Abstract
This paper presents a positional FEM formulation to deal with geometrical nonlinear dynamics of shells. The main objective is to develop a new FEM methodology based on the minimum potential energy theorem written regarding nodal positions and generalized unconstrained vectors not displacements and rotations. These characteristics are the novelty of the present work and avoid the use of large rotation approximations. A nondimensional auxiliary coordinate system is created, and the change of configuration function is written following two independent mappings from which the strain energy function is derived. This methodology is called positional and, as far as the authors' knowledge goes, is a new procedure to approximated geometrical nonlinear structures. In this paper a proof for the linear and angular momentum conservation property of the Newmark algorithm is provided for total Lagrangian description. The proposed shell element is locking free for elastic stress-strain relations due to the presence of linear strain variation along the shell thickness. The curved, high-order element together with an implicit procedure to solve nonlinear equations guarantees precision in calculations. The momentum conserving, the locking free behavior, and the frame invariance of the adopted mapping are numerically confirmed by examples.
Highlights
An accurate analysis of structures that exhibit large deflections is of great importance for structural design
In this work a proof alternative to the one given by Kane et al 21 that the Newmark β method conserves linear momentum and angular momentum for any adopted time step is given. This proof is restrict to total Lagrangian formulation not corotational and is trivially extended to energy conserving property for rigid bodies
The Newmark β method is angular momentum conserving for γ 1/2, despite the adopted time step, angular velocity or β parameter
Summary
An accurate analysis of structures that exhibit large deflections is of great importance for structural design. 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, free of large rotation descriptions and locking. There exists another kind of rotation-free elements as proposed in the works of Onate and Zarate and Flores and Onate , developed for thin shells and based on curvature considerations. All required features of the formulation as: locking free, frame invariance, and momentum conserving linear and angular are checked in the numerical examples section
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