變形巖石應變分析

第四章

變形巖石應變分析基礎(chǔ)1本章主要內(nèi)容

變形、位移和應變的概念旋轉(zhuǎn)應變與非旋轉(zhuǎn)應變遞進變形、全量應變與增量應變巖石的變形階段2變形和位移

變形、位移和應變的概念

當?shù)貧ぶ袔r石體受到應力作用后，其內(nèi)部各質(zhì)點經(jīng)受了一系列的位移，從而使巖石體的初始形狀、方位或位置發(fā)生了改變，這種改變就稱為變形。

變形3

位移物體內(nèi)部各質(zhì)點的位移是通過其初始位置和終止位置的變化來表示，質(zhì)點的初始位置和終止位置的連線叫位移矢量。

從幾何學角度來看，研究物體的變形需要比較物體內(nèi)各質(zhì)點的位置在變形前后的相對變化，為此，我們引入位移的概念。4平移旋轉(zhuǎn)(虛線為可能的路徑）形變體變P0P1P0P0P0P1P1P1巖石發(fā)生變形的四種形式5平移轉(zhuǎn)動形態(tài)變化或形變

體積變化或體變變形物體內(nèi)部各質(zhì)點相對位置無變化**變形概念的進一步理解使物體內(nèi)部各質(zhì)點之間發(fā)生了相對位移6應變：巖石變形的度量，即巖石形變和體變程度的定量表示;物體變形時內(nèi)部各質(zhì)點的相對位置發(fā)生變化;變化的兩種方式：線段長度的變化，稱為線應變;兩線間的角度變化，稱為剪應變;一般通過線應變和剪應變定量說明物體的變形程度

應變7DeformationandStrain8Deformationdescribesthecollectivedisplacementsofpointsinabody;inotherwords,itdescribesthecompletetransformationfromtheinitialtothefinalgeometryofabody.Thischangecanincludeatranslation

(movementfromoneplacetotheother),arotation

(spinaroundanaxis),andadistortion

(changeinshape).Strain

describesthechangesofpointsinabodyrelativetoeachother;so,itdescribesthedistortionofabody.9DeformationandStrainSo,strainisacomponentofdeformationandthereforenotasynonym.Inessence,wehavedefineddeformationandstrainrelativetoaframeofreference.Deformationdescribesthecompletedisplacementfieldofpointsinabodyrelativetoanexternalreferenceframe,suchastheedgesofthepaperonwhichFigure4.2isdrawn.Strain,ontheotherhand,describesthedisplacementfieldofpointsrelativetoeachother.Thisrequiresareferenceframewithinthebody,aninternalreferenceframe,liketheedgesofthesquare.Whentherotationanddistortioncomponentsarezero,weonlyhaveatranslation.Thistranslationisformallycalledrigid-bodytranslation,becausethebodyundergoesnoshapechangewhileitmoves.Whenthetranslationanddistortioncomponentsarezero,wehaveonlyrotationofthebody.Byanalogytotranslation,wecallthiscomponentrigid-bodyrotation,orsimplyspin;Whentranslationandspinarebothzero,thebodyundergoesdistortion;thiscomponentisdescribedbystrain.Summary10Deformationisdescribedby:1.Rigid-bodytranslation(ortranslation)2.Rigid-bodyrotation(orspin)3.Strain4.Volumechange(ordilation)

應變的度量——線應變——角應變——剪切應變11伸長度(Extension)：單位長度的改變量

e=(l

-l0)/l0

長度比(Stretch)：變形后的長度與原長之比

S=l

/l0=1+e平方長度比

λ=

（1+e）2倒數(shù)平方長度比

λ′=1/λ桿件的簡單拉伸變形線應變是物體內(nèi)某方向單位長度的改變量。設(shè)一原始長度為l0的桿件變形后長度為l，則其線應變e為：一般把伸長時的線應變?nèi)≌?，縮短時的線應變?nèi)∝撝怠?/p>

線應變12物體變形時，任意兩條直線間的夾角一般會發(fā)生變化。初始相互垂直的線，變形后一般不再垂直，這種直角的改變量ψ[sai]

稱為角剪應變。剪應變：角剪應變的正切

γ=tgψψγ

剪應變13AngularShear:

