變形巖石應變分析

1、第四章第四章 變形巖石應變分析基礎變形巖石應變分析基礎1本章主要內(nèi)容本章主要內(nèi)容v 變形、位移和應變的概念變形、位移和應變的概念v 旋轉(zhuǎn)應變與非旋轉(zhuǎn)應變旋轉(zhuǎn)應變與非旋轉(zhuǎn)應變v 遞進變形、全量應變與增量應變遞進變形、全量應變與增量應變v 巖石的變形階段巖石的變形階段2變形和位移變形和位移 當?shù)貧ぶ袔r石體受到應力作用后，其當?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，這種改變就稱為變形變形。q 變形變形3q 位移位移物體內(nèi)部各質(zhì)點的位移是通過其初始

2、位置和終物體內(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。位置的連線叫位移矢量。4平移旋轉(zhuǎn)(虛線為可能的路徑）形變體變P0P1P0P0P0P1P1P1巖石發(fā)生變形的四種形式巖石發(fā)生變形的四種形式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伸長度伸長度(Extension)：單位長度的改變量：單位長度的改變量 e = (l - l0) / l0 長度比長度比(Stretch)：變形后的長度與原長之比：變形后的長度與原長之比 S = l / l0

8、 = 1 + e平方長度比平方長度比 = （1 + e）2倒數(shù)平方長度比倒數(shù)平方長度比 = 1/000llllle一般把伸長時一般把伸長時的線應變?nèi)≌木€應變?nèi)≌担s短時的值，縮短時的線應變?nèi)∝撝?。線應變?nèi)∝撝?。q 線應變線應變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 .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。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，其值分別為橢圓長、短軸半徑的長度，其值分別為11/2和

25、和21/2應變橢圓軸比：應變橢圓的長、短軸比應變橢圓軸比：應變橢圓的長、短軸比Rs 11/2/21/2211 1 (X)(X)2 2 (Y)(Y)3 3 (Z)222324應變橢球體形態(tài)類型及其幾何表示法應變橢球體形態(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（來表示，常用的是弗林（Flinn）圖解，這是）圖解，這是一種用主應變比一種用主應變比a及及b作為坐標軸的二維圖解。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，如圖中的中表示為一點，

26、如圖中的P點，該點，該點的位置就反映了應變橢球體的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)形態(tài)和應變強度。橢球體的形態(tài)用參數(shù)用參數(shù)k表示，表示，k=tg=(a-1)/(b-1)K值的物理意義：相當于值的物理意義：相當于P點到原點到原點連線的斜率。點連線的斜率。25k=0k=0：軸對稱壓縮，鐵餅型；：軸對稱壓縮，鐵餅型；1k01k0：壓扁型；：壓扁型；k=1k=1： 平面應變平面應變k1k1：拉伸應變；：拉伸應變；k=k=：單軸拉伸，雪茄型：單軸拉伸，雪茄型 在形變時體積不變的條件在形變時體積不變的條件下，依據(jù)下，依據(jù)k值可分為五種形值可分為五種形態(tài)類型的應變橢球體態(tài)類型的應變橢

27、球體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：物體變形最終狀態(tài)與初始狀態(tài)對比發(fā)生的變化；物體變形最終狀態(tài)與初始狀態(tài)對比發(fā)生的變化；：物體從初始狀態(tài)變化到最終狀態(tài)

31、的過程是一個由許多：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；次微量應變的逐次疊加過程，該過程即為遞進變形；：遞進變形中某一瞬間正在發(fā)生的小應變叫增量應變；：遞進變形中某一瞬間正在發(fā)生的小應變叫增量應變；：如果所取的變形瞬間非常微小，其間發(fā)生的微量應：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。變?yōu)闊o限小應變。遞進變形遞進變形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剛 體 旋 轉(zhuǎn) 22 40簡單剪切（單剪）純剪無旋變形無旋變形， 1 1和和 3 3質(zhì)點線方向在變形前后保持不變。質(zhì)點線方向在變形前后保持不變。如果體積不變而且如果體積不變而且 2 2=0=0，則稱為純剪切。，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)點線是不斷變化的非共軸變形中，組成應變主軸的質(zhì)點線是不斷變化的3

40、6純剪切：一種均勻共軸變形，應變橢球體中主軸質(zhì)點線純剪切：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。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)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。點沿彼此平行的方向相對滑動形成。37在簡單剪切中，與剪切方向平行的方向上無線應變，三在簡單剪切中，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，所以維上剪切面上無應變，所以Y軸為無應變軸，故此簡單軸為無應變軸，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。剪切屬于平面應變。另外剪切帶的

