变形岩石应变分析

1、第四章第四章 变形岩石应变分析基础变形岩石应变分析基础1本章主要内容本章主要内容v 变形、位移和应变的概念变形、位移和应变的概念v 旋转应变与非旋转应变旋转应变与非旋转应变v 递进变形、全量应变与增量应变递进变形、全量应变与增量应变v 岩石的变形阶段岩石的变形阶段2变形和位移变形和位移 当地壳中岩石体受到应力作用后，其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，这种改变就称为变形变形。q 变形变形3q 位移位移物体内部各质点的位移是通过其初始

2、位置和终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。位置的连线叫位移矢量。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岩石发生变形的四种形式567Deformation and Strain8describes the collective displacements of points in a body; in other words, it describes the complete transformation from the initi

3、al to the final geometry of a body. This change can include a (movement from one place to the other), a (spin around an axis), and a (change in shape). describes the changes of points in a body relative to each other; so, it describes the distortion of a body. 9Deformation and StrainSo, strain is a

4、component of deformation and therefore not a synonym. In essence, we have defined deformation and strain relative to a frame of reference. Deformation describes the complete displacement field of points in a body relative to an external reference frame, such as the edges of the paper on which Figure

5、 4.2 is drawn. Strain, on the other hand, describes the displacement field of points relative to each other. This requires a reference frame within the body, an internal reference frame, like the edges of the square.When the rotation and distortion components are zero, we only have a translation. Th

6、is translation is formally called , because the body undergoes no shape change while it moves.When the translation and distortion components are zero, we have only rotation of the body. By analogy to translation, we call this component , or simply ;When translation and spin are both zero, the body u

7、ndergoes ; this component is described by .Summary 10Deformation is described by:1. Rigid-body translation (or translation)2. Rigid-body rotation (or spin)3. Strain4. Volume change (or dilation)11伸长度伸长度(Extension)：单位长度的改变量：单位长度的改变量 e = (l - l0) / l0 长度比长度比(Stretch)：变形后的长度与原长之比：变形后的长度与原长之比 S = l / l0

8、 = 1 + e平方长度比平方长度比 = （1 + e）2倒数平方长度比倒数平方长度比 = 1/000llllle一般把伸长时一般把伸长时的线应变取正的线应变取正值，缩短时的值，缩短时的线应变取负值。线应变取负值。q 线应变线应变1213Angular Shear: Measure of Change in Angles between Lines14To determine the angular shear along a given line, L, in a strained body, it is essential to identify a line that was origi

9、nally perpendicular to L. Angular shear describes the departure of this line from its perpendicular relation with L (left figure). The full description requires a sign (positive equals counterclockwise; negative equals clockwise) and a magnitude expressed in degrees.Sign conventions for angular shea

10、r. (A) Determination of the angular shear of line A requires identifying a line, in this case B, which was originally perpendicular to A. The original orientation of line B relative to line A is shown by the dash line. Angular shear of line A is the shift in angle of B original versus B final. Becau

11、se the shift is clockwise, the angular shear is negative (-). (B) In this example the angular shear of line A is 150. A counterclockwise shift is denoted by a positive (+) sign.15(A)Block containing reference circles and lines, before deformation. (B)Shape of the block after deformation. Original re

12、ference circles now are ellipses. The originally mutually perpendicular reference lines have all changed length, and most have changed orientation as well.(C) Angular shear along any line can be determined by first identifying a line originally perpendicular to it, and then measuring the angular shi

13、ft. Remember, counterclockwise shifts are positive (+); clockwise shifts are negative (-).For ellipse cd (see Figure B), the angular shear along c is +30 and the angular shear along d is -30 (see Figure C).For ellipse ed,(see Figure B), the angular shear along e is +38, and the angular shear along f

14、 is -38 (see Figure C). Finally, for ellipse gh (see Figure B) the angular shear along g is +20, and the angular shear along h is -20.Shear Strain16Let us consider how points on a line move as a response to angular shear. Points 1 to 4 on line A0 in Figure 2.52A are translated by various distances a

15、s a result of the rotation of the line on which they reside. Line A0 is the locus of points 1 to 4. Line Af is the locus of the same points in their deformed locations (Figure 2.52B). Since angular shear was systematic and deformation was homogeneous, line Af remains straight. Points 1 to 4 move a d

