The analysis of strain in folded layers

Tee tonophysics Elsevier Publishing

THE

Company,

ANALYSIS

Amsterdam

OF STRAIN

- Printed

in The Netherlands

IN FOLDED

LAYERS

B.E. HOBBS Department (Australia) (Received

of‘ Geophysicsand Geochemistry, Australian National University, Canberra, A.C. T. November

6, 1970)

ABSTRACT IIobbs,

B.E., 197 1, The analysis

of strain in folded

layers.

Tectonophysics,

11: 329-375.

The general theory of inhomogeneous finite strain is outlined. For any deformation it is possible to define two quantities, CKM and ckm, which are the deformation tensors. The principal axes of these tensors are the principal axes of strain at each point in the body and the proper numbers of these tensors are the values and reciprocals respectively of the principal quadratic elongations at each point. CKM gives the strain relative to the undeformed state and can be represented by a quadric known as the strain ellipsoid. ckm gives the strain relative to the deformed state and can be represented by a quadric known as the reciprocal strain ellipsoid. Thus, knowledge of the components of these two tensors enables the states of strain to be specified at all points in any arbitrarily deformed body. The components of C’KM also enable the changes in the angles between lines to be calculated so that the distortion of older lineations during any type of deformation may be mapped out in the deformed body. This theory is used to investigate the distributions of strain in the various types of folds delineated by Ramsay (1967, p. 365) and to examine the ways in which initially straight lineations are distorted by these folds. Two contrasted strain distributions are recognized - that which presumably arises through flexure of the deformed layer at some stage in the deformation together with arbitrary changes in dimensions and that which arises when a layer is inhomogeneously distorted to conform to a shape change dictated by neighbouring layers undergoing deformation. The analysis is applicable to any type of fold including non-plane, non-cylindrical folds. In suitable natural folds it may be possible to determine the distribution of strain throughout the fold.

INTRODUCTION

Most geological structures are the result of an inhomogeneous strain, that is, lines which were originally straight have been distorted to produce curved lines. In the past twenty years there has been a considerable volume of literature aimed at determining the strain in deformed rocks, the approach being one of dividing the deformed body into regions of homogeneous strain and applying or modifying the methods of finite strain analysis (Ramsay, 1967; Jaeger, 1969) within these regions. The purpose of this paper is to examine the general case of a three dimensional finite, inhomogeneous strain and to indicate some applications to structural geology. The general question posed by this paper is: given the deformation indicated in Fig. 1 where an initial cube is inhomogeneously disTectofzophysics,

11 (1971)

329-375

330

B.E. HOBBS

E

F

./

/”

6 F h

G D

C

Fig. 1. Finite inhomogeneo~1~ deformation

of a cube.

torted, what are the orientations and magnitudes of the principal axes of strain at each point in the deformed body? Ramsay and Graham (1970) have also developed the theory of inhomogeneous strain and applied it to discover the variations of strain that can exist in certain types of geological structures. In this paper, then, the concern is primarily with the states of strain that can exist within an arbitrarily folded layer merely because it is a certain shape. As such, the discussions and conclusions reached are independent of the mechanisms of deformation, of the stress states which exist during folding, of the mechanical properties of the material, and of the deformation path leading from the unstrained to the strained condition. There are two distinct ways of considering the distortion of a layer to produce a foId. The first way is to consider the deformation of the layer and to treat the axial surface as a purely geometrical feature defined by the geometry of the deformed state. The second way is to define a system of planes which may themselves be arbitrarily deformed as they shear past each other. The fold is then treated as a purely geometrical feature which arises during the deformation if some arbitrary layer is present to act as a marker. For simple deformatiorls these shear pfanes may become the axial surfaces of the folds formed. Although such treatments are independent of the physical mechanism of deformation, it is probable that many natural folds which are described by the first approach involve flexure of the folded layers at some stage in the formation of the fold, whereas those folds described by the second approach involve a generalized shape change prescribed by adjacent, presumably stronger, layers undergoing folding. In general, except For ideally similar folds, these two different treatments lead to different states of strain in deformed layers of identical shape. Accordingly, the following is divided into five parts: Part I. The general theory of finite inhomogeneous strain is outlined and, in so doing, the answer to the above question is indicated. Part 2. An analysis of folding, the deformation being referred to the layer undergoing deformation. Part 3. An analysis of folding, the deformation being referred to a system of shear

ANALYSIS

OF STRAIN

IN FOLDED

331

LAYERS

planes oblique to the layer to be folded. In parts 2 and 3, the theory of part 1 is applied to a number of theoretical questions which may be asked concerning the distribution of strain in plane, cylindrical and in non-plane, non-cylindrical folds. Included here is a discussion of the distortion of initially straight lineations by various types of folds. Part 4. A consideration of natural folds. This leads to a discussion of the information required to determine the strain at each point in a natural fold. Part 5. A general discussion concerning some of the geometrical aspects of folding and the significance of cleavage. Throughout this paper certain transformations and deformations relating the undeformed and deformed states are employed. These have been selected for two reasons: (1) They appear to be reasonable geologically in that they produce fold shapes which are common in nature. (2) They appear to represent the most general forms of bending and of shearing possible, there being arbitrary changes of dimensions and no restriction of constant volume deformation. There is no intention that these transformations and deformations are unique or that they are the only ones of relevance to deformed rocks. This is true particularly in part 4 which is meant solely to illustrate the type of information which can be obtained if the general nature of the deformation is postulated. It would be of interest to investigate the strain distributions which result from other geologically reasonable deformations. Although many of the results obtained here are inherent in the discussions of Ramsay (1967), the important feature of the methods outlined is the provision of an analytical tool for discussing the distribution of strain in folded layers. 1. THE GENERAL

THEORY

OF FINITE,

INHOMOGENEOUS

STRAIN

The following discussion is a brief outline of the general theory of finite strain, sufficient to answer the question posed in the Introduction and to apply the theory to various types of folds. By far the most comprehensive treatment of general finite strain is given by Truesdell and Toupin (1960) and their development is followed here. All functions considered are assumed to be single valued, continuous and as many times differentiable as is required. All transformations are assumed to possess unique inverse transformations. The convention is adopted that quantities printed as majuscules refer to the undeformed state whereas quantities printed as minuscules refer to the deformed state. Thus the coordinates XK refer to the undeformed state whereas xk refer to the deformed state. There are no restrictions on the XK or the xk, for example, one may be a system of cylindrical coordinates * whereas the other may be a system of oblique coordinates. There is no necessity for the coordinate system employed to be orthogonal. As far as geological applications are concerned it is often convenient for both XK and xk to be systems of Cartesian coordinates. However, for the solution of specific problems it may be convenient to change the xk coordinate system, either by a rotation, to another system of Cartesian *

See Eisenhart (1947, pp. 1, 63-69)

coordinate

systems.

Tectonophysics,

11 (1971)

329-375

and Eringen

(1962,

pp. 5-7, 430-434)

for a discussion

of

332

B.E. HOBBS

coordinates, or by a general transformation to a system of curvilinear coordinates. The great advantage of the formalism set up by Truesdell and Toupin (1960) is that these changes in coordinate system may be readily accomplished, so that the state of strain at each point in the body may be specified with respect to any convenient system of coordinates. Consider the deformation represented in Fig. 2 where (X’, X2, X3) and (x’ ,x2, x3) are systems of curvilinear coordinates in the undeformed and deformed bodies respectively In the deformation, the region R becomes the region r, the curve C becomes the curve c, the tangent to C, dX, becomes the tangent to c, dx, and the element of arc dS becomes the element of arc ds. The deformation may be written:

xi=xi(X’,X2,X3)

(1)

Both systems of coordinates are referred to a standard system of Cartesian coordinates (Y’ , Y2, Y3), this system being referrred to as the common frame.

Fig. 2. General deformation showing coordinates. Common frame Y’, Y’, Y3. Coordinates in undeformed state X’ , X2, X3. Coordinates in deformed state x I, x2, x3. In the deformation, the region R becomes the region r, the curve C becomes c, the element of arc dS becomes ds and the tangent to C, dX. becomes the tangent to c, dx.

The length of an element of arc in the undeformed

ds2 = GKM dxK d.P

state is given by

(2)

ANALYSIS

OF STRAIN

IN FOLDED

LAYERS

333

where the usual tensor summation convention, that repeated indices are summed, applies. G,, is the metric tensor defined for the coordinate system, XK, by the transformation which gives the XK in terms of the common frame. Thus: G,,

=6,Q--

(3)

axK axM

where 6pQ is a Kronecker

delta with components

6rr = ?jz2 = 833 = 1, and hpQ = 0 for

P # Q.For any Cartesian coordinate system, the components of the metric tensor are Grr = G22 = G3a = 1, GKM = 0 for K f M, sothat eq. 2 reduces to the familiar: dS2 = (dX’)*

+ (dX2)* + (dX3)2

In any coordinate system, knowledge of the components of the metric tensor enables lengths and angles between lines to be measured. Similarly, the length of an element of arc in the deformed state is given by: d? = gkm dx%bP

(4)

where gkm is the metric tensor for the coordinate system xk. The elements of arc dS and ds referred respectively to the deformed and undeformed states are (see Michal, 1947, p. 46):

axK axM ~ a9 axm

ds2 =gKM

~

&

axk?ff

zgkm

axK axM

dxk

dxm

uK~M

or

ds* = ckm dxk dxm (6) d.~’ = CKM dXK dXM where :

axK axM ‘km=gKM

p

G

and:

