- Email: [email protected]
Adaption of polygonal strain markers
Z’ectonophysics, 43 (1977) Tl-T10 0 Elsevier Scientific Publishing Company,
Tl Amsterdam
- Printed
in The Netherlands
Letter Section Adaption of polygonal strain markers
GREGORY
H. RODER
Department
of Scientific and industrial Research, Nelson (New Zealand)
(Submitted
June 29,1977;
accepted
for publication
July 18, 1977)
ABSTRACT Roder,
G.H., 1977. Adaption
of polygonal
strain markers.
Tectonophysics,
43: Tl-T10.
Two methods are presented whereby finite-strain data may be determined from naturally occurring irregular strain markers (poZyg0n.s) which are of unknown pre-de. formation shape and distribution, without assumptions as to the orientation of the finite-strain ellipse. The first method describes “construction” of ellipses within the polygons, these ellipses providing the basis for analysis by already developed techniques. The second method is a simple extension of Wellman’s method, which graphically establishes a strain ellipse from angle and line data.
INTRODUCTION
Direct methods of finite-strain analysis in rocks involve the use of strain markers. Geologically important markers may be loosely grouped as either naturally occurring objects included within the rock matrix, or features imposed on the rock fabric. Examples of the first group are objects of known or regular initial shape (e.g. oolites, pebbles; Cloos, 1947; Brace, 1955; Flinn, 1956; Gay, 1968), or of complex shape but known or constant angular relationships (e.g. fossils; Brace, 1960, 1961; Wellman, 1962; Ramsay, 1967; pp. 228-250). Ramsay (1967, ‘p. 195) and Mimran (1976) describe distance of separation and density techniques respectively for analysis of this group of strain markers, assuming even, or known uneven, initial distributions. Examples of the second group of strain markers are sedimentary features (e.g. Hobbs and Talbot, 1966), trace fossils (e.g. McLeish, 1971; Wilkinson et al., 1975; Paulis, 1976), crystal growth forms (e.g. Elliott, 1972; Durney and Ramsay, 1973), or tectonic structures (e.g. Matthews et al., 1976). Adaption of a greater diversity of objects or rock features for use as natural strain markers will entail removal of the limitations imposed in most of the above examples, especially for the naturally occurring included
T2
objects, by the requirements of known or regular undeformed shape, constant undeformed angular relationships, or known distribution within the rock prior to deformation. For finite-strain analysis of the group of features imposed upon the rock fabric, the orientation of the finite-strain ellipse is commonly assumed. Two methods are discussed here, whereby irregular strain markers, or polygons, which do not fit the requirements of known pre-deformation shape or distribution, may be made to yield finite-strain data, without assumptions as to the orientation of the finite-strain ellipse. METHOD 1
A common form of strain analysis utilizes strain markers of originally circular or elliptical traces on a plane, as on homogeneous deformation both form elliptical traces, for which a number of finite-strain determination techniques have been devised (Cloos, 1947; Flinn, 1956; Ramsay, 1967; Gay, 1968; Dunnet, 1969; Elliott, 1970; Dunnet and Siddans, 1971; and Matthews et al., 1974). If unique ellipses then may be constructed within the two-dimensional polygonal traces of a set of deformed irregular strain markers, these ellipses may be analyzed as a group by those now standard techniques, providing a basis for three-dimensional analysis of objects previously thought not to be useful for this purpose. Constructed ellipses need not represent the entire polygon trace, as they are not an illustration of the state of strain within the polygon, but are merely a means of attributing to a complex shape a more convenient one for analysis. Thus parallelograms, within each of which a unique ellipse may be inscribed, are defined for the irregular polygon traces by any two straight, non-parallel, line segments of each polygon, or two straight nonparallel lines joining recognizable points on a polygon (Fig. 1). In this way, more than one ellipse may be defined for each polygon. Other regular straight-sided shapes could be found within the polygon traces, but would not have the property of defining unique ellipses. Actual graphical construction of the ellipses within the parallelograms, followed by measurement of their axial ratios and orientations, is not necessary, as formulae defining the unique ellipse inscribed within a parallelogram may be utilized, and each ellipse can be defined numerically by measurement of a parallelogram. _______ Fig. 1. Establishing ellipses within polygons by the construction of parallelograms. a. Ordered array of regular polygons, with parallelogram ellipses constructed before and after deformation. Full circles and ellipses show bulk finite strain in each case. b. Close-packed array of regular polygons before and after deformation. Preferred orientations within the undeformed lattice are exemplified by the parallelogram ellipses. c. Array of irregular polygons and constructed ellipses, before and after deformation. d. Scattered irregular polygons and constructed ellipses, before and after deformation. PR and P’R’ are lines joining the same recognizable points on the polygon where appropriate straight lines are not available to form parallelograms.