MeasureofChangeinAnglesbetweenLines14Todeterminetheangularshearalongagivenline,L,inastrainedbody,itisessentialtoidentifyalinethatwasoriginallyperpendiculartoL.AngularsheardescribesthedepartureofthislinefromitsperpendicularrelationwithL(leftfigure).Thefulldescriptionrequiresasign(positiveequalscounterclockwise;negativeequalsclockwise)andamagnitudeexpressedindegrees.Signconventionsforangularshear.(A)DeterminationoftheangularshearoflineArequiresidentifyingaline,inthiscaseB,whichwasoriginallyperpendiculartoA.TheoriginalorientationoflineBrelativetolineAisshownbythedashline.AngularshearoflineAistheshiftinangleofBoriginalversusBfinal.Becausetheshiftisclockwise,theangularshearisnegative(-).(B)InthisexampletheangularshearoflineAis150.Acounterclockwiseshiftisdenotedbyapositive(+)sign.15Blockcontainingreferencecirclesandlines,beforedeformation.Shapeoftheblockafterdeformation.Originalreferencecirclesnowareellipses.Theoriginallymutuallyperpendicularreferencelineshaveallchangedlength,andmosthavechangedorientationaswell.Angularshearalonganylinecanbedeterminedbyfirstidentifyingalineoriginallyperpendiculartoit,andthenmeasuringtheangularshift.Remember,counterclockwiseshiftsarepositive(+);clockwiseshiftsarenegative(-).Forellipsecd(seeFigureB),theangularshearalongcis+30andtheangularshearalongdis-30(seeFigureC).Forellipseed,(seeFigureB),theangularshearalongeis+38,andtheangularshearalongfis-38(seeFigureC).Finally,forellipsegh(seeFigureB)theangularshearalonggis+20,andtheangularshearalonghis-20.ShearStrain16Letusconsiderhowpointsonalinemoveasaresponsetoangularshear.Points1to4onlineA0inFigure2.52Aaretranslatedbyvariousdistancesasaresultoftherotationofthelineonwhichtheyreside.LineA0isthelocusofpoints1to4.LineAfisthelocusofthesamepointsintheirdeformedlocations(Figure2.52B).Sinceangularshearwassystematicanddeformationwashomogeneous,lineAfremainsstraight.Points1to4moveadistancethatisdirectlyrelatedtotheangularshearandtothedistanceofeachpointabovethepointofintersectionwiththecomplementaryline.Ifthedistanceofeachpointabovetheintersectionisdenotedasy(Figure2.52B),thehorizontaldistanceoftranslationcanbefoundasfollows(Ramsay,1967):Thustanψisanotherwayofdescribingrelativeshiftsinorientationsoflinesthatwereoriginallyperpendicular.Itiscalledshearstrain,symbolizedbytheGreeklettergamma(γ),17Shearstrainalongaline(i.e.,alongagivendirection)maybepositiveornegative,dependingonthesenseofrotation(deflection)ofthelineoriginallyperpendiculartoit.Therangeofshearstrainiszerotoinfinity.FortheexampleshowninFigure2.52B,theshearstrainoflineBfis-tan30,or-0.58.TheshearstrainoflineAfis+tan30,or10.58.18均勻應變和應變橢球體

HOMOGENEOUSSTRAINANDTHESTRAIN

ELLIPSOID19Straindescribesthedistortionofabodyinresponsetoanappliedforce.Strainishomogeneouswhenanytwoportionsofthebodythatweresimilarinformandorientationbeforearesimilarinformandorientationafterstrain.Wedefinehomogeneousstrainbyitsgeometricconsequences:1.Originallystraightlinesremainstraight.2.Originallyparallellinesremainparallel.3.Circlesbecomeellipses;inthreedimensions,spheresbecomeellipsoids.Whenoneormoreofthesethreerestrictionsdoesnotapply,wecallthestrainheterogeneous(Figure4.3c).Becauseconditions(1)and(2)aremaintainedduringthedeformationcomponentsoftranslationandrotation,deformationishomogeneousbydefinitionifthestrainishomogeneous.strainellipseandstrainellipsoid20Inahomogeneouslystrained,two-dimensionalbodytherewillbeatleasttwomateriallinesthatdonotrotaterelativetoeachother,meaningthattheirangleremainsthesamebeforeandafterstrain.Whatisamaterialline?Amateriallineconnectsfeatures,suchasanarrayofgrains,thatarerecognizablethroughoutabody’sstrainhistory.ThebehavioroffourmateriallinesisillustratedinFigure4.4forthetwo-dimensionalcase,inwhichacirclechangesintoanellipse.Inhomogeneousstrain,twoorientationsofmateriallinesremainperpendicularbeforeandafterstrain.Thesetwomateriallinesformtheaxesofanellipsethatiscalledthestrainellipse.Analogously,inthreedimensionswehavethreemateriallinesthatremainperpendicularafterstrainandtheydefinetheaxesofanellipsoid,thestrainellipsoid.

