiscussionReferenceModeling Instructions