An Analytical Description of Slow Hysteresis in Polarized Ferroelectric Ceramic Actuators

Abstract

Recently, an existing system of electroelastic equations was extended to account for slow hysteretic effects in polarized ferroelectric ceramic plates by employing an internal variable in a thermodynamic state function. All material irreversibility was taken to be a consequence of the ferroelectric polarization-electric field irreversibility. Since only the slowest possible hysteresis is of concern, the evolution equation is ignored and the known irreversible thickness polarization-electric field saturation curve is taken as the starting curve. Since all experimental data available to us are from strain measurements on a thin wide beam forced by voltages applied to very thin polarized ferroelectric ceramic actuators on the surfaces, the linear strain, plane-stress version of the equations is employed in the description of the actuator. The new irreversible material coefficients for the case of plane-stress are obtained from data on one major hysteresis loop along with the coefficients in the saturation polarization-electric field relations. These now known coefficients are then used to calculate other hysteresis loops, which are shown to agree quite well with measurement.

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References

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2. H. F. Tiersten , "On the Nonlinear Equations of Thermoelectroelasticity," Int. J. Eng. Sci., 9:587 (1971).

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5. It would be desirable to have it done at zero frequency, but this is not possible because the material would change temperature from Joule heating, which would cause the coefficients to change drastically.

6. We use the words "switching" and "unswitching" to refer to what is happening to the domains as the rate of change of E₃ is positive or negative, respectively.

7. This is consistent with our qualitative physical reasoning based on domain switching and would certainly be required for more complicated shapes of hysteresis loops than those considered in this work.

8. IEEE Standard on Piezoelectricity, IEEE Std. 176-1978, Institute of Electrical and Electronics Engineers, New York, 1978.

9. H. F. Tiersten , "Electroelastic Equations for Electroded Thin Plates Subject to Large Driving Voltages," J. Appl. Phys., 74:3389 (1993).

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10. R. D. Mindlin , An Introduction to the Mathematical Theory of the Vibration of Elastic Plates (U.S. Army Signal Corps Engineering Laboratory, Fort Monmouth, New Jersey, 1955) Sec. 6.04.

11.

11. The experimental setup is shown and briefly discussed in D. Hughes and J. T. Wen , "Preisach Modeling and Compensation for Smart Material Hysteresis," in Active Materials and Smart Structures, G. L. Anderson and D. C. Lagoudas , Editors, Proc. SPIE, 2427:50-64 (1995).

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12. The three-dimensional linear piezoelectric constants are given in Table 1. They were calculated from data provided by the manufacturer, American Piezo Ceramics, for PZT type 8-50, which is equivalent to Navy type II or Piezoelectric Products type G-1 195. Relations in Equations (43) of Reference [8] were used in the calculation.

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13. When actuators are used on the top and bottom of plates and beams in the case of flexure, one polarization loop is always outside the initial saturation polarization-electric field curve and the other, inside. As the inside loop moves down towards depolarization under repeated operation, the outside loop moves up towards the completely switched condition. Since the distance from the starting saturation loop at zero electric field to the completely switched state is usually less than the distance to the depolarized state, it is likely that the fully switched state is reached in one actuator before the depolarized state is reached in the other. Furthermore, when the hysteretic behavior is caused by the amplitude of the driving voltage employed, if the power can be turned off at an appropriate point in the cycle, the depolarized (or completely switched) state can be avoided for a considerably longer operating period. Since the polarization-electric field loops must be traversed in a counterclockwise direction, the loops in Figure 5 indicate that the appropriate point in the cycle is just after the turning points when each loop will return to zero field at a value of polarization nearest the value on the initial saturation polarization-electric field curve.

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14. We use the word "loop" to refer to a single curve of a "loop," which is actually composed of two connected curves, because this terminology is convenient in this context.