Dynamic Localization Phenomena in Elasticity, Acoustics and Electromagnetism

260 Pages·2013·4.52 MB·English

by  Richard V. Craster

CISM International Centre for Mechanical Sciences 547
Courses and Lectures
Richard V. Craster
Julius Kaplunov
Editors
Dynamic Localization
Phenomena
in Elasticity,
Acoustics and
Electromagnetism
International Centre
for Mechanical Sciences
CISM Courses and Lectures
Series Editors:
The Rectors
Friedrich Pfeiffer - Munich
Franz G. Rammerstorfer - Wien
Elisabeth Guazzelli - Marseille
The Secretary General
Bernhard Schrefl er - Padua
Executive Editor
Paolo Serafi ni - Udine
The series presents lecture notes, monographs, edited works and
proceedings in the fi eld of Mechanics, Engineering, Computer Science
and Applied Mathematics.
Purpose of the series is to make known in the international scientifi c
and technical community results obtained in some of the activities
organized by CISM, the International Centre for Mechanical Sciences.
International Centre for Mechanical Sciences
Courses and Lectures Vol. 547
For further volumes:
www.springer.com/series/76
Richard V. Craster · Julius Kaplunov
Editors
Dynamic Localization
Phenomena in Elasticity,
Acoustics and
Electromagnetism
Editors
Richard V. Craster
Imperial College London
Julius Kaplunov
Keele University United Kingdom
ISSN 0254-1971
ISBN 978-3-7091-1618-0 ISBN 978-3-7091-1619-7 (eBook)
DOI 10.1007/978-3-7091-1619-7
Springer Wien Heidelberg New York Dordrecht London
© CISM, Udine 2013
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole
or part of the material is concerned, specifi cally the rights of translation, reprinting, reuse of
illustrations, recitation, broadcasting, reproduction on microfi lms or in any other physical
way, and transmission or information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed. Ex-
empted from this legal reservation are brief excerpts in connection with reviews or scholarly
analysis or material supplied specifi cally for the purpose of being entered and executed on
a computer system, for exclusive use by the purchaser of the work. Duplication of this pub-
lication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’s location, in its current version, and permission for use must always be obtained
from Springer. Permissions for use may be obtained through RightsLink at the Copyright
Clearance Center. Violations are liable to prosecution under the respective Copyright Law.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specifi c statement, that such names are
exempt from the relevant protective laws and regulations and therefore free for general use.
While the advice and information in this book are believed to be true and accurate at
the date of publication, neither the authors nor the editors nor the publisher can ac-
cept any legal responsibility for any errors or omissions that may be made. The publish-
er makes no warranty, express or implied, with respect to the material contained herein.
All contributions have been typeset by the authors
Printed in Italy
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)
PREFACE
A revolution is currently occurring in physics and engineering
through the manufacture and application of smart structures with de-
signer microstructure. Many of the applications: cloaking, invisibil-
ity, trapped and defect modes, resonances, ultra-refraction, all-angle-
negative refraction, wave guiding along surfaces depend upon subtle
properties ofwave localization and areubiquitous across severalfields:
examples are drawn from elasticity, acoustics and electromagnetism.
There are also numerous applications in more traditional fields such
as the non-destructive evaluation and testing of structures. For ex-
ample, prestresses or coatings on elastic media can be used to ma-
nipulate surface and edge waves, and localized modes arise in coated
or deformed waveguides and are modified by fluid flow. In addition,
surface and guided waves play a key role in crack and flaw detec-
tion and a knowledge of surface and resonant modes is invaluable.
Recent work has highlighted how localized defect modes arise in mi-
crostructured media and new homogenization theories can be used to
create continuum descriptions of micromechanical systems, even at
high frequencies.
The aim of these lecture notes is to introduce an interdisciplinary
audience to a variety of interrelated dynamic localisation phenomena
occurring in elasticity, acoustic and electromagnetism. In particular,
these involve surface and edge waves and also trapped modes localised
near defects, shape changes and the edges of elongated waveguides.
Theeffectsoflayering, prestress, anisotropy, periodicmicrostructures
as well as various multi-field phenomena are addressed with referenc-
ing to underlying industrial problems.
The course covered a wide range of subjects/techniques related
to dynamic localisation phenomena. In particular, these includes
asymptoticandperturbationmethods, modernhomogenizationmethod-
ologies, variational methods, basics of non-linear elasticity, the gen-
eral theory of surface waves, multimodal approach, and advanced ap-
plications of St Venant principle. The objective of the lectures is to
cover the essential and up to date numerical, asymptotic, and analyti-
