Optimal solid shell element for large deformable composite structures with piezoelectric layers and active vibration control

Abstract

SUMMARYIn this paper, we present an optimal low-order accurate piezoelectric solid-shell element formulation to model active composite shell structures that can undergo large deformation and large overall motion. This element has only displacement and electric degrees of freedom (dofs), with no rotational dofs, and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out-of-plane bending). The combination of the present optimal piezoelectric solid-shell element and the optimal solid-shell element previously developed allows for efficient and accurate analyses of large deformable composite multilayer shell structures with piezoelectric layers. To make the 3-D analysis of active composite shells containing discrete piezoelectric sensors and actuators even more efficient, the composite solid-shell element is further developed here. Based on the mixed Fraeijs de Veubeke-Hu-Washizu (FHW) variational principle, the in-plane and out-of-plane bending behaviours are improved via a new and efficient enhancement of the strain tensor. Shear-locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. We also present an optimal-control design for vibration suppression of a large deformable structure based on the general finite element approach. The linear-quadratic regulator control scheme with output feedback is used as a control law on the basis of the state space model of the system. Numerical examples involving static analyses and dynamic analyses of active shell structures having a large range of element aspect ratios are presented. Active vibration control of a composite multilayer shell with distributed piezoelectric sensors and actuators is performed to test the present element and the control design procedure.

Hybrid asymptotic–direct approach to finite deformations of electromechanically coupled piezoelectric shells

“…In these theories, a priori assumptions concerning the mechanical displacements and the electrical field in certain directions are made to reduce the three-dimensional continuum to a lower-order one. With respect to nonlinear formulations for piezoelectric plates and shells, we refer to the rich literature, e.g., Zheng et al [8], Tan and Vu-Quoc [9], Klinkel and Wagner [10], Marinković et al [11] and Lentzen et al [12]. In this paper, we develop a novel hybrid approach to the modeling of the nonlinear behavior of piezoelectric shells, which is here presented in a holistic form for the first time; the approach is illustrated by two numerical examples.…”

A finite element formulation for piezoelectric shell structures considering geometrical and material non‐linearities

“…In order to consider laminated structures, the above mentioned formulations include a more or less sophisticated laminate theory. References [5,10,12,13,14] point out that geometrically nonlinear characteristics can significantly influence the performance of piezoelectric systems, especially for the sensor usage. A geometrically nonlinear theory that incorporates large rotations is presented in References [1,5,6,12].…”

A piezoelectric solid shell element based on a mixed variational formulation for geometrically linear and nonlinear applications

“…One may distinguish between element formulations, which model a reference surface of the shell structure, see e.g. [1,2,3,4,5,6], and solid shell elements which model the top and bottom surfaces of structures, see e. g. [7,8,9,10,11,12]. Some of these element formulations are restricted to shallow shell structures, [5,6], where the initial shell curvature is assumed to be small.…”

“…[8,13,14] it is pointed out that geometrically nonlinear characteristics can significantly influence the performance of piezoelectric structures and systems. Geometrically nonlinear effects like buckling of plates and snap-through behavior were analyzed in [1,6,15,16].…”

“…A lot of nonlinear piezoelectric plate formulations [6,13,14,15,16] are based on von Kármán plate theory, which is a nonlinear theory of lowest order. A geometrically nonlinear theory which incorporates large rotations is presented in [1,7,8].…”