7-1 7 STRAIN INTERPRETATION We will now approach the function A(a)from an entirely different point of view, i.e we will interpret the s-fabric in terms of strain (1 )We assume that the initial fabric is defined by a random orientation of surface, i.e that the odf of the surface is uniform (2 )we assume that the surface acts as an independent strain marker, i. e. the only aspect that matters is the orientation of the line segments with respect to the principle strain axes, but not how the segments are connected (3 )we assume that the fabric anisotropy is produced by deformation only and that the deformation mechanisms produce a strain fabric Under these conditions the axial ratio of the s-fabric is equal to the axial ratio of the strain ellipse, and the preferred orientation of the fabric is equal to the direction of its long axis(see Figs. 7.1 and 7.2) In order to visualize why this is so, we apply the program ROSFIG. This program uses the surface odf and draws a figure from it by simply attaching the individual line segments one after the other in order of increasing orientation until a closed loop is formed. the outline that is thus obtained is called the characteristic shape and is always, by definition, fully convex For the surFor analysis it makes no difference whether we analyze the original fabric or the characteristic shape. Not every fabric, but every characteristic shape can be analyzed by the program PAROR We know that the a(a)of a convex shape always 2. B(a), and that the deformation of a circle yields the strain ellipse. Since the haracteristic shape of the undeformed fabric is a circle, we can conclude that the characteristic shape of the deformed fabric is the strain ellipse. Hence, the projection function A(a) of the surface fabric is equivalent to b(a)of the strain ellipse Determination of the axial ratio and the orientation of the strain ellipse is exactly analogous to the determination of the fabric ellipse shown in chapter b/a(of strain ellipse)=A(a)min/A(a)max =90°-∝min Another option would be to measure these features on a plot of the characteristic 1 Panozzo, R, 1984. Two-dimensional strain from the orientation of lines in a plane. J Structural Geology, 6, 215-22 2 Panozzo, R, 1983. Two-dimensional analysis of shape fabric using projections of lines in a plane. Tectonophysics, 95, 279-294
07 - 1 7 STRAIN INTERPRETATION We will now approach the function A(a) from an entirely different point of view, i.e., we will interpret the S-fabric in terms of strain1. (1.) We assume that the initial fabric is defined by a random orientation of surface, i.e., that the ODF of the surface is uniform; (2.) we assume that the surface acts as an independent strain marker, i.e., the only aspect that matters is the orientation of the line segments with respect to the principle strain axes, but not how the segments are connected; (3.) we assume that the fabric anisotropy is produced by deformation only, and that the deformation mechanisms produce a strain fabric. Under these conditions, the axial ratio of the S-fabric is equal to the axial ratio of the strain ellipse, and the preferred orientation of the fabric is equal to the direction of its long axis (see Figs. 7.1 and 7.2). In order to visualize why this is so, we apply the program ROSFIG. This program uses the surface ODF and draws a figure from it by simply attaching the individual line segments one after the other in order of increasing orientation until a closed loop is formed. The outline that is thus obtained is called the characteristic shape and is always, by definition, fully convex. For the SURFOR analysis it makes no difference whether we analyze the original fabric or the characteristic shape. Not every fabric, but every characteristic shape can be analyzed by the program PAROR2. We know that the A(a) of a convex shape is always 2 · B(a), and that the deformation of a circle yields the strain ellipse. Since the characteristic shape of the undeformed fabric is a circle, we can conclude that the characteristic shape of the deformed fabric is the strain ellipse. Hence, the projection function A(a) of the surface fabric is equivalent to B(a) of the strain ellipse. Determination of the axial ratio and the orientation of the strain ellipse is exactly analogous to the determination of the fabric ellipse shown in chapter. b/a (of strain ellipse) = A(a)min / A(a)max ap = 90° - amin Another option would be to measure these features on a plot of the characteristic 1 Panozzo, R., 1984. Two-dimensional strain from the orientation of lines in a plane. J. Structural Geology, 6, 215-221. 2 Panozzo, R., 1983. Two-dimensional analysis of shape fabric using projections of lines in a plane. Tectonophysics, 95, 279-294
