PIEZOELECTRIC MATERIALS
4.8 ELECTROSTRICTIVE MATERIALS
Table 4.3 Energy density of different types of piezoelectric materials in extensional and bending mode at an electric field of 1 MV/m
Ev: Ev: d33 d13 Y3E Y1E Extensional Bending
Company (pm/V) (pm/V) (GPa) (GPa) (kJ/m3) (kJ/m3)
Piezo Systems PSI-5A4E 390 190 52 66 4.0 0.7
PSI-5H4E 650 320 50 62 10.6 1.8
American Piezo APC 840 290 125 68 80 2.9 0.4
APC 850 400 175 54 63 4.3 0.5
APC 856 620 260 45 58 8.6 1.1
Kinetic Ceramics PZWT100 370 170 48 62 3.3 0.5
TRS Ceramics PMN-PT 2250 1050 12 17 30.4 5.3
TRSHK1 HD 750 360 57 65 16.0 2.4
of 1 MV/m. The energy density at other electric fields can be obtained by multiplying the value listed in the table by the square of the applied electric field in MV/m.
One material type stands out in Table 4.3, due to its high piezoelectric strain coefficients. The material, PMN-PT, is a single-crystal piezoelectricthat exhibits large piezoelectric strain coefficients and large coupling coefficients (>90%). The large strain coefficient is offset somewhat by the fact that single-crystal ceramics are softer than their polycrystalline counterparts. The energy density of single-crystal materials is generally three to five times larger than that of a conventional ceramic.
As of the writing of this book, single-crystal materials were also more expensive than conventional materials and were generally thought of as a good solution for high-end applications of piezoelectric materials where large strain (>0.5%) and good coupling properties were required.
The values listed in Table 4.3 are ideal values that do not account for certain limitations in the fabrication or operation of the material. For example, the values listed for piezoelectric stacks do not incorporate nonideal behavior introduced by inactive electrodes or insulating material. More important, these values do not reflect the inactive mass associated with important components such as the housing or preload springs. Adding the mass of inactive components can reduce the actual energy density by a factor of 3 to 5 compared to the energy density of the material itself. These issues are less important for bimorph actuators, which in many types of operation do not require preloading or casing.
As discussed in Chapter 3, the linear constitutive relationships are derived from an energy formulation which assumes a quadratic relationship in the energy function.
Another fundamental assumption is that the electromechanical coupling properties are also linear. The result of this assumption is that the coupling between electrical and mechanical domains is also modeled as a linear matrix of constants.
Electrostrictive materials are those in which the electromechanical coupling is represented by the quadratic relationship between strain and electric field. In indicial notation the strain–field relationships are written
Si j =Mi j mnEmEn. (4.169)
The variable Mi j mn is a fourth-rank tensor of electrostriction coefficients. In the case in which the applied electric field is only in a single direction, the constitutive relationships are
Si j=Mi j nE2n. (4.170)
The quadratic relationship between applied field and strain produces a response that is fundamentally different from that of a piezoelectric material. Linear coupling between strain and field produces a mechanical response that will change polarity when the polarity of the electric field is changed. For example, a piezoelectric material with a positive strain coefficient will produce positive strain when the electric field is positive and negative strain when the electric field is negative. A quadratic strain–
electric field relationship will produce strain in only a single direction. A positive electrostrictive coefficient will produce positive strain when the field is positive but will also produce positive strain when the polarity of the electric field is changed. This physical response is due to the quadratic field relationship in equation (4.170). The difference between the strain response of piezoelectric and electrostrictive materials is shown in Figure 4.32. The strain response of the piezoelectric material is, of course,
E S
d
E S
(a) (b)
Figure 4.32 Representative strain responses for (a) piezoelectric and (b) electrostrictive materials.
.
linear where the slope is equal to the strain coefficient of the material. The parabola represents the quadratic strain response of the electrostrictive material. Although the curves are only representative, the crossing of the two curves is intential since a quadratic function will always produce a higher value at some value of the applied electric field. The exact value in which the two responses are identical is a function of piezoelectric and electrostrictive coefficients.
4.8.1 One-Dimensional Analysis
To understand the basic properties of electrostrictive materials and compare them to piezoelectric materials, let us consider an analysis in which the applied electric field is in only one direction and that we are only interested in the strains in a single direction.
In this case we can drop the subscript notation in equation (4.170) and simply write the strain–electric field relationship as
S=ME2, (4.171)
whereMis the electrostrictive coefficient in the direction of interest.
Example 4.14 An electrostrictive ceramic has an electrostrictive coefficient of 8× 10−17 m2/V2. A material sample has a thickness of 0.05 mm. Compute the strain induced by the application of 100 V.
Solution The electric field induced by the applied voltage is
E= 100 V
0.05×10−3m =2 MV/m.
The electrostrictive strain is computed from equation (4.171), S=(8×10−17m2/V2)(2×106V/m)2=320µstrain.
The quadratic relationship between applied field and mechanical strain is some- times problematic for the development of devices using electrostrictive materials.
For example, in applications for motion control, it is often desirable to have a linear relationship between applied electric field and strain since it simplifies the design of actuators and motors. One method of transforming the quadratic relationship of an electrostrictive material into an equivalent linear relationship is to apply abiased electric field that consists of the sum of a direct-current (dc) value and an alternating- current (ac) component:
E=Edc+Eac. (4.172)