Wave Propagation in Linear and Nonlinear Periodic Media: Analysis and Applications

332 Pages·2012·6.167 MB·English

by  A. B. Movchan, M. Brun, N. V. Movchan (auth.), Francesco Romeo, Massimo Ruzzene (eds.)

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PREFACE
Periodicstructuralconfigurationsareubiquitous: manyheterogeneous
structures and materials, both man-made and naturally occurring,
feature geometry, micro-structural and/or materials properties that
varyperiodicallyinspace. Theclassesofperiodicmaterialsandstruc-
tures span a wide range of length scales, and a broad range of applica-
tions. Periodic trusses, periodically stiffened plates, shells and beam-
like assemblies can be found for example in many civil, aerospace,
mechanical and ship constructions. Their introduction is mostly mo-
tivated by structural strength and weight requirements, however recent
studies have shown how the periodicity can be exploited to attenuate,
isolate and localize vibrations. Such studies explore the unique ability
of periodic assemblies to impede the propagation of elastic waves over
specified frequency bands, within which strong attenuation of vibra-
tion and radiated noise can be achieved. The attenuation levels that
can be obtained through tailored structural periodicity far exceed the
performance of most energy dissipation and damping mechanisms.
For this reason, passive, active and hybrid periodic structural con-
figurations are being proposed for the reduction of vibration trans-
mission and structure-born as well as airborne noise. In addition,
the understanding of the dynamics of periodic structures is essential
for the analysis of bladed disc assemblies which are found in turbo
machinery and in turbines for energy generation where failure mech-
anisms due to localization phenomena may occur. At much smaller
scales, extensive research is being devoted to the analysis and design
of phononic metamaterials for a variety of applications. Phononic
metamaterials are essentially periodic structural configurations ob-
tained through composite designs featuring periodic modulations of
mass and stiffness properties, or elastic lattice structures. Gigahertz
communication devices, such as mobile phones, use phononic-based
systems for their low-power filtering characteristics. Many sensing
devices based on resonators, acoustic logic ports, and surface acous-
tic wave-based filters rely on the unique band gap characteristics of
periodic phononic materials. These properties are associated with
the destructive and constructive interference of acoustic waves orig-
inating at the periodic interfaces, which produce frequency band of
strong attenuation of acoustic waves (band gaps). Depending on the
inclusions, geometry, and elastic properties, one can design for spe-
cific band gaps. In photonic crystals, periodic modulations of the
dielectric properties of a medium allow guiding, focusing and steer-
ing of electromagnetic waves. Properties modulations and engineered
anisotropy in heterogeneous media can also produce negative refrac-
tive indexes, both in photonics as well as in phononic metamaterials,
which lead to super-lensing or super-focusing characteristics. Disper-
sion of waves in media with boundaries or built-in micro-structure
is a fundamental phenomenon, which occurs in problems of acous-
tics, models of water waves, simple systems involving onedimensional
harmonic oscillators, as well as complex elastic systems. Moreover,
enhanced transmission based on formation of defect modes within an
interface plays a crucial role in a wide range of practical applications
involving filtering and polarisation of waves of different physical ori-
gins. Other potential implications of the acoustic wave guiding tech-
nology include active sensing of structural integrity, smart sensing
of environment, dissipation of high frequency modes of vibration to
enhance vehicle performance or stealth, as well as applications to the
medical field for sensing or diagnostic applications. Complex dynam-
ical phenomena may be encountered in nonlinear periodic media such
as the existence of energy- depended nonlinear propagation regions.
Depending on substructure coupling, distinctly nonlinear dynamical
phenomena can arise, such as wave localization and solitary waves.
Potential applications of such phenomena are currently investigated
in the MEMS context, where arrays of coupled micro/nanoresonators
have been recently proposed. The CISM course ”Wave propagation
in linear and nonlinear periodic media: analysis and applications”
was an opportunity to combine the material issued from such a vari-
ety of applications aiming to present both the theoretical background
and an overview of the state-of-the art in wave propagation in lin-
ear and nonlinear periodic media in a consistent lecture format. The
course is intended for doctoral and postdoctoral researchers in civil
and mechanical engineering, applied mathematics and physics, aca-
demic and industrial researchers, which are interested in conducting
research in the topic. The opening chapter, by A.B. Movchan, M.
Brun and N.V. Movchan, gives an overview of some mathematical
models of wave propagation in structured media, with the emphasis on
dispersion and enhanced transmission through structured interfaces.
Chapter 2, by L. Airoldi, M. Senesi and M. Ruzzene, presents two
examples of internally resonating metamaterials whose behavior is
based on multi-field coupling. The first part of the chapter is devoted
to the analysis of a one-dimensional waveguide with a periodic array
of shunted piezoelectric patches. The second part presents the study of
piezoelectric superlattices as an additional example of an internally
resonant metamaterial. Topology optimization relevant to problems
in vibration and wave propagation is then addressed in Chapter 3 by
J. S. Jensen; in this chapter the steady-state optimization procedure
for dynamical systems is extended to a nonlinear wave propagation
problem and then the topology optimization procedure is applied to
transient dynamic simulations for which the optimized material dis-