7-2 3 ' e can use the program Spline to smooth Odfs too and to plot these as rose diagrams or regular histograms. If you compare the rose diagrams with the corresponding characteristic shapes you note that they are different and in particular, that the rose diagram of the OdF of ellipses is not itself an ellipse(see Fig 7.1) ODF OF ELLIPSE≠ ELLIPSE Since the surfor analysis is only sensitive to the orientation of surface, but not to its distribution in the x-y plane, the results obtained for the deformed L100C*** and L100D*** are exactly the same Again, for strain interpretation, you have to carefully observe the the following points (1 )check if the characteristic shape is symmetric, (2)derive the fabric ellipse, (3.) check (as best as you can)if the fabric ellipse was produced by deformation only, or whether other processes were involved (e. g, sedimentary deposition), (4 )try to find out whether the undeformed state was really one of random orientation of surface In the case of the oolithic limestone one may well assume that the initial orientation of surface was random. Note that this is not the same as assuming that the ooides were initially spherical. This is a great advantage of the SUrFoR method, because as a rule ooides are not spherical, but may display considerable flattening and furthermore they may be of a shape which is not at all elliptical. due to the random orientation of long axes(which we may assume if we can exclude cross bedding) the orientation of the surface becomes random too The deformed oolites shown in Fig.7.3. a to 7. 3.d have been obtained by axial compression in a triaxial apparatus44. The question was whether the strain measured from the flattened oolites would be a reliable measure of the bulk strain the axial ratios of the fabric ellipses derived from the projection functions of the deformed oolites is always higher, i. e. they always indicate less strain, than the applied shortening has produced. If z is the undeformed length of the sample, then 1/z is the cross sectional area normal to the axis (x y). If we assume volume constan deformation, xyz= XYZ (capital letters indicating the deformed lengths). the ratio Z/X (or Z/Y), which corresponds to the axial ratio b/a is given by the following relation 3 Schmid, S M, Panozzo, R, Bauer, S, 1987. Simple shear experiments on calcite rocks rheology and microfabric. J. Structural Geology, 9, 747-778 (experiments by S. Corbett, Texas A&M University, Center for Tectonophysics
07 - 2 shape3. We can use the program SPLINE to smooth ODFs too and to plot these as rose diagrams or regular histograms. If you compare the rose diagrams with the corresponding characteristic shapes you note that they are different, and in particular, that the rose diagram of the ODF of ellipses is not itself an ellipse (see Fig.7.1)! ODF OF ELLIPSE ¹ ELLIPSE Since the SURFOR analysis is only sensitive to the orientation of surface, but not to its distribution in the x-y plane, the results obtained for the deformed L100C*** and L100D*** are exactly the same. Again, for strain interpretation, you have to carefully observe the the following points: (1.) check if the characteristic shape is symmetric, (2.) derive the fabric ellipse, (3.) check (as best as you can) if the fabric ellipse was produced by deformation only, or whether other processes were involved (e.g., sedimentary deposition), (4.) try to find out whether the undeformed state was really one of random orientation of surface. In the case of the oolithic limestone one may well assume that the initial orientation of surface was random. Note that this is not the same as assuming that the ooides were initially spherical. This is a great advantage of the SURFOR method, because as a rule ooides are not spherical, but may display considerable flattening, and furthermore they may be of a shape which is not at all elliptical. Due to the random orientation of long axes (which we may assume if we can exclude cross bedding) the orientation of the surface becomes random too. The deformed oolites shown in Fig.7.3.a to 7.3.d have been obtained by axial compression in a triaxial apparatus44. The question was whether the strain measured from the flattened oolites would be a reliable measure of the bulk strain. The axial ratios of the fabric ellipses derived from the projection functions of the deformed oolites is always higher, i.e., they always indicate less strain, than the applied shortening has produced. If z is the undeformed length of the sample, then 1/z is the cross sectional area normal to the axis (x·y). If we assume volume constant deformation, xyz = XYZ (capital letters indicating the deformed lengths). The ratio Z/X (or Z/Y), which corresponds to the axial ratio b/a is given by the following relation: 3 Schmid, S.M., Panozzo, R., Bauer, S., 1987. Simple shear experiments on calcite rocks: rheology and microfabric. J.Structural Geology, 9, 747-778. 4 (experiments by S. Corbett, Texas A&M University, Center for Tectonophysics)
