Complementarity, Duality and Symmetry in Nonlinear Mechanics: Proceedings of the IUTAM Symposium

428 Pages·2004·13.601 MB·English

by  David Y. Gao (auth.)

COMPLEMENTARITY, DUALITY AND
SYMMETRY IN NONLINEAR MECHANICS
Advances in Mechanics and Mathematics
Volume 6
Series Editors:
David Y. Gao
Virginia Polytechnic Institute and State University, USA
RayW.Ogden
University of Glasgow, UK.
Advisory Editors:
1. Ekeland
University of British Columbia, Canada
K.R. Rajagopal
Texas A&M University, USA
W. Yang
Tsinghua University, P.R. China
COMPLEMENTARITY. DUALITY AND
SYMMETRY IN NONLINEAR MECHANICS
Proceedings of the lUTA M Symposium
David Y. Gao
Department of Mathematics
Virginia Polytechnic Institute & State University
Blacksburg, VA 24061, U.SA
Email: [email protected]
SPRINGER-SCIENCE+BUSINESS MEDIA, LLC
....
"
Library of Congress Cataloging-in-Publication
Gao, David Y.
Complementarity, Duality and Symmetry in Nonlinear Mechanics: Proceedings of
the IUTAM Symposium
ISBN 978-94-015-7119-7 ISBN 978-90-481-9577-0 (eBook)
DOI 10.1007/978-90-481-9577-0
Copyright © 2004 by Springer-Science+Business Media New York
Originally published by Kluwer Academic Publishers in 2004
Softcover reprint ofthe hardcover Ist edition 2004
AII rights reserved. No part of this publication may be reproduced, stored in a retrieval
system or transmitted in any form or by any means, electronic, mechanical, photo-copying,
microfilming, recording, or otherwise, without the prior written permission of the publisher,
with the exception of any material supplied specifically for the purpose ofbeing entered and
executed on a computer system, for exclusive use by the purchaser of the work.
Printed on acid-free paper.
Contents
List of Figures ix
Preface xiii
References xli
Mechanics and Materials: Research and Cha11enges in the Twenty-First Century
Ken P. Chong
2
Non-Convex Duality 13
Ivar Ekeland
3
Duality, Complementarity, and Polarity in NonsmoothINonconvex Dynamics 21
David Y. Gao
4
Tri-Dua1ity Theory in Phase Transformations ofFerroelectric Crystals with 67
Random Defects
David Y. Gao, Jie-Fang Li, D. Viehland
5
Mathematical Modeling of the Three-Dimensiona1 Delamination Processes of 85
Laminated Composites
Thomas C. Gasser, Gerhard A. Holzapjel
6
Newton's and Poisson 's Impact Law for the Non-Convex Case ofRe-Entrant Comers 10 1
Christoph Glocker
VI
7
Duality in Kinematic Approaches ofLimit and Shakedown Analysis of Structures 127
Nguyen-Dang Hung, Aimin Yan, Vu Duc Khoi
8
Bifurcation Analysis of Shallow Spherical Shells with Meridionally Nonuniform 149
Loading
Charles G. Lange, Frederic Y.M Wan
9
Duality for Entropy Optimization and Its Applications 167
Xingsi Li, Shaohua Pan
10
Dual Variational Principles for the Free-Boundary Problem of Cavitated Bearing 179
Lubrication
Gao-Lian Liu
11
Finite Dimensional Frictional Contact Quasi-Static Rate and Evolution Problems 191
Revisited
JA.C. Martins, A. Pinto da Costa
12
Minimax Theory, Duality and Applications 209
D. Motreanu
13
Min-Max Duality and Shakedown Theorems in Hardening Plasticity 225
Quoc Son Nguyen
14
A Fluid Problem with Navier-Slip Boundary Conditions 241
Adriana Valentina Busuioc, T. S. Ratiu
15
An Extension ofLimit Aualysis Theorems to Incompressible Material with a 255
Non-Associated Flow Ru1e
J Joachim Telega, Mohammed lljiaj, Scott W Sloan
16
Periodic Soliton Resonances 277
Masayoshi Tajiri
vii
17
Generalized Legendre-Fenchel Transfonnation 289
Claude Val/ee, Mohammed Hjia), Daniele Fortune, Gery de Saxce
18
A Robust Variational Formulation for a Rod Subject to Inequality Constraints 313
G.H.M van der Heijden
19
Computing FEM Solutions ofPlasticity Problems via Nonlinear Mixed Variational 327
Inequalities
Paolo Venini, Roberto Nascimbene
20
Finite Element Duai Analysis in Piezoelectric Crack Estimation 339
Chang-Chun Wu, Zi-Ran Li, Lei Li, G. Yagawa
21
DuaIity and Complementarity in Constrained Mechanical Systems 355
Hiroaki Yoshimura
22
Mixed Energy Method for Solution of Quadratic Programming Problems 375
Zhong Wanxie, Zhang Hongwu
List of Figures
3.1 Nonsmooth function and its smooth Legendre conjugate 24
3.2 Discontinuous constitutive law and continuous in-
verse form 24
3.3 Double-well energy and nonconvex potential 26
3.4 Unilateral buckling beam with concave obstacle 'ljJ(x). 26
3.5 Chaotic bifurcation for pre-bucked extended beam
model 28
3.6 Framework in fully nonlinear Newtonian systems 34
3.7 Structure of geometrically linear system and its polar 44
3.8 Chaos vase: A new phenomenon in chaotic vibration
of the dissipative extended beam model 52
3.9 Dual solution set and bifurcation criteria 57
3.10 Primal and dual solutions for conservative Duffing system 59