(7)

axk axm CKM=gkm-axK axM In these expressions, gKM = G, . ckm is known as Cauchy’s deformation tensor and it gives the length of the undeformed element of arc in terms of the coordinates of the deformed body. C,M is Green’s deformation tensor and it gives the length of the deformed element of arc in terms of the coordinates of the undeformed body. Both of these tensors are symmetrical so that ‘km = Cmk and CKM = CM,. In fact th ese tensors are metric tensors, ckm being the comTeclonophJ~sics, 11 (1971)

329-37s

334

B.E. HOBBS

ponents of gKM referred to a system of coordinates in the deformed state and C,, being the components of gkm referred to these same coordinates but in the undeformed state. Most treatments of this subject (see Michal, 1947, p. 77; Truesdell and Toupin, 1960, pp. 266-271) proceed to calculate (ds’ - dS2) which is a measure of the change in length of the element of arc and hence is a measure of the strain. This development leads to the definition of the Eulerian and Lagrangean strain tensors, from which the classical theories of finite and infinitesimal homogeneous strain follow (Truesdell and Toupin, 1960, pp. 28%297,303-309). However, for most geological applications it is sufficient to consider only the deformation tensors. This is so since it can be shown (Truesdell and Toupin, 1960, p. 262) that ckm and C,, are unique symmetrical tensors whose principal axes at each point are the principal axes of strain and whose proper numbers are the reciprocals and the values respectively of the principal quadratic elongations * at those points. In the literature on strain, the quadric which represents ckm is known as the reciprocal strain ellipsoid whereas that which represents CKM is known as the strain ellipsoid. In addition, there also exists a symmetric tensor F&r, the tensor inverse to ckm, which is generally more convenient to use than ck,,, . The principal axes of $, are the prmcipaf axes of strain in the deformed material (as opposed to C,, whose principal axes are the principal axes of strain in the undeformed material). Unlike ckm, however, the proper numbers of &,, are the principal quadratic elongations instead of the reciprocals of these quantities. elk, is given by: elk

(8)

where GKM is the tensor whose components

are:

CKM = (cofactor CKM)/C G being the value of the determinant

z;:, =c

-11

(9) of GKM. The components

of &,

are given by:

mglk

This is generally a more convenient form for calculation of the principal axes of strain -1 k is asymmetrical whereas that of $, is always symsince, in general, the matrix of cm metrical. Throughout this paper FAk or zirn is used depending on which is convenient. In Cartesian coordinates GKM = ?iKM and g,, = 6,,, where the 6’s are Kronecker deltas, so that eq. 8 becomes

(10) For most geological situations, the convenient tensors to use as measures of strain are CKM and $n, since both have the principal quadratic elongations as their proper numbers and both are given in terms of the derivatives of the deformed coordinates with respect to the coordinates of points in the undeformed material. CKM expresses the strain relative to the undeformed state whereas $, expresses the strain relative to the deformed state. - ._____ * Most of the literature tion is the square

on this subject uses the term stretch of this quantity (Jaeger, 1969).

to denate

(dx/dx).

The quadratic

elonga-

ANALYSIS

335

OF STRAIN IN FOLDED LAYERS

ckm

is inconvenient to use because its use involves the calculation of the inverse of eq. 1, a calculation which may be tedious. Thus, in terms of the undeformed coordinates, the principal quadratic elongations are given as the roots of the cubic equation: Cii -- XG,r

‘C,? - XGlz

C13

-

hG13

C2,

-

AC21

C22

-

hG22

c23

-

hG23

C31

-

hG31

c32

-

=32

C33

-

hG33

with a similar expression in terms of the components of &, and of &,, formed state. The orientations of the principal axes of strain may then each point by methods given by Ramsay (1967) or by Jaeger (1969). If the strain is plane, then the angle, 8, between the X’ - coordinate major principal axis of strain in the deformed state is given by (Jaeger,

sin 20 =-1 c2

(11)

, for the debe determined

at

curve and the 1969, p. 8):

z:2

(12)

-1 -cl

where the xk system is Cartesian and F:, F: are the proper numbers of F&r, with F: > Z:. Thus, the sense in which 20 is measured from the x1 coordinate curve depends only upon the sign of F:2. This result is used later to determine whether the principal planes of strain containing Ff and the fold axis form divergent or convergent fans across a given fold (see Ramsay, 1967, p. 403). All aspects of the deformation may therefore be mapped out from one point to the next in any inhomogeneously deformed body. In particular, if dV and dv are volume elements in the undeformed and deformed states respectively, then: dv =JdI’

(13)

where J = (dTm/d+) * J,. det. being the value of the determinant quantity concerned and Jx being the jacobian: ax’ __

ax’ __

of the

ax1 -

ax1 ax2 ax3

J,=

a.2 g ,,, ax1 ax2 ax3

(14)

ax3 ax3 ax3 __-ax' ax2 ax3 The condition,

then, that volumes remain constant

during the deformation

J= 1 deformations

is: (15)

for which the volume is preserved are said to be isochoric.

Tectonophysics, 11 (1971) 329-375

336

B.E. HOBBS

In principle then, the question posed in the Introduction may be answered provided the transformation which describes the deformation in Fig. 1 is known. Examples of such deformations are treated in the sections devoted to non-plane, non-cyl~drica1 folds in parts 2 and 3. 2. ANALYSIS OF FOLDS; DEFORMATION

REFERRED TO THE LAYER TO BE FOLDED

In this part, the previous theory is employed to calculate the states of strain in various types of folds. The special cases of similar and parallel folds are first considered and then more general types of folds. The various classes of fold distinguished by Ramsay (1967, pp. 365-372) appear as distinct groups in the generalization of these two special cases. A section is devoted to the shapes adopted by initially straight lineations deformed by these various groups of folds. ~rou~out this part, unless otherwise mentioned, XK are a system of Cartesian coordinates chosen such that X3 and X2 lie parallel to the plane to be folded and X’ is normal to this plane. Similar folds The most general transformation, corresponding of similar, plane cylindrical folds is: x1 = allX’

+ E(X’)

+a13X3

x2 = at,X’

+ F(X2)

+ az3X3

to eq. 1, which leads to the formation

(16)

x3 = a3,X’ +&(X2) + as3X3 where the aji are constants and E, F, ii! are non-linear functions of X2. The xk coordinate system is chosen as Cartesian as in Fig. 3. In the deformation described by eq. 16, dimensions are changed in all of the Xi, X2 and X3 directions, the resulting fold shape being shown in Fig. 3. Substitution of X’ = k, a constant, into eq. 16 gives the equations of the folded surface in terms of the “parameters” X2 and X3 : x1 =allk+E(X2)+a13X3 x2 = azlk + F(X2) + az3X3

(17)

x3 = a3] k + /#(X2) + a33X3 The plane X2 = constant remains a plane and intersects the folded surface as shown in Fig. 3. The plane X3 = constant is also folded. The elimination of X2 and X3 from eq. 17 gives as the equation the folded surface: f(x1,x2,x3,k)=

0,

and since this is to represent a cylindrical numbers, d’, such that:

fold, there exist (Eisenhart,

1947, p. 61) three

337

ANALYSIS OF STRAIN IN FOLDED LAYERS

Fig. 3. Coordinate axes for general similar fold. xk are a system of Cartesian coordinates in the deformed state. zk are another Cartesian system in the deformed state defined by eq. 18. The three SUIfaces generated by substituting X1 = const., X2 = const. and X3 = const., one at a time, into eq. 16 are shown. where the d’ are the direction cosines of the fold axis relative to the coordinate system, xk. In this manner, the orientation of the fold axis may be established. A second system of coordinates in the deformed state may be chosen such that:

zr is in the axial plane and normal to the fold axis and z2 is normal to the axial plane z3 is parallel to the fold axis

(18)

,zk is, then, a system of Cartesian coordinates (Fig. 3) related to the xk by a translation and a rotation which are functions of the aii in eq. 16. It is now possible to consider a number of theoretical questions concerning the distribution of strain in similar folds. The condition

that the deformation

be isochoric

In order that volumes should not be changed during the deformation

J=

all

E’

a13

azl

F’

a23

aa1

H’

aa3

(eq. 16):

= 1

where the primes denote differentiation with respect to X2. Thus, in order that the deformation be isochoric, E’ must be a certain linear function of F’ and of H’.

Tecronophysics, 11 (1971) 329-375

B.E. HOBBS

338

The condition that the fold axis is a principal axis of strain The components of the metric tensor, gkm, for the coordinate system xk in Fig. 3 are gkm = ‘km. Hence, the tensor zAk relative to these axes is given by eq. 10. Thus: = a:, + (E’)* + af3

-‘I cl Cl 2

* =

Cl3 3

-'I C2

ai, + (F')*t a&

=a:,

t(P)*

=a,,a2,

tai3

+ E’F’ + a,3a23 a23a33

C” = a2,a3, 3

+

F:” = a,,a,,

t H'E' + a33a,3

F’H’+

In a transformation from the axesxk to the Cartesian system zk given by eq. 18, the -Ik become the camp onents (c?;“)* defined by: components c,, ($c)*

azkaxj

= qli

(19)

axi azm (see Michal, 1947, p. SO). Since xk and zk are both systems of Cartesian coordinates, 19 may be rewritten:

eq.

where b,? is the cosine of the angle between zm and XI and is a function of the akl in eq. 16. For the fold axis to be a principal axis of strain, the tensor (zkk)* must be of the form :

(F,$,*=

. i

0

0 0

0

1

(20)

1

where the dots, in general, represent non-zero terms. That is:

($ ‘)* = 0 and:

Each of these conditions a(E’)*

+ p(F’)*

+ Ye

reduces to the form: + c(E’F’)

where the Greek letters are constants

+ {(F’H’)

+ v(H’E’)

and are functions

=X

of the aij and bj.

