Electrostrictive Disc

The electrostrictive effect describes the electric field-induced strain within a material. It is particularly important for a class of materials known as relaxor ferroelectrics. Such materials are composed of ferroelectric domains that are randomly oriented in the absence of an electric field in the material. In the presence of an electric field, these domains rotate, thereby producing strain in the material. The material extends in the direction of electric field and contracts in the direction perpendicular to the field. On applying an electric field in the reverse direction, the ferroelectric domains orient in the same direction as the field, but the material still elongates and hence electrostriction produces unidirectional strain. To produce a bidirectional strain the material can be subjected to an AC electric field superimposed on a DC bias. At very large electric fields, the electrostrictive strain saturates as all ferroelectric domains in the material align along the direction of the electric field. One of the popular electrostrictive ceramic materials is PMN-PT (lead magnesium niobate-lead titanate), which is often used with some dopant, for example, with barium titanate (BT).

Relaxor ferroelectrics have very high dielectric constants, consequently large polarizations are generated by these materials as a result of an applied electric field. There is very little hysteresis in the response of these materials to electric fields, which makes them attractive for micropositioning devices. A consequence of the absence of hysteresis is that there is negligible self-heating (dielectric heating) in dynamic applications and these materials are therefore used in sonars and ultrasonic motors. Unlike piezoelectric materials, they do not need to be poled. Since there is no residual polarization in the absence of an electric field, a mechanical stress does not change the electric field in the material. Hence these materials are in general not used in sensors.

In this tutorial, you learn how to model isotropic ferroelectroelastic materials using COMSOL’s Ferroelectroelasticity interface. This application requires either the MEMS Module or the AC/DC Module and Structural Mechanics Module.

Model Definition

This tutorial shows how to model an electrostrictive cylindrical disc surrounded by air. The geometry is axisymmetric, and consequently 2D axisymmetry is used. The geometry is shown in Figure 1.

The upper end of the disc is at electrical ground whereas the voltage at the lower end is quasi-statically varied from zero to +103 V volts by using the continuation parameter approach in the study settings.

Because of the symmetry, it is sufficient to model only the upper half of the disc using the symmetry boundary conditions.

Figure 1: Axisymmetric model geometry. The ferroelectroelastic disc is in the center of the geometry, the remain of the domain shows the surrounding air.

Electrostriction for a material of arbitrary symmetry can be represented as the following additive contribution to the strain:

(1)

which is quadratic in polarization P. However, for isotropic materials, the fourth order tensor Q has only two independent components, which are usually denoted as Q11 and Q12.

For ferroelectroelastic materials, the polarization vector is nonlinear function of the electric field and possible stress in the material. One possible choice of the polarization shape is hyperbolic tangent (Ref. 1):

(2)

where Ps is the saturation polarization, a is a material parameter called the domain wall density, and the effective electric field is given by

(3)

where E is the applied electric field, and the mechanics stress is computed assuming mechanically linear material as

(4)

where the strain is computed from the mechanical displacement

Again, for isotropic materials, the fourth-order elasticity tensor C has only two independent components. Most common choice to represent those are by specifying the Young’s modulus EYM and Poisson’s ratio ν for the material.

The effective tangential piezoelectric coupling coefficients can be computed as

where ε0,vac is the electric permittivity of free space, and

is the tangent electric susceptibility. An important observation from the above formula is that the piezoelectric coefficients should reach their maximum (or minimum) at certain strength of the applied bias field. This is because P is zero at zero applied field, while χ tends to zero at large applied field magnitudes because of saturation. The piezoelectric coupling tensor d is a third-order tensor. Due to the symmetry, it can be conventionally represented by a 3-by-6 matrix dET with only few nonzero components.

The following material data has been measured (Ref. 1) for a PMT-PT-BT relaxor-ferroelectric material that presents a ternary mixture of lead magnesium niobate with 7.7% lead titanate and 2.5% barium titanate:

Table 1: Material properties of PMT-PT-BT.
Material property	Value	Description
EYM	105 GPa	Young’s modulus
ν	0.4	Poisson’s ratio
ρ	7900 kg/m3	Density
Ps	0.2589 C/m2	Saturation polarization
a	0.86207 MV/m	Domain wall density
Q11	0.0133 m4/C2	Electrostrictive coupling coefficient
Q12	-0.00606 m4/C2	Electrostrictive coupling coefficient

The air domain is modeled with an additional layer on its periphery.