tributions may vary in time. In Chapter 4, by F. Romeo, the dynamic
behaviourofcontinuousanddiscretemodelsofbothlinearandnonlin-
ear periodic mechanical systems are dealt with by means of maps. At
first, linear problems consisting of general multi-coupled periodic sys-
tems are presented and they are handled with linear maps, namely the
transfer matrices of single units. Afterwards, a perturbation method
is applied to the transfer matrix of a chain of continuous nonlinear
beams while nonlinear maps are considered to address chains of non-
linear oscillators. In the closing chapter, by A. Vakakis, analytical
methodologies for analyzing waves in weakly or strongly nonlinear pe-
riodic media are discussed. The generation of spatial chaos in ordered
granular media, which are a special class of spatially periodic, highly
discontinuous and strongly nonlinear media is eventually highlighted.
CONTENTS
Waves and Defect Modes in Structured Media
by A.B. Movchan, M. Brun, N.V. Movchan.............. 1
Piezoelectric Superlattices and Shunted Periodic Arrays as
Tunable Periodic Structures and Metamaterials
by L. Airoldi, M. Senesi and M. Ruzzene................. 33
Topology Optimization
by J.S. Jensen........................................... 109
Map-based Approaches for Periodic Structures
by F. Romeo............................................. 161
Methodologies for Nonlinear Periodic Media
by A. Vakakis............................................ 257
Waves and defect modes in structured media
A.B. Movchana, M. Brunb, N.V. Movchana
a DepartmentofMathematical Sciences,
UniversityofLiverpool,LiverpoolL697ZL,U.K.
b DepartmentofStructural Engineering,
UniversityofCagliari,CagliariI-09123,Italy
Abstract. Thepapergivesanoverviewofsomemathematical
modelsofwavepropagationinstructuredmedia,withtheemphasis
on dispersion and enhanced transmission through structured inter-
faces. We begin with conventional definitions and classical models
of dispersive waves, discuss lattice approximations and then show
the results of the recent work on modelling of waves interacting
with a structured stack acting as a polarizer of elastic waves. Spe-
cial attention is given to the resonance modes which may enhance
transmission across the structured stack. The analytical work is
accompanied bynumerical simulations.
0.1 Introduction
Dispersionofwavesinperiodicinhomogeneousmediaisimportantin
many problems of physics, mechanics and engineering. A special mention
shouldbegiventotheoreticalandexperimentalstudiesofphotonicbandgap
materials (see, for example, Yablonovitch (1987, 1993)). Scalar models for
BlochwavesinperiodiclatticesarediscussedinBrillouin(1953),Maradudin
et al. (1963) and Kunin (1975). This theory naturally extends to vector
problems of elasticity. In particular, dispersion of elastic waves in lattice
systems of different types has been analysed in Martinsson and Movchan
(2003), Jensen (2003), Cai et al. (2005).
DynamiclatticeGreen’sfunctionsinthepropagatingfrequencyrange
were analysed in Martin (2006), whereas the stop band Green’s functions
and exponentially localised “defect modes” were studied in Movchan &
Slepyan (2007).
In wavepropagationproblems for structuredmedia, the classicalho-
mogenisationapproacheshavelimitationswhenthewavelengthiscompara-
blewiththecharacteristicsizeofelementsofthestructure,suchasatypical
2 A.B. Movchan, M. Brun, N.V. Movchan
size of the elementary cell. In particular, this appears to be important in
theanalysisofwaveinteractionwithfinitewidthstructuredinterfaces. Sev-
eral classes of non-local structured interfaces were analysed in Bigoni and
Movchan (2002) in static and dynamic configurations, which also included
semi-discrete systems incorporating both continuous parts and a lattice.
Brunetal. (2010a)haveaddressedanomaliesintransmissionofslowwaves
through dynamic structured interface of finite width. Brun et al. (2010b)
addressed the case of a shear-type structured interface excited by a plane
pressure wave at an oblique incidence, and this paper also shows that the
shear-type interface could serve as a polariser.
In the present paper we aim to discuss the dispersion, polarisation
and filtering of waves from a common perspective, with the emphasis on
the dynamicsofstructuredinterfacesandconnectionsbetweendiscreteand
high-contrast continuous systems. The structure of the paper is as follows.
We begin with the discussion of the fundamental phenomena, such as wave
dispersion, andinSection0.2we showaveryintuitive textbook exampleof
dispersive waves,derived in the framework of the linear water wave theory.
Section 0.3 includes the classical notion of Bloch-Floquet waves and an ex-
plicitanalyticalstudyofwavedispersioninone-dimensionallatticesystems.
Furthermore, this section also introduces the high-contrast continuous sys-
tems, with the emphasis on the lattice approximations of the dispersion
equations in such systems. An example of the transmission problem for a
one-dimensionalfinitewidthstructuredinterfaceisdiscussedinSection0.5,
where it is shown that an irregularity within the structure of the interface
may lead to an enhanced transmission for waves of certain frequency. A
full vector problem of elasticity is described in Section 0.6 for plane elastic
wavesinteractingwithashear-typestructuredinterface;thefinitewidthin-
terface contains severalparallelelastic bars,whosemotionis preferentialin
the directionalong the interface. This creates aneffect ofpolarization,and
we also describe an enhanced transmissionfor waves of certain frequencies.
0.2 Wave dispersion
Thisintroductorysectiondiscussessomeclassicalexamplesofdisper-
sivewaves,i.e. thewaveswhosespeedsaredifferentfordifferentfrequencies.
Here we refer to the linear water wave theory, which is very well known in
linearlisedmodelsoffluidflow(see,forexample,Billingham&King(2001),
Ockendon et al. (2003)). The second example, included in Section 0.3, is
a one-dimensional lattice system, known from classical texts (see, for ex-
ample, Kittel (1996) and Brillouin (1953)). The dispersion relations are
written in an explicit form, and formation of a stop band is discussed for