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Strain analysis of symmetric folds using slaty-cleavage
Tectonophysics, 34 (1976) 199-217 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
199
STRAIN ANALYSIS OF SYMMETRIC FOLDS USING SLATY-CLEAVAGE
P.E. MATTHEWS, R.A.B. BOND and J.J. VAN DEN BERG University
of Natal, Durban
(South
Africa)
(Submitted January 6, 1976; revised version accepted May 21, 1976)
ABSTRACT Matthews, P.E., Bond, R.A.B. and Van den Berg, J.J., 1976. Strain analysis of symmetric folds using slaty-cleavage. Tectonophysics, 34: 199-217. Despite some uncertainty concerning its use as an indicator of strain orientation in folded rocks, slaty-cleavage is generally regarded as a penetrative tectonic fabric that is related to the symmetry of inhomogeneous, finite strain within a fold system. On the basis of this concept, a quantitative method is developed for establishing the distribution of finite strain on the profiles of symmetric cylindrical folds using the relative orientations of cleavage and b,edding traces on the plane of section. The method also provides a technique for critical examination of the widely-accepted hypothesis that slaty-cleavage is formed perpendicular to the directions of maximum shortening in folded strata. The analytical procedures are demonstrated and tested by application to a mode1 of an idea1 flattened concentric fold.
INTRODUCTION
Quantitative determination of the patterns of finite inhomogeneous strain in naturally occurring folds could provide valuable information on the modes of formation of the fundamental forms of folding now recognized in structural geology. Although methods of strain analysis are available for use with dimensional markers such as deformed fossils or ellipsoidal particles, these methods are of restricted application in studies of folded strata due mainly to the scarcity of such markers in suitable structural settings. In contrast, slaty-cleavage is of common occurrence in folded rocks of low to high grade metamorphic terrains, and represents a penetrative tectonic fabric defined by a preferred dimensional orientation of mineral components. Moreover, the overall patterns of slaty-cleavage generally display a systematic relationship to the geometry of the associated fold-systems (Ramsay, 1967; Billings, 1972). On account of these features, slaty-cleavage can be regarded prima facie as a structure of considerable potential value in finite-strain analysis of natural folds. Recently, Siddans (1972) and Wood (1974) have traced the history of
200
research on slaty-cleavage, and have reviewed an impressive assemblage of published, structural evidence which suggests that slaty-cleavage is formed perpendicular to the directions of maximum shortening in deformed rocks. In other words, the cleavage is developed parallel to the XY-plane of the strain ellipsoid (X > Y > 2). However, examination of this empirical concept from a theoretical point of view (Ramsay, 1967; Matthews et al., 1971; Elliot, 1972; Bayly, 1974; Williams, 1976) indicates that under conditions of a progressive, non-coaxial (rotational) strain, as for example within the limb of a fold, the XY-plane of the strain ellipsoid will not coincide with the same material plane during successive increments of strain. Nevertheless, material planes coincident with the XY-plane at any stage of the deformation will rotate in the same sense as the XY-plane, and in general, will have a very restricted range of orientations close to that of the XY-plane of the finitestrain ellipsoid (Matthews et al., 1971, pp. 148-151). Preliminary investigation of this effect in the case of progressive, plane, non-coaxial homogeneous deformations indicates that the maximum possible angular deviation is only about five degrees. In view of this it seems possible either, that recrystallization during syntectonic growth of a mica fabric will allow slaty-cleavage to develop parallel to the XY-plane or, that the degree of preferred orientation is not sufficiently well-defined to allow accurate measurement of any angular deviation from the finite XY-plane as defined for example by deformed dimensional markers. These possibilities have been discussed in some detail by Williams (1976). A more important theoretical implication, however, is that regardless of whether or not slaty-cleavage is developed parallel to the XY-plane, the cleavage orientation is related to the symmetry of the finite-strain distribution within a fold. This concept is supported by much published structural evidence which shows, as mentioned already, that in general, patterns of slaty-cleavage have a systematic relationship to the associated fold geometry, and are symmetric about the fold-axial plane. This widely accepted concept is used in this paper to develop a quantitative method for finite-strain analysis of symmetric cylindrical folds using the measured orientations of the traces of slaty-cleavage and bedding exposed on any normal profile-section of the folds. In this context, the symmetry conceptof slaty-cleavage implies that any angular divergence between the cleavage and the finite XY-plane in the limbs of a symmetric fold will decrease to zero towards the fold-axial plane, where cleavage formation will have been restricted to the same material plane coincident with the XY-plane throughout the deformation. In order to demonstrate and test the analytical procedures, the method is applied to a mathematically derived profile-section of an ideal, flattened concentric flow-fold with an associated pattern of slaty-cleavage (Fig. 2). It should be emphasized, that although the cleavage pattern on this fold-profile was constructed using the current hypothesis of cleavage formation normal to the direction of maximum shortening within a fold, the method of strain
201
analysis presented in this paper is not dependent on this hypothesis. On the contrary, this hypothesis was used in the construction of the model foldprofile (Fig. 2) simply because no alternative hypothesis of cleavage-formation is available, and in addition it conforms to the symmetry concept of slaty-cleavage which alone forms the basic assumption in the method. Derivations of the equations used to establish the geometric and strain parameters of the model fold-profile (Fig. 2) are presented in an Appendix to this paper. PRINCIPLE
OF THE METHOD
In developing the theory of the method, attention is directed initially to symmetric cylindrical folds formed from originally parallel layers, by a volume conserving, planar deformation acting normal to the fold-axis. It is also assumed initially, that there are no ductility contrasts within the fold-system. These assumptions imply that the finite, inhomogeneous strain produced by the fold-forming deformation will be confined to normal profile planes of the folds. For the purpose of analysis, a symmetric synformal fold-profile is referred to an external xy-coordinate frame with the y-axis along the trace of the axial plane, and the x-axis tangential to the fold-hinge (Figs. 1 and 2). In accord with the usual approach to analysis of inhomogeneous strain, the pattern of strain is examined in terms of elements that are small enough to be regarded as units of finite homogeneous strain. This implies that within each elemental unit parallel straight lines have remained straight and parallel throughout the deformation. An example involving two elements, each derived from a unit square is shown in Fig. 1. In this example, the base line of each elemental unit is aligned along the bedding-trace which of course, is a visible material line before and after the deformation. In a previous study (Matthews et al., 1974) we have shown that selection of the bedding-trace as the X’-axis of an internal x’y’-coordinate system for each elemental unit, allows description of any planar finite homogeneous deformation in terms of two algebraic parameters (Yand y. The cu-parameter is a strain factor defining a change of dimension in the $-direction of any elemental unit, and represents an extension when 01> 1 and a shortening when (y.< 1. By implication, the a-parameter records any finite change in the length of line segments along the arc of a folded bedding-trace. The y-parameter is a factor of shear in the x’-direction and is related by y = tan $ to the shear angle $, which records a rotation of lines initially parallel to the y’-axis of the relevant elemental unit. It may be noted, that prior to deformation the $-axis of each elemental unit of a symmetric fold will be parallel to the external y-coordinate axis. The relationships under discussion are shown diagrammatically in Figs. 1A and B. We have also derived equations that relate the algebraic parameters CYand y of any finite homogeneous deformation to the axialratio (R) and the
4
INITIAL
B
FINAL
Fig. 1. Diagrammatic comparison of two elemental units (ae and al) of finite homogeneous strain within a symmetric fold before (A) and after (B) the deformation. Observable geometric parameters at a point are 8 (orientation of bedding-trace), t (orientation of cleavage-trace), t (orthogonal bedding-thickness), and ‘I (axial-plane bedding-thickness). In Fig. lB, (Yand y are algebraic strain parameters for elemental units of initial unit dimension, in relation to the internal coordinate axes (x’, Y’) of each unit.
orientation (4) of the major axis of the strain ellipse in relation to the x’axis of the relevant elemental unit (Matthews et al., 1974, pp. 38-39). Two of these equations are of interest in the present context. The first is:
(1) where y = tan $ ; and the second is: cos $I + y cos*f#~ R2 = sin C#J sin C#J cos 4 - y sin’@ which can also be expressed as:
Under the symmetry concept of slaty-cleavage, the orientation of the cleavage-trace on the profile of a symmetric cylindrical fold will be a continuous function of the fold-geometry, and in particular, the cleavage-trace along the axial plane will be parallel to the XY-plane of the finite-strain ellipsoid.
Use of these hypothetical relationships together with measurements of the relative orientation and spacing of the bedding-traces, allows interpolative determinations of the CYand y parameters of indivudual elemental units. This leads to direct evaluation, by means of eqs. 1 and 2, of the orientation (4) and the magnitude of the finite strain, as expressed by the axial ratio (R) of the strain ellipse, for the associated elemental unit. From a determination of the CYand y parameters at a suitable number of points, it should be possible to establish the overall pattern of finite strain for the fold-profile. An attempt is made in the following discussion to draw a distinction
Bedding
trace
----------Cleavagetrace
I
01
I
10
I
,
I
I
20
30
40
50
-
X
Fig. 2. Traces of bedding and slaty-cleavage (dashed lines) on the profile of an ideal, symmetric, flattened concentric flow-fold. It is assumed that the cleavage traces are aligned with directions of maximum extension; however, as explained in the text, this assumption is not necessary to the method of finite-strain analysis under discussion. Equations for derivation of the geometric and strain parameters are presented in an Appendix.