0
0 -i3 -
-
0
1’4
The unique
inscribed
ellipse for the parallelogram
The equations for the unique are given by Becker (1904) as: (a + b)’
and :
(a -
= a’* + b”
ellipse
inscribed
within
a parallelogram
+ 2a’b’ sin 6
b)* = a’* + b’* -
2c ‘b’ sin 19
where a and b are the semi-major and semi-minor axes of the ellipse respectively, 2a’ and 2b’ are the lengths of the parallelogram long and short sides respectively, and 6 is the acute angle between them (Fig. 2a). The possibilities of simplifying the measurements made on each parallelogram to define the parameters of the individual unique ellipses are explored in Fig. 2b, but as is seen from the graph of a/b vs. 0 (Fig. 3a), all of the parameters a’, b’, and 0 of the parallelograms must be taken into account to define a unique el.lipse. From Fig, 3a the ellipse with the maximum axial ratio (a/b) will be inscribed within the parallelogram with maximum a’/b’ and minimum 6. In the deformed state this will occur where the undeformed parallelogram (of maximum al/b’ and minimum 0 ) lay with its longest side in what became the direction of maximum elongation on deformation. Hence the simple measurement of al/b’ for a set of deformed parallelograms, ignoring 0, will yield accurate information as to the orientation of the strain ellipse in a plane, even though it is of a false size ratio. The orientation (4) of each unique ellipse “constructed” within the set of polygon traces may be found by correcting the orientation of each parallelogram longest side (or diagonal), measured from an arbitrary reference line, according to the equation (Becker, 1904): tan 2e =
sin 28 ~~ ~~ cos 20 + (a’,&‘)*
Fig. 2. a. Parameters involved in the construction of the unique ellipse inscribed within a parallelogram (after Becker, 1904). b. Possible methods of measuring the parallelogram to directly determine the parameters of individual ellipses. For 1 and 3 the ellipse is envisaged as being inscribed within the special parallelogram with 0 = 90”) and on deformation either the sides are measured where 0 remains 90” (l’), or the long diagonal, and the overall width of the parallelogram in the direction perpendicular to this, are measured (3’). Only the special case of 1’ will yield an accurate measurement of this ellipse. For 4 the ellipse is inscribed. In the deformed state the length of the longest side and thickness are measured, which do not parallel, and are not equal to, the ellipse axes. In 2 and 5 the ellipse is circumscribed. In the deformed state the long diagonal and width perpendicular to it would be measured, which would not accurately represent the deformed ellipse. In 5 and 7 the possibility of measuring the sides of the parallelogram and treating the acute angles (or opposite apices if t? = 90”) as foci of the ellipse is shown. Such a relationship does not always hold after deformation (compare 6’ and 7’).
T5
(al
7
F
6
6 F
7’
T6
(a)
t
a
I
b
10
20
30
LO
50
60
70
60
90
e---t
e=o
es0 I
,’
a’
4!fczd ’
/’
I
I
,’
I
I’
u
9
\- .
‘I,“
-.-__
E , ii ’\ -
ct
/’
‘,I
I
r,
1x,.
