Mechanics: From Theory to Computation: Essays in Honor of Juan-Carlos Simo

532 Pages·2000·37.852 MB·English

by  S. S. Antman (auth.)

Mechanics: From Theory to Computation
Springer Science+Business Media, LLC
Mechanics: From Theory to Computation
Essays in Honor of Juan-Carlos Simo
Papers Invited by iaumal of Nanlinear Science Editors
With 62 Illustrations
Springer
Library of Congress Catalogi,tlg-in-Publieation Data
Meehanics : from theory to eomputation : essays in honor of Juan-Carlos Simo.
p.em.
At head of title: Papers invited by Journal of non linear scienee editors.
lncludes bibliographieal referenees.
ISBN 978-1-4612-7059-1 ISBN 978-1-4612-1246-1 (eBook)
DOI 10.1007/978-1-4612-1246-1
1. Meehanics, Applied. 2. Simo, J.-c. (Juan Carlos), 1952-
1. Simo, J.-c. (Juan-Carlos), 1952-11. Journal of nonlinear seienee.
TA350.3 .M43 1999
620.1-de21 98-46020
Printed on acid-free paper.
© 2000 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 2000
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ISBN 978-1-4612-7059-1 SPIN 10698180
Juan-Carlos Simo (1952-1994)
Preface
Starting in 1996, a sequence of articles appeared in the Journal of Nonlinear Science
dedicated to the memory of one of its original editors, Juan-Carlos Simo, Applied Me
chanics, Stanford University. Sadly, Juan-Carlos passed away at an early age in 1994.
We lost a brilliant colleague and a wonderful person.
These articles are collected in the present volume. Many of them are updated and
corrected especially for this occasion. These essays are in areas of scientific interest
of Juan-Carlos, including mechanics (particles, rigid bodies, fluids, elasticity, plastic
ity, etc.), geometry, applied dynamics, and, of course, computation. His interests were
extremely broad-he did not see boundaries between computation, mathematics, me
chanics, and dynamics, and, in that sense, he ideally reflected the spirit of the journal
and many of the most exciting areas of current scientific interest.
Juan-Carlos was one of those select and gifted people who could cross interdisci
plinary boundaries with extremely high quality and productive interactions of lasting
value. His contributions, ranging from concrete engineering problems to fundamental
mathematical theorems in geometric mechanics, are remarkable. In current conferences
as well as in scientific books and articles, and over a wide range of subjects, one frequently
hears how his ideas as well as specific results are often used and quoted-this is one
indication of just how profound and fundamental his work has impacted the community.
On top of his brilliance was a generous and caring soul. He did not engage in those
petty academic arguments we too often see that divert lesser people from productivity
and kindness. However, his standards were high, and those with the intellectual gift and
diligence to rise to them were amply and fairly rewarded.
It is hoped that the articles contained herein honor some of Juan-Carlos' spirit and
achievement.
Contents
Preface Vll
1. E. Marsden
Dynamical Problems for Geometrically Exact Theories of
Nonlinearly Viscoelastic Rods
S.S. Antman
The Limits of Hamiltonian Structures in Three-Dimensional
Elasticity, Shells, and Rods 19
Z. Ge, H.P Kruse, and i.E. Marsden
The Membrane Shell Model in Nonlinear Elasticity:
A Variational Asymptotic Derivation 59
H. Le Dret and A. Raoult
Gravity Waves on the Surface of the Sphere 85
R. de la Llave and P Panayotaros
A Nonlinear Extensible 4-Node Shell Element Based on
Continuum Theory and Assumed Strain Interpolations 107
P Betsch and E. Stein
Multilayer Beams: A Geometrically Exact Formulation 139
L. Vu-Quoc, H. Deng, and I.K. Ebcioglu
Obstructions to Quantization 171
M. 1. Gotay
x Contents
An Impetus-Striction Simulation of the Dynamics of an Elastica 217
D.l. Dichmann and 1.H. Maddocks
A Symplectic Integrator for Riemannian Manifolds 239
B. Leimkuhler and G. W Patrick
Time Integration and Discrete Hamiltonian Systems 257
O. Gonzalez
Problems and Progress in Microswimming 277
1. Koiller, K. Ehlers, and R. Montgomery
Symmetry Methods in Collisionless Many-Body Problems 313
I. Stewart
Mathematical Analysis of Sideband Instabilities with
Application to Rayleigh-Benard Convection 335
A. Mielke
KAM Theory Near Multiplicity One Resonant Surfaces in
Perturbations of A-Priori Stable Hamiltonian Systems 379
M. Rudnev and S. Wiggins
Constrained Euler Buckling 413
G. Domokos, P. Holmes, and B. Royce
Continuity Properties and Global Attractors of Generalized
Semiftows and the Navier-Stokes Equations 447
1.M. Ball
Stacked Lagrange Tops 475
D. Lewis
On the Bifurcation and Stability of Rigidly Rotating Inviscid
Liquid Bridges 515
H.-P. Kruse and 1. Scheurle
Dynamical Problems for Geometrically Exact Theories
of Nonlinearly Viscoelastic Rods
S.S. Antman
Department of Mathematics and Institute for Physical Science and Technology.
University of Maryland, College Park, MD 20742, USA
Received October 5, 1995
Communicated by Jerrold Marsden and Stephen Wiggins
This paper is dedicated to the memory of Juan-Carlos Simo
Summary. This paper surveys recent results and open problems for the equations of mo
tion for geometrically exact theories of nonlinearly viscoelastic and elastic rods. These