(21)

ANALYSIS

OF STRAIN IN FOLDED LAYERS

Although probably of the type: xi = alax’

not the general solution to eq. 21, folds given by transformatio~~s

+ E(X2)

X-J =

339

1

a&P

x3 =

(22, @33X3

I

have (Zkk) already in the form of eq, 20 and hence have the x3-axis (the fold axis) as a principal axis of strain. A genera1 solution to this problem is given in part 3. ~~~~~~t~~n that the ~~~~ plane is e~~~yw~~rep~r~~~~~ to a ~~~~~~a1pact ~~~tr~in For the axiat plane to be everywhere parallel to a principal pIane of strain, the tensor (FAk)* must be of the form:

(zg>”

=

0 i. i

0

.

*

0

0

.~

(23)

or: ($ f )* = (Fj p* cc0

(24)

These conditions again are of the same form as eq. 2 1. A generaf solution to eq. 21 has not been found but if deformations of the type given in eq. 22 are considered, then eq. 24 reduces to E(X”) = constant and hence, no fold exists the deformation being a homogeneous pure shear. Intuitively, it is felt that a simiiar result is true for the general case and certainly: E(X2)=

&X2 + br

F(X2) = a*X2 + bz H(X’) = a3X2 + bj the ds and b’s being constants, is a solution of eq, 21, the def~r~l~tion (eq. lo) then being a homogeneous simple shear with no folds produced. The general solution to this probIem is given in part 3 where It is shown that indeed there exists no similar fold in which the axial plane is everywhere parallel to a principal plane of strain.

Special cases of the defor~~ations represented by eq. I6 are those situations where there is no eIongation parallel to the fold axis and the strain becomes plane. Tfre problem is then reduced to one in two dimensions and a number of,such folds h treated below to illustrate the distributions of strain. The coordinate system. xk, in the deformed state, is cartesian. Similar results follow if the fold axis is a principal axis of strain and the strain is non-plane.

340

B.E. HOBBS

Parabolic curves. Consider the deformations

given by:

x1 = aX’ x2 = (X’)2 + bX2 where a and b are constants. The strain relative to the undeformed respectively may be obtained from:

and deformed states

The principal axes of strain are parallel to the coordinate axes only at the origin (X’ = 0, x1 = 0) where the principal strains are a and b. The deformation is isochoric only if a = l/b and the distribution of strain for isochoric deformations with l/a = 1.0 through to 2.0 is illustrated in Fig. 4.

B

A

c

Fig. 4. Distribution of strain in ideally similar, parabolic folds with various amounts of homogeneous shortening normal to the axial plane. The orientation and magnitude of the maximum principal axis of strain is shown at each point by the solid line. The initial length of each line is equal to that at the hinge of Fig. 4(A). The undeformed shape of the layer is shown blank and is given by X2 = constant: A. No shortening; B. 25% shortening; and C. 50% shortening.

Cubic curves. An exampleSof folds which are cubits in profile are the deformations by: X 1 zaJy’ x2

=

(X’)3 + 3(X’)’

- 10(X’) + bX*

given

341

ANALYSIS OF STRAIN IN FOLDED LAYERS

where a and b are constants. respectively is given by:

CKM=

[;‘+a2

The strain relative to the undeformed

;];

?Ak=

[a

and deformed states

;+b2;

where E = 3(X’)2 f 6(X’) - 10. The defo~ation is isochoric only if a = I/b and the distribution of strain for isochoric deformations with l/a = 1.O through to 2.0 is illustrated in Fig. 5

96 41

~ 17

\ 3 \i

3s 7

~ A

Fig. 5. Distribution of strain in ideally similar, cubic folds with varsous amounts of homogeneous shortening normal to the axial plane. The orientation of the maximum principal strain is shown at each point by the solid line. The undeformed layer is given by X2 = constant. The ratio of the principal strains is given at various places: A. No shortening, B. 50% shortening.

Sine curves. Consider the deformations x1 =izx’

Tecronophysics,

11 (1971) 329-375

given by:

B.E. HOBBS

342

where a and b are constants. respectively is given by:

%t

=

[;c+;os;

The strain relative to the undeformed

“,“1;

Gk=

and deformed states

‘::..,*,~~~~~s~~~~~,~I

The principal axes of strain are parallel to the coordinate curves only at X1 = n/2; x1 = ai/2 ivhere the principal strains are a and b. The deformation is isochoric only if a = I/b and the distribution of strain for isochoric deformations with l/a = 1.0 through to 2.0 is illustrated in Fig. 6.

I:ig. 6. Distribution of strain in ideally similar, sine-curve folds. The undeformed X2 = constant. A. No shortening; B. 20% shortening; and C. .9@%Shortening.

layer is given by

It appears from these examples that the major principal axis of strain in similar folds may be almost parallel to the folded layer throughout the fold for small amounts of overall

ANALYSIS

OF STRAIN

IN FOLDED

343

LAYERS

shortening of the deformed body. For large amounts of overall shortening, this axis is almost coincident with the trace of the axial plane on the profile plane throughout the fold although the principal planes of strain parallel to this axis and to x3, always form a “divergent fan” in the terminology of Ramsay (1967, pp. 403-405). This result follows from the use of eq. 12, since F:, = a,,E’ (from eq. 22). This is referred to as a divergent principal plane fan in the rest of the paper. The reverse situation is referred to as a convergent principal plane fan. Parallel folds In the case of parallel folds, it is convenient to choose the xk as a system of cylindrical coordinates (Fig. 7) given in terms of the common frame by: x’ = [(Y’)2 + (Y’)2]” x2 = tan-’ [Y2/Y’] x3

=

(25)

y3

The xk coordinate surfaces are shown in Fig. 7; for x1 = constant, the surfaces are cylinders coaxial with Y3, for x2 = constant, the surfaces are planes passing through Y3 and for x3 = constant, the surfaces are planes normal to Y3. Consider the deformation of a block of material with faces parallel to the surfaces XK = constant (Fig. 7). If the deformation is given by: x1 = E(X’) x2 = b-(X2)

(26)

x3 = H(X3) then the surfaces X1 = constant are distorted into parts of circular cylinders, the surfaces X2 = constant, become planes, all passing through a common line, the fold axis, and the surfaces X3 = constant become planes normal to the fold axis. The deformation therefore generates a parallel fold. Since GKM = gKM = 6 KM andg,,=g33=l,g22=(x1)2,gkt=Ofork#l,thestrainis given by:

where the primes denote differentiation. Since eq. 27 is in diagonal form, the directions of the coordinate curves in both the undeformed and deformed states are principal axes cf strain and the principal strains are E’, x’F’, If’. The deformation is isochoric only (Truesdell and Toupin, 1960, p. 301) for: x1 =JzAX’

Tectonophysics,

+B,x’

11 (1971)

=CJF2,x3

329-375

=DX3,ACD=+

1

(28)

344

B.E. HOBBS

A

Fig. 7. Deformation of a block by bending about a cylinder. A. Coordinates in the undeformed xK; B. Coordinates in the deformed state xk. The point P in A becomes the point p in B.

state

where A, B, C, D are constants. An example of the distribution of strain for such deformations is illustrated in Fig. 8. Folds of this type are discussed by Ramsay (1967, pp. 391-403) and by Jaeger (1969, pp. 248-251). If relative shearing of the surfaces to be folded occurs then an example of the deformations is given by *: x’ =J2Ax’+B x2 = X2 [C-E(P)]

(29)

x3 = X3 [D- F(X’)] where the deformation is not isochoric unless E(X’) = E&Y’) = 0, and ACD = + 1. The distribution of strain throughout the fold may be obtained from: AZ/(x’)* q& L.

AE’x’M

AF’N/x’

(x’)~ [(E’M)* + (C-E)*]

E’F’MN(x’)* (FIN)2 +(0-F)*

(30) I

where : M= (x*+E)/C and: N= (x3+F)/D * Other deformations of this type, including the isochoric ones, are considered by Truesdell and Toupin, 1960, p. 301. See also Jaeger (1969, p. 250) and Ramsay (1967, p. 391).

ANALYSIS OF STRAIN IN FOLDED LAYERS

345

Fig. 8. Distribution of strain in a bar deformed by bending according to eq. 29 with A = 2.0, B = 0.0, C = 0.5and D = .l.O. The surface. of no-finite strain, given by M = (x')~ + (X~)~ = 4, is shown dotted.

Fig. 9. Distribution of strain in a concentric fold produced by bending and shearing paralIe1 to the folding layers. The deformation is given by eq. 29 with A = 2.0, B = 0.0, C = 0.5, D = 1.0 and E(X’) = Xl. The numbers quoted are the ratios of the principal strains.

An example of the distribution of strain in folds of this type is given in Fig. 9 where the strain is plane, that is, F’(X’ ) = 0 and D - F = 1.

