3.4 Acoustic Waves in Anisotropic Crystals
3.4.1 Stress and Strain Definitions: Elastic Constants
To begin, we write down a generalization of Hooke’s law, F = −Kx, for a three- dimensional,continuousmedium,asfollows:
˜σ = ẳC˜ε, (3.4.1)
thatis,
σij=ả
lm
Cijlmεlm. (3.4.2)
Here,thetermthatcorrespondstotheforceis ˜σ ,whichisa3 × 3matrixcalledthe stress tensor,which playsthe roleof theforce inHooke’s law.It is related tothe forceona surfacebytherelation
±F = ˜σ ãˆnA, (3.4.3)
where ˆn isaunitvectornormaltothesurfaceand A istheareaofthesurface.Sinceweare dealingwithacontinuousmedium,thestresshasunitsofpressure,thatis,forceperunit area.
Figure3.10showshowthestressesapplytoasmallvolumeelement.Thevolumeele- menthereis notthe sameastheunit cell ofalattice; hereweare assumingcontinuum mechanics,inwhichweassumethat avolumeelementismuchlargerthanaunitcellof the underlying lattice, butstill smallcompared to thewavelength of an acoustic wave.
z y
x
xz
xz zz
±Fig.3.10 Stressesonavolumeelement. zz
173 3.4 AcousticWavesinAnisotropicCrystals
Thefirstindexofthestressreferstothefaceofthevolumeelementtowhichtheforceis applied,whilethesecondindexreferstothedirectionoftheforce.Eachforceisassumed tobeaccompaniedbyanequalandoppositeforce,asshowninFigure3.10.Thus,anele- ment σzzofthestresstensorcorrespondstotwoequalandoppositeforcesalongthe z-axis, whichtendstosqueezethecrystal,whileastress σxz correspondstotwoforcesalongthe z-axisbutdisplacedalongthe x-axis,leadingtoatwistingofthevolumeelement.Ifthese forceswereunbalanced,thevolumeelementwouldhaveanettorqueandwouldhaveto spin;thereforeitisnormallyassumedthat σij = σji;forexample,ifthereisa σxy stress, theremustbeanequalandopposite σyxthatcancelsthetorque.Forshortperiodsoftime, however,thetorquecanbeunbalanced,leadingtorotationalmotion.
Thedisplacement ofthemediumin responsetothe stressis another3 × 3matrix, ˜ε, calledthe strain tensor.Thestrainmatrixgivesthefractionalchangeofthedimensionsof avolumeelement,andisthereforeunitless.Relatingthesetwoin(3.4.2)isa3 × 3 × 3 × 3 double tensor ẳC,which consists ofthe crystal elastic constants, which dependonthe springconstantsof themedium.To matchtheunits ofstress, the elasticconstantshave unitsofpressure.
Stressesandstrainscanbecategorizedintwotypes.Thefirstisa hydrostatic stressor strain,whichhasequaltermsalongthediagonal:
˜σ = σ
⎛
⎝ 1 0 0
0 1 0 0 0 1
⎞
⎠ . (3.4.4)
Thiscorresponds to anequal pressurein alldirections, as wouldbeexperienced byan objectimmersed inpressurizedwater. By contrast,a shear stress orstrainmatrixhasa traceofzero.
Thestrainmatrixisdefinedintermsofthedisplacement ±u ofthelocalmedium,which weusedinthespringmodelsofSections3.1.2and3.1.3.Inprinciple,onecouldhavean unbalancedstrain,oftheform
˜ε = τ
⎛
⎝ 0 1 0
0 0 0 0 0 0
⎞
⎠, (3.4.5)
with
εlm= ∂ ul
∂xm. (3.4.6)
Thisisknownasa simple shear.Itcanbedecomposedintothesumofasymmetricand anantisymmetricstrain,
⎛
⎝ 0 1 0
0 0 0 0 0 0
⎞
⎠ =
⎛
⎝ 0 12 0
12 0 0 0 0 0
⎞
⎠ +
⎛
⎝ 0 12 0
−12 0 0
0 0 0
⎞
⎠ . (3.4.7)
Ashearstraincorrespondingtoasymmetricmatrixiscalleda pure shear.Anantisym- metricpureshearimpliesrotationofthemedium,asillustratedinFigure3.11.Sincewe
dux=εxydy
duy=εyxdx
duy=εyxdx dux= –εxydy
y
(a)
(b)
x
y x
±Fig.3.11 Changeofacubicvolumeelementby(a)apureshear,and(b)theantisymmetricpartofasimpleshear.
normallyassume,asdiscussedabove,thatthemediumisirrotational,weenforcesymmetry byusingthedefinition
εlm = 1 2
²∂ul
∂xm+ ∂um
∂xl
³
. (3.4.8)
Thisdefinitionimplies εij= εji,asforthestresses.