16、istance that is directly related to the angular shear and to the distance of each point above the point of intersection with the complementary line. If the distance of each point above the intersection is denoted as y (Figure 2.52B), the horizontal distance of translation can be found as follows (Ra

17、msay, 1967):Thus tan is another way of describing relative shifts in orientations of lines that were originally perpendicular. It is called shear strain, symbolized by the Greek letter gamma (),17Shear strain along a line (i.e., along a given direction) may be positive ornegative, depending on the s

18、ense of rotation (deflection) of the line originallyperpendicular to it. The range of shear strain is zero to infinity. For the exampleshown in Figure 2.52B, the shear strain of line Bf is -tan 30, or -0.58. The shear strain of line Af is +tan 30, or 10.58.1819Strain describes the distortion of a bo

19、dy in response to an applied force. Strain is homogeneous when any two portions of the body that were similar in form and orientation before are similar in form and orientation after strain.We define homogeneous strain by its geometric consequences:1. Originally straight lines remain straight.2. Ori

20、ginally parallel lines remain parallel.3. Circles become ellipses; in three dimensions, spheres become ellipsoids.When one or more of these three restrictions does not apply, we call the strain heterogeneous (Figure 4.3c). Because conditions (1) and (2) are maintained duringthe deformation component

21、s of translation and rotation, deformation is homogeneous by definition if the strain is homogeneous.strain ellipse and strain ellipsoid 20In a homogeneously strained, two-dimensional body there will be at least two that do not rotate relative to each other, meaning that their angleremains the same

22、before and after strain. What is a material line? A material line connects features, such as an array of grains, that are recognizable throughout abodys strain history. The behavior of four material lines is illustrated in Figure 4.4 for the two-dimensional case, in which a circle changes into an el

23、lipse. In homogeneousstrain, two orientations of material lines remain perpendicular before and after strain. These two material lines form the axes of an ellipse that is called .Analogously, in three dimensions we have three material lines that remain perpendicular after strain and they define the

24、axes of an ellipsoid, . The lines that are perpendicular before and after strain are called the .应变椭圆：二维变形中初始单位圆经变形形成的椭圆应变椭圆：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。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，其值分别为椭圆长、短轴半径的长度，其值分别为11/2和

25、和21/2应变椭圆轴比：应变椭圆的长、短轴比应变椭圆轴比：应变椭圆的长、短轴比Rs 11/2/21/2211 1 (X)(X)2 2 (Y)(Y)3 3 (Z)222324应变椭球体形态类型及其几何表示法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（来表示，常用的是弗林（Flinn）图解，这是）图解，这是一种用主应变比一种用主应变比a及及b作为坐标轴的二维图解。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任意一种形态的椭球体都可在图中表示为一点，如图中的中表示为一点，

26、如图中的P点，该点，该点的位置就反映了应变椭球体的点的位置就反映了应变椭球体的形态和应变强度。椭球体的形态形态和应变强度。椭球体的形态用参数用参数k表示，表示，k=tg=(a-1)/(b-1)K值的物理意义：相当于值的物理意义：相当于P点到原点到原点连线的斜率。点连线的斜率。25k=0k=0：轴对称压缩，铁饼型；：轴对称压缩，铁饼型；1k01k0：压扁型；：压扁型；k=1k=1： 平面应变平面应变k1k1：拉伸应变；：拉伸应变；k=k=：单轴拉伸，雪茄型：单轴拉伸，雪茄型 在形变时体积不变的条件在形变时体积不变的条件下，依据下，依据k值可分为五种形值可分为五种形态类型的应变椭球体态类型的应变椭

27、球体26 Pancake shaped ellipsoid leads to S tectonites (strong schistosity, no lineation), cigar shaped ellipsoid leads to L tectonites (strong lineation, no schistosity). L=S tectonites are produced by plane strain. When strain is homogeneous it transforms an imaginary sphere into an ellipsoid (3 perp

28、endicular axes 123) called the Finite Strain Ellipsoid from which it is easy to characterize the style of strain and its intensity. When strain is heterogeneous we are stuffed as the characterization of a potatoid is extremely difficult. Fortunately it is always possible to define a scale at which s

