the morphology and development of folds a

THE MORPHOLOGY AND DEVELOPMENT OF FOLDS

A thesis submitted for the degree of Ph.D.

of the University of London

by

Peter John Hudleston

May 1969

2

ABSTRACT

This thesis is concerned with the description, classification, analysis

and interpretation of folds based on their morphologic properties.

Existing methods of geometrical fold analysis are critically examined.

Many are found to be impracticable. Two new analytical techniques are

presented, one based on the use of dip isogons and the other based on

harmonic analysis. A Visual method of rapid harmonic analysis is described

that involves no measurements.

Theories of fold development are discussed with particular reference

to folds developed by buckling in isolated competent layers embedded in

a less competent matrix. The geometric form of folds predicted by theory

are examined. Emphasis is placed on the geometric forms taken up by

buckle folds and by buckle folds modified by compression.

A series of buckling experiments in single viscous layers embedded

in a less viscous matrix at low viscosity contrasts are described.

Progressive development of the shape of the experimentally formed folds is

analysed and interpreted in terms of buckling theory.

Detailed analyses of minor folds, in small parts of the Moine rocks

in Scotland, the basement gneisses in the Swiss Alps and the Culm sediments

on the Cornish coast are described. In the rocks of each region the

geometric forms of the folds are shown to be different in each fold phase,

and are also shown to be related to differences in layer composition.

By analogy with the forms of folds predicted theoretically and observed

experimentally the geometric forms of the natural folds are accounted

for by processes of buckling and 'flattening'. Estimates of 'viscosity'

contrast and bulk deformation within the profile planes of the folds are

made in two instances.

3

CONTENTS Page

CHAPTER 1 INTRODUCTION

1.1 General Statement 7 1.2 Soto Definiticns of Terms 9 1.3 Symbols Used in the Text 10

CHAPTER 2 DESCRIPTIVE FOLD GEOMETRY

2.1 Introduction 12

2.2 General Geometry 12

2.3 Folded Layer Geometry 15 2.3.1 Thickness Parameters 16

2.3.2 Isogons 16 2.3.3 Isogon Plot - k against a 21 2.3.4 Relationship between ta l 0a, and a 22 2.3.5 Fold Classification 26 2.3.6 Errors in Measurement and Datum Fixing 35 2.3.7 Discussion 36

2.4 The Geometry of Single Folded Surfaces 39 2.5 Harmonic (Fourier) Analysis of Folds 42

2.5.1 Fourier Analysis in Geology 45 2.5.2 Fold Analysis using Harmonic Analysis 45 2.5.3 Theory 46 2.5.4 Selection of Coordinates for Analysis 50 2.5.5 Procedure for Analysis 51 2.5.6 Representation of Computed Coefficients 52 2.5.7 Visual Harmonic Analysis 63 2.5.8 Errors and Reproducibility 68

2.6 Techniques of Natural Fold Measurement 71

CILIPThii 3 THE THEORIES OF FOLD DEVELOPMENT AND GEOMETRIC FORM OF FOLDS.

3.1 Introduction 73

3.2 Passive Folds 74

3.3 Exact Mathematical Treatments 75

4

Page

3.4 The Shape of Buckled Layers 84

3.5 Homogeneous Flattening of Folds 86

3.5.1 Oblique Flattening in the Profile.Plane 99

3.6 Simultaneous Buckling and Flattening 99

CHAPTER 4 EXPERIMENTS ON BUCKLING

4.1 Introduction 107

4.2 Model Study Problems 109

4.3 Apparatus and Materials 110

4.4 Experimental Methods 118

4.5 Homogeneity of Strain and Boundary Effects 122

4.6 Results 127

4.6.1 Changes in Arc Length 127

4.6.2 Thickness Variation in the Buckled Layers 134

4.6.3 Wavelength/Thickness Ratios 135

4.6.4 Amplification 135

4.6.5 Harmonic Analysis of Fold Shape 138

4.6.6 aperimental Simultaneous Buckling and Flattening. 143

4.7 Interpretation 150

4.8 Discussion 154

4.9 Interpretation of Naturally Formed Folds 161

4.10 Conclusions 163

CHAPTER 5 AN ANALYSIS OF MINOR FOLDS IN THE MOINIAN ROCKS OF

MONAR, INVERNESS—SHIRE.


5.2 Lithology 164

5.3 Metamorphism 170

5.4 Structural Geology 170

5.4.1 Rock Fabric 171

5.5 Descriptive Geometry 171

5

Page_

5.5.1 Size of Folds 174

5.5.2 Fold Order and Asymmetry 177

5.5.3 Isogon Patterns 180

5.5.4 Interlimb Angle Variation 180

5.5.5 Thickness/Dip Variations 185 5.5.6 Harmonic Analysis of Fold Shape 202

5.5.6.1 Analysis of b1 205

5.5.6.2 Analysis of b3/b1 208


5.6.1 Discussion 228

5.7 A Study of Wavelength/Thickness in Ptygmatic Folds in

Pegmatitic Veins 230

5.8 An Analysis of Deformed Lineations 239

5.9 Conclusions 248

CHAPTER 6 AN ANALYSIS OF MINOR FOLDS IN PART OF THE MAGGIA

NAPPE„ TICINO, SWITZERLAND


6.1.1 Lithology and Mineralogy 253

6.1.2 Metamorphism 253


6.2.1 Mineral Fabric 261


6.3.1 Isogon Plots and Thickness/Dip Relationships 262


6.3.3 Refolded Folds 281


6.4.1 Discussion 289

6.5 A Wavelength/Thickness Study of F2 Folds 291

6.6 Conclusions 294

Page

CHAPTER 7 AN ANALYSIS OF MINOR FOLDS IN THE CULM MEASURES

AT BOSCASTLE, CORNWALL


7.1.1 Lithology 297


7.2.1 Fabric 306


7.3.1 Isogon Plots and Thickness/Dip Relationships 308


7.3.3 Deformed Lineation Loci 321

7.3.4 Interlimb Angles and Limb Length Ratios 321

7.3.5 Miscellaneous Features 328

7.4 Interpretation and Discussion 328

7.5 Conclusions 336

CHAPTER 8 SYNTHESIS

8.1 A Comparison of the Results of the Three Detailed 337

Fold Studies 337

8.2 Summary and Conclusions 340

ACKNOWLEDGMENTS 342

REFERENCES 343

APPENDIX 352

6

CHAPTER 1

INTRODUCTION

1.1 GENERAL STATEMENT

In many zones where the earth's crust has been deformed the litho—

logical layers take up folded forms. For many years structural geologists

have speculated as to the processes by which solid rock may become deformed

by ductile flow without fracture, to form these familiar structures; and

in recent years this problem has received increasing attention. Interest

in the subject has followed several lines of approach, all of which have

led to a better understanding of folds and folding processes.

One approach has used mathematical analysis to provide a basis for

predicting the geometrical features of folds (and involved making many

simplifying assumptions for rock). A second approach has studied the

experimental development of folds, usually in materials with properties

scaled to simulate what are believed to be the properties of rock in the

earth's crust. A third approach has investigated the forms of natural

folds, using various methods of geometrical analysis in order to elucidate

the fold geometry. Perhaps the most significant progress in this

direction in recent years has been the elucidation of fold geometry in

zones of complex and repeated deformation (as in the Moine and Dalradian

rocks of Scotland).

It is an unfortunate fact that in natural folds it is seldom possible

to compute the sta- es of strain in the folded layers, and in most folds all

that is available to the structural geologist is the geometric form of

the bedding or foliation surfaces and tectonic structures such as cleavage

or schistosity. Even in the most favourable situation where the state

of strain may be evaluated in natural folds, it is impossible to know the

'strain path' by which the final fold form was attained (Ramsay, 1967,

P.343).

7

8

In any classification of folds their geometric form (in terms of the

shape of both folded layers and individual surfaces) is important, because

it is one feature of folds that is always available for study. Methods

of accurately describing these geometric features are necessary if analysis

and classification of folds are to be made. By use of methods of

accurate fold description a link between the three approaches to folding

referred to above may be made.

Some workers (e.g. Donath & Parker, 1964) consider that fold

classification should be placed on a genetic basis. Whilst this seems

to be the best way of classifying folds it is not practical because fold

genesis in rocks is imperfectly understood at the present time, and has to

be inferred from the end products of folding processes. It seems

advisable to distinguish between classifications based on fold geometry

and those based on folding processes.

In the past geologists have often described the geometrical forms of

folds in general impressions of fold shape and style. This practice has

never been very satisfactory because of the absence of agreed nomenclature,

and because no detailed analysis of fold shape can be made on this basis.

This thesis is primarily concerned with fold morphology and concerns

the problems of fold analysis and classification based upon the geometrical

properties of folds. An attempt is made to link theoretical and experim—

ental work to the interpretation of natural folds by means of detailed

geometrical analysis. Several new analytical techniques are presented

that have proved useful both in the determination of natural fold geometry

and as an aid in the interpretation of folds in terms of folding processes.

No attempt will be made to criticise or analyse the bulk of the

previous work on the subject of folding; the relevant work on each aspect

of folding considered in this thesis will be discussed at the beginning

9

of each Chapter. For general reviews and criticisms of the substantial

literature on folding the reader is referred to the works of Fleuty (1964),

Rast (1964), Whitten (1966a) and Ramsay (1967).

A critical review of the various methods of geometrical fold analysis

is given in Chapter 2, and two new analytical techniques are described.

A technique of harmonic analysis of fold shape is developed and a simple

method of visual harmonic analysis is presented that should be useful to

field geologists.

Theories of fold development and the geometric forms of folds predicted

by these theories are discussed in Chapter 3. Particular emphasis is

placed on the development of folds in a single competent layer embedded

in a less competent matrix. A series of buckling experiments at low

viscosity contrast are described in Chapter 4. By means of geometrical

analysis the results are interpreted in terms of the theories discussed

in Chapter 3. Chapters 5, 6 & 7 concern detailed studies cf minor folds in small parts of major fold belts of Caledonian, Alpine and Hercynian

ages respectively. In these studies systematic differences in fold

geometry are shown to exist between folds of different generations, and

are also shown to be related to differences in layer composition. In

all instances the geometric forms of the folds are shown to be consistent

with simple processes of fold development.

1.2 SOME DEFINITIONS OF TERMS

Many of the terms used in this thesis have generally accepted

meanings and will not be defined here.. The reader is referred to the

works of Turner & Weiss (1963), Fleuty (1964) and Ramsay (1967) for

definitions of these terms.

The domain of a single fold is given by Turner & Weiss (1963, p.105) to

include part of a folded surface between two adjacent inflexion lines

10

(see Turner & Weiss, 1963, fig. 4-15), and this is extended here to include

successive surfaces in a stack of folded layers. The following are

practical definitions of terms relating to the geometry of individual folds

(or two adjacent folds) in profile section.

Limb Length is defined as the distance measured along a single folded

surface between two adjacent hinge points.

Arc Length is defined as the distance measured along a single folded

surface between two inflexion points so as to include a third (this

length is roughly equivalent to two limb lengths). In Chapter 4 arc

length is taken to include several folds in a single folded surface.

Wavelength. A half-wavelength is taken as the straight line distance

between two adjacent inflexion points. In Chapters 4, 5 and 6, to comply

with current usage, the term wavelength is used for arc length in analyses

of arc length/thickness ratios.

Fold Size is taken as the mean of two adjacent limb lengths.

Tightness is defined by the interlimb angle; the angle between the

tangents to the fold surfaces at the inflexion points.

Asymmetry is defined as the ratio of two adjacent limb lengths.

The term strain-slip cleavage is used in Chapter 7 without genetic 4 m-1,1 4 eu,4-4 ..4.1041-vcd.va.v"00

1.3 SYMBOLS USED IN THE TEXT

t = orthogonal thickness of a folded layer.

T = layer thickness parallel to a fold axial surface

. limb dip

angle between an isogon and the normal to the parallel tangents

to the folded surfaces of a layer.

11

an = harmonic coefficients of a cosine series.

bn = harmonic coefficients of a sine series.

1-1 viscosity.

VR = viscosity ratio

X1 = maximum principal

X2 = minimum principal

V1/112

quadratic elongation.

quadratic elongation.

S strain ratio of h. 1 2

R = apparent strain ratio ofiX /f X j 2 1

A = fold amplitude

VT = fold wavelength

d dominant wavelength

natural logarithm

Xd = non—dimensional wavenumber

XI Y, Z axes of finite strain ellipsoid X Y).Z

CHAPTER 2

DESCRIPTIVE FOLD GEOMETRY

2.1 INTRODUCTION

Increasing attention is being paid by structural geologists to the

problem of accurately describing rock structures, and the last few years

have seen the publication of several new techniques of fold analysis.

It is the purpose of this Chapter to examine geometrical methods of fold

analysis and classification.

Section 262 deals briefly with the general features of fold geometry, and the main part of the Chapter is concerned with the geometrical features

of folds in profile section. Two new analytical techniques are

presented in sections 2.3 and 2.5, involving 'dip isogons' and harmonic

analysis respectively. These are discussed in relation to existing

techniques, and a critical appraisal of some of the more recent approaches

to the subject is made. The Chapter ends (sect. 2.6) with a short

account of the techniques used in the practical application of analytical

methods to natural folds.

2.2 GENERAL GEOMETRY

MUch of modern regional structural analysis is based upon the

assumption that folds are either cylindrical or can be - split into sub-

sections that are. The dimensions of most natural (e.g. Campbell, 1958;

Wilson, 1967) and experimentally produced (e.g. Ghodh & Ramberg, 1968)

non-cylindrical folds are usually much greater along the hinge lines than

in a direction normal to the axial surfaces, and adjacent hinge lines are usually within a few degrees of mutual parallelism. Conical folds are

considered to be rare in nature, and on geometrical grounds alone the

'ends' of folds are unlikely to be conical in form (Wilson, 1967), but

rather of a more complex non-cylindrical nature.

12

13

Many natural folds appear to be approximately cylindrical in form (i.e.

they persist with little changes in profile attitude or form, for

distances along their hinge lines that are large compared to their other

dimensions), and this is true for all the folds studied in the later

chapters of this thesis. The geometry of folds may now be considered

in two parts:

a) The attitude of the fold axis, axial surface. and bedding/

foliation surfaces in space.

b) The geometry of the fold in profile.

a) Spatial Attitude of Folds

The attitudes of lines (fold axes, lineations) and planes (axial

surfaces, bealinefoliation surfaces) are amenable to exact measurement

and have for a long time been used in structural geology (and petrofabrics)

as a basis for analysis in which the stereographic projection plays an

important role. Recently, statistical techniques have been established

to calculate the mean attitudes of lines and planes (e.g. Ramsay, 1967,

p.15), the best fit fold axis of cylindrical folds and best fit cone for

conical folds (Loudon, 1964; Whitten, 1966a; Ramsay, 1967, pp.18-27*;

Cruden, 1968; Kelley, 1968) and tedious statistical operations have been

made practical by use of computers (Loudon, 1964). The problem of

testing the significance of orientation data has been discussed by Flinn

(1958) and Stauffer (1966) on an empirical basis, and the methods of

testing observed data against theoretical models is reviewed by Watson

(1966).

Fleuty (1964) proposes terms to define the attitude of folds (i.e.

of axial surface and fold axis).

b) Fold Profile Geometry

Fundamental differences in fold shape have been recognised for some

14

time. Van Hise (1896) was the first to distinguish parallel and

similar folds, and these two types have since re—appeared in practically

all classifications of folds based on fold shape.

Ramsay (1967, pp.359-372) has shown that these are two special

classes of fold in an infinite field of possible shapes. The reason why,

until recently, little systematic study has been made on fold shape is

two—fold. First, most folds except the smallest are exposed in scattered

outcrops and rarely in anything near profile section. Secondly, until

recently few suitable methods have been available to use in such a study.

Considering the first reason, early workers, taking most folds to be

parallel used constructions involving concentric arcs (Busk, 1929),

evolutes and involutes (Mertie, 1940) or similar methods to construct

fold profiles from scattered data. Recently a method of establishing

profile geometry of proved cylindrical folds from scattered data has been

proposed that does not involve initial assumptions about geometry (Phillips

& Byrne, 1968). This problem will not be treated further here; all folds

considered in this thesis are of hand specimen or outcrop size in which

the whole ..fold is accessible for measurement. However, the methods

described in this chapter are all applicable to large folds with irregular

outcrop using similar techniques to those developed by Phillips and Byrne.

Coming to the second reason, several methods have now been proposed

(e.g. LouCon, 1964; Ramsay, 1962a, 1967) for quantitative description of

fold shapes in profile. These will be discussed in the sections below

and several new techniques will be introduced.

Ramsay (1967, ch.7) gives methods of description and classification

of fold geometry in profile and much of the following develops further

his approach. A fundamental and important distinction is made by Ramsay

between the geometry of a single folded surface or form surface (Turner &

15

Weiss, 1963, p.111) and the geometrical features of a layer (i.e. the

geometrical relationship between two or more form surfaces). The two

are to a large extent independent and should not be confused. For

instance, a parallel fold may be bounded by surfaces of an infinite

number of shapes. It should be noted that, except for similar folds

and folds in which the outer and inner arcs are the same shape but differ

in scale (non-congruent similar folds of Mertie, 1959), the shapes of

the inner and outer arcs of a folded layer must differ.

2.3 FOLDED LAYER GEOMETRY

Apart from the methods developed by Ramsay (1967, pp.359-372) few ways of accurately describing folded layer shape have appeared in the

literature. Mukhopadhyay (1964, 1965a) employs methods essentially the same as those of Ramsay. Mertie (1959) presents a classification of folds

based on the use of elliptical arcs in which he accounts for both similar

(his definition of the term 'similar' is looser than that generally

accepted and followed here) and parallel folds in a general treatment

that includes many complex layer shapes. His methods are however more

pertinent to a discussion of single folded surface geometry and are

difficult to interpret in terms of layer shape. They are also difficult

to apply. Williatis (1965, 1967) suggests that many folds are concentric

and may become modified to form folds with elliptical inner and outer

arcs; and Powell (1967) presents an analysis of fold shape on the assum-

ption that folds are either concentric ,or modified concentric. These

methods are restrictive, since they tie layer geometry to a special kind

of single surface geometry.

The most useful ways of describing the geometry of folded layers is

found to be by use of thickness parameters and by study of dip isogons

(Elliott, 1965). Both these topics are discussed at length by Ramsay

(1967, pp. 359-372). Use of Ramsay's thickness parameters, and the new

parameter Oa described below, allow layer geometry of folds to be

analysed separately from single surface geometry.

2.3.1 Thickness Parameters

Thickness of a folded layer is measured between tangents to the

bounding surfaces of the layer at apparent dip a (fig. 2.1a). The •

thickness can either be measured normal to the tangent, orthogonal thickness

t a , or parallel to the axial surface of the fold, Ta . The

relationship between these two is:

Ta cos a = to To = to is the thickness at the fold hinge, and the ratio

-Oa = to /to or

T'Ta /To may be plotted against a (Ramsay, 1967, p.361). For comparison with the 'isogon plot' introduced below, the fold in fig. 2.1a

is represented in fig. 2.2a on a t cl, /a graph.

2.3.2 Isogons

Dip isogons 1965, 1968) are very useful in the analysis of

fold profiles. They are lines of equal apparent dip on a fold section

and their relationship to folded layers and more specifically to relative

curvatures of the layer surfaces is explained in detail by Ramsay (1967,

p.363 et seq.). A discussion of the more general case of isogonic

surfaces is given by Elliott(1968) who uses isogons of pitch of lineation

in a plane in order to elucidate the geometry of 'early' cylindrical

folds that have been overprinted by a later deformation. Ickes (1923),

in determining the geometry of parallel, similar and neutral-surface

folding uses isogons in the same way as described here, although he does

not use the term isogon (see Ickes, fig. 8.5).

A most important feature of Ramsay's approach is his classification

of folded layer shape on the isogon pattern (Ramsay, 1967, p.363-372,

fig. 7.24). This classification is used throughout this thesis and the

main categories are given here:

16

Fig. 2.1

a) Fold profile to show the definition of parameters

ta T

a and 0

a

Line AA is the isogon at dip a.

N.B. The datum in this example is the axial surface

trace. This is not necessarily so (see text).

b) Sign convention for 0.

i = inner arc

o = outer arc

AA is the isogon.

a Datum

1 POSITIVE

4.) NEGATIVE

b

18

Elg. 2.2

Comparison of plots of -0o variation with dip, and 0o. variation with dip.

The positions of plot for the fundamental fold classes of Ramsay are

marked. Plots for the two limbs of the fold drawn in fig. 2.1 are

also marked.

b 0

ANGLE OF DIP a

CLASS 3

30 60 ANGLE OF DIP a

90

20

a

-90

3

90 -90

CLASS 1 - Curvature

CLASS 2 - Curvature

of Inner Arc > Curvature of Outer Arc

Convergent Isogons

of Inner Arc = Curvature of Outer Arc

Parallel Isogons

21

CLASS 3 - Curvature of Inncr Arc < Curvature of Outer Arc

----4 Divergent Isogons

Class 1 is divided into three:

Subclass 1A - Strongly convergent isogons

Subclass 1B - Parallel folds

Subclass IC - Weakly convergent isogons

The five categories, 1A, 1B, 1C, 2 and 3 are found to occupy specific fields or lines on a graph of t'a or T'a against a (Ramsay, 19679 p.366). These are marked on the V/(0, graph of fig. 2.2a9 and may be

summarised here:

t'a N 1.0 IA / t:), = 1.0 IB parallel

cos a / -0 ; . a 1.0 1C t'a =cos a 2 similar

to "cos a 3

2.3.3 Isogon Plot - a against a

A useful parameter, has been developed which derives from and is

used in conjunction with dip isogons. 0a for a folded layer is defined as the angle between the normal to the tangents drawn to either fold

surface at angle of (apparent) dip, a 9 and

going from inner to outer arc the isogon is

clockwise sense relative to the normal, 0a,

and if 'deflected' in a clock-wise sense it 2.1b). a is taken as positive for the

the isogon (fig. 2.1a). If,

'deflected' in an anti-

is taken as pDsitive in sign,

is taken as negative (fig.

right limbs of antiforms and

22

the left limbs of synforms, and negative for the other limbs. With

these sign conventions the fold in fig. 2.1a is represented in fig. 2.2b,

a plot of Oa against a . This should be compared with fig. 2.2a, a

plot of tla against a for the same fold.

Folds of the different geometrical classes 1A, 1B, 1C, 2 and 3 occupy

distinct fields or lines on this graph that are analogous to their

representation on a t( /c, graph. These fields are marked on fig. 2.2b

and are defined by values of 00, summarised below (for positive a):

4 0 1A

= 0 1B parallel

> 0 1C

= a 2 similar

>a 3

For practical purposes the signs of a and cla are reversed for plotted

points on the left half of fig. 2.2b and both limbs of a fold are

represented on the right hand side of this graph.

2.3.4 Relationship between -On , Om and a

It is clear (fig. 2.1a) that there is no general relationship between

single values of tta and 0a, for a given value of a and for an arbitrary fold shape; therefore, knowing only one point on a -Oa /a, plot it is impossible to compute tha corresponding point on a Oa/ a, plot. However, the relationship between the variables may be investigated by considering

the geometry of a small part of a fold limb (fig. 2.3). Tangents to

the fold surfaces (not shown themselves) are drawn at angles of dip, a and a +60,, where nsLis a small increment of dip. The thicknesses of the fold for these dips are tta and t'a + Zstt respectively. Zst t is

a small increment of thickness. The line AB joins the intersections of

the tangents. From the geometry of this figure:

a >

00. 00. 00,

00. Oa

rig.. 2.3

To show the relationship between t and 0 a in a general case. Tangents

to the folded surfaces of a layer are drawn at two closely spaced values

of dip.

tt = orthogonal thickness at dip a

to -0 = 1 and so

a

cos 0a = tie, /AB

cos (0a, +.0a) (t'a + At`)/AB

cos (00L +Au) = t b, + cos 0 t ' a a

expanding and simplifying this equation gives:

cos.61— tan Oa, sin Aa = 1 4- 6,-0/t 10,

IfAa is very small and in radians, cos &a 1;

sin 61 ---> . Equation 2.1 then simplifies to:

At' = -t& . tan 0 a 6a

In the limit as At' and eol approach zero

dt' . tan Oa du

Or Oa = tan-1 — 1 1 ' V • ddd'

) 2.3 a

In the limit, as 6.-0/6,a ---) dtlida , the line AB becomes the

isogon at angle of dip a, and O u in these expressions is the angle

between the normal to the tangents at dip a , and the isogon.

From expression 2.3, it is clear that 00. is a function of both t'cl,

and dti/da , and in the general case of t = t( a ) there will be no

simple expression for dtl/da . For a plotted function t = t(a) it

is possible to compute the corresponding function 0 91r( a) by first

differentiating t = t( a) with respect to a (either graphically or

numerically).

Ramsay (1967, p.369-370) describes how complex changes of dip and

curvature of the fold surfaces may be revealed by constructing graphs of

25

2.1

2.2

26

tt a , dtt/da and d2tt/da 2 against a and shows that segments of the plots

for a complex fold will occupy fields of different fold classes in the

three graphs. In fig. 2.4 a comparison is made between plots of tta

dtt/da and 0a against a for a small scale natural fold. There is a close similarity between the functions dtidct, and 0a ; this is predicted from equation 2.2 because 95 = 0 when dtt/do, = 0. This fold could be

classified as irregular class 2 (similar) with a modification to class 1

for high dips. This fold is only of geometrical interest as the complex

geometry is due to irregular thickness variations in the folded layer.

2.3.5 Fold Classification

The fold classification proposed by Ramsay (19679 p.359 et seq.) and

used in this thesis implicitly requires two conditions to be met that

restricts the application of the descriptive methods employed. Fig. 2.5a

depicts the special case (Case 1) in which the axial surface trace is

normal to both fold surfaces in the hinge. In fig. 2.5b (Case 2) the

axial surface trace makes an angle of 00 with both fold surfaces at the

hinge points, and the more general case (Case 3) is shown in fig. 2.5c

in which the axial surface trace makes different angles with either fold

surface at the hinge points. The folds considered until this juncture

have met the special requirements of case 1. Although most natural

folds described in this thesis are geometrically very close to case 1,

a process of simple modification of a parallel fold that results in a case 3 fold is presented in Chapter 3 (sect. 3.5), and natural examples

of both case 2 and case 3 folds are common.

The properties of folds belonging to these three cases may be listed:

Case 1.

T = t 0 0

°o = 0

includes all parallel folds

°I = 02 = 90 The axial surface trace is parallel to the zero

Fig. 2.4

Comparison of various plots for a single fold.

a) Natural fold profile with isogons drawn at 5° intervals of dip

(except for a = 50).

Numbers refer to dip values for particular isogons

Datum line = Axial surface trace

b) c) and d) Various plots for the fold in a).

o > C \

t.

1.0

1A

0 9° b a

1A dt' da

901-- o a

C d 90

1.0

.\

N 90

70

a

Fig, 2.5

Three cases of fold profile.

. angle between axial surface trace and tangents through the hinge

points.

Dots represent hinge points.

= isogon(s) through the hinge points of adjacent surfaces.

a) CASE 1

0 = 0, 2 6 1 = 90

b) CASE 2

0 = 90 —6, 62=R1 90

c) CASE 3. 82 1

To,to and 0o cannot ba defined

I CASE 1

\ CASE 2

\ \ \

Axial Surface Trace

... (1).= 0

Axial Surface Trace

CASE 3

\

Axial Surface

Trace

30

isogon and perpendicular to the tangents at

the hinge points.

Case 2.

T o = t o sec O

0./0 1 = 0, ' 90

The axial surface trace is parallel to the zero

isogon and is oblique to the tangents at the hinge

points.

Case 3.

61 / 62

A zero isogon, To, to and 910 cannot be defined.

The tangents at the hinge points are not parallel.

The axial surface trace intersects the isogons.

To record changes in ta , T a and Oa with a it is necessary to select a datum tangent line, for which a = 0. For case 1 folds this is simple, the datum tangent is normal to the axial trace.

For case 2 folds, consider a similar fold (fig. 2.6) in which the

isogons are parallel, yet the hinge point tangents are not normal to the

axial surface. Taking the direction of these tangents as the datum line,

00 = 90-0 and To = to cosec 0 . Plots of tie. and O a against a are

also shown in fig. 2.6. It is clear that the graph of t'a against a gives a misleading representation of the true fold style, since the plots

for either limb are different in both shape and position. The plots of

Oa against a show a similar difference in position for either fold

limb, but the two plots are the same shape. A plot of Tia against a (not figured) in this particular case would give a true representation of

31

Fig. 2,6

Plots for a Case 2 'similar' fold, with the datum taken as the hinge

point tangents. Isogons are drawn at 20° intervals of dip on the fold

profile.

Note: the sign changes between limbs on the 0 Al plot

3

90

L

L

0 1- 0

1.0

1C t o

i- 90

ANGLE OF DIP a 90

0

34

the similar fold shape (11 = 1 for all a ). By taking as datum the

tangents to the fold surfaces for which Ta = tamax and 0a = 0, the fold

plots would be identical for either limb and would coincide with the plot

for a true similar fold on both the 0a / a and t'a / a graphs.

Considering case 3 folds it is impossible to define a single datum tangent line at the hinge of a fold (see fig. 2.5c). In the same way

as an alternative datum was found for a case 2 similar fold, a datum may

be defined where yf a, = 0 and ta is a maximum (or more rarely a minimum). By defining the datum in this way for natural folds it is found empirically

that plots of t'a against a for either limb of a fold match more closely than they do by defining a reference line in any other way. This

procedure of datum fixing has therefore been used in the fold studies

described in this thesis.

An important difference between a t'a / a and a Ora / a representation becomes apparent here. Because to /to is a ratio it is dependent

upon to , the datum value of ta ; so by varying the position of the

datum and hence the value of t o , the plots of against 04 constructed

for the various datum positions will differ in both shape and position.

00, on the other hand will take the same value whatever tangent plane is

taken for the datum, and so the shape of a 001a, plot will remain constant

however this datum is chosen. A change in datum merely means a

constant addition or subtraction to each a value and is reflected by

a horizontal displacement of the plot of 00. against a. on a Oa/ a graph. For the similar fold in fig. 2.6, the straight line plots of 00. against a

for either limb indicate the true geometry; the plots may be displaced

horizontally (bearing in mind the sign change for the left hand limb)

until the point 0 = 0 coincides with the origin of the graph. It is

clear that an arbitrary datum may be taken to construct a graph of 0a

against a .

All parallel folds must be case 1, even though the datum may be

35

difficult to define unambiguously. True similar folds must be either

case 1 or 2.

The situation may be complicated further by considering multiple

hinge folds (Ramsay, 1967, p.347). For a Oa / a representation these

folds present no problem because no fixed datum is required. If the

curvature of one hinge is much greater than that of the other(s), a datum

tangent line can usually be drawn in the way described above for single

hinged folds and for the purposes of measurement the subsidiary hinge(s)

may be ignored, and a plot of Volt against a may be made. However, if

the curvatures in each of the hinges are of the same order of magnitude,

then to define any single datum may be difficult. In this case a plot

of 00, against a should be made with an arbitrary datum.

The inner and outer arcs of folds should really be. treated as single

folded surfaces when considering curvature and in fact both the inner and

outer arcs of the fold drawn in fig. 2.4 are multiple hinged. However,

the curvature in the main.hing-e is much greater than the minor maxima of

curvature found on the limbs; also the main hinge is the only one for

which a datum tangent may be drawn (where 0 = 0). For practical purposes

this fold may be quite validly considered as a single hinged fold of case 1.

2.3.6 Errors in Measurement and Datum Fixing

If the bounding surfaces of a fold are both smooth, the plots of

and 0a against u for this fold should also be smooth curves. Slight non-systematic variations of plotted points from such smooth curves may be

predicted as a result measurement errors.

There is one kind of systematic error that can appear in these plots

that must be recognised, since when present it may indicate a false

difference in fold geometry between either limb of a fold for which a plot

36

of t'0, against cx has been constructed. The error is in the wrong

selection of the datum tangent and usually arises when the fold has a low

hinge curvature. Neglecting errors in thickness measurement the effect

of this error will be identical to that of choosing the hinge point

tangent as the datum for the fold in fig. 2.6. For a symmetrical fold it

causes the plots of t'a against a for either limb to be of different

shape and position. A realistic size of error (5°) is included in the

plots of t'a against for an undrawn fold (fig. 2.7). The true plots

of either limb are identical, and are represented by the centre curve on

the graph. The features of this graph may be noted:

a) For an error of 5° in datum positioning, a 100 horizontal separation

of the plots for the two limbs appears.

b) The greater the change of t'ci. with a (i.e. the steeper the slope of

the curve) the greater the apparent separation of the plots for the

two limbs.

c) For small errors, the shapes of the plots for the two limbs are

similar to one another, and to the 'parent' or true plot. (cf.

the plots in fig. 2.7 with those of fig. 2.6).

Errors of this kind may be distinguished from real differences in

geometry between the limbs of a fold by plotting 0 a against a, . If the shapes of the plots for either limb differ on this graph, this indicates

a true difference in limb geometry.

2.3.7 Discussion

Layer geometry of folds has been considered in terms of two main

descriptive parameters, 0a and -Oa . T' has been omitted because it

depends directly upon t'a . Lot us consider the relative merits

of 0a/a and th /0, representations of fold geometry. Listing the

advantages of a 0a /a, representation:

Fig. 2.7

To show the effect on the plots of t' against a of wrongly selecting a

the datum for a symmetrical fold with identical thickness/dip variations

in either limb.

Y is the true plot for either limb

X and Z are the plots for either limb that result when the datum is

taken at an angle of 5° to the 'true' datum.

(The plot Y is for a parallel fold flattened by an

amount f x = 0.6).

30 60 ANGLE OF DIP .

90

c

38

39

a) The most distinct advantage to be gained by use of fei a rather than t'a is the independence of the shape of the Oala. plot on the datum

a used; whilst the correct location of the datum is most important

in constructing a t'a /a plot.

b) Because t'a is a ratio, measured values of ta must be converted to

. This takes time and introduces the possibility of errors.

The measured 0 0, are plotted directly against a.

c) The plotted function of Oa against a is more sensitive to changes in geometry than that of t'a against a (see fig. 2.4) because it

is more or less equivalent to the function dtio, /d a, .

The advantages of a -Oa / a representation are:

a) to is easier to measure than 95awhich requires careful positioning

of tangent points.

b) Errors in O mare larger than in -Oa , and are 'picked up' by the

more sensitive nature of the plot.

c) At high limb dips 0a becomes most inaccurate because the curvature

of the folded surfaces is low and the tangent points are difficult

to locate accurately; on the other hand the most useful part of a

t'a /a plot is that for the highest limb dips.

