On the role of frame-invariance in structural mechanics models at finite rotations

doi:10.1016/S0045-7825(02)00442-5

Computer Methods in Applied Mechanics and Engineering

Volume 191, Issue 45, 25 October 2002, Pages 5159-5176

https://doi.org/10.1016/S0045-7825(02)00442-5 Get rights and content

Abstract

In this work we re-examine so called geometrically exact models for structures, such as beams, shells or solids with independent rotation field, with respect to invariance under superposed rigid body motion. A special attention is given to clarifying the issues pertaining to the finite element implementation which guarantees that the invariance of the continuum problem is preserved by its discrete approximation. Several numerical simulations dealing with finite rotations of structural models are presented in order to further illustrate and confirm the given theoretical considerations.

Introduction

Pioneering works of Reissner [22] and Antman [1] were the first to provide beam theories capable of describing with no approximation arbitrary large displacements and rotations in 2D setting. A 3D extension of this kind of beam theory for a straight beam was later given in Simo [24], who introduced the terminology ‘geometrically exact’ to indicate that no restriction is introduced on either displacements nor rotations. Subsequently, it was shown how to generalize these kinds of theories to space-curved beams [12], shells with drilling rotations [11] and 3D solids with independent rotation field [13], which provided a unified approach for modeling arbitrarily complex structural systems by using for model ingredients the finite elements which all share the same configuration space in a fully nonlinear framework.

Another line of works which contributed significantly to the success of the geometrically exact structural models pertains to clarifying the computational aspects of 3D finite rotations; After the seminal contribution of Argyris [2], a number of works followed, such as Simo and Vu-Quoc [25], Buechter and Ramm [6], Simo [27], Hassenplug [8], or Ibrahimbegovic et al. [14], [15], among others, who all contributed to establishing a very firm theoretical basis for dealing with this class of problems. The versatility of the geometrically exact structural models was further demonstrated by their ability to furnish novel approaches to several application domains, such as nonlinear instability (vs. classical buckling) analysis [16], flexible multibody dynamics [17] or yet structural shape optimization in nonlinear analysis [19].

The only exception to these positive developments on geometrically exact structural models and their applications have been very recently reported findings (see Crisfield and Jelenic [7] or Jelenic and Crisfield [20]) that none of the different discrete approximations of the geometrically exact beam model presented previously can guarantee the frame-indifference of the obtained results. The latter has even prompted further contributions on the subject (e.g. see Romero and Armero [23] or Betsch and Steinmann [5]) proposing to completely eliminate any explicit reference to finite rotation tensors and to reformulate the geometrically exact beam model by making use of two director vectors. While the developments of this kind might be of interest for providing beam models easy to combine with the models used for smooth shells without drilling rotations (e.g. see Simo and Fox [26] or Ibrahimbegovic et al. [18]), they ought not, we believe, be presented as the only ones capable of satisfying the invariance requirements.

In this respect, our goal in this work is to clarify some subtle points and to present a number of novel implementation results which ensure the frame-invariance of any geometrically exact structural mechanics model. In particular, the implementation details for an applied support finite rotation are provided, along with the corresponding modifications of the residual force vector and tangent matrix introduced by the follower forces and moments.

The outline of the paper is as follows. In the next section we briefly recall some facts about 3D finite rotations of interest for this work. The governing equations of the chosen model problem, the geometrically exact beam, are presented in Section 3, along with the pertinent discussion of its invariance. The discrete approximation of the model problem and the corresponding conditions guaranteeing the invariance of the finite element model are discussed in Section 4. In Section 5 we present the results of several numerical simulations. Closing remarks are stated in Section 6.

Section snippets

Some facts about 3D finite rotations

The classical subject of 3D finite rotations, thoroughly studied by mathematicians ever since the classical work of Euler, has attracted the attention of the computational mechanics community only since recent time since the geometrically exact structural mechanics theories have become available. In these kind of problems the finite rotation tensor appears in a natural way when using the nondeformable section hypothesis (e.g. see Reissner [22] or Antman [1]) to describe the motion of a local

Model problem: geometrically exact beam theory and its invariance

We summarize in this section the governing equations of the chosen model problems of 3D geometrically exact beam theory. (For a more thorough discussion of the chosen model we refer to Simo et al. [24], [25], [27], [28] or Ibrahimbegovic et al. [14], [15], among others.) One can derive (e.g. see Reissner [22], Antman [1]) the exact equilibrium equations for statics as $n^{′} + p = 0,$ $m^{′} + ϕ^{′} × n + l = 0,$ where n and m are stress resultants and couples and p and l are externally applied distributed forces and

Discrete approximation of model problem and its invariance

In this section we discuss the computational procedure pertaining to the discrete approximation of the chosen model problem based on the finite element method employing the standard isoparametric interpolations; several implementation details important for this work, such as the consistent linearization in the presence of follower forces and moments and imposed support rotations, are also presented.

The discrete approximation of the presented beam model is constructed by using the beam finite

Numerical simulations

In this section we present the results of a couple of numerical simulations in order to further illustrate and confirm the given theoretical results. In particular, the first example deals with the problem of superposing a large rigid body rotation of support on a given deformed configuration, whereas the second example illustrates the influence of a particular loading program in a closed loading cycle. All the computations are performed by using the computer program FEAP, which is developed at

Conclusions

In trying to achieve the main goal of this work to ensure the frame-invariance of the geometrically exact structural models and their discrete approximation, we were led to clarify several subtle points of finite element implementation. In particular, the proposed quaternion-based rotation update deserves a special attention, since it is capable of eliminating the computationally sensitive step of extracting a quaternion from a given orthogonal tensor. Another implementation detail clarified

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^☆: Dedicated to the memory of Prof. Michael Crisfield.

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On the role of frame-invariance in structural mechanics models at finite rotations☆

Abstract

Introduction

Section snippets

Some facts about 3D finite rotations

Model problem: geometrically exact beam theory and its invariance

Discrete approximation of model problem and its invariance

Numerical simulations

Conclusions

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Comput. Meth. Appl. Mech. Engrg.

Comput. Meth. Appl. Mech. Engrg.

Comput. Meth. Appl. Mech. Engrg.

Comput. Meth. Appl. Mech. Engrg.

Comp. Meth. Appl. Mech. Engrg.

Comput. Meth. Appl. Mech. Engrg.

Comp. Meth. Appl. Mech. Engrg.

Ordinary differential equations of one-dimensional elasticity: foundations of the theories of nonlinearly elastic rods and shells

Arch. Rational Mech. Anal.

Large displacement analysis of three-dimensional beam structures

Int. J. Numer. Meth. Engrg.

Shell theory versus degeneration––a comparison in large rotation finite element analysis

Int. J. Numer. Meth. Engrg.

Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation

Proc. R. Soc. Lond. A