Mathematics of Multiscale Materials

289 Pages·1998·12.562 MB·English

by  Michael Aizenman (auth.), Kenneth M. Golden, Geoffrey R. Grimmett, Richard D. James, Graeme W. Milton, Pabitra N. Sen (eds.)

The IMA Volumes
in Mathematics
and its Applications
Volume 99
Series Editors
Avner Friedman Robert Gulliver
Springer Science+Business Media, LLC
Institute for Mathematics and
its Applications
IMA
The Institute for Mathematics and its Applications was estab
lished by a grant from the National Science Foundation to the University
of Minnesota in 1982. The IMA seeks to encourage the development and
study of fresh mathematical concepts and quest ions of concern to the other
seien ces by bringing together mathematicians and scientists from diverse
fields in an atmosphere that will stimulate discussion and collaboration.
The IMA Volumes are intended to involve the broader scientific com
munity in this process.
Avner Friedman, Director
Robert Gulliver, Associate Director
**********
IMA ANNUAL PROGRAMS
1982-1983 Statistical and Continuum Approaches to Phase Transition
1983-1984 Mathematical Models for the Economics of Decentralized
Resource Allocation
1984-1985 Continuum Physics and Partial Differential Equations
1985-1986 Stochastic Differential Equations and Their Applications
1986-1987 Scientific Computation
1987-1988 Applied Combinatorics
1988-1989 Nonlinear Waves
1989-1990 Dynamical Systems and Their Applications
1990-1991 Phase Transitions and Free Boundaries
1991-1992 Applied Linear Algebra
1992-1993 Control Theory and its Applications
1993-1994 Emerging Applications of Probability
1994-1995 Waves and Scattering
1995-1996 Mathematical Methods in Material Science
1996-1997 Mathematics of High Performance Computing
1997-1998 Emerging Applications of Dynamical Systems
1998-1999 Mathematics in Biology
1999-2000 Reactive Flows and Transport Phenomena
Continued at the back
Kenneth M. Golden Geoffrey R. Gritnmett
Richard D. James Graeme W. Milton
Pabitra N. Sen
Editors
Mathematics of
Multiscale Materials
With 75 Illustrations
Springer
Kenneth M. Golden Geoffrey R. Grimmett
Department of Mathematics Statistical Laboratory
University of Utah University of Cambridge
Salt Lake City, UT 84112, USA Cambridge, CB2 ISB, England
Richard D. James Graeme W. Milton
Department of Aerospace Department of Mathematics
Engineering and Mechanics University of Utah
University of Minnesota Salt Lake City, UT 84112, USA
Minneapolis, MN 55455, USA
Pabitra N. Sen
Schlumberger-Doll Research Center
Old Quarry Road
er
Ridgefield, 06877, USA
Series Editors:
Avner Friedman
Robert Gulliver
Institute for Mathematics and its Applications
University of Minnesota
Minneapolis, MN 55455, USA
Mathematics Subject Classifications (1991): 60K35, 62M4O, 73B27, 73B35, 73B4O,
73K20,73V30,76T05,78A30,82B43,82C43,82D25,82D30
Library of Congress Cataloging-in-Publication Data
Mathematics of multiscaIe materials / Kenneth M. Golden ... [et al.].
p. cm. - (The IMA volumes in mathematics and its
applications ; 99)
Includes bibliographicaI references.
ISBN 978-1-4612-7256-4 ISBN 978-1-4612-1728-2 (eBook)
DOI 10.1007/978-1-4612-1728-2
I. MateriaIs-MathematicaI models. 2. Length measurement.
3. Continuum mechanics. I. Golden, Kenneth M. 11. Series: IMA
volumes in mathematics and its applications; v. 99.
TA405.M395 1998
620.1 'I 'OI5118-dc21 98-18394
Printed on acid-free paper.
\!:) 1998 Springer Seienee+Business Media New YOIk
OriginaIly published by Springer-Verlag New Yo rk, Ine. in 1998
Softcover reprint of the hardcover 1s t edition 1998
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission ofthe publisher (Springer Science+Business Media, LLC), except for brief
excerpts in connection with reviews or scholarly analysis. Use in connection with any form of
information storage and retrievaI, electronic adaptation, computer software, or by similar or
dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if
the former are not especiaIly identified, is not to be taken as a sign !hat such names, as
understood by the Trade Marks and Men:handise Marlcs Act, may accordingly be used freely
byanyone.
