Tangent stiffness of polar shells undergoing large displacements

Abstract

The paper deals with the definition and evaluation of the tangent stiffness of hyper-elastic polar shells without drilling rotations. The ambient space for such bodies is a non-linear differentiable manifold. As a consequence the incremental equilibrium must be expressed as the absolute time derivative of the non-linear equilibrium condition expressing the balance between the elastic response and the applied forces. In the absolute time derivative the classical directional derivative is replaced by the covariant derivative according to a fixed connection on the manifold. The evaluation of the tangent stiffness requires us to take the second covariant derivative of the finite deformation measure and this in turn requires an extension of the virtual displacement field in a neighborhood of the given configuration of the shell. It is explicitly shown that different choices of this extension lead to the same tangent stiffness, which is symmetric since the chosen connection is torsionless.