07-3 b/a=ZX=Z/Y=V(Z/z)3=v(1+8z where ez is the axial strain This points to a process of deformation whereby the matrix is more strongly deformed than the ooides the latter deform as stiffer particles in a softer matrix. a further point worth noticing is the difference between the strain measured at the center of the sample and that measured near the top ends the same effect is observe in almost all compression experiments. Towards the ends of the sample there is a cone of low deformation, a region of"stress shadow". This reveals a considerable inhomogeneity of the stress and strain fields in the sample and the relevant measurements are therfore always taken at the center Table 7.1: Experimental conditions of Corbett's experiments sample axial strain strain rate strength at 2. 5%b/a SURFOR SURFOR (%) (MPa) calculated center COR1 28.42 4.10-4 380(200) 61 69 COR2-29.36 210-5340(200) COR3 31.85 410-4395(250)* 63 COR4|-30.51 210-5335(250) 58 73 in brackets: confining pressure The axial ratios (of COR1, COR2, COR3, CoR4) determined by SURFOR analysis indicate less strain than was applied, and thus it has to be inferred that the ooides are stiffer than the matrix note that thi rtains to deformations at given strain rates and at confining pressures that correspond to a depth of 1 km roughly. The deformation mechanisms of the sparry matrix were reported to be twinning, and translation gliding and that of the ooides were assumed to be the same now turn to deformed particle fabrics. We have already seen that sedimentary processes may introduce a strong anisotropy into fabrics. In other words, we know that not every fabric anisotropy is due to deformation We must be aware of the fact that what we see in thin sections are fabrics, not strain. this distinction is very
07 - 3 b/a = Z/X = Z/Y = Ö(Z/z)3 = Ö(1+ez) 3 where ez is the axial strain. This points to a process of deformation whereby the matrix is more strongly deformed than the ooides, the latter deform as stiffer particles in a softer matrix. A further point worth noticing is the difference between the strain measured at the center of the sample and that measured near the top ends. The same effect is observe in almost all compression experiments. Towards the ends of the sample, there is a cone of low deformation, a region of "stress shadow". This reveals a considerable inhomogeneity of the stress and strain fields in the sample, and the relevant measurements are therfore always taken at the center. Table 7.1: Experimental conditions of Corbett's experiments sample axial strain strain rate strength at 2.5% b/a SURFOR SURFOR (%) (s-1) (MPa) calculated center top COR1 -28.42 4·10-4 380 (200)* .61 .65 .69 COR2 -29.36 2·10-5 340 (200)* .59 .64 .67 COR3 -31.85 4·10-4 395 (250)* .56 .59 .63 COR4 -30.51 2·10-5 335 (250)* .58 .63 .73 * in brackets: confining pressure The axial ratios (of COR1, COR2, COR3, COR4) determined by SURFOR analysis indicate less strain than was applied, and thus it has to be inferred that the ooides are stiffer than the matrix. Note that this is pertains to deformations at given strain rates and at confining pressures that correspond to a depth of 1 km roughly. The deformation mechanisms of the sparry matrix were reported to be twinning, and translation gliding, and that of the ooides were assumed to be the same. We now turn to deformed particle fabrics. We have already seen that sedimentary processes may introduce a strong anisotropy into fabrics. In other words, we know that not every fabric anisotropy is due to deformation. We must be aware of the fact that what we see in thin sections are fabrics, not strain. This distinction is very
07-4 Important FABR|C≠ STRAIN We will say that every rock has a fabric because the absence of a preferred orientation, i.e. of an anisotropy, is still the presence of an (isotropic) fabric. In the entire course we will never measure or determine strain ! (The name of the course reminds us of that). We will always measure or determine a fabric and only in cases where the necessary conditions(random initial orientation of surface surface acts as strain marker, symmetric ODF)are fulfilled, will we interprete the fabric in terms of strain As has been mentioned, not every deformation induces a strain fabric in the rock hence we have to be careful to distinguish the concepts of deformation and strain as given by classical continuum mechanics from the physical process(es)of deformation STATE OF DEFORMATION PROCESS OF