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A simple algebraic method for strain estimation from deformed ellipsoidal objects. 1. Basic theory
Tectonophysics, 36 (1976) 315-337 0 Elsevier Scientific Publishing Company,
315
Amsterdam
- Printed
in The Netherlands
Research Papers A SIMPLE ALGEBRAIC METHOD FOR STRAIN ESTIMATION DEFORMED ELLIPSOIDAL OBJECTS. 1. BASIC THEORY
TOSHIHIKO
SHIMAMOTO
and YUKIO
IKEDA
Center for Tectonophysics, Texas A&M University, College Fukuyama High School of Hiroshima University, Fukuyama
(Submitted
April 27, 1976; accepted
FROM
for publication
Station, Texas (U.S.A.) City, Hiroshima (Japan)
June 15, 1976)
ABSTRACT Shimamoto, deformed
T. and Ikeda, Y., 1976. A simple algebraic method ellipsoidal objects. 1. Basic theory. Tectonophysics,
for strain estimation 36: 315-337.
from
A new algebraic method is developed to determine the shape and orientation of the strain ellipsoid by using deformed ellipsoidal objects as strain markers. It is assumed that objects are of truly ellipsoidal shape with random orientation in the undeformed state, and that they deform homogeneously with their matrix. This part presents basic theories for: (1) the determination of the strain ellipse on a plane section from deformed elliptical objects; (2) determination of the strain ellipsoid from measurements of lengths and orientations of all principal axes of deformed ellipsoidal objects; and (3) construction of the strain ellipsoid from the two-dimensional analysis on three mutually orthogonal planes. General, finite, homogeneous deformation is treated, and the analysis gives the deviatoric natural strains in the principal directions of the Eulerian finite-strain tensor. The simplicity and usefulness of our method is demonstrated by the two-dimensional, pure-shear deformat.ion of model objects. Evaluation of error and optimum sample size, and geological applications of the method will be discussed in detail in subsequent papers.
INTRODUCTION
Determination of finite strain in rocks from distorted objects like ooids, pebbles and fossils involves certain fundamental difficulties that arise from : (1 j variable initial shape and orientation and (2) differences in mechanical properties between the objects and their surrounding matrix (Ramsay, 1967). This paper is intended to analyze the first problem; that is, the estimation of strain from deformed ellipsoidal objects which are not necessarily spherical in shape before deformation. The first significant contribution to this problem was made by Ramsay (1967) who devised a graphical method (R,/& diagram technique; see Notation I) to determine the strain ellipse on a plane section from deformed elliptical objects. His method has been further developed and applied to naturally deformed rocks by Dunnet (1969) and Dunnet and Siddans (1971). Elliott
316
(1970) has also developed a graphical method (shape-factor grid technique). They all assume initially elliptical shapes and no ductility contrast between the objects and their matrix. Initial random orientation is assumed by Ramsay (1967) and Dunnet (1969), but not by Elliott (1970) and Dunnet and Siddans (1971). The main disadvantage of these graphical methods is that a subject error tends to be introduced through curve-fitting. Moreover, their basic equations are formulated only for pure-shear deformation, so that it is still uncertain whether these methods are applicable to general deformations that include rotational strain. An algebraic method has been recently proposed by Matthews et al. (1974). It is simple and deals with general deformation, but it requires an independent knowledge of the orientations of the principal-strain axes in order to calculate strain magnitude. This obviously is a severe limitation, since the principal axes cannot always be determined a priori. Finally, in all these previous methods, equations are formulated as two-dimensional problems for which one of the principal axes of finite strain is normal to the plane of deformation. No complete theories have as yet been worked out to show that these two-dimensional methods are applicable to the determination of the strain ellipse on any arbitrary plane sections (G. Oertel, personal communication, 1975). Our major objective is to develop a simple algebraic method to determine the shape and orientation of the strain ellipsoid from deformed ellipsoidal objects. We make two basic assumptions that the objects are: (1) initially truly ellipsoidal in shape and randomly oriented; and (2) strained homogeneously together with their matrix. Two-dimensional theory is developed first in order to illustrate clearly the simplicity and accuracy of our method. We prove that the method gives the shape and orientation of the strain ellipse precisely if an infinite number of elliptical objects are measured. The practical usefulness of this two-dimensional analysis is demonstrated by the computer-simulated, pure-shear deformation of model objects. Our method is then extended to three dimensions. This is natural and straightforward, since the basic equations are formulated in matrix form. Analysis of threedimensional strains based on the two-dimensional analysis on plane sections is also emphasized for its practical applications. We prove that our twodimensional method can be applied for any arbitrary plane sections. General finite ho~~ogeneous deformation is treated, so that the method is applicable to both rotational and irrotational strains in exactly the same manner. Applicability of our two-dimensional method to geological problems is considered briefly from the measurements of undeformed oijids from the Cool Creek Formation of Oklahoma. We argue that our method would be sufficiently accurate for the analysis of geologic strain, For geological applications, any initial non-random orientation and ductility contrast should be often taken into account (Gay, 1968; Elliott, 1970; Dunnet and Siddans, 1971). The extension of our method to include these effects is also discussed briefly in the context of the two-dimensional analysis. Detailed discussions of errors in our methods, and further geological applications will be given in subsequent papers.
317
MATRIX REPRESENTATION
OF ELLIPSES AND ELLIPSOIDS
Consider an ellipse centered at the origin of the coordinate system with major and minor axes a, b (Fig. 1). The ellipse is represented by an equation:
(1) where x’ and y’ are taken parallel to the principal axes of the ellipse. Let the axial ratio be R = a/b > 1, and let a and b be normalized such that ab = 1. * Then a = JR and b = l/v’R; and eq, 1 can be written in matrix form:
Let 8 be the angle between the CCand x’ axes (0 < 0 < a). The coordinate transformation from (x, y) to (x’, y’) is (e.g., Fung, 1965):
(3)
Fig. I. An ellipse centered at the origin. The major and minor axes of the ellipse are (I and b, respectively, and 3 is the angle between the major axis and coordinate axis x. * Generality is not lost by this normalization, because we deal only with the shape (not size) of the object.
318
which on substitution
into eq. 2 yields: (4)
1
f =Rcos20+Rsin20 + R cos20
g = isin
(5)
Equation 4 is a matrix representation of an ellipse, and this form is employed throughout our analysis. Note that this matrix (called the shape matrix) is symmetric. Consider an ellipsoid centered at the origin with principal axes al, a2 and a3 (~1~2~3 = 1). Using the summation convention, we represent the ellipsoid as (e.g., Franklin, 1968, p. 94-98): BijXiXj = 1
(6)
where : (7)
The coordinate axes are taken parallel to the principal directions of the ellipsoid. The matrix [B:j] obeys the transformation law for second-order Cartesian tensors. Thus, eq. 6 can be written in terms of fixed coordinates (XI, ~2, x3) as: BijXiXj Bij
= 1
(8)
= P/z iPljBk[
(9)
Observe that Bij = Bji. The symbol /3ij denotes the cosine of the angle, Gij, between i-th primed and j-th unprimed coordinate axes (0 < ~ij < n): Pij
= COS ~ij
(10)
and: PkiPhj
=
&ij
where 6ij is the Kronecker symbol. Note that only three components are independent because of the orthogonality conditions (eq. 11).
(11)
of Pij
319
The principal axes of an ellipse or ellipsoid are the eigenvectors of the shape matrix (see Franklin, 1968). In two dimensions an eigenvalue X satisfies the characteristic equation :
f---A
h
h
g---x
=
0
which has two solutions Xi and 1s. Then the lengths of the principal axes are I/&, and l/d&. The principal direction 8 is given by the orientation of the corresponding eigenvector (Fig. 1). A simple equation can be derived for the two-dimensional case : tan28
=2h f-g
(13)
The lengths and orientations of the principal axes of an ellipsoid can be determined in a similar manner from the shape matrix [B,j] , FINITE
HOMOGENEOUS
DEFORMATION
OF ELLIPSES
AND ELLIPSOIDS
Consider an ellipse with major and minor axes ei and bi with Uibi = 1 in the initial undeformed state. The initial axial ratio Ri is a,/bi (21). Let (X, Y) be coordinates of a point before deformation, and let the major axis make an angle Qi with the coordinate axis X (0 < 4i < E). From eqs. 4 and 5 the ellipse is represented by :
[X
“Ei
HJx]=’
where the components Fi
=$
Gi =i
(14)
of the initial shape matrix are:
(($+Ri) +(k-Ri)COS2Oi) ’ ((~+Ri~ -($-Ri) COS26ij
(15)
,
This is simply eq. 5 modified in form for later convenience. Finite homogeneous deformation in two dimensions can be generally pressed as a linear transformation :
ex-
320
Here the Eulerian formulation (e.g., Mase, 1970, p. 79) is used; (x, y) are coordinates of a point after deformation, and CY,/3, y and 6 are constants (/3 # y in general). Substitution of eq. 16 into eq. 14 yields: (17)
where:
(18) Equation 17 is the matrix representation of the ellipse after the deformation. Lengths and orientations of the principal axes can be determined from eqs. 12 and 13. Observe that the final-shape matrix in eq. 17 is symmetric. For the three-dimensional extension of the analysis, let an ellipsoid before deformation be represented in the summation notation by: C~XiXj = 1
(19)
The initial state is denoted sional, finite homogeneous
by the superscript o. A general, three-dimendeformation can be expressed as:
Xi = dijXj
(26)
where dij # djj in general, and (Xi, Xs, Xs) and (xl, x2, x3) are coordinates before and after deformation, respectively, referred to fixed coordinate axes. Substituting eq. 20 into eq, 19, we derive an equation for the ellipsoid after deformation : CFjXiXj
= 1
(21)
with: Ci’j= d,id,jC,“,
)
TWO-DIMENSIONAL
f f Cij = Cji
STRAIN
(22)
ANALYSIS
We now develop an algebraic method to determine the shape and orientation of the strain ellipse from the deformed elliptical objects on a plane section. Our ultimate purpose is to analyze the state of strain in three dimensions, and the two-dimensional theory can be treated as a special case. However, two- and three-dimensional theories are developed separately, because the essence of our method is clearly illustrated in two dimensions. Moreover, strains in three dimensions can be estimated from two-dimensional analyses on plane sections. In this respect, the two-dimensional analysis provides a basis for the estimate of three-dimensional strains.