From these lists it appears that most of the disadvantages of a k /a plot are practical ones. If fold boundaries cannot be accurately defined

by a smooth curve, t'a is best used. is most accurate for folded

layers with clearly defined boundaries of high curvature, and must be used

for multiple hinged folds or folds with obscured hinges.

2.4 THE GEOMETRY OF SINGLE FOLDED SURFACES

Several different approaches have been made by a number of authors

40

to the problems of description of single surface fold morphology, and

these are discussed in this section. Most descriptions of fold style in

the literature are based upon a number of well known terms found in many

structural geology text books (e.g. Hills, 1963, p.215; Turner & Weiss,

1963, p.111). These terms have been reviewed by Fleuty (1964) and

Whitten (1966a). These authors make no distinction between the geometry

of layers and single surfaces.

The geometrical features of single folded surfaces may be considered

as a collection of several attributes that may be taken singly or together.

These include size, shape, tightness and asymmetry. Size is clearly an

independent attribute, shape may include both tightness and asymmetry;

their degree of interdependence depends upon the definitions used. Most

techniques involve more than one of these attributes and it is not

practical to treat them all separately. Working definitions of size,

asymmetry and tightness appear in section 1.2.

Ramsay (1967- P.347).considers the curvature-variations In graphed-

form, across a foldei1 surface; atechnique that. clearly brings out

features-of shape,--tightnessand asymmetry. He proposes the use of two

descriptive parameters, both functions of fold shape and tightness (see

Ramsay, p.350). Curvature is difficult to measure in practice and this

restricts the use of these parameters.

Mertie (1959) proposes the classification of fold shapes based on a

representation by elliptical arcs. A large number of fold shapes may be

represented by varying the eccentricity of the ellipses used and by forming

composite curves from ellipses of different eccentricities. However,

Mertie's method is concerned with fitting surfaces to scattered data

along a profile; for accurate representation of fold shape his methods

are unsuitable and many common fold styles (e.g. chevron and box fold)

find no place in his classification.

m1 = 1

2: cos2e. i 1 1 m2

a measure of attitude

2.4

a measure of tightness 2.5

fold The first two moments are: attribute.

. 1cose. 1

41

A quantitative description of fold shape based on statistical

techniques is proposed by Loudon (1963, 1964). He shows how information

on fold style may be obtained by taking statistical moments of the poles

to bedding, expressed as direction cosines (Loudon, 1964 and Whitten,

1966b). Loudon suggests that each moment is a measure of a particular

where O.1 is the angle between the line joining inflexion

points and a bedding pole.

N is the number of equi —spaced readings of ei.

Other moments and combinations of moments are suggested to give

measures of asymmetry, shape, skewness and kurtosis. These moments are

a series of scalar quantities, potentially useful for classification

purposes and for regional analyses of folds (see Whitten, 1966b; and

Whitten & Thomas, 1965). The biggest drawback of this technique lies

in its application. Sampling, as realised by Loudon, is of prime

importance and considerable care in data collection would be necessary

to get any meaningful results from regions of folded rooks. The only

application of these methods to date has been to a hypothetical region

of folded rocks (Whitten, 1966b; and Whitten & Thomas, 1965).

Although neither Loudon nor Whitten state this as being a necessary

restriction, the examples they consider are all folds of one or two

whole wavelengths; clearly any other sampled length (unless a multiple

of the wavelength or very large) would give very different results.

42

Another problem arises in interpretation of the statistical moments. For

instance, it can be shown that the measure of tightness, m2, is not

solely dependent on interlimb angle, but is closely related to asymmetry

as well. In fig. 2.8 eight chevron folds are drawn with the same wave—

length and interlimb angle, but with varying limblength ratios. For each

fold, m2 may be calculated according to eq. 2.5 (in which N = 47) and its

value is plotted against limblength ratio in fig. 2.8b. It is clear

that the moment, m2, is strongly related to asymmetry and its use as a

valid measure of tightness alone is considered unsound (cf. Whitten,

1966a, fig. 483, in which values of tightness are given for a number of

folds).

Loudon's method is thaqght to be of considerable potential in

structural analysis. However, considerable attentionnocdo to bo 'paid to the

problem of sampling and to the significance of the statistical moments

before this potential will be realised.

In a statistical analysis of fabric data, Kelley (1968) describes a

technique of finding the best fit fold axis to a number of bedding poles

by a trial and error method. He calculates the variance of the observed

measurements (poles to bedding) from the great circle normal to each trial

axis, for enough of such axes to enable him to draw a contoured map of

the variance on a stereogram. The shape of the contours will reflect

the shape and tightness of the folds and might be used for representing

fold shape. However, the method does not have the quantitative advantage

of Loudon's approach, whilst having the same problems of sampling.

The final method of analysis considered concerns the use of Fourier

(or harmonic) analysis, and the next section is devoted to this means of

analysis.

2.5 HARMONIC (FOURIER) ANALYSIS OF FOLDS

Harmonic, or Fourier, analysis is the representation of a function by

Fig. 2.8

Geometrical analysis of folds according to Loudon.

a) To show how the datum line is fixed (parallel to the line joining

the inflexion points), wavelength, W, defined and how 8 is measured.

b) 8 folds with the same interlimb angle but different amounts of

asymmetry.

c) Plots of Loudon's measure of Itightnessl

m2 = z_cos2 u.

i=l1=.1.

against limb length ratio (longest over shortest) for the folds

in b).

(N = 47 in the computations)

44

a

b

06

• 6 \.5

C

.7 .4

\•3

to

— 0.2

0

8 • ------,..2

3 5 7 9 11 13 LIMBLENGTH RATIO

45

the sum of a number of sine and cosine harmonics, and it provides a useful

and practical way of analysing and classifying single folded surfaces (here

in profile section) in a quantitative manner. A new method of application

of harmonic analysis to folds is described in this section; it presents

an alternative approach to those discussed in section 2.4, and has been

applied in natural fold studies described in the later chapters of the thesis.

2.5.1 Fourier Analrais in Geology

Fourier analysis is not new in geology and has been used mainly as

a tool in statistical studies. The technique finds wide application in

geophysics (e.g. Barber, 1966) including studies of gravity, seismic

and magnetic profiles. Recently, Fourier analysis has received much

attention as an alternative to the use of polynomials in calculating trend

surfaces or best fit surfaces to data showing areal variation (Krumbein,

1966; Agterberg, 1967). Fourier analysis of resistivity profiles in

stratigraphic correlation is described by Preston and Henderson (1964), its use in studying river meanders by Speight (1965) and in a study of

microrelief by Stone and Dugundji (1965).

In structural geology, until recently, no description of fold

morphology using Fourier analysis had been described.

2.5.2 Fold Analysis using Harmonic Analysis

Considering the obvious periodic nature of many folds it is perhaps

surprising that harmonic analysis has received such little attention in the

past. Norris (1963) noted the potential of the method, and in recent

years Chapple (1964, 1968), Harbaugh & Preston (1965) and Stabler (1968)

have described methods of fold analysis based on this technique. Other math-

ematical functions could be found to represent fold shapes, such as polynom-

ials or Bessel functions; however a Fourier representation seems intuitively

more useful as it is naturally periodic. Two possible uses of harmonic analysis

may be distinguished that are fundamentally different. The first is in the

46

study of single folds, to gain information about their 'Shape content'

(Chapple„ 1968), and the second is in the study of periodicity, involving

the analysis of long lengths of profile to include many folds. The

second approach is not taken here, but could prove useful in a study of

fold order, by the calculation of power spectra. The approach of Stabler

(1968) and of the writer restricts the analysis to single folds or segments

of folds. Chapple (1968), too, restricts his study to single folds,

but the function analysed is one of inclination (dip) against arclength.

2.5.3 Theory

The mathematical theory of Fourier analysis may be found in any

standard mathematics textbook (e.g. Heading, 1963; Kreyszig, 1967). If

a function, f(x), is single valued, finite and periodic, it can be

represented by a function, F(x), that is the sum of an infinite number of

sine and cosine functions called a Fourier series. If the period of

f(x) is W (W arbitrary), then F(x) is given by: cys.

F(x) . 1/2 a + a cos 2nnx

o n=1 n n=1 nx

bn sin 2n 2.6

where an and bn are constant coefficients.

For a single folded surface and a single fold W is the wavelength.

In order to eliminate the effect of scale from the analysis, the

wavelength, W, is always taken as 2n and measurements of fold amplitude

are scaled accordingly. With W = 2n equation 2.6 becomes:

F(x) iso + CY4N -747 z-- a cos nx + b sin nx n=1 n n=1 n

2.7

The function, f(x), need not be periodic, but if defined in the interval

0 ‘ x 42n;

f(x + n2n) = f(x)

where n is a positive or negative integer it becomes periodic with

a 2n period (see fig. 2.9).

Selection of Coordinate Frame for Harmonic Analysis.

a) Profile of a single folded surface. Hinge points (h) and inflexion

points (i) are marked by dots. Tangents to the surface at the

hinge points are also marked.

0 . Origin of coordinate frame

The three indicated schemes are based on:

1) a 'W/2 unit' oblique axes

2) a 'W/2 unit' rectangular axes

3) a 'W/4 unit' rectangular axes

b) W/2 fcld of scheme 2) expanded to one period (2n ).

Sampling points of x for recording y at intervals of 11/2N+1 are

shown for N = 5.

c) W/4 fold of scheme 3) expanded to one period (2u ).

In all cases W is equated to 2u .

48

W/ 2

Y n

10 II

-,.

b

C X

{-W/4->

Y A

m - 2N+1 f(xp) sin

p=1

2n pm 2141

2 2.9b

The coefficients in 2.7 are given by:

amn C2nf(x) cos mx dx 2.8a m)0

bm — 1

f (x)sin mx dx 2.8b 1 0

Where f(x) is a fold surface its analytical form is unknown and these

integrations cannot be performed. However, f(x) may be measured and

represented at a discrete number of points (x - values) and approximate

numerical methods may be used to find these coefficients. Let values of

f(x) be taken at 2N 1 poihts over the range 0 - 2n, at equal spacing of

width 2u/2N 1 (fig. 2.9b). Using the trapezoidal rule of numerical

integration the equations 2.8 become:

2N f(xp ) cos

49

2 a = m 2N+1

2.9a 2N+1

where am and am are estimates of am and bm respectively.

Note: m N and m'> 0

In practice if Er .,51 am and Om are very good estimates of am and bm, and

the latter symbols are used henceforth, rather than the symbols for their

estimates.

Simple examples of the evaluation of am and bm by this numerical method

aro given in Heading (op.cit., p.426) and Kreyszig (op.cit., p.458).

Stabler (1968) derives simple expressions for evaluating the first few

coefficients, bm. A computer programme has been developed to calculate the coefficients from equations 2.9, enabling rapid evaluation to any

number of terms (subject to m<N). A listing of the programme is given

50

in the appendix. Read into the computer are values of N, the 2N+1 values

of f(x•) and the wavelength W of the fold.

2.5.4 Selection of Co-ordinates for Analysis

An important requirement of an analysis is the selection of an

unambiguous frame of reference axes. Considering a section of a single

folded surface (fig. 2.9a) of a general kind it is clear that the invariant

points on the surface, the inflexion and hinge points, must be used as

reference points in any co-'ordinate scheme (this is not necessarily so for

polyclinal folds - see later). Considering any join of two adjacent

inflexion points to be a ,'half wavelength', AW, it is evident that if the

unit of fold for analysis were greater than -1-W, it would in general be

difficult to define an unambiguous frame of axes. Three reference axis

schemes are proposed (fig. 2.9).

1) & 2) Based on a -W unit with x-axis as the join of two inflexion

points.

1) oblique axes, y-axis normal to the tangent at the hinge point.

2) orthogonal axes, y-axis normal to x (usually y is not parallel

to the axial surface trace).

3) Based on a i--14 unit with y-axis normal to the tangent at the hinge,

and x-axis normal to y through the inflexion point.

In fig. 2.9b & c the examples of 2) and 3) are drawn with the unit lengths

of W/2 and W/4 respectively reproduced two and four times to constitute

one period of length W. In each case W would be equated to the period

2n to eliminate the difference in size. The three suggested schemes

have been tried and the third scheme based on the unit, W/4, is preferred

for the following reasons:

a) The largest number of natural folds cam be analysed this way; all

folds with limb dips 4 90°.

b) The origin, 0, is unambiguous. For both schemes 1 and 2 there are

two possible origins. (The origin could, however, always be

situated on the short limb).

c) Asymmetry is more or less separated from 'shape' and 'tightness'

(cf. fig. 2.9b & c), and may be evaluated by a comparison of co-

efficients for either 'half' of a fold.

d) The same segment of a fold is analysed as in a -00., or 0 a analysis.

All analyses described in this thesis are based on a w/4 unit - 'quarter

wavelength unit'.

For all three schemes described above, the functions analysed become

odd functions (i.e. f(x) = -f(-x) ); all sine waves are odd functions and

all cosine waves even (f(x) = f(-x) ). An odd function can only be

represented by odd harmonics and so am = 0 for all m. Further, in scheme

3, the even sine terms in the series will vanish as they are asymmetric

about the axial surface, and so b2m = 0 (m = 1,2,3 etc)., and only odd

terms remain in the series.

2.5.5 Procedure for Analysis

To analyse a single fold of a quarter wavelength, the following steps

should be taken (see fig. 2.9);

a) Locate hinge and inflexion points on the surface.

b) Draw a normal to the tangent line at the hinge through the hinge

point. This line, the y-axis, is usually parallel to the axial

surface trace.

c) Draw a normal to this line through the inflexion point, to form the

x-axis.

51

52

d) Divide the length, W/4, on the x-axis into N + 1 equal parts, to give

N + 1 sample points (see fig. 2.10).

e) Measure f(x) at each sample point. Because of the symmetry of the

fold, the 2N + 1 values of f(x) over the range O-W may be found from

the N + 1 values in the range 0 -11/4 (see fig. 2:10).

f) Measure W/4.

From the 211 + 1 values of f(x) and the value of W/4, the coefficients,

bm

may be calculated on the computer.

2.5.6 Representation of Computed Coefficients

The most diagnostic features of fold shape are brought out by the

first two coefficients, b1 and b3, and a plot of b3 against b1 (Stabler,

1968) proves a useful way of recording these (fig. 2.11). For comparison

of several coefficients a plot of log bm against log m, called here a

'spectral graph', is useful (fig. 2.12). This consists of a number of

discrete points of bm for each m.

To investigate the significance of these plots it is instructive

to see how several ideal functions (Table 2.1) are broken down into their

harmonic components. The functions given in Table 2.1 are made periodic,

with period 2n. The chevron & semi-circular functions are drawn in fig.

2.13 and also the first 3 harmonic components of each, b1, b3 and b5.

First consider the spectral graphs for these functions (fig. 2.12),

for which specific values of the constants in Table 2.1 have been

introduced. (Note that the semi-circle is a special case of the semi-

ellipse where the constant a = 1). For each graph in fig. 2.12, and for

the similar graphs that may be constructed (but are not illustrated) for

each of the other functions in Table 2.1, straight line envelopes to the

odd values of bm may be drawn, irrespective of sign. The slope of

Fig. 2.10

Repetition of a W/4 unit between inflexion point (i) and hinge point (h)

to form one whole wavelength, W (equated to 2n ). The technique of

'sampling' discrete values of f(x) is illustrated. Values of f(x) are

read at intervals of W/2N+1 in the range O-W. In the figure N = 4.

Sampling points are shown by dots. Due to symmetry, the measured

values of f(x) are all taken in the first quadrant, and are represented

by vertical lines.

y

54

4 5 6 7 4 3 6 9 7 8

8 9

Fig. 2.11

Rerresentation of Harmonic Coefficients.

Plot of the third coefficient b3 against the first bl.

Straioht lines radiating from the origin are lines of equal 'shape', the

--,ear vertical lines are lines of equal 'amplitude'.

Letters and numbers refer to the 'shapes' and 'amplitudes' of the folds

?hewn in fig. 2,14. Dots represent the plotted positions of these

folds.

Me shaded section marks a field of double hinged folds.

a, b and c are plots for the functions drawn in fig. 2.13.

Multiple

Folds 1-0

0.5

o,o.r).5

SINE WAVES

—0.5

56

Cuspate c(si,

Folds

A•s O!

—1.0

0

5.0 b, 10.0

Fig, 2.12

Spectral Graph of log bra against log m for the ideal functions drawn in

fig. 2.13.

N.B. The subscript n is shown in the figure, in place of m used in the

text.

Negative values of bra are shown by dashed lines.

58 19 17 15 13 SEMI-CIRCLE II

9

7

3

FN opt

i t

0.01

3

0.1 b 1

1.0 n

SINE WAVE

0.01 0.1 " I

1.0 bn

11

9

7 CHEVRON FOLD

5

3

4 0p

-- 0.01 0- 1 +b, 1-0

Y = +

Y = -

1 2 f nx - (0< x< n)

/arm - x2 - 2n 2 (n < x 2n)

Y = + iax2 - anx (o 4 x ( n)

y = - y k y = -k

ax2 3anx + 2an

(0 x n) x 2n)

(lc x 2n)

N.B. a is any constant < 0 k 18 any constant > 0

Box 0.333

TABLE 2.1

Function Function f(x) defined for the period 2n = b3 Type 1 wavelength.

Chevron y = kx (0 4', x < 114 y = kx - 2sc (221<x(2n) -0.111 Y = k(n - x) < x 2)

59

Sine Wave

Parabola

Semi-Circle

Semi-Ellipse

y k sin x

y ax2 - anx (0 ,( x < n) = -(ax2 - 3nax + 2an2) x A 210

0

0'037

0.165

0.165

60

these envelopes is independent of the constants a and k in Table 2.1. The

sign reversals in the case of a chevron function are systematic (b2 3 are

all —ve for m = 0,1,2,3 etc). The effect of the size and sign of the

third and fifth harmonics (and by inference the higher harmonics) in

determining the fold shape may be seen in fig. 2.13. With all harmonics

positive (fig. 2.13a), they all tend to steepen and add to the limbs and

alternately to add and subtract from the hinge, to give an overall effect

of hinge roundness. The larger b3 /b1 9

the more pronounced will this

effect be. With b3 negative and higher harmonics alternating in sign

(fig. 2.13c), all the harmonics will add to the hinge and alternately

add and subtract from the limbs, to give an overall effect of a sharp

hinge and straight limbs.

Any of these functions may be completely represented by two parameters,

b1

and b3/b1'

and may be derived from these. Variation in a or k affects

the amplitude or tightness of the function and b1, but not b3/b1. In

fig. 2.11 all functions of the same kind, or shape, lie on straight

lines or rays, radiating from the origin, away from which the amplitude

increases.

There must exist a continuous series of shapes between the chevron

and box end members, with continuous variation in b3/b1 and corresponding

variation in slope of the straight line envelope on a spectral graph.

All the functions given here are members of this series. Hyperbolas

also fit in the series but are not shown.

Lines of equal amplitude are plotted in fig. 2.11, and an area of

double hinged folds is marked. If b3/b1 is greater than that for a box

fold the fold breaks down into several smaller ones, and if less than that

for a chevron fold the fold becomes cuspate.

All natural folds may be represented on a graph of b3 against b1

.

Fig, 2,13

Three ideal functions broken down into their

harmonic components.

a) Semi—Circle

(i) Solid curve Semi—circle y = +I Tu x — x2

Dotted curve is the sum of the first three odd harmonics,

y . 1.78 sin x + 0.29 sin 3x + 0.13 sin 5x.

(ii) The harmonics drawn separately.

b) Sine Wave

y = sin x

c) Chevron Fold

(i) Solid curve, y = x (0<x<1;/2), y x ( Tc/2<x(T‘ )

Dotted carve is the sum of the first three odd harmonics,

y = 1.27 sin x 0.14 sin 3x 0.05 sin 5x.

(ii) The harmonics drawn separately,

Y A

C (i)

Y

TT/ 2

0

(i)

a

)

b

63

Spectral graphs constructed for natural folds described in this thesis

(e.g. fig. 5.13 ) seem to display envelopes to the plotted values of bm

that are very nearly straight lines. These folds will be closely

matched in shape by a member of the series of ideal functions described

above.

This observation leads to a practical and rapid application of

harmonic analysis that is now to be described.

2.5.7 Visual Harmonic Analysis

On the basis of the continuous series of ideal functions described

above, 30 idealised fold forms have been constructed for 6 different 'shapes' (values of b

3/b1) in the series including the two end members, each at 5

/amplitudes' (see fig. 2.14). The position that each of these forms

occupies on a plot of b3 against b

1 is shown in fig. 2.11, and is given

by the intersection of a ray for a particular 'shape' with a line of a

particular 'amplitude'.

In order to compare natural fold shapes with these ideal forms, they

have been reproduced on a perspex sheet. A comparison is made in the

following way. The fold is observed in profile and estimates of the

positions of hinge and inflexion points are made. Each quarter wavelength

unit of the fold from inflexion to hinge point is compared with the forms

on the perspex sheet by looking through the sheet at the fold, and the

closest match is found. Folds may occupy intermediate positions between

the ideal forms.

Fig. 2.15 shows an-example and a simple graph on which results may be

plotted for a number of folds.

The method provides a rapid alternative to that described above and

involves no measurements or calculations. It does, however, require that

Fig. 2.14

Visual Harmonic Analysis.

30 ideal fold forms, defined between inflexion and hinge points.

6 categories of 'shape' A — F

5 categories of 'amplitude 1 — 5

Plots of the coefficients b3 against b1 for these fold forms are given

in fig. 2.11.

s

1

c

Z

L

d 8 V a D 3

Fig. 2.15

a) Profile section of a single folded surface split into 'quarter

wavelength units' for classification.

inflexion point

h = hinge point

b) Box diagram of 'shape' against 'amplitude' for representation of

results.

The two limbs of Fold X are represented on this diagram.

, +2 +

A

B

w C a. < = (I)

D

E

F

b

67

AMPLITUDE 1

2 3 4

5

natural folds be closely represented by these ideal functions.

2.5.8 Errors and Reproducibility

Errors in the calculation of harmonic coefficients may arise in

several ways. Three kinds of error may be considered present in the values

of bm calculated for a single fold. These are:

1) Errors in measurement, with hinge & inflexion points, reference axes

and value of N fixed

2) Errors resulting from inadequate size of N.

3) Errors in the location of hinge and inflexion points and the reference

axes.

With N 4 errors in 1) and 2) are negligible compared with those in 3).

In fig. 2.16 are shown results of repeated measurements on several folds;

the hinge and inflexion points and reference axes being re-estimated for

each analysis. Significant variation is apparent in the plots of b3

against b1. Although not very clear from this figure, the variation

increases with b1. The apparently greater variation in b3 than in b1

in these plots is due to a scale difference of a factor of 10.

Because of this source of error, there is little point in computing

more than the first few harmonics for natural folds. A value of 5

for N is suggested, that allows three odd harmonics to be found.

For ideal functions such as chevron, sine or box, exact coefficients

can be calculated using equations 2.8. Analysing these functions by the

numerical means used for natural folds, it may be noted that the sizes of

the even coefficients, bra, theoretically zero, are of the some order of

magnitude as the errors in the calculated values of the odd bm. This

gives an indication of the size of errors involved in 1) and 2), which

are usually much smaller than those arising in 3). In the calculations

68

Fig, 2.16

Reproducibility of Harmonic Analysis.

The results of repeated harmonic analysis of two natural fold profiles

are shown on plots of b3 against b1. New estimates of hinge and

inflexion points were made for each analysis. One configuration of

coordinate axes and estimated hinge and inflexion points are shown on

each profile.

7 0 0.2

0.1

b3

0

Limb 2

- 0.1

02

Limb 1

. I

• • S.

••••

1.0 2.0 3.0 b1 0•l

b3

- 02 0

0

01 0 2.0 3.0

02

0.1

b3

0.1

02

1.0

b1

-r

Limb 3

• • •

• -

• s So

0 1.0 2.0 3.0 4.0

b1

7 1

for natural folds, the sizes of the even bm are regarded as an indication

of the size of error included in the values of the odd coefficients, for

a fixed co-ordinate frame.

2.6 TECHNIQUES OF NATURAL FOLD MEASUREMENT

The techniques outlined in this section were those used in the studies

of natural folds described in the later chapters of the thesis.

Material consisted of photographs of outcrops or of cut and polished

specimens in which the cut face was normal to the fold axis. The

photographs were all taken with the axis of the camera lens parallel to the

fold axis to within a few degrees. Outcrops were only photographed if

the outcrop surface was fairly smooth and nearly normal to the fold axis.

Measurements were made on the photographs themselves or on tracings

in which the fold surfaces were drawn as smooth curves.

A drawing machine was used in the construction of isogons and

thickness measurements were made with a small offset rule in conjunction

with this machine.

Arc lengths and curVed lines were measured with dividers or a map

measurer. Hinge points and inflexion points were estimated visually,

usually with the aid of isogons. The hinge points were taken as the

points of closest spacing of the isogons, and the inflexion points were

found by bisecting the lengths of arc in the fold limbs that were

approximately straight.

In the construction of plots of tL and against a the datum

tangent was taken in the way described in section 2.3.5 and values of t'a

and 00. were usually recorded at 100- dip intervals.

Harmonic analysis was carried out numerically or visually on well

defined single folded surfaces. For the numerical calculations N was

usually taken between 5 and 9.

72

CHLPTER

THE THEORIES OF FOLD DEVELOPMENT .AND GEOMETRIC FORM OF FOLDS

3.1 INTRODUCTION

There is considerable controversy in the literature concerning the

processes responsible for fold development, largely because the mechanical

behaviour of rocks over long periods of time under stress conditions that

exist in the earth's crust is unknown, and because the state of strain

within folded layers is usually indeterminate. Processes that have been

suggested attempt to explain the observed geometric form of folds and

related features such as cleavage, schistosity and lineations of various

kinds. For the most part they involve layer behaviour of one or more

of the following kinds:

a) Buckling (Timoshenko, 1960: Ramberg, 1963a).

Single layers embedded in a relatively more ductile medium, or a

stack of layers of varying ductilities may become mechanically unstable

when loaded parallel to the layering and this instability may lead to

the initiation of folds.

b) Passive Folding (Donath, 1963).

The layering plays no mechanical part in the deformation and acts

solely as a passive marker.

c) Kinking (Paterson & Weiss, 1962; Dewey, 1965; Ramsay, 1967, p.436 —

456).

Rocks with very well—developed layering (e.g. bedding, slaty cleavage,

schistosity) may develop an instability during deformation to form folds

in discrete zones with straight limbs and angular crests.

Ramberg (1963a), and Ramsay (1967) in particular, discuss the fold

geometry, strained state, and genesis of buckles and passive (Ramberg

uses the term bending) folds in detail. Kinking is not discussed further

7 3

74

and the reader is referred to the works listed above and to the papers

on conjugate folds by Johnson (1956) and Ramsay (1962b).

A. brief discussion of passive folds is given in section 3.2. This

is followed in section 3.3. by a discussion of the mathematical theories of buckling, with an erliphasis on those concerned with isolated layers.

The geometry of buckled layers is considered in section 3.4, the

modification of parallel folds by a homogeneous 'flattening' in section

3.5, and the effect of simultaneously buckling and flattening a layer is discussed in section 3.6.

3.2 PASSIVE FOLDS

One of the commonest situations in which passive folds develop is

in the zones of contact strain around buckled layers (Ramberg, 1963a),

where isogons alternately converge and diverge in adjacent hinges to give

folds of class 1 and class 3 geometries respectively (see Ramsay, 1967, P.416). Ramberg (1963a) describes similar kinds of folds that form

around boudins or conglomerate pebbles and refers to this type of folding

as bending.

'Similar' (class 2) folds aro usually considered to indicate passive

layer behaviour.

Where 'similar' folds persist over a large number of layers they

become difficult to account for mechanically, and the usual explanation

invoking heterogeneous simple shear acting parallel to the axial surfaces

of the folds is open to criticism on this account (Flinn, 1962, p.425). The near periodicity of many similar folds is particularly difficult to

account for by this hypothesis or by a hypothesis of differential flat-

tening producing differential shear that is transmitted through the rock-

mass to produce similar folds (Ramsay, 1962a). The explanation most

favoured by the writer is that suggested by Flinn (1962, p.425) and

75

Mukhopaahy.:.y (1965a) involving a large shortening component of finite

homogeneous strain across the axial surfaces of gently buckled layers or

initial irregularities in the layering. Nizichopadhyay shows how

similar folds may effectively be produced from originally parallel folds

in this way (see also Rancay, 1967, fig. 7-102). Slight systematic

departures from true similar geometry might be expected with this

hypothesis, and these have been observed by . Mukt_rildhyny(1964, 1965a)

and the present writer (Ch.5).

3.3 MOT MTHEMATICAL TREATMENTS

Precise mathematical analyses of folding are restricted to the case

of buckling, and a number of workers have approached the problem. The

early work in this field, discussed by Biot (1961) and Ramberg (1961a)

treated the problem as one of elasticity. Biot (1957, 1959, 1961, 1965a,

1965b) derives expressions for the buckling instability of layered

systems of general viscoelastic materials based on a principle of

correspondence t.j expressions obtained for true elastic materials. In

a series of papers, Ramberg (1961a, 1963b, 1964a) analyses the buckling behaviour of layered viscous (Newtonian) materials using the methods of

fluid mechanics. Currie, Patnode & Trump (1962) study the buckling of

elastic media, and Price (1967) extends an elastic buckling theory to

account for asymmetrical, straight—limbed folds.

Both Biot and Ramberg discuss the case of a single layer embedded

in a relatively more ductile medium, and various types of multilayered

sequences built up from layers of different thicknesses and different

ductilities. From the analytical equations expressing the deformation

behaviour of these different cases comes the concept of a dominant

wavelength (Biot, 1957); that is the wavelength in a single or multi—

layered system, of folds most likely to develop. It is the wavelength

of small sinusoidal irregularities that grow at the fastest rate.

76

In the case of an extensive, thin, isolated viscous layer embedded

in a less viscous medium, Biot and Ramberg both derive the following

expression for the dominant wavelength, Wd, of the layer when subjected

to a buckling load:

Wa = 2nt Iµ1/6 2 3.1

where t is the thickness of the layer, j1 its viscosity and 42 that of

the medium. The assumptions built into this equation are:

a) Body forces are negligible.

b) Both layer and medium are Newtonian substances.

c) The folds are sinusoidal.

d) The compression is parallel to the layer.

e) The problem is one of plane strain.

f) The amplitude of the folds is very small.

_Ramberg (1961a) assumes perfect adherance between layer and medium in

his derivation, whereas Biot (1961.) assumes perfect-slip.. However,. Biot

(1959, p.398) derives a precise expression for an adhering layer and shows

that there is only a slight difference between the cases of slip and

no-slip.

The most important prediction of the theory describing folding in

a single layer embedded in a matrix of infinite extent is the appearance

of a dominant wavelength. This has been shown experimentally-by Biot,

Ode & Roover (1961) both for the present viscous case and for an elastic

layer in .a viscous medium for which the dominant wavelength is dependent

on strain rate. Ramberg (1963b) finds tha.relatianship of eq. 3.1 holds

for.elastic layers in.an elastic medium, predicted by exchanging 1-and

42 for elastic shear moduli Gi .and G2.

In applying eq. 3.1 to folds of finite amplitude it is implicitly

assumed that the wavelength is fixed during the 'infinitesimal.' develop-

ment of the buckling and that during further fold development this wave-

length is 'frozen-in' as.the arclength of the mature folds.

77

This is valid for high viscosity contrasts, but where the contrast is small

(approx. < 100:1), Biot (1961,13.1606) predicts that layer shortening without

much folding will become significant (see also Ramsay, 1967, p.379). Sherwin

& Chapple (1968) take this into account and modify Biot's theory accordingly.

They show that the dominant wavelength changes with shortening.

Considering the growth of a sinusoidal fold in a viscous layer cont-

ained in a viscous medium, Sherwin & Chapple (1968) modify Biot's 'thin

plate' theory (Biot, 1965a, p.426), effectively replacing time in Biot's

expressions by the ratio of the quadratic elongations, S =Jhl /N2 of the

homogeneous deformation on which the buckling may be considered superimposed.

They derive an expression for the fold amplification:

A In f ho dS — 3.2

ilo 3 ...!1 241/U2 + 3

where A = Fold amplitude.

N = Non-dimensional wavenumber = 2nt/w

subscript 0 refers to initial values

subscript refers to final values

Thickness, t, and wavelength, W, are functions of S, and are given by:

t tog and W WoS

For a fixed value of S, the expression 3.2 varies with ho and passes through

a maximum value. The vsJue of Ao for which this occurs is found by differ-

entiating the right hand side of 3.2 with respect to ho and setting the

derivative to zero. Sherwin & Chapple do this, and thus derive the expres-

sion for the wavenumler whose cumulative amplification is largest, in terms

of the final wavelength and thickness. This is: x f3 _ 12112 S2

5 +1 For comparison with eq. 3.1 this may be written:

Wd

2nt y .1 S 1 6u, 2 S2 3.3

The only difference between this equation and 3.1 is the additional factor

under the cube root sign, and from 3.3 the variation of the dominant

wavelength/thickness ratio with shortening may be calculated. Biot

(1965a, p.427) concludes that for small finite shortening, the dominant

7 8

wavelength does not alter greatly due to a compensatory effect in the

changes of thickness and wavelength. However, the ratio of the dominant

wavelength/thickness, important when only the final product is observed,

changes more rapidly than does the dominant wavelength alone, and the

effects are clear from the analysis of Sherwin & Chapple and cannot be

ignored. The relationship between amplification, dominant wavenutber

and shortening for different viscosity contrasts is shown in fig. 3.1,

reproduced from Sherwin & Chapple.

Biot (1961, p.1606) introduces the concept of explosive amplification,

Showing for a particular viscosity contrast that amplification increases

enormously at some stage in the shortening process. This is well illus-

trated by Ramsay's fig. 7-37 (1967, p.379) which is plotted from Biot's expression 5-15. Biot considers this effect will only be marked for

viscosity contrasts greater than about 100:1. However, it is the fold

initially amplified by the greatest amount (i.e. the fold whose-wavelength

is given by eq. 3.3 with S =.1,0) whose progressive amplifioation.is- recorded in.Biot's analysis. in fact at. successive stages of the.

deformation, folds of progressively changing wavelength/thickness ratios

will become those most amplified. Maximum amplification, for a variety

of viscosity contrasts, has bean evaluated at different values of S

by solving eq. 3.2 for a number of closely spaced values of xo ; the

highest value of these computed amplifications has then been taken as

an approximate maximum. Fig. 3.2 records the variation of maximum

amplification with shortening (measured as a percentage to enable

comparison with Ramsay's fig. 7-37), for a range of viscosity ratios.

