Asymptotic Construction of a Reissner-like Composite Shell Model with Accurate Three-Dimensional Recovery

Abstract

The focus of this paper is to develop an asymptotically correct theory for composite laminated shells that can be implemented in standard finite element programs. The analysis is restricted to the case when each lamina exhibits monoclinic material symmetry about its own mid-surface. The development starts with formulation of the three-dimensional, anisotropic elasticity problem in which the deformation of the reference surface is expressed in terms of intrinsic twodimensional variables. The Variational Asymptotic Method is then used to rigorously split this threedimensional problem into a linear one-dimensional normal-line analysis and a nonlinear two-dimensional “shell”analysis accounting for transverse shear deformation. The normal-line analysis is solved by onedimensional finite element method and provides a constitutive law between the generalized, two-dimensional strains and stress resultants as well as recovery relations to approximately express the three-dimensional displacements, strains and stresses fields in terms of shell variables calculated in the “shell”analysis. It is known that more than one theory that is correct to a given asymptotic order may exist. This nonuniqueness is used to cast a strain energy functional that is asymptotically correct through the second order into a simple “Reissner-like”shell theory. The initial curvatures of the shell structure affect the constitutive law and consequently the recovery relations. A first-order geometrical correction is obtained for the strain energy and the recovered three-dimensional field also includes modification by the initial curvatures. Although it is not in general possible to construct an asymptotically correct Reissner-like composite shell theory, an optimization procedure is used to drive the present theory