A matrixcanhaveatrace ofzeroeither byhaving termsonthediagonalthatsum to zero,forexample,
˜ε = τ
⎛
⎝ 2 0 0
0 −1 0
0 0 −1
⎞
⎠ , (3.4.9)
orbyhavingonlyoff-diagonalterms,forexample,
˜ε = τ
⎛
⎝ 0 1 0
1 0 0 0 0 0
⎞
⎠ . (3.4.10)
Whentheaxesofasystemarerotatedtoadifferentdirection,apureshearmatrixalways continuestohaveatraceofzero.Thegeneralmatrixforarotationaroundtheunitvector ˆu byanangle θ is
175 3.4 AcousticWavesinAnisotropicCrystals
R(θ) =
⎛
⎝ c + (1 − c)u2x (1 − c)uxuy− suz (1 − c)uxuz+ suy (1 − c)uyux + suz c + (1 − c)u2y (1 − c)uyuz− sux
(1 − c)uzux− suy (1 − c)uzuy+ sux c + (1 − c)u2z
⎞
⎠ ,
(3.4.11) where c = cos θ and s = sin θ.Supposethatwehaveapureshearalongthe x-axis,given by(3.4.9).Wecanrotatethisintothe[111]axisbyperforminga35.26◦rotation around the y-axisandthena45◦rotationaroundthe z-axis.Theproductofthesetworotationsis therotationmatrix
R =
⎛
⎜⎜
⎜⎜
⎜⎜
⎝
√1
3 −√12 −√16
√1 3
√1
2 −√16
√1
3 0 à
23
⎞
⎟⎟
⎟⎟
⎟⎟
⎠
. (3.4.12)
Transformingtheshear(3.4.9)bythis,weobtain
R˜εR−1 =
⎛
⎝ 0 1 1
1 0 1 1 1 0
⎞
⎠ . (3.4.13)
Anystressorstraincanalwaysbewrittenasasumofahydrostaticplusashearterm.
Forexample,a uniaxial strain,whichconsistsofacompressionalongonlyoneaxis,can bedecomposedasfollows:
⎛
⎝ 1 0 0
0 0 0 0 0 0
⎞
⎠ = 13
⎛
⎝ 1 0 0
0 1 0 0 0 1
⎞
⎠ + 13
⎛
⎝ 2 0 0
0 −1 0
0 0 −1
⎞
⎠ . (3.4.14)
Therefore, in practice we oftendo notneed torefer tothe full strainmatrices; we can simplytalkofthemagnitudeofthehydrostaticandtheshearstrains.Forapurelyharmonic medium,thatis,onewhichobeysthegeneralizedHooke’slaw(3.4.2),ahydrostaticstress leadstoahydrostaticstrain,whichcorrespondstoavolumechangeoftheunitcell,butno changeofthecrystalsymmetry.2Ontheotherhand,apureshearstraindoes notchange theunitcellvolume,butitdoeschangethecrystalsymmetry.
Exercise3.4.1 Show explicitly that the rotation matrix (3.4.12) corresponds to the two successiverotationsgiveninthetext,thatis, thatitistherotationthat transforms the[100]-axisintothe[111]-axis.
2 As we will discuss in Section 5.4, solid state phase transitions that correspond to a change of crystal symmetry can occur under extremely high hydrostatic stress, due to anharmonic terms in the Hamiltonian.
Table3.1 Elasticconstantmatrixinreducednotationforamedium withcubicsymmetry
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎝
C11 C12 C12 0 0 0
C12 C11 C12 0 0 0
C12 C12 C11 0 0 0
0 0 0 C44 0 0
0 0 0 0 C44 0
0 0 0 0 0 C44
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎠
Becausethe3 ì 3 ì 3 ì 3doubletensor ẳC isanunwieldyfour-dimensionalmatrix,itis standardtouse reduced notationforthestressandstrainasfollows:
ε1= εxx σ1= σxx
ε2= εyy σ2= σyy
ε3= εzz σ3= σzz
ε4= 2εyz σ4= σyz ε5= 2εxz σ5= σxz ε6= 2εxy σ6= σxy.
(3.4.15)
As discussedabove,wedonotneednineterms,because εij = εjiand σij = σji.Note that ε4, ε5,and ε6aremultipliedby2,whilethestressesarenot,totakeintoaccountthat therearetwoterms εijand εjithatcontributeinthematrixmultiplicationtoeachterm σij.
The81possibletermsin ẳC arethereforecutdownto36.Wecanthenwritethegener- alizedHooke’slawintermsofa6 × 6 C-matrixoperatingonsix-componentvectors,as follows:
σI =ả
J
CIJεJ. (3.4.16)
Furthersimplification is possibleif we assumethat the potential energy is quadraticin allpositionvariables,whichimpliesthatthe C-matrix mustbesymmetric,thatis, CIJ = CJI.(SeeExercise3.4.2.)AsdiscussedinSection3.1.1,forlow-amplitudedisplacements fromanenergyminimum,thisapproximationisalwaysvalid.Inthiscase,insteadof36 constants,wehavejust21(= 6 + 5 + 4 + 3 + 2 + 1).