DEFORMATION The first is a static concept, an abstraction: the description of a geometry relative to a o-called undeformed state, without any consideration of how this geometry is produced or whether the undeformed state has ever existed. The second is something observed in nature: the change of shape of a material by mass diffusion, crystal plastic mechanisms, solution/precipitation, etc In the mathematical sense, deformation implies a transformation of points from the undeformed state into the deformed state: a mapping procedure, which has to obey very restrictive rules (for example: two distinct point may not be mapped into one) Of course, such a mathematical transformation appears to be the very model of natural deformations. and maybe it is. but how can we check, how do we now whether the combined effect of the natural processes of deformation corresponds to a mathematical transformation? What we need are strain markers, shapes of which we know what they looked like in the undeformed state. and we have to assume(a) tha these markers existed before deformation started, ( b) that they deformed conformably with the rest of the material, and (c) that they retained their identity throughout the entire deformation process. We also have to decide on the scale at which we want to look at things(at the atomic level, the continuum mechanic concept of strain breaks The coincidence of terms(deformation as geometry and deformation as process)leads to the confusing situation where a deformed rock may be undeformed .. as for example
07 - 4 important. FABRIC ¹ STRAIN We will say that every rock has a fabric because the absence of a preferred orientation, i.e., of an anisotropy, is still the presence of an (isotropic) fabric. In the entire course, we will never measure or determine strain ! (The name of the course reminds us of that). We will always measure or determine a fabric. And only in cases where the necessary conditions (random initial orientation of surface, szrface acts as strain marker, symmetric ODF) are fulfilled, will we interprete the fabric in terms of strain. As has been mentioned, not every deformation induces a strain fabric in the rock, hence we have to be careful to distinguish the concepts of deformation and strain as given by classical continuum mechanics from the physical process(es) of deformation. STATE OF DEFORMATION ¹ PROCESS OF DEFORMATION The first is a static concept, an abstraction: the description of a geometry relative to a so-called undeformed state, without any consideration of how this geometry is produced or whether the undeformed state has ever existed. The second is something observed in nature: the change of shape of a material by mass diffusion, crystal plastic mechanisms, solution/precipitation, etc. In the mathematical sense, deformation implies a transformation of points from the undeformed state into the deformed state: a mapping procedure, which has to obey very restrictive rules (for example: two distinct point may not be mapped into one). Of course, such a mathematical transformation appears to be the very model of natural deformations. And maybe it is. But how can we check, how do we now whether the combined effect of the natural processes of deformation corresponds to a mathematical transformation ? What we need are strain markers, shapes of which we know what they looked like in the undeformed state. And we have to assume (a) that these markers existed before deformation started, (b) that they deformed conformably with the rest of the material, and (c) that they retained their identity throughout the entire deformation process. We also have to decide on the scale at which we want to look at things (at the atomic level, the continuum mechanic concept of strain breaks down). The coincidence of terms (deformation as geometry and deformation as process) leads to the confusing situation where a deformed rock may be undeformed... as for example
7-5 in the case of a mylonite, which is usually highly deformed(by the process of deformation) but does not show any deformation(by its geometry). To distinguish, we will say that a certain fabric is a strain fabric if it complies with the continuum mechanics concept of strain. We will distinguished"strained"from"deformed fabrics 55 In this lesson we will use the program SMPLo to deform four fabrics that we have created in previous exercises. E36NAT and E36SYN are two isotropic particle fabrics therefore we know that the anisotropy of the particle fabric of the deformed versions E36NATP0266 and E36SYNP02, is entirely due to the deformation In these cases the anisotropy ratio, b/a, and the preferred orientation, ap, of the fabric are exactly equal to the axial ratio and the orientation of the