204
between the theoretical and empirical aspects of the method. In particular the sequence of steps and the computational formulae required in an application of the method are described under the respective sub-headings of Analytical procedures I-V. EVALUATION
OF THE STRAIN FACTOR (a)
Theoretical considerations The spacing of bedding-traces on a fold-profile can be recorded by measuring either the orthogonal (or bedding-normal) thickness, denoted by t, or the thickness parallel to the axial plane, denoted by T. In the case of a symmetric fold, these spacings for any particular layer are equal along the trace of the axial plane (external y-axis), and this may be expressed as to = To. By analogy, the t and 2’ parameters may be used for geometrical comparison of elemental units located along the same bedding-trace, with the assumption that these units had initial unit dimensions. It is of interest to make such a comparison between an element (a,,) located on the trace of the axial plane and any other element (at) on the same bedding-trace (a), as indicated in Fig. 1B. On account of conservation of the initial unit area of each element; it is evident from Fig. lB, that: t fl0 -=t al
&Xl %o
However: t 00
=
Tao
t al
=
Ta,
~0s
001
where 19~~ is the dip of the bedding-trace at a I* It follows, that the infinitesimal T,,/T,, ratio denoted by pal at the point a,, is related to the relevant aparameters, and this can be expressed as: T PO1=<=“o
_ %l
1
i COStfal
1
In order to solve for ar,, in this expression, it is necessary to establish values for pa1 and eao. In both cases, this can be achieved by inte~olative procedures using measurable data from the fold-profile, as explained in the following discussion. Analy tieal procedure I It is clear that when any two adjacent bedding-traces of the fold-profile are condensed together in a direction parallel to the y-axis, that is to say as 2’
205
and To go to zero, the T/T, ratio (p) at any fixed point will tend to a limit. This may be expressed as:
Empirical estimation of this limit, which may be referred to as Analytical procedure I, simply involves measurement of the differential T-spacing (AT) of a number of bedding-traces from a fixed point, together with measurements of the corresponding differential To-spacings (AT,) of these beddingtraces along the trace of the axial plane (y-coordinate axis). Then a plot of AT/AT, against AT will produce a trend line that cuts the line AT = 0 at the required interpolative estimate of p, which is applicable to the particular point on the fold-profile. An application of this procedure with reference to the point CZ on the fold-profile of Fig. 2 is illustrated by the numerical data given in Table 1, together with the plot of AT/AT, against AT shown in Fig. 3A. This plot gives a value of p = 1.270, which is obviously a good estimate of the calculated theoretical value of p = 1.272. (eqs. 8 and 18, Appendix). In order to facilitate strain analysis of the fold-profile, this procedure is repeated for different points along individual bedding-t~ces. The results can then be used to establish curves for p against arc length(s) measured from the y-axis, or alternatively, to establish curves for p against the x-ordinate along each bedding-trace (Fig. 3B). Analytical procedure II It is of interest now, to consider an interpolative scheme for evaluation of the strain factor ((Y,,)at the point of intersection of any bedding-trace
TABLE I Data used in estimation of p at point c2 in Fig. 2 Bedding trace
y-value forx = 0
(‘)AT~ from y = 21.3
y-value for x = 19.3
(‘)AT from y = 31.5
AT ATo i._
: c b a
36.8 29.0 21.3 13.1 4.8
15.5 7.7 0 -6.2 -16.5
41.9 55.1 31.5 21.5 12.0
(I) ATO from point co where y = 21.3. (2) AT from point c2 where y = 31.5.
23.6 10.4 0 -10.0 -19.5
1.522 1.351 1.220 1.182
206
P
x - ordlnote
AT
Fig. 3. Graphical aspects of Analytical procedure 1. A. Interpolative estimation of p at the point c2 on the fold-profile in Fig. 2, using values given in Table I. B. Variation of p along bedding-traces a-e on the fold-profile in Fig. 2.
with the y-coordinate axis. This scheme relies on the fact that, on the profile of a symmetric fold, the shear angle (J/) of each elemental unit along a bedding-trace will decrease to zero as these units are traced in succession towards the trace of the axial plane (y-coordinate axis). This is significant on account of the relation of Jt to Q!in eq. 1 for a given value of #, with the implication that an estimate of a0 can be obtained by a limiting procedure. Two steps are involved. In the first step, an estimate of a! is obtained by means of eq. 1, for a number of points along a bedding-trace, using the generally erroneous assumptions that at each point: (1) the angle of the cleavage-trace from the bedding-trace (t) gives the orientation of the strainellipse major-axis (~$1;and (2) the dip of the bedding-trace (8) is an exact reflection of the angular shear strain (li/) for the associateci elemental unit. On this basis, each estimate of CY,denoted by &, is computed from a modified form of eq. 1, which is: 8;=
tan22i t (tan% tan
[ 2 tan
-1 *‘4
1)
In the second step of this procedure, the ratio a& at each point is determined by means of eq. 3, using the relevant value of p provided by Analytics procedure I, together with the measured angle of dip (0) of the bedding-trace at each point. In each case, the computed values of two/aand 6 immediately lead to estimates of CY~, denoted by Go, in the form: 010
(1
ii0 cef -
a
ii
N
p
cos
eii
(5)
207 TABLE II Data used in estimation of tlls at point c2 in Fig. 2.
Cl
cz c3
x
t=ql”
8”
P ____.~
&
aola
go
10.2 19.3 25.8
49.7 30.7 22.2
28.3 50.8 66.5
1.058 1.272 1.781
0.854 0.956 0.792
0.932 0.804 0.710
0.796 0.768 0.562
From previous considerations, it follows that a plot of Go against the xcoordinate of the relevant reference point will define a curve which will be symmetric about x = 0, and will cut this line at a point representing the required interpolative dete~ination of eo. To illustrate these aspects of Analytical procedure II, data from points along the bedding-trace labelled C in Fig. 2, are presented in Table II, and the resulting graph of go versus x is shown in Fig. 4A. This plot gives e. = 0.8, which is in fact the value that was used in the construction of the fold-profile. Analytical procedure III This is essentially a computational procedure for establishing values for the strain factor (cx) at points along individual bedding-traces on the foldprofile. The value of Q at each point is determined, using eq. 3 in the form: a: = &)/f/J cos 8)
(6)
x- ordinate arc- length
(s)
arc - length
(sd
1
Fig. 4. Graphical aspects of Analytical procedures II and IV. A. Interpolative estimation of the strain factor (a~) along bedding-trace C (Fig. 2) at its intersection point with the axial-plane (y-coordinate axis, x = 0). The curve of&e against x is symmetric about x = 0. B. Variation of strain factor ff along bedding-traces a-e on the fold-profile in Fig. 2. C. Plot of estimated initial x-ordinate (xi) against arc-length (sd) along bedding-trace d on the fold-profile in Fig. 2. The arc length related to Xl = x,2 allows determination of the shear factor (7) at the point cs, as part of Analytical procedure IV.
208
by substitution of the values obtained previously for p and CX,,by means of Analytical procedures I and II, respectively, together with the previously measured angle of dip (0) of the bedding-trace at that point. In the case of the point C, on the fold-profile in Fig. 2, the relevant values for use of eq. 6 are p = 1.270, czo= 0.8 and 0 = 50X’, so that the empirical value obtained for the strain factor is (Y= 0.9967, whereas the calculated theoretical value is (Y= 0.9959 (Table III, Appendix). It may be noted from eq. 6 that a0 is a constant factor in an evaluation of the longitudinal strain (a) at points on an individual bedding-trace. Moreover, the values of p and 8 required in such an evaluation could be readily obtained from previously established curves for these parameters against either the arc length or the x-ordinate along the bedding-trace (Fig. 3B). Data derived by the method are presented as curve c in Fig. 4B, together with similar curves recording the variation of finite strain (a) along various bedding-traces of the fold-profile in Fig. 2. As explained later, curves of this type will also allow determination of the original x-ordinate of individual points on the bedding-traces of the fold profile. EVALUATION
OF THE SHEAR
FACTOR
( y)
Theore tical considerations It will be recalled that the shear factor 7, for each elemental unit is defined as y = tan $ where $ defines the final orientation of a line that was initially parallel to the y-coordinate axis. This implies that before the deformation this line (L in Fig. 1A) would have intersected adjacent bedding-lines at points with the same x-ordinate (Xi), and hence, would have defined straight bedding-line segments of equal length as measured from the y-axis, After the folding, these segments will in general have been transformed to arcs of unequal length, but nevertheless, with a definite relationship governed by the angle of shear ($) of the original orthogonal line (L’ in Fig. 1B). From the previously established variation of the strain factor (a) along particular bedding-traces it is possible to devise a scheme (Analytical procedure IV) for establishing arc-lengths along adjacent bedding-traces that have been derived from initial segments of equivalent length. These arc-lengths will then indicate a value for the shear angle ($) and hence, the shear factor y = tan $ for the associated elemental unit. Analytical
procedure
IV
As mentioned previously, this procedure is concerned with the determination of arc-lengths of adjacent bedding-traces that have been derived from initial line-segments of equivalent length.
