Consistency Issues in Shell Elements for Geometrically Nonlinear Problems
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.
KeywordsMicropolar 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
Unable to display preview. Download preview PDF.
- 1.Angeles, J.: The application of dual algebra to kinematic analysis. In: Angeles, J., Zakhariev, E. (eds.) Computational Methods in Mechanical Systems, vol. 161, pp. 1–31. Springer, Heidelberg (1998)Google Scholar
- 3.Badur, J., Pietraszkiewicz, W.: On geometrically non-linear theory of elastic shells derived from Pseudo-Cosserat continuum with constrained micro-rotations. In: Pietraszkiewicz, W. (ed.) Finite rotations in structural mechanics, vol. 161, pp. 19–32. Springer, Berlin (1986)Google Scholar
- 14.Merlini, T.: Variational formulations for the helicoidal modeling of the shell material surface. Scientific Report DIA-SR 08-06, Aracne Editrice, Roma (2008). ISBN: 978-88-548-1887-3Google Scholar
- 16.Merlini, T., Morandini, M.: The helicoidal modeling in computational finite elasticity. Part II: Multiplicative interpolation. Int. J. Solids Struct. 41, 5383–5409 (2004). Erratum on Int. J. Solids Struct. 42, p. 1269 (2005)Google Scholar
- 18.Merlini, T., Morandini, M.: Helicoidal shell theory. Scientific Report DIA-SR 08-07, Aracne Editrice, Roma (2008). ISBN: 978-88-548-1889-7Google Scholar
- 19.Merlini, T., Morandini, M.: Computational shell mechanics by the helicoidal modeling. Part I: Theory. J. Mech. Mater. Struct. 6 (to appear)Google Scholar
- 20.Merlini, T., Morandini, M.: Computational shell mechanics by the helicoidal modeling. Part II: Shell element. J. Mech. Mater. Struct. 6 (2011) (to appear)Google Scholar
- 21.Merlini, T., Morandini, M.: The helicoidal modelling in the approximation of shell structures mechanics. In: Pietraszkiewicz, W., Kreja, I. (eds.) Shell Structures: Theory and Applications, vol. 2, pp. 269–272. CRC Press, Taylor & Francis, London (2010)Google Scholar