COMSOL 6.4 - Piezoelectricity in a Layered Shell

This tutorial is intended as a simple example showing how to model piezoelectric devices using the Layered Shell functionality.Two cases of material orientation are investigated. In the first case, the pole axis is normal to the shell surface, which results in a change in thickness of the deformed shell. In the second case, the pole axis is tangential to the shell, which leads to the shell bending.

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Introduction to the Composite Materials Module.

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Modeling Piezoelectricity: Which Module to Use?

This model requires Composite Material Module.

Model Definition

In this tutorial, you model a thin structure with three layers. The top and bottom layers are electric conductors, and the middle one is a piezoelectric material. The geometry of the structure is shown in Figure 1.

Figure 1: Layered shell geometry representation in 3D.

The modeling approach in this tutorial is based on the layered shell technology available in COMSOL Multiphysics. Thus, you represent the geometry by a surface in 3D space, which gives the base of the layered structure; see Figure 2.

Figure 2: Model geometry representation as a surface.

You also use the surface for meshing, which gives the in-plane part of the discretized representation of the model; see Figure 3.

Figure 3: Meshed base selection.

The out-of-plane structure of the shell is modeled using an extra dimension attached to the base selection. The extra dimension together with the base selection span a local 3-dimensional space in which the equations are solved.

The extra dimension is set up using a special node, which allows you to change the number of layers as well as their thickness and in-plane rotation, and to change the number of mesh elements used in each layer in the extra dimension. You can also select the materials to be used in each layer.

Materials

The structure in this tutorial has three layers. The top and bottom ones represent 1 mm thick conductors made of aluminum. The layer in the middle has a thickness of 4 mm and is composed of a piezoelectric material called Lead Zirconate Titanate (PZT-5H).

For a piezoelectric material, the inelastic contributions to strain due to applied electric field E can be written in the strain-charge form as

For materials with tetragonal symmetry (class 4mm) such as PZT-5H, the coupling matrix has the following structure (Ref. 1):

(1)

Here, the pole direction is along the third coordinate axis (the z-axis). Such orientation is assumed for all the piezoelectric material data available in COMSOL Material Library. For PZT-5H, the following values are used: d31 = −2.74·10−10 C/N, d33 = 5.93·10−10 C/N, and d15 = 7.41·10−10 C/N.

When the electric field has only one component, Ez, the only nonzero contributions to the strain are: εxx = εyy = d31Ez and εzz = d33Ez.

In this example, the electric field component Ez is positive. Hence, the piezoelectric layer will tend to stretch in the z direction and shrink in the other two directions.

When the pole direction is changed to the x-axis, the first and last lines in the coupling matrix will be swapped. As a result, the only nonzero contribution to the strain will be εxz = 0.5d15Ez so that the piezo layer will be sheared in the xz-plane.

Boundary Conditions

At x = 0, the whole cross section of the layered shell (the yz-plane) is mechanically fixed; see Figure 1. The ground condition is applied to the top conducting interface, and a fixed electric potential of 20 V is applied to the bottom interface. All other boundaries are free.

Results and Discussion

Two different cases of poling directions of piezoelectric materials are considered: the pole axis in the z direction and the pole axis in the x direction. In both cases, the electric potential distribution is similar; see Figure 4.

Figure 4: Electric potential distribution.

In the first case, the material has a default orientation with the z-axis as the pole direction, which is normal to the shell. The structure deformation is shown in Figure 5.

In the second study, the material orientation is changed so that the pole direction is in-plane along the x-axis. The nature of resulting structural deformation is significantly different; see Figure 6.

The results show that using a correct choice of the material orientation is essential when modeling piezoelectric applications.

Figure 5: Vertical displacement for the case where the z-axis is the pole direction in the piezoelectric layer.

Figure 6: Vertical displacement for the case where the x-axis is the pole direction in the piezoelectric layer.

Notes About the COMSOL Implementation

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Modeling a composite laminate as a layered shell requires a surface geometry, in general referred to as a base surface, and a node which adds an extra dimension (1D) to the base surface geometry in the surface normal direction. You can use the functionality to model several layers stacked on top of each other having different thicknesses, material properties, and fiber orientations. You can optionally specify the interface materials between the layers, and control the number of through-thickness mesh elements for each layer.