For any viscosity contrast and for any amount of shortening, the value

of maximum amplification given by this graph is higher than that given

by Ramsay's graph; and so explosive amplification, defined by Blot to

occur at a value 2141o of about 1000, will take place at slightly lower

values of shortening than is calculated by Blot, if progressive changes

in dominant/thickness ratios are taken into account.

The effect of homogeneous shortening obviously becomes important at

Fig. 3,1

Amplification of the Dominant Wavelength Wa as a function of the

Dominant Wavenumber A d for various values of the viscosity ratio

µ 2 and the ratio S of the quadratic elongations of the uniform

shortening.

(After Sherwin & Chapple, fig. 5, 1968).

Pig. 3.2

Amplification of the Dominant Wavelength as a function of the uniform

shortening (expressed as % Compressive Strain), for various values of

the viscosity ratio.

100

40

1 30

1

1 20 1

[12

105

< <

4 10

3 0 10

co

2 10

a

10

80

1 2 Dominant

Wavenumber

3

x d t

"d

10

20

30

40

50

60

70

% Compressive Strain

81

low viscosity ratios, and this is further discussed in the follow ingsChapter.

It is not possible, with the above theory, to predict changes in

layer thickness or changes in shape of the folded surfaces as the fold

develops.

Recently, Chapple (1964, 1968) extended the viscous infinitesimal

theory of Biot to an analysis of folds with large finite amplitudes.

Considering the competent layer to be thin (he took thickness as 1/40 of

the wavelength), inextensible and initially folded into a low—dip

sinusoidal shape, he was able to calculate progressive changes in finite

strain, strain rate and fold shape accompanying progressive shortening

across a block containing a single layer in a more ductile medium, defined

between two adjacent axial surfaces that remain plane and parallel

throughout the deformation (see Chapple, 1968, fig.1). He showed that

the 'path' of fold development depended only on the ratio of the actual

wavelength, W , to the predicted or dominant wavelength, Wd, and calculated

the changes in fold shape for three values of this ratio (fig. 3.3).

Shape and changes in shape were recorded by means of a harmonic analysis

of inclination (dip) against arclength. By representing shape this way,

Chapple was able to argue that beyond limb—dips of 150 the fold shape

is relatively independent of the initial irregularities and strongly

dependent on the ratio W/Wd and limb dip. Below limb dips of 150, the

wavelength selection mechanism described by Blot (1961, p.1604) operates.

Although in Chapple's treatment, a harmonic analysis of dip against

arclength proves more useful than the type described by the present writer

(section 2.5), the latter has boon found more nractical in the description

of natural fold shapes. Chapplo's results are presented on a plot of

third against first coefficients (fig. 3.3) of a harmonic analysis of

the kind described in Chapter 2, to enable comparison with results of

Fig, 3.3

a) Successive shapes of folds developed by buckling in an isolated

viscous layer embedded in a less viscous medium. Limb dip values

are marked at the hince of each fold shape.

(After Chapple2 figs. 2, 3 & 4, 1968).

b) Plots of The third harmonic coefficient b3

against the first b1 for

the fold shapes in a).

N.B. Intermediate shapes not shown in a) are represented in b) by

unnumbered points.

90 •

•

71 •

• 55 •

33 I •-• • •

1-0

b3

0.5

0•5

0 1 0 3.0 0 1.0 3.0 b1

0.5

b3

89 •

•

•

70 •

•

b3

56• 36 I •

0 ••

•

53 33• •

• • •

89 •

69 •

•

W = 4 • 6 Wd W = Wd W < Wd

a

0 1-0 3-0 5.0 b1

84

the studies on natural and experimentally produced folds. The plots of

fig. 3.3 indicate the same features as do Chapple's; a progressive departure of fold shape from sinusoidal, the crests becoming more rounded

as deformation progresses.

Chapple suggests that because the shapes of natural folds show a lack

of correspondence with his theoretical model, real rock materials may

follow non-linear rheological laws. This might alternatively be due to

the invalidity of the assumptions of inextensibility and thin-plate

theory applied to most natural folds. Studies of folds developed in

single competent layers made by Sherwin & Chapple (1968) and by the

writer (section 5.7) suggest that wavelength/thickness ratios of

most naturally formed folds are very much less than the figure used by

Chapple in his calculations and that layer parallel shortening must have

occurred during folding. Moreover, if the wavelength/thickness ratio

is small the shapes of the inner and outer arcs of the buckles will

follow different paths of progressive change.

The multilayer theories of Ramberg and Biot are quite different

from one another, and are more difficult to interpret physically than the

single layer case in terms of wavelength selection and amplification.

Potter (1967) has analysed confined folding in a micaeous bed in terms

of Biot's theory (1964, 1965b) of the buckling of confined multilayers.

Experiments in layered viscoelastic materials made by Ramberg (1963b,

1964a) and Ghosh (1968) show complex variations in shape of the members

of a multilayered sequence, ranging from chevron folds to folds with

rounded crests. These variations appear to depend upon layer spacing

and ease of slip between layers.

3.4 THE SHAPE OF BUCKLED L.YERS

Chapple's (1968) analysis predicts the shape of a buckled layer to

85

high amplitudes only where the layer, besides being thin and inextensible,

takes on a state of strain in which lines initially normal to the layer

remain as normals, and a concentric longitudinal strain (Ramberg, 1961b)

varies linearly across the layer. This is the kind of strain geometry

predicted for and observed in low amplitude elastic buckles; it is

called tangential longitudinal strain by Ramsay (1967, p. 397).

As the wavelenjth/tkickness ratio of folds decreases, however, there

is increasing likelihood of a shearing strain developing parallel to the

layering in the fold limbs (Ramberg, 1961b), and if a rock layer under-

going folding is well-laminated, the strain may be taken up entirely by

such a shearing strain, when the fold becomes a flexural slip or flexural

flow fold (Donath & Parker, 1964). The thickness of all layers remains

constant and the shearing strain is related directly to the apparent dip

(Ramsay, 1967, p.392).

Ramsay (1967, p.391 et seq.) discusses the geometry of the two types

of layer behaviour described above in considerable detail and considers

folds with composite behaviour, and Ramberg (1961b) also discusses the

relationship between the two kinds of strain within a sinusoidal low-

amplitude fold in a viscous material.

There is no analytical prediction offInitefold shape for thick

buckled layers, and models of the two idealised fold types are now

considered assuming a sinusoidal shape for the 'central' surface in the

case of the flexural flow fold, and for the neutral surface in the case

of the tangential longitudinal strain fold. For the latter the model of

Ramsay (1967y p.398) is employed.

It is probable in nature that the layer shape would not be sinusoidal

at the amplitudes considered here, but it is argued that the relative

differences in surface and layer shape apparent here between the two

86

models will be similar to those found where folds are of a more general

shape.

The folds are drawn in fig. 3.4 and changes in shape of the surfaces across the folded layers are compared by a harmonic analysis, using a

plot of the coefficients b3 against b1. From this plot it is clear that

the difference in surface shapes between the two models is slight, and that

a better distinction between them is made by a comparison of isogon

patterns, or of plots of t aagainst a . For thin layers the difference

between the two models is very slight.

It is apparent that unless the curvature remains constant, the

'normals' drawn for a tangential longitudinal strain fold are only normal

to the neutral surface, although for a thin layer they are almost normal

to all surfaces, and the thicknesses of the layers stay almost constant.

The tangential longitudinal strain model is probably only realistic

for thin layers, and the flexural flow model for layers with a very well-

developed layer-parallel lamination. Thick buckled layers will probably

accommodate the strain in a more complex manner that may or may not be

treated a a combination of these two ideal models.

3.5 HOMOGENEOUS FLATTENING OF FOLDS

Campbell (1951), de Sitter (1964) and Ramberg (1964b) briefly consider,

and Ramsay (1962a, 1967) and Ailkhopadhyay . (1965a) discuss in some detail,

the effect of a superposed finite homogeneous strain (referred to here

loosely as a 'flattening') on an initially parallel fold in terms of

thickness modification in the profile plane. The latter two authors

derive a family of curves on graphs of the thickness parameters, t'a , and T114 , against dip, a , that represent the plotted positions taken up by

an originally parallel fold modified by different amounts of flattening.

These curves enable determination of this flattening strain for natural

Fig. 3.4

Shapes of Buckled Layers.

a) Tangential Longitudinal Strain Model.

(i) Fold profile between hinge and inflexion points.

N.S. = Neutral Surface = Sine Function

Isogons drawn at 5° intervals.

(ii) Plots of harmonic coefficients for fold surfaces 1-5.

(iii) Thickness/Dip variations for folds in layers A, B & C.

Circles Layer C

Crosses .... Layer B

Dots Layer A

b) Flexural Slip Model.

(i) Fold profile between hinge and inflexion points.

M.S. = Middle Surface = Sine Function

Isogon drawn at 5° intervals.

(ii) Plot of harmonic coefficients for fold surfaces 1-5.

M S

0

a

1.0 b1

40 50 20 30

a 10

0.051-

N 14

13 b3 1

5 ••

4 • 12

3 a 1.1 •

2

10 - 0.02

1 0.9

0 80

0

o • 0 •

9 • •

• •

• •

0-04

b3

0

- 0.04 0

• 5

•4 3

0.5 1-0 b1

2

•1

89

folds, in two dimensions only within the profile plane, providing that

the original fold was parallel and that one of the principal directions

of the three dimensional strain is parallel to the fold axis, a condition

that in most general cases may not be met (Flinn, 1962). If the fold

axis is not a principal direction of strain, the strain within the profile

plane cannot be determined from the layer geometry (Ramsay, 1967, p.415; -Nilkope.dhyan 1965a). MUkhopadhyay shows how a complete determination

of the strain may be made if one of the principal directions of strain

lies in the axial plane of the folds and if the fold axis varies in

orientation within the axial plane. The method involves computations for

three profile planes with different orientations.

Subject to the same

2.3.3), can also be used

by an initially parallel

values of the dip, a .

restrictions of use, the parameter, 0(section

to determine the amount of flattening undergone

folcl. For such a fold 0. 0 initially, for all

Lfter flattening 0. 0 at a = 0 and a = 90, but

for all other values of a 10 will be chapged by an amount that depends

on the dip, a , and on the degree of flattening involved. The

geometrical changes that occur at one value of a are shown in fig. 3.5.

the superposed

dip respectively

Using Wettstein's

instance from

A l and X2 are the principal quadratic elongations of

strain; ao and a are the initial and final angles of

and 0 is the angular shear, y, , of the deformation.

formula ffor the changes in angles involved (derived for

eq. 3-34 of Ramsay, 1967)%

ILL tan( a - 0) and / X

tan a

2 0

2 tan( 90 - ) X tan(90 - ao) 1

Multiplying these equations together and simplifying gives: X2

tan ( a - 0) tan a

3.4

Fig. 3.5

Modification of a parallel fold shape by a homogeneous 'flattening'

strain ix 2/x1 . 0.25.

a) Initial State AA = isogon at dip °, 0

b) Final State A'A' = deformed isogon at dip a

92

By substituting different values of X2/ xi into this equation curves

relating 0 and a for different amounts of flattening have been drawn

(fig. 3.6) which are directly comparable to those drawn by Ramsay (1962a; 1967, figs. 7-79 & 7-80) for the variation of t'a and T'a with a in

flattened parallel folds. The graph, fig. 3.6, is the mirror image

of one constructed by Breddin (1957, Table 5 fig.4) for strain determination

in fossils.

Because the value of /X2/ X1, determined from a plot of tta or

0 against a for a natural fold may not be atrue measure of strain, I will follow Mukhopadhyay (1965a) and refer to this value as an apparent

strain ratio, R (Fiukhopadhyay uses a symbol, a', the reciprocal of R).

R can be conaiclerad as an empirical parameter of fold shape that is only

directly related to a superposed strain in certain circumstances.

It is convenient to discuss hero a modification of the graphs of

and t'a against a that transform the family of curves of various x2/x1

values into straight lines. From equation 3.4 it is clear that a plot of tan( a - 0) against tan a will produce this transformaticn (fig. 3.7).

The relevant equation relating t'a , a and x2/x1

is given in Ramsay

-0 a 2

= xl A

X2 ( c _ X21 2 os a 3.5

J plot of t'a2 1

against cos2a will give a family of straight lines of

different slope for various values of X2/X1 (fig. 3.8).

The usefulness of a straight line representation lies in the fact

that plotted data for natural folds may be approximately re--resented by

best—fit straight lines using the familiar techniques of linear

regression by least squares. single fold may then be represented by

(1967, P.412):

Fig. 3.6

Variations of 0 with dip in flattened parallel folds for various values

of /2""l •

0 10 20 30 40 50 60 70 80 90

94

TT v 0 . 9 ----

!._

. - -- /1/

- - , / •

\ N...

---,

N ----..,„___________

--------__ 0 .8 .-----------

—__, 07

/'..-1.---'--------

___-------

I

. "

--, 0 6 ____-------

,,-

----------____ 0 5 , .,,, / r

.

\"-------„, o .3

N.Ns'''..\,.0 2

11441\ , 0 1

Angle of Dip a

0

10

2

3

4

5

6

7

8

90

Fig. 3.7

Straight line relationship in the variations of tan (a — 0) with tan a in

flattened parallel folds for various amounts of flattening.

50

1 0-4 co 1- 40

30

20

10

0

10

- 20

- 30

60

96

A2 9 0

7 1 0 8 All

o7

1 I I WO Pr "

. _I-- 0 4 -

1------ - 0 3

0.2

1 :.0.036.--'...11111111111 0

0 10 20 30 40 50 60

70

Angle of Dip a

0

10 Tan a 2 0

70

I

2 0-

Fig. 3.8

Linear relationship in the variations of -02 with cos2o, in flattened

parallel folds for various amounts of flattening.

1.1

1 • 0 --I 10

to

09

0.8

ta =

07

06

05

O 4

O 3

O 2 O 1

0-5 -1

98

r

I L

I

1

-1-

,.. A , -4..

`..-- , , ',..

_l____ ----............„L ----..„..........

1--

A I 0-9 ' -- 1 ----i-, .....

-4--

4. .,

.., , J ---.9.1•<...,„

,,,, \ 1. .

1.-----1--

N NN

'N.

1-

1-

`‘

o.

1

, ....

010 20 30 40 50

60

70

80 90

Angle of Dip a

10 0 5

COS 2 a.

a single parameter, either the slope or the intercept of the best-fit

straight line

Plots of nearly all the natural folds measured by the writer are

closely approximated by straight lines on these graphs; the derived

parameters of slope and intercept, in the same way as R, may be considered

empirical and need not be related to flattening at all.

Since a graph of tan( a. -0) against tan a is infinite in two directions its use is best restricted to values of a less than 70°.

3.5.1 Oblique Flattening in the Profile Plane.

A single example of the effect of flattening a parallel fold will be

considered, to illustrate the general case in which two principal directions

of the superposed strain lie in the profile plane obliquely to the axial

surface trace. In fig. 3.9 a fold is drawn undeformed and at two finite

states of flattening,A2/xi. 0.5 and 0.25 the angle between the

principal axes of strain and the initial axial surface trace is 45°. In

the deformed states the originally 'symmetrical' fold with a straight

axial surface trace is transformed into an asymmetrical fold with a

curved axial surface trace (of the general case 3, section 2.3.5). This

is because the folded surfaces are different in shape, so that the hinge

points move around each deforming surface at a different rate.

Geometrically the effect is similar to that described by Schryver (1966),

of obliquely sectioning' cylindrical folds.

In nature, this kind of effect is likely to occur in parasitic folds

on the rotatinf, limbs of lar-er structures (Ramberg, 1963c).

3.6 SIMULTANEOUS BUCKLING AND FLATTENING.

Although de Sitter (1964) and Ramsay (1967, p.411) recognise that in

general unlimited shertenin of an actively folding layer maintaining a

99

Fig. 3.9

Oblique Flattening of a Parallel Fold.

a) Undeformed state

b) Deformed state JA 2/ A l = 0.25

c) Deformed state IA 2/ A= 0.5

a) Sketch of axial surface traces for a), b) & c)

A.S. = Axial Surface Trace

Isogons are drawn at 15° intervals in a), b) & c)

101

a

A

HINGE POINTS

c d

3.6 0n N

102

parallel form is impossible, the production, by a buckling process, of

a parallel fold which then becomes homogeneously flattened seems equally

unlikely, since the initial buckling necessitates a competency difference

between the layer and the enclosing medium which must disappear at some

later stage for the fold to flatten in a true homogeneous manner.

It seems likely that where the ductility contrast is low an actively

folding layer will deform by a combination of a buckling and a flattening

process. L. simple process that simulates the simultaneous buckling and

flattening of a competent layer is described below.

The fold is represented by a large number, N, of discrete sections

between hinge and inflexion points (fig. 3.10a), each section defined

solely by its dip, a , and its thickness, t. Folding is initiated with

a buckling increment to give the nth of the N sections a dip of:

a (n,k) =

and a thickness of:

t(n,k) 1.0 3.7

where k = 1 for the first increment.

= the maximum rotation of the limb for each buckling increment.

This is now followed by the first increment of flattening, and then

alternate increments of buckling and flattening are successively added.

k increases by 1 for each increment of either buckling or flattening.

Each flattening increment of pure

formula (e.g. Ramsay, 1967, eq.3-34):

—1 a (n,k) = tan

and thickness according to eq. 7-30

t(n,k) = t(nlk-1)

shear changes a according

1

( Al/A2 )2 tan

of Ramsay (1967): 1

( A1)2 cos (n,k)

to a standard

3.8 (a,k-1)

3.9 cos (n,k-1)

Fig. 3.10

Simultaneous Buckling and Flattening of Folds,

a) 3 of the N discrete sections of the fold model, after k increments

of buckling and flattening (see text). The quadratic elongation

directions are those of both incremental and finite strains in the

hinge section, and of incremental strains only in the other sections.

b) Curves A, B & C — thickness variations with dip in parallel folds

flattened by strains of IA 2/ 1 = 0.75, 0.5 & 0.25 respectively.

Curves 1-7 — thickness variations with dip in folds formed by

simultaneous buckling and flattening.

Curve No.

Flatt. Incr.

Max. Buckle Incr.

K Total Buckl. Rot.

Finite Flatt. Hinge

1 0.998 1o 173 86° 0.84

2 0.990 2° 83 80° 0.66

3 0.990 10 153 750 0.47

4 0.900 5° 27 6o° 0.25

5 0.990 0.25° 437 54.5° 0.113

6 0.990 0.125° 655 41° 0.038

7 0.900 0.5° 95 23° 0.007

Flattening values given as X2/ X1

Max. Buckl. Incr. = Increment of rotation by buckling of the limb

section.

Total. Buckl. Rot. = Total rotation by buckling of the limb section.

a

t (N,k

INFLEXION POINT SECTION

n N

IN a k I Max

A

HINGE SECTION

n =1

a(l.k) = 0

0 90

PARALLEL -7 1-0

0.5 -

0

Angle of Dip a,

t (Lk) tMax

A

105

where X1 and X2 are the incremental quadratic elongations.

Each buckling increment changes a by an amount given by:

A (n k ) = a (n,k-1) 6 3.10 a (N k-1)

and so a (n, k) = a ) Au (n,k) 3.11

Thickness is kept constant, and so:

t(n,k) t(n,k_.1) 3.12

In the calculations, N was taken as 50, and a range of values of 0 and

of NIX1A2 were used to vary the relative amounts of buckling and

flattening. Values of 0.9, 0.99 and 0.998 were taken for fx2/xi ,

and for etch value 0 was varied between 0.125° and 20o. For each pair

of IX2A1 increments until the limb dip reached approximately 90°; the total

number of increments, K, the finite strain at the hinge and the final 0(1 K) and values of = t(n,K, 9

a(n,K) were recorded. The calcul-

ations were dune on a digital computer.

On plots of t' against a , the resultant fold geometries vary

between parallel (almost pure buckling) and similar (involving large

components of flattening); the plots form a family of non-intersecting

curves that differ systematically from those relating t'a to a for

flattened parallel folds: the difference between these two sets of

curves is most marked at very high limb dips, and very slight at low dips.

Representative curves are plotted in fig. 3.10b where this comparison

may be made.

This folding process is independent of the shape of the layer

bounding surfaces, which are arbitrary: the state of strain within the

layer is subject only to the 'geometrical restraints imposed by the

and 0 the equations 3.6 - 3.12 wem solved for successive

106

values of din and thickness and can therefore be accommodated in any

number of ways (not all of which are equally likely).

It must be stressed that there is no mechanical basis for choosing

this model, and the actual process will certainly be more complex.

However, the model proves useful, in a qualitative way, in the interpret—

ation of plots of tla against a for natural and experimentally produced folds (sections 5.7 and 4.8).

The reason why many natural folded competent layers appear to show

a geometrical relationship between t'a and a expected of ideal flattened

parallel folds may be due to:

a) Tho rheological contrast decreasing or vanishing during deformation

b) The resistance to continued buckling in the layer, imposed by the

enclosing medium, increasing as the fold tightens and the material

of the medium becomes extruded from the inner arc regions between

the limbs of the buckling layer (see Chapple, 1968). The

competent layer may then be forced to take up further deformation by

flattening.

c) A "tickling and flattening' process resulting in a fold whose geometry

is scarcely distinguishable from that produced by homogeneously

flattening a parallel fold.

The model proposed here is compatible with c) for small dips.

a) is thought to be unlikely as a general rule, and some combination of

the reasons given in b) and c) may account for the observed geometry of

many natural folds.

107

GRAPIER 4

EXPERIMENTS ON BUCKLING

4.1 INTRODUCTION

The existence of a buckling instability in many kinds of layered

systems (isolated beams, single layers set in a more ductile medium,

several layers of different ductilities forming a multilayered sequence)

shortened parallel to the layering has been well established experimentally

by a number of workers using a great variety of materials of a visco-

elastic or plastic nature. A brief summary of the relevant work is given

here.

The state of stress within and around buckled (less ductile) layers

at finite amplitudes has been studied using photoelastic techniques by

Bell & Currie (1964) and Currie, Patnode & Trump (1962), and the state

of finite strain within buckled layers by Ramberg (1963a).

Ramberg (1959) and Ghosh (1966) have studied the development of

buckles in layered systems undergoing simple shear. Experiments to

determine the relationship between the axial surface and fold axis and

the axes of bulk finite strain, where the buckling layers lie oblique to

the principal directions of strain, have produced ambiguous and controversial

results (see McBirney & Best, 1961; Ghosh, 1966, 1967; Singh, 1967).

Buckling experiments usually involve a maximum shortening parallel to

the layering in a bulk deformation of 'pure shear' type, resulting in more

or less symmetrical folds. The most important outcome of this work has

been the verification of the theory of Biot and Ramberg that predicts the

dominant wavelength of folds developed in a single layer embedded in a

relatively more ductile medium, where layer and medium are both Newtonian,

elastic or general viscoelastic bodies (see Section 3.3). The experimental

checks on the theory have treated the cases of viscous layer and medium,

108

elastic layer and viscous medium (Blot, Ode &Roever, 1961) and elastic

layer and medium (Ramberg, 1963b).

Ramberg (1963b), in a series of buckling experiments on elastic

multilayers has found that the observed wavelength/thickness ratios are the

same as the dominant wavelength/thickness ratios predicted theoretically for

viscous multilayers, by substituting elastic for viscous moduli in the

theoretical expressions. This might be expected from the rule of correspon—

dence between expressions derived for elastic and general viscoelastic

materials (Riot, 1957).

A series of experiments on the buckling of single layers of a viscous

material embedded in a less viscous medium, with low viscosity contrasts,

has been undertaken by the writer. These experiments are described in this

Chapter. This work was prompted by the apparent lack of correspondence between

a conclusion reached by Biot (1961, p.1607), who considered that no

significant folding would occur for a viscous layer embedded in a viscous

medium with a viscosity contrast of less than 100:1 (equivalent to a

dominant wavelength/thickness ratio of about 16:1), and the observed fact

that wavelength/thickness ratios of natural single layered folds are

frequently very much smaller than 16:1 (Sherwin & Chapple, 1968; sections

5.7 & 6.5 this thesis). At low viscosity contrasts shortening within the

competent layer becomes imnortant (Biot, 1961) and although the effects of

this have recently been theoretically evaluated in some detail by Sherwin

& Chapple (1968), they have not been verified experimentally.

The results of the experimonts described in this chapter will be

discussed in terms of the theories of Biot (1961) and Sherwin & Chapple (1968).

It was not found possible to determine accurately the viscosity contrasts

in the experiments, because of diffusional effects taking place between

layer and medium. The results could not therefore be used as a proper

109

quantitative check to the theory.

4.2 MODEL STUDY PROBLEMS

The problems of modelling tectonic processes are discussed by Hubbert

(1937), Donath (1963), Grovsky (1959), Beloussov (1960), Bell & Currie (1964)

and Ramberg (1967)f, the need to maintain similarity between a model and the

original (its prototype) by means of dimensional analysis is stressed by these

authors. Grovsky (1959) and Ramberg (1967) have attempted to simulate large

scale tectonic processes by employing fairly rigorous dimensional analyses, as

a basis for experimentation.

The present problem is the simulation of small scale folding in thin

isolated single layers in rock.

Geometrically the model will be about the same size as the prototype

folds. Beyond this, however, accurate modelling is impossible because the

rheological nature of the rock layer and its medium are unknown, the time

factor is indeterminate and the boundary conditions are also largely unknown

(see Heard, 1968).

Although rocks exposed at the earth's surface are clearly not viscous

substances, that rocks are able to deform very considerably by some kind of

flow over geological periods of time is indisputable from evidence of deformed

objects of known original shape and of folds themselves.

The treatment of rocks as Newtonian viscous bodies has several

advantages; mathematical analyses are generally simplified, and buckling

is theoretically independent of both absolute values of viscosity and rate

of deformation, Gay (1966) discusses at some length the validity of an

assumption of Newtonian viscosity for rocks, and he concludes that such an

assumption may be reasonably valid where stresses are applied for long periods

of time or where the rate of strain is slow. Flinn (1965) discusses the

nature of flow in rocks and the possible mechanisms by which it can be

110

accomplished. He considers diffusion to control deformation in metamorphism,

but thinks a true Newtonian viscosity unlikely. Reiner (1960, p.11)

considers all materials to possess all rheological properties at once, and

if stresses are maintained for a sufficient length of time Carey (1954)

considers that the effects of the viscous component of strain will exceed

all others.

Following Gay, I will assume that rocks probably behave as Newtonian

bodies (i.e. the stress/strain rate equations are linear) to a first

approximation, under conditions that are most likely to be met in areas of

regional metamorphism.

4.3 APPARATUS AND MATERIALS

The simplest type of experiment that will produce more or less

symmetrical folds is one in which the principal direction of shortening

is initially parallel to the layering, and remains parallel to the enveloping

surface of the folds as they develop. If in addition the fold axis is a

direction of no change the problem reduces to one of two dimensions. These

are the conditions usually met with in theoretical work; they can be

satisfied experimentally by use of a shear box that maintains plane strain

in pure shear. For the present series of experiments, a shear box was

designed and built to meet those requirements.

The shear box is shown in plate 1, and a plan is drawn in fig. 4.1.

It has four rigid L-shaped corner pieces made from Tufnel plate. From

the base of each of these protrudes a short steel peg that is free to

rotate, and which moves in a groove cut in the metal base plate of the box.

On two opposite sides of the box the corner units are connected by smooth

lubricated metal rods, free to slide through close-fitting tubes rigidly

attached to the corner units, and on the other two sides similar rods and

tubes are threaded to allow for a screw movement only. The connecting

rods ensure that the construction maintains a rectangular shape, and keeps

Plate .1 Shear box ;Jet up for an experiment.

Ethyl cellulose solution is in position

inside the box. On the left is a prepared

layer in a metal mould.

Fig. 4.1

Simplified plan of the shear box. Only two of the connecting rods are

shown. The box is symmetrical about the x axis.

A Base Plate

B Hyperbolic Groove

C Corner Piece

D Side Plate

E Connecting Rods

F Operating Handle

113

A

'7=

1

F

6 INCHES

114

the corner units in mutual alignment. They also allow for changes in

overall length of the box sides. Part of each side is formed by the

corner pieces, and the central part is a thin flat metal plate that rests

against and is able to slide over the corner pieces to allow for changes

in length of sidle. These plates are held in position by the material

in the box. The box is 4" deep. By turning the handles attached to the two threaded connecting rods the corners of the box move simultaneously along

their grooves, and the overall lengths of side progressively change, whilst

maintaining a rectangular fora. The grooves were cut in the base plate

such that the inside corners of the corner pieces move along hyperbolic

paths given by;

x z = k

where x and z are coordinates parallel to the sides of the box

with the cen-Le as origin, and k is a constant.

and so the area contained 'within' the box remains constant as the lengths

of side change.

The limiting ratios of lengths of two adjacent sides are 2:1 and 1:2

respectively.

If, during deformation, each material point in the box moves according

to:

x z - k' 4.

where k' and c are constants; k 1

Y = c varies with distance from the centre

of the box, k' k. c varies

ITith distance from the base.

The conditions of plane strain and pure shear would be fulfilled. The

maximum strain ratio (iX1/X2) expected would be 4:1.

Because the base plate is rigid and each side is compounded from three

rigid parts, considerable boundary effects may be expected to modify the

ideal behaviour of material deformed within the box. Their effects are

discussed in the next section.

115

The materials used for the experiments were all solutions of ethyl

cellulose (marketed as "Ethocel" by the Dow Chemical Company and of

standard ethoxy content, viscosity grade 100) in benzyl alcohol (B.P. grade).

These materials were chosen because solutions show a rapid onset of viscous

flow in deformation (Grovsky, 1959 Bell & Currie, 1964) and because

viscosity can be varied by varying the concentration of solute in the

solution. At less than about 15% concentration, solutions are almost

true Newtonian fluids, whilst at higher concentrations slight non—Newtonian

effects of hysteresis and thixotropy may be observed (Osakina et . 196 0) . These effects may be reduced by keeping the strain rates small. Gay (1966)

has used solutions of these materials in a series of experiments on the

deformation of rigid and viscous particles in a viscous medium.

The viscosity rises exponentially with solution concentration, and

decreases exponentially with rise in temperature (Osakina et al., 1960).

Gay (1966) describes how the solutions may be made up, and how their

viscosities may be measured. Solutions varying in concentration between

15P and 40% solute by weight were used in the present study. Fig. 4.2 is a

graph of viscosity against concentration determined for these solutions

(cf. Gay, 1966, fig. 13).

The concentrations and viscosities of solutions used as layers and

medium are tabulated below:

1) MEDIUM 20% soln. 1-12

2.5 x 103 poises

Approx. Conc. %

28.5

30

LAYERS

P1 poises

14 x 103

41 x 103

Apparent

11

Fig. 4.2

Relationship between viscosity and concentration of solutions of ethyl

cellulose (viscosity grade 100) in benzyl alcohol.

(cf. Gay, fig. 13, 1966).

117

15

20

25

30

35

40

0/0 Concentration

/4

/4

/

•

L /

/ , I ,

/

t-•

-

50

100

2

1

r

20

5

118

2) MEDIUM 15% soln. 112 0.8 x 103 poises

LAYERS

Approx. Conc. % 1 Ideal u 1212 Apparent pl/j1,2

30 41 x 103 51 24

35 136 x 103 170 50

40 375 x 103 470 100

A diffusional effect that increased with viscosity contrast was found

to take place between the layer and medium, tending to reduce the effective

viscosity ratio. Unfortunately it was not found possible to evaluate this

effect accurately, due to the difficulties in separating a layer from the

medium. The ideal viscosity ratios were computed from the viscosity values

of the pure solutions; the apparent viscosity ratios were derived from en

interpretation of the ex1Derimental results (see section 4.7). The relation-

ship between ideal and apparent viscosity ratios is shown in fig. 4.15.

The apparent viscosity ratios are those referred to throughout the text.

The dye ethyl blue was used to colour the layer solutions.

4.4 EXPERIMENTAL METHODS

Before carrying out any experiments, the sides and base of the shear

box were lubricated with vaseline; care was taken to ensure that the contacts

of the side plates (held in place only by the medium) and the base plates

with the rest of the box were well sealed, and that there was no leakage of

the contents.

The box was set up with dimensions 2:1 and filled with the appropriate

medium solution to a depth of about 2". The solution was allowed to stand

for several hours to allow any trapped air bubbles to clear.

Layer solutions were pressed into open moulds made up from discrete

metal strips: the cross-sections of the moulds (and hence of the layers)

were square and either 1/A" ore thick and their lengths were 8", 11" or 15".

119

Care was taken to ensure that no bubbles were included in the layers, and

the moulds were left to allow the solution to settle and take up the mould

shape (see plate 1).

A layer was emplaced in the medium in the following manner. Mould

plus layer were immersed in a freezing mixture of dry ice and acetone. The

large drop in temperature caused the viscosity of the solution to increase

considerably without freezing. Whilst very cold, the metal strips were 'cut

away' from the layer with a scalpel. The layer was rigid enough to be

handled and was fist carefully wiped with a tissue to remove any acetone;

it was then very carefully placed in position on one of its sides on the

surface of the matrix solution in the centre of the box, where it slowly

sank under its own weight until its top surface became flush with that of

the matrix.

A film cassette holder (for circular marks), or a wire mesh, were

lightly sprayed with paint and pressed gently against the surface of the

material within the box to form markers to record the strain set up during

the course of the experiments. 40 to 90 minutes were allowed to elapse

between emplacing the layer and beginning an experiment.

A camera was set up on a tripod over the box and photographs were taken

after successive increments of deformation, until each run was completed.

The duration of the runs varied between 30 and 90 minutes. The strain rate

was neither constant nor rigidly controlled.

The number of runs made for each viscosity contrast were as follows:

5.5:1 — 3; 11:1 — 13; the rest 5 runs each.

The process of very great cooling and immersion in acetone did not

permanently effect the viscosity of the solutions, and it was found that

after removal from the freezing mixture, the temperature of the layers

increased rapidly at first, to within a degree or two of the matrix solution

temperature (equivalent to room temperature) after about 30 minutes (fig.4.3)•

Fig. 4.3

Temperature increase with time in a layer taken from a freezing mixture

(of dry ice and acetone), and placed in a host medium. (The bulb of a

thermometer was encased in a blob of a solution of 30% concentration so

that the solution formed a shell about 1/8" thick. Both blob and

thermometer were placed in the freezing mixture, left for 5 minutes, and

were then removed and placed in a solution of 20% concentration at room

temperature.).

20 .7- — Room Temperotur ,!

121

30

10

—10

20 0

0

TEM

PER

ATU

RE

10 20 30 40 50 60

TIME Mans

122

The rate of further increase in temperature was very slow.