321
Strain calculation We propose the following simple procedures for the estimation of strain (see Notation I for major symbols used): (1) First measure the axial ratio and orientation of the major (long) axis for a certain number (N) of elliptical objects on a plane section. Then calculate the final-shape matrix from eq. 5 for each object. (2) Take an average of each component of the final-shape matrix. Then the strain ellipse of the system is given by the averaged final ellipse:
Lx
Yl
Ii-x
1 IL’ f
h
gi
=l
(23)
yl
with:
ii=&-,+ hi
I1
(24)
The principal axes of the strain ellipse can be calculated by eqs. 12 and 13. We shall prove that eq. 23 represents the strain ellipse of the system if the objects have initially random orientations. Since the deformation is assumed
NOTATION Major
I
symbols
used for two-dimensional
strain
analysis
Symbol
Quantity
Ri
Axial ratio of an object before deformation Arithmetic mean of the initial axial ratio Ri Orientation of major (long) axis of an object before deformation Components of initial-shape matrix (eqs. 14, 15) Components of averaged initial-shape matrix (eqs. 26, 27) Axial ratio of the averaged initial ellipse Orientation of major axis of the averaged initial ellipse Axial ratio of an object after deformation Orientation of major axis of an object after deformation Components of final-shape matrix (eqs. 17, 18) Components of averaged final-shape matrix (eqs. 23, 24) Axial ratio of the averaged final ellipse Orientation of major axis of the averaged final ellipse Axial ratio of the strain ellipse Orientation of major axis of the strain ellipse Deviatoric natural strain (2~~ = In R,) Number of measurements for strain analysis Error in percent for the estimation of the axial ratio, R,, of the strain (Er = 100 pf - R, B,)
RT 43
El, _Gi, -Hi F, G, H g; Rf @f
fi, gi, hi f; 2, h Ef @f
2 Es
N Er
ellipse
322
to be homogeneous throughout the system, (Y,/3, y and 6 in eq. 16 are constants for all elliptical objects. Then we derive from eqs. 18 and 24:
(25)
(26) Equation 25 relates the averaged final-shape matrix to the averaged initialshape matrix in terms of the transformation matrix in eq. 16. Obviously, the averaged final ellipse (eq. 23) represents the strain ellipse if and only if the averaged initial ellipse :
LX
i?
Yl F rH
(27)
GY Ih
is a circle centered F=G,
xl=1
at the origin. Thus we need to show:
H=O
(23) It can be assumed on physical grounds that the initial axial ratio Ri has a finite range: 1 < Ri < R,,,, where R,,, is the maximum initial axial ratio. We derive from eqs. 15 and 26 the components of the averaged initial-shape matrix in an integral form (e.g., Arley and Buch, 1950):
9 ((i +Ri)+(&-Ri)cos 2h)$(Ri,@i)dRidA 0
I
&$
JRmax1
((i
+ Ri) - ($
- Ri) COS20,) $(Riy$i)dRid@i
(29)
i HZ i 7”“” I
j
((& - Ri) sin 2@i] $(Ri,@i)dRid@i
0
where the frequency distribution ll = 1 j_= s $(Ri,&)dRid& 1
+(Rip pi) is normalized
(30)
0
It follows from eq. 29 that eq. 28 is satisfied if: II, = sx”“” I
i
((k
- Ri)
COS
such that:
2@i) ~(Ri,~i)dRid~i
=0
323
(31) Because the initially random orientations of the objects are assumed, Ri and @i are independent. Therefore, the frequency distribution $(Ri, Cpi)can be assumed to be a product of a function of Ri only and a function of $1 only (see Arley and Buch, 1950, p. 47): ~(~i,~i)
= ~~(Ri)~~~~i)
Here the frequency
(32)
distributions
are normalized
as: (33)
Equation
31 then becomes:
(34)
For a random
distribution
sp
li,@(9+h)COS2@id@i = J
ti
which on substitution proof.
of #i, tcl~(~i) = l/n, Thus: $e(@i) sin 2$id& = 0
(35)
0 into eq. 34, yields II, = n, = 0. This completes
the
We have proved that the averaged initiaI ellipse (eq. 27) represents precisely a circle centered at the origin for any distribution of the initial axial ratio Ri,if an infinite num~r of objects are included. This is only approximately valid when the sample size N is finite. The practical usefulness of our method will be discussed in detail in a subsequent paper (Part 2). Here we briefly discuss simple examples in order to illustrate the applicability and accuracy of our method. We use a computer-simulated pure-shear deformation of initially randomly oriented ellipses. The initial shape of an elliptical object can be specified by its axial ratio Ri and the orientation of its major axis @i. Ri and +i are generated automatically in the computer. For the generation of the random distribution of #it the multiplicative-congruence method (Harbaugh and Bonham-Carter, 1970,
324
p. 65-70) is used to obtain quasi-random numbers in the range of 0 to 1. These sequences of numbers are converted to those in the desired range of 0 to 7r simply by multiplying by 71.The axial ratio Ri can be obtained in a similar way, given the frequency distribution of Ri.The distribution of Ri is subdivided into a finite number of subgroups. The same procedures as used to generate pi are repeated for each subgroup to obtain any desired distribution of Ri.Only a unimodal distribution of Ri is considered here. These model objects are subjected to a pure-shear deformation :
(36)
Note that the axial ratio of the strain ellipse is R,,and its major axis is parallel to the coordinate axis x (i.e., 4, = 0). The final-shape matrix of each object is calculated by use of eq. 18. Our method is then applied to these simulated deformations, and the results are compared with exact values. Consider first three models A, B and C (Fig. 2 and Table I). Each model
5
F
Models A,BhC
n
n
Fig. 2. Frequency histograms of initial axial ratio Ri (a-c) and initial orientation of major (long) axis @i (d) for three models. F is the frequency in number of objects. Total number of objects is 50 for all models.
325 TABLE
I
Initial data of elliptical
model
objects
in Fig. 2
Model
N
Ry
o2 of Ri
F
G
H
Ri
5%
A B C
50 50 50
1.6 2.0 2.8
0.06 0.06 0.06
1.097 1.206 1.490
1.144 1.302 1.670
-0.043 -0.063 -0.100
1.045 1.066 1.089
120.3’ 116.2O 114.0°
u 2 ._ -- variance
of Ri. See Notation
___.-__
~__
I for other symbols.
consists of 50 ellipses with initial normal distribution of Ri (Fig. 2a-c). The arithmetic mean, RF, of Ri increases from model A to C, and the same distribution of pi (Fig. 2d) is used for the three models. Randomness of $i has been checked by the vector method of Curray (1956). Vector magnitude of the orientation data in Fig. 2d is 11.2%. According to the Raleigh test (Curray, 1956, pp. 124-127), the probability of obtaining a greater preferred orientation than that in Fig. 2d by pure-chance combination of random phases is larger than 50%. Thus the initial orientations $i are not significantly different from random distribution. These models have been subjected to the pure-shear deformation defined by eq. 36, and our method has been applied throughout the deformation process. The results are shown in Fig. 3. Percentile error, Er, for the estimate of the axial ratio of the strain ellipse, R,, is less than 10% in all cases (Fig. 3a). It is notable that Er is almost constant during deformation except at the very early stage. The orientation of the major axis of the strain ellipse can also be estimated with a fair degree of accuracy (Fig. 3b); the difference between the true and calculated values is less than 6 to 7” when R, > 1.2 (c, > 10%). It is clear from eq. 25 that the accuracy of our method depends largely upon how closely the averaged initial ellipse (eq. 27) represents a circle. Therefore, the accuracy is expected to increase as the axial ratio of the averaged initial ellipse Ri approaches 1. Indeed, Ri increases and the accuracy decreases from model A to C (Table I, column 8; Fig. 3). The ratio Ri decreases, in general, with a decrease in the initial axial ratio Ri for the same distribution of pi and for the same sample size N (cf., eq. 34). As expected, the accuracy is in fact reduced as the arithmetic mean, Ry, of the initial axial ratio increases (Figs. 2 and 3). Results from other analogous models are shown in Fig. 4. The number of measurements N ranges from 10 to 300. In all cases, the initial axial ratio Ri has normal distribution, and the major axes are nearly randomly oriented in the initial undeformed state. Arithmetic mean, Rp , of Ri is in the range of 1.5 and 2.0, and the variance of Ri ranges from 0.012 to 0.334. Irrespective of the sample size N, the mean initial axial ratio Rim , or the variance of Ri, the accuracy is closely related to the axial ratio of the averaged initial ellipse
326
Fig. 3. Errors for the axial ratio (a) and the orientation of major axis (b) of the strain ellipse for the three models. See Notation I for the symbols used..
Fig. 4. Error of the computed strain Er versus the axial ratio of the averaged initial ellipse Ri when the axial ratio of strain ellipse R, is 1.5, i.e., when the strain f, is about 20%. Dashed line indicates an upper limit of Er for a given value of Ri.