29、train is, in first approximation, homogeneous. The strain, as geometrically characterized by an ellipsoid, is so easy to assess that only two parameters K and D completely define the style of strain (shape of ellipsoid) and the amount of strain (ellipsoidicity, ie how far it is from a perfect sphere

30、) respectively. As shown on the right these two parameters are both function of the ratio 1/2 and 2/3. K and D do not request knowledge of the radius of the initial sphere only knowledge of the principal axes of the finite strain ellipsoid.27：物体变形最终状态与初始状态对比发生的变化；物体变形最终状态与初始状态对比发生的变化；：物体从初始状态变化到最终状态

31、的过程是一个由许多：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；次微量应变的逐次叠加过程，该过程即为递进变形；：递进变形中某一瞬间正在发生的小应变叫增量应变；：递进变形中某一瞬间正在发生的小应变叫增量应变；：如果所取的变形瞬间非常微小，其间发生的微量应：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。变为无限小应变。递进变形递进变形28COAXIAL AND NON-COAXIALSTRAIN ACCUMULATION29In the general case for strain, the principal incremental

32、 strain axes are not necessarily the same throughout the strain history.The principal incremental strain axes rotate relative to the finite strain axes, a scenario that is called The case in which the same material lines remain the principal strain axes at each increment is called coaxial strain acc

33、umulation. So, with coaxial strain accumulation there is no rotation of the incremental strain axes with respect to the finite strain axes.The case in which the same material lines remain the principal strain axes at each incrementis called .Simple shear，pure shear and general shear30The component d

34、escribing the rotation of material lines with respect to the principal strain axes is called the , which is a measure of the degree of non-coaxiality.If there is zero internal vorticity, the strain history is coaxial (as in Figure 4.6b), which is sometimes called .The non-coaxial strain history in F

35、igure 4.6a describes the case in which thedistance perpendicular to the shear plane (or the thickness of our stack of cards) remains constant; this is also known as . In reality, a combination of simple shear and pure shear occurs, which we call (or general non-coaxial strain accumulation; Figure 4.

36、7). kinematic vorticity number31Internal vorticity is quantified by the kinematic vorticity number, Wk, which relates the angular velocity and the stretching rate of material lines.For pure shear Wk = 0 (Figure 4.8a), for general shear 0 Wk 1 (Figure 4.8b), and for simple shear Wk = 1 (Figure 4.8c).

37、 Rigid-body rotation or spin can also be described by the kinematic vorticity number (in this case, Wk = ; Figure 4.8d), but remember that this rotational component of deformation is distinct from the internal vorticity of strain. 32Using Figure 4.6 as an example, the deformation history shown in Fi

38、gure 4.6a represents non-coaxial, nonrotational deformation. The orientation of the shear plane does not rotate between each step, but the incremental strain axes do rotate. The strain history in Figure 4.6b represents coaxial, nonrotational deformation, because the incremental axes remain parallel.

39、Types of strain3334ACDBOdabccAbOO56 2033 4040刚 体 旋 转 22 40简单剪切（单剪）纯剪无旋变形无旋变形， 1 1和和 3 3质点线方向在变形前后保持不变。质点线方向在变形前后保持不变。如果体积不变而且如果体积不变而且 2 2=0=0，则称为纯剪切。，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非共轴变形中，组成应变主轴的质点线是不断变化的3

40、6纯剪切：一种均匀共轴变形，应变椭球体中主轴质点线纯剪切：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。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。点沿彼此平行的方向相对滑动形成。37在简单剪切中，与剪切方向平行的方向上无线应变，三在简单剪切中，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，所以维上剪切面上无应变，所以Y轴为无应变轴，故此简单轴为无应变轴，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。剪切属于平面应变。另外剪切带的

41、厚度也保持不变。剪切面剪切面剪切方向剪切方向剪切带厚度剪切带厚度38STRAIN PATH39The measure of strain that compares the initial and final configuration is called, identified by subscript f, which is independent of the details of the steps toward the final configuration. When these intermediate strain steps are determined they are c

42、alled , identified by subscript i.(1) 持续拉伸区持续拉伸区(2) 先压缩后拉伸，变形先压缩后拉伸，变形 后长度超过原长后长度超过原长(3) 先压缩后拉伸，变形先压缩后拉伸，变形 后长度未达到原长后长度未达到原长(4) 持续压缩区持续压缩区40有限应变：岩石变形程度的量度有限应变：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有限应变（状态）的表示：应变椭球的主轴长度 比（比（RsRs）和主轴方向）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 状态的标志性物体状态的标志性物体41砾石、砂粒、气孔