4.5 HOMOGENEITY OF STRAIN & BOUNDARY EFFECTS

I series of experimental runs were carried out with one solution only

in the box to determine the degree and extent of homogeneous deformation.

Fig. 4.4 is a drawing (traced from a photograph) showing the deformation

of originally circular markers) to form 'ellipses' with axial ratios of

about 4:1 near the centre of the box. The variations of strain near the

boundaries of the box (in the blank regions of fig. 4.4) were mite complex.

These variations involved increases in surface area along the two approaching

sides of the box and decreases along the receding sides. The strain at

the centre of the box (in an area bounded by lengths roughly half those of

the total length and breadth of the box at any one time), was however fairly

homogeneous, and measurements show that there was no surface area change

in this region. Moreover no significant differences in strain pattern could

be detected when the depth of material was varied between 1" and 2i".

It was concluded that the strain at the centre of the box for depths

of material between 1" and 2" was close to a two dimensional pure shear.

The relationship between the observed strain ratio (S =iX11x2) at the

centre of the box and that predicted from dimensional changes of the box

(the strain that would exist if deformation progressed according to

equations 4.1) was found to be consistent for all the runs mado5 data for

one run is shown in fig. 4.5a; measurements were recorded at equal

increments of predicted strain. The relationship between the observed

strain ratio and the predicted strain ratio was clearly not linear, and

the maximum observed strain ratio was almost twice that predicted.

The non—linear relationship is to be expected if the predicted

incremental strain ratio, IR 9 and the observed incremental strain ratio

IRo, are linearly related by:

ER = k1Ro

k = constant

FiE, 4- 4

Deformed circular narks on the surface of material in the shear box

after a strain of about jA,2/A l .--- 0.25.

124

,

Relationship between the strain, So, observed at the centre of the shear

box and the strain, Sp, predicted from shape changes of the box.

In a) each numbered dot marks each increment of deformation,

6.0 7.0 2.0 3.0 4.0 5.0

9

8

7

6

n 5

4

3

2

1

0 0.6 0.8 0

0.2 0.4

En so — In sp

126

So OBSERVED RATIO

1.0 1.0

a 2.0 U)

8.0

a

PRED

ICTED

RA

TIO

5.0

4.0

3.0

I

9

8•

7•

6• 5 •

4

3 26 1

•

127

For finite strains, S and So respectibely, of n increments:

P = (IR P)n and So = (IRdn

and the finite strains are related by:

S = kn So 4.2

This is not linear; 4.2 may be written:

n ln k = ln So — ln S p 4.3

The plotted points of n against In So In Sp, drawn in fig. 4.5b for the data

of fig. 4.5a, lie close to a straight line.

Fold development was only investigated within layers situated in the

central part of -box in *the region of homogeneous strain.

With a layer in position in the matrix and for a given value of S the

value of the strain computed from the straight line distance between two

points on the layer (near the centre) was found to be slightly lower than that

observed (So above) when one solution only was in the box. The difference

between these two values of strain was reduced by making the layers as long

and as thin as possible. Bulk strain was calculated from the changes in

the straight line distance between two points on the central part of the

layer.

4.6 RESULTS

The shapes of typical folded layers at the final stage of deformation

are shown in fig. 4.6.

4.6.1 Changes in Arc Length

Arc length was measured along the middle line of a layer (see fig.4.7)

between two inflexion points, so as to include several folds. The overall

shortening(strain ratio S =1X1A2 ) was calculated from the changes in the

straight line distance between these inflexion points (see fig. 4.7) and

this has been plotted against arc length (taken as unity before deformation)

in fig. 4.8a for selected experiments at each of the viscosity contrasts

Fig. 4.6

Examples of Experimentally Produced Buckles.

a) Viscosity ratio of 24:1 bulk deformation S = 6.

b) Deformed rectangular grid constructed for a). N.B. The initial

spacing of the grid lines between the three discrete segments of

grid shown in a) is irregular.

c) Viscosity ratio of 100:1 , bulk deformation S = 6.

d) Viscosity ratio of 5,5:1 2 bulk deformation S = 6.5.

(the scale is the sane as that in b)).

1

I

1" a

Fi. 4.7 To illustrate measured quantities in folded layers.

h — hinge point

i = inflexion point

t = orthogonal thickness

A = amplitude

W . true wavelength

ihihi = arc length = "wavelength" in w/t analysis.

131

!<

w

Fig. 4.8

Changes in arc length with total (bulk) shortening.

a) Data for selected experiments at each viscosity contrast.

o viscosity ratio 100:1

50:1

24:1

dots It 11:1

b) Smoothed curves fitted to the data in a).

Heavy curve represents uniform shortening with no folding.

AR

CLE

NG

TH

AR

CLE

NG

TH

O

RIG

INA

L A

RC

LEN

GT

H

OR

IGIN

AL

AR

C L

EN

GT

H

9

0

0

+ 0

0

0

•

0

•

• •

0

•

134

(with the exception of the smallest). Smoothed curves for the same data

have been drawn in fig. 4.8b, together with a curve representing length

changes due to homogeneous shortening with no folding.

From this figure it is apparent that the initial shortening of a layer

or change of arc length is that of a homogeneous compressive strain.

However, a stage is reached where little further changes in arc length occur.

The stage (i.e. bulk strain ratio) at which the arc length becomes 'stable'

is different for each viscosity contrast. The maximum percentage shortening

undergone by the layer has been estimated in each experiment. The values

are listed below.

Viscosity ratio

shortening Mean

100 13 10 13 12 12.5 12

50 23 20 23.5 22 23 22

24 34 33 36.5 33 33 34

11 ? 50 — 60 for all expts.

The degree of correspondence between values of shortening in several

experiments at any one viscosity contrast is fairly close. Since the amount

of shortening possible in the shear box is limited to about 640, the estimates

for the lowest viscosity contrast,11:1, may be too small.

It was found that at the stages where the arc lengths became

'stabilised' (corresponding to the points of divergence of the experimentally

derived curves from the curve representing uniform shortening in fig. 4.8b),

the folds had mean limb dips of about 15° (varying between 5o and 30°) and

this value was found to be independent of viscosity contrast.

4.6.2 Thickness Variation in the Buckled Layers

No significant thickening in the hinges or thinning in the limbs was

observed in any of the folds developed in the experiments, for the whole

135

range of viscosity contrasts (see fig. 4.6). At the maximum deformation

attainable in the box, folds developed at a viscosity ratio of 11:1 rarely

attained limb dips greater than 30°. One experiment at this viscosity

contrast was continued beyond the limits imposed by the box, by crudely

compressing the folded layer between two plates to produce near isoclinal

folds (at a bulk strain ratio of about 20:1). These folds did show a

Certain amount of thickening in the hinges.

4.6.3 Wavelength/Thickness Ratios

The range in values of fold wavelengths developed at each viscosity

contrast was considerable.

The 'wavelength' (the arc length of mature folds, see section 1.2),

taken as twice the measured distance between adjacent hinges along the

middle line of the folded layer(fig. 4.7), and layer thickness were recorded

for each fold in each experiment, at a stage of the deformation when the arc

length had reached its stable value. For the viscosity contrast of 11:1,

these parameters were measured at a bulk strain of about S = 5.0. Wavelength/

thickness ratios were calculated, and frequency histograms of this ratio

have boon plotted in fig. 4.9 for each viscosity contrast. Examination

of this figure shows:

a) a considerable spread in w/t ratios for every viscosity contrast, and a

substantial overlap between the ranges of ratios observed for the different

viscosity contrasts.

b) mean values of W/t of 3.8, 6.8, 10.5, and 13.3 for viscosity contrasts

of 11, 24, 50 and 100:1 respectively.

c) similar relative spreads of the distributions as measured by the ratio

of standard deviation/mean.

4.6.4 Amplification

Progressive changes in amplitude during fold development were recorded

with increasing strain ratio, S, of the overall shortening. Amplitude, A,

FiF. 4.9

Frequency histograms of the values of W/t.

Is/ 1/ R 2 Mean

Standard Deviation

Mean/S.D.

a) 11 3.8 1.3 2.9

b) 24 6.8 1.5 4.5

c) 50 10.5 2.3 4.6

d) 100 13.3 3.3 4.0

mean = ungrouped arithmetic mean

137

50 -

a 68 Folds F %

M = MEAN

F = FREQUENCY 2

I 10 12

50-

F% 39 Folds b

W T

1 . 1 6 8 10 12 14

w T

4 6 8 10 12 14

30-

F%

2

C 32 Folds

w T

34 Folds d

2 4 8 10

M

, I

WAVELENGTH

THICKNESS

1.".'"...-1

W T

12 " 14 16 18 20 I r - 1

22 24 26

30-

F%

138

of a single fold, was taken as half the measured distance between the

enveloping surfaces to one of the layer boundaries (see fig. 4.7) and its

value was divided by thickness to give a dimensionless measure.

Plots of amplitude against S were essentially similar for all the folds

measured, and selected plots for individual folds, developed in experiments

at each viscosity contrast, have been drawn in fig. 4.10 to show the slight

differences in amplification behaviour involved. From this figure we can

note that:

a) if points on each plot are joined by a smooth curve, A . f(S), in all

cases the second derivative, f"(S), would pass through a maximum

value. This represents the point at which the amplitude is growing

fastest.

b) the point of fastest amplitude growth occurs at progressively smaller

values of S with increase of viscosity contrast.

c) greater amplitudes are reached for progressively greater viscosity_

contrasts..

4.6.5 Harmonic Analysis of Fold Shape

The progressive development of fold shape may be studied using the

harmonic analysis described in section 2.5. A plot of the third harmonic

coefficient, b3, against the first, b1, for successive shapes in the

deformation progress of a single 'quarter wavelength unit' (see p. 51 )

of a folded layer will represent a 'deformation path' of fold shape.

Most individual folds analysed followed similar paths irrespective of

viscosity contrast and typical plots of b3 against b

1 have been drawn in

fig. 4.11. Both inner and outer arcs were analysed and have been

distinguished in this figure; from which may be noted:

a) The earliest measurable fold shape in any of the experiments is close

to sinusoidal (b3 is almost zero).

Fig. 4.10

Amplitude growth with total (bulk) shortening.

a) Data for individual folds at each viscosity contrast.

o viscosity ratio 100

50

dots If 24

11

b) Smoothed curves fitted to the data in a)

Arrows indicate the points of fastest amplitude growth.

0 0 0

0 x

O x

•

• •

0 x o•

x

0

x

• +

2.0 3.0 4.0

SHORTENING S =, J A2

0

11

U)

z

0

SHORTENING

0 1-0

4

3

2

AM

PLITU

DE

T H

ICKN

ES

S

1

6.0 5.0

x

Fig. 4.11

Harmonic Analysis of Individual Folds through progressive deformation.

a) Successive shapes in the development of a single fold (p,i/ µ 2 = 100),

from the point where folds are first visible. Inflexion and hinge

points are marked by dots.

b) Plots of b3 against b1 for the left hand limbs of the folds in a).

c) d) Plots of b3 against b1 for successive shapes of folds developed at

a viscosity contrast of 50:1.

Note: Tie lines between two points relate inner and outer arcs

of a single fold at each stage of fold development.

h

h

2 ----- 3 ----._

0-1.

1" -J

0.3 x

b3 0.2

0

x Inner Arcs

0 Outer Arcs

a

b 3

b

3.0 4 0

b

0.1 0 5(

x

0 x

-0.05 0 1.0 2.0 3.0

b1

b 3

-005 0

0-/.

0.2

0.5

03

01

0

1 0 2 b1

3 0

x

x

x

0

o-Y

x

0/

0

143

b) The paths of fold shape development are curved, with b3 increasing at

a greater rate than b1. This implies that the folds become progres-

sively broader in the hinges as they develop (see fig. 2.11).

c) Inner and outer arcs are usually similar in shape at any stage of

deformation, and follow similar paths of change. Inner arcs, however,

tend to attain higher values of b1 and b3/b1.

The middle line of the folded layer has been traced out in four of the

experiments, for the highest viscosity ratio, 100:1, at the final stage of

deformation, and every 'quarter wavelength unit' in the fold trains has been

analysed. Fig 4.12 is a plot of the values of b3 against b1 for the

results. On this plot the various fold shapes define a markedly-elongate

field, the trend of which is very similar to that of the path-traced out

during the development of a single fold. This suggests that - all the folds

have followed similar paths of development of shape, but at different rates,

so that at any one stage a spectrum of shapes exists that occupy_different

positions along the path.

Several individual 'quarter-folds' wereanalysed.to obtain- mare

coefficients in the .harmonic sine -series. • Spectral--gamaphs(see. fig. 2.12)

for those fold shapes have been plotted in fig. 4.13: the envelopes to the

plotted points on these graphs are nearlylinearl signifying that these fold

shapes will lie in the spectrum of ideal shapes described and figured in

section 2.5.

4.6.6 acperimental Simultaneous Buckling and Flattening

Layers with the low viscosity-contrast.of 5.5:1 -were -placed in the

medium so as to form several open folds: the experiments were carried

out in the normal way.

Fig. 4.14a & b shows the initial and final states_of one experiment.

The initial folds are almost parallel and sinusoidal in.shape, and after

deformation the folds have become thickened in the hinges-and-thinned in the

Fig. 4.12

Plots of b3

against b1 for 50 folds at the final stage of deformation

(S = 5.0 — 6.0) at a viscosity ratio of 100:1.

145

b1

•

• • •

b3 0.5

- 0.05 0

0.4

0.9

0.6

0.3

0.2

07

1.0

0-8

0.1

0

1.0 2.0 3.0 4.0 5.0 6.0

•

• •

•

•

•

•

. • : •

•

•

• .„

mr

0

•

•

•

•

•

•

•

..5 • • •

• •

Fig. 4.13

Spectral Graphs for individual folds (quarter wavelength units) at the

final stage of deformation.

a) Viscosity ratio = 50:1

b) = 100:1

c) 11 = 50:1. This is the outer arc of the fold whose

progressive deformation is traced in fig. 4.11d.

Note: each fold shape is an outer arc.

i = inflexion point

h = hinge point

1 0.1 b„

147

13 11 9

n 7

5

3

1

0.01

a

b b, 1 10 0-01 0 1

0 - 01 0.1

3

n 5

1

13 11 9 7

b,

9 7 5

3

1 1

Fig. 4.14

Experiment to show both buckling and flattening in a competent layer.

(Viscosity ratio 5.5:1).

a) Initial fold shape

b) Final fold shape after a total (bulk) shortening of about S = 6.

c) Plots of b3 against bl for progressive changes in shape of the

segment ih of the folded layer.

i = inflexion point

h = hinge point

d) Thickness variations with dip in the fold limbs C and D. Dotted

curve represents thickness/dip variations in a parallel fold

flattened uniformly by a strain ps. 2/ Xi = 0.6

e) Thickness variations with dip in the fold limbs A and B. Dotted

curve represents thickness/dip variations in a parallel fold

flattened by a strain IX 2/ Al = 0.5.

0 b, 50 9 0

• 0

0

0

• 0

•

0

o Limb C • D

I 0

05

b3

b

0 x

a

-0

O Oiitei Arc

• I ni,ei

0 x

O 8

07

O 6

80 70 90 0 10 20 30 40 50 60

ANGLE OF DIP a,

• •

• 0 N

• + `,0

o limb A 0

90 60 30

a

02 0

e

10

09

0.5

04

0.3

O 2

,0

In

0 6

d

150

limbs. Plots of thickness against dip are constructedfor two of the

deformed folds (fig. 4.14d & e); and shape, for one 'quarter-fold' is

harmonically analysed at successive stages of the deformation. A plot

of the coefficients b3 against b

1 (fig. 4.14c) shows that

b3/b1 increases

more rapidly than does bi; but this effect is less pronounced for this

fold than for the 'naturally developed' folds (see fig. 4.11).

4.7 ItITERPRETAT ION

It will now be shown that the experimental results may be interpreted

in terms of the theory of Sherwin & Chapple (1968), considering the

viscosity contrasts to be less than the ideal values given in section 4.3. Approximate' values of the viscosity ratios will be derived whose relation-

ship to the ideal values for undiffused solutions is systematic (see fig..

4.15).

It is assumed that the- mean value of the W/t ratios for each viscosity

contrast is that of the 'dominant wavelength folds'.

As a-first-approximation it is considered valid to take-these.

experiments to be. equivalent to ones carried out with immiscible liquids

at significantly lower viscosity contrasts than those ideally existing

between the layers and medium in these experiments were diffusion absent.

Two observations support this:

a) 'Values of maximum. layer shortening and mean -W/t are consistent for

all experiments at any one viscosity contrast, despite variations in

both the length of time allowed to elapse before starting each

experiment, and the length of duration of each.once started.

b) Grid lines crossing the limbs of folded layers show a fairly sharp

transition in trend at the edges of the layers-(see.fig, 4.6). A

gradual transition of solution concentration (and hence viscosity)

across the layer/matrix interface as a result of diffusion, would

Fir.;. 4.15

Relationship between values of Apparent Viscosity Ratio and Ideal

Viscosity Ratio.

152

• •1111 111v

•

•

•

10 • • 11111 111•1

10 100 1000 Ideal PA

2

153

show in the experiments as a gradual transition in the trend of the

deformed grid lines.

Using the mean value of the observed w/t ratios (fig. 4.9) and the mean

limiting value of layer shortening, for each viscosity contrast used, an.

estimate may be made of the effective (apparent) viscosity ratio by

substituting these values into equation 3.3. In this way the figures in

row 5 of the table below were derived. Now, taking each of these estimates

of viscosity ratio and each mean limiting value of layer shortening, estimates

of amplification (of the dominant wavelength) may be read from the graph

in fig. 3.2. Those estim?.tes appear in row 6 of the table.

For all but the lowest viscosity contrast used, it appears that

shortening in the layers ceases when the limb dips of developing folds

reaches about 15°. Assuming the initial irregularities of all the layers

used in the experiments to be in the same order of size, the mean

amplification required to produce folds with 15° limb dips will be the same.

The values of amplification in row 6 of the above vary between 20 and

80. The mean value is 50, and taking this to be the best estimate of

amplification for all the viscosity contrasts used, a new estimate of each

effective viscosity ratio may be read from fig. 3.2, using the mean limiting

value of layer shortening and a value of 50 for the amplification.

The final estimate of the effective viscosity ratio is an average

of the first two estimates. (The values in row 9 of the table are the means of the values in rows 5 and 8). The final estimates of these ratios

are plotted against the ideal values for undiffused solutions in fig. 4.15.

see over

154

Table 4.1

Ideal V R 16 51 170 470

Mean W/t 3.8 6.8 10.5 13.3

Limiting S Shortening 5.0 2.3 1.6 1.3

Limiting Shortening % 55 34 22 12

1st est. V R 11 24 56 82

Amplification 50 50 80 20

Mean Amp. 50 50 50 50

2nd est. V R 11 24 45 120

Final Estimated Viscosity Ratio 11 24 50 100

4.8 DISCUSSION

The discussion which follows on the significance of the experimental

results in terms of buckling theory, is to a large extent independent of an

exact knowledge of the viscosity contrasts.

One of the most interesting features of the experiments is the apparent

cessation of changes in both the arc length and the thickness of the buckling

layers when the folds attain limb dips of about 1509 so that furthrx fold

development is not accompanied by thickening in the fold hinges or thinning

on the limbs. This confirms Sherwin & Chapple's (1968) postulate that

major changes in the length of layers occur only in the early stages of

deformation.

Whether these results can be applied to conditions of very low viscosity

contrast (say < 10) is uncertain: much larger deformations than those

attainable in the present shear box would be required to test this, since the

fold amplification at these contrasts is very small until considerable

shortening has occurred (see fig. 3.2).

155

It seems likely, however, with continuing deformation at very low

viscosity contasts, that further changes in arc length and thickness will

occur. These will be unequally distributed around the folded layer, and

may involve shortening along the layer in the hinge regions and extension

in the limbs (see Ramberg & Ghosh, 1968- ).

The large spread in wavelength/thickness ratios (fig. 4.9) suggests

that folding in these experiments is of low selectivity (see Biot, 1961,

p.1604). This feature is dependent upon the 'degree of flatness' of the

layers before deformation; the flatter the layers the more regular the

folds that develop (Biot, 1961, p.1605). assuming the amplification of

the folds (i.e. amplification of the dominant wavelength) up to the '15°

limb dip stage', to be the same for all the viscosity contrasts, selectivity

of the folding should also be the same, since it depends only upon

amplification to a first approximation (Biot, 1961, p.1605). For the

wavelength/thickness histograms of fig. 4.9, a measure of selectivity is given by the ratio of standard deviation/mean. Values of these ratios are

of the same order of size for the four distributions.

The final distribution of ?•d/t ratios, and therefore selectivity, is

probably 'fixed' at or before the 15° limb dip stage, and the amplification

required to attain this stage will depend upon the initial flatness of the

layer.

The relationship between'amplification and selectivity is illustrated

in fig. 4.16., where the variation of amplification with wavelength, at

different amounts of shortening, for a viscosity ratio of 16 is graphically

recorded. Equation 3.2 was used to derive these curves, and at each value

of shortening the maximum amplification has been set equal to unity to enable

comparison of the curve shapes.

Analysis of the experimental data shows that the change of amplitude

Fig. 4.16

To illustrate selectivity of folding.

Variations of amplification with W/t at various values of total

shortening, S, for a viscosity ratio of 16 between layer and matrix.

(Based on the theory of Sherwin & Chapple, 1968). In each graph the

maximum amplification has been scaled to 100 to enable comparison of

the shapes of the graphs. A measure of selectivity is given by

Wd/(W2—W) (for unit thickness), where W1 and W2 are wavelengths

amplified by half the amount of the dominant wavelength.

a) S = 2.0 Wd/(W2—W1) = 0.69

b) S = 4.0 Wd/(V—W1) = 0.82

c) S = 10.0 Wd/(W2—W1) = 0.72

100 0 0

, a < I

50

0

0 0 100

lz z o o I- , a <

u_ =.1.1

5 0 u_ 71;

.‹

wi 5 Wd 10

w2 15

20 WAVELENGTH W THICKNESS

0 WI vvd 5 w2 10 15 20

100 0

0

< 50

"wd 5 W2 10

15 20

158

with shortening (fig. 4.10) is initially an exponential increase (as is

predicted by the Biot theory), but the rate of increase reaches a maximum

value and then declines. Beyond the initial section of exponential increase

neither Biot's theory nor that of Sherwin & Chapple are valid and are unable

to predict further fold development.

The harmonic analysis shows the kind of fold shape development predicted

theoretically by Chapple (1968) for thin inextensible layers of original

sinusoidal shape. The progressive changes in fold shape observed in these

experiments (figs. 4.11., 4.12) are almost identical to those undergone by

the folds predicted by Chapple, in which the wavelength is either equal to

or very much loss than the dominant wavelength (cases L = Ld and L << Ld of

Chapple). The difference between these two cases of Chapple is not very

marked (see fig, 3.3).

The assumptions of inextensibility and constant thickness in Chapple's

analysis are reasonably valid here beyond the 15° limb dip stage.

Although no detailed study of the state of strain within and around the

buckled layers has been made, it was apparent that grid lines initially

normal to the layers tended to remain normal throughout fold development,

where the viscosity contrast between layer and matrix was high (see fig.4.6).

lit lower viscosity contrasts a certain amount of concentric shearing strain

(Ramberg, 1961b) caused the grid lines to deflect from the normal position.

Contact strain in the matrix was observed to die out rapidly away from the

buckled layers as predicted by Ramberg (1961a).

In the attempt to produce simultaneous buckling and flattening, the

folds artificially induced in the layers before the start of the experiment

have wavelength/thickness ratios greater than that predicted for the dominant

wavelength folds using equation 3.1. The progressive changes in fold shape recorded in fig. 4.14c are

159

different for the inner and outer arcs, but are generally very similar to

those undergone by the 'weak plate' fold of Chapple (1968), for which the

wavelength is several times that predicted for the dominant wavelength

(L = 4.6 La in Chapple's analysis). However Chapple's assumptions of an

inextensible layer with no thickness changes do not hold for the case

considered here. Thickening of the hinges and thinning of the limbs have

occured during deformation and the relationship between thickness and dip

(fig. 4.14d & e) observed for a number of folds is very similar to that

expected for a fold formed by a combination of buckling and flattening

according to the model proposed in section 3.6. Other than by inspection

of this graph, the non-passive behaviour of these folds is indicated by the

rotation of the grid lines in the limbs, and by the progressive changes in

fold shape indicated in fig. 4.14c. In homogeneous flattening,

progressive changes in fold shape would trace out a straight line path on

this graph, radiating from the origin. For an isolated viscous layer in

a less viscous matrix, the effect of progressive layer shortening during the

early stages of buckling will be to alter the predicted value of the

dominant wavelength/thickness ratio according to equation 3.3. By

assuming layers to have initial irregularities of the same order of size,

and by assuming a particular value for the amplification needed to bring

folds to the stage at which shortening ceases (i.e. the 150 limb dip stage),

the degree to which the predicted value of the dominant wavelength/thickness

ratio will change from its initial value (given by equation 3.1) may be

evaluated. This has been done for several viscosity contrasts, taking a

value of 100 for the amplification. The results are shown in table 4.2.

see over

160

Table 4.2

Viscosity % Initial W /t Wd/t Final

Contrast Shortening • (Biot) d Change

1,000 3.7 34.6 33.3 3.8

200 10.5 20.2 18.2 9.9

100 16.5 16.0 13.5 15.6

50 25.0 12.7 9.7 23.6

20 42.5 9.4 5.7 39.5

10 61.5 7.5 3.3 56.0

Equations 3.1 and 3,3., and the graph in fig. 3.2 were used to make the

calculations; the figures in columns 2, 4 and 5 are approximate because

they depend on values of shortening read from fig. 3.2. The percentage

change in the Wd/t ratio is an indication of the error that would be involved

if Biot's formula (eq. 3.1) were used to calculate Wd/t values.

In arriving at the conclusion that a viscosity contrast of 100 is the

minimum for which significant buckling can occur, Biot(1961) is considering

initial irregularities of a much smaller magnitude than those inferred in

the present experiments, and consequently considers far larger amplifications

necessary to produce significant folding (e.g. he takes a value of 1000 to

mark the point of explosive amplification). In the present stuay,

irregularities initially present in the layers lead to amplifications of

20 — 100, which are large enough to produce distinct folds (see also

Sherwin & Chapple, 1968)

Biot considers that the buckling instability will be masked by

passive shortening at low viscosity contrasts. However, it may be shown

theoretically that a buckling instability is likely to be more effective

than passive shortening in the development of folds, where the viscosity

contrast is much less than 100:1. This is shown below for a viscosity

ratio of 16:1.

161

Considering a sinusoidal fold of unit amplitude at a finite state of

shortening, S = 7, an initial value of the fold amplitude can be found by

assuming the fold to have formed by

a) buckling (taking the fold to be the dominant wavelength fold at

S = 7, and applying Sherwin & Chapple's theory) and

b) passive flattening. Calculations show that the initial amplitude

needed to form the fold by passive shortening is over three orders of

magnitude larger than that needed to produce the fold by buckling, for a

viscosity ratio of 16. In this case buckling is 1000 times more effective

in fold development than passive deformation.

Sherwin & Chapple (1968), although recognizing that shortening occurs

in the early stages of buckling, extend their analysis to treat cases where

the amplitudes are very large, where the assumption of a flat plate, on which

their analysis depends, must be invalid.. , They estimate the amplification

of natural folds that have limb dips of 10°-20° upwards. Using this estimate

and the mean of the measured W/t ratios they obtain a rough value of

viscosity contrast by reference to a graph relating amplification, dominant

wavenumbor, shortening and viscosity ratio (see fig. 3.1, this thesis).

However, in the light of the present experimental results, it seems likely

that the W/t ratios become fixed when folds attain limb dips of about 15°.

The value of the mean W/t ratio should therefore be combined with an

estimate of the amplification required to produce folds with limb dips of

about 150, in order to estimate the viscosity contrast. Using this,rather

than Sherwin & Chapple's approach, slightly lower estimates of viscosity

contrast will be obtained.

4.9 INTERPRETATION OF NATURALLY FORT 2D FOLDS

Assuming that it is valid to compare naturally formed folds with those

developed experimentally (see section 4.2), several important geological

implications follow from the experimental results described 'above.

162

One implication is that the arc length of a folded layer cannot in

general be taken as the original length, even in the case where the orthog—

onal tlickness remains constant around the folds (see also Sherwin &

Chapple, 1968). Estimates of deformation based on the assumption that the

arc length is the initial lverbngth will in general be too small.

Slightly modified, the methods of Sherwin & Chapple (1968) will enable

estimates of total shortening (within the profile plane of the folds)

and viscosity contrast tolp made where 'suitable' natural folds exist.

Suitable folds are those that appear to meet the assumptions of the Biot

theory (see section 3.2).

The layers in these experiments are considered to have been amplified

by a factor of about 50 to attain limb dips of about 15°. Thin rock

layers or veins will probably have initial irregularities several times

larger than those of the layers used in these experiments, and values of

between 10 and 100 are suggested as amplifications likely to be required

to develop folds to the stage whore layer shortening effectively ceases.

Sherwin & Chapple's methods have been applied, in a modified form, in

natural fold studies described in sections 5.7 and 6.5.

In order to experimentally develop folds with significant amounts of

thickening in the hinge regions, it was found necessary to employ a low

viscosity contrast between layer and matrix, and to induce wavelengths in

the layer greater than the predicted dominant wavelength. Many natural

folds possess thickened hinges (see also section 3.6.), and this may be due

to:

a) a decreasing or vanishing viscosity contrast during deformation.

b) greater deformation occuring in nature than attained in these experiments,

reaching a stage of deformation not seen in the experiments.

163

c) an effect operating on individual competent layers within multilayered

sequences, in which some control of the development of folds is

exerted by other competent layers within the sequence. Such layers

will tend to form folds of larger wavelengths when part of a multi—

layered sequence, than when 'isolated' (Ramberg, 1961a; Biot, 1961),

and fold flattening (see section 3.5) may be facilitated in a multilayer.

d) non—linear rheological behaviour of rock.

4.10 CONCLUSIONS

A) Buckling may occur for values of viscosity contrast much smaller than

that of 100, suggested by Biot to be the minimum for which distinct

folding would occur.

B) Buckling in these experiments is adequately explained by a theory

based on those of Biot (1961), Chapple (1968) and Sherwin & Chapple

(1968).

C) Layer shortening occurs at the onset of deformation, but effectively

ceases when folds have attained limb dips of.about 150.

D) Deformation of folds at low viscosity contrasts, with induced wave—

lengths greater than the predicted dominant wavelength, appears to

take place by a combination of buckling and flattening.

E) The range of fold shapes present in a buckled layer at a given stage

of deformation appears to reflect the shape changes undergone by a

single fold in the course of progressive deformation.

164

CHAPWi 5

AN ANALYSIS OF MINOR FOLDS IN THE MOINIAN ROCKS OF MONAR, INVERNESS-pSHIRE

5.1 INTRODUCTION

Loch Monar is situated about 30 miles west of Inverness in the

Central Highlands of Scotland, the Moinian rocks exposed here forming a

central part of the Caledonia° fold belt. These rocks are metamorphosed

sediments; siliceous and micaceous granulites. The general area was

first studied by Peach &Othero (1913), and more recently the structural

and metamorphic history of the Moines has been summarised by Ramsay

(1963) and Johnson (1965). The area around Loch Monar has been mapped

in detail by Ramsay (1954, 1958). The geology of part of this area is

shown in fig. 5.1, and within this the locality under study is indicated.

The reason for selecting this small area is that minor folds (F2 of Ramsay)

are intensely developed here, and near-profile sections of these folds

are superbly exposed on glaciated pavement that has been washed clean

during a dam construction scheme.

A brief account of the general geology of the area is given in

sections 5.2-5.4. This is followed by a detailed geometrical analysis

of the minor F2 folds in section 5.5. Section 5.6 is concerned with

the interpretation of the fold geometry in terms of folding processes.

Specific studies oflitY gmatic folds in pegmatitic veins (section 5.7), and

of lineations deformed around F2 folds (section 5.8) are described at the

end of the Chapter. As a result of these studies, estimates of viscosity

contrast have been made, and information as to the orientation of the 'bulk

finite strain ellipsoid' of the F2 deformation has been obtained.

5.2 LITHOLOGY

The rocks exposed in the study area are psammites belonging to the

Monar psammitic group of Ramsay (1958, p.274). They are banded siliceous,

Fiff. 5.1

a) Geological nap of the area around the eastern end of Loch 'Abner.

(After Ramsay, 1958).

d = dam

A.S. = Axial Surface

The shaded area at the end of the loch is the region selected for

detailed fold study.

b) Map to show the location of nap a).

166

PSAMMIT IC MOINE ./ FAULT .."

PELITIC MOINE ..../.. A.S. TRACE Major F2 Fold LEW ISIAN "'". A.S. TRACE Loch Monar Synform

a

b

167

semi-pelitic and pelitic granulites. The bands or layers are essentially

composed of varying proportions of quartz, plagioclase feldspar and biotite

and muscovite micas and probably represent original sedimentary layers

of variable composition. Pegmatitic veins of quartzo-feldspathic

material, either concordant or slightly discordant to the layering, are

common. These are coarse-grained and consist mainly of quartz, potash

feldspar and plagioclase. Sheets of similar material are also found

parallel or slightly oblique to the axial surfaces of the minor F2 folds

(plate 2). These pegmatitic sheets are considered to have formed either

late in the history of development of the F2 folds, or after folding had

ceased. They will not be considered further. Both types of vein are

considered metamorphic-metasomatic in origin (Ramberg, 1952).

The quartz and feldspar in the granulites exist as fairly equant

grains, the size of most lying in the range 0.1mm - 1.0mm. In the

pegmatitic veins, however, these minerals are about an order of magnitude

larger and are cormonly several mm in size.

It is convenient in describing fold geometry to distinguish four

lithologies; politic, semi-politic, psammitic and pegmatitic quartze-

feldspathic. The first three, referred to collectively as the layered

'granulites' can be considered as members of a continuous sequence of

composition variation based on dark mineral (biotite) content. The

distinction made is:-

politic - abundant biotite

psammitic - sparse biotite

semi-politic - intermediate in biotite content.

This distinction, although somewhat subjective, is significant in con-

sidering detailed fold geometry. Only the early generations of

pegmatitic veins are involved in the F2 folding, and these veins make

up the fourth category.

Plate. 2 F2 folds (300X NOW. of the little dam on the S. shore

of Loch llonar). Pegmatitic veins of one age are folded,

while those of a later age lie in sheets parallel to the

axial surfaces of the folds.

Plate 1 Folic.tion surface dcformed by F2 folds. (Road

section at E. end of little dam).