321
Ri (Fig. 4). The error Er for the estimate of R, falls on or below the dashed line in Fig. 4; in other words, there exists an upper limit in the error Er for a given value of Ri. This maximum error becomes larger as Ri increases; i.e., as the averaged initial ellipse deviates more from a circle. For instance, the accuracy is within 10% if Ri is less than 1.1, except at the very early stage when R, < 1.2 (E, < 10%). For a given Rti the accuracy depends on the orientation of the averaged initial ellipse pi. That is, Er is largest and I@,- & I is smallest when the principal directions of the averaged initial ellipse and those of the strain ellipse coincide. On the other hand, Er decreases and I@,- qr I increases as the angle between pi and I$, approaches 45”. Note that all these aspects of the estimate of error are to be readily understood through consideration of the geometrical relations among the averaged initial and final ellipses, and the strain ellipse. In general, Ri decreases with an increasing number of measurements N. The minimum sample size required for the accurate estimate of strain depends on the type of natural objects. If the relation between Ri and N is established by the study of undeformed natural objects, data in Fig. 4 could be used to determine the optimum sample size. It should be kept in mind that the accuracy of our method in Fig. 4 is established for pure-shear deformation (irrotational strain). Fortunately, the accuracy is not significantly different, even when the strain field is rotational (i.e., p # y in eq. 16). This will be discussed in more detail in the subsequent paper. THREE-DIMENSIONAL
STRAIN
ANALYSIS
The two-dimensional theory of the previous section is extended to three dimensions. Various types of strains are then reviewed briefly in connection with the strain ellipsoid. The significance of the two-dimensional analysis in the estimate of three-dimensional strains is emphasized. Strain calculation
from deformed
ellipsoidal objects
The direct extension of our two-dimensional theory is as follows (see Notation II) : (1) Measure lengths and orientations of all principal axes of deformed ellipsoidal objects. Then calculate the final-shape matrix by use of eq. 9 for each object. (2) Then take an arithmetic mean of each component of the final-shape matrix. The strain ellipsoid is given by the averaged final ellipsoid: C~jXiXj
=
1
(37)
We now prove that eq. 37 represents the strain ellipsoid if an infinite number of objects are measured. The transformation matrix [dij] in eq. 20 is constant for all deformed objects, so that we derive from eq. 22:
328 NOTATION Major
II
symbols
used for three-dimensional
strain
analysis
Quantity Normalized axial lengths of an object before deformation Arithmetic mean of l/(A,)2 Orientation of principal axes of an object before deformation (cf. eq. IO) Components of initial-shape matrix of an object (eq. 19) Components of averaged initial-shape matrix (eq. 39) Components of final-shape matrix of an object (eq. 21) Components of averaged final-shape matrix (eq. 37) Components of transformation matrix of deformation (eq. 20) Cauchy’s deformation tensor (eqs. 47, 48) Eulerian (Almansi’s) finite-strain tensor (eq. 49) Deviatoric natural strains in the principal directions of Eii(eqs. 50, 51)
Equation 37 gives the shape and orientation only if the averaged initial ellipsoid:
of the strain ellipsoid if and
C~XiXj = 1
(39)
is a sphere centered Qj = k6ij )
at the origin. Thus we need to show:
k = constant
(40)
This equation holds if the objects have initial random orientation. Here we shall prove only C& = Ci2 and c, = 0. Proof for the other cases can be done simply by the cyclic permutation of the indexes. We derive from eqs. 7-10: CL -c;,
= ___ 1 2&4J2
(cos 2&l, - cos 2&l, )
(41)
(42) where Cy,, C;, and Cy, are the components of the initial-shape matrix of an object; A, are the lengths of their principal axes before deformation (A, are normalized as A1A2A, = 1). The summation convention is used for the index m (m = 1, 2, 3). Initial random orientation of the objects has been assumed, and hence the axial lengths A, are independent of the distribution of #ii. Thus, we derive by the mathematical theorem that the arithmetic mean of the product of two independent quantities is the product of arithmetic means of these individual quantities (see Arley and Buch, 1950, p. 73):
Here&, is arithmetic mean of 1/(A,)2 krl*> il/?n are normalized as:
and the frequency
distributions
tirnl,
(45)
(46) Note that only three components of @ii are independent because of the orthogonality conditions (eq. 11). Consequently, all distributions of $ij in eqs. 43 and 44 cannot be, in general, specified independently. In the present case, however, we deal only with a random orientation. All components of $fj will, therefore, exhibit random distributions, even though all of them are not independent. Thus we have tirn1 = \il,,,* = 117~and $, = l/r2 for a random distribution of @ii. Substitution of these into eqs. 43 and 44 yields c, = c2 and cy2 = 0, and this completes the proof. We now consider briefly the limitations of our method in determining various types of strains. A sphere of radius r. before deformation changes into an ellipsoid after a finite homogeneous deformation. The equation for the ellipsoid is given by (e.g., Mase, 1970, p. 81-89): CisiXiXj
= r$
(47)
Cc = dhidhj
(48)
where CFj is called the Cauchy’s deformation tensor, and [d,] is the transformation matrix in eq. 20. The Eulerian (Almansi’s) finite-strain tensor Eij is given by: Eij = ~ (Sij - C~)
(49)
Observe that the principal directions of the Eulerian strain tensor coincide with those of the strain ellipsoid. We deal only with change in shape of the objects, and hence our analysis yields only an ellipsoid whose shape is geometrically similar to that of the strain ellipsoid (i.e., volumetric strain cannot be determined). Therefore, the Cauchy’s deformation tensor and the Eulerian finite strains cannot be estimated, unless an incompressible deformation is assumed. Lagrangian
330
(Green’s) finite strains and Biot’s strains (Biot, 1965) cannot be estimated either, because the amount of rigid-body rotation is unknown. Our analysis gives only the deviatoric natural strains (e ;, ~a, E;) in the principal directions of the Eulerian strain tensor. Let X1, X, and X3 be eigenvalues of the averaged final-shape matrix [~~j]. Then the lengths of the principal axes of the averaged final ellipsoid (eq. 37) are given by l/d/x,, l/&s and l/v%,, and we derive (e.g., Elliott, 1970, p. 2225-2226): e; = iln-,
x Xl
’ =!zln--_ x
E2
ci
=
$
x2
In
Ax3
(50)
with: h=
(h,X2X3)*‘3
(51)
These are the only strains that can be exactly evaluated in our analysis. Observe also that E; + EL + E: = 0, so that two of these strains or two of Xi, X2 and hs are independent. In other words, two parameters (say, Xl/h2 and X2/h,) are adequate to specify the shape of the strain ellipsoid. In addition, three independent components of directional cosines (eq. 10) are required to specify the orientation of the ellipsoid. Consequently, five components of the shape matrix [~~j] are independent. Strain estimation
based on two-dimensional
analysis
In principle, there should be no difficulty in the calculation of threedimensional strains from the lengths and orientations of principal axes of deformed ellipsoidal objects. However, the precise measurements of all principal axes of naturally deformed objects are often very difficult. For practical purposes, therefore, it would be more useful to analyze the strains in three dimensions from the two-dimensional strain analysis on plane sections (Ramsay, 1967, p. 142-147). Here we briefly discuss the technique in connection with our matrix formulation. We have proved that the averaged initial ellipsoid (eq. 39) is a sphere centered at the origin for a perfectly random orientation of the objects. Hence, the averaged initial ellipse on any plane represents a circle. It follows therefore that the averaged final ellipse (eq. 23) on any plane section simulates the shape of the strain ellipse on that plane. Note that our proof in the previous subsection is formulated for ellipsoidal objects centered at the origin. The proof need not be modified in the present case, because all elliptic sections of an ellipsoid cut by parallel planes are geometrically similar, and only the shape of the sections is treated in our analysis. Also, recall that the validity of all previous two-dimensional methods (Ramsay, 1967; Dunnet, 1969; Dunnet and Siddans, 1971; Elliott, 1970; Matthews et al., 1974) has been proved only for the two-dimensional deformation for which one of the principal axes of strain is normal to the plane of deformation. That these methods are valid on any arbitrary planar sections has not been proved. Our
331
method is essentially Let an ellipsoid:
free from this restriction.
(52)
be geometrically similar to the strain ellipsoid. Since the size of this ellipsoid can be specified arbitrarily and since Cij = Cji, only five components of the shape matrix [Cij] are independent. Suppose for the sake of convenience that the strain analysis has been done on three mutually perpendicular planes; i.e., on the xy, yz and zx planes. The analysis yields six quantities: axial ratio of the strain ellipse R, and orientation of its major axis 4, on the three plane sections. Thus, the matrix [Cij] is over-determined, and there is no unique way of obtaining it. We propose the following simple procedure to determine the ellipsoid statistically. The three ellipse sections of eq. 52 on xy, yz and zx planes, respectively, become : Lx
Ylpk
1C
Lx
21
r
clzjpj
21
Cl1
i c31
c22J
Cl3
=
1
(53)
=
1
(55)
LY]
x
c33 II 2
The shape and orientation of the strain ellipse are given by the averaged final ellipse (eq. 23) in the two-dimensional strain analysis. We will write the ellipses on xy, yz and zx planes, respectively, as: ]x
Yl[fX,
h.Xyl [xl
= 1
(56)
(57)
(58)
332
We shall determine [Cij] such that the ellipses given by eqs. 53-55 are geometrically similar to those of eqs. 56-58, respectively. One of the simplest procedures for this is as follows. First, let the ellipses given by eqs. 53 and 56 be the same:
1
h XY gxs
J
(59)
C22 is the common coefficient in eqs. 53 and 54, so that in order for the ellipses given by these equations to be similar to the observed ones, we must have :
Note that C22 = g,, in both eqs. 59 and 60. All coefficients except for Cl3 = Csl have been already determined. Eq. 55 has C,r in common with eq. 53 and Css with eq. 54, and hence there are two ways to obtain the coefficient: C 13
= C,,
+&
+xh,, xz
YZ
(61)
or:
(62) These two equations can be used to check the internal consistency of the results of strain analysis. The same procedures can be repeated starting from eq. 54 or 55, and there are six ways in total to determine [Cij] from the results of two-dimensional strain analysis. The arithmetic mean of [Cij] over these six possible cases could be used for the construction of the strain ellipsoid. G. Oertel (personal communication, 1975) has devised a more elaborate method employing a least-squares fitting, but this will not be discussed here. DISCUSSION
AND CONCLUSIONS
A simple algebraic method has been introduced to determine (1) the strain ellipse on a plane section from deformed elliptical objects and (2) the strain ellipsoid from deformed ellipsoidal objects. The analysis gives the deviatoric natural strains in the principal directions of the Eulerian finite-strain tensor. Otiier quantities like Lagrangian finite strains and Biot’s strains cannot be estimated by our analysis, since the method deals only with changes in shape of the objects. We prove that the shapes and orientations of the strain ellipse
333
and ellipsoid can be precisely determined by our method, if an infinite number of objects are measured. The analysis of pure-shear deformation of modelled elliptical objects reveals that the two-dimensional method is sufficiently accurate even for a finite sample size. The error has been evaluated in terms of the axial ratio of the averaged initial ellipse (Fig. 4). We have also confirmed that the result is not significantly different even when the deformation is rotational. In the light of theoretical considerations, our method is expected to be valid for the analysis of three-dimensional strains, if the averaged initial ellipsoid (eq. 39) is close to a sphere. Error estimation for the three-dimensional case is now under study, and will be reported in a subsequent paper. Detailed discussion of geological applications is beyond the scope of this paper. We suggest, however, that ooids are perhaps among the most suitable objects for the analysis of strain by our method, because they have a nearly spherical initial shape, little initial preferred orientation, and essentially the same ductility as that of their enclosing matrix. To check the applicability of the method, we have measured axial ratio Ri and orientation of major axis, pi, of the undeformed ooids from the Cool Creek Formation of southcentral Oklahoma (Chase et al., 1956). All ooids in arbitrarily selected small domains have been measured microscopically. Figure 5 is a typical result for Ri and & distributions, plotted on the polar graph. A perfectly random distribution of ellipses has a circular distribution on the graph (Elliott, 1970, p. 2223). Our data are not perfectly randomly oriented, but they are close to it. * The axial ratio of the averaged initial ellipse Ri is 1.051 for the oiiids in Fig. 5, so that this ellipse is fairly close to a circle. Data for the axial ratio Ri and the sample size N indicate that if N is larger than 10, Ri is less than 1.1; i.e., the averaged initial ellipse is close to a circle for N > 10 (Fig. 6). Accordingly, based on Fig. 4, the accuracy of our two-dimensional method would be within 10% except at very small strains (R,< 1.2 or E, < lo%), if only 10 oiiids were measured. Although this conclusion should be checked in more detail by examining undeformed odids from other localities, the minimum sample size for our method seems to be considerably smaller than that for the previous methods (Ramsay, 1967; Dunnet, 1969; Elliott, 1970; Matthews et al., 1974). Ramsay (1967, p. 187-190) list other geological objects whose initial shapes are nearly spherical. Tests of the applicability of our method to them must await future studies.