3.11 Chaotic solutions, invariant sets, and bifurcation
criterion in dissipative Duffing system 60
3.12 Chaos: numeric al results by two differential numer-
ical methods in MATLAB 61
4.1 Effects of defect on Landau's potential q;(rJ). 70
4.2 Singular elliptic curve of dual solutions for LG equa-
tion (4.8) 76
4.3 Effect of Ginzburg contribution on potential dia-
grams at constant random-field contribution. 78
4.4 Illustration of effect of imperfect ion contribution on
stability of solution at constant Ginzburg/random-
field contribution. 79
4.5 Effect of Ginzburg contribution on potential dia-
grams at constant random-field contribution for var-
ious values of B. 80
x COMPLEMENTARITY, DUALITY AND SYMMETRY
4.6 Bright field images of domain states in PMN-PT.
(a) PMN-PT 60/40 on the FEt side of the MPB,
(b) PMN-PT 65/35, (c) PMN-PT 65/35 modified
with 1 at.% La on the Pb-site, and (d) PMN-PT
65/35 modified with 5 at.% La. 82
5.1 Kinematics of a body separated by a displacement
discontinuity. 88
5.2 Reference geometry, boundary conditions and loading for the 9
dissection analysis of the middle layer of an artery. The colla
gen fibers and the interface zone are schematically indicated.
In order to initialize the crack in the middle of the specimen a
rigid component transmits the load into the strip.
5.3 3D dissection analysis of the middle layer of an artery. Load- 9
displacement response for the (a) SOS, (b) KOS and (c) SKON
formulations. Regular and distorted meshes using 20580, 7500
and 1620 elements are used. Note the stress locking effects
accompanied with the SOS formulation for distorted meshes.
With the SKON formulation and for distorted meshes no
meaningful 3D results could be achieved with the fixed load
step of 0.1 (mm).
5.4 Maximum principal stresses, in (mN/mm2), are plotted onto 9
the deformed configurations during the dissection process. The
3D computation is based on the KOS formulation. Slightly
distorted meshes with 1620 and 7500 tetrahedral elements were
used.
6.1 Gap function 9 and impulsive impact forces A. 104
6.2 Newton's impact law. 106
6.3 Poisson's impact law: Compression and decompression. 110
6.4 The geometry of impacts with global dissipation index. 113
6.5 On the kinematic compatibility of Newton's impact law. 114
6.6 On the difference of Newton's and Poisson's impact law. 117
6.7 Impact at a re-entrant corner. 118
6.8 Non-uniqueness of the impact law at re-entrant corners. 119
6.A.l The cones /Ce, Te and Ne at different points of C. 121
6.A.2 Normal and tangent cone for simple unilateral constraints. 122
6.A.3 Orthogonal pair of cones (R, Rl.), and the cone
71-0
orthogonal to the tangent cone of R at (.). 123
8.1 150
List of Figures xi
11.1 ( a) The (configurat ion and reaction dependent) sets
of admis si bIe right velocities, JC~ (X, r), and admissi
bIe right re act ion rates, JC~(X, r), for a particle cur
rently in contact with zero reaction, pE Pz. (b) The
mutually dual cones JC~(X, r) and gP(JC~(X, r)), for
pE Pz. 197
11.2 Dimensions of the finite element version of the ex-
ample of Klarbring [Klarbring, 1990]. 205
11.3 Three solutions of the rate problem, for the (unde
formed) equilibrium state of the structure in contact
with zero reaction at all contact nodes: (a) rate so
lution involving stick of all contact nodes; (b) rate
solution involving slip towards the left of all contact
nodes; (c) rate solution involving loss of contact and
slip towards the left. Angle {3 = 296.70°; coefficient
of friction J-L = 2. 205
11.4 Regions in the (in, A) plane where the first order
rate problem has three solutions for an equilibrium
state in grazing contact with f-L = 2: (a) Two degree
of-freedom example ({3 E ]296.57°, 315°[); (b) Fi
nite element version with eleven contact nodes ({3 E
]296.57°, 298.09°[). 206
13.1 Cam-clay model of geomaterials: the elastic domain
is limited by a family of ellipses in the stress space
(p x q) and represented by a cone in the force space
(Ao x A' x Ar). 235
13.2 A model for limited kinematic hardening 237
15.1 The Drucker-Prager cone 263
16.1 The sequence of snapshots of the resonant inter-
action between y-periodic soliton and line soliton
with the parameters close to the resonant condi
tion, which follows the KP equation with positive
dispersion. The are a inside the dotted lines in (a)
is shown in figures [9]. 278
16.2 The sequence of snapshots of the resonant inter
act ion between y-periodic soliton and line soliton
with the parameters close to the resonant condition,
which follows the DSI equation [10]. 280
16.3 The sequence of snapshots of the resonant interac-
tion between line soliton and growing-and-decaying
mode which follows the DSI equation [12]. 283