. ,. -;i, - . ' . :` ••'.- .,, -', \ \ - .... ., ' •

-• • - • • - .

. • • •"

• \

•-•,„

1111.'4' -144 ,

170

5.3 METAMORPHISM

Moine rocks appear to have undergone intense regional metamorphism,

accompanied by local migmatisation and the prodution of quartzo-feld-

spathic veins. Staurolite and kyanite are found in the Monar region

(Ramsay, 1954) and garnet is common in relitic rocks found to the south,

west and north of the area under discussion. The grade of metamorphism

is garnet-kyanite. There is evidence (Ramsay, 1963, p.168) of several

successive regional metamorphisms in parts of the Moines, all of about

this grade. That the grade of metamorphism stayed fairly constant through-

out the whole deformation history as seen at Monar, is suggested by the

presence of several generations of quartzo-feldspathic veins. The

earliest of these have been affected by the early fold phase movements,

whilst veins of a similar nature, parallel to the axial surfaces of the

second folds, are contemporaneous with or post-date these later fold

movements. The rocks may well have been in a near-migmatitic state

throughout this history.

5.4 STRUCTURAL GEOLOGY

Ramsay (1954, 1958) showed the existence of two major fold phases;

he demonstrated a close and consistent relationship between small-scale

folds and the larger structures throughout the region. The early phase

of deformation produced a series of major folds, overturned to the north,

with axial traces trending east-west, plunging gently to the west. The

later phase produced a series of major folds overturned in general to

the north-west, trending in a north-east south-west direction with

variable plunge. Ramsay (1954, 1958, p.291) showed how this plunge was

determined geometrically by the intersection of the second phase axial

surface (constant in orientation) with the fold limbs of the earlier

structures.

The area of interest (fig. 5.1) lies on the steeply dipping southern

limb of the Loch Monar synform, a major first phase fold. This limb has

171

been refolded by several major second folds. These folds trend north-

east south-west and their axial surfaces are near vertical, the fold axes

plunging steeply within them. The limbs and hinges of these major folds

are intensely puckered by small-scale structures, whose axes and axial

surfaces have attitudes in accord with the major structures. The fold

plunge, however, does vary locally where these folds are superimposed

on the minor folds of the early generation on the limb of the Loch Monar

synform. These second phase minor folds are the main objects of this

study.

Structural elements from the area are plotted on a stereogram in

fig. 5.2 (cf. Ramsay, 1958, figs. 3 & 4 sub-area 11).

The two fold phases of Monar are correlated with the second and

third phases of the general Moinian structural history (Ramsay, 1963).

Very tight isoclinal folds of the first Moinian deformation occur in the

Monar region. Those are not important in the present study and will

not be discussed further.

5.4.1 Rock Fabric

Ramsay (1958) showed that petrofabric girdles of quartz c-axes and

poles to basal planes of biotite closely matched the girdles of poles to

foliation planes in each of his sub-areas. The dominant visible feature

of the fabric is the well-developed mica schistosity which closely

parallels the axial surfaces of the F2 minor folds.

5.5. DESCRIPTIVE GEOMETRY

The material for this study consisted of a large number of field

photographs and specimens. devious analyses of folds from this and

adjacent areas are found in Ramsay (1962a) and Luahopadhyay (1964,

1965a). Their results will be considered later.

There is apparently no systematic variation in the spatial dis-

Fig. 5.2

Equal area stereographic plot of structural data for the area under

investigation.

0

Fl lineations (intersection of foliation

with an Fl schistosity).

F2 minor fold axes.

dots

Poles to F2 axial surfaces.

173

N

174

tribution of axial surfaces and lineations within the area, except for

the local variation of F2 axes due to their superposition on F1 minor

folds. Also, no systematic cereal variation in any of the parameters

described below could be discerned. For these reasons it is considered

valid to treat the area structurally as a single homogeneous domain. In

the third dimension individual fold axes tend to persist for considerable

distances (plato 3), and the folds can be considered cylindrical.

The geometrical analysis of these folds is quite independent of

any interpretation placed upon them in terms of folding mechanisms or

processes. It will be shown in section 5.6 that the detailed fold

geometry is consistent with a simple process of fold development.

Unless otherwise stated, the geometrical analyses that follow all

refer to profile sections of folds, which are all steeply plunging minor

F2 folds (F2 folds superimposed on the gently dipping limbs of Fl minor

folds are excluded).

5.5.1 Size of Folds

A size/frequency study of the highest order (Ramsay, 1967, p.355)

folds (i.e. the folds formed by a layer itself rather than by its

enveloping surface) has been made. Tho size measure taken was that

definod in section 1.2. Measurements wore made throughout the area and

a distinction has been made between folds in the layered granulites

(politic - psammitic layers) and thJse in the pegmatitic veins, which

commonly show ptygmatic relations to their host rock. The distributions

for those two categories are markedly asymmetrical (fir. 5.3a), but

become near-symmetrical 5y a transformation to a logarithmic size scale

(fig. 5.3b). The following deductions can be made from these histograms:

a) Both distributions (especially that for the granulites) are

approximately lor;-normal.

b) The mean fold size for the granulitic layers, 2.2", is considerably

Fig. _5.3

Frequency histograms of the size of F2 minor folds.

F = Frequency

a) & c) for 282 folds in the granulites.

b) & d) for 170 folds in the pegriatitic veins.

DI = Arithmetic (grouped) mean.

30-

F% 20-

10-

a

5 6 7 8 9 10 11 12 SIZE inches

b

4 5 6

SIZE inches

176

25-

20-

F% 15-

10-

I O1

10 SIZE inches

25-,

20-

F% 15-

10-

0

0

O 1 , I I

1

10 SIZE inches

177

greater than the mean for the pegmatitic veins, o.9".

An upper limit to the size of folds recorded is imposed by the scale of

the photographs on which measurements are made. A lower limit may also

be imnosed by the limiting visibility of the naked eye. The effect

of an upper size limit would be to truncate the 'tailing-off' of the

distribution, without otherwise affecting the overall shape. The

smallest fold sizes recorded here are of the same order of magnitude as

the grain size: folding on this scale becomes meaningless, and so the

lower limits of the distributions are the true limits of fold size.

5.5.2 Fold Order and Asymmetry.

The symmetry of the minor folds is always in accord with the geometry

of the larger structures, and between the highest order folds and the

major folds (fig. 5.1) exists a spectrum of sizes that do

form distinct fold orders of fixed wavelength magnitude,

order folds show considerable variation in symmetry about

mediate 'orders'. Asymmetry of the highest order folds,

ratio of the lengths of adjacent limbs (see section 1.2),

not seem to

The highest

these inter-

taken as the

has been

studied on folds taken more or less at random within the region. The

positions of the major folds can be seen from fig. 5.1a. A frequency

histogram of limb ratis, with 'S' and 'Z' sense of asymmetry distin-

guished, is plotted in fig. 5.4. For the sampled folds we can state:

a) 'Z' and 'S' folds are ap7)roximately equally common (45/0 'Z' and

55% 's')•

b) The distribution is fairly symmetrical about the ratio 1 : 1,

c) of all folds are asymmetrical in the range of ratios 1:1 - 3:1.

As in the study of fold size, an upper limit to the degree of asymmetry

recorded has been imposed by the scale of the photographs on which the

measurements were made. This has caused a truncation of the 'tails' of

the distribution. The shape of this distribution is largely due to

ag,__5.4 Asymmetry of F2 folds — frequency of limblength ratios.

60

50-

40-

30-

20-

10- NU

MB

ER

OF

FO

LD

S

'S '

sz

TOTAL S . Z 278

r-r- 11 10 9 8 7 6 5 4 3 2

te r, 2 3 4 5 6 7 8 9 10 11 12

0 12

RATIO OF LIMBLENGTHS

180

the existence of a continuous spectrum of fold 'orders' that results

in frequent changes in the sense and amount of asymmetry of the smallest

scale folds.

5.5.3 Isogon Patterns

For a large proportion of the minor F2 folds, the geometry is very

nearly 'similar' (class 2), and the isogons run in near parallel lines,

concentrated in the hinge zones (see fig. 5.5a & b). Zones of near—similar folds usually occur where folded pegmatitic veins are sparse

or absent. Within those zones it is usually possible to detect slight

divergencies from class 2 geometry (see fig.. 5.5b, layer B — class 1C

geometry, layer A — class 3 geometry). Where such divergencioseccur politic layers take up class 3, and psammitic layers class 1C geometry. Folded pegmatitic veins tend to have ptygmatic form and disrupt the

'similar' isogon pattern (fig. 5.5c, d & e). They usually show a

pronounced class 1C geometry, and have greater limb dips than do the folds

in the enclosing layers, causing the isogons to form closed patterns

across the folded veins. In fig. 5.5d a series of closed patterns follows

a pegmatitic vein across the fold profile. Where strongly convergent

isogons are found in pegmatitic veins, strongly divergent ones also

occur in the adjacent layers (fig. 5.50, thus tending to maintain an

overall class 2 geometry.

5.5.4 Interlimb Angle Variation

The closed patterns of isogons described above are related to

changes in interlimb angle. 11 study of the variation of interlimb

angle along axial surface traces for folds in a layered sequence

containing pegmatitic veins reveals several features of interest (see

the example drawn in fig. 5.6):

a) The lowest values of interlimb angle occur within, or at the

bounding surfaces of the pegmatitic veins.

b) There is little difference between the values of this angle for

Fie. 5.5

Isogon patterns for selected F2 folds.

Heavy Stippling Pegmatitic veins

Medium 17 Psamnitic layers

Light Semi—pelitic layers

No Pelitic layers

a) (300X north—west of the little dam, S. side of loch).

Isogons at 30° dip intervals.

b) (Loose block, S. shore of loch by little dan).


c) (200X north—west of main dam).

Isogons at 20° intervals.

d) (100X south of little dam, by the new road).


e) (Loose block, by main dam).


Fig. 5.6

Local Interlimb Jingle Variation.

a) Fold profile (part of the fold in fig. 5.5d) - pegtiatitic vein

stippled - axial surface traces narkec1 by dashed lines.

b) Plots of interlimb angle variation for the folds drawn in a) - the

"position" of the pegmatitic vein is shown by stippling.

184

a

1 % .. • • / I I \

_1 tu

Z <

0\ \ .

co • 0 /I

0

:.; 9 0 ct f • w .2 /

A

o . 180

DISTANCE ALONG AXIAL SURFACE

185

either bounding surface of a pegmatitic vein.

c) The greatest rate of change of interlimb angle occurs in the layers

immediately adjacent to the pegmatitic veins.

In the example shown in fig. 5.6, interlimb angle is taken as the minimum

angle subtended by the fold limbs for a particular surface, and is

recorded as ordinate on a graph, with abscissa as the distance of this

surface from a datum point measured along the axial surface trace. The

variations in interlimb angle are fairly symmetrical on either side of the

layer.

In folded sequences containing several pegmatitic veins, an interlimb

angle variation curve shows peaks at positions where these veins are

crossed by the curve.

When variation of interlimb angle for one particular folded surface

is studied in a direction normal to the axial surface trace, there

appears to be no systematic change in its value from one fold to the next.

5.5.5 Thickness/Dip Variations

The techniques described in Chapter 2 have been applied in the study

of thickness variation with dip for

folds' (i.e. segments between hinge

analysed in detail. The thickness

been recorded as a function of dip,

2.2a and 3.8.

the F2 folds. About 350 'quqrter—

and inflexion points) have been

parameter used was , t'a which has

on graphs of the kind shown in figs.

In order to synthesise the data it was convenient to represent each

fold by a single parameter, the dope or intercept of the best fit

stright line on a graph of t t2a against cos2a . The degree of goodness of

fit of straight lines to the data was empirically found to be very good

for practically all the measured folds. A minimum of five pairs of tVa values were used in each analysis, and the computations were done on a

186

computer with the aid of a least squares linear regression programme.

Fig. 5.7a is a frequency histogram of the intercept values for all

the measured folds, irrespective of lithelogy. A study of this figure

reveals that:

a) the distribution is symmetrical about a mean value of 0.06, with a

standard deviation of 0.02.

b) 65% of the folds have intercepts lying in the range -0.1 to +0.2.

c) the total range of intercept values is from -0.8 to +1.1.

The average measured fold therefore has a class 1C geometry which is not

far removed from that of a 'similar' fold.

Splitting up the 'parent' histogram, by distingdshing between the

intercepts for pelitic, semi-politic, psammitic and pegmatitic layers, four

separate frequency histograms may be derived (fig. 5.7b, c, d & e respec-

tively). Several features of interest may be noted in these:

a) The moan intercepts of the distributions differ systematically, with

differences in composition. The folded politic layers have a

largo negative mean intercept, -0.14- with progressive decrease in

biotite content of the layers, the mean value increases (semi-

pelitic layers, mean -0.006) reaching a value of 0.09 for the

psammitic layers. For the pegmatitic veins, which generally contain

less mica than the psammitic layers and are always coarser grained,

the mean value of the intercepts is 0.16.

b) The distributim for politic layers is distinctly asymmetrical and

skew to the left. The other distributions are more symmetric, but

that for the pegmatitic veins is slightly skew to the right.

c) There is considerable overlap between the ranges of all the

distributions, Ind the range of intercept values (covering 3 class

intervals) containing all the individual means, is relatively short

Fig. 5.7 Synthesised data for thickness variations with dip.

a) For all measured folds.

b) c) d) & e) For folds of different layer composition.

Q, F = Quartzo—feldspathic

M = Arithmetic grouped mean

Standard Mean Deviation

a) 0.06 0.22

b) —0.14 0.15

c) —0.006 0.22

d) 0.09 0.20

e) 0.16 0.18

80-

u- 60-

40-

2 Nu

mb

er

of F

old

s

M

339 Folds

-0.8 -06 -04 -02 r 0 02 04 0 6 Or 8 0 1. V. Intercept Value on a tc,/cos2a. Graph

40-

30-

F% 20- 65 Folds Pelitic Layers

10- M

-05 0

d 5 1'0

I.V. 30-

20- 57 Folds Semi -pelitic Layers F%

10- M

-0 5 0

d5 1'0 I.V.

40-

30-

64 Folds Psammitic Layers

-0.5 0 05 I.V.

F% 2

10- M

F%

30

20

151 Folds Pegmatitic 0.F Layers

10

0

M

-0-5 0 05 10

189

compared with the complete range (the smallest covers 10 class

intervals) of any one of the distributions.

It is clear that the geometry of folds defined on a tVo, graph, for any

one lithological type, shows considerable variation. It should be

pointed out that there are more data for folds in pegmatitto veins than

for any other rock typo. This reflects an observation bias in fold

measurement, and results in increasing the value of the total frequency

distribution mean (fib;. 5.7a). A distribution that gives equal weight to

all four lithological types has a mean intercept of 0.02, which is close

to the intercept for a true similar fold.

Subjective description of the lithology of individual layers has

tended to enhance the overlap of the distributions. However, errors

arising from this source are considered to be slight, and the pronounced

mutual overlap is thought to be a valid feature of these distributions.

Selected individual folds will now be described in order to bring

out geometrical features not apparent in the histograms. The first

two examples concern folds where pegmatitic veins are absent.

Example 1. (fig. 5.8, isogon pattern fig. 5.5b).

This is a typical fold in the layered granulites in which the geometry

of all the layers ap?roaches class 2, irrespective of composition. talc,

variations are plotted for layers A and B, whose geometric forms deviate

slightly from class 2. Layer A is semi—aelitic with a class 3 geometric fold form, whereas layer B is psammitic and its fold form is class 10.

Example 2. (fig. 5.9, isogon pattern fig. 5.5a).

The fold geometry in a single psammitic layer, X, and adjacent

politic layer, Y, is compared in several folds in a fold train, by

means of thickness/dip plots. From the figure we may note;

a) the overall cuspate style of folding (the 'antiforms' are tight and

Fig. 5.8

Thickness variations with dip.

a) Fold Profile (part of fig. 5.5b).

Stippling as in fig. 5.5. Dashed line is the datum for the

analysis.

b) Plots of t'a2 /cos2a for the folds in a).

Heavy line is the plot for a similar fold.

Naft. -zz-:—_____ x N •

• ' " .... ". ":" •.. N .... '. .... • " N N. N ,

". " N XN " ...., ". ... .....

S. .% ..... '`. ..... • N "•

N ' '. s N. .. 0 .8 • • N. .., ". ".

O " ' , .... N N ". N. "..,, 's .....

N ". ...

"*".. ".. N '.

, N ...., ...

▪ ... N.

' ". % N. 1) •,. • .. N "•- •• % \ ‘ ' • \

". % • N

". - % N. N N • ' , N. 0

,

o • N \ , ...s N

N. • ... N. • ♦ N. N. N.

• s• N. N • N N , N • N N. ....• ' ..... \ •' , N. N.

N N. .., \

N

., ...

0 • Layer A limb 1 o x \ "\ s• ir ss \

\ 0‘ % N . \ N

_ _ 2 0.9

x n 2 • •

• Layer B limb 1 • • 0.

t o 2 • S. 0

x O•

o • • N.• •

191

t INCH

a

1.0

0-9

0.8

Ca 0.7

0.6

0.5

0.4

0.3 0.2 0.1

0 0 10 20 30 40 50

Angle of DIp a 60 70 8090

Fig. 5,9


a) Fold Profile (part of fig. 5.5a)

Stippling as in fig. 5.5. Dashed linos are the datum lines for

the analysis.

2 b) c) Plots of t'a /cos2u for the folds in a).

C

. a

0.5

Layer X x Limb 1 Fold A + .. 2 ,. A o Limb 3 Fold B • .. 4 ,. B (I) Limb 5 Fold C ✓ u 6 .. C

b

30

60

90 a

194

the 'synforms' open)

b) the considerable differences between the positions taken up by the

plotted points for each layer in different folds.

c) the similarity of the Wu relationships for either limb of the same

fold in a given layer.

d) that the plots for both layers in the 'antiforms' (i.e. fold A) are

bettor represented by straight lines than the plots for these layers

in the 'synforms' (folds B Sc c). Both plots for layer X in fold C

move from field 3 tc field 10 at high limb dips.

Example 3. (fig. 5.10, isogon pattern fig. 5.5d).

Folds I and B die cut away from a pegmatitic vein (see fig. 5.6, where

the interlimb angle variations for these folds are recorded). Thickness/

dip plots for the pegmatitic vein (layer Y) and for semi—pelitic layers X

and Z either side of the vein, are recorded. From the figure we may note

that:

a) the variations for layer Y are almost identical on both limbs of

folds A and B (R = 0.4, see section 3.5).

b) whore themmi—pelitic layers X and Z lie beyond the outer arcs of

the folded pegmatitic vein, they take on a class 10 fold geometry

(except for limb 1 of layer Z), and where these layers lie within

the inner arcs of the folded vein, their geometric form is class 3.

Example 4. (fig. 5.11, isogon pattern fig. 5.5e).

The profile drawn is part of a sequence of folded pegmatitic veins,

alternating with pelitic rock layers. to variations with dip are plotted

for the massive pegmatitic vein X. All the plots fall within field 10

on the graph.

Considering the two pegmatitic veins together c.)nstitute a single

unit, where vein X forms the inner arc of this unit its geometric form

Fig. 5.10


a) Fold profile (part of fig. 5.5d).

Stippling as in fig. 5.5. Dashed lines are the datum lines for

the analysis.

b) c) d) Plots of t'(i /cos2a for the folds in a).

Dashed line in c) is the plot for a parallel fold flattened by a

strain of ik2/7\.1 = 0.4.

a

10

t "c,

0.5

0 0 30 60 90

a

1.0

0.5

0 0 30 60 90

t to

0 0 30 60 90

1.0

C

Limb I Fold A 2 A

Limb 3 Fold B 4 B

0-5

a a

Fig_. 5.11


a) Fold profile (part of fig. 5.50.

Stippling as in fig. 5.5. Dashed lines are the datum lines for the

analysis.

b) Plots of t'c /cos2a for the folds in a).

a

198

0 r ' 0

co •

O

• Limb 1 Fotd A o ir 2 ii A ✓ Limb 3 Fold B . ii 4 II g

30 60 a

10

0.5

•

Layer X b

NI 90

199

deviates less from that of fold class 2, than where it forms the outer arc

of the unit.

Example 5. (fig. 5.12)

Thickness/dip variations for several ptygmatic folds in a pegmatitic

vein are plotted. Three features of the fold geometry are worth noting:

a) there are c.)nsiaeralde differences in thickness/dip variations between

some of the analysed folds (e.g. compare limb 6, fold E with limb 39

fold C).

b) some of the plots of t t2 against cos are well represented by

straight lines, (e.g. the plot for limb 3, fold C), and all the plotted

points fall in the field of fold class 1C.

c) a number of plots of tot against cos2a show a systematic transgressive

relationship to straight lines passing through the point (0,1.0) on

a t'2/cos2a graph (see fig. 5.I2d). (These straight lines represent

linos of different R values, see section 3.5).

The total variation in geometric form, of all the folds analysed in this

profile (20 quarter—folds) is shown in fig. 5.12b — a histogram of the

slopes of the best fit straight lines to the plotted points on a t&/cos2a

graph.

The boundaries of the pegmatitic veins are usually irregular on the

scale of the grain size (see figs. 5.19 & 5.20). These irregularities

together with local thickness variations will introduce inaccuracies into

the determinatbn of tLand a values. For this reason, the data for

limb 6, fold E in fig. 5.12 (and data for many of the other folds described

above), proably contains spurious geometric features unconnected with the

fundamental folding process. The lower the curvatures in the hinge

zones of folds, the more likely are random thickness changes and surface

irregularities liable to affect the thickness/dip variations of those folds.

This is perhaps the reason why the 'open' synforms of fig. 5.9 give rise to

far less regular plots of tLagainst a , than do the 'tight' antiforms.

Fig. 5.12


a) Fold Profile (300X east of main dam).

Pegmatitic vein (stippled) in pelitic rock. Dashed lines are the

datum lines for the analysis.

b) Histogram of the values of the slope of the best fit straight line

2 (on a -0a/cos

2 a,graph) for 20 folds in the profile shown in a).

c) d) Plots of -02 /cos2a for the folds in a). a

Dashed lines in d) are the plots for parallel folds flattened by

strains of JX2/X1 = 0.4 & 0.5.

15

V 0

0 0 z

10

b 5-

0 2 12 0 4 10 06 08 8

SLOPE

30

tQ

• v

1.0

0-5 • Limb 3 Fold C o "

ii 4 6

I,

"

C E

N.

90 60 a a

202


About 580 F2 'quarter—fold' single surfaces have been analysed by

the methods described in Chapter 2.

Several individual 'quarter—fold' surfaces are drawn in fig. 5.13.

For each surface, the computed coefficients, bn, derived from the harmonic

analysis, are plotted on a spectral graph (see section 2.5.6) of log bn

against log n. Also drawn in this figure are computed surfaces, given by

the sum of the first and third odd harmonics (i.e. the surfaces defined

by b1 sin x + b

3 sin 3x) of the harmonic series. The fold surfaces in

fig. 5.13a, b do c are in the layered granulites, and the surface in fig.

5.13d is the outer arc of a ptygmatic fold in a 'egmatitic vein.

Inspection of the spectral graphs in fig. 5.13a, b (Si c shows that only

the coefficients b1 and b

3 are consistently above the 'noise level',

determined by the peaks of the oven coefficients (theoretically zero).

For the one example (fig. 5.13c) where the coefficients 1, 3, 5 and 7 are clear of the noise level, the sizes of the values of the odd bn are

contained by a straight line envelope, and the sign of bn changes regularly

with each increase in n. In fig. 5.13d the envelope to the odd

coefficients, b1 — b9, all positive in sign, is almost a straight line,

and is of greater negative slope than is the envelope in fig. 5.13c. The

noise level is also higher.

For the 'quarter—fold' surface in fig. 5.13a, the noise level is about

0.02. The actual fold hls'an amplitude of 3.5" and the value of b1 is

approx. 3.0. Therefore the maximum contribution of any of the noise

level harmonics to the fold shape will be about 0.024", which is in the

order of m.aguitude of measurement errors. In the hinge of this fold,

the first harwenic (b1 sin x) will account for about 94.51. of the shape,

the third (b3 sin 3x) for about 4.%, and the fifth (b

5 sin 5x) will

account for well under t; of the fold shape. For this fold, therefore,

the first and third harmonics are sufficient to define the shape very

Fig. 5.13

Individual 'quarter wavelength units' of F2 folds, and spectral graphs of

their harmonic components.

The solid curves are the actual fold shapes, and the dashed curves are

the sums of the first two odd harmonics; y = sin x + b3 sin 3x

The first of the two lithologies referred to in each case below lies on

the "inside" of the folded surfaces shown.

a) b) & c) are all folds in the granulites.

a) Folded surface between psanmitic and pelitic layers.

b) Surface between psamnitic and semi-pelitic layers.

c) Surface between semi-pelitic and pelitic layers.

d) Surface between a pegmatitic vein and a pelitic layer.

d

• 0

2 10

-bn 0 001 0 01 0'1

9 7

a

1 0

b

0 rt

2

0.001 0-'01 Oi 1 1 10

9 bn 7

n 5 3

1

0 001 0 01 0 1 10

2

C n

0 rr 2

- n

15 13 11 9

5

„, op

3

1

0 001 0.01 0.1 1 10

bn

N egative bn

5 4

3

2

9 7

5 n

1

•

205

closely. Higher harmonics have little significance, as their values are

of the same order of magnitude as the expected measurement errors.

The close match of the sums of the first and third harmonics to the

actual fold shapes is evident in fig. 5.13a, b & c. For nearly all the

folds in the layered granulites, the sum of the first two odd harmonics

were found to define the fold shape fairly accurately.

In fig. 5.13d the sum of the first and third harmonics is clearly

insufficient to define the fold shape; the coefficients decrease in size

at a much slower rate than they do for the folds in the granulites. For

this fold at least 10 harmonics would be required to define the shape with

any degree of accuracy. However, because the envelope to the computed

values of bra

is a straight line, it is valid to represent the fold shape

using only the first and third coefficients (see section 2.5.6), and the

shape will be closely matched by a member of the ideal series of fold

forms described in section 2.5.6.

Spectral graphs similar to that drawn in fig. 5.13d were obtained

for nearly all the analysed ptygmatic folds in pegmatitic veins.

In the analysis that follows only the coefficients b1 and b3 are

considered.

5.5.6.1 Analysis of bi.

The size of the coefficient bI is closely related to the fold

amplitude/wavelength ratio, or fold tightness (see section 2.5.6).

Frequency histograms of the size of b1 appear in fig. 5.14, for all the

analysed F2 folds. In drawing these histograms values of b1 for fold

surfaces in the granulites (fig. 5.14a) hwe been distinguished from those

in the pegmatitic veins (fig. 5.14b), and surfaces in those pegmatitic

veins with ptygmatic form have been treated as a separate category, (fig.

5.14c). Examination of fig. 5.14 reveals that:

Fig. 5.14

Harmonic Analysis.

Frequency histograms of the value of b1.

a) For folds in the granulites

b) For folds in pegmatitic veins

c) For ptygmatic folds in pegnatitic veins

Mean Standard Deviation

a) 2.7 2.4

b) 3.5 1.9

c) 4.9 2.2

M = Grouped mean.

30 -

4 ; . 6 7. 8 9 10 11 12 13 14

b1

b 168 Folds

5 10 '

b1

C 101 Folds

207

30-

412 Folds

M

I f

10-

M

F%

10-

M

I I I I

5 10 '

b1

20-

F%

20-

a

40-

F%

20

10 -

208

a)

all the distributions are asymmetrical.

b) the mean size

than that for

c) the mean size

of b1 for the folds in pegmatitic veins, 3.5, is greater

the folds in the granulites, 2.7. of b

1 for folds in ptygmatic pegmatitic veins is 4.9.

5.5.6.2 Analysis of b34.

The ratio of b31

is sensitive to changes in fold shape (see section

2.5.6). Several interesting features may be shown by presenting the

data in synthesised form. This has been done by constructing frequency

histograms of the value of the ratio b3/b1 (fig. 5.15). Separate histo-

grams have been drawn for folds in the granulitic layers, and for folds

in the pegmatitic veins. From this figure it is apparent that:

a) both distributions are roughly symmetrical about their means, and

the means of the distributions differ significantly in value; that

for the folds in the granulites is 0.02, whilst that for the folds

in the pegmatitic veins is 0.10.

b) the variance of the distribution for the folds in the pegmatitic

veins, 0.008, is more than twice that of the distribution for the

folds in the granulites, 0.003.

c) the range of b3/b1 ratios recorded, is greater for the folds in the

pegmatitic veins than it is for the folds in the granulitic layers.

These features imply that fold shape is far more variable in the pegmatitic

veins than it is in the granulitic layers, but that folds in the pegmatitic

veins will in general have more rounded crests than those in the

granulites, where fold shape will be nearer sinusoidal.

The data for folds in the pegmatitic veins has been replotted in

fig. 5.16 where a distinction has been made between the values of bibi

for folds with a ptygmatic form (fig. 5.16a) and the values for the

remaining folds (fig. 5.16b). Within each histogram in fig. 5,16,

further distinction between inner arcs and outer arcs has been made.

Fig. 5.15

Harmonic Analysis

Frequency histograms of b3/b1.

a) For folds in the granulites

b) For folds in the pegnatitic veins

Mean Variance

a) 0.021 0.0032

b) 0.099 0.0082

M = Arithmetic grouped mean,

0 0.04 0.08 0.12 016 0.2 0.24 0.28 -0.12 -0.08 -0.04

210

b3/

168 Folds b

F%

0 0.08 0 0.08 0.16 0-24 0.32

b3/ 0.16

20-

10-

412 Folds

10 -

30-

F% 20 -

a

Fig. 5.16

Harmonic Analysis.

Frequency histograms of b3/b1 for folds in peguatitic veins.

In a) & b) shaded section is for inner arcs of folds, unshaded section

is for outer arcs. M1 is the mean for the inner arcs and M2 the mean

for the outer arcs. M is the total mean.

a) For 101 ptygmatic folds (68 cuter arcs, 33 inner arcs).

b) For 67 non-ptygnatic folds (43 outer arcs, 24 inner arcs).

Mean Variance Mean Variance

a) M 0.134 0.0076 b) M 0.048 0.0050

Ml 0.101 0.0110 -0.005 0.0043

M2 0.150 0.0053 M2 0.079 0.0029

Fir- 5.17 Harmonic Analysis.

Frequency histograms of b3/b1 for folds in the granulites.

a) For 122 outer arcs of 'more psamnitic' layers

b) For 113 inner arcs of 'more psammiticl layers

c) For 177 folds with no distinct lithological contrast across the

folded surface.

Mean Variance

a) 0.031 0.0034

b) 0.006 0.0039

c) 0.023 0.0023

-0.08 0.08 016 0.24 0.32

- 0.08 0.08 0.16 0 24

b3/

b3 / b,

20

F% 10

0 -0 16

30

F%20

10

0

212

a

O 0.08

016

0.24

b3 / b,

30 -

20- F%

10 - a

- 0 08

30

20 - F%

10 - M b

-0:08 0 0.08 016 0.24

b3/b, 40 -

30-

F%

20 -

10-

-0.08 0 0.08 016 0.24

b3/

C

213

EXamination of these histograms reveals that:

a) both the mean value of b3/b1

and the range of the distribution are

greater for the ptygmatic folds than for the remainder.

b) in both distributions, the mean b3/b1 value is lower and the range

in values is greater for the inner arcs than for the outer arcs.

The data for folds in the granulitic layers is subdivided in the

following way, which depends upon the relative composition of the layers

either side of a particular surface. Where a folded surface is the outer

arc of a more psammitic layer, its b3/b1 value is plotted in fig. 5.17a.

Where no distinct lithological contrast exists across the surfacelthe

b3/b

1 value is plotted in fig. 5.17b, and where a folded surface is the

inner arc of a more psammitic layer, its b3/b1 value is plotted in fig.

5.17c. Inspection of these histograms shows that:

a) the distribution for the 'outer arcs' has the highest mean value,

and the distribution for the 'inner arcs' has the lowest mean value

of b3/b1.

the distribution for 'little lithological contrast' has a lower

variance than the other distributions.

The relationship between the distributions for all the categories of

folds (distinguished on lithological grounds), is best seen by plotting

the value of the mean against that of the variance for each distribution

(fig. 5.18). Summarising the data in this figure:

a) the mean b3/b1 ratio is smallest for the folds in the granulites,

and largest for the ptygmatic folds in the pegtmatitic veins.

b) in general the variance of the distributions increases with the mean.

c) the k'istributions for the inner arcs of folds have a lower mean value

Fig. 5.18

Harmonic Analysis.

Synthesis of data from figs. 5.15, 5.16 & 5.17.

For each distribution of the values of b3/b1 the arithmetic grouped mean

is plotted against the variance.

1 — For folds in the granulites

2 — For non—ptygmatic folds in pegnatitic veins

3 - For ptygmatic folds in pegmatitic veins

X Total of measured folds

triangles Inner arcs

dots Outer arcs

G No lithological contrast across the folded

surface.

0.15 0.1 0-05 MEAN

0

j

215

VA

RI A

NC

E

0-005

0-01

0

216

but higher variance than those for the outer arcs.

d) the group of folded surfaces that shows least variation in b3/b1 ratio

is that for which there is no distinct lithological contrast across

the surfaces.

In general terms of fold style, these results imply that the inner

arcs of folds are usually sharper in the hinges yet more variable in shape

than the outer arcs, for a particular category of folds, The ptygmatic

folds in the pegmatitic veins have higher amplitudes, are in general more

rounded in the crests and are much more variable in shape than the folds

in the layered granulites.

The results of analyses of a number of individual folds will now be

described in order to bring out features not aroarent in the synthesised

data. The selected examples are shown in figs. 5.19 and 5.20, where the

data are presented in plots of b3

against b1.

Shape Variations related to Variations in Composition.

a) Differences in shape between folds in the pegmatitic veins and folds

in the granulitic layers.

The values of b1 and b3/b1, for ptygmatic folds in pegmatitic veins,

tend t, be high; and several plotted points for these folds on a graph of

b3 against b

1 tend to display a pronounced 'linear' trend (fig. 5.19a,b).

Values of b3 are seldom negative.

The plotted points of b31, determined from an analysis of folds in

the granulitic layers, tend to cluster about the line b3 = 0 (see fig.

5.19d and fig. 5.20a,c): occasionally a distinct linear trend is apparent

(fig. 5.20d).

Non—ptygmatically folded veins are present in fig. 5.19c. On a

graph of b3 against b1, the data for these folds occupy a field which is

intermediate in position between the field occupied by data for ptygmatic

Fig. 5.19

Harmonic Analysis.

