Lecture 13: Strain part 2 - gps.alaska.edu
Transcript of Lecture 13: Strain part 2 - gps.alaska.edu
StrainandRotaAonTensors
• WedescribedthedeformaAonasthesumoftwotensors,astraintensorandarotaAontensor.
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∂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
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
DeformingBlockModel• BasedonGPSdatafrom44sites• FourblocksmovingrelaAvetoeachotheronmajorfaults,plusuniformstrain– BlockmoAonsarepredominantlystrike-slip
• ModelswithspaAalvariaAonsinstraindonotfitsignificantlybeYerthanuniformstrain
• Modelswithallslipconcentratedonafewfaultsfitworsethanthedeformingblockmodel.
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
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
2θ
€
0 00 ε22
⎡
⎣ ⎢
⎤
⎦ ⎥ ε22 > 0 ΔL /L = ε22 sin
2θ
€
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
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)
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