209
By definition, the incremental arc-length (ds) along any bedding-trace of the fold-profile is given by: ds = odxi so that, the initial x-ordinate (Xi) of the end-point of any specified or measured arc-length s from the y-axis, can be determined by integration of this expression in the form: Xi =
s
‘Ads
0
(7)
a
using the previously established relationship between the strain factor (cr) and the arc-length (s). As an example of this part of the analytical procedure, consider the point CZ on the bedding-trace labelled C on the fold-profile in Fig. 2, The arc-length to this point from the y-axis is s = 22.6. The variation of (x along this arc is shown in Fig. 4B. According to eq. 7, the initial x-ordinate (Xi) of the point C2 is obtained in effect by integration of the area under the curve of l/cr against s from s = 0 to s = 22.6. In this case, numerical integration using the trapezoidal rule gave xi = 25.5, whereas the calculated theoretical value is Xi = 26.0 (Table III, Appendix). The next step in procedure IV is to determine the arc length along an adjacent bedding-trace of an undeformed segment, the end point of which had the same x-ordinate (Xi = 25.5) as that already determined for the reference bedding-trace. This is achieved by first drawing up a curve of xi against arclength along the second bedding-trace, by integration of l/o with respect to s, using the already established variation of (y:against s for this bedding-trace (Fig. 4B). The curve of Xi against s, is then used to determine the arc-length that is related to the particular xi value. Figures 4B and C show data derived in this manner for the bedding-trace labelled d on the fold-profile in Fig. 2. In particular, these data indicate that an initial x-ordinate of Xi = 25.5 is related to an arc-length of s = 23.2 along this bedding-trace. A line joining the end point of this arc-length to the point CZ, has an orientation of /3= 94” as measured from the x-coordinate axis, and this indicates a shear angle of $ = 90” - (/3- 0) = 46.5” at the point Cz, where the measured angle of dip of the bedding-trace is 0 = 50.5”. This estimate of the shear angle ( $I) gives a value of y = 1.0538 for the shear factor, and is compatible with a theoretically computed value of $ = 47.9” (Table XII, Appendix). EVALUATION
OF THE STRAIN-ELLIPSE
ORIENTATION
($) AND AXIAL RATIO
(R)
A~ffly~ic~l procedure
V
This is the final step in an evaluation of the finite strain at a point on the profile-plane. It simply involves use of previous estimates of 01and y to calcu-
210
late values for the major-axis orientation (4) and the axial ratio (R) of the strain ellipse at the point, by means of eqs. 1 and 2 respectively; tan2$=2y/(cz4+Y2-1) where y = tan $, and R* = tan( 9 + @)/tan $. In the case of the point C2 on the fold-profile of Fig. 2, the relevant empirical values are cx= 0.9967 and y = 1.0538 which give R = 2.755 and 4 = 31.3”, whereas the calculated theoretical values at this point are R = 2.890 and 4 = 30.7” (Table III, Appendix). Thus, the fractional error in the determination of R is 0.0467, and in 4 it is 0.0195”. Since the procedure used in the determination of $ did not involve the assumption of alignment of the cleavage-trace with the strain-ellipse major axis, it would seem from the small error in this determination, that the method as a whole provides a technique for critical examination of the widely accepted hypothesis that slaty-cleavage is formed normal to the directions of maximum shortening in natural folds. The Analytical procedures I-V used in a determination of the strain-ratio (R) and strain-orientation (4) at a point, provide the basis for establishing the overall pattern of finite inhomogeneous strain from an array of points on the fold-profile plane. When devising a scheme for repeated application of these procedures, it would be advisable in view of Analytical procedure I, to select an array of points located at the intersections of bedding-traces with a set of lines parallel to the y-coordinate axis (axial plane trace). This type of grid would certainly allow integration of the various procedures into a systematic analytical scheme, which could be applied with the aid of a computer. FOLD GEOMETRY
AND STRAIN
PATTERNS
The method of strain analysis outlined in the preceding sections has been formulated specifically for use with symmetric folds produced by a single phase of tectonic deformation. In theoretical discussion of the kinematics of this type of folding, it is customary to refer the fold-system to a set of external, orthogonal tectonic axes (a, b, c) with the &-plane coincident with the fold-axial plane, and the a-axis representing the direction of maximum tectonic displacement. Unless there is evidence to the contrary, it may be assumed for this type of folding, that the fold axes were more or less parallel to the b-tectonic axis. It follows, that in the case of a planar deformation, the pattern of finite strain obtained by the method from any normal profile-section of the fold wiil be an exact reflection of the inhomogeneous finite deformation. Consider now, the situation where the bulk finite deformation has included a uniform strain in the direction of the fold axis (b-tectonic axis). From theoretical considerations, it can be shown that this component of the finite strain will have been compensated by a uniform dilation or change of scale
211
in the ac-plane, and consequently, it will have had no effect on dimensional ratios or angular relationships in any profile-section of the fold (Matthews et al., 1971, p. 134). The implication here, is that strain analysis of a normal profile-section will provide a complete picture of the finite, fold-forming strain, but it will not of course, indicate the precise magnitude of the strain throughout the fold. In contrast to the situations just discussed, the only other possible type of orientation for a single phase of symmetric folding is where the fold-axis is inclined to the b-tectonic axis within the a&tectonic plane. This situation may be indicated by associated structural features, such as a mineral lineation, in which case it would be possible to select for strain analysis a profile-plane that can be regarded as an ac-tectonic plane. From previous considerations it is evident that this type of profile-plane will provide a record of the foldforming finite deformation within the plane of section. The particular pattern of strain will be applicable to any ac-section of the fold if the structure does not change along the axial-plane. If variations are apparent, then of course, the pattern of strain will be applicable only to the plane of section. The method of strain analysis under discussion has been developed on the assumption that there are no ductility contrasts within the fold-system. Such contrasts are usually expressed by a refraction of slaty-cleavage across boundaries between layers of different lithology. In such cases, the method of analysis can be applied independently to individual units of uniform lithology provided there are a sufficient number of bedding-traces within each unit to allow estimation of p by means of Analytical procedure I. DISCUSSION
AND CONCLUSIONS
This paper has been concerned with the theory and application of a method for quantitative determination of finite, inhomogeneous strain in profiles of single-phase symmetric folds. The method relies only on measurements of the spacing of bedding-traces and the relative orientations of the cleavage and bedding traces exposed on the plane of section. The various empirical steps involved in the method have been described under the respective subheadings of Analytical procedures I-V. Application of these procedures to an ideal fold-model has in each case provided results that are compatible with calculated theoretical values. Determination of the magnitude and orientation of the finite-strain at any point on the fold-profile can be carried out expeditiously with the aid of a pocket calculator. When using the method to establish the overall pattern of strain from an array of points on a fold-profile it would be advantageous of course, to use a digital computer with output to a graphical plotter. It is appreciated, that the method as a whole would be enhanced if it included internal checks on the basic assumptions. In this connection, we are investigating the possibility of formulating an undeforming procedure for restoring the folded bedding-traces to their original configuration in terms
212
of the pattern of finite strain established for the plane of section together with any rigid body rotations that may have been involved. Such a procedure would at least provide an internal check on the assumption of an initial parallel orientation of the bedding-traces. It would also provide a basis for kinematical interpretation of strain-patterns established by the method. In relation to modem developments in the study of folded rocks, the potential value of the method of analysis presented here, is that it would allow quantitative comparison of the patterns of finite strain in various types of naturally occurring symmetric folds with the patterns derived from theoretical and experimental models. Finally, within the limits of error inevitably involved in measurements of geological structures, the method could be used for assessment of the empirically well-established hypothesis that slaty-cleavage is formed normal to the directions of maximum shortening in folded rocks. APPENDIX Derivation of geometric
and strain parameters of a flattened concentric
fold
The objective is to establish the fold-profile, and the distribution of finite, inhomogeneous strain for a plane, symmetric, uniformly flattened concentric flow-fold (Fig. 2). The fold i, derived from bedding-planes initially perpendicular to the y-axis of a rectangular xycoordinate frame in the profile-plane. It is assumed that the cleavage-traces are aligned with the directions of maximum elongation in the profile-plane.
Y
I
A
-
-
,A--
C'(0, K,)
e
I \
\
Po(xo. Yo)
(O,Yo)
(0, KY,)
I
-0
X
C)
X
Fig. 5. A. Geometric features of a concentric fold with circular arcs. B. Shows additional effect of flattening with an extension factor K along the trace of the axial-plane (y-coordinate axis).