‘3 “’ z
,,
/’
, I’ ,
f’ /
c
(b)
e
\\\ ?+-iy \b\ \ I’
-.
\
‘\
1
‘\ \
\
‘c-a
‘es0
\ \
e>o
where e is the angle between the long side of a parallelogram and the major axis of its inscribed ellipse (Fig. 2a). e may be positive, negative, or in the cases of 0 = 90”, or a’ = b’, zero (Fig. 3b). From the collected data on a set of parallelograms within polygons, a set of ellipse data is thus generated. From this data, the two-dimensional finite. strain parameters may be defined for the plane of measurement by the techniques of analysis of sets of deformed ellipses mentioned earlier. Prob. lems of initially non-random fabrics and original shape variations only effec the procedure described here so far as the number of polygons/parallelo. grams/ellipses measured is concerned. Original shape and fabric problems essentially belong to the next step of data treatment, not considered here (see especially Dunnet and Siddans, 1971; Elliott, 1970; Matthews et al., 1974). METHOD
2
A simple method of deriving two-dimensional strain data from a set OF deformed polygons with no initial fabric is found in an extension of Wellman’s method (Wellman, 1962). The method has previously been used fog: sets of strain markers in which original angles between line pairs were constant (Wellman, 1962; Ramsay, 1967, p.242). However, it may also be used for an initial variable angle, provided a sufficient number of variably oriented line pairs (polygon sides) are available. For a set of undeformed line pairs of constant angle, line intersection:; may be constructed from each strain marker, and projected from a chord onto a circle circumference (Wellman, 1962, and Fig. 4). For a set of strain markers in which this original undeformed angle varied, these line intersections scatter within a circle. On deformation they will give an elliptical scatter of points about which the applied tectonic ellipse may be drawn (Fig. 4).
Fig. 3. a. Plot of ellipse axial ratio (a/b) against the acute angle of the parallelograrl (0) for parallelogram length ratios of 1 :l, 2:1, 3 :l, and 4 :l. a/b ratios depend on a , b’, and 8. For any 8, the unique inscribed ellipse with the maximum axial ratio wi 1 occur where a’/b’ is a maximum. Conversely, for a set a’/b’ value a/b will increase as ? decreases. Thus there is an overlap of a/b values depending on a’, b’, and 8, and it will be necessary to consider all three of these when deriving the unique ellipse representing the parallelogram. b. Sign of the correction factor e to be applied to o depending on the nature and orientation of the parallelogram as shown. Dashed lines are the orientations of the parallelograms, full lines are the orientations of their inscribed ellipse:;. e will be added to or subtracted from 0, according to whether the rotation of the parallelogram long side into the direction of the ellipse major axis is clockwise or anticlockwise. In the special cases of 0 = 90”, e is zero, and where a’ = b’, the orientation is taken as that of the long diagonal, and e is again zero.
* .. . . *. :. * -.
(cl
‘...
-.. ,
*
.:..
*
D/ED
lb) . . ., .
. .
- .. . . . . . : * .:..
.
.
. l
.
.
.
:.
.* .
.
.a
T9
CONCLUSIONS
There are practical limits to all techniques of strain analysis, imposed by the nature of the strain markers. The two methods described here serve to extend the limitations imposed by requirements of knowledge of the pre-deformation shape, but still only to a degree. In Method 1, the greater the range of size ratios and angles of parallelo. grams derived from polygons in the set, the greater the number of then must be considered to be sure of including the entire range of ellipse axial ratios. Matthews et al. (1974) discuss sample size with regard to initiai ellipse ratios and orientations required to obtain a certain accuracy in the strain parameters. In Method 2, as the angle between line pairs approaches zero, the size of the constructed ellipse for the deformed state will approach infinity. This limits the combination of original angles and strain for which the method is practical. The initially minimum O’s in the undeformed state are crucial, as they will lie on the perimeter of the ellipse in the deformed state. As only deformed angles are involved in this method, it may prove useful where solution processes have altered length ratios, or distances between strain markers. In addition, the work involved in Method 1 may’ not be justified for many geological problems, unless information regarding original shape is required. ACKNOWLEDGEMENTS
The author gratefully acknowledges Drs. C.J.L. Wilson and M.B. Stephens for critically reading draughts of the manuscript. The work was carriec! out in the Department of Geology, School of Earth Sciences, University. of Melbourne.
