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Consistency Issues in Shell Elements for Geometrically Nonlinear Problems

  • Teodoro MerliniEmail author
  • Marco Morandini
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 15)

Abstract

Some singular concepts and non-standard practices in the FEM solution of geometrically nonlinear shell problems are highlighted and discussed. In particular, four issues are addressed. (i) The question of the drilling rotation: a shell is essentially a non-polar medium in its tangent plane, so the drilling rotation is a redundant d.o.f. to be defined by an extra stress field, and the latter ought to hold as a primary unknown field of the surface mechanics. It is shown that a proper constitutive characterization and a sound variational formulation lead to a full micropolar setting of the shell mechanics with a true three-parametric rotation tensor. (ii) The interpolation of the orientation field on the shell surface. It is shown that an interpolation scheme firmly abiding by the rules of the SO(3) group leads naturally to frame-invariant and path-independent finite elements. (iii) The linearization of the virtual functional. Again, an approach fully complying with the special orthogonal group allows an easy and correct resolution of the mixed virtual-incremental variation variables that issue in nonlinear variational formulations involving finite rotations. (iv) The question of a good discrete representation of curved surface geometries. It is shown that a pole-based kinematics built on an integral orthogonal oriento-position field leads to a fair approximation of curved geometries and allows to build low-order finite elements that are naturally locking-free.

Keywords

Micropolar variational mechanics Non-polar finite elasticity Drilling rotations Finite rotations and rototranslations Multiplicative interpolation of rotations Dual numbers and tensors Helicoidal modeling Nonlinear shell elements 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Politecnico di Milano, Dipartimento di Ingegneria AerospazialeMilanoItaly

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