Authorization to photocopy items for intemal or personal use, or the internal or persona1 use of
specific clients, is granted by Springer Science+Business Media, LLC provided !hat the
appropriate fee is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers,
MA 01923, USA (Telephone: [phone]), stating the ISBN number, the title ofthe book,
and the first and last page numbers of each article copied. The copyright owner's consent does
not include copying for general distribution, promotion, new WOlXS, or resaie. In these eases,
specific written permission must first be obtained from the publisher.
Production managed by Allan Abrams; manufacturing supervised by Jeffrey Taub.
Camera-ready copy prepared by the IMA.
9 8 7 6 5 432 1
ISBN 978-1-4612-7256-4
FOREWORD
This IMA Volume in Mathematics and its Applications
MATHEMATICS OF MULTISCALE MATERIALS
is based on the combined proceedings of the following workshops: Disor
dered Materials; Interface and Thin Films; Mechanical Response of Ma
terials from Angstroms to Meters; and Phase Transformation, Composite
Materials and Microstructure.
These workshops were integral parts of the 1995-1996 IMA program
on "Mathematical Methods in Material Science." We would like to thank
Kenneth M. Golden, Geoffrey R. Grimmett, Richard D. James, Robert
Kohn, Perry Leo, Daniel Meiron, Graeme W. Milton, Stefan Muller, Tom
Tsakalakos, Pabitra N. Sen, and Adrian Sutton for their excellent work as
organizers of the meetings. We would like to express our further gratitude
to Golden, Grimmett, James, Milton, and Sen for editing the proceedings.
We also take this opportunity to thank the National Science Foun
dation (NSF), the Army Research Office (ARO) and the Office of Naval
Research (ONR), whose financial support made the workshops possible.
A vner Friedman
Robert Gulliver
v
PREFACE
Polycrystalline metals, porous rocks, colloidal suspensions, epitax
ial thin films, rubber, fibre reinforced composites, gels, foams, granular
aggregates, sea ice, shape-memory metals, magnetic materials, electro
rheological fluids, and catalytic materials are all examples of materials
where an understanding of the mathematics on the different length scales
is a key to interpreting their physical behavior. In their analysis of these
media, scientists coming from a multitude of professions have encountered
similar mathematical problems, yet it is rare for researchers in the various
fields to meet.
The 1995-96 program at the Institute for Mathematics and its Ap
plications was devoted to Mathematical Methods in Materials Science.
This program was attended by material scientists, physicists, geologists,
chemists, engineers, and mathematicians, and many stimulating interac
tions emerged between the different groups. Four of the workshops during
the year concerned primarily materials with many scales. These were:
• Disordered Materials, November 13-17, 1995,
• Interfaces and Thin Films, February 5-9, 1996,
• Mechanical Response of Materials from Angstroms to Meters,
September 11-15, 1995,
• Phase Transformation, Composite Materials and Microstructure,
September 18-22, 1995.
The scales treated in these workshops ranged from the atomic to the mi
crostructural to the macroscopic, the microstructures from ordered to ran
dom. Although the styles of the lectures often betrayed the scientific back
ground of the speakers, we soon realised that the theory of multiscale ma
terials transcends inter-disciplinary barriers.
The papers of the present volume have emerged from these meetings;
taken together, they form a compelling and broad account of many aspects
of the science of multiscale materials. In compiling this volume, we have
tried to aid access to particular areas by the non-specialist, thereby further
promoting research across the self-imposed barriers of twentieth century
science. The work described here ranges from the 'purely' theoretical to
the applied, from (for example) the percolation phase transition to the
catalytic oxidation of carbon monoxide.
vii
viii PREFACE
We acknowledge the support of the National Science Foundation
through the Institute for Mathematics and its Applications at the Univer
sity of Minnesota for funding the workshops. We thank Av ner Friedman
and Robert Gulliver who contributed greatly on behalf of the IMA to the
success of the meetings. Finally, we thank Patricia V. Brick for her help in
producing the final version of this volume.