rods can deform in space by undergoing not only flexure and torsion, but also extension
and shear. The paper begins with a derivation of the governing equations, which for
viscoelastic rods form a quasilinear system of hyperbolic-parabolic partial differential
equations of high order. It then derives the energy equation and discusses difficulties that
can arise in getting useful energy estimates. The paper next treats constitutive assump
tions precluding total compression. The paper then discusses the curious asymptotic
problems that arise when the inertia of the rod is small relative to that of a rigid body
attached to its end. The paper concludes with discussions of traveling waves and shock
structure, Hopf bifurcation problems, and problems of control.
1. Introduction
The development of theories of rods has been intertwined with the development of the
three-dimensional theory of deformable solids for over 250 years. Although nonlinear
problems for the planar equilibrium of elastic rods were correctly formulated and solved
by Euler [18] in 1744, two hundred years before the solution of three-dimensional equi
librium problems for nonlinearly elastic bodies, it was not until after the appearance of
clean, correct, and simple formulations of three-dimensional continuum mechanics that
such formulations of geometrically exact problems for rods became available. Thus only
recently have the equations of rods attained a form readily accessible to analysis. And
only recently have methods of analysis (and computation-see [32], [33], [34], [35], and
[36]) attained a level capable of handling geometrically exact problems for rods with
nonlinear constitutive equations. It is the purpose of this paper to describe the analysis
of several such dynamical problems for nonlinearly viscoelastic rods.
Journal of Nonlinear Science, Mechanics: From Theory to Computation
© Springer-Verlag New York, Inc. 2000
2 5.5. Antman
Since they have but one independent variable, rod theories are analytically far simpler
than three-dimensional theories. On the other hand, inverse and semi-inverse problems
having but one spatial variable are virtually the only problems of the three-dimensional
theory that admit solutions with readily determined qualitative properties, and these
problems typically have a much simpler structure than do problems for rods. In mechan
ical terms, the complexity of rod problems is due to the presence of nonuniform flexural
effects.
Theories of rods can be derived systematically by regarding rods as (i) three-dimen
sional bodies so constrained that their strains depend on only one spatial variable, with
the governing equations obtained by the theory of material constraints (which gen
eralizes projection methods of numerical analysis like the method of lines), (ii) thin
three-dimensional bodies, with the governing equations obtained as the leading terms
of asymptotic expansions in a thickness parameter, or (iii) intrinsically one-dimensional
bodies, with their governing equations obtained by invoking the standard balance laws of
mechanics. In this paper, we adopt the third approach because it is the simplest and most
direct, method (i) is in complete agreement with it, and method (ii) agrees with it to the
extent permitted by the lev~1 of approximation. See [3], [8], [16], [24], [27], [28], [32],
[33], [34], [35], [36], [44]. Almost all of our methods extend to far more complicated
theories of rods, which are most easily generated by method (i).
Notation
We employ Gibbs notation for vectors and tensors: Vectors, which are elements of
Euclidean 3-space IE3, and vector-valued functions are denoted by lower-case, italic,
bold-face symbols u, v, .... The three vectors {iI, i2, i3} == {i,j, k} are assumed to form
a fixed right-handed orthonormal basis. The dot product of (vectors) u and v is denoted
by u . v. Tensors are linear transformations of IE3 into itself. They are denoted by upper
case, italic, bold-face symbols A, B, .... The value of tensor A at vector v is denoted
A . v (in place of the more usual Av) and the product of A and B is denoted A . B (in place
of the more usual AB). The transpose of A is denoted A *. We write u . A = A * . u. The
dyadic product of vectors a and b is denoted ab (in place of the more usual a 0 b). It is
the tensor defined by (ab) . u = (b . u)a for all u.
Triples of components of vectors are denoted by lower-case, bold-face roman symbols
u, v, .... We use the same notational scheme for operations involving these entities.
Twice-repeated lower-case Latin indices are summed from I to 3 and twice-repeated
lower-case Greek indices are summed from I to 2.
The (Gateaux) differential of u r-+ feu) at v in the direction his -!hf(v + th)lt=o'
When it is linear in h, we denote this differential by ~{ (v)·h or fll(v) ·h. The function J"
is tensor-valued. The partial derivative of a function f with respect to a scalar argument
t is denoted by either J, or at f. Obvious analogs of these notations will also be used. In
particular, af/au is the matrix of partial derivatives ofthe components off with respect to
the components ofu. Iff and g are three-component functions of the triples u and v, then
a( f, g) = (amu af/av)
a(u, v) - ag/au ag/av
is the 6 x 6 matrix consisting of the four indicated 3 x 3 submatrices.