Accepted Manuscript

A locking-free finite element formulation and reduced models for geometrically exact Kirchhoff rods

Christoph Meier, Alexander Popp, Wolfgang A. Wall

PII: S0045-7825(15)00093-6

DOI: http://dx.doi.org/10.1016/j.cma.2015.02.029

Reference: CMA 10574

To appear in: Comput. Methods Appl. Mech. Engrg.

Received date: 15 May 2014

Revised date: 20 February 2015

Accepted date: 25 February 2015

Please cite this article as: C. Meier, A. Popp, W.A. Wall, A locking-free finite element formulation and reduced models for geometrically exact Kirchhoff rods, Comput. Methods

Appl. Mech. Engrg. (2015), http://dx.doi.org/10.1016/j.cma.2015.02.029

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A Locking-Free Finite Element Formulation and Reduced Models for

Geometrically Exact Kirchhoff Rods

Christoph Meiera, Alexander Poppa,∗, Wolfgang A. Walla aInstitute for Computational Mechanics, Technische Universita¨t Mu¨nchen, Boltzmannstrasse 15, D–85748 Garching b. Mu¨nchen, Germany

Abstract

In this work, we suggest a locking-free geometrically exact finite element formulation incorporating the modes of axial tension, torsion and bending of thin Kirchhoff beams with arbitrary initial curvatures. The proposed formulation has been designed in order to represent general load cases and three-dimensional problem settings in the geometrically nonlinear regime of large deformations. From this comprehensive theory, we not only derive a general beam model but also several reduced formulations, which deliver accurate solutions for special problem classes concerning the beam geometry and the external loads. The advantages of these reduced models arise for example in terms of simplified finite element formulations, less degrees of freedom per element and consequently a higher computational efficiency of the corresponding numerical models. A second core topic of this publication is the treatment of membrane locking, which is a locking phenomenon predominantly occurring in highly slender curved structures, thus, exactly in the prime field of application for Kirchhoff theories. In order to address the membrane locking effect, we will propose a new interpolation strategy for the axial strain field and compare this method with common approaches such as

Assumed Natural Strains (ANS) or reduced integration. The effectiveness of this method as well as the consistency and accuracy of the general finite element formulation and the reduced beam models will be illustrated with selected numerical examples.

Keywords: Geometrically exact Kirchhoff beams, Large rotations, Finite elements, Reduced models, Membrane locking 1. Introduction

For many years, the design of geometrically exact beam elements according to the Cosserat theory (see e.g. [2]) has essentially been restricted to shear-deformable beam formulations. The starting point for these developments can be traced back to the important works of Reissner ([37], [38]) and Simo ([44], [46]). Crisfield and Jelenic were the first ones who recognized the special significance of objectivity and path-independence in the context of geometrically exact beams, see [12] and [23]. They have shown the lack of these essential properties for all existing geometrically exact formulations at that time and developed a strain-invariant and path-independent interpolation strategy for the rotational degrees of freedom of nonlinear beam elements. Many other authors followed this example and developed conceptually different interpolation methods, still preserving objectivity and path-independence (see e.g. [15], [17], [18], [39], [41], [43], [47], [52]). In his recent contributions [39] and [40], Romero pointed out the excellent performance of geometrically exact beam formulations of Simo-Reissner type, especially when designed according to the principles presented in [12]. Furthermore, [32] presents a recent overview concerning the development of different types of geometrically exact Simo-Reissner beams and possible difficulties.

However, state-of-the-art shear-free beam elements according to the Kirchhoff theory of thin rods ([25], [28]) have not reached these outstanding properties of geometrically exact Simo-Reissner beams so far. In the remainder of ∗Corresponding author

Email address: popp@lnm.mw.tum.de (Alexander Popp)

Preprint submitted to Computer Methods in Applied Mechanics and Engineering March 13, 2015 this work, the notion “shear-free” is equivalent to “vanishing shear strains”, but, of course, not to “vanishing shear stresses” for general load states. In the early works [6] and [7], numerical investigations and comparisons of the standard Kirchhoff beam elements existing at that time have been made. Accordingly, many of these first representatives showed a poor numerical accuracy, a fact that amongst others was attributed to the non-objective nature of these formulations. Different strategies in order to preserve objectivity and improve the accuracy of such elements have been developed in the following years. However, most of these strategies were limited to plane problems and simple geometries like circular arcs or based on straight “framework” or “facet” element approximations. It has been shown in [3] that these “framework” or “facet” element approximations, when applied to Kirchhoff beam formulations, may lead to a deterioration of the numerical performance as compared with corresponding geometrically exact approaches.

A further critical issue relevant in the context of thin Kirchhoff beams is membrane locking, a locking phenomenon given distinction to by Stolarski and Belytschko [48]. In general, membrane locking denotes the inability of curved structural elements, e.g. beams or shells, to represent the inextensibility constraint of vanishing membrane / axial strains. For thin Kirchhoff beams, [16] was one of the first contributions in which this effect was investigated by relating the beam slenderness ratio to the condition number of the stiffness matrix, but without explicitly using the term locking. Diverse methods have been proposed in the literature in order to avoid membrane locking of Kirchhoff rods.