Piezoelectric Shear-Actuated Beam

This example performs a static analysis on a piezoelectric actuator based on the movement of a cantilever beam, using the Piezoelectricity predefined multiphysics interface. Inspired by work done by V. Piefort (Ref. 1) and A. Benjeddou (Ref. 2), it models a sandwich beam using the shear mode of the piezoelectric material to deflect the tip.

Model Definition

Geometry

The model consists of a 100-mm long sandwiched cantilever beam (Figure 1).

Figure 1: The shear bender geometry. Note that a piezoceramic material replaces part of the foam core.

This beam is composed of a 2-mm thick flexible foam core sandwiched by two 8-mm thick aluminum layers. Furthermore, the device replaces part of the foam core with a 10-mm long piezoceramic actuator that is positioned between x = 55 mm and x = 65 mm. The cantilever beam is orientated along the global x-axis.

Boundary Conditions

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Solid Mechanics: the cantilever beam is fixed at its surfaces at x = 0; all other surfaces are free.

•

Electrostatics: The system applies a 20 V potential difference between the top and bottom surfaces of the piezoceramic domain (Figure 2). This gives rise to an electric field perpendicular to the poling direction (x direction) and thus induces a transverse shear strain.

Figure 2: Applied voltage through the piezoelectric material

Material Properties

The following table lists the material properties for the aluminum layers and the foam core:


Property	Aluminum	Foam	Piezoceramic
E	70 GPa	35.3 MPa	-
ν	0.35	0.383	-
ρ	2700 kg/m3	32 kg/m3	7500 kg/m3

Aluminum is available as a predefined material, whereas you must define the foam material manually.

The piezoceramic material in the actuator, PZT-5H, is already defined in the material library. Thus, you do not need to enter the components of the elasticity matrix, cE, the piezoelectric coupling matrix, e, or the relative permittivity matrix, εrS.

Results

The shear deformation of the piezoceramic core layer and the flexible foam layer induce a bending action. Figure 3 shows the resulting tip deflection. The model calculates this deflection as 83 nm, a result that agrees well with those of Ref. 1 and Ref. 2.

Figure 3: Tip deflection with the piezoceramic positioned at x = 60 mm.

Notes About the COMSOL Implementation

The matrix components for the piezoelectric material properties refer to a coordinate system, where the poling direction is the z direction. Because the poling direction of the piezoceramic actuator in this model is aligned with the x-axis, you need to use a local coordinate system in the material settings to rotate the piezoceramic material.

More specifically, you define a local coordinate system that is rotated 90 degrees about the global y-axis. Then, you use this coordinate system in the piezoelectric material settings to rotate the material so that the polarization direction is aligned with the x-axis (Figure 4).

Figure 4: Definition of local coordinate system to define the piezoelectric orientation. The material is poled along the local x3 direction (blue arrow).

References

1. V. Piefort, Finite Element Modelling of Piezoelectric Active Structures, Ph.D. thesis, Université Libre de Bruxelles, Belgium, Dept. Mechanical Engineering and Robotics, 2001.

2. A. Benjeddou, M.A. Trindade, and R. Ohayon, A Unified Beam Finite Element Model for Extension and Shear Piezoelectric Actuation Mechanisms, CNAM (Paris, France), Structural Mechanics and Coupled Systems Laboratory, 1997.

MEMS_Module/Piezoelectric_Devices/shear_bender

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Block 1 (blk1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 100.

In the text field, type 30.

In the text field, type 18.

Click

Block 2 (blk2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 100.

In the text field, type 30.

In the text field, type 2.

Locate the section. In the text field, type 8.

Click to expand the section. Find the subsection. Select the check box.

Clear the check box.

In the table, enter the following settings:


Layer name	Thickness (mm)
Layer 1	55
Layer 2	10

Click

Click the

button in the toolbar.

The model geometry is now complete.

Click the

button in the toolbar.

The geometry in the window should now look like that in Figure 1.

Click the

button in the toolbar.

Definitions

Define a coordinate system whose third axis is aligned with the global x-axis, that is, the polarization direction of the piezoceramic material. Choose the second axis to be parallel to the global y-axis.

Base Vector System 2 (sys2)

In the toolbar, click

and choose .

In the window for , locate the section.

In the table, enter the following settings:


	x	y	z
x1	0	0	-1
x3	1	0	0

Leave the other components at their default values. You will use this coordinate system in the piezoelectric material settings.

Find the subsection. Select the check box.

Electrostatics (es)

In the window, under click .

In the window for , locate the section.

Click

Select Domain 4 only.

Solid Mechanics (solid)

Piezoelectric Material 1

In the window, under click .

In the window for , locate the section.

Click

Select Domain 4 only.

Locate the section. From the list, choose .

Materials

For the aluminum layers, use a library material.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

Materials

Al - Aluminum (mat1)

In the window for , locate the section.

Click

Select Domains 1 and 3 only.

For the foam core, specify the material properties by hand.

Foam

In the window, right-click and choose .

In the window for , type Foam in the text field.

Select Domains 2 and 5 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	35.3[MPa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.383	1	Young’s modulus and Poisson’s ratio
Density	rho	32	kg/m³	Basic