Furtherreductionofthenumberofelasticconstantscanbedonebyknowingthesym- metryofthecrystal.Ingeneral,manyoftheelasticconstantswillbezeroorequaltoother elastic constants. The website that accompanies this book (www.cambridge.org/snoke) givestheformoftheelasticconstant C-matrixforallofthepossiblecrystalsymmetries.
Forexample,Table3.1givestheformoftheelasticconstantmatrixinreducedmatrixfor acubiccrystal.
As seeninthistable, acubiccrystalhasonly three independentelasticconstants.An isotropicmediumhasthesameformoftheelasticconstantmatrixasacubiccrystal,butin additiontheelasticconstantsaresubjecttotheconstraint C44= (C11− C12)/2.Acrystal withnorotationalsymmetries,whichhas21independentelasticconstants,isknownasa triclinic system.
177 3.4 AcousticWavesinAnisotropicCrystals
Often, weknowthestressand wanttofind thestrain. Inthiscase,itis convenientto definethe compliance tensor, S = C−1,thatis,
εI=ả
J
SIJσJ. (3.4.17)
Thenumberofindependentcomplianceconstantsisequaltothenumber ofindependent elasticconstants.
Exercise3.4.2 Prove the statement above, that if the potential energy of the system is quadratic,oftheform
U V = 1
2
ả
ij
σijεij, (3.4.18)
thentheelasticconstantmatrix Cijlmmustbesymmetricwithrespecttointerchang- ingthefirsttwoindiceswiththelasttwoindices.
Exercise3.4.3 (a)Determinethecompliancetensor S intermsoftheelementsofthecubic, orisotropic,elasticconstantmatrix C giveninTable3.1.
(b)Ifauniaxialstressisappliedalongthe[110]axis,whatstrainiscreated?
(c) Show that for anisotropic medium, if the direction of a uniaxial stress is originallyalongthe x-axis,creatingastrain,andthenthedirectionofthestressis rotatedaboutthe y-axisbyanyangle θ,thestraincreated,relativetothenewuniaxial stressaxis,isstillthesame.
Exercise3.4.4 Verifythestatementabove,thatahydrostaticstressleadstoahydrostatic strain,usingthecubic(butnotnecessarilyisotropic)compliancetensordeducedin part(a)ofthepriorexercise.Isitalsothecasethatapureshearstressleadstoapure shearstrain?
Theelasticconstantsinthe C-matrixarerelatedtotwosimple“engineering”constants definedforanisotropicmedium. Young’s modulusgivesthestrainproducedbyagiven stressinthesamedirection,thatis,thefractionalincreaseinthelengthofabeamduetoa tensilestress,asillustratedinFigure3.12(a).Foranisotropicmedium,thisisequalto
Y =σxx
εxx = C11. (3.4.19)
Fortypical solids, this isonthe orderof1011dynes/cm2 = 100kbar.In otherwords,a pressureof1kbaristypicallyneededtocreateastrainof1%,thatis,afractionalchange ofthevolumeof1%.
When a solid is stressed, it does not experience strain onlyin the same direction as thestress.AsillustratedinFigure3.12(b),acompressivestresswillalsocauseexpansion ofthe solid in the perpendicular directions as the solidis squeezedoutward. Poisson’s ratio isdefinedastheratioofthetransversestrain(fractionalexpansion)tothefractional compression,whichforanisotropicmediumisequalto
ν =−ε
ε´ =−S12
S11 = C12
C11+ C12. (3.4.20)
Thisisaunitlessratio,oftheorderof0.3fortypicalsolids.
(a) (b) 1 + 1 +
1 – ´
±Fig.3.12 (a)Young’smodulusdefinition.(b)Poisson’sratiodefinition.
Sinceanisotropicmediumhasonlytwoindependentelasticconstants,thesevaluestell usallweneedtoknow abouttheelasticdeformationsofanisotropic medium.Onecan alsodefine Lamộ coefficientsλ and à,whicharerelated totheelasticconstantsbythe following:
C11 = λ + 2à C12 = λ
C44 = à. (3.4.21)
Last,the bulk modulusisoftendefined,whichgivespressureneededperfractionalchange involume:
B = λ +2 3à = 1
3(C11+ 2C12). (3.4.22)
The Lamộcoefficient à is sometimes called the shear modulus.Inmedia thatwill not supportshear,suchaswater, à = 0.Allofthesedifferenttermsarejustdifferentnotations toreflectthebasicfactthatanisotropicmediumhasonlytwoindependentparametersfor itslinearelasticresponse.