* By measuring undeformed ooids, Boulter (1976) shows that initial fabrics cannot be identified so readily as claimed by Elliott (1970), when the fabric is weak. We agree, because the distribution of points on the polar graph often deviates from a circular pattern even if the objects have nearly random orientation (e.g., data in Fig. 5). Elliott’s method is probably not precise when the initial fabric is nearly random. Our method is very accurate for an initially random fabric, but not for those with a strong initial fabric. Various techniques should be used in combination for the analysis of strain.
334
.
900
.
.
. .
*..-• l. . .%. ‘e*.* .-ad-
. .
.
.
1 ::E:“” IO”
0’
.
.
.
.
’
. .
.
llOl”I
.
Fig. 5. Axial ratio Creek Formation,
+lnR,:?
Ri and orientation Oklahoma, plotted
“Lcro”
SCllL 0.1
0.2
of major axis @i for 50 undeformed ooids on the polar graph of Elliott (1970).
from Cool
We have assumed the initially random orientation of the objects. Perfectly random orientation, however, is infrequent in nature (Elliott, 1970; Dunnet and Siddans, 1971). If the objects have a strongly oriented initial fabric, our method is no longer valid. * We shall now show how to extend it to include non-random orientation. For simplicity, only two-dimensional analysis is treated here. Only the averaged final-shape matrix (left side of eq. 25) can be determined from naturally deformed objects. It consists of three components. But only two quantities, the axial ratio of the averaged final ellipse R, and orientation of its major axis &, are meaningful, since the volumetric strain cannot be determined. On the other hand, there are six unknowns on the right side of eq. 25: axial ratio of the strain ellipse R, and orientation of its major axis @,, volumetric strain, rigid-body rotation, and the axial ratio of the averaged initial ellipse Ri and orientation of its major axis $i. Thus the problem is undeterminate, unless these unknown quantities are reduced to two by additional simplifying assumptions. First consider the geometrical meaning of the averaging process in eqs. 23 and 27. A simple example is given in Fig. 7. In general, if two ellipses have the same axial ratio and if their major axes are symmetric about the coordinate
* There are cases in which the objects have initial preferred orientation, and yet the averaged initial ellipse still represents a circle. A simple case of interest is that the objects have an initial bimodal orientation, where the frequency and mean axial ratio of the objects are about the same in two orientation modes, and the angle between them is 90’ For example, if pi is 45O and -45O for ellipses A and B, respectively, in Fig. 7, the averaged ellipse A&B is a circle. Our method is valid for these special cases.
335
.
i
!
:
l *
?k--
sm
$0
?ET--
.6
do
N
Fig. 6. The axial ratio of the averaged initial ellipse Ri versus number of measurements N for the undeformed oiiids from Cool Creek Formation, Oklahoma. Data for the encircled dot are shown in Fig. 5.
axis x;the principal axis of the averaged ellipse is parallel to the x-axis. Observe also that the area of the averaged ellipse is larger than those of individual ellipses (Fig. 7; examine also Table I, columns 5-7). The initial fabrics of natural objects are commonly symmetric about the bedding plane, with unimodal preferred orientation parallel to the bedding as a special case (Elliott, 1970; Dunnet and Siddans, 1971). The geometric interpretation of the averaging process suggests that the averaged initial ellipse for the sym-
Y
Y
Y
a : ellipse
A
X
X
x
b : ellipse
B
C :ellipse
A&B
Fig. 7. An example of the averaging process of ellipses used in eqs. 23, 27, 31 and 39 of text. Axial ratio is 2 for both ellipses, and orientation of major axis is 30’ and -30’ for ellipses A and B, respectively. Tbe averaged ellipse A&B has major axis parallel to the x-axis and axial ratio of 1.36. The volume of eliipse increases slightly during this averaging process.
3:36
metrical fabrics may be approximated by one whose principal axis is parallel to the bedding. Thus, taking the x-axis parallel to the bedding and normalizing the lengths of principal axes, we find the averaged initial-shape matrix in terms of a single parameter:
Note that the parameter Fis not necessarily larger than 1. Moreover, eq. 63 is a poor approximation, if the initial fabric is nearly random (e.g., ooids). Changes in area act equally in all directions. Hence, the volumetric strain may be assumed to be zero without loss of generality, since only changes in shape of the objects are treated here. We will further assume u-rotational strains, although this may not be generally true. The same assumption has been made by Elliott (1970) and Dunnet and Siddans (1971). Then the transformation matrix in eq. 16 can be written as: i
CY
__ I[
P
1.7 6
=Idz,
cos2~,+ dR, - JR.)
-$s
sin2@,
c &
sin 4, cos @, &
- dR,)
sin 4, cos 4,
s sin2@, + JR,
(64) cos2A
s
where R, is the axial ratio of the strain ellipse and 4, is the orientation of its major axis. The determinant of this matrix is unity, so that the condition of no volumetric strain is satisfied. We now have only three unknowns, R,, 4, and F. If one of these were determined, eq. 25 could be solved for the rest of unknowns. The principal directions @, might be determined independently by petrofabric techniques, or the possible range of the parameter Fmight be estimated from the undeformed shapes of natural objects. Newton’s method (e.g., Sokolnikoff and Redheffer, 1966, p. 659) could be used to solve the non-linear equations numerically. The lengths of the principal axes of the averaged final ellipse must be normalized to satisfy the condition of no volume change. Note that this extended method applies only if $r and @, are different. If they are the same (6, = @, = 0 in our formulation), eq. 25 is reduced to a single equation, R, = FR,, which is indeterminate. The method of Dunnet and Siddans (1971) has the same limitation. The validity and accuracy of our extended method must be further examined in the future. Our method of strain analysis also fails when there is a ductility contrast between the objects and their enclosing matrix. Extension of our method to include this effect is exceedingly difficult at this stage, because the deformation depends on the mechanical properties of the materials and the problem has not been generally solved. The key point is whether the transformation can be written in the form of eq. 16 or 20, and the initial-shape parameters
337
such as Ri, Cpi,do not enter into the tr~sformation extension of our method is impossible.
matrix.
If they do, the
ACKNOWLEDGEMENTS
We wish to express our sincere gratitude to Professors J. Handin, M. Friedman, and D.K. Parrish of Texas A&M University and Professor G. Oertel of the University of California, Los Angeles, for their encouragement, useful suggestions and discussions, and critical review of the manuscript. The extension of our two-dimensional method to three dimensions was suggested by Professor G. Oertel. We should also like to acknowledge the encouragement and helpful suggestions of Professors J.M. Logan and C.B. Johnson of Texas A&M University, Professors C. Kojima, I. Hara and K. Hide of Hiroshima University and Professor D. Elliott of the Johns Hopkins University. This work was supported in part by the National Science Foundation, Grant DES ‘74-22954, and by the Grant-in-Aid for Scientific Researches from the Ministry of Education, Japan.
REFERENCES A&y, N. and Buch, K.R., 1950, Introduction to the Theory of Probability and Statistics. Wifey, New York, 236 pp. Biot, MA., 1965. Mechanics of Incremental Deformations. Wiley, New York, 504 pp. Boulter, C.A., 1976. Sedimentary fabrics and their relation to strain-analysis methods. Geoiogy, 4: 141-146. Chase, G.W., Frederickson, E.A. and Ham, W.E., 1956. Resume’ of the geology of the Wichita Mountains, Oklahoma. In: Petroleum Geology of Southern Oklahoma, 1: 36.55. Am. Assoc. Pet. Geol. Curray, J.R., 1956. The analysis of two-dimensional orientation data. J. Geol., 64: 117131. Dunnet, D., 1969. A technique of finite-strain analysis using elliptical particles. Tectonophysics, 7: 117-136. Dunnet, D. and Siddans, A.W.B., 197 1. Non-random sedimentary fabrics and their modification by strain. Tectonopbysics, 1‘2: 307-325. Elliott, D., 1970. Determination of finite strain and initial shape from deformed elliptical objects. Geot. Sot. Am. Bull., 81: 2221-2236. Franklin, J.N., 1968. Matrix Theory. Prentice-Hall, New York, 292 pp. Fung, Y.C., 1965. Foundation of Solid Mechanics. Prentice-Hall, New York, 525 pp. Gay, N.C., 1968. Pure shear and simple shear deformation of inhomogeneous viscous fluids. 1. Theory. Tectonophysics, 5: 211-234. Harbaugh, J.W. and Bonham-Carter, G., 1970. Computer Simulation in Geology. Wiley, New York, 575 pp. Mase, G.E., 1970. Theory and Problems of Continuum Mechanics. McGraw-Hill, New York, Schaum’s Outline Series, 221 pp. 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.C., 1967. Folding and Fracturing of Rocks. McGraw-Hiil, New York, 568 pp. Sokolnikoff, IS. and Redheffer, R.M., 1966. Mathematics of Physics and Modern Engineering. McGraw-Hill, New York, 2nd ed., 752 pp.
