Keywords: non-linear dynamics, shells, finite element method.

An accurate analysis of structures that exhibit large deflections is of great importance for structural design. The increasing search for economy and optimal material application leads to the conception of very flexible structures. As a consequence, the equilibrium analysis in the non-deformed position is no more acceptable for most applications. This affirmation is confirmed by the large amount of research regarding this subject [1,2,3,4,5].

This study is concerned with the development of an alternative finite element methodology to solve geometrical non-linear dynamics of shells. In order to achieve a robust formulation, the resulting element should be free of shear and volumetric locking. This problem is solved here by the natural presence of the transverse shear strain in the proposed kinematics. The novelty of the proposed formulation is the use of positions and generalized unconstrained vector mapping, resulting in a naturally objective continuum representation of the shell that avoids large rotation descriptions and is free of locking.

The formulation proposed here is total Lagrangian and as a result of its unconstrained vector mapping, presents a constant mass matrix. Therefore it is possible to apply the Newmark beta integrator as a momentum conserving algorithm. A simple proof of the momentum conserving property of the Newmark beta method for rigid bodies is given in this paper. The proof is restricted to the total Lagrangian formulation (not extended to co-rotational formulations) and trivially fulfills the energy conserving property for rigid bodies. All required features of the formulation such as: locking free, frame invariance and momentum conserving (linear and angular) are checked.

1

F. Baginski, K. Brakke, W.W. Schur, "Cleft formation in pumpkin balloons", Adv. Space Res., 37, 2070-2081, 2006. doi:10.1016/j.asr.2005.04.104

2

J. Bonet, R.D. Wood, J. Mahaney, P. Heywood, "Finite element analysis of air supported membrane structures", Comput. Meth. Appl. Mech. Engrg., 190(5-7), 579-595, 2000. doi:10.1016/S0045-7825(99)00428-4

3

E. Onate, F. Zarate, "Rotation-free triangular plate and shell elements", International Journal for Numerical Methods in Engineering, 47, 557-603, 2000. doi:10.1002/(SICI)1097-0207(20000110/30)47:1/3<557::AID-NME784>3.0.CO;2-9

4

S. Lopez, "Improving stability by change of representation in time-stepping analysis of non-linear beams dynamics", International Journal for Numerical Methods in Engineering, 69(4), 822-836, 2007. doi:10.1002/nme.1789

5

I. Breslavsky, K.V. Avramov, Y.U. Mikhlin, R. Kochurov, "Nonlinear modes of snap-through motions of a shallow arch", Journal of Sound and Vibration, 311, 297-313, 2008. doi:10.1016/j.jsv.2007.09.015

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