变形岩石应变分析

第四章

变形岩石应变分析基础1本章主要内容

变形、位移和应变的概念旋转应变与非旋转应变递进变形、全量应变与增量应变岩石的变形阶段2变形和位移

变形、位移和应变的概念

当地壳中岩石体受到应力作用后，其内部各质点经受了一系列的位移，从而使岩石体的初始形状、方位或位置发生了改变，这种改变就称为变形。

变形3

位移物体内部各质点的位移是通过其初始位置和终止位置的变化来表示，质点的初始位置和终止位置的连线叫位移矢量。

从几何学角度来看，研究物体的变形需要比较物体内各质点的位置在变形前后的相对变化，为此，我们引入位移的概念。4平移旋转(虚线为可能的路径）形变体变P0P1P0P0P0P1P1P1岩石发生变形的四种形式5平移转动形态变化或形变

体积变化或体变变形物体内部各质点相对位置无变化**变形概念的进一步理解使物体内部各质点之间发生了相对位移6应变：岩石变形的度量，即岩石形变和体变程度的定量表示;物体变形时内部各质点的相对位置发生变化;变化的两种方式：线段长度的变化，称为线应变;两线间的角度变化，称为剪应变;一般通过线应变和剪应变定量说明物体的变形程度

应变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)

应变的度量——线应变——角应变——剪切应变11伸长度(Extension)：单位长度的改变量

e=(l

-l0)/l0

长度比(Stretch)：变形后的长度与原长之比

S=l

/l0=1+e平方长度比

λ=

（1+e）2倒数平方长度比

λ′=1/λ杆件的简单拉伸变形线应变是物体内某方向单位长度的改变量。设一原始长度为l0的杆件变形后长度为l，则其线应变e为：一般把伸长时的线应变取正值，缩短时的线应变取负值。

线应变12物体变形时，任意两条直线间的夹角一般会发生变化。初始相互垂直的线，变形后一般不再垂直，这种直角的改变量ψ[sai]

称为角剪应变。剪应变：角剪应变的正切

γ=tgψψγ

剪应变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均匀应变和应变椭球体

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.应变椭圆：二维变形中初始单位圆经变形形成的椭圆应变主轴：应变椭圆的长、短轴方向，该方向上只有线应变而无剪切应变。最大应变与最小应变：应变主轴方向上的线应变，即应变椭圆长、短轴半径的长度，其值分别为λ11/2和λ21/2应变椭圆轴比：应变椭圆的长、短轴比Rs＝λ11/2/λ21/2应变椭圆与应变椭球21应变椭球：三维变形中初始单位球体经变形形成的椭球应变主轴：应变椭球的三主轴方向。分别称为最大、中间和最小应变主轴。记做λ1(X)

，λ2(Y)，λ3(Z)

长度分别为X＝λ11/2，Y＝λ21/2，Z＝λ31/2应变主平面：应变椭球上包含任意两个应变主轴的切面。

XY，XZ，YZ面，λ1(X)λ2

(Y)λ3

(Z)22圆切面：应变椭球上各个方向线应变均相等的两个圆形切面。它们相交于中间轴Y。平面应变：应变椭球中间轴（λ2，Y）不发生线应变的应变，其中间轴Y（λ21/2）＝1。无伸缩面（无线应变面）：平面应变椭球的圆切面23主轴、主平面的地质意义：

X方向－反映在矿物的定向排列上（拉伸线理）

XY面－压扁面：代表褶皱的轴面或劈理面的方位

YZ面—张性面：代表了张性构造的方位（张节理）24应变椭球体形态类型及其几何表示法a=X/Y,b=Y/Z,

各种应变椭球体的形态可以用不同的图解来表示，常用的是弗林（Flinn）图解，这是一种用主应变比a及b作为坐标轴的二维图解。abK=0K=∞任意一种形态的椭球体都可在图中表示为一点，如图中的P点，该点的位置就反映了应变椭球体的形态和应变强度。椭球体的形态用参数k表示，k=tgα=(a-1)/(b-1)K值的物理意义：相当于P点到原点连线的斜率。25k=0：轴对称压缩，铁饼型；1>k>0：压扁型；k=1：平面应变∞>k>1：拉伸应变；k=∞：单轴拉伸，雪茄型三维应变的弗林（Flinn）图解

在形变时体积不变的条件下，依据k值可分为五种形态类型的应变椭球体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.三维应变的弗林（Flinn）图解参考注释27有限应变（总应变）：物体变形最终状态与初始状态对比发生的变化；递进变形：物体从初始状态变化到最终状态的过程是一个由许多次微量应变的逐次叠加过程，该过程即为递进变形；增量应变：递进变形中某一瞬间正在发生的小应变叫增量应变；无限小应变：如果所取的变形瞬间非常微小，其间发生的微量应变为无限小应变。

递进变形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共轴递进变形（无旋转变形）：在递进变形过程中，各增量应变椭球体主轴始终与有限应变椭球体主轴一致，即在变形过程中有限应变主轴方向保持不变。非共轴递进变形（旋转变形）：在递进变形过程中，增量应变椭球体主轴与有限应变椭球体主轴不一致，即在变形过程中有限应变主轴方向发生变化。共轴与非共轴递进变形34有旋变形和无旋变形根据应变主轴方向的物质线在变形前后平行与否，可把变形分为有旋变形和无旋变形。简单剪切（单剪）纯剪单剪与纯剪应变有旋变形的的

