Lecture 13: Strain part 2 - gps.alaska.edu

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Lecture 13: Strain part 2 GEOS 655 Tectonic Geodesy Jeff Freymueller

Transcript of Lecture 13: Strain part 2 - gps.alaska.edu

Lecture13:Strainpart2

GEOS655TectonicGeodesyJeffFreymueller

StrainandRotaAonTensors

•  WedescribedthedeformaAonasthesumoftwotensors,astraintensorandarotaAontensor.

∂u1∂x1

∂u1∂x2

∂u1∂x3

∂u2∂x1

∂u2∂x2

∂u2∂x3

∂u3∂x1

∂u3∂x2

∂u3∂x3

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

∂u1∂x1

12∂u1∂x2

+∂u2∂x1

⎝ ⎜

⎠ ⎟ 12∂u1∂x3

+∂u3∂x1

⎝ ⎜

⎠ ⎟

12∂u1∂x2

+∂u2∂x1

⎝ ⎜

⎠ ⎟

∂u2∂x2

12∂u2∂x3

+∂u3∂x2

⎝ ⎜

⎠ ⎟

12∂u1∂x3

+∂u3∂x1

⎝ ⎜

⎠ ⎟ 12∂u2∂x3

+∂u3∂x2

⎝ ⎜

⎠ ⎟

∂u3∂x3

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

+

0 12∂u1∂x2

−∂u2∂x1

⎝ ⎜

⎠ ⎟ 12∂u1∂x3

−∂u3∂x1

⎝ ⎜

⎠ ⎟

12∂u2∂x1

−∂u1∂x2

⎝ ⎜

⎠ ⎟ 0 1

2∂u2∂x3

−∂u3∂x2

⎝ ⎜

⎠ ⎟

12∂u3∂x1

−∂u1∂x3

⎝ ⎜

⎠ ⎟ 12∂u3∂x2

−∂u2∂x3

⎝ ⎜

⎠ ⎟ 0

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

ui x0 + dx( ) = ui x0( ) +12∂ui∂x j

+∂u j

∂xi

⎝ ⎜ ⎜

⎠ ⎟ ⎟ dx j +

12∂ui∂x j

−∂u j

∂xi

⎝ ⎜ ⎜

⎠ ⎟ ⎟ dx j

ui x0 + dx( ) = ui x0( ) + εijdx j +ω ijdx j

symmetric, strain anti-symmetric, rotation

OK,SoWhatisaTensor,Anyway•  Tensor,nottonsure!è•  Examplesoftensorsofvariousranks:

–  Rank0:scalar–  Rank1:vector–  Rank2:matrix–  AtensorofrankN+1islikeasetoftensorsofrankN,likeyoucanthinkofamatrixasasetofcolumnvectors

•  Thestresstensorwasactuallythefirsttensor--themathemaAcswasdevelopedtodealwithstress.

•  ThemathemaAcaldefiniAonisbasedontransformaAonproperAes.

AndWriAngTheminIndexNotaAon

•  Wecanwritethestraintensor

•  Thestraintensorissymmetric,soonly6componentsareindependent.Symmetryrequiresthat:

•  TherotaAontensorisanA-symmetricandhasonly3independentcomponents:

εij =12∂ui∂x j

+∂u j

∂xi

⎝ ⎜ ⎜

⎠ ⎟ ⎟ =

ε11 ε12 ε13ε21 ε22 ε23ε31 ε32 ε33

⎢ ⎢ ⎢

⎥ ⎥ ⎥

ε21 = ε12ε31 = ε13ε23 = ε32

ω ij =12∂ui∂x j

−∂u j

∂xi

⎝ ⎜ ⎜

⎠ ⎟ ⎟ =

0 ω12 ω13−ω12 0 ω23

−ω13 −ω23 0

⎢ ⎢ ⎢

⎥ ⎥ ⎥

EsAmaAngStrainandRotaAonfromGPSData

•  ThisispreYyeasy.WecanesAmateallofthecomponentsofthestrainandrotaAontensorsdirectlyfromtheGPSdata.–  WecanwriteequaAonsintermsofthe6independentstraintensor

componentsand3independentrotaAontensorcomponents–  OrwecanwriteequaAonsintermsofthe9componentsofthe

displacementgradienttensor

•  WritethemoAons(relaAvetoareferencesiteorreferencelocaAon)intermsofdistancefromreferencesite:

•  Herex0istheposiAonofthereferencelocaAon,anddxisthevectorfromreferencelocaAontowherewehavedata

ui x0 + dx( ) − ui x0( ) = εijdx j +ω ijdx j

TheEquaAonsin2D

•  TheequaAonsbelowarewriYenoutforthe2Dcase(velocityandstrain/rotaAonrate):

– For2Dstrain,wehave4parameters(3strainrates,1rotaAonrate)andweneed≥2siteswithhorizontaldata.

– For3D,wewouldhave9parametersandwouldrequire≥3siteswith3Ddata.

vx =Vx + ˙ ε xxΔx + ˙ ε xyΔy + ˙ ω Δyvy =Vy + ˙ ε xyΔx + ˙ ε yyΔy − ˙ ω Δx

Chen et al. (2004), JGR

VelociAesRelaAvetoEurasia

DisplacementandStrain

•  Displacements(orrates)areacombinaAonofrigidbodytranslaAon,rotaAonandinternaldeformaAon

veastvnorth

⎣ ⎢

⎦ ⎥ =

ve,body

vn,body

⎣ ⎢

⎦ ⎥ +

˙ ε 1112 ˙ ε 12 + ˙ ω ( )

12 ˙ ε 12 − ˙ ω ( ) ˙ ε 22

⎣ ⎢

⎦ ⎥ xy⎡

⎣ ⎢ ⎤

⎦ ⎥

ε = strain tensor components ω = rotation

(x, y) = position v = velocity

UniformStrainRate

0001

1001

00

22

2

1−

+−

= εεε

ε!!

!!

DeformingBlockModel•  BasedonGPSdatafrom44sites•  FourblocksmovingrelaAvetoeachotheronmajorfaults,plusuniformstrain– BlockmoAonsarepredominantlystrike-slip

•  ModelswithspaAalvariaAonsinstraindonotfitsignificantlybeYerthanuniformstrain

•  Modelswithallslipconcentratedonafewfaultsfitworsethanthedeformingblockmodel.

5.9+/-0.7

7.4+/-0.7

4.4+/-1.1

Chenetal.DeformingBlockModel

RotaAonTensor

0 12∂u1∂x2

−∂u2∂x1

⎝ ⎜

⎠ ⎟ 12∂u1∂x3

−∂u3∂x1

⎝ ⎜

⎠ ⎟

12∂u2∂x1

−∂u1∂x2

⎝ ⎜

⎠ ⎟ 0 1

2∂u2∂x3

−∂u3∂x2

⎝ ⎜

⎠ ⎟

12∂u3∂x1

−∂u1∂x3

⎝ ⎜

⎠ ⎟ 12∂u3∂x2

−∂u2∂x3

⎝ ⎜

⎠ ⎟ 0

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

0 12∂u1∂x2

−∂u2∂x1

⎝ ⎜

⎠ ⎟ 12∂u1∂x3

−∂u3∂x1

⎝ ⎜

⎠ ⎟

12∂u2∂x1

−∂u1∂x2

⎝ ⎜

⎠ ⎟ 0 1

2∂u2∂x3

−∂u3∂x2

⎝ ⎜

⎠ ⎟

12∂u3∂x1

−∂u1∂x3

⎝ ⎜

⎠ ⎟ 12∂u3∂x2

−∂u2∂x3

⎝ ⎜

⎠ ⎟ 0

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

= βij( )x − xIn terms of our earlier coordinate rotation matrix:

RotaAonasaVector•  WecanrepresenttherotaAontensorasavector.Why?–  RotaAoncanbedescribedbyavector;thinkangularvelocityvector

– Withonlythreeindependentcomponents,thenumberoftermsaddsup

•  Wecanwriteacrossproductoperatorusingthe“permutaAontensor”(actuallyatensorofrank3,likeamatrixwith3dimensions): Ωk = − 1

2 eijkωij

eijk =e123 = e231 = e312 =1e213 = e321 = e132 = −1otherwise := 0

⎨⎪⎪

⎩⎪⎪

⎬⎪⎪

⎭⎪⎪

MoreonthePermutaAonTensor•  Wecanwritethevectorcrossproductintermsofthis

permutaAontensor

•  Rememberthisrule?