The previous discussion has considered the two special cases of similar and parallel folds. All other types of folds may be generated from these two classes by superposing Tectonophysics, 11 (1971) 329-375

346

B.E.HOBBS

another strain upon them. Thus, for example, Ramsay (1962) has shown that one type of fold (class Ic of Ramsay, 1967) may be generated by superposing a homogeneous strain upon a parallel fold. This superposition of strains is meant solely as a convenient mathematical device and is not meant to have any physical reality. All that is meant is that the final strained state can be thought of as resulting from an inhomogeneous strain together with a homogeneous strain added in that order. The term superposition, then, is not used in a sense implying two distinct deformation periods. The most convenient method of handling such deformations is to employ a coordinate system in the deformed state which has been distorted from a Cartesian system and then refer the folding to this distorted system. In this section the coordinates xk, in the deformed state, are derived from the common frame by a homogeneous transformation, thus enabling the class IA and IC folds cf Ramsay to be generated. In the following section, a second group of folds, the class III folds of Ramsay, is produced by deriving the xk from the common frame by a certain type of inhomogeneous transformation. Finally, non-plane, non-cylindrical folds may be treated if the xk are derived from the common frame by a general inhomogeneous transformation. Consider the coordinate system, x k, in the deformed state to be derived from the common frame by a transformation of the type: X ’

=a,Y’

x2 = a2 Y2

(31)

x3 =a3Y3 where the ai are constants,

and the deformation

to be given by:

x’ = (cos CX”) 4GFG x2 = (sin CX*) &ZFF

(32)

.x3 = Lfxa where A, B, C, D are constants, the XK being a system of Cartesian coordinates. Then, for 0 < a, < 1 < a2 (eq. 32) generates a family of folds labelled class IC by Ramsay (I 967, p. 365). If 0 < a2 < 1 < a, then the folds belong to class IA. If ai = a2 = 1, then the folds belong to class IB, that is, they are parallel folds. The components of the metric tensors are G,, = 6,, andg,, = l/ir:,g,, = l/a,2, = I /Q:, gkI = 0 for k # 1. Hence, the strain at each point may be obtained from: g33

y.[(g2+(
c2[(g)‘+(;)‘]

0

-

1

0

z o! D

2

ANALYSIS

OF STRAIN

IN FOLDED

347

LAYERS

and:

where A4 = (xl)’ + (.x2)‘. From eq. 15, the deformation is isochoric if ACD = f ala2a3. The distribution of strain throughout folds of this type is illustrated in Fig. 10 where the deformation is isochoric and the strain is plane (D = a3 = 1). The form of c:2, together with eq. 12 means that, in any fold of this type, when (x1)* + (x 2) 2 = IA/Cl, the principal planes of strain are parallel to the coordinate curves. When (x’)’ + (x’)’ > I A/Cl, the principal planes of strain containing the fold axis and F:, form a divergent fan; when (xl)’ + (A?)~ < 1A/Cl these principal planes form a convergent fan (see Fig. 10). In part 3, class I folds are described where these principal planes of strain form a divergent fan throughout the fold.

A

, 0.0

0.5

PO

I.5

.a2

B

0.0

IO

2.0

. s.I

Fig. 10. Class IC folds formed by bending together with various amounts of homogeneous shortening normal to the axial plane. The deformation is given by eq. 32 with the coordinates xk in the deformed state defined by eq. 31.Mis the sum: (x1)* + (x2)*, A = 2.0, B = 0.0, C = 0.5, D = 1.0, u3 = 1.0. A. 20% shortening. B. 50% shortening.

Tectonophysics,

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B.E. HOBBS

348

Folds with a superposed inhomogeneous component of strain Consider the coordinate system x k, in the deformed state to be derived from the common frame by a transformation of the type: Y’ = e(x’) Y2 = f(x2)

(33)

Y3 = h(x3) If the deformation is given by eq. 32 then folds belonging to either of class I or III of Ramsay may be produced (Fig. 11). The components of the metric tensors are G,, = SKM and g, i = (e’)2, g22 = (f ‘)2, g33 = (h')' , gkl = 0 for k # 1, the primes indicating differentiation with respect to x1 ,x2 and x3 respectively. The strain at each point may now be derived from: -A2 s [(e’x’)’

'KM

+ (f’x2)‘]

*

0

[(e’)’ - (f’)‘]

Cz [(e'x')' t (f’x’)2]

=

0 (h’D)’

and: (e’)4 [

$

(xl)’

+ C2(x2)2 ]

(erf’)2x1x2

(s2- C2)

(f’y

(xy

[$

t cyxy]

where M = (x’)~ + (x”)~. From eq. 15, the deformation is isochoric if e’f’h’ACD = f 1. However, since e, f and h are independent of each other, this condition is satisfied only if e, f and h are certain linear functions of xi, x2 and x3 respectively. The fold then becomes a class I fold. Thus, the important result follows that there are no folds of this type for which the volume remains constant during the development of the fold. This statement is not true for all class III folds as is shown in part 3. The distribution of strain throughout a fold produced by this type of deformation is illustrated in Fig. 11 where the deformation involves the volume changes indicated and the strain is plane, that is, h(x3) = x3 and D = 1. The fold is class I. As in the previous folds, the distribution of principal planes of strain depends on the sign of rir2, this component being referred to Cartesian coordinates. Referred to the common frame, Fir2 is e’f’x1x2((A2/M2) - C’), so that here the principal plane distribution depends on the signs of e’ and off’ as well as on the relative magnitudes of 1A/MI and of I Cl. In part 3 it

349

ANALYSIS OF STRAIN IN FOLDED LAYERS

M=2.4

8 dv dV 4

0 Fig. 11. Class IC fold formed by bending together with inhomogeneous shortening normal to the axial plane. The deformation is given by eq. 32 with A = 2.0, B = 0.0, C = 0.5, D = 1.0 and the coordinates in the deformed state by eq. 33 with e(x’) = 2x’ andf(xZ) = - ln(x2). The orientations of the maximum principal strain is shown by the solid lines and the numbers are the ratios of the principal strains. The volume change, dv/dV, is plotted across the fold.

is shown that there exist other folds of this type (class III in particular) pal plane fan is divergent throughout the fold. Non-plane,

non-cylindrical

If the coordinate by a transformation xk=xk(Yl,

where the princi-

folds

system xk in the deformed state is derived from the common of the type:

frame

Y2, Y3)

and the deformation is expressed by eq. 16 or by eq. 32, then a system of non-plane, cylindrical folds is produced. As an example, consider the xk defined by: x1 =3(Y2)2

t y’

x2 = y2 x3 = y3

Tectonophysics,

11 (1971) 329-375

non-

B.E.HOBBS

and the deformation

given by:

xl =X’ X 2 =x* X3

+(x1)2

=x3

where the XK are a system of Cartesian coordinates. The deformation results in a non-plane, cylindrical fold, the profile of which is shown in Fig. 12. The components of the metric tensors are GKM = &KM, and g,r = g33 = 1, g22 = 1 + (x2)‘, gr2 = x2, g,, = g2, = 0. The deformation is isochoric and the strain at each point may be obtained from:

CKM=

.

where: N=

1 + [X” +(X1)*]*,

and from: zu’ [l t(x’)‘] ,rk

m

=

2w1x2 + [l +(x2)2] 0

0

+x2 [1+4(x’)2]

0 1I

The distribution of strain is illustrated in Fig. 12. By procedures such as this, the distribution of strain in any inhomogeneously deformed body may be determined. Notice that any system of non-plane, non-cylindrical folds may be generated in a single deformation. However, not all of these deformations are isochoric. Redistribution of lineations The deformation of initially straight lineations by folding has been considered by Weiss (1959) and by Ramsay (1960) who established that an initially straight lineation is deformed so as to lie in a plane, by similar folding, and so as to lie on a cone, by parallel folding. Ramsay (1967, pp. 463-486) has extended the discussion to more general types of folds. The purpose of this section is to generalize the considerations of Weiss and of Ramsay to include any type of fold. Consider a lineation, L 1, lying in a plane, P (Fig. 13) at an angle A to another line, L2 in P. In the deformed state (Fig. 13) xk are a system of general curvilinear coordinates and P, L, and L2 have been deformed top, 1, and 12 respectively according to (I). Then, if h(L, , L2) is the decrease in the angle A during the deformation:

351

ANALYSIS OF STRAIN IN FOLDED LAYERS

Fig. 12. Distribution of strain in a non-plane fold produced in a single deformation (see text). The numbers are the ratios of the principal strains. Notice that in some areas these ratios can become very large. The undeformed Iayers are given by X2 = constant with X2 parallel to Y2.

where Ly andL.9 are the components of a unit vector parallel toL, and L, respectively in the undeformed state. Hence, from eq. 34, the change in the angle A, between two lines may be mapped out for any deformed body. In particular, if X3 is selected parallel to 15s and to a principal axis of strain in both the undeformed and deformed states with X’ normal to the layer to be folded and this line, Lz, is the axis of a plane, cylindrical fold, then: L; =o

L: = sinA

.L: = cos A

Li=O

t;=o

1;;=

I

and hence: (35) where h is the decrease in angle between the lineation principal axis of strain (eq. 35) becomes: JCT;; COSA c OS(A - X) = --~.~__.__-X_ &ZZ sin2 A + CS3 cos2 A

and the fold axis. Since X3 is a

l

(36)

The different classes of fold of Ramsay are now considered, &hefold axis in each case being a principal axis of strain in both the deformed and undeformed states. Tectcmophysics,

11 (1971)

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352

B.E.

HOBBS

Fig. 13. Deformation of an initially straight lineation by folding. Coordinates in the undeformed state (A) are xK. Coordinates in the deformed state (B) are xk. In the deformation the plane P becomes the surface p and the two lines I, 1 and L2 become the lines I, and 12. The angle between these lines changes from A to (A ~ A).

Similar folds (class II)

Consider the deformations x’ =aX’

given by:

+ E(X’)

X 2 =bX2 x3

=

X3/ah

the xk being a system of cartesian coordinates. The deformation is isochoric. If a lineation lies at A to the fold axis prior to deformation then, after deformation: cos A cos (A - h) = -~-~ [ab + sin A - (b + E’)] ’ + cos’ A

An example of the variation Fig. 14.

in the angle between a lineation

(37)

and the fold axis is given in

ANALYSIS

OF STRAIN

IN FOLDED

353

LAYERS

Fig. 14. Pattern of redistribution of initially straight lineations deformed by the ideally similar parabolic folds shown in Fig. 4. The curves are for various initial angles A between the fold axis and the lineation. (A - h) is this angle after deformation. X2 is the distance measured normal to the axial plane. A. no shortening; B. 50% shortening normal to the axial plane.