strain ellipse Strain determination by the program Paror are possible under the following conditions: (a) If the particle outlines are fully convex, (b) if the undeformed partick are spherical or if the orientation of the(elliptical)outlines is random, i. e. if the ODF uniform, (c)if the outlines deform passively with the matrix and no new outlines are created and none are destroyed. The ParoR analysis is completely unsensitive to the centerpoint distribution The projection function B(a) of E36NATP02 and E36SYNP02 are exactly the same, that is, the function B(a) cannot discriminate between the two fabrics. However, their re/p plots, which are obtained from the paror output files, *** to2, differ significantly 77. The advantage of the function b(a)is that we accurately obtain the strain ellipse for any fabric that conforms to the conditions mentioned above the disadvantage is that it provides no information about the initial axial ratios of the particles. The advantage of the re/o -plot is, that we can estimate the distribution of the initial axial ratios of the particles, the disadvantage is, that we get only rough estimates of the strain ellipse. Note that in both cases we have to relie on an initial state which is isotropic. If the undeformed fabric already has an anisotropy as in the cases of E36SYNN and E36SYNP, the analyses yield erroneous values for the strain ellipse. This we shall study in a later exercise The p-fabric of the mud pebbles and the s-fabric of the solution seams of the Ammonitico RoSSo(Fig. 7. 4)are both incompatible with an interpretation of strain (note that this is the result of a very cursory first evaluation). For the solution surfaces this is not so surprising because critertion(a) is clearly violated in that the solution 5 Means, W.D., 1976. Tensors, chap 19-22, in: Stress and Strain. Springer Verlag, New York 6 Po2 indicates Pure shear strain with axial ratio b/a =1: 2 7 Lisle, R.J., 1985. Geological strain anlysis. A manual for the R/f technique. Pergamon Press, Oxford
07 - 5 in the case of a mylonite, which is usually highly deformed (by the process of deformation) but does not show any deformation (by its geometry). To distinguish, we will say that a certain fabric is a strain fabric if it complies with the continuum mechanics concept of strain. We will distinguished "strained" from "deformed" fabrics55. In this lesson we will use the program SMPLO to deform four fabrics that we have created in previous exercises. E36NAT and E36SYN are two isotropic particle fabrics, therefore we know that the anisotropy of the particle fabric of the deformed versions, E36NATP0266 and E36SYNP02, is entirely due to the deformation. In these cases, the anisotropy ratio, b/a, and the preferred orientation, ap, of the fabric are exactly equal to the axial ratio and the orientation of the strain ellipse. Strain determination by the program PAROR are possible under the following conditions: (a) If the particle outlines are fully convex, (b) if the undefomed particles are spherical or if the orientation of the (elliptical) outlines is random, i.e., if the ODF is uniform, (c) if the outlines deform passively with the matrix and no new outlines are created and none are destroyed. The PAROR analysis is completely unsensitive to the centerpoint distribution. The projection function B(a) of E36NATP02 and E36SYNP02 are exactly the same, that is, the function B(a) cannot discriminate between the two fabrics. However, their Rf /fplots, which are obtained from the PAROR output files, ***.T02, differ significantly77. The advantage of the function B(a) is, that we accurately obtain the strain ellipse for any fabric that conforms to the conditions mentioned above, the disadvantage is that it provides no information about the initial axial ratios of the particles. The advantage of the Rf /f- plot is, that we can estimate the distribution of the initial axial ratios of the particles, the disadvantage is, that we get only rough estimates of the strain ellipse. Note, that in both cases we have to relie on an initial state which is isotropic. If the undeformed fabric already has an anisotropy, as in the cases of E36SYNN and E36SYNP, the analyses yield erroneous values for the strain ellipse. This we shall study in a later exercise. The P-fabric of the mud pebbles and the S-fabric of the solution seams of the Ammonitico Rosso (Fig.7.4) are both incompatible with an interpretation of strain (note that this is the result of a very cursory first evaluation). For the solution surfaces this is not so surprising because critertion (a) is clearly violated in that the solution 5 Means, W.D., 1976. Tensors, chap. 19 - 22, in: Stress and Strain. Springer Verlag, New York. 6 P02 indicates Pure shear strain with axial ratio b/a = 1:2. 7 Lisle, R.J., 1985. Geological strain anlysis. A manual for the Rf /f technique. Pergamon Press, Oxford
