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In this article, we are going to demonstrate the simulation of the converse piezoelectric effect by using a FEA simulation program known as COMSOL. We will apply voltage to a piezoelectric element and from that voltage the piezoelectric element will produce strain, or we can also say displacement.

Equations Describing the Converse Piezoelectric Effect

Figure 1: Piezoelectric Element

In order to determine the strain, and hence, displacement, we multiply the electric field E_3 by the piezoelectric constant d33. d33 is measured for strain in same direction as the spontaneous polarization and this property is measured under constant stress which means that there are free-stress boundary conditions on all of the adjacent sides of the piezoelectric element in question. 

Along the polarization direction, x_3, $x_3$   is the strain, d_33 is the piezoelectric charge constant, and E_3 s the electric field.

Setting up a Static Simulation in COMSOL for the Converse Piezoelectric Effect

Now let’s witness electric field induced strain (converse piezoelectric effect) by designing a simple rectangular block in COMSOL and applying voltage.

 Firstly, we will go to model wizard in COMSOL.

In our case select 3D then go to structural mechanics and select piezoelectric devices and add necessary physics for this simulation. We want to look at stationary results because we are doing a static simulation, so add stationary physics.

Change meters to millimetres because that’s primarily the scale we work in for piezoelectric devices.

In order to add the piezoelectric element geometry, go to the geometry panel and open your work plane and build a circle of radius 10mm at coordinates (0,0).

Then draw another circle of radius 0.2mm in the center and select build all. 

After that extrude the larger circle and we will get our final piezoelectric disk as shown in figure. The smaller circle will be used to add a fully contrained mechanical boundary condition which has a low impact on the final simuation, because all static simulation must have a fixed boundary as a reference. An important note here is that if we were to full constrain the entire face, the element would undergo resistance to stretching in the thickness due to the poissons ratio lateral contraction expansion that happens in tandem with extension in the thickness direction, which is the main effect and the effect we will be looking at in this simulation. If you do fully constrain the entire face, you will get less displacement (but not zero) and you will not be able to validate the equation of $d_{33}$ .

From the materials options, select the PZT-4 and add that in model. 

We will now need to add boundary conditions. We added a small circle in the model so that we can add a fixed constraint reference. Add this condition that part by selecting Fixed Constraint in the Solid Mechanics.

In order to apply voltage, go to Electrostatic and apply voltage boundary conditions on the opposing faces of the disk. We have to apply a voltage boundary condition on the bottom as well. Assign an Electric Potential boundary condition on the top face of the disk and Ground to the bottom.

Now go to the Piezoelectric Materials setting and change to “Strain-charge form” because this form of describing the piezoelectric effect allows us to use the piezoelectric charge coefficient d33. The alternative standard setting is to use the piezoelectric stress coefficient e33 which is not directly calculated from any standard experimental measurement methods. It’s is derived from measuring multiple anisotropic properties and matrix calculations; however, the piezoelectric charge coefficient can be readily determined from laboratory equipment, especially from a d33 meter. 

Change the piezoelectric coefficient value to 289 E-12, whose units are Coulombs/meter.

To more easily view the model, you can just change the extrusion distance. I changed it to 10mm here. Now, build the mesh. 

Compare Simulation Results (Displacement) to Equations

Then go to Study and Compute and go to the 2D plot Group 3. By looking at the scale from the figure, we determine the total displacement. The maximum displacement is little more than 3 E-7 mm, which is 3 nanometers.

We can clearly see the displacement varying dominantly in the thickness direction, and a slight variation along with width effect due the fixed boundary constrain we needed to put in order to allow the model to solve. When we apply an electric field to a piezoelectric material it expands to  a zero stress condition. So, although there is no stress, there is displacement. 



We can understand this from an analogy to thermal expansion. For example if we heat a material, it expands but when the material expands it’s not under stress. Internal stress is actually causing to expand. Once expansion is finished there is no more internal stress. However you will clamp it anyway you will develop stress. There it is correct that there is no stress involved here. If we did not allow the material to expand then it would be under external stress. Basically, applying an electric field resets the equilibrium dimensions of the piezoelectric material, after which the material deforms to reach that equilibrium zero stress state.

In order to find the exact displacement, make a probe by selecting definitions in Component 1 and finally Domain Point Probe in probes. Run the simulation again to get the solution. Select Probe Point 4 in results. Here we can easily see that it’s 2.9E-7mm. Therefore,

delta t  = 2.9 E-7 mm = 2.9 E-9 m

As we applied electric potential of 1V, d33 calculated from the simulation via the electric field and the strain

d33 = E_3 /  x_3 = 290E-12 C/N

We can check the coefficient value that we got by simulation simply by going to material properties again. Here we can see that it is 2.89 E-12, exactly what we put in for the d33 earlier

In microns the measurement will be 2.9e-4 $2.9\times {10}^{-4}$ μm, very small!. Because the displacement generated by the converse piezoelectric is very small, we use resonance, large voltages, and multi-layer structures to increase the displacement