Parallelfolds(class IS) If the deformation is given by eq. 28 then: cos(A-h)=

DcosA C2(x’)2

(38)

sin2 A+ D2 cos2 A

Thus, for a given x1, that is, at a constant distance from the centre of curvature of the fold, the angle between the lineation and the fold axis remains constant. However, as one moves from one layer to the next in the fold, the angle between the lineation and the fold axis changes. These changes are depicted by Ramsay (1967, Fig. 8-3) and are given graphically in Fig. 15A. Clearly, if this variation can be studied throughout a given parallel fold, then a considerable amount of information on the states of strain throughout the fold can be established. If relative shearing of the surfaces to be folded occurs then the deformations are given by eq. 39 with F(X’) = 0. Then:

‘OS

‘*

-

‘)

= &j~~~~-+~;+;2

] sin2

,,

+ D

2 cos2

,,

(39)

where M= (x2 +E)/C. Here, within a given layer in the fold (that is, for x1 = constant) the angle between the lineation and the folh axis does not stay constant, even though the fold is perfectly concentric, but depends on x2. Some aspects of these changes in angle are depicted in Fig. 1SB. Tectonophysics,

11 (1971)

329-375

354

B.E. HOBBS

0

8

4

A

M

l:ig. 15. Pattern of redistribution of initially straight lineations deformed by concentric folding. The curves are for various initial angles A between the fold axis and the lineation. (A - h) is the angle between the fold axis and the lineation in the deformed state. A. Deformation is that shown in Fig. 8 and M = (x I)* + (x2)* defines individual layers in the fold. M = 4 corresponds to the surface of no-finite strain in Fig. 8. B. Deformation is that shown in Fig. 9. Y* is the distance measured normal to the axial plane.

If the deformation

is given by eq. 32, the coordinates

xk being as in eq. 3 1, then:

. cos(A-X)= a~J&~:i;]

sin’A+$cor’A

Some aspects of the redistribution

If the deformation

of lineations

by class IC folds are indicated

is given by eq. 32 and the coordinatesxk

in Fig. 16.

by eq. 33:

I

The pattern of redistributjon 3. ANALYSIS

OF FOLDS

is similar in principle to that developed for class IC folds.

PRODUCED

BY GENERALIZED

SNEAR

The discussion of part 2 has considered that the plane to be folded is given by an expression of the form X’ = constant, and has investigated the distortion of this plane. An alternative method is to define a family of shear planes (defined below as X2 = constant) and to investigate the deformations which are produced by allowing various shears parallel

ANALYSIS OF STRAIN IN FOLDED LAYERS

3.55

Fig. 16. Pattern of redistribution of initially straight lineations across class IC folds. A is the initial angle between the lineation and the fold axis and the cases A = 20’ and A = 60° are plotted for various layers in the fold. (A - h) is this angle in the deformed state. M gives the layer within the fold. The deformation is that corresponding to 50% homogeneous shortening shown in Fig. 1OB.

to these planes together with changes in lengths normal and parallel to these planes. l’he shear planes themselves are purely mathematical devices and need have no physical reality. The surface to be folded is then considered as a passive marker which heIps to delineate some aspects of the deformation. In th? following the coordinatesxk in the deformed state are taken as Cartesian. More complicated structures may be produced by taking xk as non-Cartesian as in part 2. The deformations involved here have been considered by Truesdell and Toupin (1960, pp. 297-298) under the heading of generalized shear, and part of their discussion is repeated below. Consider the deformations given by: x J = x’ F(X2) + G(P) x2 = H(XZ)

(42)

x3 = K(X3) + L(X2) where the XK are a Cartesian coordinate system. In the deformations defined by eq. 42, the planes X2 = constant, slide parallel to each other, the direction and amount of slide varying from one plane to another. There are also arbitrary changes in dimensions in the XK directions. A stack of planar markers, therefore, initially oblique to the planes X2 = constant, becomes deformed into a system of non-plane, non-cylindrical folds by such a deformation. Notice that in the general case, the planesx’ = constant are not parallel to the axial planes of the folds that develop (see Fig. 18 for an example). in most of what follows, however, only plane, non-cylindrical folds are considered with axial planes given by x2 = constant. Tectonophysics,

11 (1971) 329-375

B.L.IIOBBS

356

Sections through a layer, initially oblique to the planes X2 = constant, are given in Fig. 17. Fig. 17A, B show the situation in the undeformed state and Fig. 17C, D show the situation in the deformed state. Fig. 17C, D show that the thickness of the layer, measured parallel

to the axial plane, is

T,-F(X*)measured

parallel

to x’ and K(Xf)

- K(Xi)

measured parallel to x3, TX being the thickness in the undeformed state measured parallel to X1 and (XT - Xi) being the thickness in the undeformed state measured parallel to X3. The folds produced by eq. 42 are therefore general folds; the thickness of the folded layer measured parallel to the axial plane varies arbitrarily from one place to another. The strain at each point may be obtained from:

0 (X'F'tG')* t/f’*+,!,‘*

C

,y’~’

(43.1)

K12 and from:

F2+M2 FL", =

MH' HI2

I:

(43.2)

.

Fig.17.Graphical representation of the generalized shear deformation given by eq. 42.A and B show sections through the undeformed layer whereas C and D show sections through the deformed layer.

ANALYSIS

OF STRAIN

357

IN FOLDED LAYERS

where M = [(x’ - G/F)F’+ G’]. The condition that the deformation be isochoric is that together with FH’ = l//3. Consider now the deformation, according to eq. 42, of a layer whose boundaries are parallel and are planes parallel to X3. The following discussion investigates the conditions that various types of folds should be developed. Similar discussions follow for a layer initially parallel to X1.

K’ = 0, a constant,

Similar

folds

From Fig. 17, the folds are ideally similar if F(X’) = constant. For isochoric deformations, this then means that H(X’) is a linear function of X2. Such folds are given by: x1 = a,,X’

f G(X’)

x2 = a22X2

+a21

x3

=

(44)

a33X3 + L(X2)

the fold axis beingx”,

which, in general, is not a principal axis of strain

Parallel folds

If the hinge of a fold produced by eq. 42 occurs at X2 = 0 and TX is the thickness of the layer measured parallel to X’ in the undeformed state, then the thickness at the hinge is TX* F(0) = TX * y, say. TL is defined by: TX - F(X2) T; = T,y. Y

being the ratio of the thickness measured parallel to X’ at X2, to the thickness at the hinge. (7r/2 -Q) is the angle between the folded layer and the axial plane. For a parallel fold (Ramsay, 1967, p. 367): TL = set cy

or: F(X’)

= y * set

tan-’ [

5 i

[X’ F’(X’)

+ G ‘(X2)] II

(45)

Since eq. 45 is to be true for all values of X’ and X2, F’(X’) = 0, in which case any folds are also similar folds. For isochoric deformations then, G(X*) is a linear function of X2 so that the only isochoric deformations of this type which produce a folded layer of constant “radial” thickness are homogeneous shears. Hence, the only parallel folds which can be produced by isochoric deformations of the type given by eq. 42 are folds with straight limbs and sharp hinges, that is, chevron folds. Class IA and IC folds

For class IA and IC folds, the thickness measured parallel to the axial plane increases Tectonophysics,

11 (1971) 329-375

358

B.E. HOBBS

away from the hinge. Hence the condition

that these folds be formed is that E”(X’) > 0.

Class III folds

For class III folds, the thickness measured parallel to the axial plane decreases away from the hinge. Hence the conditions that class III folds be formed is that F’(X*) < 0. Notice that here, as opposed to those class II1 folds considered in part 2, the deformations producing class III folds can be isochoric. An example of the various types of folds produced in a layer initially parallel to X3 by the deformations (eq. 42) is shown in Fig. 18 together with the distributions of strain within the deformed Iayer. For these deformations L(X’) = I, K(X3) = X3 and G(X’) = [ln(X2)- 11’ so that x3 is the fold axis and is also a principal axis of strain; the deformation is isochoric.

+a

at

x.1= 17.7

Fig. 18. Non-plane fold produced by the generaked shear eq. 42 with F(Xz) = X2; C(X2) = [ln(X’) - 112 N (X2) = ln(X2) - 1; K(X3) = X3 and L(X2) = 1. The stippled area ABCD becomes abed where a is off the page at x ’ = 17.7. The orientations of the maximum principal strain at each point is shown by a solid line and the numbers are the ratios of the principal strains at each point.

ANALYSIS

01: STRAIN

IN FOLDED

359

LAYERS

Since Fiz = (X’F’ + G ‘)H’ and the angle between the folded layer and x2 is tan-’ {(X'F' t G’)/(H’)2 }, it follows from eq. 12, that all principal plane fans for plane folds formed by eq. 42 are divergent, provided (H’) is positive. This is true independently of the class of fold. For the class I and class III folds treated in part 2, the deformations produced principal plane fans which are both convergent and divergent as illustrated in Fig. 8- 11. The deformations of part 2 never produce principal plane fans which are wholly divergent. Condition that a principal axis

ofstrain lie in the axial plane

In this section and the following one, two questions raised in part 2 in regard to similar folds are reconsidered, this time in relation to the general system of plane. noncylindrical folds produced by eq. 42.

