Locking alleviation in the large displacement analysis of beam elements: the strain split method

Abstract

This paper proposes a new locking alleviation technique for absolute nodal coordinate formulation (ANCF) beam and plate elements based on a strain split approach. The paper also surveys classical finite element (FE) and ANCF locking alleviation techniques discussed in the literature. Because ANCF beam elements, which allow for the cross section stretch, fully capture the Poisson effect, Poisson locking is an issue when such beam elements are considered. The two-dimensional fullyparameterized ANCF beam element is primarily used in this investigation because such an element can serve as a good surrogate model for three-dimensional ANCF beams and plates as far as membrane, bending and transverse shearing behavior is concerned. In addition to proposing the strain split method (SSM) for ANCF locking alleviation, this work assesses the ANCF FE performance in the cases of higher-order interpolation, enhanced assumed strain method, elastic line method, and the enhanced continuum mechanics approach; and demonstrates the design of the enhanced strain interpolation function by using the shape functions of higher-order ANCF elements. Additionally, a new higher-order ANCF two-dimensional beam element is proposed in order to compare its performance with other finite elements that require the use of other locking alleviation techniques proposed and reviewed in the paper. Finally, several numerical examples are shown to demonstrate the effectiveness of the locking alleviation methods applied to ANCF elements. The purpose of this investigation, apart from proposing a new locking alleviation technique, a new higher-order beam element, and comparing several existing locking alleviation techniques, is to show that dealing with locking in fully-parameterized ANCF elements is feasible and that several methods exist to effectively improve the ANCF element performance without sacrificing important ANCF element properties and features including position vector gradient continuity. Because of the use of ANCF position vector gradients as nodal coordinates, complex stress-free initially-curved geometries can be systematically obtained. Such initially-curved geometries require special attention when attempting to solve locking problems, as will be discussed in this paper.

A condensed algorithm for adaptive component mode synthesis of viscoelastic flexible multibody dynamics

“…One accounts for Poisson's ratio and the other does not. The roots of this split approach 8,[11][12][13][14][15]66 lie in the concept of classical finite elements selectively reduced integration technique. Nachbagauer et al 12,14 proposed two quadratic three-node beam elements of ANCF based on the enhanced continuum mechanics 11 and the structural mechanics of the strain energy defined by Simo and Vu-Quoc, 2,3 respectively.…”

“…Numerical examples show that these two beam elements of ANCF are all locking-alleviated 12,14 and have the same order of convergence compared with Reissner's shear deformation beam. 11,12 Henrik Ebel et al 16 and Patel et al 8 recently proposed several higher-order beam elements of locking alleviation, and Patel et al 8 pointed out that the linear interpolation considered in the transverse direction is one of the main reasons for the poor performance of the standard formulation based on continuum mechanics. Because the use of Hooke's law in this case leads to a stiff behavior in the response to nonuniform strain distribution like bending.…”

“…Because the use of Hooke's law in this case leads to a stiff behavior in the response to nonuniform strain distribution like bending. 8 Based on the enhanced continuum mechanics, the quadratic three-node beam elements of ANCF are utilized to model the FMBS with the viscous damping of Kelvin-Voigt type in this study. The classical Kelvin-Voigt model consist of a spring in parallel with a dashpot, thus the resultant stress is the sum of the stress of spring and the stress of dashpot, and the train is the same of those.…”

“…In addition, it also allows for the stretch of the cross-section of a beam, which is different from the classical finite elements. 8 Because of the use of the transverse gradient vectors, however, the beam elements of ANCF may suffer from Poisson locking, and lead to poor convergence. [9][10][11][12][13] A variety of locking mechanisms have been thoroughly studied in the community of finite elements, because many classical finite elements may also suffer from locking problems.…”

“…[9][10][11][12][13] A variety of locking mechanisms have been thoroughly studied in the community of finite elements, because many classical finite elements may also suffer from locking problems. 8 For a detailed discussion or quantification about the locking alleviation techniques for ANCF, one can refer to the work of Nachbagauer, 12,14 Bauchau et al, 13 Gerstmayr et al, 15 Henrik Ebel et al, 16 and Patel et al 8 In addition, the ANCF 11,12,14 can be modified according to the strain energy defined by Simo and Vu-Quoc with fully equivalent static and dynamics behaviors compared with Reissner's shear deformation beam. 11,12 Both formulations above have seen successful applications in the field of flexible multibody dynamics.…”