periodic laces: bandgaps and direconality
TRANSCRIPT
M.Ruzzene–Lecture4
PeriodicLa4ces:BandgapsandDirec;onality
Massimo RuzzeneD.GuggenheimSchoolofAerospaceEngineering
G.WoodruffSchoolofMechanicalEngineering
GeorgiaIns;tuteofTechnology
Atlanta,GA
6/7/10 1
WavePropaga+oninLinearandNonlinearPeriodicMedia:AnalysisandApplica+ons
June21‐25,2010
M.Ruzzene–Lecture4
Outline
• Analysisof2Dperiodicla4ces:– Generalmappinginla4cespace
– Indirectla4ceandFirstBrillouinzones
• Structural/Phononicla4ces:– Overview– AgeneralFE‐basedapproachfordispersionanalysis– Hexagonalla4ces
– Chiralla4ces
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M.Ruzzene–Lecture4
2DHoneycombgrid 2DRe‐entrantgrid
StructuralLa4ces
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Kagomegrid TriangularlaHce
ChirallaHce
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Structuralla4ces
• Objec;ve:inves;ga;onofuniqueproper;esofla4ces:– Beamingandfocusingofacous;cwaves
– Bandgaps– Frequencydependentdirec;onality– EM/Acous;ccoupling
– Acous;cnega;verefrac;on
• Mo;va;ons:– La4ces’behaviorisdefinedbygeometry,topologyandmaterialarrangement
• Richdesignspace– Proper;esrelevanttostressmi;ga;on/redirec;on
– Mul;func;onalcharacteris;cs(thermal,mechanical,EM)
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M.Ruzzene–Lecture4
Introduc;on
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• Objec;ve:analysisofgeneric2Dperiodicdomains
• Analysisrequires:– Ageneraliza;onofBlochtheorem
– Defini;onof“DirectLa4ce”and“ReciprocalLa4ce”space;
– Procedurefortheiden;fica;onoftheFirstBrillouinzone.
• Posi;onofagenericpointPincelln,m:
M.Ruzzene–Lecture4
La4cesandunitcells
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HEXAGONALLATTICE RE‐ENTRANTLATTICE CHIRALLATTICE
• Eachunitcellcanbeinscribedinapolygon• Ageneralrepresenta;onforthela4ce:
:La4cevectors
Generalanalysisframeworkisavailable
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Unitcellmapping
:La4cevectors
La4ceMatrix
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ReciprocalLa4ce
Defineasetofvectorssuchthat:
:Reciprocalla4cespace
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Reciprocalla4ce
• Expressthewavevectorinthereciprocalla4ce:
• Accordingly:
• Hence
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Reciprocalla4ce
• Finally:
• where:
• Replacein(*)
• Thesolu;onisperiodicfor:
Propaga;onvector
(*)
=1
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Example:1Dla4ce
• La4cevectors:
• Inversela4cevectors:
FirstBrillouinzone
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Example:cartesianla4ce• La4cevectors:
• Inversela4cevectors:
Unit cell
Hence:
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FirstBrillouinzone
IrreducibleBrillouinZone
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Generalprocedure
• Iden;fica;onoftheFirstBrillouinzone:– Givenla4cevectors,findreciprocalla4cevectors
– Connectoriginwithneighboringpointsinreciprocalla4ce
O
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Generalprocedure
O O
Constructbisectorsonthelines
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Hexagonalandre‐entrantla4ces
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Unitcellconfigura;onandla4cevectors
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FirstBrillouinZonesandUnitCellFEmodeling
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Banddiagrams
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Banddiagrams
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Banddiagrams
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Equivalentproper;es
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BeaminginHoneycombla4ces
0.5 1 1.5 2 2.5 3 0.5
1 1.5
2 2.5
3
8.66 17.3
26 34.6
52 60.6
69.3
77.9 95.3
104
130 147
0 0.5 1 1.5 2 2.5 3 0
0.5
1
1.5
2
2.5
3
w=44 rad/s
w=101 rad/s
Contoursof1stDispersionsurface
Re‐entrantla4cesaremoredirec;onalduetotheirhigherANISOTROPY
M.Ruzzene–Lecture4
Deformedshapesat80rad/s
Ruzzene M., Soranna F., Scarpa F., 2003 “Wave Beaming Effects in Bi‐DimensionalCellularStructures.”,SmartMaterialsandStructures,12,pp.363‐372.
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Beaminginhoneycombla4ces
q
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Bandgapsinhexagonalla4ces
Dispersionsurfaces
Band Gap
RuzzeneM.,ScarpaF.,2005“Direc;onalandBand‐GapBehaviorofAuxe;cLa4ces.”PhysicaStatusSolidiB,242,No.3,665–680.
M.Ruzzene–Lecture4
excitation 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10 4 10 -25
10 -20
10 -15
10 -10
10 -5
10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10 4 10 -30 10 -25 10 -20 10 -15 10 -10 10 -5 10 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4
10 -12
10 -10
10 -8
10 -6
10 -4
10 -2 Band gap
1
2 3
Partial band-gap
Auxetic (q=-30o)
M.Ruzzene–Lecture4
Chiralla4ces
• Chiralla4cesare:– Periodic– Richmicrostructure– Largedesignflexibility
• Chiralla4cesuniqueproper;es:– In‐planenega;vePoisson’sra;o~‐1– Largedeforma;oncapabili;es
– Mechanicalproper;esstronglyinfluencedbyafewtopologyparameters
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r
θ
R
L t
Prall,D.,Lakes,R.S.,“Proper;esofachiralhoneycombwithPoisson’sra;oof‐1,“Interna+onalJournalofMechanicalSciences,39(3),1997,pp.305‐314.