Fig. 19. Coordinate shear deformation

axes in the deformed (see text).

state for a plane, cylindrical

fold formed

by a generalized

The coordinate system xk is selected as in Fig. 19 with x1 and x3 in the axial plane of the fold system and x2 normal to the axial plane. Another Cartesian system, zk, is chosen such that zi and z3 lie in the axial plane with zr at angle 0 to x1 ; z2 is coincident with x2. If the deformation is given by eq. 42 then the strain at each point in the deformed body relative to xk is given by eq. 43 .2. The condition that a principal axis of strain -lk should reduce to the form: should lie in the axial plane is that the tensor cm

Tectonophysics,

ll(1971)

329-375

when referred to the zkx z3 being a principal axis of strain. The transformation (FAk)* is given by eq. 19. In this situation, (c$)* and &)* are given by:

of FAk to

(~~‘)*=~(~~3-C~‘)sin28+(i;:‘)cos28=0

(47)

and (~~2)*=-(~~2)sinBt(~~2)c~~O=0

(48)

For both eq. 47 and 48 to be true, it is necessary that F’(X’)= 0, that is, the folds are similar in the x*x* section, and that &(X3) = F(X2) = constant. The angie between x1 and the principal axis of strain is then given by tan @= L’/C’

(49)

Also, any fold for which L’ = 0 and K’ = constant

has x3 as a principal axis of strain.

Con&rim that the axial plane is everywhere parallel to a principal plane of strain The axial plane is everywhere paraiiel to a principal plane of strain if the tensor FAk referred to the xk coordinate system reduces to the form:

ctk = m

that is: (X’F’+G”)H’=

0

(50)

and: L’H’ = 0 The conditions

(eq. 50) are satisfied if H’ = 0, in which case no fold exists, or:

L = constant and:

Since G and F are independent of X’ , G’ = F’ = 0. These conditions again mean that no fold exists. Hence, there is no plane fold defined by eq. 42 in which everywhere the axial plane is parallel to a principal plane of strain. Redistribution of lineations The deformation of initially straight lineations may be treated as in part 2 using eq. 34 with the components of CKM given by eq. 43.1. The results are not written out here but are similar in nature to those obtained in part 2.

ANALYSIS

OF STRAIN

4. APPLICATION

IN FOLDED

TO NATURAL

361

LAYERS

FOLDS

The previous discussion indicated that a considerable amount of information on states of strain in deformed rocks is potentially available in the detailed variation of shape of deformed layers and lineations. The purpose of this part is to discuss the kind of information that needs to be available before some useful analysis of the strain can be made. The methods outlined here can only give information on the states of strain produced in layers which were initially planar; if there was a period of homogeneous shortening of the layer prior to the commencement of folding (Biot, 1961; Sherwin and Chapple, 1968) then unless there are markers present to record this straining, no information can be obtained on this homogeneous strain from the shape of the deformed layer alone.

Fig. 20. Distribution of strain in ideally similar, sine-curve folds with various shortening. A. 10Y0 shortening. B. 30% shortening. C. 50% shortening. Tectonophysics,

11 (1971)

329-375

amounts

of homogeneous

362

B.E. HOBBS

The folds shown in Fig. 20 are all ideally similar folds and all have the same shape given by x * = sin (xl), however, the distribution of strain in this group varies widely from that where a principal plane of strain is at a low angle to the folded layer throughout the fold to that where this principal plane of strain is almost parallel to the axial plane throughout the fold. Thus, the precise shape of a folded layer does not enable the state of strain to be established at each point in the fold. In fact, an infinite number of different strain distributions will be consistent with a given fold shape. Thus, additional data other than the detailed fold shape are needed, in general, in order to specify the distribution of strain throughout a given fold. However, although the possible distributions of strain throughout a given fold are infinite in number, the discussion of parts 2 and 3 indicates that the precise shape of a folded layer does impose a number of restrictions on these distributions, provided that the final strained state is independent of the deformation path which gave rise to the fold. In those ideally similar folds, for instance, for which the fold axis is a principal axis of strain, the principal plane fan must always be divergent as shown in Fig. 4-6, 20. In parallel folds, on the other hand, the principal plane fan must always be at least partly convergent on one side of the deformed neutral layer and be at least partly divergent on the other side (Fig. 8 and 9). For other classes of fold (IC and III) the principal plane fan may be convergent, divergent or a combination of both, but still the distributions are restricted in that they presumably must conform to those distributions resulting from the deformations (eq. 32), if the fan is partly convergent or from the “generalized shear” deformations (eq. 42) if the fan is wholly divergent. Thus, the classification of folds presented by Ramsay is quite fundamental in that it enables restrictions to be imposed on the possible distributions of strain which can exist in a given fold. The first step, then, in determining the distribution of strain in a folded layer is to classify the fold (or parts of the fold) according to the scheme proposed by Ramsay (1967, pp. 365-372). This enables the general form of the coordinate systems and of the transformations representing the deformation to be established. For ideally similar folds, the form of the fold must be fitted to some analytical expression in order that the transformation representing the deformation may be more closely approached. In most cases, the transformation 42 is probably the most convenient to use. For parallel folds, the transformation 29 may be adopted. For folds other than parallel or similar, it is necessary to establish whether the principal plane fan is partly convergent or divergent. Failing the presence of markers such as deformed fragments to supply this information, the slaty cleavage or schistosity associated with folding may be used, the assumption being that these structures form normal to the principal axis of shortening at each point in the fold (cf. Cloos, 1947; Dieterich, 1969). Having established the type of fan, a choice can be made between the deformations represented by eq. 32 or the generalized shear deformations (eq. 42) as expressing the general form of the deformation. If the deformations (eq. 42) are selected then both the shape of the fold and the variations in layer thickness must be expressed in some analytical form. It appears from folds figured in the literature, that a partly convergent principal plane fan is common in natural class IC folds indicating that the deformations (eq. 32) are relevant. A divergent fan is common in natural class III folds indicating that the deformations (eq. 42) are relevant; this observation means that the

ANALYSIS

01: STRAIN IN FOLDED LAYERS

363

class III folds most common in nature are capable of being formed without a volume change being involved. The details of the choice of a coordinate system depend on what additional information is available. However, the problem may be greatly simplified if one coordinate axis can be selected parallel to a principal axis of strain and the fold shape determined in the principal plane of strain normal to this axis. For many folds, the orientation of a principal axis of strain may be inferred from the preferred orientation of c-axes of quartz or of calcite throu~out the fold (for examples see Sch~ffer-Zozn~an~l, 19.55; Hobbs. 1966) or from the presence of a lineation due to elongated particles. For any analysis of strain to proceed past this stage, some knowledge of the volume changes which accompanied the deformation must be available. In general, such knowledge is lacking but it is possible to assume different reasonable volume changes and to investigate the effect this variable has on the final calculated strain distribution. In general. such postulated volume changes will not be arbitrary but must conform to eq. 13. in order to determine the nature of the tra~lsformat~on which represents the dcformation in the chosen coordinate system, the relative orientations of the undeformed and deformed layers must be known to within a rigid body deformation. In some situations where it is possible to follow the development of folding, such as across a shear zone, the relative orientation of the undeformed and deformed layering may be known including a rigid body deforlnation. In other examples it may be possible to establish that the enveloping surface of a fold system is a good approximation to the undeformed orientation of the layer. although here, information on rigid body rotations during the deformation may be lacking. For many folds, information on the undeformed state will be absent and a strain analysis must then depend on other features. Knowledge of the transformation which represents the deformation e&bles the components of the tensor 5’ to be calculated in terms of parameters which represent a horn+ geneous strain and hence the possible range of states of strain throughout the fold may be specified. Some other information on the strain must then be available somewhere in the fold in order that this range can be further reduced. This information could consist of data on the orientations of principal planes of strain (as inferred from the orientation of cleavage or from the preferred orientation of quartz), or of the determination of the state of strain at one point in the fold. An example here is the fold described by Hobbs and Talbot (1966) where a range of possible strain states was established on the basis of deformed ripple marks. This range could be further reduced by establishing the range of strain states possible on the basis of fold shape. Finally. the accurate determination of the shape of a redistributed lineation could also enable the strain distribution to be established for a given fold. The examples presented below are designed to illustrate some of these points. Example 1 A fold has a lineation defined by elongated fossil fragments parallel to the fold axis, thus identifying the profile plane as a principal plane of strain. The fold is asynlmetrical and belongs to class IC of Ramsay’s classification. The distribution of (001) for muscovitc is indicated in Fig. 21. The fold is part of an extensive system of identical folds, the enveloping surface of the system being also indicated in Fig. 21. Tecmnophysirs.

11 (1971) 329-375

B.E. HOBBS

364

x----Y

FNVELOPING SURFACE A

l:ig. 21. A. Sketch of fold system showing coordinate axes. Coordinate axes in the undeformed state taken with X’ and X2 normal and parallel respectively to the enveloping surface which is assumed to be the initial orientation of the folded layer. Coordinate axes in the deformed state taken with x1 normal to the enveloping surface and x2 parallel to the axial plane. B. Sketch of the hinge region of one of the folds in A showing the distribution of (001) of mica. The dots represent areas where the preferred orientation of mica is low.