The voltage applied at the disk lower boundary is gradually varying from zero to 4 kV. The upper boundary of the disc is electrically grounded, and it can be loaded mechanically to study the inverse electrostrictive effect. Three cases are analyzed: one without mechanical loading and two others using a vertical boundary load of −40 GPa and −80 GPa, respectively.

Results and Discussion

Figure 2 shows the distribution of the electric potential in the material and in air around it. In Figure 3, the displacement magnitude is visualized in 3D in case of no mechanical loading.

Figure 2: Surface plot of electric potential distribution in the electrostrictive material and surrounding air domain for a 4 kV applied voltage.

Figure 3: Surface plot of the mechanical displacement magnitude.

Figure 4 shows a plot of the z-component of polarization plotted against the Z-component of the electric field at the center of the disc. Figure 5 shows a similar plot of the ZZ-component of electrostrictive strain at the same position. A series of both negative and positive voltages applied on the top surface of the disc lead to a change in the sign and hence the direction of the electric field. However, the strain and displacement is always unidirectional.

Figure 4: Axial polarization vs. axial electric field at the center of the disc.

Figure 5: Axial electrostrictive strain vs. axial electric field at the center of the disc.

Finally, Figure 6 shows the effective tangent piezoelectric coefficients computed at a given value of the applied electric field.

Figure 6: Tangent piezoelectric coupling coefficients vs. axial electric field at the center of the disc.

Reference

1. C.L. Horn and N. Shankar, “A finite element method for electrostrictive ceramic devices,” Int. J. Solids Structures, vol. 33, pp. 1757–1779, 1995.

MEMS_Module/Actuators/electrostrictive_disc

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

Global Definitions

Create parameters to define the geometry, material properties, and applied loads.

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
t	0[s]	0 s	Time parameter
H0	2[mm]	0.002 m	Disc thickness
R0	5[mm]	0.005 m	Disc radius
Q11	0.0133[m^4/C^2]	0.0133 m4/C²	Electrostrictive coupling coefficient
Q12	-0.00606[m^4/C^2]	-0.00606 m4/C²	Electrostrictive coupling coefficient
E1	105[GPa]	1.05E11 Pa	Young's modulus
nu1	0.4	0.4	Poisson's ratio
rho1	7900[kg/m^3]	7900 kg/m³	Density
Ps	0.2589[C/m^2]	0.2589 C/m²	Saturation polarization
a	0.86207 [MV/m]	8.6207E5 V/m	Domain wall density
Vmax	2[MV/m]*H0	4000 V	Maximum applied voltage
F0	-80[MPa]	-8E7 Pa	F0

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
V0	Vmaxsin(2pi*t[1/s])	V	Applied voltage

This variation of the potential with respect to the parameter at one of the disc boundaries will cause the electric field within the material to gradually change between -Vmax and Vmax.

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type R0.

In the text field, type H0/2.

Rectangle 2 (r2)

Right-click and choose .

In the window for , locate the section.

In the text field, type 3*R0.

In the text field, type 5*H0/2.

Click

Solid Mechanics (solid)

In the window, under click .

Select Domain 1 only.

Symmetry Plane 1

In the toolbar, click

and choose .

Select Boundary 2 only.

Boundary Load 1

In the toolbar, click

and choose .

Select Boundary 4 only.

In the window for , locate the section.

Specify the FA vector as


0	r
F0	z

Electrostatics (es)

In the window, under click .

Ground 1

In the toolbar, click

and choose .

Select Boundary 4 only.

Electric Potential 1

In the toolbar, click

and choose .

Select Boundary 2 only.

Because of the symmetry, the voltage at the horizontal symmetry plane equals to a half of that applied at the bottom surface.

In the window for , locate the section.

In the V0 text field, type V0/2.

Charge Conservation, Ferroelectric 1

In the window, click .

Select Domain 1 only.

In the window for , locate the section.

Find the subsection. From the list, choose .

Find the subsection. From the α list, choose . In the associated text field, type 0.

Multiphysics

Electrostriction 1 (efe1)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the Q11 text field, type Q11.

In the Q12 text field, type Q12.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Right-click and choose .

In the toolbar, click

to close the window.

Materials

PMT-PT-BT

In the window, under right-click and choose .

In the window for , type PMT-PT-BT in the text field.

Select Domain 1 only.

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


Property	Variable	Value	Unit	Property group
Young’s modulus	E	E1	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	nu1	1	Young’s modulus and Poisson’s ratio
Density	rho	rho1	kg/m³	Basic
Saturation polarization	Psat	Ps	C/m²	Ferroelectric
Domain wall density	aJAe_iso ; aJAeii = aJAe_iso, aJAeij = 0	a	V/m	Ferroelectric