Modeling Piezocomposite Actuators Embedded in Slender Structures

This work presents a comprehensive methodology for dimensional reduction of anisotropic slender structures with embedded anisotropic piezocomposite materials. The analysis is based on a variational-asymptotic formulation, and provides cross-sectional stiffness, inertia and actuation forces for a beam modeling of the structure. It can retain higher-order information corresponding to non-classical deformation modes. The particular case of the active anisotropic Timoshenko-like beam formulation is considered separately. Some results for typical active beam configurations are included and compared with three-dimensional shell finite element solutions using thermal analogy for the distributed actuation. Introduction Numerical simulation of structures through threedimensional (3-D) Finite Element Analysis has become widely accessible to engineers using the power of modern computers. However, one should not underestimate the value of engineering understanding of physical problems to identify essential features that could simplify their mathematical models. Such approach can improve the feasibility and efficiency of the structural analysis, particularly within the context of a multidisciplinary design environment. This work presents one of such particular scenarios: slender active structures, defined as systems with one dominant spatial dimension; typical examples in aircraft applications are helicopter rotor blades and high-aspectratio wings. The basic feature of these systems is that their structural modeling process can be significantly simplified through the dimensional reduction of the 3-D structure to a (1-D) beam model One can clearly distinguish two steps in a general beam theory [17]: first is the process of dimensional reduction, which is performed through a 2-D analysis in the beam cross sections; second, the computed equivalent stiffness and inertia properties are used in the 1-D analysis of the loaded beam. Finally, the 3-D stress/strain and displacement fields can be determined based on the combination of these two previous steps. This paper focuses on the dimensional reduction of general nonhomogeneous anisotropic active beams with arbitrary cross sections and initial pre-twist and curvature. The concept of active beam refers here to beams with distributed actuators embedded within the composite structure. The actuation is assumed to derive from the electroelastic response of piezoelectric-based materials. Typical anisotropic piezocomposite actuators are the active fiber composites described by Bent [1], and the macro fiber composites of Wilkie et al. [20]. Notice that the present work encompasses also hygrothermal effects in a composite beam constitutive relation. Several different formulations have been proposed for the analysis of anisotropic beams, among the leading efforts are the works of Giavotto et al. [10], Kosmatka [13] and Cesnik and Hodges [4]. The formulation proposed in [4], known as the Variational-Asymptotic Beam CrossSectional (VABS) analysis, will set the basic framework for this active beam development. VABS is the application to anisotropic beams of the variationalasymptotic method of Berdichevsky [2], which is based on the asymptotic solution of the 3-D cross-sectional warping field corresponding to a set of general 1-D strain measures through the minimization of the associated strain energy. Asymptotic solutions of anisotropic beam cross sections have been pursued in the last decade by the first author and his co-workers. Successive contributions have presented the solution to the passive problem for prismatic beams [12], initially curved and twisted beams [5], beams with non-perpendicular cross-sectional planes [15], beams with arbitrary deformation modes [7], beams with transverse shear effects [14], [21], and, more recently, active cross-sectional formulation for thinwalled beams [8] and general beams [6]. In particular, the latter is restricted to a classical beam formulation (EulerBernoulli-like), which, although sufficient for most applications, does not address short composite beams (where the slenderness property is not only associated with its dimensions but also with the material anisotropy in the structure) and the estimation of the net shear loads induced by certain bending actuation modes. Finally, of practical interest is the capability to model general deformations that are not included in the classical or Timoshenko-like formulations, e.g., airfoil camber * Associate Professor. Associate Fellow, AIAA ([email protected]) . Graduate Research Assistant. F.X. Bagnoud Fellow ([email protected]) Copyright  2003 by Carlos E.S. Cesnik and Rafael Palacios. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS Page 1 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Confere 7-10 April 2003, Norfolk, Virginia AIAA 2003-1803 Copyright © 2003 by Carlos E. S. Cesnik and Rafael Palacios. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. deformation in a slender wing. These deformation modes would require additional beam degrees of freedom. ( ) ij ji ij ij A A δ − + = Γ 2 1 (1) This paper presents the most general active beam crosssectional analysis within the VABS framework. The formulation accounts for arbitrary cross-sectional geometries without restrictions in the active and passive material distribution. It also includes the effect of initial twist and curvature. Finally, finite-section effects, which can require user-defined deformation modes beyond the four classical ones (extension, twist, and two bending modes), are included in the formulation. Special attention is given to the Timoshenko-like formulation and the calculation of the active forcing constants that are part of the active beam constitutive relation. The methodology presented here has been implemented in a computer code named UM/VABS. Aij are the components of the deformation gradient tensor, A, resolved along mixed bases: ( )( ) j k k i b g G B ⋅ ⋅ = ij A (2) {Bi} is the deformed beam reference triad, {g} is the contravariant form of the tangent base vectors to the undeformed beam, and {Gk} is the covariant base of the deformed beam.