An efficient one-dimensional model is developed for the statics of piezoelectric sandwich beams. Third-order zigzag approximation is used for axial displacement, and the potential is approximated as piecewise linear. The displacement field is expressed in terms of three primary displacement variables and the electric potential variables by satisfying the conditions of zero transverse shear stress at the top and bottom and its continuity at layer interfaces. The deflection field accounts for the piezoelectric transverse normal strain. The governing equations are derived using a variational principle. The present results agree very well with the exact solution for thin and thick highly inhomogeneous simply supported hybrid sandwich beams. The developed theory can accurately model open and closed circuit boundary conditions.
IntroductionComposite laminates and sandwich structures with some piezoelectric layers, acting as sensors and actuators to achieve desired control, form part of a new generation of adaptive structures. Sandwich beams have high ratio of flexural stiffness to weight resulting in lower deflection, higher buckling load and higher natural frequencies compared to beams of other constructions. Piezoelectric layers are incorporated in sandwich beams for active control. Sandwich structures offer advantage of placement of the electrodes for the piezoelectric layers. A review of threedimensional (3D) continuum-based approaches, 2D theories for plates and shells and 1D theories for beams, along with their comparative study for plates under static loading, has been presented in [1]. Analytical 3D solutions are available only for some specific shapes and boundary conditions, [2, 3]. The 3D finite element (FE) analysis, [4], results in large problem size which may become computationally costly for practical dynamics and control problems. Hence, efficient accurate electromechanical coupled 2D plate and 1D beam models are required without too much loss of accuracy compared to 3D models. Early works used elastic beam models, [5][6][7], with effective forces and moments due to induced strain of actuators. A discrete layer theory with layerwise approximation of displacements was developed for elastic laminated beams with induced actuation strain in [8]. Classical laminate theory (CLT), [9], firstorder shear deformation theory (FSDT), [10], and the refined third-order theory, [11,12] have been applied without electromechanical coupling to hybrid beams and plates. Coupled CLT, FSDT,[13][14][15], and third-order, [16,17], solutions for hybrid beams and plates including the charge equation of electrostatics and electromechanical coupling have been reported. In [18], coupled discrete layer theory (DLT) was presented using layerwise approximation for displacement and potential, which yields accurate results for thin and thick beams. However, it is expensive for practical problems since the number of displacement unknowns depends on the