Thelinesthatareperpendicularbeforeandafterstrainarecalledtheprincipalstrainaxes.應變橢圓：二維變形中初始單位圓經(jīng)變形形成的橢圓應變主軸：應變橢圓的長、短軸方向，該方向上只有線應變而無剪切應變。最大應變與最小應變：應變主軸方向上的線應變，即應變橢圓長、短軸半徑的長度，其值分別為λ11/2和λ21/2應變橢圓軸比：應變橢圓的長、短軸比Rs＝λ11/2/λ21/2應變橢圓與應變橢球21應變橢球：三維變形中初始單位球體經(jīng)變形形成的橢球應變主軸：應變橢球的三主軸方向。分別稱為最大、中間和最小應變主軸。記做λ1(X)

，λ2(Y)，λ3(Z)

長度分別為X＝λ11/2，Y＝λ21/2，Z＝λ31/2應變主平面：應變橢球上包含任意兩個應變主軸的切面。

XY，XZ，YZ面，λ1(X)λ2

(Y)λ3

(Z)22圓切面：應變橢球上各個方向線應變均相等的兩個圓形切面。它們相交于中間軸Y。平面應變：應變橢球中間軸（λ2，Y）不發(fā)生線應變的應變，其中間軸Y（λ21/2）＝1。無伸縮面（無線應變面）：平面應變橢球的圓切面23主軸、主平面的地質(zhì)意義：

X方向－反映在礦物的定向排列上（拉伸線理）

XY面－壓扁面：代表褶皺的軸面或劈理面的方位

YZ面—張性面：代表了張性構(gòu)造的方位（張節(jié)理）24應變橢球體形態(tài)類型及其幾何表示法a=X/Y,b=Y/Z,

各種應變橢球體的形態(tài)可以用不同的圖解來表示，常用的是弗林（Flinn）圖解，這是一種用主應變比a及b作為坐標軸的二維圖解。abK=0K=∞任意一種形態(tài)的橢球體都可在圖中表示為一點，如圖中的P點，該點的位置就反映了應變橢球體的形態(tài)和應變強度。橢球體的形態(tài)用參數(shù)k表示，k=tgα=(a-1)/(b-1)K值的物理意義：相當于P點到原點連線的斜率。25k=0：軸對稱壓縮，鐵餅型；1>k>0：壓扁型；k=1：平面應變∞>k>1：拉伸應變；k=∞：單軸拉伸，雪茄型三維應變的弗林（Flinn）圖解

在形變時體積不變的條件下，依據(jù)k值可分為五種形態(tài)類型的應變橢球體26PancakeshapedellipsoidleadstoStectonites(strongschistosity,nolineation),cigarshapedellipsoidleadstoLtectonites(stronglineation,noschistosity).L=Stectonitesareproducedbyplanestrain.Whenstrainishomogeneousittransformsanimaginarysphereintoanellipsoid(3perpendicularaxesλ1≥λ2≥λ3)calledtheFiniteStrainEllipsoidfromwhichitiseasytocharacterizethestyleofstrainanditsintensity.Whenstrainisheterogeneousweare"stuffed"asthecharacterizationofa"potatoid"isextremelydifficult.Fortunatelyitisalwayspossibletodefineascaleatwhichstrainis,infirstapproximation,homogeneous.Thestrain,asgeometricallycharacterizedbyanellipsoid,issoeasytoassessthatonlytwoparametersKandDcompletelydefinethestyleofstrain(shapeofellipsoid)andtheamountofstrain(ellipsoidicity,iehowfaritisfromaperfectsphere)respectively.Asshownontherightthesetwoparametersarebothfunctionoftheratioλ1/λ2andλ2/λ3.KandDdonotrequestknowledgeoftheradiusoftheinitialsphereonlyknowledgeoftheprincipalaxesofthefinitestrainellipsoid.三維應變的弗林（Flinn）圖解參考注釋27有限應變（總應變）：物體變形最終狀態(tài)與初始狀態(tài)對比發(fā)生的變化；遞進變形：物體從初始狀態(tài)變化到最終狀態(tài)的過程是一個由許多次微量應變的逐次疊加過程，該過程即為遞進變形；增量應變：遞進變形中某一瞬間正在發(fā)生的小應變叫增量應變；無限小應變：如果所取的變形瞬間非常微小，其間發(fā)生的微量應變?yōu)闊o限小應變。

遞進變形28COAXIALANDNON-COAXIAL

STRAINACCUMULATION29Inthegeneralcaseforstrain,theprincipalincrementalstrainaxesarenotnecessarilythesamethroughoutthestrainhistory.Theprincipalincrementalstrainaxesrotaterelativetothefinitestrainaxes,ascenariothatiscallednon-coaxialstrainaccumulation.Thecaseinwhichthesamemateriallinesremaintheprincipalstrainaxesateachincrementiscalledcoaxialstrainaccumulation.

