A hybrid/mixed model for non‐linear shell analysis and its applications to large‐rotation problems

Abstract

SUMMARYAdopting an updated Lagrange approach, the general framework for the fully non-linear analysis of curved shells is developed using a simple quadrilateral Co model (HMSHS). The governing equations are derived based on a consistent linearization of an incremental mixed variational principle of the modified Hellinger/Reissner type with independent assumptions for displacement and strain fields. Emphasis is placed on devising effective solution procedures to deal with large rotations in space, finite stretches and generalized rate-type material models. In particular, a geometrically exact scheme for configuration update is developed by making use of the so-called exponential mapping algorithm, and the resulting element was shown to exhibit a quadratic rate of (asymptotic) convergence in solving practical shell problems with Newton-Raphson type iterative schemes. For the purpose of updating the spatial stress field of the element, an 'objective' generalized midpoint integration rule is utilized, which relies crucially on the concept of polar decomposition for the deformation gradient, and is in keeping with the underlying mixed method. Finally, the effectiveness and practical usefulness of the HMSHS element are demonstrated through a number of test cases involving beams, plates and shells undergoing very large displacements and rotations.

Numerical Analysis of Geometrically Non-Linear Behavior of Functionally Graded Shells

“…The problem has been considered in Refs. (Kim et al 2008;Sze et al 2004Sze et al , 2002aSze et al , 2002bBuechter and Ramm 1992;Sansour and Kollmann 2000;Park et al 1995;Saleeb et al 1990;Simo et al 1990aSimo et al , 1990bLee et al 1984;Basar et al 1992;Sansour and Bufler 1992;Betsch and Stein 1995;Hauptmann and Schweizerhof 1998;Providas and Kattis 1999;Hong et al 2001 andKim et al 2003), among others. In one hand, we consider the case of isotropic material.…”

On the use of an enhanced transverse shear strain shell element for problems involving large rotations

“…Although possessing an initial small-value character, the discretized rotational degrees of freedom employed in equation (4) can be employed within the nodal frame update procedure stated in (25) and (26) (implicitly involving large discrete rotations) without loss of generality, as demonstrated in [43]. Also benefiting from the previous study, in the present work the secondorder rotation terms in (4) are not taken into account in the derivation of an ''additional'' geometric stiffness contributions.…”

An isogeometric continuum shell element for non-linear analysis

“…The problem of a pinched cylinder with a rigid diaphragm at the ends has been studied by several authors [28,29,30] in order to test the convergence behaviour and non-linear performance of shell elements. Since large rotations occur the problem provides a test for the finite rotation capability of the shell formulation.…”