This work presents a fully nonlinear sixparameter (3 displacements and 3 rotations) shell model for finite deformations together with a triangular shell finite element for the solution of the resulting static boundary value problem. Our approach defines energetically conjugated generalized cross-sectional stresses and strains, incorporating first-order shear deformations for an inextensible shell director (no thickness change). Finite rotations are treated by the Euler-Rodrigues formula in a very convenient way, and alternative parameterizations are also discussed herein. Condensation of the three-dimensional finite strain constitutive equations is performed by applying a mathematically consistent plane stress condition, which does not destroy the symmetry of the linearized weak form. The results are general and can be easily extended to inelastic shells once a stress integration scheme within a time step is at hand. A special displacement-based triangular shell element with 6 nodes is furthermore introduced. The element has a nonconforming linear rotation field and a compatible quadratic interpolation scheme for the displacements. Locking is not observed as the performance of the element is assessed by several numerical examples, which also illustrate the robustness of our formulation. We believe that the combination of reliable triangular shell elements with powerful mesh generators is an excellent tool for nonlinear finite element analysis. IntroductionMost of the research carried out over the past two decades regarding nonlinear finite shell elements deals essentially with quadrilateral elements. As a result of locking phenomena triangular domains were very often avoided, despite their greater flexibility for mesh generators. With respect to shell kinematics, a large number of these works commonly employ only two components to entirely describe the shell rotation field. The so-called drilling degreeof-freedom is frequently ignored as it has no inherent stiffness during the shell motion. Six-parameters -3 displacements and 3 rotations -shell models can however be very convenient for engineering applications since no special connection scheme is necessary at the shell edges and intersections, and no particular care needs to be taken when coupling shell and rod elements.In this work we review and extend the geometricallyexact six-parameter shell formulation of Pimenta [17] (which is one of the existing shell models undergoing large strains and finite rotations, see [2, 9-11, 15, 24, 32] to name just a few others), and introduce a special triangular shell finite element for the solution of the resultant static boundary value problem.Although it may be not necessary, our approach defines energetically conjugated cross sectional stresses and strains, based on the concept of shell director with a standard Reissner-Mindlin kinematical assumption. Appealing is the fact that both the first Piola-Kirchhoff stress tensor and the deformation gradient appear as primary variables. Due to the use of cross sectional quantitie...

This work presents a fully nonlinear multi-parameter shell formulation together with a triangular shell finite element for the solution of static boundary value problems. Our approach accounts for thickness variation as additional nodal DOFs, using a director theory with a standard Reissner-Mindlin kinematical assumption. Finite rotations are exactly treated by the Euler-Rodrigues formula in a pure Lagrangean framework, and elastic constitutive equations are consistently derived from fully three-dimensional finite strain constitutive models. The corresponding 6-node triangular shell element is presented as a generalization of the T6-3i triangle introduced by the authors in [3].A good number of nonlinear shell models which explicitly account for thickness change has been developed in the past years (see ½1; 2; 7; 8; 11; 16; 19). Most of these works incorporate the extra thickness strain terms at the element level via the enhanced assumed strain concept, leading very often to hybrid-mixed formulations. A few others like [11] provide only translational degrees-of-freedom in a continuum basis, but ill-conditioned systems are generated and additional locking effects have to be taken care of.Rotation variables are also avoided in [16] but a difference displacement vector needs to be introduced then.The main purpose of this work is to present a different geometrically-exact multi-parameter shell model and its related finite element implementation. In a pure displacement-oriented approach, we account for thickness variation at the level of the shell theory, based on the concept of shell director. A standard Reissner-Mindlin kinematical assumption is employed. The model constitutes an extension of our earlier work in [3], in the sense that the restriction to an inextensible director field is now removed from the theory. Appealing is the fact that both the first Piola-Kirchhoff stress tensor and the deformation gradient appear as primary variables. Another attractive aspect is that the fundamental equations are totally written in terms of power conjugated cross-sectional stresses and strains.Finite rotations are exactly treated by the EulerRodrigues formula in a pure Lagrangean framework. A plane reference configuration is assumed for the shell mid-surface, but initially curved shells can also be considered if regarded as a stress-free deformed state from the plane position. The use of convective non-Cartesian coordinate systems is avoided here so that only components related to orthogonal frames are employed.As thickness deformation is incorporated within the shell kinematics, the usual plane-stress approximation is unnecessary and fully three-dimensional constitutive models are used without condensation. We adopt here a hyperelastic finite-strain material law but extension to inelastic shells is straightforward, once a 3-D stress integration scheme within a time step is at hand.A complete set of shell local equilibrium equations and the related boundary conditions are consistently derived in terms of stress resultants, s...

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A fully conserving algorithm is developed in this paper for the integration of the equations of motion in nonlinear rod dynamics. The starting point is a re-parameterization of the rotation field in terms of the so-called Rodrigues rotation vector, which results in an extremely simple update of the rotational variables. The weak form is constructed with a non-orthogonal projection corresponding to the application of the virtual power theorem. Together with an appropriate time-collocation, it ensures exact conservation of momentum and total energy in the absence of external forces. Appealing is the fact that nonlinear hyperelastic materials (and not only materials with quadratic potentials) are permitted without any prejudice on the conservation properties. Spatial discretization is performed via the finite element method and the performance of the scheme is assessed by means of several numerical simulations.