M.Ruzzene–Lecture46/7/10 27
Theore;calBackground
harmonic wave propagation
Bloch Theorem
First Brillouin Zone
Spadoni,A.,Ruzzene,M.,Gonella,S.,Scarpa,F.,“PhononicProper;esofHexagonalChiralLa4ce”WaveMo+on,2009.46(7):p.435‐450
M.Ruzzene–Lecture4
Theore;calBackground
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Bloch Theorem
Periodicity Periodicity + equilibrium
FE representation
Periodicity + equilibrium + harmonic solution E.V.P.
dispersion surfaces
dispersion relations to study: 1. band gaps;
2. directionality.
M.Ruzzene–Lecture4
DispersionSurfaces&BandStructure
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M.Ruzzene–Lecture4Spadoni,A.,Ruzzene,M.,Gonella,S.,Scarpa,F.,“PhononicProper;esofHexagonalChiralLa4ce”WaveMo+on,2009.46(7):p.435‐450
BanddiagramsandbandgapsFreq
uency
Freq
uency
Wavenumber Wavenumber
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Bangaps
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Bandgaps&experimentalvalida;on
0 500 1000 1500 2000 25000102030405060708090
100
w
O
A
B
O
FRFfromfinite‐elementsimula;on
0 500 1000 1500 2000 250010-10
10-8
10-6
10-4
10-2
100
FRF
M.Ruzzene–Lecture4
ExperimentalResults
• Experimentalset‐up
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DAQ &
Signal processing Post‐
Processing
Piezo Exciter
MATLAB
Scanninghead(PolytecPSV400M2)
Piezoelectricactuator
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ExperimentalResults
2 4 6 8 10 12 14 16x 10-5
2
4
6
8
10
12
14
16
18
20
FRF
x 103x 103Banddiagram Measuredaveragedspectrum
Gonella S., Spadoni A., Ruzzene M. Scarpa F., “Wave Propaga;on And Band‐Gap Characteris;cs Of Chiral La4ces”,ProceedingsofIDETC/CIE2007ASME2007September4‐7,2007,LasVegas,NV.
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Bandgapmaps
L/R = 0.60
L/R = 0.90
t/tco = 0.2
t/tco = 1
M.Ruzzene–Lecture4
Acous;cWaveguide
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θ
R
L t
tc
Freq
uency
M.Ruzzene–Lecture4
Acous;cWaveguide
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θ
R
L t
tc
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Groupveloci;es
L/R=0.902ndMode
L/R=0.901stMode
Caus;cs
• Caus;cscorrespondtodirec;onsofstrongfocusingoftheacous;cenergy
• Resultofinteferencewithwavespropaga;ngindifferentdirec;ons
• Caus;cshavebeenobservedincrystal
J.P.Wolfe,ImagingPhonons:Acous+cWavePropaga+oninSolids(CambridgeUniversityPress,1998).
M.Ruzzene–Lecture4 38
Acous;cfocusing
M.Ruzzene–Lecture4
Acous;cfocusingthroughbeamingGroupVelocity@500Hz GroupVelocity@2500Hz
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Somenotes
• Nega;veslopeisanindica;onofnega;vemass
• Nega;vemassresultsinnega;veacous;crefrac;on
Posi;verefrac;on Nega;verefrac;on
Focusingeffect
• Modeisalmostflat:verylowspeedofpropaga;on
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SomeNotes• Ques;onsregardingtheequivalentmechanicalproper;esofthe
hexagonalchiralla4cearisefromthedispersionanalysis– Banddiagramforanhexagonalla4ce
K*‐equiv.Bulkmod.G*‐equiv.Shearmod.r*‐rela;vedensity
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SomeNotes
• Thismaybeassociatedtothefactthat• Shearvelocityisnotveryhigh,whichconfutestheno;onthat
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PreviousApproachvs.currentanalysis
• Assump;onsofini;alinves;ga;ons(*):– nodesarerigid;– noaxial/sheardeforma;ons;
– internalforces();– nodesmovealongR;
– slenderligaments;
– smalldeforma;ons.
• Currentanalysis:– Phase1:
• Includeaxialandsheardeforma;ons
– Phase2:• Includenode(circle)deforma;ons
(*) Prall, D., Lakes, R.S., "Properties of a chiral honeycomb with a Poisson's ratio of - 1.” International Journal of Mechanical Sciences, v 39, n 3, 1997, pp. 305-314
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Phase1:axialandsheardeforma;on
equivalent Young’s modulus
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Phase1:axialandsheardeforma;on
previous work
Euler-Bernoulli + axial
Timoshenko + axial
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FEResults:StrainEnergyDistribu;on
strain energy distribution
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MacroFEAnalysis
35 x 59
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Resultsofrefinedanalysis
previous work
Euler-Bernoulli + axial
Refined analysis
Numerical“correc;on”
factor
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Resultsofrefinedanalysis
Poisson’sRa;o
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Comparisonwithotherla4cesKumar, R., McDowell, D. L., “Generalized continuum modeling of 2D periodic cellular solids.” International Journal of Solids and Structures, 41(26). 2004