So,withcoaxialstrainaccumulationthereisnorotationoftheincrementalstrainaxeswithrespecttothefinitestrainaxes.Thecaseinwhichthesamemateriallinesremaintheprincipalstrainaxesateachincrementiscalledcoaxialstrainaccumulation.Simpleshear，pureshearandgeneralshear30Thecomponentdescribingtherotationofmateriallineswithrespecttotheprincipalstrainaxesiscalledtheinternalvorticity,whichisameasureofthedegreeofnon-coaxiality.Ifthereiszerointernalvorticity,thestrainhistoryiscoaxial(asinFigure4.6b),whichissometimescalledpureshear.Thenon-coaxialstrainhistoryinFigure4.6adescribesthecaseinwhichthedistanceperpendiculartotheshearplane(orthethicknessofourstackofcards)remainsconstant;thisisalsoknownassimpleshear.Inreality,acombinationofsimpleshearandpureshearoccurs,whichwecallgeneralshear(orgeneralnon-coaxialstrainaccumulation;Figure4.7).kinematicvorticitynumber31Internalvorticityisquantifiedbythekinematicvorticitynumber,Wk,whichrelatestheangularvelocityandthestretchingrateofmateriallines.ForpureshearWk=0(Figure4.8a),forgeneralshear0<Wk<1(Figure4.8b),andforsimpleshearWk=1(Figure4.8c).Rigid-bodyrotationorspincanalsobedescribedbythekinematicvorticitynumber(inthiscase,Wk=∞;Figure4.8d),butrememberthatthisrotationalcomponentofdeformationisdistinctfromtheinternalvorticityofstrain.32UsingFigure4.6asanexample,thedeformationhistoryshowninFigure4.6arepresentsnon-coaxial,nonrotationaldeformation.Theorientationoftheshearplanedoesnotrotatebetweeneachstep,buttheincrementalstrainaxesdorotate.ThestrainhistoryinFigure4.6brepresentscoaxial,nonrotationaldeformation,becausetheincrementalaxesremainparallel.Typesofstrain33共軸遞進變形（無旋轉(zhuǎn)變形）：在遞進變形過程中，各增量應變橢球體主軸始終與有限應變橢球體主軸一致，即在變形過程中有限應變主軸方向保持不變。非共軸遞進變形（旋轉(zhuǎn)變形）：在遞進變形過程中，增量應變橢球體主軸與有限應變橢球體主軸不一致，即在變形過程中有限應變主軸方向發(fā)生變化。共軸與非共軸遞進變形34有旋變形和無旋變形根據(jù)應變主軸方向的物質(zhì)線在變形前后平行與否，可把變形分為有旋變形和無旋變形。簡單剪切（單剪）純剪單剪與純剪應變有旋變形的的

1和3質(zhì)點線方向?qū)淖儭Ｗ畹湫偷那闆r是簡單剪切，體變?yōu)榱愕钠矫鎽?；是由物質(zhì)中質(zhì)點沿著彼此平行的方向相對滑動而成。無旋變形，

1和3質(zhì)點線方向在變形前后保持不變。如果體積不變而且2=0，則稱為純剪切。35共軸與非共軸遞進變形中應變主軸物質(zhì)（質(zhì)點）線的變化共軸變形中，組成應變主軸的物質(zhì)（質(zhì)點）線不變非共軸變形中，組成應變主軸的質(zhì)點線是不斷變化的36純剪切：一種均勻共軸變形，應變橢球體中主軸質(zhì)點線在變形前后保持不變且具有同一方位。簡單剪切：一種無體應變的均勻非共軸變形，由物體質(zhì)點沿彼此平行的方向相對滑動形成。純剪切與簡單剪切37在簡單剪切中，與剪切方向平行的方向上無線應變，三維上剪切面上無應變，所以Y軸為無應變軸，故此簡單剪切屬于平面應變。另外剪切帶的厚度也保持不變。剪切面剪切方向剪切帶厚度38STRAINPATH39Themeasureofstrainthatcomparestheinitialandfinalconfigurationiscalledthefinitestrain,identifiedbysubscriptf,whichisindependentofthedetailsofthestepstowardthefinalconfiguration.Whentheseintermediatestrainstepsaredeterminedtheyarecalledincrementalstrains,identifiedbysubscripti.(1)持續(xù)拉伸區(qū)(2)先壓縮后拉伸，變形后長度超過原長(3)先壓縮后拉伸，變形后長度未達到原長(4)持續(xù)壓縮區(qū)應變歷史及應變橢圓分區(qū)40有限應變：巖石變形程度的量度有限應變（狀態(tài)）的表示：應變橢球的主軸長度比（Rs）和主軸方向應變標志體：變形巖石中可用于測量和計算應變狀態(tài)的標志性物體巖石有限應變測量（課外閱讀材料）41礫石、砂粒、氣孔、鮞粒、放射蟲、還原斑等原始形狀規(guī)則的標志物：變形化石和變形晶體等與變形有關(guān)的小型構(gòu)造標志物：壓力影、生長礦物纖維、石香腸構(gòu)造、線理、面理、節(jié)理等已知原始形狀的其它標志物原始為圓球或橢球的標志體應變標志體