Kenneth M. Golden
Geoffrey R. Grimmett
Richard D. James
Graeme W. Milton
Pabitra N. Sen
CONTENTS
Foreword v
0000000000000000000000000000000000000000000000000000 0 0 0 00 0 000
Preface vii
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Scaling limit for the incipient spanning clusters 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Michael Aizenman
Bounded and unbounded level lines in
two-dimensional random fields 25
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Kenneth So Alexander
Transversely isotropic poroelasticity arising from thin
isotropic layers 37
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
lames Go Berryman
Bounds on the effective elastic properties
of martensitic polycrystals 51
0 0 0 0000000000000000000000000000000000000000
Oscar Po Bruno and Fernando Reitich
Statistical models for fracture 63
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Lucilla de Arcangelis
Normal and anomalous diffusions in random flows 81
0 0 0 0 0 0 0 0000000000000
Albert Fannjiang
Calculating the mechanical properties of materials
from interatomic forces 101
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • • • • • • • • • • • • • • ••
Roger Haydock
Granular media: some new results. 109
0 •••••••• 00.00 ............. 0 ......
Holo Herrmann
Elastic freedom in cellular solids and composite
materials ...................................................... 129
0 • • ••
Roderic Lakes
Weakly nonlinear conductivity and flicker noise
near percolation .............. 155
0 •• 0 ••••••••••••••••••••••••••••••••••
Ohad Levy
ix
x CONTENTS
Fine properties of solutions to conductivity equations
with applications to composites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 179
V. Nesi
Composite sensors and actuators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 209
Robert E. Newnham
Bounding the effective yield behavior of mixtures .................. " 213
Tamara Olson
Upper bounds on electrorheological properties ... " ............. , .... 223
Ping Sheng and Hongru Ma
On spatiotemporal patterns in composite reactive media ............. 231
S. Shvartsman, A.K. Bangia, M. Bar, and I.G. Kevrekidis
Equilibrium shapes of islands in epitaxially strained solid films ...... 255
Brian J. Spencer and J. Tersoff
Numerical simulation of the effective elastic properties
of a class of cell materials.. . .. .. . . .. . . . . . .. .. .. . . . .. .. .. . . . . .. .. . . .. 271
Pierre Suquet and Herve Moulinec
SCALING LIMIT FOR THE
INCIPIENT SPANNING CLUSTERS
MICHAEL AIZENMAN*
Abstract. Scaling limits of critical percolation models show major differences be
tween low and high dimensional models. The article discusses the formulation of the
continuum limit for the former case. A mathematical framework is proposed for the di
rect description of the limiting continuum theory. The resulting structure is expected to
exhibit strict conformal invariance, and facilitate the mathematical discussion of ques
tions related to universality of critical behavior, conformal invariance, and some relations
with a number of field theories.
Key words. Percolation, critical behavior, scaling limit, incipient spanning clusters,
fractal sets, conformal invariance. random fields.
AMS(MOS) subject classifications. 82B43. 82B27. 60D05, 82-02
1. Iniroduction. Incipient percolation clusters have attracted atten
tion as objects of interesting physical and mathematical properties, and
potential for applications. An example of a setup in which they play a
role is an array of conducting elements, placed at random in an insulating
medium, with the density adjusted to be close to the percolation threshold.
In such arrays the current is channelled through fractal-like sets. The con
centration of the current, or stress/strain in other similar setups, may result
in high amplification of non-linear effects. The phenomenon is of techno
logical interest, and plays a role in high-contrast composite materials and
non-linear composites, utilized in thermistors and other devices [1, 2, 3].
Studies of the relevant random geometry have yielded interesting geometric
concepts such as the celebrated (but often misunderstood) Incipient Infi
nite Cluster (IIC). The topic was reviewed from a physics perspective in
an article (Stanley [4]) which appeared in Volume 8 of this series, in the
proceedings of a workshop held at IMA in 1986.
It is somewhat surprising that percolation threshold phenomena are
still a source of delightful and new observations, since the subject seemed
to be reaching its maturity already ten years ago. Nevertheless, the subject
has recently enjoyed renewed attention; in part because it was realized that
some entrenched notions need correction (ref. [5-13]), and in part because
it was realized that the scaling (continuum) limit has interesting proper
ties, e.g., conformal invariance, and its construction presents an interesting
mathematical challenge [14-16, 10].
This article focuses on issues related to the scaling limit of the Incipient
* Departments of Physics and Mathematics, Jadwin Hall, Princeton University,
Princeton, NJ 08544-0708.
© Copyrights rest with the author. Faithful reproduction for non-commercial pur
pose is permitted.
1
K. M. Golden et al. (eds.), Mathematics of Multiscale Materials
© Springer-Verlag New York, Inc. 1998