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Piezoceramic Tube
Introduction
This example involves a static 2D axisymmetric analysis of a piezoelectric actuator using the Piezoelectricity multiphysics 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
Geometry
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. Length units on x- and y-axes are shown in μm.
Boundary Conditions
The 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 1 — Direct Piezoelectric Effect:
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Structural mechanics boundary condition — constrain the bottom surface from moving axially (in the z direction), but also add an internal fluid pressure of 0.1 MPa.
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Electrostatics boundary condition — ground the inner and outer surfaces.
Case 2 — Inverse Piezoelectric Effect:
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Structural mechanics boundary condition — constrain the bottom surface from moving axially (in the z direction).
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Electrostatics boundary condition — apply a 1 V potential difference between the tube’s inner and outer surfaces.
Material Orientation
The material library data in COMSOL Multiphysics is entered in a form that assumes that the crystal polarization is aligned with the global coordinate 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 coordinate system in the Piezoelectric Material feature. By selecting the coordinate 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 material’s x direction is aligned with the model’s 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
Figure 2 shows the radial displacement due to the applied pressure in Case 1, and Figure 3 shows the corresponding induced electric potential. Both the radial displacement and potential are shown along a cut line 300 μm above the base of the tube in Figure 4 and Figure 5, receptively.
Figure 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).
Figure 4: Radial displacement as a function of r-coordinate at a height of 300 μm above the base of the tube. The results are for Case 1 — the direct piezoelectric effect.
Figure 5: Electric potential as a function of r-coordinate at a height of 300 μm above the base of the tube. The results are for Case 1 — the direct piezoelectric effect.
Figure 6: Deformed shape and radial displacement of the piezoceramic-tube actuator due to the radial electric field (Case 2 — Inverse Piezoelectric Effect).
Figure 7: Electric potential applied to the tube to induce the displacements shown in Figure 6 (Case 2 — Inverse Piezoelectric Effect).
Figure 8: Radial displacement as a function of r-coordinate at a height of 300 μm above the base of the tube. The results are for Case 2 — the inverse piezoelectric effect.
Figure 9: Electric potential as a function of r-coordinate at a height of 300 μm above the base of the tube. The results are for Case 2 — the inverse piezoelectric effect.
Figure 6 shows the radial displacement resulting from the applied potential shown in Figure 7. The radial displacement and potential are shown along a cut line 300 μm above the base of the tube in Figure 8 and Figure 9, respectively.
These results show good agreement with those from Peelamedu and others (Ref. 1).
Reference
1. S.M. Peelamedu and others, “Numerical Approach for Axisymmetric Piezoceramic Geometries towards Fluid Control Applications,” Proc. Institution of Mech. Engineers, Part I: J. Systems and Control Engineering, vol. 214, no. 2, pp. 87–97, 2000.
Application Library path: MEMS_Module/Piezoelectric_Devices/piezoceramic_tube
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
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In the Model Wizard window, click  2D Axisymmetric.
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In the Select Physics tree, select Structural Mechanics > Electromagnetics–Structure Interaction > Piezoelectricity > Piezoelectricity, Solid.
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Click Add.
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Click  Study.
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In the Select Study tree, select General Studies > Stationary.
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Click  Done.
Geometry 1
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In the Model Builder window, under Component 1 (comp1) click Geometry 1.
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In the Settings window for Geometry, locate the Units section.
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From the Length unit list, choose µm.
Create the tube by adding an off-axis rectangle in the axisymmetric geometry.
Rectangle 1 (r1)
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In the Geometry toolbar, click  Rectangle.
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In the Settings window for Rectangle, locate the Size and Shape section.
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In the Width text field, type 240.
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In the Height text field, type 620.
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Locate the Position section. In the r text field, type 380.
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Click  Build All Objects.
Add a PZT 5H to the model.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
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Go to the Add Material window.
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In the tree, select Piezoelectric > Lead Zirconate Titanate (PZT-5H).
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Click the Add to Component button in the window toolbar.
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In the Materials toolbar, click  Add Material to close the Add Material window.