315
Amsterdam
- Printed
in The Netherlands
Research Papers A SIMPLE ALGEBRAIC METHOD FOR STRAIN ESTIMATION DEFORMED ELLIPSOIDAL OBJECTS. 1. BASIC THEORY
TOSHIHIKO
SHIMAMOTO
and YUKIO
IKEDA
Center for Tectonophysics, Texas A&M University, College Fukuyama High School of Hiroshima University, Fukuyama
(Submitted
April 27, 1976; accepted
FROM
for publication
Station, Texas (U.S.A.) City, Hiroshima (Japan)
June 15, 1976)
ABSTRACT Shimamoto, deformed
T. and Ikeda, Y., 1976. A simple algebraic method ellipsoidal objects. 1. Basic theory. Tectonophysics,
for strain estimation 36: 315-337.
from
A new algebraic method is developed to determine the shape and orientation of the strain ellipsoid by using deformed ellipsoidal objects as strain markers. It is assumed that objects are of truly ellipsoidal shape with random orientation in the undeformed state, and that they deform homogeneously with their matrix. This part presents basic theories for: (1) the determination of the strain ellipse on a plane section from deformed elliptical objects; (2) determination of the strain ellipsoid from measurements of lengths and orientations of all principal axes of deformed ellipsoidal objects; and (3) construction of the strain ellipsoid from the two-dimensional analysis on three mutually orthogonal planes. General, finite, homogeneous deformation is treated, and the analysis gives the deviatoric natural strains in the principal directions of the Eulerian finite-strain tensor. The simplicity and usefulness of our method is demonstrated by the two-dimensional, pure-shear deformat.ion of model objects. Evaluation of error and optimum sample size, and geological applications of the method will be discussed in detail in subsequent papers.
INTRODUCTION
Determination of finite strain in rocks from distorted objects like ooids, pebbles and fossils involves certain fundamental difficulties that arise from : (1 j variable initial shape and orientation and (2) differences in mechanical properties between the objects and their surrounding matrix (Ramsay, 1967). This paper is intended to analyze the first problem; that is, the estimation of strain from deformed ellipsoidal objects which are not necessarily spherical in shape before deformation. The first significant contribution to this problem was made by Ramsay (1967) who devised a graphical method (R,/& diagram technique; see Notation I) to determine the strain ellipse on a plane section from deformed elliptical objects. His method has been further developed and applied to naturally deformed rocks by Dunnet (1969) and Dunnet and Siddans (1971). Elliott
316
(1970) has also developed a graphical method (shape-factor grid technique). They all assume initially elliptical shapes and no ductility contrast between the objects and their matrix. Initial random orientation is assumed by Ramsay (1967) and Dunnet (1969), but not by Elliott (1970) and Dunnet and Siddans (1971). The main disadvantage of these graphical methods is that a subject error tends to be introduced through curve-fitting. Moreover, their basic equations are formulated only for pure-shear deformation, so that it is still uncertain whether these methods are applicable to general deformations that include rotational strain. An algebraic method has been recently proposed by Matthews et al. (1974). It is simple and deals with general deformation, but it requires an independent knowledge of the orientations of the principal-strain axes in order to calculate strain magnitude. This obviously is a severe limitation, since the principal axes cannot always be determined a priori. Finally, in all these previous methods, equations are formulated as two-dimensional problems for which one of the principal axes of finite strain is normal to the plane of deformation. No complete theories have as yet been worked out to show that these two-dimensional methods are applicable to the determination of the strain ellipse on any arbitrary plane sections (G. Oertel, personal communication, 1975). Our major objective is to develop a simple algebraic method to determine the shape and orientation of the strain ellipsoid from deformed ellipsoidal objects. We make two basic assumptions that the objects are: (1) initially truly ellipsoidal in shape and randomly oriented; and (2) strained homogeneously together with their matrix. Two-dimensional theory is developed first in order to illustrate clearly the simplicity and accuracy of our method. We prove that the method gives the shape and orientation of the strain ellipse precisely if an infinite number of elliptical objects are measured. The practical usefulness of this two-dimensional analysis is demonstrated by the computer-simulated, pure-shear deformation of model objects. Our method is then extended to three dimensions. This is natural and straightforward, since the basic equations are formulated in matrix form. Analysis of threedimensional strains based on the two-dimensional analysis on plane sections is also emphasized for its practical applications. We prove that our twodimensional method can be applied for any arbitrary plane sections. General finite ho~~ogeneous deformation is treated, so that the method is applicable to both rotational and irrotational strains in exactly the same manner. Applicability of our two-dimensional method to geological problems is considered briefly from the measurements of undeformed oijids from the Cool Creek Formation of Oklahoma. We argue that our method would be sufficiently accurate for the analysis of geologic strain, For geological applications, any initial non-random orientation and ductility contrast should be often taken into account (Gay, 1968; Elliott, 1970; Dunnet and Siddans, 1971). The extension of our method to include these effects is also discussed briefly in the context of the two-dimensional analysis. Detailed discussions of errors in our methods, and further geological applications will be given in subsequent papers.
317
MATRIX REPRESENTATION
OF ELLIPSES AND ELLIPSOIDS
Consider an ellipse centered at the origin of the coordinate system with major and minor axes a, b (Fig. 1). The ellipse is represented by an equation:
(1) where x’ and y’ are taken parallel to the principal axes of the ellipse. Let the axial ratio be R = a/b > 1, and let a and b be normalized such that ab = 1. * Then a = JR and b = l/v’R; and eq, 1 can be written in matrix form:
Let 8 be the angle between the CCand x’ axes (0 < 0 < a). The coordinate transformation from (x, y) to (x’, y’) is (e.g., Fung, 1965):
(3)
Fig. I. An ellipse centered at the origin. The major and minor axes of the ellipse are (I and b, respectively, and 3 is the angle between the major axis and coordinate axis x. * Generality is not lost by this normalization, because we deal only with the shape (not size) of the object.
318
which on substitution
into eq. 2 yields: (4)
1
f =Rcos20+Rsin20 + R cos20
g = isin
(5)
Equation 4 is a matrix representation of an ellipse, and this form is employed throughout our analysis. Note that this matrix (called the shape matrix) is symmetric. Consider an ellipsoid centered at the origin with principal axes al, a2 and a3 (~1~2~3 = 1). Using the summation convention, we represent the ellipsoid as (e.g., Franklin, 1968, p. 94-98): BijXiXj = 1
(6)
where : (7)
The coordinate axes are taken parallel to the principal directions of the ellipsoid. The matrix [B:j] obeys the transformation law for second-order Cartesian tensors. Thus, eq. 6 can be written in terms of fixed coordinates (XI, ~2, x3) as: BijXiXj Bij
= 1
(8)
= P/z iPljBk[
(9)
Observe that Bij = Bji. The symbol /3ij denotes the cosine of the angle, Gij, between i-th primed and j-th unprimed coordinate axes (0 < ~ij < n): Pij
= COS ~ij
(10)
and: PkiPhj
=
&ij
where 6ij is the Kronecker symbol. Note that only three components are independent because of the orthogonality conditions (eq. 11).
(11)
of Pij
319
The principal axes of an ellipse or ellipsoid are the eigenvectors of the shape matrix (see Franklin, 1968). In two dimensions an eigenvalue X satisfies the characteristic equation :
f---A
h
h
g---x
=
0
which has two solutions Xi and 1s. Then the lengths of the principal axes are I/&, and l/d&. The principal direction 8 is given by the orientation of the corresponding eigenvector (Fig. 1). A simple equation can be derived for the two-dimensional case : tan28
=2h f-g
(13)
The lengths and orientations of the principal axes of an ellipsoid can be determined in a similar manner from the shape matrix [B,j] , FINITE
HOMOGENEOUS
DEFORMATION
OF ELLIPSES
AND ELLIPSOIDS
Consider an ellipse with major and minor axes ei and bi with Uibi = 1 in the initial undeformed state. The initial axial ratio Ri is a,/bi (21). Let (X, Y) be coordinates of a point before deformation, and let the major axis make an angle Qi with the coordinate axis X (0 < 4i < E). From eqs. 4 and 5 the ellipse is represented by :
[X
“Ei
HJx]=’
where the components Fi
=$
Gi =i
(14)
of the initial shape matrix are:
(($+Ri) +(k-Ri)COS2Oi) ’ ((~+Ri~ -($-Ri) COS26ij
(15)
,
This is simply eq. 5 modified in form for later convenience. Finite homogeneous deformation in two dimensions can be generally pressed as a linear transformation :
ex-
320
Here the Eulerian formulation (e.g., Mase, 1970, p. 79) is used; (x, y) are coordinates of a point after deformation, and CY,/3, y and 6 are constants (/3 # y in general). Substitution of eq. 16 into eq. 14 yields: (17)
where:
(18) Equation 17 is the matrix representation of the ellipse after the deformation. Lengths and orientations of the principal axes can be determined from eqs. 12 and 13. Observe that the final-shape matrix in eq. 17 is symmetric. For the three-dimensional extension of the analysis, let an ellipsoid before deformation be represented in the summation notation by: C~XiXj = 1
(19)
The initial state is denoted sional, finite homogeneous
by the superscript o. A general, three-dimendeformation can be expressed as:
Xi = dijXj
(26)
where dij # djj in general, and (Xi, Xs, Xs) and (xl, x2, x3) are coordinates before and after deformation, respectively, referred to fixed coordinate axes. Substituting eq. 20 into eq, 19, we derive an equation for the ellipsoid after deformation : CFjXiXj
= 1
(21)
with: Ci’j= d,id,jC,“,
)
TWO-DIMENSIONAL
f f Cij = Cji
STRAIN
(22)
ANALYSIS
We now develop an algebraic method to determine the shape and orientation of the strain ellipse from the deformed elliptical objects on a plane section. Our ultimate purpose is to analyze the state of strain in three dimensions, and the two-dimensional theory can be treated as a special case. However, two- and three-dimensional theories are developed separately, because the essence of our method is clearly illustrated in two dimensions. Moreover, strains in three dimensions can be estimated from two-dimensional analyses on plane sections. In this respect, the two-dimensional analysis provides a basis for the estimate of three-dimensional strains.