1和3质点线方向将会改变。最典型的情况是简单剪切，体变为零的平面应变；是由物质中质点沿着彼此平行的方向相对滑动而成。无旋变形，

1和3质点线方向在变形前后保持不变。如果体积不变而且2=0，则称为纯剪切。35共轴与非共轴递进变形中应变主轴物质（质点）线的变化共轴变形中，组成应变主轴的物质（质点）线不变非共轴变形中，组成应变主轴的质点线是不断变化的36纯剪切：一种均匀共轴变形，应变椭球体中主轴质点线在变形前后保持不变且具有同一方位。简单剪切：一种无体应变的均匀非共轴变形，由物体质点沿彼此平行的方向相对滑动形成。纯剪切与简单剪切37在简单剪切中，与剪切方向平行的方向上无线应变，三维上剪切面上无应变，所以Y轴为无应变轴，故此简单剪切属于平面应变。另外剪切带的厚度也保持不变。剪切面剪切方向剪切带厚度38STRAINPATH39Themeasureofstrainthatcomparestheinitialandfinalconfigurationiscalledthefinitestrain,identifiedbysubscriptf,whichisindependentofthedetailsofthestepstowardthefinalconfiguration.Whentheseintermediatestrainstepsaredeterminedtheyarecalledincrementalstrains,identifiedbysubscripti.(1)持续拉伸区(2)先压缩后拉伸，变形后长度超过原长(3)先压缩后拉伸，变形后长度未达到原长(4)持续压缩区应变历史及应变椭圆分区40有限应变：岩石变形程度的量度有限应变（状态）的表示：应变椭球的主轴长度比（Rs）和主轴方向应变标志体：变形岩石中可用于测量和计算应变状态的标志性物体岩石有限应变测量（课外阅读材料）41砾石、砂粒、气孔、鲕粒、放射虫、还原斑等原始形状规则的标志物：变形化石和变形晶体等与变形有关的小型构造标志物：压力影、生长矿物纤维、石香肠构造、线理、面理、节理等已知原始形状的其它标志物原始为圆球或椭球的标志体应变标志体

确定岩石内的有限应变状态及其分布规律的一个方法，就是测量和统计变形岩石内已知原始形状的标志物在变形后的形态变化，然后加以对比分析。

根据变形标志物中已知长度或相对长度比的线性标志物发生的长度变化，可以计算伸缩线应变。

根据两条直线之间原始角度的变化可以计算角剪应变和剪应变。应变测量概述42原理：应变标志体变形前为球体或某一截面上的圆，变形后为椭球体或椭圆。如砾石、鲕粒和还原斑等为球体，而海百合茎的截面为圆，它们变形后的形态代表应变状态1.长短轴法431.寻找三轴及主平面方向；2.在XZ、XY和YZ面上测量标志体的长、短轴；3.投图；4.求斜率得X/Z、X/Y和Y/Z。5.还可用线性回归及最小二乘法进行计算机处理测量步骤：44原理：应变标志体变形前并非球体，而是随机分布的具有原始轴比（Ri

）的椭球体，变形后形态和长轴方位均发生变化。其最终的形态（轴比，Rf

）和方位（长轴方向，φ）取决于测量标志初始轴比（Ri）、初始长轴方向（θ）、及应变椭圆轴比（Rs），关系如下：θφRiRsRf测量标志体：砾石、鲕粒、还原斑矿物颗粒等2.Rf/φ法4550％资料线：变形前长轴与应变主轴成±45°的不同轴比的椭球变形后所在的方向与轴比。RfφRfφ46472）在透明纸上画上左上图的Rf和φ轴并标上刻度，同时标上参考方向3）测量标志体的长短轴比（Rf）及其与参考方向的夹角（φ）4）将测量数据投到透明纸上5）将带有测量数据的透明纸蒙在如左上图那样的曲线图上，使透明纸和曲线图中的φ轴重合，对不同Rs的曲线图逐个套用，直到找到一个曲线图，其上的50％资料线和主轴将所有数据点四等分。此时该曲线图的Rs即为测量值6）透明纸上的参考轴与曲线图主轴的夹角即为参考轴与实际应变主轴的夹角测量方法：1）根据应变标志体长轴的统计方位，在测量面上标一参考的应变主轴方向。48DePaor的Rf/φ网49要求：应变标志体变形后可辨认变形前相互垂直的标志线。3.摩尔圆法502αα2θθψ1ψ2ψ1ψ2514.心对心法－Fry法521.Means,W.D.,1976,StressandStrain,Spring–VerlagNewYork,Inc中文译本：《应力与应变》，[美]W.D.米恩斯，淮南煤炭学院译，煤炭工业出版社出版，1980.102.Thetechniquesofmodernstructuralgeology.v.1,strainanalysis/JohnG.R...中文译本：《现代构造地质学方法.第一卷应变分析》徐树桐主译1991年，参考书籍53ADDITIONALREADING154Elliott,D.,1972.Deformationpathsinstructuralgeology.GeologicalSocietyofAmericaBulletin,83,2621–2638.Erslev,E.A.,1988.Normalizedcenter-to-centerstrainanalysisofpackedaggregates.JournalofStructuralGeology,10,201–209.Fry,N.,1979.Randompointdistributionsandstrainmeasurementinrocks.Tectonophysics,