–  ThepermutaAontensorislikeashort-handforthatrule

•  ThepermutaAontensorcanbewriYenintermsoftheKroneckerdelta(thisissomeAmesuseful):

eijkeist = δ jsδkt −δ jtδks

a × b( )k = eijkaib j

Ωandωij

•  WecanwriteΩintermsofu:

Ωk = − 12 eijkω ij = − 1

2 ekijω ij

Ωk = − 12

12 ekij

∂ui∂x j

− 12 ekij

∂u j

∂xi

⎝ ⎜ ⎜

⎠ ⎟ ⎟

Ωk = 12

12 ekji

∂ui∂x j

+ 12 ekij

∂u j

∂xi

⎝ ⎜ ⎜

⎠ ⎟ ⎟

Ωk = 12 ekji

∂ui∂x j

⎝ ⎜ ⎜

⎠ ⎟ ⎟

Ω = 12 ∇ × u( )

Notice the permutation

These two terms are equal. Why?

It is the curl of u

∇ =∂∂x

ˆ i + ∂∂y

ˆ j + ∂∂z

ˆ k

ωijandΩ

•  WecanturnthisaroundandwriteωijintermsofΩ.Startwith:

•  MulAplybothsidesbyemnk

Ωk = − 12 eijkω ij

emnkΩk = − 12 emnkeijkω ij

emnkΩk = − 12 δmiδnj −δmjδni( )ω ij

emnkΩk = − 12 δmiδnjω ij −δmjδniω ij( )

emnkΩk = − 12 ωmn −ωnm( )

emnkΩk = − 12 ωmn +ωmn( )

emnkΩk = −ωmn

ω ij = −eijkΩk

OneFinalManipulaAon

•  Let’sgobacktotheoriginaltermintheTaylorSeriesexpansion:

–  ThismakesitabitmoreclearthatthisisarotaAon.

ui(rot ) =ω ijdx j = −eijkΩkdx j = eikjΩkdx j

u(rot ) =Ω× dx

StrainTensor

∂u1∂x1

12∂u1∂x2

+∂u2∂x1

⎝ ⎜

⎠ ⎟ 12∂u1∂x3

+∂u3∂x1

⎝ ⎜

⎠ ⎟

12∂u1∂x2

+∂u2∂x1

⎝ ⎜

⎠ ⎟

∂u2∂x2

12∂u2∂x3

+∂u3∂x2

⎝ ⎜

⎠ ⎟

12∂u1∂x3

+∂u3∂x1

⎝ ⎜

⎠ ⎟ 12∂u2∂x3

+∂u3∂x2

⎝ ⎜

⎠ ⎟

∂u3∂x3

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Shear Strains Axial Strains

ExamplesofStrains

uniaxial strains

An example Including shear strain

PrincipalAxesofStrain•  Foranystrain,thereisa

coordinatesystemwherethestrainsareuniaxial–noshearstrain.

•  Calledtheprincipalaxesofthestraintensor.–  StrainsinthosedirecAonsare

theprincipalstrains.

•  Principalaxesaretheeigenvectorsofthestraintensorandprincipalstrainsaretheeigenvalues.

ε1

ε2

StraininaParAcularDirecAon•  Let’slookathowwecanusethestraintensortogetaline

lengthchangeinanarbitrarydirecAon.Startwithasimpleexamplein2D:

•  Ifwestartwithacircle,itwillbedeformedintoanellipse.