Fig. 4. a. Positioning of intersection points from line pairs via Wellman’s method. CD/EL’ and FG/HG are two intersecting line pairs drawn from irregular polygons. An arbitrary chord A’B’ is drawn, and through its ends are traced the orientations of the line pairs This must be done in a regular fashion, for example considering each line pair clockwise, across the acute angle from one line to the other, the starting line being drawn through the same arbitrary point A’ for each line pair. If this is not done, two elliptical scatters will be formed as in Fig. 412. b. Scatter of points derived from a set of undeformed randomly oriented line pairs, with variable angles, derived from irregular polygons. c. Scatter of points derived from the line pairs used for Fig. 4b after deformation. The two possible ellipses, of identical axial ratios and orientations, as described in Fig. 4a, are shown.
TlO
REFERENCES Becker, G.F., 1904. Experiments on schistosity and slaty cleavage. U.S. Geol. Surv. Bull., 241: l-34. Brace, W.F., 1955. Quartzite pebble deformation in Central Vermont. Am. J. Sci., 253: 129-145. Brace, W.F., 1960. Analysis of large two-dimensional strain in deformed rocks. 21st Int. Geol. Congr., Copenhagen, Rept. Sess., 18: 261-269. Brace, W.F., 1961. Mohr construction in the analysis of large geological strain. Bull. Geol. Sot. Am., 72: 1059-1080. Cloos, E., 1947. Oolite deformation in South Mountain Fold, Maryland. Bull. Geol. Sot. Am., 58: 843-918. Dunnet, D., 1969. A technique of finite-strain analysis using elliptical particles. Tectonophysics, 7: 117-136. Dunnet, D. and Siddans, A.W.B., 1971. Non-random sedimentary fabrics and their modification by strain. Tectonophysics, 12: 307-325. Durney, D.W. and Ramsay, J.G., 1973. Incremental strains measured by syntectonic crystal growths. In: K.A. de Jong and R. Scholten (editors), Gravity and Tectonics. Wiley Interscience, New York, pp. 67-96. Elliott, D., 1970. Determination of finite strain and initial shape from deformed elliptical objects. Geol Sot. Am. Bull., 87: 2221-2236. Elliott, D., 1972. Deformation paths in structural geology. Geol. Sot. Am. Bull., 83: 2621-2638. Flinn, D., 1956. On the deformation of the Funzie Conglomerate, Feltlar, Shetland. J. Geol., 64: 480-505. Gay, N.C., 1968. Pure-shear and simple-shear deformation of inhomogeneous viscous fluids: 2. The determination of the total finite strain in rock from objects such as deformed pebbles. Tectonophysics, 5: 295-302. Hobbs, B.E. and Talbot, J.L., 1966. The analysis of strain in deformed rocks. J. Geol., 74: 500-513. 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. 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. McLeish, A.J., 1971. Strain analysis of deformed pipe rock in the Moine Thrust zone, northwest Scotland. Tectonophysics, 12: 469-503. Mimran, Y., 1976. Strain determination using a density-distribution technique and its application to deformed Upper Cretaceous Dorset chalks. Tectonophysics, 31: 175-192. Paulis, R.C., 1976. A method of strain measurement using initially orthogonal planar and linear markers. Tectonophysics, 34: T29-T36. Ramsay, J.G., 1967. Folding and Fracturing of Rocks. McGraw-Hill, New York, 568 pp. Wellman, H.W., 1962. A graphical method for analysing fossil distortion caused by tectonic deformation. Geol Mag., 99: 348-352. Wilkinson, P., Soper, N.J. and Bell, A.M., 1975. Skolithus pipes as strain markers in mylonites. Tectonophysics, 28: 143-157.
