Dynamic Localization Phenomena in Elasticity, Acoustics and Electromagnetism PDF
Preview Dynamic Localization Phenomena in Elasticity, Acoustics and Electromagnetism
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
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