PIEZOELECTRIC MATERIALS
4.1 ELECTROMECHANICAL COUPLING IN PIEZOELECTRIC DEVICES: ONE-DIMENSIONAL MODEL
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PIEZOELECTRIC
T [N/m2]
Y [N/m2]
S [m/m]
linear elastic
softening hardening
applied stress
T
elongation
Figure 4.1 Representative stress–strain behavior for an elastic material.
elongation divided by the original length of the specimen. At low values of applied stress the strain response will be linear until a critical stress, at which the material will begin to yield. In the linear elastic region the slope of the stress–strain curve is constant. The slope of the line is called themodulus, orYoung’s modulus, and in this book the modulus is denotedY and has units of units N/m2. The stress–strain relationship in this region is
S= 1
YT=sT, (4.1)
wheres, the reciprocal of the modulus, is called themechanical compliance(m2/N).
Above the critical stress the slope of the stress–strain curve changes as a function of applied load. A softening material will exhibit a decreasing slope as the stress is increased, whereas a hardening material will exhibit an increasing slope for stress values above the critical stress.
Now consider the case when a piezoelectric material is being subjected to an applied stress. In addition to elongating like an elastic material, a piezoelectric material will produce a charge flow at electrodes placed at the two ends of the specimen. This charge flow is caused by the motion ofelectric dipoleswithin the material. The application of external stress causes the charged particles to move, creating an apparent charge flow that can be measured at the two electrodes. The charge produced divided by the area of the electrodes is the electric displacement, which has units of C/m2. Applying an increasing stress level will produce an increase in the rotation of the electric dipoles and an increase in the electric displacement. Over a certain range of applied mechanical stress, there is a linear relationship between applied stress and measured electric displacement. The slope of the curve, called thepiezoelectric strain coefficient(Figure 4.2), is denoted by the variabled. Expressing this relationship in
T [N/m2] D [C/m2]
d [C/N]
applied stress
T electrode
electrode
charge flow
- + -
+ -
+ -
+ -
+ - +
-
+ +
- linear
response
saturation
Figure 4.2 (a) Direct piezoelectric effect; (b) relationship between stress and electric displace- ment in a piezoelectric material.
a proportionality, we have
D=dT, (4.2)
where D is the electric displacement (C/m2) anddis the piezoelectric strain coefficient (C/N). At sufficient levels of applied stress, the relationship between stress and electric displacement will become nonlinear due to saturation of electric dipole motion (Figure 4.2). For the majority of this chapter we concern ourselves only with the linear response of the material; the nonlinear response is analyzed in the final section.
4.1.2 Converse Effect
The direct piezoelectric effect described in Section 4.1.1 is the relationship between an applied mechanical load and the electrical response of the material. Piezoelectric materials also exhibit a reciprocal effect in which an applied electric field will produce a mechanical response. Consider the application of a constant potential across the electrodes of the piezoelectric material as shown in Figure 4.3. Under the assumption that the piezoelectric material is a perfect insulator, the applied potential produces an electric field in the material, E, which is equal to the applied field divided by the distance between the electrodes (see Chapter 2 for a more complete discussion of ideal capacitors). The units of electric field are V/m. The application of an electric field to the material will produce attractions between the applied charge and the electric dipoles. Dipole rotation will occur and an electric displacement will be measured at the electrodes of the material. At sufficiently low values of the applied field, the relationship between E and D will be linear and the constant of proportionality, called thedielectric permittivity, has the unit F/m. The relationship between field and electric
E [V/m]
D [C/m2]
ε [F/m]
electrode
electrode
charge flow
- + -
+ -
+ - + -
+ - +
-
+ +
- linear
response
saturation +
-
Figure 4.3 Relationship between applied electric field and the electric displacement in piezo- electric material.
displacement in the linear regime is
D=εE. (4.3)
As is the case with an applied stress, the application of an increasingly high electric field will eventually result in saturation of the dipole motion and produce a nonlin- ear relationship between the applied field and electric displacement. The converse piezoelectric effect is quantified by the relationship between the applied field and mechanical strain. For a direct piezoelectric effect, application of a stress produced dipole rotation and apparent charge flow. Upon application of an electric field, dipole rotation will occur and produce a strain in the material (Figure 4.4). Applying suf- ficiently low values of electric field we would see a linear relationship between the applied field and mechanical strain. Remarkably enough, the slope of the field-to- strain relationship would be equal to the piezoelectric strain coefficient, as shown in
E [V/m]
S [m/m]
charge flow
- + -
+ - + -
+ -
+ - +
-
+ + -
d [m/V]
linear response
elongation elongation
Figure 4.4 Relationship between electric field and strain in a piezoelectric material.
Figure 4.4. Expressing this as an equation, we have
S=dE. (4.4)
In this expression, the piezoelectric strain coefficient has the unit m/V. Equation (4.4) is an expression of theconverse effectfor a linear piezoelectric material.
Example 4.1 Consider a piezoelectric material with a piezoelectric strain coefficient of 550×10−12m/V and a mechanical compliance of 20×10−12 m2/N. The material has a square geometry with a side length of 7 mm. Compute (a) the strain produced by a force of 100 N applied to the face of the material when the applied electric field is zero, and (b) the electric field required to produce an equivalent amount of strain when the applied stress is equal to zero.
Solution (a) Compute the stress applied to the face of the material,
T= 100 N
(7×10−3m)(7×10−3m)=2.04 MPa. The strain is computed using equation (4.1),
S=(20×10−12m2/N)(2.04×106Pa)=40.8×10−6m/m.
The units of×10−6m/m are often calledmicrostrain; a stress of 2.04 MPa produces 40.8 microstrain in the piezoelectric material.
(b) The electric field required to produce the same strain in the material is computed using the equations for the converse effect. Solving equation (4.4) for E yields
E= S
d = 40.8×10−6m/m
550×10−12m/V =74.2 kV/m.
This value is well within the electric field limits for a typical piezoelectric material.
4.2 PHYSICAL BASIS FOR ELECTROMECHANICAL COUPLING