Tl Amsterdam
- Printed
in The Netherlands
Letter Section Adaption of polygonal strain markers
GREGORY
H. RODER
Department
of Scientific and industrial Research, Nelson (New Zealand)
(Submitted
June 29,1977;
accepted
for publication
July 18, 1977)
ABSTRACT Roder,
G.H., 1977. Adaption
of polygonal
strain markers.
Tectonophysics,
43: Tl-T10.
Two methods are presented whereby finite-strain data may be determined from naturally occurring irregular strain markers (poZyg0n.s) which are of unknown pre-de. formation shape and distribution, without assumptions as to the orientation of the finite-strain ellipse. The first method describes “construction” of ellipses within the polygons, these ellipses providing the basis for analysis by already developed techniques. The second method is a simple extension of Wellman’s method, which graphically establishes a strain ellipse from angle and line data.
INTRODUCTION
Direct methods of finite-strain analysis in rocks involve the use of strain markers. Geologically important markers may be loosely grouped as either naturally occurring objects included within the rock matrix, or features imposed on the rock fabric. Examples of the first group are objects of known or regular initial shape (e.g. oolites, pebbles; Cloos, 1947; Brace, 1955; Flinn, 1956; Gay, 1968), or of complex shape but known or constant angular relationships (e.g. fossils; Brace, 1960, 1961; Wellman, 1962; Ramsay, 1967; pp. 228-250). Ramsay (1967, ‘p. 195) and Mimran (1976) describe distance of separation and density techniques respectively for analysis of this group of strain markers, assuming even, or known uneven, initial distributions. Examples of the second group of strain markers are sedimentary features (e.g. Hobbs and Talbot, 1966), trace fossils (e.g. McLeish, 1971; Wilkinson et al., 1975; Paulis, 1976), crystal growth forms (e.g. Elliott, 1972; Durney and Ramsay, 1973), or tectonic structures (e.g. Matthews et al., 1976). Adaption of a greater diversity of objects or rock features for use as natural strain markers will entail removal of the limitations imposed in most of the above examples, especially for the naturally occurring included
T2
objects, by the requirements of known or regular undeformed shape, constant undeformed angular relationships, or known distribution within the rock prior to deformation. For finite-strain analysis of the group of features imposed upon the rock fabric, the orientation of the finite-strain ellipse is commonly assumed. Two methods are discussed here, whereby irregular strain markers, or polygons, which do not fit the requirements of known pre-deformation shape or distribution, may be made to yield finite-strain data, without assumptions as to the orientation of the finite-strain ellipse. METHOD 1
A common form of strain analysis utilizes strain markers of originally circular or elliptical traces on a plane, as on homogeneous deformation both form elliptical traces, for which a number of finite-strain determination techniques have been devised (Cloos, 1947; Flinn, 1956; Ramsay, 1967; Gay, 1968; Dunnet, 1969; Elliott, 1970; Dunnet and Siddans, 1971; and Matthews et al., 1974). If unique ellipses then may be constructed within the two-dimensional polygonal traces of a set of deformed irregular strain markers, these ellipses may be analyzed as a group by those now standard techniques, providing a basis for three-dimensional analysis of objects previously thought not to be useful for this purpose. Constructed ellipses need not represent the entire polygon trace, as they are not an illustration of the state of strain within the polygon, but are merely a means of attributing to a complex shape a more convenient one for analysis. Thus parallelograms, within each of which a unique ellipse may be inscribed, are defined for the irregular polygon traces by any two straight, non-parallel, line segments of each polygon, or two straight nonparallel lines joining recognizable points on a polygon (Fig. 1). In this way, more than one ellipse may be defined for each polygon. Other regular straight-sided shapes could be found within the polygon traces, but would not have the property of defining unique ellipses. Actual graphical construction of the ellipses within the parallelograms, followed by measurement of their axial ratios and orientations, is not necessary, as formulae defining the unique ellipse inscribed within a parallelogram may be utilized, and each ellipse can be defined numerically by measurement of a parallelogram. _______ Fig. 1. Establishing ellipses within polygons by the construction of parallelograms. a. Ordered array of regular polygons, with parallelogram ellipses constructed before and after deformation. Full circles and ellipses show bulk finite strain in each case. b. Close-packed array of regular polygons before and after deformation. Preferred orientations within the undeformed lattice are exemplified by the parallelogram ellipses. c. Array of irregular polygons and constructed ellipses, before and after deformation. d. Scattered irregular polygons and constructed ellipses, before and after deformation. PR and P’R’ are lines joining the same recognizable points on the polygon where appropriate straight lines are not available to form parallelograms.