cal techniques as well as relevant continuum theories that are required
to make progress in, and understand, wave localization and allied ef-
fects. A major focus is on the qualitative physical insight into the
mechanisms of dynamic localisation.
The lectures were chosen to appeal to researchers, primarily but
not exclusively graduate students and postdoctoral researchers, from
Mechanical, Aerospace and Civil Engineering programs and should
naturally also be of interest to Physicists and Applied Mathematicians
and focus on recent work in localized modes and waves that are un-
likely to appear in traditional university graduate courses; the lectures
are also suitable for industrial researchers who encounter resonant or
localised waves. The topics explore the applications in Engineering
and Physics, notably in photonics, showing the interconnections with
acoustics and elasticity that are normally treated independently. Both
theoreticians and experimentalists are expected to gain useful knowl-
edge from these lecture notes.
Richard Craster and Julius Kaplunov
CONTENTS
Modelling microstructured media: periodic systems and
effective media
by R. V. Craster and M. Makwana....................... 1
Multiscale models of electromagnetic and plasmonic meta-
materials
by S. Guenneau.......................................... 19
Explicit models for surface, interfacial and edge waves
by J. Kaplunov and D. Prikazchikov ..................... 73
Elastodynamic end effects in structural mechanics
by B. Karp and D. Durban .............................. 115
Trapped modes and edge resonances in acoustics and elas-
ticity
by V. Pagneux........................................... 181
Surface waves in elastic half spaces coated with crystalline
films
by D. J. Steigmann...................................... 225
Modelling microstructured media: periodic
systems and effective media
R. V. Craster & M. Makwana ∗
Abstract My aim in these lectures is to give a broad overview of
the Mathematics and Physics of perfectly periodic systems, draw-
ing heavily upon the literature of solid-state physics: it is essential
to understand how structure on a micro-scale affects longer scale
macro-scale behaviour and periodic systems are a naturally place
to begin. Periodic systems are, on one hand, quite special and the
constructive interference created by periodicity leads to strong ef-
fectsthatweshallseelater,butontheotherhandmanynaturaland
man-madestructuresexhibit,atleastsome,generalperiodicstruc-
ture. Afterdevelopingthelanguageofperiodicsystemswewillturn
our attention to the development of asymptotic “effective” media
that are posed entirely upon the macro-scale. Importantly we will
developasymptotictheoriesvalidathighfrequencies. Ageneralap-
proach valid for continua, semi-discrete (frame) and fully discrete
(mass-spring)systemswillbedeveloped. Iftimeallowswewillthen
look further into some of the remarkable physics that can be seen
whenwavesmovethroughstructuredmedia: defectstates,all-angle
negative refraction and ultra-refraction.
1 Motivation
Periodic, oralmost periodic, structuressurroundusandareofconsiderable
technological importance. One of the most talked about materials at the
momentisgraphene,analmostperfectmaterialconstructedfromahexago-
nal lattice and graphene has truly remarkable properties some of which are
related to the properties of the waves that pass through it. Many atomistic
systems are remarkably regular in their structure mainly due to the ener-
geticargumentsthatforcethematerialtoadoptsuchregularpatterns. The
attraction of one atom to its neighbour, or neighbours, can be modelled by
discrete mass-string models with the physics of the attraction lumped into
some effective string constants. Historically the study of perfect lattice-like
systems originated in solid-state physics and a huge amount of effort and
∗
DepartmentofMathematics,ImperialCollege,London,SW72AZ,U.K.
R. Craster, J. Kaplunov (Eds.), Dynamic Localization Phenomena in Elasticity, Acoustics
and Electromagnetism, CISM International Centre for Mechanical Sciences,
DOI 10.1007/978-3-7091-1619-7_1, © CISM, Udine 2013
2 R. V. Craster and M. Makwana
Figure 1: Photographs of cellular solids: (a) open-cell polyurethane (b)
closed-cell polyurethane (c) nickel (d) copper (e) zirconia (f) mullite (g)
glass (h) a polyether foam with both open and closed cells. Taken from the
book of Gibson and Ashby [8].
scientific progress was made in that area: This is fortunate as we can then
use that accumulated knowledge! The books of Kittel [11] and Brillouin
[4] are the classical texts in this area and we will draw upon them in these
lectures. It is also notable that considerable effort went into the properties
of atomic systems with defects, i.e extra atoms or disruption/ disorder in
the atomic structure [1].
Shiftingto, yet, anotherarea, thatis, structuralmechanicsanddesigner
materials one finds that the subjects of solid mechanics also abound with
structured media. Cellular solids, engineering foams, or panels, created
from honeycomb lattices are popular in industry for their lightweight, but
strong, properties. A typical range of engineering foams are shown in Fig.
1, taken from [8], and although not perfectly regular, they still retain some
periodic and regular structure. Once again waves passing through such a