43、、鲕粒、砾石、砂粒、气孔、鲕粒、放射虫、还原斑等放射虫、还原斑等原始形状规则的标志物：原始形状规则的标志物：变形化石和变形晶体等变形化石和变形晶体等与变形有关的小型构造标志物：与变形有关的小型构造标志物：压力影、生长矿物纤维、石香肠压力影、生长矿物纤维、石香肠构造、线理、面理、节理等构造、线理、面理、节理等已知原始形状的已知原始形状的其它标志物其它标志物原始为圆球或原始为圆球或椭球的标志体椭球的标志体应变标志体应变标志体42431.寻找三轴及主平面方向；寻找三轴及主平面方向；2.在在XZ、XY和和YZ面上测量标志体的长、短轴；面上测量标志体的长、短轴；3.投图；投图；4.求斜率得求斜率得X/Z

44、、X/Y和和Y/Z。5.还可用线性回归及最小二乘法进行计算机处理还可用线性回归及最小二乘法进行计算机处理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（轴比（ Ri ）的椭球体，变形后形态和长轴方位均发生变化。其）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，最终的形态（轴比， Rf ）和方位（长轴方向，）和方位（长轴方向，）取决于测量取决于测量标志初始轴比（标志初始轴比（Ri）、初始长轴方向（）、初始长轴方向（）、及应变椭圆轴比）、及应变椭圆轴比（Rs），关系如下：），关系如下：RiRsRf) 1)(1()

45、 1(2) 1)(1(2cos22222sfifsisfiRRRRRRRRR测量标志体：测量标志体：砾石、鲕粒、还原斑矿物颗粒等砾石、鲕粒、还原斑矿物颗粒等4550资料线：变形前长轴与应变主轴成资料线：变形前长轴与应变主轴成45的的不同轴比的椭球变形后所在的方向与轴比。不同轴比的椭球变形后所在的方向与轴比。RfRf46472）在透明纸上画上左上图的）在透明纸上画上左上图的Rf和和轴并标上刻度，同时标上参考方向轴并标上刻度，同时标上参考方向3 3）测量标志体的长短轴比（）测量标志体的长短轴比（RfRf）及其与参考方向的夹角（）及其与参考方向的夹角（ ）4 4）将测量数据投到透明纸上）将测量数据投

46、到透明纸上5 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）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图中的轴重合，对不同轴重合，对不同RsRs的曲线图逐个套用，直到找到一个曲线图，其上的的曲线图逐个套用，直到找到一个曲线图，其上的5050资料线和主轴将所有数据点四等分。此时该曲线图的资料线和主轴将所有数据点四等分。此时该曲线图的RsRs即为测量值即为测量值6 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）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：测量方法：1）根据应变标志体长轴的统计方

47、位，）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。在测量面上标一参考的应变主轴方向。4849要求：应变标志体变形后可辨认变形前相互垂直的标志线。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. 摩尔圆法摩尔圆法50122/1122121251521.Means, W.D.,1976,Stress and Strain, Spring Verlag New York, Inc中文译本：中文译本：应力与应变应力与应变，美美 W.D.米恩斯，淮南米恩斯，淮南煤炭学院译，煤炭工业出版社出版，煤炭学院译，煤炭工业出版社出版，1980.102.The techniques of

48、 modern structural geology. v.1,strain analysis / John G. R. 中文译本：中文译本：现代构造地质学方法现代构造地质学方法.第一卷应变分析第一卷应变分析徐树桐主译徐树桐主译 1991年，年，参考书籍参考书籍53ADDITIONAL READING 154Elliott, D., 1972. Deformation paths in structural geology. Geological Society of America Bulletin, 83, 26212638.Erslev, E. A., 1988. Normalized center-to-center strain analysis of packed aggregates. Journal of Structural Geology, 10, 201209.Fry, N., 1979. Random point distributions and strain measurement in rocks. Tectonophysics, 60, 89104.Groshong, R. H., Jr., 1972. Strain calculated from twining in calcite. Geological S