periodic laces: bandgaps and direconality

M.Ruzzene–Lecture4

PeriodicLa4ces:BandgapsandDirec;onality

Massimo RuzzeneD.GuggenheimSchoolofAerospaceEngineering

G.WoodruffSchoolofMechanicalEngineering

GeorgiaIns;tuteofTechnology

Atlanta,GA

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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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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:

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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.

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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.

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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

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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

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Acous;cWaveguide

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θ

R

L t

tc

Freq

uency

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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).

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Acous;cfocusing

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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