Geometrically exact displacement-based shell theory

Abstract

The deformed geometry often is the most important information for applications of highly flexible plates/shells, and a geometrically exact shell theory should be displacement-based in order to directly and exactly describe any greatly deformed geometry. The main challenges of modeling a shell undergoing large deformation are how to describe its deformed reference plane and its differential element's large rotations and how to derive objective strains in terms of global displacements and rotations that contain both elastic straining and rigid-body movement. This paper presents a truly geometrically exact displacement-based shell theory without singularity problems. The theory fully accounts for geometric nonlinearities, all possible initial curvatures, and extensionality by using Jaumann strains and stresses, exact coordinate transformation, and orthogonal virtual rotations. Moreover, transverse shear deformations are accounted for by using a high-order shear deformation theory. The derived fully nonlinear strain–displacement relations enable geometrically exact forward analysis (obtaining the deformed geometry under a set of known loads) and inverse analysis (obtaining the required loads for a desired deformed geometry). Several numerical examples are used to demonstrate the accuracy and capabilities of the geometrically exact shell theory. Moreover, different theoretical and numerical problems of other geometrically nonlinear shell theories are shown to be mainly caused by the use of Mindlin plate theory to account for transverse shears, Green–Lagrange strains to account for geometric nonlinearities, and/or Euler and Rodrigues parameters to model large rotations.