0
0 -i3 -
-
0
1’4
The unique
inscribed
ellipse for the parallelogram
The equations for the unique are given by Becker (1904) as: (a + b)’
and :
(a -
= a’* + b”
ellipse
inscribed
within
a parallelogram
+ 2a’b’ sin 6
b)* = a’* + b’* -
2c ‘b’ sin 19
where a and b are the semi-major and semi-minor axes of the ellipse respectively, 2a’ and 2b’ are the lengths of the parallelogram long and short sides respectively, and 6 is the acute angle between them (Fig. 2a). The possibilities of simplifying the measurements made on each parallelogram to define the parameters of the individual unique ellipses are explored in Fig. 2b, but as is seen from the graph of a/b vs. 0 (Fig. 3a), all of the parameters a’, b’, and 0 of the parallelograms must be taken into account to define a unique el.lipse. From Fig, 3a the ellipse with the maximum axial ratio (a/b) will be inscribed within the parallelogram with maximum a’/b’ and minimum 6. In the deformed state this will occur where the undeformed parallelogram (of maximum al/b’ and minimum 0 ) lay with its longest side in what became the direction of maximum elongation on deformation. Hence the simple measurement of al/b’ for a set of deformed parallelograms, ignoring 0, will yield accurate information as to the orientation of the strain ellipse in a plane, even though it is of a false size ratio. The orientation (4) of each unique ellipse “constructed” within the set of polygon traces may be found by correcting the orientation of each parallelogram longest side (or diagonal), measured from an arbitrary reference line, according to the equation (Becker, 1904): tan 2e =
sin 28 ~~ ~~ cos 20 + (a’,&‘)*
Fig. 2. a. Parameters involved in the construction of the unique ellipse inscribed within a parallelogram (after Becker, 1904). b. Possible methods of measuring the parallelogram to directly determine the parameters of individual ellipses. For 1 and 3 the ellipse is envisaged as being inscribed within the special parallelogram with 0 = 90”) and on deformation either the sides are measured where 0 remains 90” (l’), or the long diagonal, and the overall width of the parallelogram in the direction perpendicular to this, are measured (3’). Only the special case of 1’ will yield an accurate measurement of this ellipse. For 4 the ellipse is inscribed. In the deformed state the length of the longest side and thickness are measured, which do not parallel, and are not equal to, the ellipse axes. In 2 and 5 the ellipse is circumscribed. In the deformed state the long diagonal and width perpendicular to it would be measured, which would not accurately represent the deformed ellipse. In 5 and 7 the possibility of measuring the sides of the parallelogram and treating the acute angles (or opposite apices if t? = 90”) as foci of the ellipse is shown. Such a relationship does not always hold after deformation (compare 6’ and 7’).
T5
(al
7
F
6
6 F
7’
T6
(a)
t
a
I
b
10
20
30
LO
50
60
70
60
90
e---t
e=o
es0 I
,’
a’
4!fczd ’
/’
I
I
,’
I
I’
u
9
\- .
‘I,“
-.-__
E , ii ’\ -
ct
/’
‘,I
I
r,
1x,.
‘3 “’ z
,,
/’
, I’ ,
f’ /
c
(b)
e
\\\ ?+-iy \b\ \ I’
-.