321
Strain calculation We propose the following simple procedures for the estimation of strain (see Notation I for major symbols used): (1) First measure the axial ratio and orientation of the major (long) axis for a certain number (N) of elliptical objects on a plane section. Then calculate the final-shape matrix from eq. 5 for each object. (2) Take an average of each component of the final-shape matrix. Then the strain ellipse of the system is given by the averaged final ellipse:
Lx
Yl
Ii-x
1 IL’ f
h
gi
=l
(23)
yl
with:
ii=&-,+ hi
I1
(24)
The principal axes of the strain ellipse can be calculated by eqs. 12 and 13. We shall prove that eq. 23 represents the strain ellipse of the system if the objects have initially random orientations. Since the deformation is assumed
NOTATION Major
I
symbols
used for two-dimensional
strain
analysis
Symbol
Quantity
Ri
Axial ratio of an object before deformation Arithmetic mean of the initial axial ratio Ri Orientation of major (long) axis of an object before deformation Components of initial-shape matrix (eqs. 14, 15) Components of averaged initial-shape matrix (eqs. 26, 27) Axial ratio of the averaged initial ellipse Orientation of major axis of the averaged initial ellipse Axial ratio of an object after deformation Orientation of major axis of an object after deformation Components of final-shape matrix (eqs. 17, 18) Components of averaged final-shape matrix (eqs. 23, 24) Axial ratio of the averaged final ellipse Orientation of major axis of the averaged final ellipse Axial ratio of the strain ellipse Orientation of major axis of the strain ellipse Deviatoric natural strain (2~~ = In R,) Number of measurements for strain analysis Error in percent for the estimation of the axial ratio, R,, of the strain (Er = 100 pf - R, B,)
RT 43
El, _Gi, -Hi F, G, H g; Rf @f
fi, gi, hi f; 2, h Ef @f
2 Es
N Er
ellipse
322
to be homogeneous throughout the system, (Y,/3, y and 6 in eq. 16 are constants for all elliptical objects. Then we derive from eqs. 18 and 24:
(25)
(26) Equation 25 relates the averaged final-shape matrix to the averaged initialshape matrix in terms of the transformation matrix in eq. 16. Obviously, the averaged final ellipse (eq. 23) represents the strain ellipse if and only if the averaged initial ellipse :
LX
i?
Yl F rH
(27)
GY Ih
is a circle centered F=G,
xl=1
at the origin. Thus we need to show:
H=O
(23) It can be assumed on physical grounds that the initial axial ratio Ri has a finite range: 1 < Ri < R,,,, where R,,, is the maximum initial axial ratio. We derive from eqs. 15 and 26 the components of the averaged initial-shape matrix in an integral form (e.g., Arley and Buch, 1950):
9 ((i +Ri)+(&-Ri)cos 2h)$(Ri,@i)dRidA 0
I
&$
JRmax1
((i
+ Ri) - ($
- Ri) COS20,) $(Riy$i)dRid@i
(29)
i HZ i 7”“” I
j
((& - Ri) sin 2@i] $(Ri,@i)dRid@i
0
where the frequency distribution ll = 1 j_= s $(Ri,&)dRid& 1
+(Rip pi) is normalized
(30)
0
It follows from eq. 29 that eq. 28 is satisfied if: II, = sx”“” I
i
((k
- Ri)
COS
such that:
2@i) ~(Ri,~i)dRid~i
=0
323
(31) Because the initially random orientations of the objects are assumed, Ri and @i are independent. Therefore, the frequency distribution $(Ri, Cpi)can be assumed to be a product of a function of Ri only and a function of $1 only (see Arley and Buch, 1950, p. 47): ~(~i,~i)
= ~~(Ri)~~~~i)
Here the frequency
(32)
distributions
are normalized
as: (33)
Equation
31 then becomes:
(34)
For a random
distribution
sp
li,@(9+h)COS2@id@i = J
ti
which on substitution proof.
of #i, tcl~(~i) = l/n, Thus: $e(@i) sin 2$id& = 0
(35)
0 into eq. 34, yields II, = n, = 0. This completes
the
We have proved that the averaged initiaI ellipse (eq. 27) represents precisely a circle centered at the origin for any distribution of the initial axial ratio Ri,if an infinite num~r of objects are included. This is only approximately valid when the sample size N is finite. The practical usefulness of our method will be discussed in detail in a subsequent paper (Part 2). Here we briefly discuss simple examples in order to illustrate the applicability and accuracy of our method. We use a computer-simulated pure-shear deformation of initially randomly oriented ellipses. The initial shape of an elliptical object can be specified by its axial ratio Ri and the orientation of its major axis @i. Ri and +i are generated automatically in the computer. For the generation of the random distribution of #it the multiplicative-congruence method (Harbaugh and Bonham-Carter, 1970,
324
p. 65-70) is used to obtain quasi-random numbers in the range of 0 to 1. These sequences of numbers are converted to those in the desired range of 0 to 7r simply by multiplying by 71.The axial ratio Ri can be obtained in a similar way, given the frequency distribution of Ri.The distribution of Ri is subdivided into a finite number of subgroups. The same procedures as used to generate pi are repeated for each subgroup to obtain any desired distribution of Ri.Only a unimodal distribution of Ri is considered here. These model objects are subjected to a pure-shear deformation :
(36)
Note that the axial ratio of the strain ellipse is R,,and its major axis is parallel to the coordinate axis x (i.e., 4, = 0). The final-shape matrix of each object is calculated by use of eq. 18. Our method is then applied to these simulated deformations, and the results are compared with exact values. Consider first three models A, B and C (Fig. 2 and Table I). Each model
5
F
Models A,BhC
n
n
Fig. 2. Frequency histograms of initial axial ratio Ri (a-c) and initial orientation of major (long) axis @i (d) for three models. F is the frequency in number of objects. Total number of objects is 50 for all models.
325 TABLE
I
Initial data of elliptical
model
objects
in Fig. 2
Model
N
Ry
o2 of Ri
F
G
H
Ri
5%
A B C
50 50 50
1.6 2.0 2.8
0.06 0.06 0.06
1.097 1.206 1.490
1.144 1.302 1.670
-0.043 -0.063 -0.100
1.045 1.066 1.089
120.3’ 116.2O 114.0°
u 2 ._ -- variance
of Ri. See Notation
___.-__
~__
I for other symbols.
consists of 50 ellipses with initial normal distribution of Ri (Fig. 2a-c). The arithmetic mean, RF, of Ri increases from model A to C, and the same distribution of pi (Fig. 2d) is used for the three models. Randomness of $i has been checked by the vector method of Curray (1956). Vector magnitude of the orientation data in Fig. 2d is 11.2%. According to the Raleigh test (Curray, 1956, pp. 124-127), the probability of obtaining a greater preferred orientation than that in Fig. 2d by pure-chance combination of random phases is larger than 50%. Thus the initial orientations $i are not significantly different from random distribution. These models have been subjected to the pure-shear deformation defined by eq. 36, and our method has been applied throughout the deformation process. The results are shown in Fig. 3. Percentile error, Er, for the estimate of the axial ratio of the strain ellipse, R,, is less than 10% in all cases (Fig. 3a). It is notable that Er is almost constant during deformation except at the very early stage. The orientation of the major axis of the strain ellipse can also be estimated with a fair degree of accuracy (Fig. 3b); the difference between the true and calculated values is less than 6 to 7” when R, > 1.2 (c, > 10%). It is clear from eq. 25 that the accuracy of our method depends largely upon how closely the averaged initial ellipse (eq. 27) represents a circle. Therefore, the accuracy is expected to increase as the axial ratio of the averaged initial ellipse Ri approaches 1. Indeed, Ri increases and the accuracy decreases from model A to C (Table I, column 8; Fig. 3). The ratio Ri decreases, in general, with a decrease in the initial axial ratio Ri for the same distribution of pi and for the same sample size N (cf., eq. 34). As expected, the accuracy is in fact reduced as the arithmetic mean, Ry, of the initial axial ratio increases (Figs. 2 and 3). Results from other analogous models are shown in Fig. 4. The number of measurements N ranges from 10 to 300. In all cases, the initial axial ratio Ri has normal distribution, and the major axes are nearly randomly oriented in the initial undeformed state. Arithmetic mean, Rp , of Ri is in the range of 1.5 and 2.0, and the variance of Ri ranges from 0.012 to 0.334. Irrespective of the sample size N, the mean initial axial ratio Rim , or the variance of Ri, the accuracy is closely related to the axial ratio of the averaged initial ellipse
326
Fig. 3. Errors for the axial ratio (a) and the orientation of major axis (b) of the strain ellipse for the three models. See Notation I for the symbols used..
Fig. 4. Error of the computed strain Er versus the axial ratio of the averaged initial ellipse Ri when the axial ratio of strain ellipse R, is 1.5, i.e., when the strain f, is about 20%. Dashed line indicates an upper limit of Er for a given value of Ri.
