Continuous electric field modeling of Macro-Fiber Composites for actuation and energy harvesting

Abstract

This paper presents a new low-order electric field model for Macro-Fiber Composite devices with interdigitated electrodes. Specifically, the paper proposes a continuous electric field model, where the differential form of Gauss's law is used to obtain spatial terms of the electric field, and the integral form of Gauss's law is used to obtain temporal term of the electric field. By neglecting three-dimensional effects, the method of matched asymptotic expansion is employed to obtain the electric field between two pairs of interdigitated electrodes under quasi-electrostatic conditions. By averaging the solution along with the thickness of piezoceramic fibers and expanding it to the device's entire geometry by Fourier series, the low-order model of the electric field between the interdigitated electrodes is obtained. The matched asymptotic expansion and the low-order model agree well with finite element results. Generalized Hamilton's principle is employed to obtain the coupled electromechanical equations of motion for a beam with the Macro-Fiber Composite device. The Euler-Bernoulli assumptions are used to approximate strain distribution. Under current assumptions, equivalent capacitance and electromechanical coupling terms are obtained without using discontinuous functions (e.g., Heaviside or piecewise functions) for electric field terms. The equivalent capacitance is within 2% of the values reported by the manufacturer. Frequency response functions are obtained for the output voltage, tip displacement, and optimum working load of the energy harvester, and the current and the tip displacement for the actuator by assumed modes solution for a cantilever beam. It is shown that the actuator's current is related to the optimum working load of the energy harvester by Ohm's law. In the limited case, the proposed continuous electric field model reduces to the constant electric field model. When the constant electric field model is compared to the continuous electric field model, it is shown that the constant electric field model overpredicts the equivalent capacitance and electromechanical coupling. The relative difference between the two models for equivalent capacitance is around 41%, and for electromechanical coupling is around 22%. These relative differences are in agreement with the empirical correction factors defined in the earlier constant field models.