Solid Mechanics (solid)
Piezoelectric Material 1
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In the Settings window for Piezoelectric Material, locate the Coordinate System Selection section.
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From the Coordinate system list, choose Material ZX-plane System (comp1_zx_sys).
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.
This example comprises two studies: the direct effect and inverse effect. All loadings for both studies are defined together and then a selection of relevant features will be done in the study settings.
Add a pressure follower load to the inner surface of the cylinder.
Boundary Load 1
1
In the Physics toolbar, click  Boundaries and choose Boundary Load.
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Select Boundary 1 only.
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In the Settings window for Boundary Load, locate the Force section.
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From the Load type list, choose Pressure.
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In the p text field, type 0.1[MPa].
Constrain the lower surface of the tube with a roller boundary condition.
Roller 1
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In the Physics toolbar, click  Boundaries and choose Roller.
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Select Boundary 2 only.
Electrostatics (es)
Ground both the inner and outer surfaces of the cylinder.
Ground 1
1
In the Physics toolbar, click  Boundaries and choose Ground.
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Select Boundaries 1 and 4 only.
Add an electric potential feature on the outer boundary. This will override the existing Ground feature.
Electric Potential 1
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In the Physics toolbar, click  Boundaries and choose Electric Potential.
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Select Boundary 4 only.
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In the Settings window for Electric Potential, locate the Electric Potential section.
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In the V0 text field, type 1.
Mesh 1
Create a mapped mesh.
Mapped 1
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In the Mesh toolbar, click  Mapped.
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In the Settings window for Mapped, click  Build All.
Study 1
The first study simulates the direct effect. All mechanical loads are kept and the electric potential feature is disabled in solver settings. It is automatically replaced by the ground feature that was previously overridden.
Step 1: Stationary
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In the Model Builder window, under Study 1 click Step 1: Stationary.
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In the Settings window for Stationary, locate the Physics and Variables Selection section.
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Select the Modify model configuration for study step checkbox.
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In the tree, select Component 1 (comp1) > Electrostatics (es) > Electric Potential 1.
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Right-click and choose Disable.
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In the Study toolbar, click  Compute.
Results
Radial Displacement (Direct Effect)
The default plot groups show stress in the tube and the induced electric potential. Adapt these for comparison with Ref. 1. First replace stress plot by radial displacement.
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In the Settings window for 2D Plot Group, type Radial Displacement (Direct Effect) in the Label text field.
Surface 1
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In the Model Builder window, expand the Radial Displacement (Direct Effect) node, then click Surface 1.
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In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Solid Mechanics > Displacement > Displacement field - m > u - Displacement field, R-component.
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In the Radial Displacement (Direct Effect) toolbar, click  Plot.
Stress, 3D (Direct Effect)
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In the Model Builder window, under Results click Stress, 3D (solid).
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In the Settings window for 3D Plot Group, type Stress, 3D (Direct Effect) in the Label text field.
Electric Potential (Direct Effect)
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In the Model Builder window, under Results click Electric Potential (es).
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In the Settings window for 2D Plot Group, type Electric Potential (Direct Effect) in the Label text field.
Electric Potential, 3D (Direct Effect)
Change the dataset of the potential plot in order to see a 3D cut view of the potential.
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In the Model Builder window, under Results click Electric Potential, Revolved Geometry (es).
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In the Settings window for 3D Plot Group, type Electric Potential, 3D (Direct Effect) in the Label text field.
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In the Electric Potential, 3D (Direct Effect) toolbar, click  Plot.
Create a cross section through the geometry to use for line plots of the electric potential and displacement.
Cut Line 2D 1
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In the Results toolbar, click  Cut Line 2D.
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In the Settings window for Cut Line 2D, locate the Line Data section.
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In row Point 1, set r to 380 and z to 300.
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In row Point 2, set r to 620 and z to 300.
Visualize the cross section line.
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Click  Plot.
Add line plots of the radial displacement and the potential along the cross section.
Radial Displacement, Cut (Direct Effect)
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In the Results toolbar, click  1D Plot Group.
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In the Settings window for 1D Plot Group, type Radial Displacement, Cut (Direct Effect) in the Label text field.