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The third direction for the selected coordinate system in the , , or represents the normal direction in the Layered Shell and Shell interfaces. This is also the direction in which the layer stacking is interpreted from bottom to top, and therefore, it is crucial to know it during modeling. There are two ways to achieve this:

Using physics symbols: Go to the physics settings, find the section, and select the checkbox. Then go to the material feature, for instance, , to see the normal direction represented by green arrows in the geometry.

Using result templates: When a solution dataset is available, use the result template to plot the normal direction.

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The device is modeled in 3D using the predefined multiphysics interface . Two physics interfaces, structural Layered Shell and Electric Currents in Layered Shells, will automatically be added to the model together with a multiphysics coupling feature called .

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The Layered Shell interface will contain a node, where you can select geometry boundary for the base surface, and also select certain layers within the layered material. The Electric Currents in Layered Shells interface will contain a node , where a similar selection should be made. It is important to have the same selections under both interfaces. All settings for the material properties and orientation can be found under the node within the Layered Shell interface. This includes the structural, dielectric, and coupling properties.

References

1. A.L. Kholkin, N.A. Pertsev, and A.V. Goltsev, “Piezoelectricity and Crystal Symmetry,” Piezoelectric and Acoustic Materials for Transducer Applications, A. Safari and E.K. Akdogan, eds., Springer, Boston, MA, 2008.

MEMS_Module/Piezoelectric_Devices/piezoelectric_layered

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 .

Work Plane 1 (wp1)

In the toolbar, click

In the window for , click

Work Plane 1 (wp1) > Plane Geometry

In the window, click .

Work Plane 1 (wp1) > Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 50.

In the text field, type 30.

Array 1 (arr1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

In the text field, type 2.

Locate the section. In the text field, type 50.

Click

Click the

button in the toolbar.

Next, add the materials, aluminum and PZT-5H from the .

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Right-click and choose .

In the tree, select > .

Right-click and choose .

In the toolbar, click

to close the window.

Add a layered material node and defined a stacking sequence. Here you can also change the number of mesh elements used in the thickness direction.

Global Definitions

Layered Material 1 (lmat1)

In the window, under right-click and choose .

In the window for , locate the section.

Click

twice.

In the table, enter the following settings:


Layer	Material	Rotation	Value	Thickness	Mesh elements
Layer 1	Aluminum (mat1)	0.0	0 rad	1[mm]	2
Layer 2	Lead Zirconate Titanate (PZT-5H) (mat2)	0.0	0 rad	4[mm]	4
Layer 3	Aluminum (mat1)	0.0	0 rad	1[mm]	2

You can preview the laminate cross section.

Click to expand the section. In the text field, type 8/30.

Locate the section. Click in the upper-right corner of the section.

Add a layered material link node to make the layered material available in . Using this node, you can also select a coordinate system defining the orientation. Note that only boundary coordinate systems can be selected.

Materials

Layered Material Link 1 (llmat1)

In the window, under right-click and choose > .

Next, select only the middle layer of the laminate to use the piezoelectric material.

Electric Currents in Layered Shells (ecis)

Piezoelectric Layer 1

In the window, under > click .

In the window for , locate the section.

Clear the checkbox.

In the table, clear the checkboxes for and .

Layered Shell (lshell)

Piezoelectric Material (Z Pole Axis)

In the window, under > click .

In the window for , type Piezoelectric Material (Z Pole Axis) in the text field.

Locate the section. Click

Select Boundary 1 only.

Locate the section. Clear the checkbox.

In the table, clear the checkboxes for and .

Locate the section. From the list, choose .

Add a new feature and change the orientation of the piezoelectric layer.

Piezoelectric Material (X Pole Axis)

Right-click and choose .

In the window for , type Piezoelectric Material (X Pole Axis) in the text field.

Locate the section. Click

Select Boundary 2 only.

Click to expand the section. The layered material always operates with a boundary coordinate system on the base surface (laminate system). For such systems, the third base vector direction is always normal to the surface.

Use a special control if you need to change the out-of-plane orientation of the material.

From the list, choose .

Next, set up the boundary conditions for the structure and the charge balance. Note that you need to operate with edges of the base surface.

Fixed Constraint 1

In the toolbar, click