A Model for the Analysis of Beams with Embedded Piezoelectric Layers

Abstract

This paper provides a basic mechanics model for the flexural analysis of beams with embedded piezoelectric layers. The Euler model for a long and thin beam is employed and two different ways of connecting the electrodes on the surfaces of the piezoelectric layers, namely, closed circuit (Case I) and open circuit (Case II) are considered. The distribution of the piezoelectric potential in the longitudinal direction is derived by including Maxwell equation in the formulation and assuming a half-cosine distribution for the potential in the thickness direction of the piezoelectric layer for Case I. The validity of this assumption is investigated first theoretically and then numerically by FEM for a pure piezoelectric beam subjected to uniform moment. A semi-analytical analysis is carried out for Case II where an electric potential function is assumed. The resonant frequencies of the structure for the models presented are first validated by the FEM software, ABAQUS, for simply-supported and propped cantilever boundary conditions. Based on the results of vibration analysis, it is shown that the dynamic characteristics of the entire structure can be significantly affected by piezoelectric effects, especially for the open circuit case. More importantly, the mode shapes of the electric potential in the piezoelectric layer are different for Cases I and II. For the closed circuit case, the potential shape function is related to the transverse displacement, or more accurately the curvature of the beam. For the open circuit case, the potentials at the boundaries in the longitudinal direction are directly related to the slope of the deflection of the beam. Hence, the commonly adopted assumption of uniform electric potential needs to be carefully re-examined.