Piezoelectric actuation of a compliant semi-infinite beam

Abstract

Piezoelectric materials (PZT) are commonly used as actuators and sensors for vibration suppression in flexible metal or composite substrates. There are well-established techniques for modeling the actuation of PZTs when they are bonded to these structures. However, if the substrate material is much softer than the piezoelectric actuator/sensor, a higher level of modeling is needed to predict the local deformations at the interface. In this research, a finite-length piezoelectric element bonded perfectly to an infinite elastic strip is modeled. The specific goal was to quantify the actuation and sensing mechanics of piezoelectric devices on substrates potentially much softer than the piezoelectric element. Previous works have addressed membranes or plates bonded to an elastic half-space subjected to mechanical or thermal loads. Euler-Bernoulli beam theory is used to derive equations of equilibrium for the piezoelectric beam. These equations are then recast as integral equations for the interface displacement gradients and equated to the equivalent quantities for an elastic layer subject to distributed shear and normal tractions. The resulting singular integral equations are solved by expanding the interface tractions using a series of Chebyshev polynomials. First, certain sanity checks are performed to confirm the validity of the model by choosing a stiff substrate for which Euler-Bernoulli beam-assumptions holds good. For certain combinations of geometrical and material parameters, the substrate has a positive curvature, whereas the piezoelectric has a negative curvature and vice versa. After analyzing the forces acting on both piezoelectric and the substrate, the reasons for this behavior in soft substrates are justified here. Finally, the range of geometric parameters where the reversal of bending occurs in the piezoelectric is given.