The coordinate system, XK, in the undeformed state is selected as in Fig. 21 with X2 and X3 parallel to the undeformed layer and X’ normal to the layer. The coordinate system, xk, in the deformed state (Fig. 21) is chosen such that x1 and x2 lie in the profile plane of the fold with x1 parallel to X’ and x2 parallel to the trace of the axial plane at 60” to x1. x3 is parallel to the fold axis and to X3. The xk are given in terms of the common frame by X ’

=a,X’

.- a2X2/fi

x2 = 2a2X2/fi

(51)

x3 = a3X3

where aI, a2, a3 are constants to be determined coincident with the XK coordinate system.

and the common

frame has been taken as

ANALYSIS

365

OF STRAIN IN lyOLDE.D LAYERS

From eq. 5 1 the components

&,,, of the metric tensor are

Since the fold is a class IC fold and the preferred orientation of mica has the form shown in Fig. 21, the deformation is assumed to be given by eq. 32. These deformations are isochoric if:

The strain at each point relative to the deformed state for isochoric deformations given by: r A2 I . -- [(x’)*+;x’x*] [(x2)2

is

+c’

PI*

a:

A2 -._- .

r [ x’x*+;(x*)*]

M*

a:

a:

--

C-2

[x,x* _f(x’)*]

a:

+g22(x*)* 1 -- c* [

x’x2 ---g**(x’)2 2af

where M = (x’)’ + (x2)‘. The distribution of strain for l/a, = 1.O through to 2.0 is shown in Fig. 22 where D/a3 is taken as unity. This is equivalent to saying that only the ratio c,‘/Fi can be determined; the absolute values of 7: and 7;: could be determined if the elongation parallel to the fold axis was known. It can be seen that the case I/a, = 1.4 most closely duplicates the pattern of preferred orientation of (001) of muscovite, assuming this orientation to be everywhere parallel to a principal plane of strain. It is emphasized that this type of analysis assumes that there was no period of homogeneous shortening prior to the initiation of the fold. It is possible that in some folds with suitable deformed markers, information on such a prior history may be obtained by analysing the fold as in this example, thus supplying important information on the mechanism of folding. The values of A, B and C were selected in this example to give the deformed “neutral layer” as shown in Fig. 22. More work is needed on natural folds in order to gain information on what are geologically reasonable values for these constants. Example 2 Folds are developed in a narrow “shear zone” which crosses layered granulites Tectonophysics,

11 (1971) 329-375

at right

366

B.E. HOBBS

Fig. 22. Distribution of strain in asymmetrical folds developed by bending together with a homogeneous shortening. The deformation is given by eq. 32 with A = 2.0, B = 0, C = 0.5, D = 1.0. The coordinates in the deformed state are given by eq. 5 1. The value of LI, in eq. 5 1 is given by each fold. The orientation of the maximum principal strain at each point is given by the solid lines. The bar at the top of the figure represents the undeformed state and the orientations of strain have been recorded at the corners of the squares outlined in this bar. The “deformed neutral layer” is the central line in each fold.

angles to the layering (Fig. 23A). A lineation due to elongated aggregates of garnet is present at 60” to the fold axis (Fig. 23B), and a section across the zone, normal to the lineation is shown in Fig. 23C, the section plane being taken as a principal plane of strain. The folds are ideally similar in style, their shapes (in profile) being simple parabolas when referred to the coordinate system x k, shown in Fig. 23B. The orientation of schistosity is indicated in Fig. 23C, the schistosity fanning slightly about the lineation. The coordinate systems XK and xk in the undeformed and deformed states respectively are indicated in Fig. 23B with both X3 and x3 being selected parallel to the fold axis. Both systems are Cartesian. Taking the deformation to be given by eq. 44, since the folds are simple parabolas, this can be rewritten x’ = a,,X’ X’ =

a2,X2

+ (0,*X2 +a2,)2 + azl

(52)

x3 = a3,X3 + L (X2) Since there is a principal axis of strain at 30” to x1, by substitution L = (a,,X’

+a2,)’ /fi

in eq. 49. we obtain

ANALYSIS

OF STRAIN

IN FOLDED

367

LAYERS

‘4, , \ .\

I I

\

I \

I 1.0

I

1.6

I I I

1.8

I

I

I

A

\

\

I

I

D

Fig. 23. A. Sketch of deformed zone. B. Block diagram of fold from deformed zone showing coordinate axes used in the calculations. C. Sketch of fold from deformed zone showing orientation of (001) of mica in a section normal to the lineation. D. Distribution of strain throughout folds for various amounts of shortening. The deformation is given by eq. 52 and the value of ~7,~in eq. 52 is given near each fold.

or: L = - fi(a,,P

Hence,

+a,,)*

the strain relative

to the xk is given by

4d2(x212

-----_-+3

or by: Tectonophysics,

11 (1971)

329-375

l 2 2 alla22

B.E. HOBUS

a;, + 4a;,(x2)2 elk In

=

20:,x2

-4fia;,(x2)2

2 a22

_

I .

1

-2fiai,x’

124, (x2)2 + -f, allaz2 1

the deformation being considered isochoric so that a, I (122a33= 1. If zk is a cartesian coordinate system chosen such that z ’ is parallel to the lineation, z2 is normal to the axial plane and z3 is normal to the lineation and in the axial plane, then the strain relative to these new coordinate axes is given by:

r /

($)‘=

1

1

0

0

2 a22

4a:,x21fi

4,42

.

or by:

-;’

+ 16aZ,,(~~)~

4ai,x*

u11a22 I 2 a22

I i_.

.

0

)

o:iz 1

provided that a,, = u33. Of these two possible deformation tensors, the first is selected since it defines z’ as a principal axis of strain. The distribution of strain is plotted in the section normal to the lineation in Fig. 23D for values of a,, ranging from 1.O to 1.8. If the preferred orientation of mica is assumed to be everywhere parallel to a principal plane of strain then the value 0 ,r = 1.2 gives the closest approximation to the mica distribution shown in Fig. 23C. S.SUMMARY

ANDDISCUSSION

The following serves as a summary of the previous parts of this paper and extends the discussion to consider some aspects of cleavage in folds.

ANALYSlS01~STR.4ININFOLDED

LAYERS

369

For any body which has undergone an inhomogeneous finite deformation, it is possible to define two quantities, CKM and ckm, which are the deformation tensors. The principal axes of these tensors are the principal axes of strain at each point in the body and the proper numbers of these tensors are the values and reciprocals respectively of the principal quadratic elongations. C,, gives the strain relative to the undeformed state and it can be represented by a quadric known as the strain ellipsoid. ckm gives the strain relative to the deformed state and it can be represented by a quadric known as the reciprocal strain ellipsoid. thence, knowledge of the components of these two tensors enables the state of strain to be specified at all points in the deformed body. The components of C,, also enable the changes in the angles between lines to be calculated so that the distortion of older lineations during any type of deformation may be mapped out in the deformed body. In practice, instead of using the tensor ck,,, , it is generally more convenient to use the inverse of this tensor , elk m , which again has the principal axes of strain as its principal axes but has the principal quadratic elongations as its proper numbers. It appears that no folds can exist for which the axial plane is everywhere parallel to a principal plane of strain. For folds with a principal axis of strain everywhere parallel to the fold axis it is convenient to use the terms convergent fan and divergent fan to describe the distribution of principal planes of strain which contain the fold axis and E: The terms convergent and divergent are used in the sense of Ramsay (I 967, p. 403) and these fans are referred to as principal plane fans. It is found that folds may be divided into two broad groups; those in which the principal plant fan is partly convergent, and those in which this fan is divergent. Those of the first group (for fold classes IB, IC and II1 of Ramsay) are developed by the transformations 29 or 32, the coordinates in the deformed state being variously selected to produce the various classes of fold defined by Ramsay (1967, pp. 365--372). Such folds presumably always involve flexure of the folded layer at some stage in the deformation and include those folds considered by Dieterich (1969). The class III folds produced by these deformations always involve a volume change which may explain why they appear to be relatively rare in nature. Other than ideally parallel folds, all others in this group involve the deformations (32). This is an assumption in this paper based on the observations of Ramsay ( 1962, 1967). Whether natural folds of classes IA, IC and 111always conform to this transformation needs to be investigated. The strain at each point in the fold can be calculated from cl:,” and the change in the angle between the fold axis and a distorted older lineation is given by eq. 34. An important feature of some parallel folds produced by these deformations is that the angle between an initially straight lineation and the fold axis need not remain constant within a given layer across the fold. Folds belonging to the second group are developed by deformations of the type 41 called generalized shear by Truesdell and Toupin (1960). These deformations involve arbitrary shears parallel to specific planes together with arbitrary changes in dimensions normal and parallel to these planes. Of these deformations, the most general one produces a non-plane, non-cylindrical fold in a layer initially oblique to the coordinate axes in the undeformed state, even if the coordinatesxk are cartesian. The fold is a general one in that the thickness measured parallel to the axial plane, as well as the shape of the profile of the fold, varies arbitrarily. For various special cases of these deformations, classes lA, IC, 11 and 111of Ramsay may be produced; parallel folds cannot be produced by this type Tcctotropll.'~sic's. I1 (1971)329~-375

B.E. HOBBS

370

of‘deformation. In particular, the class II and III folds produced resemble those of nature, the class III folds being important in that they can be produced by constant volume deformations. The strain at each point in the fold is given by eq. 43, and the change in the angle between the fold axis and an older lineation is again given by eq. 34. One of the most important aspects of the various strain distributions figured in the previous parts is the presence of quite large strains even in material which has suffered very little overall shortening. Thus, in the similar folds shown in Fig. 4-6, even with no overall shortening, the local shortenings in the general hinge region can be as high as 50% and at 30% overall shortening the local shortenings close to the hinge can be greatly in excess of 50%. Similarly, in class IC and III folds (Fig. 10, 18) the strains in the vicinity of the hinge can be very large. Further away from the hinge, these local shortenings can be even greater. The possibility of such high strains does not appear to be generally appreciated in the literature. This observation is relevant to the genesis of slaty cleavage and like structures for it means that in any folded material that has experienced an overall shortening in excess of about 30% local shortenings in excess of 50% must be common, except perhaps in the immediate vicinity of the hinge. Hence the classical question: does cleavage form parallel to a principal plane of strain or parallel to a plane of no-finite strain? - is largely academic since the angle between the two will generally be less than 10” and often be of the order of 1”. In Fig. 24 a situation which may be very common in nature is illustrated. The fold consists of three layers A, B and C of which only the boundaries of B have been shown. It is supposed that layers A and C have been deformed to produce class IC folds and that the weak material comprising layer B has been forced to conform to the shapes adopted by these two stronger layers. Layer B thus undergoes a generalized shear to form a class III fold. The transformation which produces the fold in layer B is then:

X

‘=x’[a,{~~-Jr:-oZ}tb]ta*~~ (53)

x2 = azX2 where rr , rz , al , a2 and b are constants. The coordinates xk are Cartesian although the deformation is not isochoric. The distribution of strain is plotted in Fig. 24 for rl = 6, r2 = 4 and b = 4. The overall distribution of principal plane fans is identical to the cleavage fans observed in many natural examples. This example, together with the other situations treated in this paper, emphasizes the fact that the distribution of maximum principal strains is similar in detail to the traces of slaty cleavages and of schistosities on the profile planes of folds. This reinforces the hypothesis that preferred orientations of (001) of mica in deformed rocks form parallel to principal planes of strain. It is apparent also, given this hypothesis, that the detailed distribution of (001) of mica throughout a given fold, together with the detailed variation of shape and thickness of the folded layer, may be used to give an indication of the distribution of strain throughout the fold and of the overall shortening that has accompanied the formation of that structure.