213 Coordinates of a point before and after the deformation Consider first, the partial profile of a concentric flow-fold as shown in Fig. 5A. The common centre for the circular arcs of the bedding-traces is located on the y-axis at C (0, r). Any point Pe(xe,y~) on a bedding-trace initially parallel to the x-axis, is located after the folding at the point P’ with coordinates: x’ = (r - ya) sin X y’ = r-(r-ya)cosX where x is the angle between CP’ and the y-axis. Since concentric volve any change of dimension along the bedding-traces: x = x&r
folding does not in-
- yo) in radians
Now, consider conservation and fold axial-plane). point P originally
(3)
the effect of a superimposed uniform flattening (pure shear), with area the direction of maximum extension parallel to the y-axis (trace of the If the extension factor is K, then it is clear from Figs. 5A and B that the at (xe,y~) will have coordinates:
x=K-‘(r-ye)sinX
(9)
y = K[r - (r - ya) cos X]
It follows that the fold-profile can be constructed if K and r are specified. In the case of the profile-fold in Fig. 2, r = 56, and K = 1.25, or l/K = 0.8 which implies a 20% shortening parallel to the x-axis. Angle of dip (8) at a point As indicated in Figs. 6A and B the angle of dip (8) of any line segment centred at a point, say P, is by definition given by: 8 = ar~tan(dy~~~
(10)
In order to determine 8 from given values of K, r and the initial coordinates of P(xe,ye), the above expression can be written as: dyldx tan 0 = d&dX which, on evaluation using eq. 9 gives: tan0 =
K’(r-ye)sinx
(r-y0)cosx
or: tan0=K2
tanx
(11) (12)
so 0 can be determined from the given value of K and the value of x obtained by means of eq. 8. Determination
of (Y and p
By definition, CY,at any point on a bedding-trace of the fold-profile, is the ratio of the final to the initial length of any infinitesimal line segment initially parallel to the x-axis.
214 Using the notation given in Figs. 6A, B: a: = AB/AaBa
= AB/dxa
(13)
and: AB* = dx* + dy’ or: AB = Jl
+ (dy/dx)*dx
(14)
Now, from eqs. 9 and 8 respectively: dx = K-‘(r - ya) eos xdx dx = dxa /(r - ya ), . . . (ya constant) so: dx = K-’ cos xdxa This expression combined with eqs. 10, 13 and 14 gives: QL= ABfdxa = v/l + tan*B(K-’ cos x) and simplification, using 1 + tan20 = l/cos2@, gives:
xicose)
Q: = ~~~~~~
At any point on the trace of the axial plane (y-coordinate eq. 15 reduces to:
cto=K-’
05) axis), x = 0, and 8 = 0, hence,
(16)
From eqs. 15 and 16 it follows that: 01 -=ue
cos x
07)
case
and using eq. 3:
p = (cos
x)-l
~eter~inat~o~
(18) of the shear factor ‘y, and the shear angle tt/
As noted previously y = tan 9, and $ describes the rotation of lines initially parallel to the y’-coordinate axis of an elemental unit of finite homogeneous deformation, with the X’S& aligned with the bedding-trace before and after the deformation (Fig. 1). As indicated in Figs. 6C-E the angular relationships of these line elements can be examined in terms of two vectors attached to a material point (P). If the linear transformation from initial (xe, ye) to final (x, y) coordinates is expressed as: x = f(x0. Yo) Y = g@o, Yo) then the initial vectors: (ckc~, O>,and (0, dya) are transformed to: (f,o,g,o)
Got and (fyo, g,a) d,a
(19)
215
a
40
--~.,---_
00
PO
__-/
dx
I
4
dxo
I
?
I
X
0
X
c \\
ho, 0)
L
&
E
$
\
‘\ 90’
‘1 --
(dxo,O)
A
--
e --
PO
x
x
0
0
X
Fig. 6. Initial (A) and final (B) configurations of a line element on a folded-bedding trace. C-E, transformation of two initially perpendicular line elements by a finite homogeneous deformation. These vectors represent the sides of an elemental unit (cf. Fig. 1). Note that vector (dxe, 0) is aligned with the same bedding-trace before and after the deformation. $ is the shear angle, and 0 is the orientation of the bedding-trace at the point P.
where (a, b) = ai + bj Now, from Fig. 6E it is evident that: tan J/ = tan[?T/:! - (A - 0)] = l/tan(A=
0)
1 + tan A tan 8
(20)
tan A - tan 8
But: tan A = gyolfyo
tan 0 so
=gxOlfxO
substitution
tan $ =
in eq. 20 and simplification,
fxofyo + &dyo fxogyo- fYogX0
gives:
(21)
216 However, the denominator in this expression is the determinant of the transformation matrix of eq. 19, and is unity for an area conserving transformation, so eq. 21 reduces to: tan \Ei=
fXOfyO +gXOgyO
The functions
f
(221
and g in this expression are defined by eq. 9 as:
f=K’(r-yo)sinx (231
g=Kr-K(r-ya)cosX We note from eq. 8 that: x=x0/@-~0)
so :
X0
3X _-xv
aYO (r ax _-=ax0
yo)2 (24)
I r-y0
These expressions may be used to establish the derivatives of f and g in eq. 22, which are obtained from eq. 23 by partial differentiation of x with respect to xe or ye, which yields: f,0 =K-'tr-Yo)cosx(r-y0)-'
=iT’cosx fyo =-K'
=- K1
sin X + K-l(r-ya) sin X + K-‘&d
cosXxa(r-Y~)-~ Cos X
g,e = K(r - ye) sin X(r - ya)‘-l = K sin X g,,o = K cos X + K(r - yo) sin xxo(r - y(j)_2 =KcosX+KXdsinX Substitution of these expressions in eq. 22 with simplification, leads finally to: y = tan J, = KM2X,adcos’x + K2Xrad sin’x + (K2 - K2) Strain ellipse axial-ratio (R) and orientation
sin X cos X
(25)
(4)
The values of @ and R for a particular elemental unit are determined by means of eqs. 1 and 2 respectively, using the atready established values of CY,‘y and $. Table III lists the computed values of various geometric and strain parameters for an initially square grid of points defined by the initial positions of the points co-c4 and the undeformed bedding-trace d of the fold-profile in Fig. 2. These data were used in the construction of this fold-profile and the trajectories of the maximum extension directions.
217 TABLE III Geometric and strain parameters at points on bedding-traces c and d of fold-profile
x0
Yo
x
Y
e
a
7
0.800 0.859 0.996 1.128 1.226 0.800 0.862 1.053 1.190 1.250
0 0.529 1.108 1.668 2.193 0 0.636 1.342 1.983 2.447
0 CO Cl c2 c3 c4 d0
dr dz da d4 -
0 12.95 25.96 38.00 51.00 0 12.95 25.95 38.00 51.00
17.00 17.00 17.00 17.00 17.00 23.30 23.30 23.30 23.30 23.30
0 10.17 19.27 25.81 30.13 0 10.09 18.65 24.01 26.16
21.25 23.91 31.66 42.62 57.32 29.13 32.29 41.34 53.75 69.54
0 28.3 50.8 66.5 80.2 0 33.2 57.8 74.5 89.6
-+ (“) 0 27.9 47.9 59.1 65.5 0 32.5 53.3 63.2 67.8
in Fig. 2
Q, (“1
R
90 49.7 30.7 22.2 18.0 90 46.0 26.5 19.4 16.7
1.563 1.964 2.890 4.002 5.197 1.563 2.179 3.339 4.676 5.884
x0, YO = initial coordinates of the point. x, Y = final coordinates of the point. 0 = angle of dip of bedding-trace. (Y = strain factor along bedding-trace. = shear factor along bedding-trace. = shear angle = arctan 7. = orientation of strain-ellipse major axis from bedding-trace. ! R = axial ratio of strain ellipse.
REFERENCES Bayly, B., 1974. Cleavage not parallel to finite-strain ellipsoid’s XY-plane. Discussion. Tectonophysics, 23: 205-207. Billings, M.P., 1972. Structural Geology. Prentice-Hall, New Jersey, 606 pp. Elliot, D., 1972. Deformation paths in structural geology. Geol. Sot. Am. Bull., 83: 2621-2638. Matthews, P.E., Bond, R.A.B. and Van den Berg, J.J., 1971. Analysis and structural implications of a kinematic model of similar folding. Tectonophysics, 12: 129-154. Matthews, P.E., Bond, R.A.B. and Van den Berg, J.J., 1974. An algebraic method of strain analysis using elliptical markers. Tectonophysics, 24: 31-67. Ramsay, J-G., 1967. Folding and Fracturing of Rocks. McGraw Hill, New York, 568 pp. Siddans, A.W.B., 1972. Slaty-cleavage - a review of research since 1815. Earth-Sci. Rev., 205-232. Williams, P-F., 1976. Relationships between axial-plane foliations and strain. Tectonophysics, 30: 181-196. Wood, D.S., 1974. Current views of the development of slaty cleavage. Ann. Rev. Earth Planet. Sci., 2.
