Spectral element formulation for an electrostrictive material embedded in a beam

Abstract

In this paper, a spectrally formulated finite element in frequency domain for an electrostrictive material embedded in a composite beam is derived. The presence of quadratic non linearity of the electric field in the constitutive equation poses some challenge in the formulation of the model and we use convolution in the frequency domain. The formulation is carried out using Hamilton’s theorem and this is subsequently applied to a bimorph beam which has electrostrictive wafer mounted on the top as well as bottom surfaces of the beam. The bimorph beam is chosen so that we can control the beam by applying different and independent electric potentials for the top and bottom surfaces of the beam. The non linearity in the constitutive model results in the generation of additional nonlinear waves. The formulated spectral element elegantly captures these nonlinear waves. The efficiency of electrostrictive actuator for both single and multi modal open loop control is studied in detail. This aspect is demonstrated through a number of numerical examples. The studies also show that the control efficiency in electrostrictive actuator is limited by the amplitude of the electric field. When the electric field is beyond a certain level, it can be seen that although the amplitude of the primary wave decreases, non-linearity in the model causes sign reversal and splitting of incident and reflected waves. It can also be seen that the amplitude of nonlinear wave tend to increase with increase in applied electric field. The paper initially investigates the use of electrostrictive material as an actuator and this idea is gradually expanded to the use of electrostrictive material as a sensor. Various numerical experiments are carried out to demonstrate this aspect and its applications in the detection of the crack. It can be seen that the spectral element derived elegantly captures the location and size of the crack.