ε11 00 0⎡

⎣ ⎢

⎦ ⎥ ε11 > 0

StraininaParAcularDirecAon

ε11 00 0⎡

⎣ ⎢

⎦ ⎥ ε11 > 0

r

ux

ux = xε11 = r ⋅ cosθ( )ε11uy = 0

ΔL = ux cosθ = rε11 cos2θ

ΔL /L = ε11 cos2θ

ux r

x

ΔL

θ

θ

StraininaParAcularDirecAon•  ForastrainonlyinthexdirecAon,

•  IfwedothesameforastraininonlyintheydirecAon,wegetsomethingsimilar,

•  Ingeneral,wegetanequaAonforanellipse:

ε11 00 0⎡

⎣ ⎢

⎦ ⎥ ε11 > 0 ΔL /L = ε11 cos

0 00 ε22

⎣ ⎢

⎦ ⎥ ε22 > 0 ΔL /L = ε22 sin

err = ε11 cos2θ + ε22 sin

2θ + ε12 sin2θ

SupposewehavemulAplelines

•  IfwemeasurelinelengthchangesforseverallinesofdifferentorientaAons,θi

•  IfwehavelinesofthreeormoredifferentorientaAons,wecanesAmatethethreehorizontalstrains.

err( )i = ε11 cos2θi + ε22 sin

2θi + ε12 sin2θi

θi

SupposewehavemulAplelines•  ThisisthebasisforesAmaAngstrainfromEDMnetworks

•  Thelinesdon’thavetohavethesameorigin.Ijustdrewitthatwayforconvenience.

•  ThemorelinesofdifferentorientaAons,thebeYertheesAmatebecomes

err( )i = ε11 cos2θi + ε22 sin

2θi + ε12 sin2θi

ChenandFreymueller(2002)

•  Evaluatedstrainratesforfourdifferentnear-faultEDMnetworksalongtheSanAndreasfault

•  Allsitesweresoclosetothefaultweassumeuniformstrainwithinthenetwork

EsAmatedStrainRateTensors

EsAmaAngStrainfromLineLengths•  Obviously,whatwelookedatisa2Dproblem.Weassumed

thestateofplanestrain,inwhichalldeformaAonishorizontal.

•  LinelengthsorlinelengthchangesareinvariantunderrotaAons,sowecanlearnnothingabouttherotaAontensorfromthem–  ButwealsocanesAmatestrainwithoutworryingaboutrotaAon

•  Wehavetoassumeuniformstrain,whichmightbethewrongassumpAon.

•  AnalternaAvemightbetoassumesimpleshear(noaxialstrains),suchasforglacialfloworsomestrike-slipfaultmoAonproblems.

StrainandAngleChanges

•  Ifwestrainanobject,lineswithintheobjectwill,ingeneral,rotate:

•  Butauniformstrain(uniformscalingupordowninsize)doesnotrotateanylines.Sothatmeanswecan’tesAmateallthreecomponentsofthestraintensorfromanglechanges.

EsAmaAngStrainfromAngleChanges

•  Wecan’tesAmatethedilataAon(Δ).Rewriteourthreehorizontalstrainsthisway:

•  Thegammasarecalledtheengineeringshearstrains(theepsilonsarethetensorstrains).Thefactorof2isfromhistoricalpracAce.–  Olderpapersooenusethegammas.Newerworkusuallyusestensorstrains.Becarefulofthefactorof2!

Δ = ε11 + ε22γ1 = ε11 −ε22γ 2 = 2ε12

Dilatation Pure shear with N-S contraction, E-W extension Pure shear with NW-SE contraction, NE-SW extension (or simple shear with displacement along x-axis)

IllustraAonsofEngineeringShears

γ1 = ε11 −ε22 > 0

γ 2 = 2ε12 > 0

WhatCanBeDeterminedFromtheData?

•  Thereisagenerallessonhere:–  YoumightwanttoknowallcomponentsofthestrainandrotaAontensors.

– Whenyourdataonlydeterminesomepartsofyourparameters,itmakessensetore-parameterizetheproblemsothatyoudirectlyesAmatewhatyourdatacandetermine

•  ThealternaAveistoputpossiblyarbitraryconstraintsonyourparameters

–  ThediscussionofwhatyoucanlearnaboutaplaterotaAonfromasinglesitewasanotherexample.

– Whatwearedoingisbreakingupourparametersintotwosets(determinedandimpossibletodetermine)thatare(ideally)orthogonalinsomeway.