199
STRAIN ANALYSIS OF SYMMETRIC FOLDS USING SLATY-CLEAVAGE
P.E. MATTHEWS, R.A.B. BOND and J.J. VAN DEN BERG University
of Natal, Durban
(South
Africa)
(Submitted January 6, 1976; revised version accepted May 21, 1976)
ABSTRACT Matthews, P.E., Bond, R.A.B. and Van den Berg, J.J., 1976. Strain analysis of symmetric folds using slaty-cleavage. Tectonophysics, 34: 199-217. Despite some uncertainty concerning its use as an indicator of strain orientation in folded rocks, slaty-cleavage is generally regarded as a penetrative tectonic fabric that is related to the symmetry of inhomogeneous, finite strain within a fold system. On the basis of this concept, a quantitative method is developed for establishing the distribution of finite strain on the profiles of symmetric cylindrical folds using the relative orientations of cleavage and b,edding traces on the plane of section. The method also provides a technique for critical examination of the widely-accepted hypothesis that slaty-cleavage is formed perpendicular to the directions of maximum shortening in folded strata. The analytical procedures are demonstrated and tested by application to a mode1 of an idea1 flattened concentric fold.
INTRODUCTION
Quantitative determination of the patterns of finite inhomogeneous strain in naturally occurring folds could provide valuable information on the modes of formation of the fundamental forms of folding now recognized in structural geology. Although methods of strain analysis are available for use with dimensional markers such as deformed fossils or ellipsoidal particles, these methods are of restricted application in studies of folded strata due mainly to the scarcity of such markers in suitable structural settings. In contrast, slaty-cleavage is of common occurrence in folded rocks of low to high grade metamorphic terrains, and represents a penetrative tectonic fabric defined by a preferred dimensional orientation of mineral components. Moreover, the overall patterns of slaty-cleavage generally display a systematic relationship to the geometry of the associated fold-systems (Ramsay, 1967; Billings, 1972). On account of these features, slaty-cleavage can be regarded prima facie as a structure of considerable potential value in finite-strain analysis of natural folds. Recently, Siddans (1972) and Wood (1974) have traced the history of
200
research on slaty-cleavage, and have reviewed an impressive assemblage of published, structural evidence which suggests that slaty-cleavage is formed perpendicular to the directions of maximum shortening in deformed rocks. In other words, the cleavage is developed parallel to the XY-plane of the strain ellipsoid (X > Y > 2). However, examination of this empirical concept from a theoretical point of view (Ramsay, 1967; Matthews et al., 1971; Elliot, 1972; Bayly, 1974; Williams, 1976) indicates that under conditions of a progressive, non-coaxial (rotational) strain, as for example within the limb of a fold, the XY-plane of the strain ellipsoid will not coincide with the same material plane during successive increments of strain. Nevertheless, material planes coincident with the XY-plane at any stage of the deformation will rotate in the same sense as the XY-plane, and in general, will have a very restricted range of orientations close to that of the XY-plane of the finitestrain ellipsoid (Matthews et al., 1971, pp. 148-151). Preliminary investigation of this effect in the case of progressive, plane, non-coaxial homogeneous deformations indicates that the maximum possible angular deviation is only about five degrees. In view of this it seems possible either, that recrystallization during syntectonic growth of a mica fabric will allow slaty-cleavage to develop parallel to the XY-plane or, that the degree of preferred orientation is not sufficiently well-defined to allow accurate measurement of any angular deviation from the finite XY-plane as defined for example by deformed dimensional markers. These possibilities have been discussed in some detail by Williams (1976). A more important theoretical implication, however, is that regardless of whether or not slaty-cleavage is developed parallel to the XY-plane, the cleavage orientation is related to the symmetry of the finite-strain distribution within a fold. This concept is supported by much published structural evidence which shows, as mentioned already, that in general, patterns of slaty-cleavage have a systematic relationship to the associated fold geometry, and are symmetric about the fold-axial plane. This widely accepted concept is used in this paper to develop a quantitative method for finite-strain analysis of symmetric cylindrical folds using the measured orientations of the traces of slaty-cleavage and bedding exposed on any normal profile-section of the folds. In this context, the symmetry conceptof slaty-cleavage implies that any angular divergence between the cleavage and the finite XY-plane in the limbs of a symmetric fold will decrease to zero towards the fold-axial plane, where cleavage formation will have been restricted to the same material plane coincident with the XY-plane throughout the deformation. In order to demonstrate and test the analytical procedures, the method is applied to a mathematically derived profile-section of an ideal, flattened concentric flow-fold with an associated pattern of slaty-cleavage (Fig. 2). It should be emphasized, that although the cleavage pattern on this fold-profile was constructed using the current hypothesis of cleavage formation normal to the direction of maximum shortening within a fold, the method of strain
201
analysis presented in this paper is not dependent on this hypothesis. On the contrary, this hypothesis was used in the construction of the model foldprofile (Fig. 2) simply because no alternative hypothesis of cleavage-formation is available, and in addition it conforms to the symmetry concept of slaty-cleavage which alone forms the basic assumption in the method. Derivations of the equations used to establish the geometric and strain parameters of the model fold-profile (Fig. 2) are presented in an Appendix to this paper. PRINCIPLE
OF THE METHOD
In developing the theory of the method, attention is directed initially to symmetric cylindrical folds formed from originally parallel layers, by a volume conserving, planar deformation acting normal to the fold-axis. It is also assumed initially, that there are no ductility contrasts within the fold-system. These assumptions imply that the finite, inhomogeneous strain produced by the fold-forming deformation will be confined to normal profile planes of the folds. For the purpose of analysis, a symmetric synformal fold-profile is referred to an external xy-coordinate frame with the y-axis along the trace of the axial plane, and the x-axis tangential to the fold-hinge (Figs. 1 and 2). In accord with the usual approach to analysis of inhomogeneous strain, the pattern of strain is examined in terms of elements that are small enough to be regarded as units of finite homogeneous strain. This implies that within each elemental unit parallel straight lines have remained straight and parallel throughout the deformation. An example involving two elements, each derived from a unit square is shown in Fig. 1. In this example, the base line of each elemental unit is aligned along the bedding-trace which of course, is a visible material line before and after the deformation. In a previous study (Matthews et al., 1974) we have shown that selection of the bedding-trace as the X’-axis of an internal x’y’-coordinate system for each elemental unit, allows description of any planar finite homogeneous deformation in terms of two algebraic parameters (Yand y. The cu-parameter is a strain factor defining a change of dimension in the $-direction of any elemental unit, and represents an extension when 01> 1 and a shortening when (y.< 1. By implication, the a-parameter records any finite change in the length of line segments along the arc of a folded bedding-trace. The y-parameter is a factor of shear in the x’-direction and is related by y = tan $ to the shear angle $, which records a rotation of lines initially parallel to the y’-axis of the relevant elemental unit. It may be noted, that prior to deformation the $-axis of each elemental unit of a symmetric fold will be parallel to the external y-coordinate axis. The relationships under discussion are shown diagrammatically in Figs. 1A and B. We have also derived equations that relate the algebraic parameters CYand y of any finite homogeneous deformation to the axialratio (R) and the
4
INITIAL
B
FINAL
Fig. 1. Diagrammatic comparison of two elemental units (ae and al) of finite homogeneous strain within a symmetric fold before (A) and after (B) the deformation. Observable geometric parameters at a point are 8 (orientation of bedding-trace), t (orientation of cleavage-trace), t (orthogonal bedding-thickness), and ‘I (axial-plane bedding-thickness). In Fig. lB, (Yand y are algebraic strain parameters for elemental units of initial unit dimension, in relation to the internal coordinate axes (x’, Y’) of each unit.
orientation (4) of the major axis of the strain ellipse in relation to the x’axis of the relevant elemental unit (Matthews et al., 1974, pp. 38-39). Two of these equations are of interest in the present context. The first is:
(1) where y = tan $ ; and the second is: cos $I + y cos*f#~ R2 = sin C#J sin C#J cos 4 - y sin’@ which can also be expressed as:
Under the symmetry concept of slaty-cleavage, the orientation of the cleavage-trace on the profile of a symmetric cylindrical fold will be a continuous function of the fold-geometry, and in particular, the cleavage-trace along the axial plane will be parallel to the XY-plane of the finite-strain ellipsoid.
Use of these hypothetical relationships together with measurements of the relative orientation and spacing of the bedding-traces, allows interpolative determinations of the CYand y parameters of indivudual elemental units. This leads to direct evaluation, by means of eqs. 1 and 2, of the orientation (4) and the magnitude of the finite strain, as expressed by the axial ratio (R) of the strain ellipse, for the associated elemental unit. From a determination of the CYand y parameters at a suitable number of points, it should be possible to establish the overall pattern of finite strain for the fold-profile. An attempt is made in the following discussion to draw a distinction
Bedding
trace
----------Cleavagetrace
I
01
I
10
I
,
I
I
20
30
40
50
-
X
Fig. 2. Traces of bedding and slaty-cleavage (dashed lines) on the profile of an ideal, symmetric, flattened concentric flow-fold. It is assumed that the cleavage traces are aligned with directions of maximum extension; however, as explained in the text, this assumption is not necessary to the method of finite-strain analysis under discussion. Equations for derivation of the geometric and strain parameters are presented in an Appendix.