Plots of b3 against bl for individual folds. In a) inflexion and hinge

points are marked on the fold profile and each 'quarter wavelength unit'

is numbered. In b) d) the analysed folds are not marked on the

photographs.

o Outer arc of fold in pegmatitic vein or

psammitic layer.

x Inner arc of fold in pegmatitic vein or

psammitic layer.

dot No lithological contrast across the folded

surface.

a) Pegmatitic vein in pelitic rock. (700x east of main dam, in the

river valley).

b) Pegmatitic veins in semi—politic granulites. (Below the little

dam). Straight lines on the b5/b1 plot join the plots of inner and

outer arcs of a single fold.

c) Non—ptygmatic folds in pegmntitic veins folded with granulitic

layers. (Below the main dam).

d) Folds in layered granulites, (500x N.W. of the little dam, S. side

of the loch). Data for folds to the left of the pegmatitic sheet,

P, are marked by large circles on the b5/bI plot.

Fig. 5.20

Harmonic Lnalysis.

Plots of b3

against b1 for individual folds. In a) inflexion and hinge

points are narked on the fold profile and each 'quarter wavelength unit'

is numbered. In b) — d) the analysed folds are not narked on the

photographs.

o Outer arc of 'more psammitic' layer.

x Inner arc of 'more psanmitic' layer.

dot No lithological contrast across the folded surface.

a) b) c) d) Folds in layered granulites.

a) (Lt eastern end of the little dam). Large circles mark the plots

for folds from the left limb of fold A. Large crosses mark the

plots for folds from the right limb of fold A.

b) (part of fig. 5.5a).

c) (100x east of the main dam, in the river valley). The plotted

points enclosed by the dashed line are for the Isynfornall fold in

the photograph.

d) (Loose block, below main dam).

221

folds in pegmatitic veins (see fig. 5.19a,b), and that occupied by data

for the folds in the granulites.

b) Differences of shape of the inner and outer arcs of folded layers.

An analysis of folds in pegmatitic veins shows that the inner arcs are

far more variable in shape than the outer arcs (see the plots of b3

against

b1 in fig. 5.19a,b,c). This feature is either absent or slightly shown in

the folds in the granulites (see fig. 5.20a,b,c,d). In most of the plots

in figs. 5.19 & 5.20 both u1 b3

attain more extreme values

for the inner arcs of folds than they do for the outer.

Other Local Shape Variations.

a) Differences in shape between the limbs of the same fold.

In successive fold surfaces in fig. 5.20a, b1 takes on greater values

on ono limb of the fold A, than on the other.

b) Differences in shape between adjacent folds.

In fig. 5.20c, data for folds in the open 'synform' define a restricted

field on a graph of b3

against b1,

in contrast to the wide field defined

by the data for the tight adjacent tantiforms'.

In fig. 5.19d, there are distinctive differences between the positions,

on a graph of b3 against b1, taken up by the plotted points for fold

surfaces on either side of the pegmatitic sheet, P.

c) Progressive changes in fold shape along axial surfaces.

In the fold profile drawn in fig. 5.20b, there are progressive

changes in fold amplitude along the _axial surface traces. These changes

are reflected in the plots of b3

against b1.for a number of fold surfaces

in this profile. The plotted points for these surfaces on a graph of b3

against b1, fall in a broad field which displays an arcuate trend. For

222

low values of b1,

values of b3 are small, and either positive or negative

in sign: as the value of b1 increases,b

3 takes on positive values and

increases at a greater rate than b1.

d) Differences in shape between a number of ptygmatic folds in a single

layer.

In fig. 5.19a & b, the results are shown of analyses of a number of

folded surfaces in single layers with ptygmatic fold form. Picts of

the values of b3 against b1 for these folds fall in fields which show a

distinct linear trend. An increase in the values of both b1 and b

3 away

from the origin of the graph is accompanied by an increase in the value of

the ratio b3/b1. The trends do not appear to project through the origin

of the graph.

A summary of the features of folded surface geometry, brought out

by the harmonic analyses of individual folds drawn in figs. 5.19 & 5.20,

will now be made.

a) The existence of trends in some of the plots of b3 against b

1 has

been demonstrated: All the trends lead to an increase of b3/b1 with

b1'

implying that fold crests bocome more rounded as fold amplitude

increases.

b) Local differences in shape often exist between limbs of single folds

or between adjacent folds. The cuspate style of folding in fig.

5.20c has been 'brought out' by the harmonic analysis.

c) In any of the figured examples the spread in fold shapes present in

the analysed folded surfaces is large (i.e. the range in b3/b1

ratios is large).

d) The fold shape in the granulitic layers is usually fairly close to

being sinusoidal.

223

5.6 INTERPREMTION

The results of the geometrical analyses described above, all show

systematic variations in fold shape (both for layers and for single

surfaces) that can be related to variations in composition of the folded

layers. The layering in the rocks has not therefore behaved in a truly

passive manner (see section 3.2) throughout the deformation, and the folds

that have developed are not truly similar.

Nearly all the features brought out in the geometrical analysis are

compatible with a folding process that involves buckling of the more

competent layers in the rock, and a considerable amount of 'flattening'

(see section 3.5) of the whole folded sequence.

By analogy with buckling theory and the results of experiments on

folding (see Chapter 3), the relative competencies of the various rock

types may be determined by observation of isogon patterns (fig. 5.5), or

by inspection of the histograms of intercept values derived from -002,/cos2a

plots (fig. 5.7). The more competent a layer is with respect to its

neighbours in general the less will its geometric form depart from that 2 On a plot of -0a against cos

2a, , the most competent

layers will in general have intercept values closest to 1.0, and the least

competent layers values of intercept furthest removed from a value of 1.0.

Using these criteria we may rank the rock types according to competency:

pegmatitic veins > psammites > semi-pelites > pelites

Examination of fig. 5.5 shows that an overall class 2 fold geometry often results from alternating class 1C and class 3 fold geometries in individual layers. Ramsay (1967, p.432) describes this feature in a

folded phyllite.

The folds in the pegmatitic veins show features expected of buckle

folds that have been modified by a flattening deformation (section 3.5).

The geometric form of the folds in these veins is class 10, and on a plot

of

a parallel fold.

224

of a-02 against costa , the folds show a mean intercept of 0.16 (fig. 5.7).

Assuming that the profile plane of the folds is a principal plane of the

'bulk strain ellipsoid', that the folds were originally parallel and that

a flattening deformation followed the buckling, an estimate of ITT; = 0.4

for the mean value of flattening may be obtained from the value of the

mean intercept on a te/cos2a graph (see fig. 3.8).

Thickness variations with dip are for the most part very similar to

those predicted for parallel folds modified by a uniform flattening (i.e.

straight lines on a tDcos2a, plot; e.g. fig. 5.10). However, in fig.

5.12d several ptygmatic folds in a pegmatitic vein give rise to curves,

relating thickness to dip on a t'2/cos2 e, graph, that systematically trans—

gress the straight lines representing thickness/dip variations in flattened

parallel folds. The thickness/dip variations in these ptygmatic folds are

very similar to those theoretically predicted for a fold formed by a

process of combined buckling and flattening (section 3.6). It is not

possible to compute the relative amounts of buckling to flattening from

this data.

Adjacent to the folds in the pegmatitic veins, and especially around

the ptygmatic folds, are found geometrical features typical of the zones

of contact strain around buckled layers (Ramberg, 1961a; Ramsay, 1967,

P.416). The most dinstinctive feature of these zones is the increase

of interlimb away from the buckled layer. This is shown in isogon

patterns by the presence of closed isogonic lines (fig. 5.5c & d), and is

clearly brought out in the studies of interlimb angle variation (fig.5.6).

This feature is also apparent in the results of the harmonic analysis,

where the mean value of the first coefficient' b1, is greater for the folds

in the_pegmatitic veins than for the_folds in the granulites (fig. 5.14).

A second feature, typical of contact strain zones, is the difference

in the geometric form taken up by the incompetent layers near the inner

225

and outer arcs of the buckled layer (see Ramsay, 1967, P.416). In the

example described by Ramsay (1967, p.416), the folded incompetent layers

'within' the inner arcs of the buckled layer (itself a fold of class 1B

geometry) take up a class 3 fold geometry, whilst 'outside' the outer arcs of the buckled layer, these same layers take up a class 1A fold

geometry. A similar situation exists in the folds in fig. 5.10, except

that the 'buckled layer' itself shows a class 10 fold geometry and the

fold form of the incompetent layers outside the outer arcs of the 'buckled

layer' is also class 10. This is similar to the modified geometry that

would be produced by uniformly flattening a buckled layer and its contact

strain zone, where the original fo:..d geometry was of the kind described by

Ramsay.

There are features, more or less identical to those found in contact

strain zones, that occur whore no single competent (buckled) layer can be

identified as 'controlling' the fold geometry. Many cuspate folds (e.g.

fig. 5.9), often with different layer thicknesses in adjacent hinges (cf.

Ramsay, 1962a, figs. 10 & 11), are an example of such a feature. This

style of folding is nearly always associated with large local variations

in interlimb angle, and both features may result from the mutual inter-

ference of buckling layers within their zones of contact strain. A more

probable explanation of the cuspate style of folding, where apparently

outside the contact strain zone of an individual competent layer, is as

follows less competent layers in the contact strain zone of a competont

buckling lar,r will take up a cuspate fold style impised by the behaviour

of the more competent layer, and at the same time tend to buckle themselves.

In buckling, they may help to propagate the cuspate fold style away from

the contact strain zone of the original buckled layer.

With increasing amplitude, Chap le (1968) predicts that isolated

buckled layers will change their shape, to become more rounded in the fold

crests. This feature is observed in the ptygmatic folds in the pegmatitic

veins (fig. 5.19a & b), whore the ratio of harmonic coefficients b3rip1

226

increases with the value of b1 in the group of fold surfaces analysed (cf.

fig. 3.3). The same feature is observed in folds in the granulitic layers

in fig. 5.20b, where fold amplitude changes progressively along the axial

surface traces. In this case the analysed fold surfaces are in not one,

but several layers of different composition. It appears that the

different fold shapes observed in the ptygmatic folds of different

amplitude (fig. 5.19a & b), and observed in the folds of varying amplitude

in the multilayered granulites (fig. 5.20b), are very similar to the

sequence of fold shapes predicted in the progressive development of folds

by buckling in a single isolated layer. (Compare the b3Al1 plots in

figs. 5.19a & b and fig. 5.20b with those in fig. 3.3 and fig. 4.11 ).

Chapple (1968) predicts three different 'paths' of fold shape development

in buckled layers; the 'path' followed depends upon the ratio of the fold

wavelength to the predicted dominant wavelength, for a particular viscosity

contrast. The difference between the three paths is slight, and the

considerable spread in fold shape recorded in these natural folds precludes

detailed comparison of data for these folds with the fold shapes in any

one of the paths of shape development predicted by Chapple.

The folds in the layered granulites most nearly exhibit a true

'similar' geometric form. Those folds have fairly regular wavelengths

(i.e. are near periodic), and are regular in size (fig. 5.3). These

features are typical of folds developed by buckling, and are difficult

to account for by other folding processes (see section 3.2). Although

the geometric form of all the folded layers does not depart greatly from

that of fold class 2, slight but distinct overall differences in geometry

exist between folds in psammitic, semi-pelitic and pelitic layers. These

differences in geometry are similar to those that would be produced by

intensely flattening a gently folded multilayer, with low competence

contrast between individual layers. The competent layers would take on

a class 1C fold form (cf. the fold geometry in the psammites), and the

incompetent layers a class 3 fold form (cf. the fold geometry in the

227

pelites). With increasing flattening the geometric form of all the layers

would approach the similar fold model (Ramsay, 19679 p.434).

The initial folds produced by buckling are usually sinusoidal in

form (see Chapters 3 & 4), and the near—sinusoidal single surface fold geometry that exists in the granulites (b3 is close to zero on plots of b

3 against b1, see figs. 5.19 & 5.20), is the kind of fold geometry that would

be produced by uniformly flattening gently buckled layers. (i.e. the

amplitude of the folds would increase, whilst the 'shape' would remain

unchanged).

It is concluded that the geometric forms of the folds in both the

pegmatitic veins and the granulitic layers are adequately explained by

a hypothesis of fold development that involves shortening parallel to the

layering setting up buckling instabilities that load to the initiation

of folds. Further fold development involves processes of both buckling

and flattening. The more competent layers control the fold development

by buckling, whilst the least coEpetent layers accommodate themselves to

fold forms imposed by the buckled layers.

Because of the different thicknesses and compositions of the layers,

it is not possible to analyse the folding in terms of the multilayer

theories of Biot (1965a) or Ramberg (1961a, 1963b).

There is much greater variation in fold shape in the pegmatitic veins

than there is in the granulitic layers. This is brought out in the

harmonic analysis (see fig. 5.18), and the reason for this difference is

almost certainly due to the greater size of the irregularities (in layer

thickness and in smoothness of the layer boundaries), that must have been

initially present in the pegmatitic veins. The size of the surface

irregularities of layers will be in the order of magnitude of the grain

size; and the grain'size in the pegmatitic veins is about an order of

228

magnitude larger than that in the granulites. Chapple (1968) shows that

the final fold shape is to an extent independent of the size of the

initial layer irregularities, although the degree of dependence of final

fold shape on these irregularities must increase with their size. The

greater the size of the initial irregularities in a buckled layer, the more

variable will the final fold shape be.

The large variation in geometric form of folded layers in a particular

lithology, recorded by thickness/dip variations, is brought out in

fig. 5.7, Among the factors that will contribute towards causing this

variation will be:

a) Local variations in the intensity of flattening.

b) Differences in relative competency contrast.

c) Local thickness variations.

d) For incompetent layers, the geometric form of the folds will differ

near the inner and outer arcs of buckled competent layers.

Very asymmetrical folds of low amplitude (e.g. the folds on the left hand

side of fig. 5.5b) are common on the long limbs of larger folds. These

folds are considered to have developed as 'drag folds' in the sense. .of

Ramberg (1963c).

5.6.1 Discussion.

The interpretation given above, based upon a hypothesis involving

both buckling and flattening, is essentially the same as one proposed by

MukhoPad-hYaY (1964, 1965a) to account for folds of the same generation

in an adjacent area. However, Mukhopadhyay considers that all the, layers,

irrespective of lithology, initially took on parallel fold forms and that

in the course of progressive deformation, the folds in the less competent

layers became more flattened than those in the more competent layers.

This interpretation is considered unlikely in the light of buckling

229

theory and buckling experiments on multilayers (e.g. Ramberg, 1964a), which

predict that the geometric form of folds in competent and incompetent

layers will differ at the initiation of folding. The competent layers

will take up a class 1 fold form, and the incompetent layers a class 3 fold form.

Mukhopadhyay (1965a) does show, however, for a given value of flat-

tening, superimposed on an initially parallel fold, and for a given

percentage error in the measurement of thickness, that a critical angle

of limb dip exists for that fold; at angles of dip less than this critical

value the fold is indistinguishable from a similar fold. He interprets

the 'similar' folds in the Moines in this way.

Ramsay (1962a, p.317) describes folds from Monar in which maximum

and minimum values of thickness of several layers of different composition,

occur in the axial surfaces of the folds. This he ascribes (1962a, 1967,

p.434) to differential flow giving rise to local maximum and minimum

extensions within the fold axial surfaces. Similar geometrical features

have been interpreted by the writer (p.225) as having formed either

directly in the contact strain zones of buckled competent layers, or

indirectly by propagation through a multilayer, of the fold style

initiated within the contact strain zone of a single layer or group of

competent layers. No evidence could be found for inhomogeneities of

the kind envisaged by Ramsay (1967, p.434) to be required to sot up

differential flow.

The classical concept of shear folding, involving laminar flow parallel

to the axial surfaces of the folds (e.g. Carey, 1954; de Sitter, 1964),

is mechanically an unrealistic process (Flinn, 1962), and is inadequate

to account for many of the geometrical features of the F2 folds at Monar.

In particular it does not explain the ptygmatic fold form of many of the

pegmatitic veins. Accepting the view, now generally held, that the

form of ptygmatic folds is the result of tectonic deformation (Rem-

230

berg, 1959) r:.thor than the result of the mode of vein emplacement

(e.g. Wilson, 1952), these folds are incompatible with a classical shear

hypothesis of fold formation, without invoking a strong shortening

parallel to the layering to cause buckling in the pegmatitic veins.

Ramsay (1967, p.430) discusses a modified differential shear hypothesis

in which flow lines converge in the development of similar folds. In

this process, a shortening along the layering is involved that could set

up buckling instabilities which might lead to the initiation of folds by

buckling in the more competent layers in the rock mass. However, the

development of buckles in this way obviates the need to invoke a differential

shear hypothesis in the first place.

If folds in the layered granulites were solely the result of a very

intense shortening along initially non-planar layering, causing accent-

uation of the irregularities present (Flinn, 1962), no regularity in the

size of the folds or in fold wavelength would be exnected. Irregular

folds of the kind described by Ramberg (1964b) would result.

It is concluded that alternative processes of fold development by

differential shear or by finite homogeneous flattening with no buckling,

are inadequate to account for many of the geometrical features of the

Monar F2 folds.

5.7 A STUDY OF WAVELENGTH/THICKNESS IN PPYGMATIC FOLDS IN PEGMiaITIC VEINS

If the ptygmatic folds in the pegmatitic veins are treated as folds

formed in isolated viscous layers in a less viscous matrix, then estimates

of the viscosity contrast between the veins and their matrix, and the total

deformation within the profile plane of the folds may be made. The folds

are assumed to have developed according to a theory based on those of Biot

(1961)t Sherwin & Chapple (1968) and Chapple (1968). These theories

have been discussed in some detail in sections 3.3. and 4.8.

The assumptions on which the basic Biot theory depends are listed in

231

section 3.3, and the degree to which these assumptions are met in the present

study will now be discussed. As the folds are small body forces may be

ignored. It is considered valid to treat rocks undergoing regional meta-

morphism as Newtonian viscous bodies to a first approximation (see section

4.2 ). The assumptions of sinusoidal fold shape and small amplitude concern

the infinitesimal development of folding and may be relaxed in considering

the final form of folds (see Sherwin & Chapple, 1968 ; Chapple, 1968). The

ptygmatic folds analysed are all symmetrical (i.e. the axial surfaces are

almost normal to the enveloping surfaces) and so the maximum compression is

considered to have acted parallel to the layering (or enveloping surface)

throughout fold development. The assumption of plane strain is not wholly

valid (see section 5.8). Lastly, none of the folded veins are truly 'isolates

Some are 'confined' between more massive layers (fig. 5.22), whilst others

are really part of a multilayer (see fig. 5.19c). Compared with the predic-

ted ratio of Wd/t for folds in an isolated layer, the Wd/t ratios of folds

in a 'confined' situation are theoretically reduced (Ramberg, 1963b), whilst

the Wd/t ratios of folds that are part of a multilayer are theoretically

increased (Ramberg, 1961a). However, no systematic differences of this kind

could be detected in the analysed folds, and closely confined folded veins,

and those forming part of a multilayer are not included in the analysis.

Where ptygmatic folds are developed in the pegmatitic veins, the matrix

is invariably pelitic (more rarely semi-pelitic) granulite, and is assumed

to be uniform in composition for the analysed folds.

The 'wavelength' considered is the arc length of the mature folds (see

section 1.2). The mean of the wavelength/thickness (w/t) ratios for a number

of folds is assumed to be the dominant wavelength/thickness (Wd/t) ratio.

The procedure of analysis followed is similar to that described by Sherwin

& Chapple (1968), with slight differences that will be described below.

A total of 157 'half-folds' (segments of a folded vein between

adjacent hinges) were analysed. For each 'half-fold', the arc length

(w/2), measured along the mid-line of the folded layer, and the thickness,

measured at several points along the layer were recorded. The mean value

of thickness was found. Folds wore measured in 17 specimens or photo-

graphs, and mean values of W/2 and t for the folds in each of these were

computed. Mean W/2 is plotted against mean t in fig. 5.21a, and the 157

individual W/t ratios are plotted in histogram form in fig. 5.21b (i).

Fin. 5.21

a) Relationship between mean values of 'wavelength' (the arc length of

nature folds) and thickness in ptygmatic folds in pegmatitic veins.

b) Frequency histograms of WA values for ptygmatic folds in pegmatitic

veins.

(i) For 157 folds. Mean = 3.7

(ii) For 35 individually 'unflattened' folds. Mean = 4.65.

Note: F = No. of folds

Mean, M = Arithmetic ungrouped mean.

1•0

THI

CK

NE

SS

a

b

20

in u .0 u e

0

233

• • i

1 . I . 1 ' -r i .

• —

•

• •

•

• •

1..

• • • • •

•

I I • I • • I I . • II I • 1 1 1 I • 1 •

0

110 2.0 1/2 WAVELENGTH inches

50 -

40 -

F 30-

20-

10-

1 2 3 4

10- F

6. 7 8

• W/T

7

M

1

2 3 4

W/T

Fig. _ „5.22

To illustrate the confinement of a thin ptygmatically folded pegmatitic

vein, A, between more massive pegmatitic sheets.

Heavy stippling Pegnatitic veins

Pelitic rock.

Fig. 5.23

Theoretical variation of amplification with w/t for a viscosity ratio of

10 (between a single viscous layer and its less viscous matrix) at a

value of shortening given by S = 3.0.

No

?I

20

1"

235

2 4 6 8 10 12 14 16 18 20

WAVELENGTH / THICKNESS

236

These figures show that the averaged relationship between W/2 and t is

near linear (fig. 5.21a), although the variation displayed by the

individual W/t ratios is large. A linear relationship between W and t is

predicted in equations 3.1 and 3.3 (where S is constant).

The mean W/t ratio is 3.7 (fig. 5.21b (i)). An estimate of ii1t12

= 1.2 is obtained by substituting Wd/t = 3.7 into equation 3.1. This

viscosity contrast is so low that very considerable layer parallel

shortening must have occurred (Blot, 1965a, p.425) before any significant

folds developed.

The folded veins are all thickened in the hinges and most appear to

be geometrically similar to parallel folds, flattened by an average

amount of strain jX2/ = 0.4 (fig. 5.7). It is assumed that a

flattening component of strain followed the buckling. Although flattening

and buckling probably proceeded together (see section 3.6), it is likely

that flattening became important when the folds tightened up and the

matrix material became extruded from the inner arc regions (see Chapple,

1968, p.62).

35 'half-folds' in 7 specimens were individually 'unflattened' by

graphical means (the value of the flattening strain was determined from

a tj/a graph). The W/t ratios measured in the unflattened folds (fig.

5.21b (ii)) have a mean value of 4.6. By substituting Wd/t = 4.6 into

equation 3.3 an estimate of 111/112 = 2.4 is obtained. Thus 'unflattening'

has a slight effect in increasing the mean Wit value, and hence the

estimate of the viscosity contrast.

The average limb dip of the unflattened folds is 53°, and values of

dip range from 20° to 90°. It is assumed that the folds developed from

sine waves of 15° limb dips to their present shape (see fig. 5.19a & b)

without change in arc length (see section 4.6 ). It is further assumed

that during progressive deformation the shapes of the folds followed the

'path' of progressive changes predicted by Chapple (1968) for folds of

2nt X - d w

1.35

111412 (Sherwin &

Chapple)

237

the dominant wavelength. The development of the 'average' fold, involving

an increase in limb dip from 15o to 53

o would be accomplished by a bulk

shortening of JX2A i = 0.65.

It remains to estimate the amplification required to produce folds

with 15° limb dips. Applying the theory of Sherwin & Chapple (1968), and

using a similar argument to theirs, it is considered that initial

irregularities in the veins would have had limb dips in the order of 1°,

and that amplifications of 10 to 20 would have been necessary in order to

develop folds with 15° limb dips (see the discussion in section 4.8 ).

Taking the value of fold amplification to lie between 10 and 20, and

taking the value Wa/t . 4.6, estimates of both viscosity contrast (between

the pegmatitic veins and the matrix granulites) and of shortening (up

until the 115° limb dip stage') may be read from the graph drawn in fig.

3.1, relating amplification, viscosity contrast, shortening and dominant

wavenum'Jer. The data and results are tabulated below.

T.,..BLE 5.1 D 1N V.J.LTES Tim UNFLTTENED FOLDS

Strain W/t Ratio - 3 Unflattened Flattening Folds

0.42 4.65

Strain Ratio - 2 Assumed Shortening Amp.

150 53°

Limb Dip 111/112 Unflattened (Biot) Folds

2.4 530

Strain Strain Ratio - 1 Ratio - Shortening Total -7x,-15° 1 x 2 x 3

0.65 10 - 20 10 0.33 0.09

238

The strain ratio is defined ast%.2A

1. The value of the total shortening

(equivalent to a strain ellipse ratio of ca. 11:1) of the bulk strain

in the profile plane of the folds, is found by multiplying together the

values of shortening for the three stages of fold development distinguished.

The estimated value of the viscosity contrast, read from the graph in

fig. 3.1, is about four times the value estimated when layer shortening

is not taken into account. By closely following the procedure of analysis

described by Sherwin and Chapple (without considering the stage of fold

development where the arc length remains unchanged), an amplification

of about 100 would be considered necessary in order to form folds with limb

dips of about 50°. The estimated viscosity contrast would then be about

16 and the total shortening aboutjA2A1 = 0.1.

The large variation in the measured values of Wit (fig. 5.21b (i))

indicates the low selectivity of the folding. This is consistent with

the low values of amplification considered necessary to produce folds

with 15° limb dips, since Biot (1961) shows that selectivity depends only

upon amplification to a first approximation. The theoretical variation

of amplification with W/t for a viscosity contrast of 10 at a value of

shortening of/A.2A I = 0.33 (values estimated in the present analysis)

is shown in fig. 5.23. Sherwin & Chapple consider that a frequency

histogram of N/t values should reflect the broad features of a graph

relating amplification to Vt. The histograms in fig. 5.21b (i) & (ii)

may be compared with fig. 5.23.

The mainly untestable nature of the assumptions on which the theory

is based, and the variability of the measured factors both indicate the

limitations of the analysis. The results are therefore considered to

give rough estimates only of the viscosity contrast between the pegmatitio

veins and the politic granulites, and the total amount of shortening

within the profile plane of the F2 folds.

239

5.8 l T ANALYSIS OF DEFORMbn LINEATIONS.

There are a series of linear structures developed parallel to the axes

of the early folds (the first folds of Ramsay, 1958) at Monar, that are

deformed about the minor F2 folds. The geometry of these deformed

lineations provides useful information concerning progressive fold develop—

ment. The commonest linear feature is a striping on Toliation surfaces

(plates 4 & 5). This is an intersection effect of the early cleavage with

the foliation.

The attitudes of deformed lineations were measured at several points

around a number of minor folds. Data for two examples has been plotted

on a stereogram in fig. 5.24a. It was not found possible to measure the

loci accurately enough to compare them with theoretical loci related to

various folding processes (Ramsay, 1967, Chapter 8). However, individual

loci are almost contained within planes. In the field, it was possible

to visually 'fit' a lineation locus into a plane (see Ramsay, 1967, p.472),

where the locus crossed the two limbs of a single fold, but this was not

possible where the lineation crossed several folds (see plate 5).

Assuming, as a first approximation, that individual lineation loci

are contained within planes, the intersections of the fold axial surfaces

with these planes may be found. These intersections are equivalent to

the 'a' directions of a classical simple shear hypothesis. They are

referred to here as 'a' directions without any genetic implications. On

a stereogram the 'a' directions show as a cluster of points within the

field containing the F2 minor fold axes (fig. 5.24b). Two distinct

types of lineation locus may be distinguished, depending on the relative

orientation of the F2 fold axis to the 'lineation locus plane' (see fig.

5.25.). The first type is the commonest, where the fold axis is near

vertical; and the second type is local, and is only found where F2 minor

folds are superimposed on the gentle limb of an F1 minor fold. Viewed

towards the north—east, the sense of relative displacement of the lineation

Plate 4 Coarse Fl lineations deformed around F2 folds.

(Road section, N. end of main dam).

Plate 5 Fl lineations on a foliation surface deformed by

F2 folds. (Road section at E. end of little dam).

The camera is pointed such that the lineations in

the centre of the photograph most nearly 'fit' into

planes.

Kils2.JJA

Deformed Fl lineations (plotted on a Wulff stereographic net).

a) Plots of lineations deformed around two examples of F2 minor folds.

Dashed lines refer to ex. 2, and solid lines to ex. 1. The fold axis,

F, is common to both F2 folds.

A.S. = Axial Surface

The large circles are the intersections of the 'best fit' great

circles (through the plotted lineations) with the axial surfaces in

each example.

b) 'Best fit' great circles for several individual examples of Fl

lineations deformed around F2 folds. The intersections of these

great circles with the axial surfaces in each case are marked by open

circles.

A.S. = Average axial surface

Great circles 1 & 2 are for the data shown in a).

Great circle 3 is the only example of deformed lineations in a type 2

situation (see fig. 5.25).

The dashed line encloses the field of plot of the F2 fold axes. X,

Y and Z are the inferred positions of the axes of finite bulk strain.

243

` • \ /

\ / • o

2 ‘ \o + • o. e F •, z • ... /

a

b

Fig. 5.25

Two types of deformed lineation locus.

Type 1 Steep F2 fold plunge

Type 2 Gentle F2 fold plunge.

a) Sketches of the observed patterns of lineation locus.

B = Axial surface of the F2 folds.

b) View normal to the axial surface (B), towards 3200.

AA = Trace of the 'lineation locus plane' on the axial

surface.

F = Trace of the F2 fold axis.

Arrows indicate the sense of rotation of the traces in the axial

surfaces of the folds.

A A

245

Type 1

Type 2

a

A A

Type 1 Type 2

246

between the antiformal and synformal hinges is reversed (fig. 5.25) in

these two types.

On lineation geometry alone these patterns could be consistently

interpreted according to the classical simple shear hypothesis of fold

development. However, an alternative explanation will now be given

which is compatible with the interpretation of fold profile geometry

given in section 5.6.

It must be assumed that the F2 axial surfaces are parallel to a

principal plane of the bulk strain ellipsoid (considered for a volume of

rock that is much larger than the size of individual folds). This is

not generally agreed upon for folds formed by buckling (e.g. Ghosh, 1966,

1967; Singh, 1967); but the symmetry of fold geometry about the axial

surfaces, in terms of thickness/dip variations, and the presence of an

axial surface schistosity suggest that this assumption holds for the F2

minor folds at Monar.

The maximum shortening direction (Z) of the bulk strain ellipsoid

must be perpendicular to the axial surfaces of the folds. The deformed

lineations lie at low angles to F2 in the limbs of the folds but are

almost normal to F2 in the hinges. The lineations must therefore have

been normal to F2 in the hinges initially, or they would have rotated away

from this position in the course of the deformation (Flinn, 1962). The

'instability' of lines lying close to Z in progressive deformation may

account for the considerable s7)read in trend of the 'lineation locus planes' ,

(fig. 5.24h). The present pattern of relative displacement of the

lineations in adjacent synformal and antiformal hinges must be due to the

rotation of the minor F2 fold axes within their axial surfaces during

deformation to produce a shear component across the fold profiles. The

traces of the 'lineation locus planes' and the F2 fold axes on the axial

surfaces may be treated as line elements that have undergone homogeneous

247

strain within the XY plane of the bulk strain ellipsoid. The attitudes

of the traces of the 'lineation locus planes' and fold axes on the axial

surfaces are shown in fig. 5.25b, for the two types of lineation locus

distinguished. In either case both line elements must have rotated

towards the X direction of the bulk strain ellipsoid, which therefore lies

in the shaded zones of each of the figures in fig. 5.25b. The trace

of the 'lineation locus plane', or 'a' direction, is the only line in

common to both shaded zones. Assuming that the bulk deformation within

the region is homogeneous, the 'a' direction must be approximately

parallel to the X axis of the bulk strain ellipsoid.

Examination of the stereograms in figs. 5.24b and 5.2 shows that .

the 'at directions lie in the centre of the field in which the F2 fold axes

plot. Fold axes plotting either side of the 'a' direction cluster, are

those of the two types distinguished in fig. 5.25. The near-vertical

fold axes (type 1) lie very close to X. In one of the measured examples,

the angle between the fold axis and the 'a' direction was virtually

zero, and the deformed lineation lay on an almost unfolded surface (see

Ramsay, 1967, p.473).

Only a small buckling component would be necessary in order to initiate

folding, and thus form the fold axes and 'lineationlocueplanes t , which would subsequently behave more or less as passive elements throughout

progressive deformation. A buckling process equivalent to that of

oblique flexural slip (Ramsay, 1967, p.396) could initiate the lineation

locus in a plane or surface nearly parallel to the X direction of the

bulk strain ellipsoid, and at an acute angle to the fold axis. This

would account for the parallelism of the 'a' directions, irrespective of

the attitude of the fold axes.

From the data available, it is not possible to find the k- value

(Flinn, 1962) of the finite bulk strain. However, because lines have

rotated in the XY plane, k 0; and because the axial surfaces are

248

constant in orientation and there is a well-developed schistosity, k

The interpretation of the geometry of the deformed lineation loci

is more or less consistent with the interpretation of the geometric foxqn

of the fold profiles. All the folds analysed in profile section were in

the group with uear-vertical axes. It is apparent that the mean profile

plane of these folds is not a principal plane of the finite bulk strain

ellipsoid. Therefore the estimates of the deformation in the fold profile

planes made in sections 5.6 and 5.7 will be invalid (Makhopadhyay, 1965a;

Ramsay, 1967, p.415). It is impossible to compute the true strain in

the profile planes from the fold geometry, but because the profile planes

are within a few degrees of being normal to X, it is suggested that the

estimates of deformation made in sections 5.6 and 5.7 give a qualitative

indication of the strain in the YZ plane of the finite bulk strain ellipsoid.

5.9 CONCLUSIONS

A) A detailed geometrical analysis of the minor F2 folds shows that

systematic overall differences in the geometric forms of folds can

be related to differences in layer composition.

B) The total variation in the fold geometry of both single surfaces

and layers is large.

C) The fold geometry is more variable in the pegmatitic veins than in

the granulitic layers.

D) The relative competencies of different lithological types is found

to be: pegmatitic veins > psammites semi-pelites pelites.

E) The fold geometry is adeauately explained by a hypothesis of fold

development involving processes of buckling and flattening.

F) The viscosity ratio of the most competent rock type (pegmatitic

qualtzo-feldspathic veins) to the politic granulites is considered

to be about 10:1.

249

G) The steeply inclined F2 fold axes are almost parallel to the

principal extension axis (X) of the finite bulk strain ellipsoid,

and the profile planes of the folds are nearly parallel to the YZ

plane of the finite bulk strain ellipsoid.