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Locate the Data section. From the Dataset list, choose Cut Line 2D 1.
Line Graph 1
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Right-click Radial Displacement, Cut (Direct Effect) and choose Line Graph.
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In the Settings window for Line Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Solid Mechanics > Displacement > Displacement field - m > u - Displacement field, R-component.
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Locate the y-Axis Data section. From the Unit list, choose nm.
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Locate the x-Axis Data section. From the Parameter list, choose Expression.
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In the Expression text field, type r.
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In the Radial Displacement, Cut (Direct Effect) toolbar, click  Plot.
Voltage, Cut (Direct Effect)
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In the Model Builder window, right-click Radial Displacement, Cut (Direct Effect) and choose Duplicate.
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In the Settings window for 1D Plot Group, type Voltage, Cut (Direct Effect) in the Label text field.
Line Graph 1
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In the Model Builder window, expand the Voltage, Cut (Direct Effect) node, then click Line Graph 1.
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In the Settings window for Line Graph, locate the y-Axis Data section.
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In the Expression text field, type abs(V).
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In the Voltage, Cut (Direct Effect) toolbar, click  Plot.
Finally add a new study to compute the results for the inverse effect.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
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Go to the Add Study window.
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Find the Studies subsection. In the Select Study tree, select General Studies > Stationary.
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Click the Add Study button in the window toolbar.
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In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
The second study simulates the inverse effect. All electrical loads are kept and the pressure load feature is disabled in solver settings. It is automatically replaced by the Free boundary feature that was previously overridden.
Step 1: Stationary
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In the Model Builder window, under Study 2 click Step 1: Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
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Select the Modify model configuration for study step checkbox.
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In the tree, select Component 1 (comp1) > Solid Mechanics (solid) > Boundary Load 1.
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Click  Disable.
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In the Study toolbar, click  Compute.
Results
Radial Displacement (Inverse Effect)
In the Settings window for 2D Plot Group, type Radial Displacement (Inverse Effect) in the Label text field.
Surface 1
1
In the Model Builder window, expand the Radial Displacement (Inverse Effect) node, then click Surface 1.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Solid Mechanics > Displacement > Displacement field - m > u - Displacement field, R-component.
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In the Radial Displacement (Inverse Effect) toolbar, click  Plot.
Stress, 3D (Inverse Effect)
1
In the Model Builder window, under Results click Stress, 3D (solid).
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In the Settings window for 3D Plot Group, type Stress, 3D (Inverse Effect) in the Label text field.
Electric Potential (Inverse Effect)
1
In the Model Builder window, under Results click Electric Potential (es).
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In the Settings window for 2D Plot Group, type Electric Potential (Inverse Effect) in the Label text field.
Electric Potential, 3D (Inverse Effect)
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In the Model Builder window, click Electric Potential, Revolved Geometry (es).
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In the Settings window for 3D Plot Group, locate the Data section.
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From the Dataset list, choose Revolution 2D 2.
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In the Label text field, type Electric Potential, 3D (Inverse Effect).
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In the Electric Potential, 3D (Inverse Effect) toolbar, click  Plot.
Create a second Cut Line 2D for the new solution.
Cut Line 2D 2
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In the Model Builder window, under Results > Datasets right-click Cut Line 2D 1 and choose Duplicate.
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In the Settings window for Cut Line 2D, locate the Data section.
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From the Dataset list, choose Study 2/Solution 2 (sol2).
Radial Displacement, Cut (Inverse Effect)
1
In the Model Builder window, right-click Radial Displacement, Cut (Direct Effect) and choose Duplicate.
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In the Settings window for 1D Plot Group, type Radial Displacement, Cut (Inverse Effect) in the Label text field.
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Locate the Data section. From the Dataset list, choose Cut Line 2D 2.
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In the Radial Displacement, Cut (Inverse Effect) toolbar, click  Plot.
Voltage, Cut (Inverse Effect)
1
In the Model Builder window, right-click Voltage, Cut (Direct Effect) and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Voltage, Cut (Inverse Effect) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Cut Line 2D 2.
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In the Voltage, Cut (Inverse Effect) toolbar, click  Plot.