204
between the theoretical and empirical aspects of the method. In particular the sequence of steps and the computational formulae required in an application of the method are described under the respective sub-headings of Analytical procedures I-V. EVALUATION
OF THE STRAIN FACTOR (a)
Theoretical considerations The spacing of bedding-traces on a fold-profile can be recorded by measuring either the orthogonal (or bedding-normal) thickness, denoted by t, or the thickness parallel to the axial plane, denoted by T. In the case of a symmetric fold, these spacings for any particular layer are equal along the trace of the axial plane (external y-axis), and this may be expressed as to = To. By analogy, the t and 2’ parameters may be used for geometrical comparison of elemental units located along the same bedding-trace, with the assumption that these units had initial unit dimensions. It is of interest to make such a comparison between an element (a,,) located on the trace of the axial plane and any other element (at) on the same bedding-trace (a), as indicated in Fig. 1B. On account of conservation of the initial unit area of each element; it is evident from Fig. lB, that: t fl0 -=t al
&Xl %o
However: t 00
=
Tao
t al
=
Ta,
~0s
001
where 19~~ is the dip of the bedding-trace at a I* It follows, that the infinitesimal T,,/T,, ratio denoted by pal at the point a,, is related to the relevant aparameters, and this can be expressed as: T PO1=<=“o
_ %l
1
i COStfal
1
In order to solve for ar,, in this expression, it is necessary to establish values for pa1 and eao. In both cases, this can be achieved by inte~olative procedures using measurable data from the fold-profile, as explained in the following discussion. Analy tieal procedure I It is clear that when any two adjacent bedding-traces of the fold-profile are condensed together in a direction parallel to the y-axis, that is to say as 2’
205
and To go to zero, the T/T, ratio (p) at any fixed point will tend to a limit. This may be expressed as:
Empirical estimation of this limit, which may be referred to as Analytical procedure I, simply involves measurement of the differential T-spacing (AT) of a number of bedding-traces from a fixed point, together with measurements of the corresponding differential To-spacings (AT,) of these beddingtraces along the trace of the axial plane (y-coordinate axis). Then a plot of AT/AT, against AT will produce a trend line that cuts the line AT = 0 at the required interpolative estimate of p, which is applicable to the particular point on the fold-profile. An application of this procedure with reference to the point CZ on the fold-profile of Fig. 2 is illustrated by the numerical data given in Table 1, together with the plot of AT/AT, against AT shown in Fig. 3A. This plot gives a value of p = 1.270, which is obviously a good estimate of the calculated theoretical value of p = 1.272. (eqs. 8 and 18, Appendix). In order to facilitate strain analysis of the fold-profile, this procedure is repeated for different points along individual bedding-t~ces. The results can then be used to establish curves for p against arc length(s) measured from the y-axis, or alternatively, to establish curves for p against the x-ordinate along each bedding-trace (Fig. 3B). Analytical procedure II It is of interest now, to consider an interpolative scheme for evaluation of the strain factor ((Y,,)at the point of intersection of any bedding-trace
TABLE I Data used in estimation of p at point c2 in Fig. 2 Bedding trace
y-value forx = 0
(‘)AT~ from y = 21.3
y-value for x = 19.3
(‘)AT from y = 31.5
AT ATo i._
: c b a
36.8 29.0 21.3 13.1 4.8
15.5 7.7 0 -6.2 -16.5
41.9 55.1 31.5 21.5 12.0
(I) ATO from point co where y = 21.3. (2) AT from point c2 where y = 31.5.
23.6 10.4 0 -10.0 -19.5
1.522 1.351 1.220 1.182
206
P
x - ordlnote
AT
Fig. 3. Graphical aspects of Analytical procedure 1. A. Interpolative estimation of p at the point c2 on the fold-profile in Fig. 2, using values given in Table I. B. Variation of p along bedding-traces a-e on the fold-profile in Fig. 2.
with the y-coordinate axis. This scheme relies on the fact that, on the profile of a symmetric fold, the shear angle (J/) of each elemental unit along a bedding-trace will decrease to zero as these units are traced in succession towards the trace of the axial plane (y-coordinate axis). This is significant on account of the relation of Jt to Q!in eq. 1 for a given value of #, with the implication that an estimate of a0 can be obtained by a limiting procedure. Two steps are involved. In the first step, an estimate of a! is obtained by means of eq. 1, for a number of points along a bedding-trace, using the generally erroneous assumptions that at each point: (1) the angle of the cleavage-trace from the bedding-trace (t) gives the orientation of the strainellipse major-axis (~$1;and (2) the dip of the bedding-trace (8) is an exact reflection of the angular shear strain (li/) for the associateci elemental unit. On this basis, each estimate of CY,denoted by &, is computed from a modified form of eq. 1, which is: 8;=
tan22i t (tan% tan
[ 2 tan
-1 *‘4
1)
In the second step of this procedure, the ratio a& at each point is determined by means of eq. 3, using the relevant value of p provided by Analytics procedure I, together with the measured angle of dip (0) of the bedding-trace at each point. In each case, the computed values of two/aand 6 immediately lead to estimates of CY~, denoted by Go, in the form: 010
(1
ii0 cef -
a
ii
N
p
cos
eii
(5)
207 TABLE II Data used in estimation of tlls at point c2 in Fig. 2.
Cl
cz c3
x
t=ql”
8”
P ____.~
&
aola
go
10.2 19.3 25.8
49.7 30.7 22.2
28.3 50.8 66.5
1.058 1.272 1.781
0.854 0.956 0.792
0.932 0.804 0.710
0.796 0.768 0.562
From previous considerations, it follows that a plot of Go against the xcoordinate of the relevant reference point will define a curve which will be symmetric about x = 0, and will cut this line at a point representing the required interpolative dete~ination of eo. To illustrate these aspects of Analytical procedure II, data from points along the bedding-trace labelled C in Fig. 2, are presented in Table II, and the resulting graph of go versus x is shown in Fig. 4A. This plot gives e. = 0.8, which is in fact the value that was used in the construction of the fold-profile. Analytical procedure III This is essentially a computational procedure for establishing values for the strain factor (cx) at points along individual bedding-traces on the foldprofile. The value of Q at each point is determined, using eq. 3 in the form: a: = &)/f/J cos 8)
(6)
x- ordinate arc- length
(s)
arc - length
(sd
1
Fig. 4. Graphical aspects of Analytical procedures II and IV. A. Interpolative estimation of the strain factor (a~) along bedding-trace C (Fig. 2) at its intersection point with the axial-plane (y-coordinate axis, x = 0). The curve of&e against x is symmetric about x = 0. B. Variation of strain factor ff along bedding-traces a-e on the fold-profile in Fig. 2. C. Plot of estimated initial x-ordinate (xi) against arc-length (sd) along bedding-trace d on the fold-profile in Fig. 2. The arc length related to Xl = x,2 allows determination of the shear factor (7) at the point cs, as part of Analytical procedure IV.
208
by substitution of the values obtained previously for p and CX,,by means of Analytical procedures I and II, respectively, together with the previously measured angle of dip (0) of the bedding-trace at that point. In the case of the point C, on the fold-profile in Fig. 2, the relevant values for use of eq. 6 are p = 1.270, czo= 0.8 and 0 = 50X’, so that the empirical value obtained for the strain factor is (Y= 0.9967, whereas the calculated theoretical value is (Y= 0.9959 (Table III, Appendix). It may be noted from eq. 6 that a0 is a constant factor in an evaluation of the longitudinal strain (a) at points on an individual bedding-trace. Moreover, the values of p and 8 required in such an evaluation could be readily obtained from previously established curves for these parameters against either the arc length or the x-ordinate along the bedding-trace (Fig. 3B). Data derived by the method are presented as curve c in Fig. 4B, together with similar curves recording the variation of finite strain (a) along various bedding-traces of the fold-profile in Fig. 2. As explained later, curves of this type will also allow determination of the original x-ordinate of individual points on the bedding-traces of the fold profile. EVALUATION
OF THE SHEAR
FACTOR
( y)
Theore tical considerations It will be recalled that the shear factor 7, for each elemental unit is defined as y = tan $ where $ defines the final orientation of a line that was initially parallel to the y-coordinate axis. This implies that before the deformation this line (L in Fig. 1A) would have intersected adjacent bedding-lines at points with the same x-ordinate (Xi), and hence, would have defined straight bedding-line segments of equal length as measured from the y-axis, After the folding, these segments will in general have been transformed to arcs of unequal length, but nevertheless, with a definite relationship governed by the angle of shear ($) of the original orthogonal line (L’ in Fig. 1B). From the previously established variation of the strain factor (a) along particular bedding-traces it is possible to devise a scheme (Analytical procedure IV) for establishing arc-lengths along adjacent bedding-traces that have been derived from initial segments of equivalent length. These arc-lengths will then indicate a value for the shear angle ($) and hence, the shear factor y = tan $ for the associated elemental unit. Analytical
procedure
IV
As mentioned previously, this procedure is concerned with the determination of arc-lengths of adjacent bedding-traces that have been derived from initial line-segments of equivalent length.