確定巖石內(nèi)的有限應變狀態(tài)及其分布規(guī)律的一個方法，就是測量和統(tǒng)計變形巖石內(nèi)已知原始形狀的標志物在變形后的形態(tài)變化，然后加以對比分析。

根據(jù)變形標志物中已知長度或相對長度比的線性標志物發(fā)生的長度變化，可以計算伸縮線應變。

根據(jù)兩條直線之間原始角度的變化可以計算角剪應變和剪應變。應變測量概述42原理：應變標志體變形前為球體或某一截面上的圓，變形后為橢球體或橢圓。如礫石、鮞粒和還原斑等為球體，而海百合莖的截面為圓，它們變形后的形態(tài)代表應變狀態(tài)1.長短軸法431.尋找三軸及主平面方向；2.在XZ、XY和YZ面上測量標志體的長、短軸；3.投圖；4.求斜率得X/Z、X/Y和Y/Z。5.還可用線性回歸及最小二乘法進行計算機處理測量步驟：44原理：應變標志體變形前并非球體，而是隨機分布的具有原始軸比（Ri

）的橢球體，變形后形態(tài)和長軸方位均發(fā)生變化。其最終的形態(tài)（軸比，Rf

）和方位（長軸方向，φ）取決于測量標志初始軸比（Ri）、初始長軸方向（θ）、及應變橢圓軸比（Rs），關(guān)系如下：θφRiRsRf測量標志體：礫石、鮞粒、還原斑礦物顆粒等2.Rf/φ法4550％資料線：變形前長軸與應變主軸成±45°的不同軸比的橢球變形后所在的方向與軸比。RfφRfφ46472）在透明紙上畫上左上圖的Rf和φ軸并標上刻度，同時標上參考方向3）測量標志體的長短軸比（Rf）及其與參考方向的夾角（φ）4）將測量數(shù)據(jù)投到透明紙上5）將帶有測量數(shù)據(jù)的透明紙蒙在如左上圖那樣的曲線圖上，使透明紙和曲線圖中的φ軸重合，對不同Rs的曲線圖逐個套用，直到找到一個曲線圖，其上的50％資料線和主軸將所有數(shù)據(jù)點四等分。此時該曲線圖的Rs即為測量值6）透明紙上的參考軸與曲線圖主軸的夾角即為參考軸與實際應變主軸的夾角測量方法：1）根據(jù)應變標志體長軸的統(tǒng)計方位，在測量面上標一參考的應變主軸方向。48DePaor的Rf/φ網(wǎng)49要求：應變標志體變形后可辨認變形前相互垂直的標志線。3.摩爾圓法502αα2θθψ1ψ2ψ1ψ2514.心對心法－Fry法521.Means,W.D.,1976,StressandStrain,Spring–VerlagNewYork,Inc中文譯本：《應力與應變》，[美]W.D.米恩斯，淮南煤炭學院譯，煤炭工業(yè)出版社出版，1980.102.Thetechniquesofmodernstructuralgeology.v.1,strainanalysis/JohnG.R...中文譯本：《現(xiàn)代構(gòu)造地質(zhì)學方法.第一卷應變分析》徐樹桐主譯1991年，參考書籍53ADDITIONALREADING154Elliott,D.,1972.Deformationpathsinstructuralgeology.GeologicalSocietyofAmericaBulletin,83,2621–2638.Erslev,E.A.,1988.Normalizedcenter-to-centerstrainanalysisofpackedaggregates.JournalofStructuralGeology,10,201–209.Fry,N.,1979.Randompointdistributionsandstrainmeasurementinrocks.Tectonophysics,