Geometrically-exact sandwich shells: The static case L. Vu-Quoc * , H. Deng, X.G. Tan Department of Aerospace Engineering, Mechanics and Engineering Science, University of Florida, 231 Aerospace Building, P.O. Box 116250, Gainesville, FL 32611-6250, USA Received 25 June 1998; received in revised form 20 May 1999 Abstract The present formulation oers a general method for analyzing the static response of geometrically-exact sandwich shells undergoing large deformation. The layer directors at a point in the reference surface are connected to each other by universal joints, and form a chain of rigid links. Finite rotations of the directors in every layer are allowed, with shear deformation independently accounted for in each layer. The thickness and the length of each layer can be arbitrary, thus allowing the modeling of an important class of multilayer structures having ply drop-os. The present formulation is thus suitable to model shell structures with patches of constrained vi- scoelastic materials or of piezoelectric materials. The nonlinear weak form of the governing equations of equilibrium is constructed here, and then the linearization of the weak form and the associated inextensible directors update are derived, leading to a symmetric tangent stiness matrix. A Galerkin ®nite element projection of the linearized equilibrium equations results in a system of nonlinear algebraic equations, in which the interpolation accounts for the ®nite rotations of the directors. We present extensive numerical ex- amples, including sandwich shells with three identical layers and ply drop-os, to illustrate the applicability and versatility of the proposed formulation. Ó 2000 Elsevier Science S.A. All rights reserved. 1. Introduction Sandwich structures have played an important role in several areas of engineering. Many background references were cited in [13], and will not be repeated in the present follow-up paper, except for particularly relevant ones. We refer to review papers such as [4±6,8], and the references therein for various aspects on formulations for multilayer structures. The accuracy of layerwise theory, as compared to single-layer theory with a shear correction factor, has been amply demonstrated in [6], where a comparison of transverse shear stress with 3-D elasticity solution is provided; see also [7]. We describe in this paper a continuation of the results reported in [15], where we derive the equations of motion for geometrically-exact sandwich shells. Focusing on the static case in the present work, we develop a Galerkin projection of the resulting nonlinear governing equations of equilibrium. In the present formulation, each layer in a sandwich shell structure can have dierent thickness and side lengths. As such, the present formulation can be employed to model an important class of multilayer structures having ply drop-os. Another important application of the present formulation is the modeling of shell structures having patches of constrained viscoelastic materials and/or patches of piezoelectric materials. No restriction is imposed on the magnitude of the displacement ®eld, whose continuity across the layer interfaces is exactly enforced. Finite rotations of the directors in each layer are allowed, with shear deformation independently accounted for in each layer. The layer directors at a point in the reference surface are connected to each other by universal joints, and form a chain of rigid links. The overall www.elsevier.com/locate/cma Comput. Methods Appl. Mech. Engrg. 189 (2000) 167±203 * Corresponding author. Tel.: +1-352-392-6227; fax: +1-352-392-7303; URL: www.aero.u¯.edu/~vql/. E-mail address: vu-quoc@u¯.edu (L. Vu-Quoc). 0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 2 9 4 - 7