209
By definition, the incremental arc-length (ds) along any bedding-trace of the fold-profile is given by: ds = odxi so that, the initial x-ordinate (Xi) of the end-point of any specified or measured arc-length s from the y-axis, can be determined by integration of this expression in the form: Xi =
s
‘Ads
0
(7)
a
using the previously established relationship between the strain factor (cr) and the arc-length (s). As an example of this part of the analytical procedure, consider the point CZ on the bedding-trace labelled C on the fold-profile in Fig. 2, The arc-length to this point from the y-axis is s = 22.6. The variation of (x along this arc is shown in Fig. 4B. According to eq. 7, the initial x-ordinate (Xi) of the point C2 is obtained in effect by integration of the area under the curve of l/cr against s from s = 0 to s = 22.6. In this case, numerical integration using the trapezoidal rule gave xi = 25.5, whereas the calculated theoretical value is Xi = 26.0 (Table III, Appendix). The next step in procedure IV is to determine the arc length along an adjacent bedding-trace of an undeformed segment, the end point of which had the same x-ordinate (Xi = 25.5) as that already determined for the reference bedding-trace. This is achieved by first drawing up a curve of xi against arclength along the second bedding-trace, by integration of l/o with respect to s, using the already established variation of (y:against s for this bedding-trace (Fig. 4B). The curve of Xi against s, is then used to determine the arc-length that is related to the particular xi value. Figures 4B and C show data derived in this manner for the bedding-trace labelled d on the fold-profile in Fig. 2. In particular, these data indicate that an initial x-ordinate of Xi = 25.5 is related to an arc-length of s = 23.2 along this bedding-trace. A line joining the end point of this arc-length to the point CZ, has an orientation of /3= 94” as measured from the x-coordinate axis, and this indicates a shear angle of $ = 90” - (/3- 0) = 46.5” at the point Cz, where the measured angle of dip of the bedding-trace is 0 = 50.5”. This estimate of the shear angle ( $I) gives a value of y = 1.0538 for the shear factor, and is compatible with a theoretically computed value of $ = 47.9” (Table XII, Appendix). EVALUATION
OF THE STRAIN-ELLIPSE
ORIENTATION
($) AND AXIAL RATIO
(R)
A~ffly~ic~l procedure
V
This is the final step in an evaluation of the finite strain at a point on the profile-plane. It simply involves use of previous estimates of 01and y to calcu-
210
late values for the major-axis orientation (4) and the axial ratio (R) of the strain ellipse at the point, by means of eqs. 1 and 2 respectively; tan2$=2y/(cz4+Y2-1) where y = tan $, and R* = tan( 9 + @)/tan $. In the case of the point C2 on the fold-profile of Fig. 2, the relevant empirical values are cx= 0.9967 and y = 1.0538 which give R = 2.755 and 4 = 31.3”, whereas the calculated theoretical values at this point are R = 2.890 and 4 = 30.7” (Table III, Appendix). Thus, the fractional error in the determination of R is 0.0467, and in 4 it is 0.0195”. Since the procedure used in the determination of $ did not involve the assumption of alignment of the cleavage-trace with the strain-ellipse major axis, it would seem from the small error in this determination, that the method as a whole provides a technique for critical examination of the widely accepted hypothesis that slaty-cleavage is formed normal to the directions of maximum shortening in natural folds. The Analytical procedures I-V used in a determination of the strain-ratio (R) and strain-orientation (4) at a point, provide the basis for establishing the overall pattern of finite inhomogeneous strain from an array of points on the fold-profile plane. When devising a scheme for repeated application of these procedures, it would be advisable in view of Analytical procedure I, to select an array of points located at the intersections of bedding-traces with a set of lines parallel to the y-coordinate axis (axial plane trace). This type of grid would certainly allow integration of the various procedures into a systematic analytical scheme, which could be applied with the aid of a computer. FOLD GEOMETRY
AND STRAIN
PATTERNS
The method of strain analysis outlined in the preceding sections has been formulated specifically for use with symmetric folds produced by a single phase of tectonic deformation. In theoretical discussion of the kinematics of this type of folding, it is customary to refer the fold-system to a set of external, orthogonal tectonic axes (a, b, c) with the &-plane coincident with the fold-axial plane, and the a-axis representing the direction of maximum tectonic displacement. Unless there is evidence to the contrary, it may be assumed for this type of folding, that the fold axes were more or less parallel to the b-tectonic axis. It follows, that in the case of a planar deformation, the pattern of finite strain obtained by the method from any normal profile-section of the fold wiil be an exact reflection of the inhomogeneous finite deformation. Consider now, the situation where the bulk finite deformation has included a uniform strain in the direction of the fold axis (b-tectonic axis). From theoretical considerations, it can be shown that this component of the finite strain will have been compensated by a uniform dilation or change of scale
211
in the ac-plane, and consequently, it will have had no effect on dimensional ratios or angular relationships in any profile-section of the fold (Matthews et al., 1971, p. 134). The implication here, is that strain analysis of a normal profile-section will provide a complete picture of the finite, fold-forming strain, but it will not of course, indicate the precise magnitude of the strain throughout the fold. In contrast to the situations just discussed, the only other possible type of orientation for a single phase of symmetric folding is where the fold-axis is inclined to the b-tectonic axis within the a&tectonic plane. This situation may be indicated by associated structural features, such as a mineral lineation, in which case it would be possible to select for strain analysis a profile-plane that can be regarded as an ac-tectonic plane. From previous considerations it is evident that this type of profile-plane will provide a record of the foldforming finite deformation within the plane of section. The particular pattern of strain will be applicable to any ac-section of the fold if the structure does not change along the axial-plane. If variations are apparent, then of course, the pattern of strain will be applicable only to the plane of section. The method of strain analysis under discussion has been developed on the assumption that there are no ductility contrasts within the fold-system. Such contrasts are usually expressed by a refraction of slaty-cleavage across boundaries between layers of different lithology. In such cases, the method of analysis can be applied independently to individual units of uniform lithology provided there are a sufficient number of bedding-traces within each unit to allow estimation of p by means of Analytical procedure I. DISCUSSION
AND CONCLUSIONS
This paper has been concerned with the theory and application of a method for quantitative determination of finite, inhomogeneous strain in profiles of single-phase symmetric folds. The method relies only on measurements of the spacing of bedding-traces and the relative orientations of the cleavage and bedding traces exposed on the plane of section. The various empirical steps involved in the method have been described under the respective subheadings of Analytical procedures I-V. Application of these procedures to an ideal fold-model has in each case provided results that are compatible with calculated theoretical values. Determination of the magnitude and orientation of the finite-strain at any point on the fold-profile can be carried out expeditiously with the aid of a pocket calculator. When using the method to establish the overall pattern of strain from an array of points on a fold-profile it would be advantageous of course, to use a digital computer with output to a graphical plotter. It is appreciated, that the method as a whole would be enhanced if it included internal checks on the basic assumptions. In this connection, we are investigating the possibility of formulating an undeforming procedure for restoring the folded bedding-traces to their original configuration in terms
212
of the pattern of finite strain established for the plane of section together with any rigid body rotations that may have been involved. Such a procedure would at least provide an internal check on the assumption of an initial parallel orientation of the bedding-traces. It would also provide a basis for kinematical interpretation of strain-patterns established by the method. In relation to modem developments in the study of folded rocks, the potential value of the method of analysis presented here, is that it would allow quantitative comparison of the patterns of finite strain in various types of naturally occurring symmetric folds with the patterns derived from theoretical and experimental models. Finally, within the limits of error inevitably involved in measurements of geological structures, the method could be used for assessment of the empirically well-established hypothesis that slaty-cleavage is formed normal to the directions of maximum shortening in folded rocks. APPENDIX Derivation of geometric
and strain parameters of a flattened concentric
fold
The objective is to establish the fold-profile, and the distribution of finite, inhomogeneous strain for a plane, symmetric, uniformly flattened concentric flow-fold (Fig. 2). The fold i, derived from bedding-planes initially perpendicular to the y-axis of a rectangular xycoordinate frame in the profile-plane. It is assumed that the cleavage-traces are aligned with the directions of maximum elongation in the profile-plane.
Y
I
A
-
-
,A--
C'(0, K,)
e
I \
\
Po(xo. Yo)
(O,Yo)
(0, KY,)
I
-0
X
C)
X
Fig. 5. A. Geometric features of a concentric fold with circular arcs. B. Shows additional effect of flattening with an extension factor K along the trace of the axial-plane (y-coordinate axis).