\
‘\
1
‘\ \
\
‘c-a
‘es0
\ \
e>o
where e is the angle between the long side of a parallelogram and the major axis of its inscribed ellipse (Fig. 2a). e may be positive, negative, or in the cases of 0 = 90”, or a’ = b’, zero (Fig. 3b). From the collected data on a set of parallelograms within polygons, a set of ellipse data is thus generated. From this data, the two-dimensional finite. strain parameters may be defined for the plane of measurement by the techniques of analysis of sets of deformed ellipses mentioned earlier. Prob. lems of initially non-random fabrics and original shape variations only effec the procedure described here so far as the number of polygons/parallelo. grams/ellipses measured is concerned. Original shape and fabric problems essentially belong to the next step of data treatment, not considered here (see especially Dunnet and Siddans, 1971; Elliott, 1970; Matthews et al., 1974). METHOD
2
A simple method of deriving two-dimensional strain data from a set OF deformed polygons with no initial fabric is found in an extension of Wellman’s method (Wellman, 1962). The method has previously been used fog: sets of strain markers in which original angles between line pairs were constant (Wellman, 1962; Ramsay, 1967, p.242). However, it may also be used for an initial variable angle, provided a sufficient number of variably oriented line pairs (polygon sides) are available. For a set of undeformed line pairs of constant angle, line intersection:; may be constructed from each strain marker, and projected from a chord onto a circle circumference (Wellman, 1962, and Fig. 4). For a set of strain markers in which this original undeformed angle varied, these line intersections scatter within a circle. On deformation they will give an elliptical scatter of points about which the applied tectonic ellipse may be drawn (Fig. 4).
Fig. 3. a. Plot of ellipse axial ratio (a/b) against the acute angle of the parallelograrl (0) for parallelogram length ratios of 1 :l, 2:1, 3 :l, and 4 :l. a/b ratios depend on a , b’, and 8. For any 8, the unique inscribed ellipse with the maximum axial ratio wi 1 occur where a’/b’ is a maximum. Conversely, for a set a’/b’ value a/b will increase as ? decreases. Thus there is an overlap of a/b values depending on a’, b’, and 8, and it will be necessary to consider all three of these when deriving the unique ellipse representing the parallelogram. b. Sign of the correction factor e to be applied to o depending on the nature and orientation of the parallelogram as shown. Dashed lines are the orientations of the parallelograms, full lines are the orientations of their inscribed ellipse:;. e will be added to or subtracted from 0, according to whether the rotation of the parallelogram long side into the direction of the ellipse major axis is clockwise or anticlockwise. In the special cases of 0 = 90”, e is zero, and where a’ = b’, the orientation is taken as that of the long diagonal, and e is again zero.
* .. . . *. :. * -.
(cl
‘...
-.. ,
*
.:..
*
D/ED
lb) . . ., .
. .
- .. . . . . . : * .:..
.
.
. l
.
.
.
:.
.* .
.
.a
T9
CONCLUSIONS
There are practical limits to all techniques of strain analysis, imposed by the nature of the strain markers. The two methods described here serve to extend the limitations imposed by requirements of knowledge of the pre-deformation shape, but still only to a degree. In Method 1, the greater the range of size ratios and angles of parallelo. grams derived from polygons in the set, the greater the number of then must be considered to be sure of including the entire range of ellipse axial ratios. Matthews et al. (1974) discuss sample size with regard to initiai ellipse ratios and orientations required to obtain a certain accuracy in the strain parameters. In Method 2, as the angle between line pairs approaches zero, the size of the constructed ellipse for the deformed state will approach infinity. This limits the combination of original angles and strain for which the method is practical. The initially minimum O’s in the undeformed state are crucial, as they will lie on the perimeter of the ellipse in the deformed state. As only deformed angles are involved in this method, it may prove useful where solution processes have altered length ratios, or distances between strain markers. In addition, the work involved in Method 1 may’ not be justified for many geological problems, unless information regarding original shape is required. ACKNOWLEDGEMENTS
The author gratefully acknowledges Drs. C.J.L. Wilson and M.B. Stephens for critically reading draughts of the manuscript. The work was carriec! out in the Department of Geology, School of Earth Sciences, University. of Melbourne.