7-6 surfaces were formed during and not before the deformation the asymmetry of the ODFS points to the "asymmetry of the deformation process",i.e. to rotational deformation. It is difficult to infer the sense of shear from the ODFs. Compared with the only published evidence it would be sinistral shear. But this is probably a useless comparison because the deformation of the published oblique ODFs have been obtained from high temperature deformation of calcite. Furthermore, the shape of the mud pebbles is the result of at least to distinct processes, one being plastic deformation as evidenced by tha flattening and the other one solution at the surfaces which are more or less normal to the prinicpal shortening direction PROGRAM DESCRIPTION Program ROSFIG This program is used to plot characteristic shapes. It uses files*R. as input. These files are created by program SURFOR and contain the surface ODF of a fabric run rosing WHICH FILE (***R.?> mist r05 DICATE ROTATION OF PLOT (DEG CL WISE)>0 WANT NAME OF FILE PLOTTED (Y/N)>y 4 INDICATE ADDITIONAL TEXT (OPTIONAL) PLOTFILE: mist-r plt Line Answer 1 The name fo the file is mist vax 2 Typing o leaves the plot unaltered 3 Yes the file name should be plotted 4 No additional text is provided 5 The name of the plot file is mist-r. plt. If no name is provided the plot is immediately presented on the Tektronix 4014 screen, but not saved
07 - 6 surfaces were formed during and not before the deformation. The asymmetry of the ODFs points to the "asymmetry of the deformation process", i.e., to rotational deformation. It is difficult to infer the sense of shear from the ODFs. Compared with the only published evidence it would be sinistral shear. But this is probably a useless comparison because the deformation of the published oblique ODFs have been obtained from high temperature deformation of calcite. Furthermore, the shape of the mud pebbles is the result of at least to distinct processes, one being plastic deformation as evidenced by tha flattening and the other one solution at the surfaces which are more or less normal to the prinicpal shortening direction. PROGRAM DESCRIPTION Program ROSFIG This program is used to plot characteristic shapes. It uses files *.R.. as input. These files are created by program SURFOR and contain the surface ODF of a fabric. 0 $ run rosfig 1 WHICH FILE (***.R..) ? > mist.r05 2 INDICATE ROTATION OF PLOT (DEG.CL.WISE) > 0 3 WANT NAME OF FILE PLOTTED (Y/N) > y 4 INDICATE ADDITIONAL TEXT (OPTIONAL) > 5 PLOTFILE: mist-r.plt Line Answer 1 The name fo the file is mist.vax. 2 Typing 0 leaves the plot unaltered. 3 Yes, the file name should be plotted. 4 No additional text is provided 5 The name of the plot file is mist-r.plt. If no name is provided the plot is immediately presented on the Tektronix 4014 screen, but not saved
07-7 Figure 7.1 Rose diagram of ODF of strain ellipses. Smoothed outlines of strain ellipses with axial ratios of o1. 2. 1.00 are shown
07 - 7 Figure 7.1 Rose diagram of ODF of strain ellipses. Smoothed outlines of strain ellipses with axial ratios of 0.1, 0.2,... 1.00 are shown
07-8 Figure 7. 2 Strain interpretation (a) Randomly oriented lines b) Corresponding characteristic shape= circle (c) Projection function A(a) of random lines= projection function B(a)of circle (d) Deformed random lines (e) Corresponding characteristic shape strain ellipse (f) Projection function A(a)of deformed random lines= projection function B(a)of strain ellipse
07 - 8 Figure 7.2 Strain interpretation. (a) Randomly oriented lines (b) Corresponding characteristic shape = circle (c) Projection function A(a) of random lines = projection function B(a) of circle (d) Deformed random lines (e) Corresponding characteristic shape = strain ellipse (f) Projection function A(a) of deformed random lines = projection function B(a) of strain ellipse
07-9 Figure 7. 3 SURFOR analysis of eperimentally deformed oolithic limestone. See text
07 - 9 Figure 7.3 SURFOR analysis of eperimentally deformed oolithic limestone. See text
07-10 igure 7. 4 SURFOR analysis of a natural fabric. See text (a) Outcrop of Ammonitico Rosso (b) P-and S-fabric of natural fabric: top: case where data base for PAROR and SURFOR analysis are the same, bottom: case where data base for PAROR and sUrFoR analysis are not the same
07 - 10 Figure 7.4 SURFOR analysis of a natural fabric. See text. (a) Outcrop of Ammonitico Rosso. (b) P- and S- fabric of natural fabric: top: case where data base for PAROR and SURFOR analysis are the same, bottom: case where data base for PAROR and SURFOR analysis are not the same