An estimate of the two dimensional strain in the F2 fold deformation,

in the mean profile plane (roughly equivalent to the YZ plane of the

finite bulk strain ellipsoid) of the steeply inclined folds is

1- 2/ x - 0.09 •

250

CHAPTER 6

AN ANALYSIS OF MINOR FOLDS IN PART OF THE MiGGIA NAPPE, TICINO, SWITZERLAND

6.1 INTRODUCTION

An investigation of fold geometry was carried out in a small part of

one of the Lower Pennine nappes of the Swiss Alps on the same lines as the

study of folded Moinian rocks in the Scottish Caledonides described in

Chapter 5. The rocks in both areas are highly deformed and metamorphosed.

Unlike the situation at Monar, the structure and exposure of the rocks

studied here enable a geometric analysis of several successive generations

of folds to be made. A geometrical comparison of these fold phases forms

the main part of this chapter.

The selected area for study lies just north-east of Lago Sambuco and

below the ridge separating Valle Leventina from Val Sambuco (fig. 6.1),

where the rocks are superbly exposed on glaciated surfaces.

The rocks in this area are basement gneisses, forming part of the

Maggia nappe, which lies near the base cf the exposed nappe pile of the

whole mountain belt.

Hasler (1949) has described the geology of the area around Sambuco,

including the site of the present study, mainly from a petrographical/

petrological viewpoint; his tectonic analysis is brief and does not provide

an adequate frame of reference for the present work. The work of Heim (1921)

still provides one of the best descriptions of the major tectonics of the

Alps.

The area was chosen because several fold phases are developed more or

loss co-axially, about an almost vertical axis, to give near profile sections

on the glaciated outcrop surfaces.

The general geology of the area is described in this section (6.1) and

section 6.2. A detailed geometrical fold analysis is described in section

6.3, and the fold geometry is interpreted in terms of folding processes in

section 6.4. On the basis of an analysis of wavelength/thickness ratios,

in thin buckled layers estimates of viscosity contrasts and deformation in

the F2 folding episode are made in section 6.5.

Fir. 6.1

a) Geological nap of the Sambuco area

(After Hasler, 1949). The region selected for detailed study is

narked by diagonal shading.

b) Map to show the location of nap a).

252

„.... ..,. ...... .."- ....

r- e... - • ...- ,••••• -... •-.- ... ..... ..... .,. ,..... ,.._

P. Cor no •-•

r A ,- • \ s -....

( • .... M. .4. . S.

< " C

( r' -,... ... "I ' 1- -1 1.

.., , ---'-<,,•,) ' r tc ( . -1 " ‘ N 1 • .-... Ala • - , t ''s

1 / P. Scheggia

1-

99/0 k t N • .. ). \ t I 1A 1 .‘ ''.7. . A

1 t- 1.-. • ) 1 -N. • 1

.. '.•..., .4p.- N-. .. A. ., .‘: _ \. \ .N* • N ' - ..,:‘• A -,,,„ 1.. 1. N..

1.. I_ • I .

A. k . ' . N N.:• • . A.,

-. ‘ -'-• * N

.. -,

COVER

Bundnerschiefer

Trios

FUSIO 0

+ 4- 4- +

P. Sciresa •

+

.... • .

.. 1 .. logo Tremorgto •

• • -•••• •

• • Pso di Campolungo

•

+ +

BASEMENT

Undifferentiated Gneisses arid Schists

Banded Gneisses

Granite Gneiss

a

253

6.1.1 Lithology and Mineralogy.

The basement rocks of this area are included in the group of 'banded

gneisses' recognised and mapped by Hasler (1949). They are variable

in composition and for the most part strongly banded; typically dark layers

of amphibolitic rock alternate with white layers composed of quartz and

feldspar.

Quartz, plagioclase and hornblende are the three principal minerals

that, in varying proportions, go to make up bands of different composition

and colour index. Chlorite and more especially apidote-clinozoisite

frequently make up a significant .proportion of the dark bands, and appear

also in the light bands. Biotite, usually rare, locally becomes an

important constituent of the dark layers. Garnets, and small crystals

of zircon and opaque minerals also occur.

The texture of the quartz-feldspar layers is granular and that of

the darker layers usually schistose.

For the purposes of fold analysis, the only practical division of

layer types is into 'dark', 'intermediate' and 'light', and in terms of

composition this is generally equivalent to:

dark predominantly hornblende epidote, biotite.

intermediate - between 'light' and 'dark' in dark mineral content.

light predominantly quartz and plagioclase.

Several cross-cutting basic dykes and other intrusive bodies appear

to have been emplaced between the first and second phases of deformation

described below.

6.1.2 Metamorphism.

The rocks in the Sambuco area have been affected by a strong Alpine

metamorphism (the metamorphic culmination of the Alpine orogeny was

254

centred in southern Ticino), the main part of which postdated the

formation of the nappes (Wenk, 1962), attaining garnet Iganite grade

over the general region of north Ticino. The Mesozoic sediments

(Bundnerschiefer), incorporated in the nappe movements, mantle the Maggie.

nappe (see fig. 6.1) and are also strongly metamorphosed.

The preferred orientation of hornblende in the banded gneisses is

for the most part parallel to the axial surfaces of the first phase folds

recognised here; and an axial surface schistosity is locally developed

in the second folds. The granular texture of the quartz and feldspar,

and the more or less strain-free nature of all minerals shows that

recrystallisation had gone on after deformation had ceased.


Three distinct phases of Alpine folding in the banded gneisses have

been recognised on the basis of the interference relationships of small

scale structures (plates 6 & 7). The interference patterns are all

Type 3 of Ramsay (1967, p.530).

The observable relationship between the basement gneisses and the

Bundnerschiefer around the Sambuco region and to the west around

Cristallina is complicated, suggesting a complex major structural

pattern. For present purposes, the relationship between the small scale

folds analysed in this study and the major structures is relatively

unimportant, and no proper attempt to analyse the major structures has

been made. It is not known whether the first generation folds observed

here were developed during or later than the nappe forming movements.

In the Lebendun nappe and Bundnerschiefer exposed near Basodino

(about 15 km west of Sambuco) Higgins (1964) has established a succession

of fold episodes similar to those described here; and in the Bundner-

schiefer and Mesozoic cover rocks of the Gotthard massif to the north of

Plate 6 Near—isoclinal P1 folds refolded by F2.

(Grid. Ref. 69491478).

Plate 7 P2 folds refolded by P3.

(Grid. Ref. 69441479).

256

the region, polyphase folding has been described by Chadwick (1968) and

Sibbald (personal communication).

Orientation data for axes and axial surfaces of the three fold phases,

measured in the study area, are plotted on an equal area stereogram

in fig. 6.2. The fields in which the fold axes of each phase plot are

coincident, and the folds of all three phases arc therefore more or less

coaxial about a rather variable, near vertical axis. Taken together,

the axial surfaces of the fold phases show considerable variation in trend,

from Eql through SE-NW to S-N.

The first episode of deformation led to the production of tight

isoclinal folds, F1 (plate 8), a preferred orientation of hornblende

(± biotite) parallel to the axial surfaces of these folds and an inten-

sification of the banded structure of the gneisses, with boudinage of

some of the amphibolitic bands.

The folds most prominently developed in the region are those of the

second phase of deformation, F2 (plate 9), and these deform the Fl

isoclines (plate 6). The axial surface trends of the F2 folds are

somewhat variable on a small scale, partly an intrinsic feature of the

fold development and partly due to refolding during the later deformation;

and on a larger scale show slightly greater variation due mainly to

refolding. The overall trend is roughtly 1300 ± 15°. No indication

of the presence of a single major structure is given by the symmetry of

the minor folds; there are frequent reversals of symmetry, and a

considerable number of folds with 'MI symmetry.

The third generation folds, F3 (plate 10), are developed locally,

and display symmetry or are asymmetrical with an 'S' sense throughout

the area. The axial surfaces trend between 155° and 180°. Their size

(see section 1.2) is small (few larger than several cm) and the effect

Fig. 6.2


investigation.

Poles to Axial surfaces.

triangles

dots

circles

Fold Axes.

0

Fl

F2

x F3

Fl

F2

F3

co an N

Plate 8 Isoclinal Fl folds (Grid. Ref. -9461480).

Plate 9 F2 folds (Grid. Ref. 69471481). Crenulation

cleavage is developed in the dark hornblnHe rich

layers.

2 6 0

Plate 10 F3 folds (Grie. Ref. 69441479)•

261

they have on reorientating the earlier structures appears to be limited.

There is evidence of a further phase of folds with axial surfaces

roughly normal to those of F3. These structures are very local and their

relationship to F3 could not be determined. They do not enter into the

present discussion.

6.2.1 Mineral Fabric

The hornblende fabric parallel to the axial surfaces of Fl folds has

been mentioned above. A penetrative axial surface schistosity, formed

by the alignment of hornblende (and biotite when present) is present only

locally in the F2 folds. A common fabric, parallel to the F2 axial

surfaces, is a crenulation cleavage of kinking. It is only found in

the dark hornblende rich layers; tightly packed grains of hornblende

crystals initially parallel to the banding are responsible for its

localisation.

The development of F1 and F2 schistosities and the coaxial refolding

give rise to a common penetrative intersection lineation that is often

strongly developed parallel to the fold axes. The longest dimensions

of hornblende crystals often lie in this direction.

6.3 DESCRIPTIVE GEOMETRY

In many respects the variations in fold shape within and between

individual folded layers brought out in this study by use of harmonic

analysis, isogon plots and plots of thickness variation with dip, are

similar to those described in some detail in the Monar study. For this

reason individual examples of analysed folds are kept to a minimum, and

most data presented in this section, are in a synthesised form to Dhow

most clearly the overall differences in fold geometry between the fold

phases.

262

The direction of the 'common' fold axis is fairly constant, and no

systematic areal changes in either fold shape or fabric development were

observed within the region. For these reasons it is considered valid to

treat the area as a single homogeneous structural domain, although on

a smaller scale inhomogeneities may be apparent (e.g. the local development

of P3 folds).

Because many of the folds are small, irregularities in layer surfaces

on the scale of the grain size are significant; and so t a, rather

than0 a is used to record changes in layer shape with dip, being a less

sensitive parameter (see section 2.5.5).

Best fit straight lines to the values of -00,2 and cos

2a were

calculated for every measured fold and the value of the intercept of the

best fit line is used here as a parameter of fold shape (see section 3.5).

6.3.1 Isogon Plots and Thickness/Dip Relationships

Examples of F2 folds are taken to illustrate the irregular and

apparently inconsistent variations in fold geometry related to composition.

The same phenomena can be observed in all three fold phases; F2 however

affords the best examples.

Isogon plots are constructed for selected F2 folds in figs. 6.3a and

6.4a. In the former, the pattern of the isogons in the dark layers is

consistently convergent from outer to inner arc (class 10 geometry) and

in the light layers consistently divergent (class 3 geometry). In fig.

6.4a the patterns are reversed; the geometry of the isogons in the

light layers is class 10 and in the dark layers class 3.

These geometrical differences are equally apparent in plots of t'a

against cos2a . Fig. 6.5b is such a plot for both limbs of a fold in

adjacent light and dark layers. The two plots relating to either limb

Fig. 6.5

F2 folds (grid ref. 69491476).

a) Isogons at 30° dip intervals on the fold profile.

Heavy stipples • • • • Massive hornblende layer.

Light stipples • • • . Hornblende rich layer.

No stipples • • • • Quartz—feldspar layer.

2 b) Frequency histogram of the intercept values (on a t'a /cos

2 a graph)

for 35 folds in the profile shown in a).

Shaded section • • • • For folds in hornblende rich layers.

Unshaded section For folds in iCti-F layers.

F = Frequency

Note: Grid references for this and subsequent figures will be

found on the National Maps of Switzerland.

04 06 -04 -02 0 02

INTERCEPT VALUE

264

b

F

0

10-

F2 folds (G.R. 69501480).

a) Isogons at 30° dip intervals on the fold profile.

Stipples Hornblende rich layers.

No Stipples Quartz-feldspar layers.

(The isogons are drawn at 120° intervals of dip on layer S ).

b) Adjacent folds to those in a).

c) Plots of t'2 against costa for various folds in a) and b). a

d) Frequency histogram of intercepts (on a tla2 /cos2a graph) for a

number of folds in a) and b).

Cross-ha±ched section ... Folds in hornblende rich layers.

Diagonal-lined section ... Folds in layers Y and Z.

Unshaded section Folds in Q-F layers other than Y and Z.

e) Plot of thickness t against half wavelength W/2 for folds in

quartz-feldspar layers.

10

0 -02 0

1/4"

b

N.NNN.1 0-5

INTERCEPT VALUE 1-0

d

1/4"

• Limb 1 0, I; 2 + Limb 3 x I; 4 ✓ Limb 5 4, ii 6

0

C

03 inches

e 0.1 0-2

W/2

10

t "c,

0-5

Inches

0 2

T

0.1

0

•

0

0

0

0 0 21

30

60

90

ANGLE OF DIP a,

6

F2 folds (G.R. 69441480).

a) Fold profile.

Stippling Hornblende rich layer

No stippling Quartz—feldspar layer.

2 b) Plots of t'a against cos

2a for various folds in a).

O 30 60 90

" 268

a ,

10

0.5

Fold A \.

\ \c, \\ • \

° \ \ • -

+ Limb 1 Layer X • it 2 • Limb 3 Layer Y

4 •

269

of the fold in the dark layer have slopes of less than 1.0 (fold class 1C),

whilst plots for the two limbs of the fold in the light layer have slopes

greater than 1.0 (fold class 3)• In fig. 6.4c the position of the

plots for folds in light and dark layers (layers W and X respectively)

is reversed. The plots imply a class 1C fold geometry in the light

layers and a class 3 fold geometry in the dark layers. This situation

holds for most of the other folds in this specimen. However, data for

the light layer Z, indicates a class 3 fold geometry that is out of keeping

-with the general pattern of fold geometry in this particular .sspecimen.

For most folds analysed the variation of t'a 2 with cos

2 a is almost

linear (figs. 6.4c, 6.5b), and thickness/dip variations for either limb

of a fold in a single layer are usually aimilar.

The folds in a single layer of a particular composition are usually

consistent in the type of isogon pattern displayed (i.e. all are class 1C

or all class 3). However, in some layers, such as layer Y in fig. 6.4a,

the type of isogon pattern observed is inconsistent between folds. In

layer Y (a light quartzo-feldspathic band) the outer arcs of the folds are

mostly rather rounded compared with the inner arcs, but in several folds

the outer arcs are very sharp in the crests and indistinguishable in

shape from the inner arcs.

A thin section study of several folds with dissimilar geometry,

in both light and dark layers, revealed very slight differences in

mineralogy and texture that could be associated with the geometrical

inconsistencies described above. Where biotite is present in the dark

layers the folds invariably take on a class 3 geometry. The light layers

in which the folds show a class 3 geometry tend to be richer in feldspar•

than the light layers in which folds display a class 10 geometry. Folds

in the dark layers with a granular texture more commonly take on a class

1C geometry than do those in dark layers having a good schistose texture.

270

Also, folds in dark layers with a very large proportion of hornblende

usually take up a class 1C geometry.

Thickness/dip measurements were made on a total of about 260 F2 folds,

130 F3 folds and 80 F1 folds.

The range in values of the intercepts of the best fit straight lines

on a a2/cos2a graph is shown for the F2 folds in figs. 6.3 and 6.4.

Treating the area as a single homogeneous domain, it is considered

valid to construct frequency distributions of intercept values for all

the measured folds to find the general nature of the fold geometry. This

has been done in fig. 6.6 for three categories of layer; dark, intermediate

and light, for each of the fold phases. There is no consistency in

either sense or amount of asymmetry in these distributions, and most

are fairly symmetrical about means that are close to zero. In the case

of F2 for which most data is available, the distribution for folds in

the light layers has a greater variance than the distributions for folds

in the dark or intermediate layers.

The apparent lack of any systematic relationship between the

mean -0a2/cos2a intercept values and composition, reflects the inconsistent

behaviour of individual fold geometry described above.

Comparison of the Three Fold Phases

The fold phases may be compared by computing the total frequency

distributions of intercept values for folds in each phase, neglecting

compositional differences. This has been done in fig. 6.7. The three

distributions are all fairly symmetrical about small positive means.

Their variances are, however, strikingly different. The distribution for

F1 folds, with very low variance, indicates a close approach to true

similar fold (class 2) geometry in all the measured folds. The highest

variance is recorded in the distribution for F3 folds, and an intermediate

Fig. 6.6

Histograms of intercept values (on a 2 /cos2ia graph) for all measured

folds, distinguished according to fold phase and layer 'composition'.

a) F3 b) F2 c) Fl

In each fold phase:

(i) for 'dark'layers (ii) for 'intermediate' laters

(iii) for 'light' layers

F =

IV =

No. of Folds

Frequency

Intercept Value

Mean Variance

a) (i) 46 0.035 0.127

(ii) 44 —0.048 0.080

(iii) 40 0.193 0.120

b) (i) 90 0.070 0.017

(ii) 50 0.010 0.008

(iii) 114 0.095 0.066

e) (i) 12 0.017 0.002

(ii) 20 0.007 0.004

(iii) 45 0.045 0.004

Short thick lines on each histogram locate the position of the

(grouped) mean.

10 F

rI 1.0

5

- 1.0 - 0.5 0 0.5

I

F

0 I-1 - 1.0

r - 05 0 05 1.0

101

IV

F 10

30

F 20

10

0 -0.2 0

IV

0.2 - 0.2

IV

( i i i)

71--1 0.2

-10 -0.5 0 0.5

1.0 IV

b 50

40-

30-

F

20-

10-

0

-1 0.5 1.0

IV

0.5 1.0 I

- 0- 5

20.

F

10-

0 - 0 5

20-

F

10-

0 - 0.5 0 0.5 1.0

I

F 10

0

C

-02 0 2 0 IV

)

0.2

Fig. 6.7

2 Total frequency histograms of intercept values (on a tta /cost graph)

for folds in each fold phase.

No. of Folds Mean Variance

P3 130 0.052 0.114

F2 254 0.069 0.037

Fl 77 0.032 0.004

10- M I 0

-1.0 - 0.5 0 0:5 1.0

1

30-

F% 20-

F3

IV

40

F% F2 20-

0 - 0:5

M 1

0 0.5

p IV

11=111!11111111

1.0

60-

F% 40-

F1

20 -

o - 05 0:5

IV

275

variance in the distribution for F2 folds.

The value of a mean intercept close to zero in each case indicates

an average shape that approaches a true similar form. This is a

necessary characteristic if folds are to persist for any distance (Ramsay,

1967, p.433) without radical changes in overall shape.

A slight bias towards measuring folds with class 1C geometries

existed, accounting for the small positive means of these distributions.

6.3.2 Harmonic Analysis .of Fold Shape.

Single folded surface shapes of a number of folds in all three fold

phases, were studied using the visual method of harmonic analysis described

in section 2.5. The same material was used, and approximately the same

number of folds were analysed, as in the thickness/dip study described

above. Individual 'quarter wavelength units' were analysed separatelyl-

and a distinction was made between inner and outer arcs of folds in

both light and dark layers. Natural fold shapes were matched against

the ideal shapes in fig. 2.14 , and a dot was recorded in the appropriate

cell of a box diagram of 'shape' against 'amplitude' (fig. 2. 15 ).

Folds whose 'shape' or 'amplitude' was estimated to lie between those of

two of the figured forms in fig. 2.14 were represented by points on the

lines between boxes. In the case of F1 folds an additional category

of amplitude, 6, has been introduced, in which the ideal fold shapes have twice the amplitude of those in category 5.

An example of the visual analysis of several F3 folds is shown in

fig. 6.8.

Ignoring differences in composition and the distinction between

inner and outer arcs, the combined plots for all the analysed folds should

give an indication of the type and variability of fold shape within each

Fig. 6.8

Ln example of visual harmonic analysis.

The fold profiles are of F3 folds (G.R. 69491475).

i . inflexion point

h = hinge point

Stipples • • • • Hornblende/biotite rich rock

No Stipples • •• • Quartz—feldspar layer

In the plot of 'shape' against 'amplitude':

0 .... Outer arc of Q-F layer

X .... Inner arc of Q—F layer

5

AMPLITUDE

2 3 4

277

1/2

1

A

B

w EL C

D (I)

E

F

(Ds

9 11 X ° 70 10

80 X

X 2 X

1

6X

X 3

13 X

12 X

4

278

fold phase. This synthesised data is best presented in frequency histograms

of both 'shape' (categories A - E) and 'amplitude' (categories 1 - 6)

(fig. 6.9). The terms 'mean' and 'variance' cannot properly be applied

in this case, and with this in mind the histograms drawn in fig. 6.9 show:

a) the 'spread' in the distributions of both 'shape' and 'amplitude'

is greatest for the F1 folds, and least for the F3 folds.

b) the peak category of 'shape' is either D or E for all three fold

phases. The sharpest peak occurs in category D, in the F3 'shape'

distribution.

c) more rounded fold shapes become increasingly common in progressively

older fold phases (i.e. the frequency of fold shapes plotting in

categories C, B and A increases in the older folds).

d) the overlap between the 'shape' distributions is large, that between

the 'amplitude' distributions is considerably less.

e) the 'mean' fold amplitude (and the peak category in the distributions)

increases in progressively older fold phases.

The differences in geometry between inner and outer arcs of folds is

illustrated with reference to F2 folds. With this distinction made, data

for 'shape' of folds in both dark and light layers are presented in

histogram form in fig. 6.10. It is apparent that there is little overall

difference in fold shape between inner and outer arcs for either type of

layer. Neither are there any overall differences in fold shape between

folds in dark and light layers. This reflects again on the inconsistent

fold geometry in both dark and light layers. There is virtually no

difference between the amplitude distributions (not shown) for inner and

outer arcs of folds, or for folds in dark and light layers.

Fig. 6.9

Harmonic Analysis.

Frequency histograms of 'shape' and 'amplitude' of folds in each fold

phase. The totals of folds are:

Fl 81

F2 261

F3 154

Fig. 6.10

Harmonic Analysis.

Couparison between 'shape' distributions for inner and outer arcs of

folds in 'dark' layers (a) and 'light' layers (b). All are F2 folds.

Numbers of folds are:

Inner arcs Outer arcs

a) 72 92

b) 89 90

F%

50- Outer Arcs

\\.

F%

280

F3

Fl

3 4 5 AMPLITUDE

ABCD

SHAPE

AB C D

SHAPE

a

281

6.3.3 Refolded Folds

The more or less coaxial refolding enables a study to be made of the

variation of interlimb angle of an early fold deformed about a later

structure.

Several examples of F1 folds refolded about F2 have been analysed,

and two of these have been selected to illustrate different effects. In

fig. 6.11a the interlimb angle of the F1 fold appears to attain a maximum

value of about 70° near the F2 fold crest, and a minimum value of about

10° on the left hand limb of the F2 fold. In fig. 6.11b there is little

change in the value of the interlimb angle of the F1 fold from one limb

of the F2 fold to the other. The maximum value of the interlimb angle is

about 25°.

Strikingly different effects in the modification of F1 folds at

different positions within a medium sized F2 fold are apparent in

plates 11 & 12. In plate 11 the axial surfaces of the Fl folds are

almost normal to that of the F2 fold; the F1 folds are open, very

irregular and complicated by the development of small F2 folds. In

contrast, the axial surfaces of the Fl folds in plate 12, are almost

parallel to that of the F2 folds,and the F1 folds have an exceedingly

tight and near similar form.

Unfortunately, measurement of Fl folds in zones where they have been

'opened out' by the later deformation, as in plate 11, is extremely

difficult, and for this reason the Fl folds selected for the detailed

analysis described in the previous section are those whose axial surfaces

are almost parallel to the axial surfaces of the F2 folds. These are

the only Fl folds suitable for analysis.

6.4 INTERPRETATION

In the isogon patterns (figs. 6.3a & 6.4a) and plots of the variation

FiR. 6.11 Examples of Fl interlimb angle variation around F2 folds.

The folding is coaxial. Fl axial surface traces are shown by solid

lines, F2 axial surface traces by broken lines.

a) (G.R. 69471479)

b) (G.R. 69501477)

0 90 60 30

left Q 30 60

right

Fl

80

INTE

RLI

MB

AN

GLE

a 90

a

•

•

• •

•

• •

•

• • •

. • • • •

• •

• • •

•

• •

• • • .

........ 1

. ..

. • . • •• • • ••• • • .. . . • .

• •

• ..

40

20

0 90 60 30 0 30 60

ANGLE OF APPARENT DIP a. 90

•

b F2 I

Fl

Plates 11 & 12

Fl folds at different positions in a single F2 structure.

(Grid. Ref. 69501478).

Plate 11 The Fl fold axial surfaces are almost normal to that of

the F2 structure.

Plate 12 The Fl fold axial surfaces are almost parallel to that

of the F2 structure which is visible at the top of the

plate.

Plate 11

Plate 12

286

of -0112 with cos2a (figs. 6.4c & 6.5.b) there is ample evidence of

non-passive layer behaviour in the F2 folding. Similar evidence can be

produced for the Fl and F3 folds, and so the layering is considered

to have behaved non-passively in all three phases of deformation.

Isogon patterns give an indication of the relative ductility contrast

between adjacent folded layers (Chapter 5; Ramsay, 19679 p.416 & 432) and

in the present case this leads to the conclusion that in some situations

the dark amphibolitic layers have behaved in a more competent manner than

the light quartzo-feldspathic layers (e.g. fig. 6.3, 6.5). In other

situations the relative competency has been reversed, and the light layers

have behaved in a more competent manner (e.g. fig. 6.4). Rarely,

reversals in relative competence contrast appear to have taken place between

adjacent folded layers in neighbouring folds.

That these 'anomalous' variations in relative competence existed

throughout all three folding episodes is reflected in the lack of any

systematic differences between the overall -0(12/cos2a intercept distrib-

utions fcr dark, intermediate and light layers (fig. 6.6) in all three

fold phases. This is further shown for the F2 folds, by the similarity

of the 'shape' distributions for inner and outer arcs of folds in both

dark and light layers.

Using similar arguments to those developed in Chapter 5 it is

considered that buckling was the most important fold-forming process in

all three deformation episodes; the banded gneisses are thought to have

behaved as a complex multilayer in which either quartzo-feldspathic or

amphibolitic layers behaved as tile more competent units. As an

individual example of evidence for buckling, consider the F2 folds in

fig. 6.4. The folds in the thin light layer S can only be satisfactorily

explained by a buckling hypothesis. Fold wavelength in the light layers

in fig. 6.4a increases with layer thickness (fig. 6.4e). A direct

287

relationship between fold wavelength and layer thickness is characteristic

of buckling processes.

A component of flattening (see section 3.5), normal to the axial

surfaces, is inferred to have either followed or accompanied the buckling

in each episode of folding. This component has been greatest in the case

of the Fl folding, and least in the case of the F3 folding, accounting

for the differences between fold phases in the overall distributions of

both layer shape, based upon -02/ces2a intercept values (fig. 6.7) and

of 'amplitude' in the harmonic analysis (fig. 6.9). The larger the

flattening component of deformation the smaller the shapes of folded

layers depart from a class 2 geometry, and the higher the amplitude (taken

as the amplitude/wavelength ratio) becomes.

The variation of t' with a. observed in fold B in layer W in fig.

6.4c is similar to that predicted from a process of simultaneous

buckling and flattening (section 3.6).

Progressive changes in single surface fold shape during the

buckling of an isolated layer are discussed in Chapters 3 and 4. The

most distinct change observed is that the folds become progressively more

rounded in the crests with increasing deformation. In terms of the

visual harmonic analysis this implies changes in the closest matching

'shape' category from D or E through C to B, with increasing deformation.

In the overall distributions of 'shape' for the three fold phases (fig.

6.9), the frequencies of fold shapes in the categories C, B and A

increase in progressively older fold phases, together with progressively

higher amplitudes (and increasingly large components of inferred flat—

tening). By analogy with the isolated layer case, this is interpreted

to be the result of buckling of the more competent layers during the

deformation.

288

The frequency of folds with sharp crests (categories D and E) is

large in the distributions relating to F1 and F2, and this is considered

to be due to the flattening part of the deformation. In passive

flattening there is no change in fold 'shape' with increase in amplitude

(section 3. 5 ), and fold shapes initially near sinusoidal (categories D or E) would remain near sinusoidal throughout deformation.

The single surface fold 'shape' distributions are therefore taken

as evidence for simultaneous buckling and flattening in at least F1 and

F2 folding episodes.

Some of the deformation undergone by the Fl folds will have taken

place during the F2 phase of deformation (and perhaps during that of F3

as well). Since the measured F1 folds have axial surfaces almost

parallel to those of F2 and F3 (see sect. 6.3.3 and fig. 6.2), this

additional deformation will have been in the form _of a continued

shortening almost normal to the axial surfaces. Thus the measured F1

folds record rather more deformation than occurred in the Fl phase alone.

The extent to which F2 folds have been affected by F3 deformation outside

the zones of F3 fold development, is notthought to be great.

It is not possible to interpret the variations of the interlimb

angle of F1 folds, refolded by F2, unnnbiguously in terms of Ramsay's

(1967, p.491 et. seq.) idealised models of the modification of dihedral

angles in various types of refolding. However, the observed interlimb

angle variations (fig. 6.11) are compatible with a theory of buckling and

flattening. As a result of the flattening component the F1 folds would

tend to become opened out in the hinge regions of the F2 folds (figs.

6.11a & plate 11). Differences in the value of the maximum interlimb

angle attained near the F2 fold crests (cf. figs. 6.11a & b, and see

Ramsay, 1967, Chapter 9), would result from slight differences in

initially small interlimb angles becoming accentuated during the F2

289

flattening.

The kinking, or crenulation cleavage, developed in the amphibolitic

layers in F2 folds is thought to have played a small part in determining

fold size and geometry on the scale of the kinks themselves. There is

no evidence to suggest that these folds developed from conjugate kinks in

the manner described by Paterson & Weiss (1966). Paterson & Weiss (1968)

show how kinking can control the folding in more 'competent' layers which

take on fold geometries similar to those observed in buckling. However,

in this case there is no dependence of fold wavelength on the thickness

of the 'competent' layers, and zones of contact strain would not be present.

Buckling is thought to-be the primary process responsible for F2 fold

development, and kinking to be a secondary phenomenon.

F3 folds in some ways resemble the kinking in the amphibolitic bands

in the F2 folds; the whole stack of thin folded layers appear similar to

the kinks within a single amphibolitic band (compare plates 9 & 10). However, no sign of conjugate structures is apparent.

6.4.1 Discussion.

One of the biggest problems presented by the analysis, arises from

the conclusion that amphibolitic layers may behave either in a more or a

less competent manner than quartzo-feldspathic layers. If this conclusion

is correct it seems likely that the competency contrast is always very

small, and that very small differences in composition or texture, of the

kind described in section 6.3.1, will be sufficient to reverse the relative

competence of 'dark' and 'light layers.

It was not found possible to analyse the buckling, in any fold phase,

in terms of the precise multilayer theories of Biot (1964; 1965a; 1965b;

1965c) or Ramberg (1961a; 1963b; 1964a). Biot's (1964; 1965b) analysis

of the internal buckling of confined multilayers is probably the

290

theoretical treatment most applicable to the kind of folding observed here,

but it is not valid to very large deformations nor does it account for the

development of more than one order of folds. Ramberg (1964a) does

qualitatively discuss the simultaneous development of several fold orders

and is able to show that this phenomenon is a characteristic of buckling

in multilayers, where layers are of different ductility and thickness.

In the light of the series of experiments on multilayers made by

Ramberg (1963b) and Ghosh (1968) using viscoelastic materials, the

comparison made in tie interpretation above, between the development of

fold shape in isolated layers and multilayers, may not be valid. By

considering such factors as layer spacing, ease of slip between layers

(Ghosh) and the degree of confinement of the multilayer (Ghosh), these

authors show that a multilayer can behave as a single or a composite unit

and that folds may develop with rounded or angular crests. Ghosh shows

that kinking may occur under certain conditions (with a limited amount of

slip between layers).

Under metamorphic conditions, slip between layers (and therefore

kinking) is probably unimportant in fold development, and in the present

case it is considered that progressive changes in fold shape (in the

competent units) will be of the same relative kind as those undergone

during the buckling of a single isolated layer.

The fact that three phases of folding have developed coaxially is of

interest. Wynne—Edwards (1963) considers this kind of folding can occur

in a single continuous deformation involving fluid flow with convolution

of the flow lines. However, applied to the Sambuco folds, such a theory

does not account for the systematic nature of the refolding and has the

disadvantages of any theory involving heterogeneous simple shear (see

section 3.2). MUkhopadhyay (1965b) considers that examples of coaxial

refolding in the Moinian rocks of the Scottish Highlands are due to the

291

geometrical restraints imposed by the early structures on a subsequent

refolding which contains a strong flexural component. Such a restraint

would be expected to affect any type of superposed deformation that

involved buckling, and this may partly explain the coaxial refolding in

the Sambuco rocks.

6.5 A WAVELENGTH/THICKNESS STUDY OF F2 FOLDS

Certain thin layers have developed folds of the kind produced by

the buckling of competent layers isolated in a less competent medium, in

which a zone of contact strain dies out within a short distance of the

buckled layer. The layers may be either 'dark' amphibolitic bands embedded

in 'light' quartzo—feldspathic material, or 'light' quartz rich layers

surrounded by 'dark' hornblende and/or biotite rich rock (fig. 6.12a & b).

An analysis of these folds (all of the F2 phase) is made here, treating

them as buckles developed according to a modified theory of Sherwin &

Chapple (1968) (see Chapter 4 and section 5.7). Estimates of viscosity contrast and the amount of F2 deformation within the profile planes of the

folds will be made.

The procedure of analysis is identical to that described in section

5.79 and will not be repeated here.

The bulk F2 strain is assumed to be homogeneous within the area

studied; the effects of F3 are local and are assumed to be unimportant

away from the zones of F3 fold development. The basic assumptions of

the buckling theory (section 3.3) are for the most part difficult or

impossible to test. The validity of an assumption of Newtonian viscosity

for rocks undergoing metamorphism is discussed in section 4. 2 . The folded layers are never truly isolated and two effects are observed: one

is tLat of unfolded neighbouring layers acting as a confinement, tending

to reduce the dominant wavelength/thickness ratio (Ramberg, 1963b); and

Fig. 6.12

a) & b) Examples of folds measured in an analysis of Wt.

Stippling Hornblende rich layers

No stippling Quartz—feldspar layers

Layer A — competent quartz—feldspar layer

Layer B — competent hornblende layer

c) Frequency histogram of W/t values for 114 folds in competent 'light?

layers. Mean = 5.8.

Frequency histograms of W/t ratios for 40 folds in competent 'light'

layers.