Fig. 4. a. Positioning of intersection points from line pairs via Wellman’s method. CD/EL’ and FG/HG are two intersecting line pairs drawn from irregular polygons. An arbitrary chord A’B’ is drawn, and through its ends are traced the orientations of the line pairs This must be done in a regular fashion, for example considering each line pair clockwise, across the acute angle from one line to the other, the starting line being drawn through the same arbitrary point A’ for each line pair. If this is not done, two elliptical scatters will be formed as in Fig. 412. b. Scatter of points derived from a set of undeformed randomly oriented line pairs, with variable angles, derived from irregular polygons. c. Scatter of points derived from the line pairs used for Fig. 4b after deformation. The two possible ellipses, of identical axial ratios and orientations, as described in Fig. 4a, are shown.
TlO
REFERENCES Becker, G.F., 1904. Experiments on schistosity and slaty cleavage. U.S. Geol. Surv. Bull., 241: l-34. Brace, W.F., 1955. Quartzite pebble deformation in Central Vermont. Am. J. Sci., 253: 129-145. Brace, W.F., 1960. Analysis of large two-dimensional strain in deformed rocks. 21st Int. Geol. Congr., Copenhagen, Rept. Sess., 18: 261-269. Brace, W.F., 1961. Mohr construction in the analysis of large geological strain. Bull. Geol. Sot. Am., 72: 1059-1080. Cloos, E., 1947. Oolite deformation in South Mountain Fold, Maryland. Bull. Geol. Sot. Am., 58: 843-918. Dunnet, D., 1969. A technique of finite-strain analysis using elliptical particles. Tectonophysics, 7: 117-136. Dunnet, D. and Siddans, A.W.B., 1971. Non-random sedimentary fabrics and their modification by strain. Tectonophysics, 12: 307-325. Durney, D.W. and Ramsay, J.G., 1973. Incremental strains measured by syntectonic crystal growths. In: K.A. de Jong and R. Scholten (editors), Gravity and Tectonics. Wiley Interscience, New York, pp. 67-96. Elliott, D., 1970. Determination of finite strain and initial shape from deformed elliptical objects. Geol Sot. Am. Bull., 87: 2221-2236. Elliott, D., 1972. Deformation paths in structural geology. Geol. Sot. Am. Bull., 83: 2621-2638. Flinn, D., 1956. On the deformation of the Funzie Conglomerate, Feltlar, Shetland. J. Geol., 64: 480-505. Gay, N.C., 1968. Pure-shear and simple-shear deformation of inhomogeneous viscous fluids: 2. The determination of the total finite strain in rock from objects such as deformed pebbles. Tectonophysics, 5: 295-302. Hobbs, B.E. and Talbot, J.L., 1966. The analysis of strain in deformed rocks. J. Geol., 74: 500-513. 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. 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. McLeish, A.J., 1971. Strain analysis of deformed pipe rock in the Moine Thrust zone, northwest Scotland. Tectonophysics, 12: 469-503. Mimran, Y., 1976. Strain determination using a density-distribution technique and its application to deformed Upper Cretaceous Dorset chalks. Tectonophysics, 31: 175-192. Paulis, R.C., 1976. A method of strain measurement using initially orthogonal planar and linear markers. Tectonophysics, 34: T29-T36. Ramsay, J.G., 1967. Folding and Fracturing of Rocks. McGraw-Hill, New York, 568 pp. Wellman, H.W., 1962. A graphical method for analysing fossil distortion caused by tectonic deformation. Geol Mag., 99: 348-352. Wilkinson, P., Soper, N.J. and Bell, A.M., 1975. Skolithus pipes as strain markers in mylonites. Tectonophysics, 28: 143-157.