Distributed transfer function-based unified static solutions for piezoelectric short/open-circuit sensing and voltage/charge actuation of beam cantilevers

Abstract

Closed-form unified solutions using the distributed transfer functions (DTFs) method are presented for the first time for the static short/open-circuit sensing and voltage/charge actuation of moderately thick beam cantilevers with co-localized surface-bonded piezoelectric patches. For this purpose, the smart beam is divided into three segments, of which the clamp and free sides parts are elastic, while the middle one is made of an elastic core sandwiched between two electroded piezoelectric patches. The latter can be different in material properties and thickness but should have the same length, and their widths can be different from the host elastic beam. The theoretical formulation is based on Timoshenko’s first-order shear deformation theory for the kinematics and piezoelectric constitutive equations and the principle of virtual works for the variational equations. The latter integrate explicitly the physical equipotential constraints on the patches electrodes. The balance equations and boundary conditions are derived for the three segments independently and then connected at their interfaces by the equilibrium equations and continuity conditions. The unified static solutions for the resulting four problems are derived analytically in closed form using the DTF approach. These are validated against only open literature benchmarks having tabulated results or analytical formulas in order to avoid curves induced inherent additional deviations. Very good correlations were obtained in comparison with the found reference two-dimensional (2D) plane strain/stress analytical and 2D plane strain/stress and three-dimensional finite element results.