321
Ri (Fig. 4). The error Er for the estimate of R, falls on or below the dashed line in Fig. 4; in other words, there exists an upper limit in the error Er for a given value of Ri. This maximum error becomes larger as Ri increases; i.e., as the averaged initial ellipse deviates more from a circle. For instance, the accuracy is within 10% if Ri is less than 1.1, except at the very early stage when R, < 1.2 (E, < 10%). For a given Rti the accuracy depends on the orientation of the averaged initial ellipse pi. That is, Er is largest and I@,- & I is smallest when the principal directions of the averaged initial ellipse and those of the strain ellipse coincide. On the other hand, Er decreases and I@,- qr I increases as the angle between pi and I$, approaches 45”. Note that all these aspects of the estimate of error are to be readily understood through consideration of the geometrical relations among the averaged initial and final ellipses, and the strain ellipse. In general, Ri decreases with an increasing number of measurements N. The minimum sample size required for the accurate estimate of strain depends on the type of natural objects. If the relation between Ri and N is established by the study of undeformed natural objects, data in Fig. 4 could be used to determine the optimum sample size. It should be kept in mind that the accuracy of our method in Fig. 4 is established for pure-shear deformation (irrotational strain). Fortunately, the accuracy is not significantly different, even when the strain field is rotational (i.e., p # y in eq. 16). This will be discussed in more detail in the subsequent paper. THREE-DIMENSIONAL
STRAIN
ANALYSIS
The two-dimensional theory of the previous section is extended to three dimensions. Various types of strains are then reviewed briefly in connection with the strain ellipsoid. The significance of the two-dimensional analysis in the estimate of three-dimensional strains is emphasized. Strain calculation
from deformed
ellipsoidal objects
The direct extension of our two-dimensional theory is as follows (see Notation II) : (1) Measure lengths and orientations of all principal axes of deformed ellipsoidal objects. Then calculate the final-shape matrix by use of eq. 9 for each object. (2) Then take an arithmetic mean of each component of the final-shape matrix. The strain ellipsoid is given by the averaged final ellipsoid: C~jXiXj
=
1
(37)
We now prove that eq. 37 represents the strain ellipsoid if an infinite number of objects are measured. The transformation matrix [dij] in eq. 20 is constant for all deformed objects, so that we derive from eq. 22:
328 NOTATION Major
II
symbols
used for three-dimensional
strain
analysis
Quantity Normalized axial lengths of an object before deformation Arithmetic mean of l/(A,)2 Orientation of principal axes of an object before deformation (cf. eq. IO) Components of initial-shape matrix of an object (eq. 19) Components of averaged initial-shape matrix (eq. 39) Components of final-shape matrix of an object (eq. 21) Components of averaged final-shape matrix (eq. 37) Components of transformation matrix of deformation (eq. 20) Cauchy’s deformation tensor (eqs. 47, 48) Eulerian (Almansi’s) finite-strain tensor (eq. 49) Deviatoric natural strains in the principal directions of Eii(eqs. 50, 51)
Equation 37 gives the shape and orientation only if the averaged initial ellipsoid:
of the strain ellipsoid if and
C~XiXj = 1
(39)
is a sphere centered Qj = k6ij )
at the origin. Thus we need to show:
k = constant
(40)
This equation holds if the objects have initial random orientation. Here we shall prove only C& = Ci2 and c, = 0. Proof for the other cases can be done simply by the cyclic permutation of the indexes. We derive from eqs. 7-10: CL -c;,
= ___ 1 2&4J2
(cos 2&l, - cos 2&l, )
(41)
(42) where Cy,, C;, and Cy, are the components of the initial-shape matrix of an object; A, are the lengths of their principal axes before deformation (A, are normalized as A1A2A, = 1). The summation convention is used for the index m (m = 1, 2, 3). Initial random orientation of the objects has been assumed, and hence the axial lengths A, are independent of the distribution of #ii. Thus, we derive by the mathematical theorem that the arithmetic mean of the product of two independent quantities is the product of arithmetic means of these individual quantities (see Arley and Buch, 1950, p. 73):
Here&, is arithmetic mean of 1/(A,)2 krl*> il/?n are normalized as:
and the frequency
distributions
tirnl,
(45)
(46) Note that only three components of @ii are independent because of the orthogonality conditions (eq. 11). Consequently, all distributions of $ij in eqs. 43 and 44 cannot be, in general, specified independently. In the present case, however, we deal only with a random orientation. All components of $fj will, therefore, exhibit random distributions, even though all of them are not independent. Thus we have tirn1 = \il,,,* = 117~and $, = l/r2 for a random distribution of @ii. Substitution of these into eqs. 43 and 44 yields c, = c2 and cy2 = 0, and this completes the proof. We now consider briefly the limitations of our method in determining various types of strains. A sphere of radius r. before deformation changes into an ellipsoid after a finite homogeneous deformation. The equation for the ellipsoid is given by (e.g., Mase, 1970, p. 81-89): CisiXiXj
= r$
(47)
Cc = dhidhj
(48)
where CFj is called the Cauchy’s deformation tensor, and [d,] is the transformation matrix in eq. 20. The Eulerian (Almansi’s) finite-strain tensor Eij is given by: Eij = ~ (Sij - C~)
(49)
Observe that the principal directions of the Eulerian strain tensor coincide with those of the strain ellipsoid. We deal only with change in shape of the objects, and hence our analysis yields only an ellipsoid whose shape is geometrically similar to that of the strain ellipsoid (i.e., volumetric strain cannot be determined). Therefore, the Cauchy’s deformation tensor and the Eulerian finite strains cannot be estimated, unless an incompressible deformation is assumed. Lagrangian
330
(Green’s) finite strains and Biot’s strains (Biot, 1965) cannot be estimated either, because the amount of rigid-body rotation is unknown. Our analysis gives only the deviatoric natural strains (e ;, ~a, E;) in the principal directions of the Eulerian strain tensor. Let X1, X, and X3 be eigenvalues of the averaged final-shape matrix [~~j]. Then the lengths of the principal axes of the averaged final ellipsoid (eq. 37) are given by l/d/x,, l/&s and l/v%,, and we derive (e.g., Elliott, 1970, p. 2225-2226): e; = iln-,
x Xl
’ =!zln--_ x
E2
ci
=
$
x2
In
Ax3
(50)
with: h=
(h,X2X3)*‘3
(51)
These are the only strains that can be exactly evaluated in our analysis. Observe also that E; + EL + E: = 0, so that two of these strains or two of Xi, X2 and hs are independent. In other words, two parameters (say, Xl/h2 and X2/h,) are adequate to specify the shape of the strain ellipsoid. In addition, three independent components of directional cosines (eq. 10) are required to specify the orientation of the ellipsoid. Consequently, five components of the shape matrix [~~j] are independent. Strain estimation
based on two-dimensional
analysis
In principle, there should be no difficulty in the calculation of threedimensional strains from the lengths and orientations of principal axes of deformed ellipsoidal objects. However, the precise measurements of all principal axes of naturally deformed objects are often very difficult. For practical purposes, therefore, it would be more useful to analyze the strains in three dimensions from the two-dimensional strain analysis on plane sections (Ramsay, 1967, p. 142-147). Here we briefly discuss the technique in connection with our matrix formulation. We have proved that the averaged initial ellipsoid (eq. 39) is a sphere centered at the origin for a perfectly random orientation of the objects. Hence, the averaged initial ellipse on any plane represents a circle. It follows therefore that the averaged final ellipse (eq. 23) on any plane section simulates the shape of the strain ellipse on that plane. Note that our proof in the previous subsection is formulated for ellipsoidal objects centered at the origin. The proof need not be modified in the present case, because all elliptic sections of an ellipsoid cut by parallel planes are geometrically similar, and only the shape of the sections is treated in our analysis. Also, recall that the validity of all previous two-dimensional methods (Ramsay, 1967; Dunnet, 1969; Dunnet and Siddans, 1971; Elliott, 1970; Matthews et al., 1974) has been proved only for the two-dimensional deformation for which one of the principal axes of strain is normal to the plane of deformation. That these methods are valid on any arbitrary planar sections has not been proved. Our
331
method is essentially Let an ellipsoid:
free from this restriction.
(52)
be geometrically similar to the strain ellipsoid. Since the size of this ellipsoid can be specified arbitrarily and since Cij = Cji, only five components of the shape matrix [Cij] are independent. Suppose for the sake of convenience that the strain analysis has been done on three mutually perpendicular planes; i.e., on the xy, yz and zx planes. The analysis yields six quantities: axial ratio of the strain ellipse R, and orientation of its major axis 4, on the three plane sections. Thus, the matrix [Cij] is over-determined, and there is no unique way of obtaining it. We propose the following simple procedure to determine the ellipsoid statistically. The three ellipse sections of eq. 52 on xy, yz and zx planes, respectively, become : Lx
Ylpk
1C
Lx
21
r
clzjpj
21
Cl1
i c31
c22J
Cl3
=
1
(53)
=
1
(55)
LY]
x
c33 II 2
The shape and orientation of the strain ellipse are given by the averaged final ellipse (eq. 23) in the two-dimensional strain analysis. We will write the ellipses on xy, yz and zx planes, respectively, as: ]x
Yl[fX,
h.Xyl [xl
= 1
(56)
(57)
(58)
332
We shall determine [Cij] such that the ellipses given by eqs. 53-55 are geometrically similar to those of eqs. 56-58, respectively. One of the simplest procedures for this is as follows. First, let the ellipses given by eqs. 53 and 56 be the same:
1
h XY gxs
J
(59)
C22 is the common coefficient in eqs. 53 and 54, so that in order for the ellipses given by these equations to be similar to the observed ones, we must have :
Note that C22 = g,, in both eqs. 59 and 60. All coefficients except for Cl3 = Csl have been already determined. Eq. 55 has C,r in common with eq. 53 and Css with eq. 54, and hence there are two ways to obtain the coefficient: C 13
= C,,
+&
+xh,, xz
YZ
(61)
or:
(62) These two equations can be used to check the internal consistency of the results of strain analysis. The same procedures can be repeated starting from eq. 54 or 55, and there are six ways in total to determine [Cij] from the results of two-dimensional strain analysis. The arithmetic mean of [Cij] over these six possible cases could be used for the construction of the strain ellipsoid. G. Oertel (personal communication, 1975) has devised a more elaborate method employing a least-squares fitting, but this will not be discussed here. DISCUSSION
AND CONCLUSIONS
A simple algebraic method has been introduced to determine (1) the strain ellipse on a plane section from deformed elliptical objects and (2) the strain ellipsoid from deformed ellipsoidal objects. The analysis gives the deviatoric natural strains in the principal directions of the Eulerian finite-strain tensor. Otiier quantities like Lagrangian finite strains and Biot’s strains cannot be estimated by our analysis, since the method deals only with changes in shape of the objects. We prove that the shapes and orientations of the strain ellipse
333
and ellipsoid can be precisely determined by our method, if an infinite number of objects are measured. The analysis of pure-shear deformation of modelled elliptical objects reveals that the two-dimensional method is sufficiently accurate even for a finite sample size. The error has been evaluated in terms of the axial ratio of the averaged initial ellipse (Fig. 4). We have also confirmed that the result is not significantly different even when the deformation is rotational. In the light of theoretical considerations, our method is expected to be valid for the analysis of three-dimensional strains, if the averaged initial ellipsoid (eq. 39) is close to a sphere. Error estimation for the three-dimensional case is now under study, and will be reported in a subsequent paper. Detailed discussion of geological applications is beyond the scope of this paper. We suggest, however, that ooids are perhaps among the most suitable objects for the analysis of strain by our method, because they have a nearly spherical initial shape, little initial preferred orientation, and essentially the same ductility as that of their enclosing matrix. To check the applicability of the method, we have measured axial ratio Ri and orientation of major axis, pi, of the undeformed ooids from the Cool Creek Formation of southcentral Oklahoma (Chase et al., 1956). All ooids in arbitrarily selected small domains have been measured microscopically. Figure 5 is a typical result for Ri and & distributions, plotted on the polar graph. A perfectly random distribution of ellipses has a circular distribution on the graph (Elliott, 1970, p. 2223). Our data are not perfectly randomly oriented, but they are close to it. * The axial ratio of the averaged initial ellipse Ri is 1.051 for the oiiids in Fig. 5, so that this ellipse is fairly close to a circle. Data for the axial ratio Ri and the sample size N indicate that if N is larger than 10, Ri is less than 1.1; i.e., the averaged initial ellipse is close to a circle for N > 10 (Fig. 6). Accordingly, based on Fig. 4, the accuracy of our two-dimensional method would be within 10% except at very small strains (R,< 1.2 or E, < lo%), if only 10 oiiids were measured. Although this conclusion should be checked in more detail by examining undeformed odids from other localities, the minimum sample size for our method seems to be considerably smaller than that for the previous methods (Ramsay, 1967; Dunnet, 1969; Elliott, 1970; Matthews et al., 1974). Ramsay (1967, p. 187-190) list other geological objects whose initial shapes are nearly spherical. Tests of the applicability of our method to them must await future studies.