213 Coordinates of a point before and after the deformation Consider first, the partial profile of a concentric flow-fold as shown in Fig. 5A. The common centre for the circular arcs of the bedding-traces is located on the y-axis at C (0, r). Any point Pe(xe,y~) on a bedding-trace initially parallel to the x-axis, is located after the folding at the point P’ with coordinates: x’ = (r - ya) sin X y’ = r-(r-ya)cosX where x is the angle between CP’ and the y-axis. Since concentric volve any change of dimension along the bedding-traces: x = x&r
folding does not in-
- yo) in radians
Now, consider conservation and fold axial-plane). point P originally
(3)
the effect of a superimposed uniform flattening (pure shear), with area the direction of maximum extension parallel to the y-axis (trace of the If the extension factor is K, then it is clear from Figs. 5A and B that the at (xe,y~) will have coordinates:
x=K-‘(r-ye)sinX
(9)
y = K[r - (r - ya) cos X]
It follows that the fold-profile can be constructed if K and r are specified. In the case of the profile-fold in Fig. 2, r = 56, and K = 1.25, or l/K = 0.8 which implies a 20% shortening parallel to the x-axis. Angle of dip (8) at a point As indicated in Figs. 6A and B the angle of dip (8) of any line segment centred at a point, say P, is by definition given by: 8 = ar~tan(dy~~~
(10)
In order to determine 8 from given values of K, r and the initial coordinates of P(xe,ye), the above expression can be written as: dyldx tan 0 = d&dX which, on evaluation using eq. 9 gives: tan0 =
K’(r-ye)sinx
(r-y0)cosx
or: tan0=K2
tanx
(11) (12)
so 0 can be determined from the given value of K and the value of x obtained by means of eq. 8. Determination
of (Y and p
By definition, CY,at any point on a bedding-trace of the fold-profile, is the ratio of the final to the initial length of any infinitesimal line segment initially parallel to the x-axis.
214 Using the notation given in Figs. 6A, B: a: = AB/AaBa
= AB/dxa
(13)
and: AB* = dx* + dy’ or: AB = Jl
+ (dy/dx)*dx
(14)
Now, from eqs. 9 and 8 respectively: dx = K-‘(r - ya) eos xdx dx = dxa /(r - ya ), . . . (ya constant) so: dx = K-’ cos xdxa This expression combined with eqs. 10, 13 and 14 gives: QL= ABfdxa = v/l + tan*B(K-’ cos x) and simplification, using 1 + tan20 = l/cos2@, gives:
xicose)
Q: = ~~~~~~
At any point on the trace of the axial plane (y-coordinate eq. 15 reduces to:
cto=K-’
05) axis), x = 0, and 8 = 0, hence,
(16)
From eqs. 15 and 16 it follows that: 01 -=ue
cos x
07)
case
and using eq. 3:
p = (cos
x)-l
~eter~inat~o~
(18) of the shear factor ‘y, and the shear angle tt/
As noted previously y = tan 9, and $ describes the rotation of lines initially parallel to the y’-coordinate axis of an elemental unit of finite homogeneous deformation, with the X’S& aligned with the bedding-trace before and after the deformation (Fig. 1). As indicated in Figs. 6C-E the angular relationships of these line elements can be examined in terms of two vectors attached to a material point (P). If the linear transformation from initial (xe, ye) to final (x, y) coordinates is expressed as: x = f(x0. Yo) Y = g@o, Yo) then the initial vectors: (ckc~, O>,and (0, dya) are transformed to: (f,o,g,o)
Got and (fyo, g,a) d,a
(19)
215
a
40
--~.,---_
00
PO
__-/
dx
I
4
dxo
I
?
I
X
0
X
c \\
ho, 0)
L
&
E
$
\
‘\ 90’
‘1 --
(dxo,O)
A
--
e --
PO
x
x
0
0
X
Fig. 6. Initial (A) and final (B) configurations of a line element on a folded-bedding trace. C-E, transformation of two initially perpendicular line elements by a finite homogeneous deformation. These vectors represent the sides of an elemental unit (cf. Fig. 1). Note that vector (dxe, 0) is aligned with the same bedding-trace before and after the deformation. $ is the shear angle, and 0 is the orientation of the bedding-trace at the point P.
where (a, b) = ai + bj Now, from Fig. 6E it is evident that: tan J/ = tan[?T/:! - (A - 0)] = l/tan(A=
0)
1 + tan A tan 8
(20)
tan A - tan 8
But: tan A = gyolfyo
tan 0 so
=gxOlfxO
substitution
tan $ =
in eq. 20 and simplification,
fxofyo + &dyo fxogyo- fYogX0
gives:
(21)
216 However, the denominator in this expression is the determinant of the transformation matrix of eq. 19, and is unity for an area conserving transformation, so eq. 21 reduces to: tan \Ei=
fXOfyO +gXOgyO
The functions
f
(221
and g in this expression are defined by eq. 9 as:
f=K’(r-yo)sinx (231
g=Kr-K(r-ya)cosX We note from eq. 8 that: x=x0/@-~0)
so :
X0
3X _-xv
aYO (r ax _-=ax0
yo)2 (24)
I r-y0
These expressions may be used to establish the derivatives of f and g in eq. 22, which are obtained from eq. 23 by partial differentiation of x with respect to xe or ye, which yields: f,0 =K-'tr-Yo)cosx(r-y0)-'
=iT’cosx fyo =-K'
=- K1
sin X + K-l(r-ya) sin X + K-‘&d
cosXxa(r-Y~)-~ Cos X
g,e = K(r - ye) sin X(r - ya)‘-l = K sin X g,,o = K cos X + K(r - yo) sin xxo(r - y(j)_2 =KcosX+KXdsinX Substitution of these expressions in eq. 22 with simplification, leads finally to: y = tan J, = KM2X,adcos’x + K2Xrad sin’x + (K2 - K2) Strain ellipse axial-ratio (R) and orientation
sin X cos X
(25)
(4)
The values of @ and R for a particular elemental unit are determined by means of eqs. 1 and 2 respectively, using the atready established values of CY,‘y and $. Table III lists the computed values of various geometric and strain parameters for an initially square grid of points defined by the initial positions of the points co-c4 and the undeformed bedding-trace d of the fold-profile in Fig. 2. These data were used in the construction of this fold-profile and the trajectories of the maximum extension directions.
217 TABLE III Geometric and strain parameters at points on bedding-traces c and d of fold-profile
x0
Yo
x
Y
e
a
7
0.800 0.859 0.996 1.128 1.226 0.800 0.862 1.053 1.190 1.250
0 0.529 1.108 1.668 2.193 0 0.636 1.342 1.983 2.447
0 CO Cl c2 c3 c4 d0
dr dz da d4 -
0 12.95 25.96 38.00 51.00 0 12.95 25.95 38.00 51.00
17.00 17.00 17.00 17.00 17.00 23.30 23.30 23.30 23.30 23.30
0 10.17 19.27 25.81 30.13 0 10.09 18.65 24.01 26.16
21.25 23.91 31.66 42.62 57.32 29.13 32.29 41.34 53.75 69.54
0 28.3 50.8 66.5 80.2 0 33.2 57.8 74.5 89.6
-+ (“) 0 27.9 47.9 59.1 65.5 0 32.5 53.3 63.2 67.8
in Fig. 2
Q, (“1
R
90 49.7 30.7 22.2 18.0 90 46.0 26.5 19.4 16.7
1.563 1.964 2.890 4.002 5.197 1.563 2.179 3.339 4.676 5.884
x0, YO = initial coordinates of the point. x, Y = final coordinates of the point. 0 = angle of dip of bedding-trace. (Y = strain factor along bedding-trace. = shear factor along bedding-trace. = shear angle = arctan 7. = orientation of strain-ellipse major axis from bedding-trace. ! R = axial ratio of strain ellipse.
REFERENCES Bayly, B., 1974. Cleavage not parallel to finite-strain ellipsoid’s XY-plane. Discussion. Tectonophysics, 23: 205-207. Billings, M.P., 1972. Structural Geology. Prentice-Hall, New Jersey, 606 pp. Elliot, D., 1972. Deformation paths in structural geology. Geol. Sot. Am. Bull., 83: 2621-2638. Matthews, P.E., Bond, R.A.B. and Van den Berg, J.J., 1971. Analysis and structural implications of a kinematic model of similar folding. Tectonophysics, 12: 129-154. Matthews, P.E., Bond, R.A.B. and Van den Berg, J.J., 1974. An algebraic method of strain analysis using elliptical markers. Tectonophysics, 24: 31-67. Ramsay, J-G., 1967. Folding and Fracturing of Rocks. McGraw Hill, New York, 568 pp. Siddans, A.W.B., 1972. Slaty-cleavage - a review of research since 1815. Earth-Sci. Rev., 205-232. Williams, P-F., 1976. Relationships between axial-plane foliations and strain. Tectonophysics, 30: 181-196. Wood, D.S., 1974. Current views of the development of slaty cleavage. Ann. Rev. Earth Planet. Sci., 2.