A Geometrically Nonlinear Shear Deformation Theory for Composite Shells

A geometrically nonlinear shear deformation theory has been developed for elastic shells to accommodate a constitutive model suitable for composite shells when modeled as a two-dimensional continuum. A complete set of kinematical and intrinsic equilibrium equations are derived for shells undergoing large displacements and rotations but with small, two-dimensional, generalized strains. The large rotation is represented by the general finite rotation of a frame embedded in the undeformed configuration, of which one axis is along the normal line. The unit vector along the normal line of the undeformed reference surface is not in general normal to the deformed reference surface because of transverse shear. It is shown that the rotation of the frame about the normal line is not zero and that it can be expressed in terms of other global deformation variables. Based on a generalized constitutive model obtained from an asymptotic dimensional reduction from the three-dimensional energy, and in the form of a Reissner-Mindlin type theory, a set of intrinsic equilibrium equations and boundary conditions follow. It is shown that only five equilibrium equations can be derived in this manner because the component of virtual rotation about the normal is not independent. It is shown, however, that these equilibrium equations contain terms that cannot be obtained without the use of all three components of the finite rotation vector.