* By measuring undeformed ooids, Boulter (1976) shows that initial fabrics cannot be identified so readily as claimed by Elliott (1970), when the fabric is weak. We agree, because the distribution of points on the polar graph often deviates from a circular pattern even if the objects have nearly random orientation (e.g., data in Fig. 5). Elliott’s method is probably not precise when the initial fabric is nearly random. Our method is very accurate for an initially random fabric, but not for those with a strong initial fabric. Various techniques should be used in combination for the analysis of strain.
334
.
900
.
.
. .
*..-• l. . .%. ‘e*.* .-ad-
. .
.
.
1 ::E:“” IO”
0’
.
.
.
.
’
. .
.
llOl”I
.
Fig. 5. Axial ratio Creek Formation,
+lnR,:?
Ri and orientation Oklahoma, plotted
“Lcro”
SCllL 0.1
0.2
of major axis @i for 50 undeformed ooids on the polar graph of Elliott (1970).
from Cool
We have assumed the initially random orientation of the objects. Perfectly random orientation, however, is infrequent in nature (Elliott, 1970; Dunnet and Siddans, 1971). If the objects have a strongly oriented initial fabric, our method is no longer valid. * We shall now show how to extend it to include non-random orientation. For simplicity, only two-dimensional analysis is treated here. Only the averaged final-shape matrix (left side of eq. 25) can be determined from naturally deformed objects. It consists of three components. But only two quantities, the axial ratio of the averaged final ellipse R, and orientation of its major axis &, are meaningful, since the volumetric strain cannot be determined. On the other hand, there are six unknowns on the right side of eq. 25: axial ratio of the strain ellipse R, and orientation of its major axis @,, volumetric strain, rigid-body rotation, and the axial ratio of the averaged initial ellipse Ri and orientation of its major axis $i. Thus the problem is undeterminate, unless these unknown quantities are reduced to two by additional simplifying assumptions. First consider the geometrical meaning of the averaging process in eqs. 23 and 27. A simple example is given in Fig. 7. In general, if two ellipses have the same axial ratio and if their major axes are symmetric about the coordinate
* There are cases in which the objects have initial preferred orientation, and yet the averaged initial ellipse still represents a circle. A simple case of interest is that the objects have an initial bimodal orientation, where the frequency and mean axial ratio of the objects are about the same in two orientation modes, and the angle between them is 90’ For example, if pi is 45O and -45O for ellipses A and B, respectively, in Fig. 7, the averaged ellipse A&B is a circle. Our method is valid for these special cases.
335
.
i
!
:
l *
?k--
sm
$0
?ET--
.6
do
N
Fig. 6. The axial ratio of the averaged initial ellipse Ri versus number of measurements N for the undeformed oiiids from Cool Creek Formation, Oklahoma. Data for the encircled dot are shown in Fig. 5.
axis x;the principal axis of the averaged ellipse is parallel to the x-axis. Observe also that the area of the averaged ellipse is larger than those of individual ellipses (Fig. 7; examine also Table I, columns 5-7). The initial fabrics of natural objects are commonly symmetric about the bedding plane, with unimodal preferred orientation parallel to the bedding as a special case (Elliott, 1970; Dunnet and Siddans, 1971). The geometric interpretation of the averaging process suggests that the averaged initial ellipse for the sym-
Y
Y
Y
a : ellipse
A
X
X
x
b : ellipse
B
C :ellipse
A&B
Fig. 7. An example of the averaging process of ellipses used in eqs. 23, 27, 31 and 39 of text. Axial ratio is 2 for both ellipses, and orientation of major axis is 30’ and -30’ for ellipses A and B, respectively. Tbe averaged ellipse A&B has major axis parallel to the x-axis and axial ratio of 1.36. The volume of eliipse increases slightly during this averaging process.
3:36
metrical fabrics may be approximated by one whose principal axis is parallel to the bedding. Thus, taking the x-axis parallel to the bedding and normalizing the lengths of principal axes, we find the averaged initial-shape matrix in terms of a single parameter:
Note that the parameter Fis not necessarily larger than 1. Moreover, eq. 63 is a poor approximation, if the initial fabric is nearly random (e.g., ooids). Changes in area act equally in all directions. Hence, the volumetric strain may be assumed to be zero without loss of generality, since only changes in shape of the objects are treated here. We will further assume u-rotational strains, although this may not be generally true. The same assumption has been made by Elliott (1970) and Dunnet and Siddans (1971). Then the transformation matrix in eq. 16 can be written as: i
CY
__ I[
P
1.7 6
=Idz,
cos2~,+ dR, - JR.)
-$s
sin2@,
c &
sin 4, cos @, &
- dR,)
sin 4, cos 4,
s sin2@, + JR,
(64) cos2A
s
where R, is the axial ratio of the strain ellipse and 4, is the orientation of its major axis. The determinant of this matrix is unity, so that the condition of no volumetric strain is satisfied. We now have only three unknowns, R,, 4, and F. If one of these were determined, eq. 25 could be solved for the rest of unknowns. The principal directions @, might be determined independently by petrofabric techniques, or the possible range of the parameter Fmight be estimated from the undeformed shapes of natural objects. Newton’s method (e.g., Sokolnikoff and Redheffer, 1966, p. 659) could be used to solve the non-linear equations numerically. The lengths of the principal axes of the averaged final ellipse must be normalized to satisfy the condition of no volume change. Note that this extended method applies only if $r and @, are different. If they are the same (6, = @, = 0 in our formulation), eq. 25 is reduced to a single equation, R, = FR,, which is indeterminate. The method of Dunnet and Siddans (1971) has the same limitation. The validity and accuracy of our extended method must be further examined in the future. Our method of strain analysis also fails when there is a ductility contrast between the objects and their enclosing matrix. Extension of our method to include this effect is exceedingly difficult at this stage, because the deformation depends on the mechanical properties of the materials and the problem has not been generally solved. The key point is whether the transformation can be written in the form of eq. 16 or 20, and the initial-shape parameters
337
such as Ri, Cpi,do not enter into the tr~sformation extension of our method is impossible.
matrix.
If they do, the
ACKNOWLEDGEMENTS
We wish to express our sincere gratitude to Professors J. Handin, M. Friedman, and D.K. Parrish of Texas A&M University and Professor G. Oertel of the University of California, Los Angeles, for their encouragement, useful suggestions and discussions, and critical review of the manuscript. The extension of our two-dimensional method to three dimensions was suggested by Professor G. Oertel. We should also like to acknowledge the encouragement and helpful suggestions of Professors J.M. Logan and C.B. Johnson of Texas A&M University, Professors C. Kojima, I. Hara and K. Hide of Hiroshima University and Professor D. Elliott of the Johns Hopkins University. This work was supported in part by the National Science Foundation, Grant DES ‘74-22954, and by the Grant-in-Aid for Scientific Researches from the Ministry of Education, Japan.
REFERENCES A&y, N. and Buch, K.R., 1950, Introduction to the Theory of Probability and Statistics. Wifey, New York, 236 pp. Biot, MA., 1965. Mechanics of Incremental Deformations. Wiley, New York, 504 pp. Boulter, C.A., 1976. Sedimentary fabrics and their relation to strain-analysis methods. Geoiogy, 4: 141-146. Chase, G.W., Frederickson, E.A. and Ham, W.E., 1956. Resume’ of the geology of the Wichita Mountains, Oklahoma. In: Petroleum Geology of Southern Oklahoma, 1: 36.55. Am. Assoc. Pet. Geol. Curray, J.R., 1956. The analysis of two-dimensional orientation data. J. Geol., 64: 117131. Dunnet, D., 1969. A technique of finite-strain analysis using elliptical particles. Tectonophysics, 7: 117-136. Dunnet, D. and Siddans, A.W.B., 197 1. Non-random sedimentary fabrics and their modification by strain. Tectonopbysics, 1‘2: 307-325. Elliott, D., 1970. Determination of finite strain and initial shape from deformed elliptical objects. Geot. Sot. Am. Bull., 81: 2221-2236. Franklin, J.N., 1968. Matrix Theory. Prentice-Hall, New York, 292 pp. Fung, Y.C., 1965. Foundation of Solid Mechanics. Prentice-Hall, New York, 525 pp. Gay, N.C., 1968. Pure shear and simple shear deformation of inhomogeneous viscous fluids. 1. Theory. Tectonophysics, 5: 211-234. Harbaugh, J.W. and Bonham-Carter, G., 1970. Computer Simulation in Geology. Wiley, New York, 575 pp. Mase, G.E., 1970. Theory and Problems of Continuum Mechanics. McGraw-Hill, New York, Schaum’s Outline Series, 221 pp. 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.C., 1967. Folding and Fracturing of Rocks. McGraw-Hiil, New York, 568 pp. Sokolnikoff, IS. and Redheffer, R.M., 1966. Mathematics of Physics and Modern Engineering. McGraw-Hill, New York, 2nd ed., 752 pp.