Moreexamplesofthis

•  IfyouhaveGPSdisplacement/velocitydata,youcandetermineallcomponentsofthedisplacementgradienttensor(orstrain+rotaAon)

•  Ifyouhavelinelengthchanges,youcandetermineallcomponentsofstrain,butnothingaboutrotaAon–  Sodon’ttrytoesAmatethedisplacementgradienttensor;esAmatethestraintensorinstead

•  Ifyouhaveanglechanges,youcan’tdeterminerotaAonordilataAon–  Sodon’ttrytoesAmatethestraintensor;esAmatethegammasinstead.

AxisofMaximumShear

•  Oneotherthingyoucangetoutofthegammasistheaxisofmaximumshear

–  Ifγ1=0,themaximumshearisalongthex-axis.–  Ifγ2=0,themaximumshearis45°fromthex-axis.€

tan2Θs = γ1 γ 2Θs = 1

2 tan−1 γ1 γ 2( )

γ = γ12 + γ 2

2 Magnitude of maximum shear

(QuanAtaAve)RotaAonofaLineSegment

•  DefinetherotaAonvectorofthelinebytheequaAonatleo.InindexnotaAonitis:

•  Becauseξi=xi+ui,

•  Note:curlydandstraightdarenotthesame(parAalvs.totalderivaAves).

•  AlsothemagnitudesofdxanddξareequalforapurerotaAon.

x+dx

x

ξ+dξ

ξ

θ

Θ =dx × dξdx ⋅ dξ

Θ = sinθ ≈ θ

Define the rotation vector Θ€

Θ i =eijkdx jdξkdxndxn

dξk =∂ξk∂xm

dxm = δkm +∂uk∂xm

⎝ ⎜

⎠ ⎟ dxm

Total vs. Partial Derivative •  There is a subtlety to partial derivatives that you might or

might not know. •  Assume a function of space and time, f(x,y,t), where the

variables x and y also depend on time. –  Example: A function that depends on the angle of the sun, which

depends on your position (as a function of time if you are moving), and with time directly

–  The function may have a direct dependence on time, and an indirect dependence that comes from x(t) and y(t).

•  The total derivative is

dfdt

=∂f∂t

+∂f∂x

dxdt

+∂f∂y

dydt

(QuanAtaAve)RotaAonofaLineSegment

•  DefinetherotaAonvectorofthelinebytheequaAonatleo.InindexnotaAonitis:

•  Becauseξi=xi+ui,

•  Note:curlydandstraightdarenotthesame(parAalvs.totalderivaAves).

•  AlsothemagnitudesofdxanddξareequalforapurerotaAon.

x+dx

x

ξ+dξ

ξ

θ

Θ =dx × dξdx ⋅ dξ

Θ = sinθ ≈ θ

Define the rotation vector Θ€

Θ i =eijkdx jdξkdxndxn

dξk =∂ξk∂xm

dxm = δkm +∂uk∂xm

⎝ ⎜

⎠ ⎟ dxm

RotaAonofLineSegment•  SubsAtuAngthisingives:

Θ i =1

dxndxn

eijkdx j δkm +∂uk

∂xm

⎝ ⎜

⎠ ⎟

⎣ ⎢

⎦ ⎥ dxm

Θ i =1

dxndxn

eijkdx j δkmdxm +∂uk

∂xm

dxm

⎝ ⎜

⎠ ⎟

⎣ ⎢

⎦ ⎥

Θ i =1

dxndxn

eijkdx jdxk + eijkdx j∂uk

∂xm

dxm

⎣ ⎢

⎦ ⎥

Θ i =dx jdxm

dxndxn

0 + eijk εkm +ωkm( )[ ]

Θ i = eijk εkm +ωkm( )dˆ x jdˆ x m

This term is just x x x = 0 This term is the velocity gradient tensor, or strain + rotation

These are just unit vectors

WhatDoesThisMean?•  Here’stheequaAon:

•  ThisequaAontellsusthatalinesegmentcanrotatebecauseofeitherastrainorarotaAon.–  TherotaAonpartisobvious–  Thestrainpartwesawearlier(graphically).Dependingonthevalues

ofthestraincomponents,therotaAonmightbezeroornon-zero.–  Also,thetwotermsmightcancelout!

Θ i = eijk εkm +ωkm( )dˆ x jdˆ x m