ANALYSIS

OF STRAIN

IN FOLDED

LAYERS

371

f 2247 t

A

Fig. 24. Distribution of strain in class III folds formed by the generalized shear eq. 53. In each example the stippled block EFGH is deformed to produce efgh The orientation of the maximum principal strain is given at each point by the solid line. The numbers represent the ratios of the principal strains at each point. The rate of volume change, du/dC’, is plotted across each fold. A. nl = 1.0. B. IZ, = 1.3.

ACKNOWLEDGEMENTS I would like to thank D.E. Anderson and P.F. Williams for discussions which lead to this paper. Prof. J.C. Jaeger, W.D. Means, M.J. Rickard, R.J. Twiss and P.F. Williams read the paper and offered many useful comments. In particular R.J. Twiss corrected many errors and I am indebted to him for stimulating discussion whilst the paper was being written. G.T. Milburn not only patiently drafted the figures but materially helped in their construction. A considerable amount of assistance was rendered by the careful typing of Elizabeth Clement.

APPENDIX

ON NOTATION

In most elementary applications of tensor analysis it is usual to employ orthogonal co-ordinate systems, that is, co-ordinate systems where the co-ordinate axes intersect at right angles. However, there are many applications, such as when considering asymmetrical or non-plane folds, where it is convenient to use non-orthogonal co-ordinate systems. The convenience arises not only from the simplicity with which the transformations expressing Tectonophysics, 11 (1971) 329-375

the deformation may be written but also from the compact form taken by the matrix of the deformation tensor. Thus, the matrix of FAk m example I (part 4) of this paper is cumbersome enough, the components being referred to the oblique axes shown in Fig. 21 A If the components of this deformation tensor were to be referred to cartesian co-ordinates then many pages would be required to write down the matrix. It is never necessary to write such matrices down since once Fm Ik has been obtained, any transformation to cartcsian axes can be handled on a computer. The introduction of non-orthogonal co-ordinate systems does, however, introduce some complexity in the ways in which the components of vectors and of tensors may be defined and the notation employed in this paper is designed to deal with this complexity. Let the cartesian co-ordinates of a point P(x, ,x z ~.x3) be expressed as functions of (If,. 112)Uj) so that:

where it is assumed that the unique inverse: Ui

=U*(X(,X?.X3) (55)

112 =~~2(x,,xz,x3)

143 =u3(x,,x2,x3)

exists at all points such as P. Then eq. 55 gives a unique set of co-ordinates (U 1, u2, 11~) which are the curvilinear co-ordinates of P. The surfaces U, = c, , u2 = c2 and 11~= c~, where the c’s are constants, intersect in three curves (Fig. 25A) which are the II,, u2, and 113 co-ordinate curves. Along the u1 co-ordinate curve only U, varies and u2 = c2, u3 = c,~. For a detailed discussion of co-ordinate systems see Eisenhart ( 1947, pp. 63--69) and Eringen (1962, pp. 5-7, 430-434).

“3

3

Q2 “fC,

U2ZC2

P

u,=

c3

“2

VI

Fig, 25. A. Curvilinear

coordinates

of a point

P. H. Unit vectors

at P.

ANALYSIS

OF STRAIN IN I:OLlIEI)

LAYERS

373

Once the co-ordinate curves at P have been defined. there are two convenient ways of defining unit vectors at P. It is this ambiguity which gives rise to the notation used in this paper. Fig. 25B illustrates these two ways of choosing unit vectors at P. On the one hand, unit vectors may be chosen tangent to the co-ordinate curves at P. These are the vectors e’, e*, e3 in Fig. 25B. Alternatively, unit vectors may be chosen normal to the surfaces = c2, u3 = c3 at P. These are the vectors e, 1e,, e3 in Fig. ZSB. The superu, =c,,u* script and subscript notation is employed to distinguish between these two triads of unit vectors. Since the unit normal to the surface II , =c, is in the direction of grad u,, a vector V can be written: V= 0’ grader, + 0’ gradu? + 0” gradt13 = 8’ grad 11~

(56) (57)

where in eq. 57 the summation convention that any repeated index is automatically summed for i = 1 to i = 3 is used. The components 0 ’ , O*. 0’ are known as the contravariant components of V. If r is the position vector of P in the cartesian system (x, ,x2, x3), then the tangent to the U, co-ordinate curve at P is in the direction &/all, . Hence, the vector V can also be written:

v = 0,

to*

;;I

$ t o3 g 2

(58) 3

=(j.ar

(59)

1 ati,

The components 0, , 02, O3 are known as the covariant components of V. For a detailed discussion of covariant and contravariant components see Eisenhart (1947, pp. 74688). The superscript and subscript notation is therefore employed to distinguish between contravariant and covariant components. Similarly, the components cv are the contravariant components of the tensor c whereas the components C’ijare the covariant components of C. In addition, because the components of a second order tensor are associated with two directions in space, it is possible to have components which are part contravariant and part covariant. These are referred to as mixed components and are written in the form ci. The mixed components are often far more compact to write down than the covariant or contravariant components. It is clear that in situations where Cartesian or other orthogonal co-ordinate systems are used, there is no distinction between the contravariant, covariant or mixed components and this is why no discussion of such components is included in most elementary texts on vector or tensor analysis. In a transformation of co-ordinates given by:

Tectonophvsics,

11 (1971) 329~-375

314

B.E. HOBBS

x1 =x’(X’,X2,X3)

x2 =x2(X’,X2,X3)

(60)

x3 =x3(X’,X2,X3) a second order tensor C(X’ , X2, X3) may transform following three rules:

to c(x’ , x2, x3) according to the

(1) Covariant tensor field:

cij=cKM

axK axM __ axi axi

__

(2) Contravariant $j

= cKM

tensor field:

axiaXi

axK axM

(3) Mixed tensor field:

Discussions of these transformations are given by Eisenhart (1947, pp. 89-97), Eringen (1962, pp. 434-439) and by Michal(l947, pp. 48-53).

by

REl:ERENCES

Biot, M.A., 1961. Theory of folding of stratified viscoelastic media and its implications in tectonics and orogenesis. Geol. Sot. Am. Bull., 12: 1595-1620. Cloos, E., 1947. Oolite deformation in the South Mountain fold, Maryland. Bull. Geol. Sot. Am., 58: 843-918. Dieterich, J.H., 1969. Origin of cleavage in folded rocks. Am. J. Sci., 267: 155-165. Eisenhart, L.P., 1947. An Introduction to Differential Geometry. Princeton University Press, Princeton, N.J., 304 pp. Eringen, A.C., 1962. Nonlinear Theory of ContinuousMedia. McGraw-Hill, New York, N.Y., 470 pp. Hobbs, B.E., 1966. Microfabric of tectonites from the Wyangala Dam area, N.S.W., Australia. Geol. Sot. Am. Bull., 77: 685-106. Hobbs, B E. and Talbot, J.L., 1966. The analysis of strain in deformed rocks. J. Geol., 74: 500-513. Jaeger, J.C., 1969. Elasticity. Fracture and Flow. Methuen, London, 3rd ed., 268 pp. Maxwell, J.C., 1962. Origin of slaty and fracture cleavage in the Delaware Water Gap area, New Jersey and Pennsylvania. In: A.&J. Engle, H.L. James and B.L. Leonard (Editors), Petrologic Studies (Buddington Volume). Geol. Sot. Am. New York, N.Y. pp. 281-311. Michal, A.D., 1947,Matrix and Tensor Calculus. Wiley, New York, N.Y. 132 pp. Ramsay, J.G., 1960. The deformation of early linear structures in areas of repeated folding. J. Geol., 68: 75-93. Ramsay, J.G., 1962. The geometry and mechanics of formation of “similar” type folds. J. Geol., 70: 309-327.

ANALYSIS

01: STRAIN

IN FOLIjED

LAYERS

315

Ramsay, J.G., 1967. Folding and Fracruring qfliocks. McGraw-IIill, New York, N.Y. 560 pp. Ramsay, J.G. and Graham, R.H., 1970. Strain variation in shear belts. Can. J. Earth Sci., 7: 786 813. SchBffer-Zozmann, I., 1955. Gefiigeanalysen an Quartzfalten. hTrues Jahrh. Mineral. Ahhattdl., 87: 321 350. Sherwin, J.A. and Chapple, WM., 1968. Wavelengths of single layer folds: a comparison between theory and observation. Am. J. Sci., 266: 167-179. Truesdell, C. and Toupin, R., 1960. The classical field theories. In: S. Fliigge (Editor), Etzcyclopedia 01 Physics, 3 (I). Springer-Verlag, Berlin. pp. 226-793. Weiss, L.E., 1959. Geometry of superposed folding. G‘rol. Sot. Am. Bull., 70: 91-106.

Tectonophysics,

11 (1971)

329-375