41、厚度也保持不變。剪切面剪切面剪切方向剪切方向剪切帶厚度剪切帶厚度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) 持續(xù)拉伸區(qū)持續(xù)拉伸區(qū)(2) 先壓縮后拉伸，變形先壓縮后拉伸，變形 后長度超過原長后長度超過原長(3) 先壓縮后拉伸，變形先壓縮后拉伸，變形 后長度未達到原長后長度未達到原長(4) 持續(xù)壓縮區(qū)持續(xù)壓縮區(qū)40有限應變：巖石變形程度的量度有限應變：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有限應變（狀態(tài)）的表示：應變橢球的主軸長度 比（比（RsRs）和主軸方向）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)的標志性物體狀態(tài)的標志性物體41礫石、砂粒、氣孔

43、、鮞粒、礫石、砂粒、氣孔、鮞粒、放射蟲、還原斑等放射蟲、還原斑等原始形狀規(guī)則的標志物：原始形狀規(guī)則的標志物：變形化石和變形晶體等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：與變形有關(guān)的小型構(gòu)造標志物：壓力影、生長礦物纖維、石香腸壓力影、生長礦物纖維、石香腸構(gòu)造、線理、面理、節(jié)理等構(gòu)造、線理、面理、節(jié)理等已知原始形狀的已知原始形狀的其它標志物其它標志物原始為圓球或原始為圓球或橢球的標志體橢球的標志體應變標志體應變標志體42431.尋找三軸及主平面方向；尋找三軸及主平面方向；2.在在XZ、XY和和YZ面上測量標志體的長、短軸；面上測量標志體的長、短軸；3.投圖；投圖；4.求斜率得求斜率得X/Z

44、、X/Y和和Y/Z。5.還可用線性回歸及最小二乘法進行計算機處理還可用線性回歸及最小二乘法進行計算機處理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（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）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，最終的形態(tài)（軸比， Rf ）和方位（長軸方向，）和方位（長軸方向，）取決于測量取決于測量標志初始軸比（標志初始軸比（Ri）、初始長軸方向（）、初始長軸方向（）、及應變橢圓軸比）、及應變橢圓軸比（Rs），關(guān)系如下：），關(guān)系如下：RiRsRf) 1)(1()

45、 1(2) 1)(1(2cos22222sfifsisfiRRRRRRRRR測量標志體：測量標志體：礫石、鮞粒、還原斑礦物顆粒等礫石、鮞粒、還原斑礦物顆粒等4550資料線：變形前長軸與應變主軸成資料線：變形前長軸與應變主軸成45的的不同軸比的橢球變形后所在的方向與軸比。不同軸比的橢球變形后所在的方向與軸比。RfRf46472）在透明紙上畫上左上圖的）在透明紙上畫上左上圖的Rf和和軸并標上刻度，同時標上參考方向軸并標上刻度，同時標上參考方向3 3）測量標志體的長短軸比（）測量標志體的長短軸比（RfRf）及其與參考方向的夾角（）及其與參考方向的夾角（ ）4 4）將測量數(shù)據(jù)投到透明紙上）將測量數(shù)據(jù)投

46、到透明紙上5 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）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圖中的軸重合，對不同軸重合，對不同RsRs的曲線圖逐個套用，直到找到一個曲線圖，其上的的曲線圖逐個套用，直到找到一個曲線圖，其上的5050資料線和主軸將所有數(shù)據(jù)點四等分。此時該曲線圖的資料線和主軸將所有數(shù)據(jù)點四等分。此時該曲線圖的RsRs即為測量值即為測量值6 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）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：測量方法：1）根據(jù)應變標志體長軸的統(tǒng)計方

47、位，）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。在測量面上標一參考的應變主軸方向。4849要求：應變標志體變形后可辨認變形前相互垂直的標志線。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. 摩爾圓法摩爾圓法50122/1122121251521.Means, W.D.,1976,Stress and Strain, Spring Verlag New York, Inc中文譯本：中文譯本：應力與應變應力與應變，美美 W.D.米恩斯，淮南米恩斯，淮南煤炭學院譯，煤炭工業(yè)出版社出版，煤炭學院譯，煤炭工業(yè)出版社出版，1980.102.The techniques of

48、 modern structural geology. v.1,strain analysis / John G. R. 中文譯本：中文譯本：現(xiàn)代構(gòu)造地質(zhì)學方法現(xiàn)代構(gòu)造地質(zhì)學方法.第一卷應變分析第一卷應變分析徐樹桐主譯徐樹桐主譯 1991年，年，參考書籍參考書籍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