(i) Before individually unflattening mean = 5.1

(ii) After individually unflattening — mean = 5.9

Note: all means are ungrouped

1/2-

293

a

vi

b

30

M

20- F

10-

C

5

F

F

10

0 0

10

0 0

1 I

I

10 15 W/T

d (i)

5 10

m I-1 , , 1 1

1 ( ii) 5 10

W/T

294

the other is a multilayer effect in which all the competent layers behave

in a similar fashion (Ramberg, 1961a) and the dominant wavelength/thickness

ratio tends to increase for a single competent member of the sequence.

These two effects are considered to 'neutralize' one another in the

calculation of mean wavelength/thickness ratios.

A frequency distribution of Wit ratios for isolated buckle folds in

'light' layers is drawn in fig. 6.12c. Nearly all the measured folds were

thickened in the hinges, and it was assumed that a component of homogeneous

flattening followed the buckling for the purposes of the analysis (but

see p.106)• To eliminate the effect of flattening, 35 folds in 'dark'

buckled layers and 40 folds in the 'light' buckled layers were individually

unflattened. Measurements of wavelength, thickness and limb dip were

made on each unflattened fold. The results are given in table 6.1.

The total shortening should be the same in the analyses of folds in

both 'dark' and 'light' layers. However, the calculated values of

Shortening are not quite the same, and are equivalent to strain ratios

(Jh1/ X2) of about 6:1 and. 5:1 respectively.

Because of the difficulties in both testing and meeting the assum—

ptions on which the theory is based, the results are only considered to

give rough estimates of the 'viscosity' contrast and the total amount of

shortening within the profile plane of the F2 folds.

6.6. CONCLUSIONS

A) Geometrical analyses of folds in all three fold phases show that

no systematic differences in fold geometry can be related to

different layer types (i.e. 'dark', 'intermediate' and 'light').

295

TABLE 6.1

MEAN VALUES FOR UNFLATTENED FOLDS

'Light' Layers

i

Flattening A

Component 2 0.64 of Strain X,

\ -1.

'Dark' Layers

0.52

W/T Unflattened Folds

2nT X=

5.9

1.05

5.0

51°

0.68

10 - 20

14 - 18

0.45

0.20

4.9

1.28

2.8

38°

0.82

10 - 20

10 - 12

0.36

0.16

d W

111/112 (Blot)

Limb Dip of Unflattened Folds,

Component of ix Shortening 15 - a Dip

Assumed Amplification

P1/112 (Sherwin

& Chapple)

Shortening Component 0 - 15° Dip

Total X2 Shortening x

1

296

B) Distinct differences in overall (synthesised) fold geometry exist

between the three fold phases.

C) The relative competence of the dark amphibolitic layers to the

light quartz-feldspar layers depends upon very slight differences

in composition and texture, and varies locally in all three fold

phases.

D) The fold geometry is adequately explained in all three folding

episodes by a hypothesis involving both buckling and flattening.

E) The geometrical differences between the three fold phases may be

accounted for by considering the folds in each fold phase to represent

different stages in progressive fold development by buckling and

flattening.

F) The 'viscosity' ratio of dark amphibolitic layers to light quartz-

feldspar layers is estimated to vary between about 11:1 to 1:16.

G) The bulk F2 deformation in the profile planes of the F2 folds is

estimated to be about JX2/X 1 = 0.18

297

CHAPTER

AN ANALYSIS OF MINOR FOLDS IN THE CULM MEASURES AT BOSCASTLE, CORNWALL.

7.1 INTRODUCTION

The sedimentary rocks of the Culm are generally unmetamorphosed,

although a few have slaty cleavage developed and show the effects of

extremely low grade metamorphism.

A small area was selected for study at Boscastle (fig. 7.1). just

within the southern margin of the extensive Culm outcrop. Good exposures

of rock are found here on Penally Hill and Penally Point, at the mouth

of the small harbour.

The rocks at Boscastle are a series of unfossiliferous alternating

dark slates and sandstones, generally considered to be Carboniferous in

age. Ashwin (1957) referred to them as the Boscastle Measures. Selwood

(1961) considered them to be part of the Upper Carboniferous, and placed

them in the Upper Culm.

At least two generations of folds have developed in the Boscastle

Measures, as a result of Hercynian deformation which has affected Devonian

and Carboniferous rocks throughout south-west England. Recent attempts

to elucidate the regional structural pattern have been made by Ashwin

(1957), Zwart (1964) and Freahney et al. (1966).

The general geology of the area is briefly discussed in sections

7.1 and 7,2, and this is followed in section 7.3 by a description of detailed geometrical analyses of the minor folds. The significance of

the fold geometry in terms of folding processes is discussed in section 7.4.

7.1.1 Lithology

In the area of interest around Boscastle harbour, the rocks consist

a) Geological nap of the area around Boscastle (After Dearnan &

Feshney, 1966).

b) Enlarged nap of Boscastle to show the area selected for detailed

fold investigation. This area is narked by diagonal shading.

299

Loca ion of Enlarged Map

Penally iH II

300

of alternating slates, siltstones, sandstones and graywackes; their

sedimentary features have been described by Ashwin (1957). The rock

layers are in the form of continuous sheets, and the coarser sandstone

beds rarely exceed an inch in thickness. Within the coarser layers,

bedding laminations are often conscpicuous. Most layer boundaries are

well-defined surfaces and graded junctions are uncommon.

A simple division of layers into two types has been made for the

purpose of fold analysis:

'sandy' - sandstones, graywackes

'shaly' - slates

Nearly all the beds in which folds have been analysed, fall distinctly

into one or other of these categories.


Two distinct generations of minor folds were first recognised at

Boscastle by Ashwin (1957); and these have recently been described in some

detail by Dearman & Freshney (1966).

Fig. 7.2 is a stereographic plot of poles to axial surfaces and of

fold axes for both fold phases in the selected area.

Data for the early folds, F1, were collected on the long flat limbs

of the later folds, F2: the following description applies to F1 folds

situated on these long limbs of the F2 folds. The Fl folds (plate 13)

usually 11,1ve sharp crests and small interlimb angles. They are frequently

isoclinal. Their axial surfaces are either horizontal or dip gently

towards the north. The dominant axial trend is NNW-SSE, but locally E-W

trends are found, and some intermediate between these directions (see

fig. 7.2; and Dearman & Freshney, 1966, fig. 7a). The mutual relation-

ship between folds with different trends could not be determined. There

is, however, no evidence for the existence of more than a single fold

Fig. 7.2


investigation.

0 Fl fold axes

X Poles to Fl axial surfaces

dots .. F2 fold axes

Poles to F2 axial surfaces

0 Poles to 'kink—like' F2 axial surfaces

302

N

303

Plate 13 Fl folds with well—developed slaty cleavage.

(Penally Point).

Plated F2 folds with strain—slip cleavage developed

preferentially in the steel, limbs. (S. side

of harbour near harbour arm).

-;'''",1A01071116

4)A

304

Plate] F2 folds (penally Point).

Plate 16 F2 folds with well—developed strain—slip

cleavage in the slate bands. (Below the

footpath opposite the harbour arm).

305

phase (see Dearman & Freshney, 1966), and it appears that the different

trends are simply due to large variation of the hinge lines within the

flat axial surfaces. Viewed to the west or north, the symmetry of folds

with E-W or N-S axial trends respectively is either IS' or IZI, and Dearman

& Freshney consider all the long limbs to be the right way up, irrespective

of symmetry, implying a rather peculiar structural pattern (see Dearman

& Freshney, 1966, fig.3d).

The second folds, F2, refold the F1 structures and fold both bedding

and an F1 slaty cleavage. They are of a more open zig zag style, and

are developed locally in zones. Their axial surfaces and axes are

variable in orientation (see fig. 7.2). The axes are nearly horizontal

and trend between NE-SW and ESE-WNW. The axial surfaces have a variable

dip (between 0 and 60°) towards the SE or S. Variations in axial trend

of up to 50°, that are not th result of superimposed deformation, may be

measured along a single fold hinge in the space of a few feet (cf. Dearman

& Freshney, 1966, fig. 4e) where the folds have developed an 'en echelon'

pattern. The overall impression of fold symmetry is of 'SI type

viewed along the axes from west to east, although most folds occur in

zones of near 'M' symmetry. There is no evidence of large scale folds.

The general appearance of the F2 folds depends on the thickness of the

coarser sandstone layers involved in the folding (see plates 14, 15 & 16).

Individual superimposed structures have been described by Dearman &

Freshney; the only distinctive interference patterns observed are Type 3 of Ramsay (19679 p.530), where F2 folds refold Fl folds with an E-W trend.

The major structural pattern and interpretation of folds developed

in south-west England is complicated by the presence of northward-dipping

low-angle normal faults (see for e.g. Freshney et al. 1966). Two

recent attempts have been made to incorporate the structures observed

along the coastline between north Devon and Tintagel in Cornwall, into a

306

unified structural scheme.

Zwart (1964) correlates the two sets of structures observed at

Boscastle with two sets, both developed about E41 trending axes, occuring

in the main Culm outcrop to the north. He explains the regional structure

in terms of 'stockwerk' tectonics, according to which the folds occurring

at Boscastle and Tintagel were developed at deeper tectonic levels (forming

an infrastructure) than those occurring in the Culm to the north (forming

a suprastructure). Freshney et al. (1966) suggest an alternative

explanation, they consider that the major part of the Culm outcrop to the

north has been affected by a single phase of folding deformation only (and

that fanning of the axial surfaces of observed folds about major structures

accounts for the two fold phases of Zwart), and that these folds correlate

with the F1 folds of Boscastle. They consider that the F2 Boscastle

folds are associated with the development of the low-angle normal faults.

For a fuller discussion of the nroblem of major structures the reader is

referred to the works of Ashwin (1957), Zwart (1964) and Freshney et al.

(1966).

Neither Zwart's explanation nor that of Freshney et al. gives any

clear indication of the type of major structure to be expected in the region

of Boscastle; and despite the presence there of the minor Fl folds, there

appears to be a simple stratigraphic sequence from older beds to the

south-west of Boscastle through the Boscastle Measures to the Upper Culm

Measures appearing at Rusey to the north-east (see Selwood, 1961).

The nature of the major structure, if any, associated with the minor Fl

folds at Boscastle remains obscure.

The few F1 folds suitable for analysis and described in this Chapter

are all amongst those with axes trending nearly north-south.

7.2.1 Fabric

An intense slaty cleavage has developed in the shaly bands parallel

307

to the axial surfaces of the F1 folds (see plate 13), and because the

folds are often isoclinal, over large areas the cleavage is virtually

parallel to the bedding. It is in this situation that the F2 folds are

best developed.

Cleavage refracted through the coarser sandstone layers is a common

phenomenon. The intersection of bedding and cleavage produces a lineation

parallel to the F1 fold axes, and a mineral lineation parallel to this is

also developed on some cleavage faces and more particularly on bedding

surfaces. All these lineations are dominantly N-S in orientation.

A strain-slip cleavage has developed in the slates, either parallel

to, or fanning symmetricallyabutthe axial surfaces of the F2 folds

(plates 14 & 16). The intensity of development is variable; it is

always high in the crest regions of the folds, may be either high or low

in the limbs and often appears to be greater in the steeply dipping

short limbs of the folds than in the long limbs (plate 14).

7.3 DESCRIPTIVE GEOMETRY

Most of the data is presented in synthesised form and individual

examples have been selected to illustrate specific features of the fold

geometry.

Because F2 folds at Boscastle are frequently irregular and often en

echelon, care was taken to ensure that all the folds analysed were

approximately cylindrical, and could be measured in profile section.

The local variation of both F1 and F2 fold axis and axial surface

orientation is of the same order of magnitude as the total variation within

the region, and it is considered valid to treat the region as a single

homogeneous structural domain for both phases of deformation.

308

In F1 folds the layer boundaries are often ill-defined, and only in

a few examples could the:profile geometry be measured.

7.3.1 Isogon Plots and Thickness /Dip Relationships.

The variation of thickness with dip was measured in a total of 22 Fl

folds and 160 F2 folds; and for each fold the intercept of the best fit

straight line through the data plotted on a graph of tic_2 against costa

was computed. At least five pairs of t'a ,a values were used in each

calculation.

The synthesised data is presented in fig. 7.3a & b in the form of

frequency histograms of the value of the best fit straight line intercepts.

The intercept values for the Fl folds (fig. 7.3a) are all very close to

zero, indicating that the individual fold geometry is almost similar

(class 2) in all cases. The mean intercept is 0:005. In contrast the

range in values of intercept for the F2 folds is very large (from -0.8 to

1.0), reflecting a wide variation in folded layer geometry. In

addition, the distribution of intercepts is distinctly bimodal, one peak

greater than zero and the other less. By making a distinction between

intercepts for F2 folds in 'sandy' and 'shaly' layers, two separate

distributions may be constructed (fig. 7.3c) which are unimodal, with

virtually no mutual overlap. The intercepts relating to the folds in

tsandy' layers are all greater than 0 (fith a mean of 0.48), reflecting

a class 1 (and mostly 10) fold geometry; whilst those relating to the

folds in 'shaly' layers are predominantly less than 0 (with a mean of -0.15)

indicating a class 3 fold geometry.

Layers of sandstone and slate, with geometric fold forms of class 1C

and class 3 respectively, alternate in sequence to give an overall

similar (class 2) form to the F2 folds. Because the slate beds are

consistently thicker than the sandstones the separate distributions of

fig. 7.3c do not 'balance one another out', and the mean intercept of

Fig. 7,3

Frequency histograms of intercept values (on a t'a2 /cos2a graph) for

Boscastle folds.

a) For all measured Fl folds

b) For all measured F2 folds

c) (i) For 81 F2 folds in 'shaly' layers

(ii) For 79 F2 folds in 'sandy' layers.

F = Frequency

IV = Intercept Value

Mean Variance

a) 0.005 0.003

b) 0.162 0.140

c) (i) -0.153 0.027

(ii) 0.480 0.053

310

20

F 22 Fl Folds a

6.3 0

— 0:3 0 IV

160 F2 Folds

b 20F

-

— 0-5 0 0.5 1.0

IV

F%

20

40]

M

t a

0.5 —0:5 1.0 IV

20-

F%

— 0.5 0 0.5 I 1-0

IV

311

the combined distribution of fig. 7.3b is considerably greater than zero.

(mean 0.16).

It was not found -2-7actical to distinguish between 'shaly' and 'sandy'

layers in the case of the F1 folds.

An example of an Fl fold profile is shown in fig. 7.4. The isogon

plot and a graph of t 'a2 against cos 2a both reflect the very slight

differences in fold geometry that exist between the slate and the sandstone

layers. The fold geometry approaches class 2 in all layers and the tL2/

cost plot for the folded sandstone layer has a slightly smaller slope

than that for the slate layer.

In the case of F2 it is of interest to examine the variation of

geometry within the folded sandstone layers. Two examples are considered.

In fig. 7.5 the layer of interest is a massive sandstone bed within which there is little visible sign of bedding laminations. Isogons are

2 constructed on the fold profile, and a plot of t'a against cost a is

shown for a single fold in the massive layer. Both this graph and the

isogon pattern show that there is a considerable difference in geometry

between the inner and outer arc segments of the fold; the inner part is

nearly 'similar' (class 2) in geometry, whilst the outer part takes up

a class 1C fold form.

A similar situation exists in fig. 7.6, but here the sandstone layer

X, has a well-developed internal bedding structure. The isogons and the

tDcos2a plots for the selected fold in layer X show similar, though

less pronounced differences in geometry between the inner and outer parts

of the folded layer in the right-hand limb of the fold, and very little

difference in the left-hand limb. The complete fold has a class 10 fold

form.

t a2 /cos2a, data for a slate layer with a distinct class 3 geometry is also plotted in fig. 7.6c and the t'a2/cos2o, intercept values for folds

Fig. 7.4

Fl fold (between Penally Hill and Penally Point).

a) Isogons drawn at 30° intervals of dip on the fold profile.

Heavy stippling Quartz vein

Lighter stippling Sandstone beds

No stippling Slates

2 b) Plots of t'a against cos2a for folds in a).


a.

313

1" I I

a

b

t a

FiF7. 7.5 F2 folds below the footpath opposite the harbour arm.

a) Isogons drawn at 20° intervals of dip on the fold profile.

Stippling Sandstone beds (except beds Y & Z that are

siltstones)

No stippling Slates.

b) Plots of -P02, against cos2a for fold A in the massive sandstone X.

Note: in fold A layer XI i is the inner arc segment and 0 is the

outer arc segment.

31 5

a

t b

30

60

90 a

Fig. 7.6

F2 folds (Penally Point).

a) Isogons at 20° intervals of dip on the fold profile.

Stippling Sandstone beds

No stippling Slates

b) Frequency histogram of intercept values (on a t 2'a /coAgraph) for

16 folds (in a) and adjacent folds).

Shaded section For folds in sandstones

Unshaded section For folds in slates.

c) Plots of tla2 against cos2a for fold A in the sandstone bed X.


t o C

30 60 90

2-

317

a

5

F

b

- 0 3

0 09 Iv

318

in several "sandy' and 'shaly' layers from the same specimen are shown in

fig. 7.6b.

Most folds are well—represented by straight lines on a plot of t'a2

against costa (see figs. 7.4, 7.5, 7.6), and the thickness variations

with dip in the two limbs of a single fold in a layer are usually similar.


Single surface fold geometry was investigated using the techniques

of visual harmonic analysis described in section 2.5 and utilised in

Chapter 6. The analysis is based on matching 'quarter wavelength units'

of folds with a series of idealised fold forms (see section 2.5). 35

and 166 quarter wavelength units of folds were analysed in the F1 and F2

fold phases respectively.

The synthesised data is presented in the form of frequency histograms

of 'shape' and 'amplitude' of both Fl (fig. 7.7a) and F2 (fig. 7.7b &c)

folds. In the case of F2 a distinction has been made between the inner

and outer arcs of folds in sandstone layers (which is roughly equivalent

to making the distinction between the outer and inner arcs respectively,

of folds in slaty layers). This distinction could not be made in

most F1 folds. In fig. 7.7c the data for folds in sandstone layers

whose thickness is small compared with their lidblengths (roughly with

a limblength/thickness ratio of greater than 20:1) have been separated

from the distribution in fig. 7.7b. The points of interest in these

distributions may be listed:

a) F1 and F2 folds all have sharp crests: in F2 'shape' categories

D, E and F predominate, and :in F1, E and F.

b) In F2 there is a distinct difference in the 'shape' distributions

between the inner and outer arcs of the folds in sandstone layers.

The outer arcs have more rounded crests (with a peak in the

'shape' category D) than the inner arcs (with a peak in category F).

Fif. 7.7

Harmonic Analysis.

Frequency histograms of 'shape' and 'amplitude'.

a) For 166 F2 folds (shaded distribution) and 37 Fl folds (unshaded).

b) For 87 inner arcs (shaded) and 79 outer arcs (unshaded) of F2 folds

in sandstone layers.

c) For 26 inner arcs (shaded) and 26 outer arcs (unshaded) of F2 folds

in thin sandstone layers.

a F 2

F1

40 40-

F% F%

20-

0 1 2 3 4 5 6

320

0 60- 60-

40-

F% -

A B C D E F

20-

0

20F-

A BC DE F 1 2 3 4

SHAPE

AMPLITUDE

C

321

c) The distribution of outer arc 'shapes' in F2 is more restricted (and

the mode frequency is greater) than that of the inner arcs.

d) The distributions of 'shape' relating to the thin sandstones (fig.

7.70) in F2 show little difference between inner and outer arcs.

e) The 'amplitudes' of the F2 folds fall mainly in categories 2 and 3 (and the inner arcs tend to have slightly higher smplitudes than the

outer). The 'amplitudes' of the Fl folds fall mostly in the higher

categories 3, 4 and 5, and are more variable. (Note that the extra

'amplitude' category 6 has been included here as defined on page 275).

An example of data for several F2 folds in plate 16, is presented in

box diagram form in fig. 7.8, where the differences in position of plot

between outer and inner arcs of folds in sandstones layers is marked.

7.3.3 Deformed Lineation Loci

Several examples of an Fl mineral lineation deformed around individual

F2 folds of hand specimen size, were carefully measured to determine the

pattern of the deformed lineation loci. Data for two examples is

presented in fig. 7.9. In both these the measured Fl lineation lies

on the outer arc surface of a folded well-laminated sandstone bed, and in

both cases the angle between the F2 fold axis and the F1 lineation is

about 70°9 remaining faily constant about the F2 fold. The deformed

lineations appear to follow small circle paths on a stereogram.

7.3.4 interlimb Angles and Limb Length Ratios

About 160 F2 interlimb angles were measured on several folds in a

large number of specimens, and the results are presented in histogram

forb in fig. - 7.10a.. Tho mean interlimb angle is 61°, and the range is

rather large, from isoclinal to about 160°, with 85% of the values lying

in a 50° range between 350 and 850.

Fig. 7.8

Visual Harmonic Analysis.

Plots of 'shape' against 'amplitude' for a number of folds in plate 16.

0 Outer arcs of folds in sandstone beds.

X Inner arcs of folds in sandstone beds.

323

B

E

F

0 0 0 0

0 0

0 0 o

0 x C o 0

0

0

x x x x

x

x

x

x

x

•

x

1

2 3 4 5

AMPLITUDE

w C a_

U) D

aa2.2

Fl lineations (plotted on a Wulff stereographic net) deformed around

individual F2 minor folds.

L - Measured point on lineation locus

F F2 minor fold axis

A.S. - Axial surface

325 N a

N b

FiF. 7.10

a) Frequency histogram of interlimb angles of 160 F2 folds.

Grouped mean = 61°.

b) Frequency histogram of limb length ratios for 86 F2 folds.

(Note: the tails of this histogram have been truncated).

1.11

F2 folds (Penally Point), viewed towards the west.

327

-40

F

M

II 165

r-1

145 125 105 8'565 4.5 INTERLIMB ANGLE

25 5 -5

a

20

I- 0

STEEP > FLAT FLAT > STEEP 20- F

65 4.5 25 25 4.5 6.5 8.5 105 12,5

1 1 LIMBLENGTH RATIOS

328

The ratio of adjacent limb lengths (see section 1.2) was calculated

for 88 F2 folds, 'randomly' chosen from a number of photographs. Of these folds, 26 had steep limbs longer than flat limbs (i.e. possessing

a 'Z' symmetry viewed to the east), and 60 had flat limbs longer than

steep limbs (with an 'S' symmetry viewed to the east). A histogram

of the ratios (fig. 7.10b) shows a peak category at the value 1:1 with a

decrease in frequency at progressively higher limb ratios of either

symmetry sense; although there are a,sTeater number ofISI =rise folds thnn

The maximum limb lengths (and hence limb length ratios) recorded are

limited by the size of the photographs, and so the tails of the

distribution, particularly for the predominant 'S' sense folds, will

probably extend to much larger values of limb length ratios. The

maximum ratio recorded was 16:1.

7.3.5 Miscellaneous Features

The size of the F2 folds appears to a large extent to be independent

of the thickness of the sandstone layers involved in the folding, and the

general appearance of the folds in profile section depends on the relation—

ship between the thickness of the sandstone beds and the limb lengths (or

wavelengths) of the folds (see plates 14, 15 & 16). However, in many

instances, and particularly in regions with folds of 'VP symmetry, the

F2 fold size does appear to be related to the thickness of the sandstone

beds (e.g. the folds in the centre of plate 17).

The orientation of the F2 axial surfaces frequently varies in the

manner illustrated in fig. 7.11. The folds with the more steeply

inclined axial surfaces are 'kink—like' in nature, whereas those with the

more gently inclined axial surfaces are more 'normal' zig zag folds.

7.4 INTERPRELITION AND DISCUSSION

On the basis of isogon patterns and thickness/dip relationships

329

Plate 17 F2 folds (Penally Point).

330

(e.g. fig. 7.4), and using the criteria discussed in Chapter 5, there is

consistent evidence of slight non-passive layer behaviour during the Fl

deformation, the sandstone beds having behaved in a more competent manner

than the slates. The intensely developed slaty cleavage (usually

considered to form normal to the direction of maximum finite shortening

(see Ramsay, 1967, p.180)), the near-similar fold geometry and the tight

or isoclinal fold style are all consistent with a considerable shortening

normal to the axial surfaces of the folds. It is impossible to estimate

the amount of shortening that has occurred, but this seems almost

certainly greater than the 3C% compressive strain considered by Cloos

(1947) to be necessary in order to form slaty cleavage. The folds may

have initiated as buckles, and if so the angular fold style (indicated

in fig. 7.7) suggests that the shortening was not accompanied by further

buckling, which would have produced folds with more rounded crests.

The isogon patterns, thickness/dip plots and harmonic analysis of

the F2 folds provide evidence to suggest that the sandstone beds have

behaved in a much more competent fashion than the slates. Difficulties

in interpreting these folds arise because features of both buckling and

kinking are present. However, the folds are thought to have formed

essentially by a process of buckling in which slip between layers has

played an important role, accounting for the kink-like features observed.

This interpretation , and the problems that arise from it will now be

discussed. 'Let us first consider the features that are consistent

with a buckling hypothesis.

The fold geometry of the sandstone and slate layers is consistent

with class 1C and class 3 tyres respectively. This is a charactertistic

of buckled multilayers made up of alternating competent and incompetent

units, that have suffered a small amount of flattening (see section 3..5)

normal to the fold axial surfaces. Interpreted in this way, the mean

value of the flattening component affecting the F2 folds would be

331

,A2/.1' 1 = 0.7. This is determined for the folds in the sandstone beds

by taking the mean of the intercept distribution derived from tt(ficos20,

plots (fig. 7.3).

In the massive sandstone layer of fig. 7.5, the internal fold geometry

is that expected of a buckled layer in which deformation has been taken up

by tangential longitudinal strain, and modified by a flattening component.

(Compare fig. 7.5 with fig. 3.4a). In the well—laminated sandstone layer

of fig. 7.6, the internal geometry is closer to that expected of a buckled

layer in which the strain has beenactoommodatedby flexural slip or flow,

and again modified by a flattening component. (Compare fig. 7.6 with

fig. 3. 4b).

The loci of the early lineations deformed around folds in well—

laminated sandstone beds (fig. 7.9) are of a type produced by flexural

slip folding i.e. the angle between the F2 fold axis and the lineation

remains constant around the fold. A small component of superimposed

flattening would tend to increase this angle slightly in the fold limbs

and decrease it in the crests. This effect is seen in fig. 7.9a„ but

is not noticeable in fig. 7.9b.

In some zones of near—'M' symmetry the relationship between thickness

and wavelength in the sandstone layers suggests that fold development

was controlled by buckling.

The style of the folds is compatible with the buckling hypothesis

of Price (1967), who presents an analysis based on elastic theory that

accounts for the formation of asymmetrical folds with straight limbs and

angular or rounded crests (chevron and 'plastica' styles respectively).

Failure in the hinge regions of the buckles, in tension or in compression

(see Price, 1967, fig. 6), and subsequent 'rigid' rotation of the limbs

is the reason for the angular fold style. Price considers that this

332

process will operate in the deformation of competent and incompetent

sediments in the upper non—metamorphic parts of the crust. There is no

evidence of failure under tension in the hinges of the F2 folds, the fold

style in the competent sandstone layers is of the 'plastica' type,

produced by failure under compression in the fold hinges. In multilayers

slip between layers is a corollary of this type of buckling. In

assuming that the work done in rotating either limb of a single fold is

the same, the theory predicts greater rotation of the short limbs of

asymmetrical folds, and hence a greater slip between the layers in these

limbs. Price also predicts that the direction of the maximum principal

stress (of an 'average' stress field) will be normal to the axial surfaces

of the folds. The fold geometry within the competent beds will be

similar to that produced by tangential longitudinal strain. This is in

accord with the observations made above for the massive sandstone

beds, but not with those made for the well—laminated sandstones. The

fold geometry within the well—laminated sandstones is more consistent

with the development of chevron folds according to Ramsay (1967, Pp.440

447), who assumes that the strain is accommodated by flexural slip. Ramsay shows that chevron folds developed by flexural slip alone are

likely to 'lock' at interlimb angles of about 60° (the mean interlimb

angle of the F2 folds).

Let us now discuss the features of the F2 folds that are suggestive

of kinking (Dewey, 1965; Ramsay, 1967, p.447), and of kinking leading

to chevron folding (Paterson & Weiss, 1966).

The overall angularity of the folds, their asymmetry and development

in discrete zones are all characteristic of kinking. The occasional

occurrence of open kink structures related to more typical chevron folds

are compatible with the process of fold development described by Paterson

& Weiss (1966), by which conjugate folds form initially and with

progressive deformation interfere to form chevron folds whose axial

333

surfaces are normal to the direction of maximum compression. However,

only one of the two sets of initial kinks predicted by the theory is

actually observed among the F2 folds (see fig. 7.11)9 and there is no

evidence of interfering structures. Paterson & Weiss (1968) show how

kinking can control folding in quartz—rich layers in a phyllite. The

folds that develop in these layers show some charasteristics of buckled

competent bands, possessing rounded outer arcs and sharp inner arcs.

:according to this mode of formation, the fold size will be more or less

independent of the thicknesses of the 'competent' layers. There is often

little systematic relationship between F2 fold size and the thickness of

the 'competent' sandstone layers (plates 14, 15 & 16).

The folding process described by Paterson & Weiss(1966) provides

an alternative mode of chevron fold development to that proposed by Price

(1967). Paterson & Weiss predict that chevron folds formed in an 'ideal

foliated body' will have interlimb angles of 60°. (60° is also the

interlimb angle at which chevron folds developed, without kinking (Ramsay

1967, p.444), are prone to "lock").

It is apparent that many of the geometrical features of F2 folds

are compatible with theories of both buckling and kinking. It is

pertinent to consider the essential differences between buckling and

kinking in multilayers. In buckling the /theological contrast of the

layers is the cause of the folding instability, whereas in kinking it is

the anisotropic nature of the layering as a direction of weakness and

potential slip that can lead to a folding instability; there need be no

rheological contrast between the layers (as in kinking in a crystal).

In the case where the maximum compression acts parallel to the layering,

buckling would produce symmetrical folds with axial surfaces normal to the

compression; whereas kinking would produce asymmetrical folds with axial

surfaces inclined to the direction of maximum compression, and deformation

would be confined to the short limbs (kink zones). If multilayers pos—

sess both a rheological contrast between layers and a direction of

334

potential slip parallel to the layering, characteristics of both buckling

and kinking may appear when the multilayer is deformed. This is the case

in the series of experiments made by Ghosh (1968), and in those made by

Paterson & Weiss (1968). In both series of experiments the competent

members of the sequence displayed features of buckle folds (with rounded

outer arcs and sharp inner arcs) although the primary process of folding

was kinking leading to chevron folding.

The relationship between the orientation of the strain-slip cleavage

and the strain developed in the slates is critical to the interpretation

of these folds. The divergent fans of the strain-slip cleavage in slate

layers confined between folded sandstone beds (plates 14 & 16) are

identical to slaty cleavage fans that occur in incompetent material

sandwiched between buckled layers (see Ramsay, 1967, p.404-405). This

leads to the conclusion that the strain-slip cleavage formed normal to the

direction of maximum finite compression. The intensity of strain-slip

cleavage development, however, may not be valid indicator of the intensity

of deformation, since the presence of a slaty cleavage at a suitable

orientation to the deforming stresses is a prerequisite of strain-slip

cleavage development. !.t the stage of deformation when strain-slip

cleavage formed in the F2 folds, the short limbs may have been.in a more

suitable position than the long limbs for the development of this cleavage.

With this interpretation of the significance of the strain-slip

cleavage, the possibility that the F2 folds represent a single set of

kinks formed by shear deformation within the short limbs only, is ruled

out. The kinking/chevron folding process (Paterson & Weiss, 1966) is

unable to account for very asymmetric folds, since only one set of kinks

is likely to develop initially. It is possible that the folds developed

from a single set of kinks formed initially in discrete zones of localised

deformation, and that further deformation affected the whole rock mass so

that the direction of maximum finite shortening became normal to the

axial surface of the folds. This is the kind of interpretation made by

335

Roberts (1966) for somewhat similar structures in the Scottish Dalradian.

The interpretation for the development of the F2 folds and strain—

slip cleavage suggested here involves asymmetrical buckling (Price, 1967)

in the rock mass, and continued folding controlled in part by the

rheological contrast of the sandstones and slates, and in part by the planes

of weakness and potential slip parallel to the layering, to result in

chevron folds with axial surfaces normal to the direction of the maximum

finite compression. The strain in the competent layers (sandstones)

has been accommodated by flexural slip, tangential longitudinal strain or

in some less regular manner caused by local failure in the hinges of the

folds. In the slates, strain has been accommodated by slip on the

early slaty cleavage (mainly in the long limbs), or by some kind of

shortening along the slaty cleavage, associated with the production of

strain—slip cleavage (microfolding?). Slip between layers has been of

major importance in the long fold limbs. A small component of flattening

is thought to have affected the whole sequence. By ignoring the

'flattening', and assuming that the total length of the beds has remained

unchanged during folding, in the regions of best developed chevron folds

with interlimb angles of about 600, a shortening of about 50% normal

to the axial surfaces may be inferred.

The biggest problem with this interpretation is the degree of

asymmetry of some of the observed folds. This is certainly a primary

feature (see Breddin & Furtak, 1962), and strongly asymmetric folds are

difficult to account for by Price's theory.

Some of the difficulties in interpreting these folds would be

resolved through a better understanding of the folding processes in

multilayers with a rheological contrast between the layers, and a

direction of weakness and potential slip parallel to the layering.

336

7.5 CONCLUSIONS

A) There are systematic differences in fold geometry (slight in the Fl

folds, and very pronounced in the F2 folds) that can be related to

differences in layer composition.

B) Sandstones have consistently behaved in a more competent manner

than slates, in both folding episodes.

C) The geometric forms of the folds show features of both buckling

and kinking.

D) The primary process responsible for fold development is suggested

to be asymmetrical buckling.

E) Progressive folding has taken place by a combination of buckling

in the sandstones, and by slip between layers.

F) Strain-slip cleavage is considered to have developed normal to the

direction of the maximum finite compression in the rocks.

337

CHAPThli 8

SYNTHESIS

8.1 A COMPARISON OF RESULTS OF THE TRTTP. DETAILED FOLD STUDIES

Most of the features brought out in the geometric fold analyses

described in Chapter 5, 6 & 7 are common to folds in all three regions.

In each instance distinct overall differences in fold geometry can be

related to differences in layer composition. In the Moine rocks of Monar,

and more especially in the Culm sediments of Boscastle, differences in

fold geometry are related to distinct differences in rock composition.

In the gneisses of Sambuco, however, significant differences in the

geometric form of folds appear to be related to more subtle differences

in both composition and texture of the layers.

Both in the gneisses of Sambuco, and in the Culm of Boscastle overall

differences in the geometric form of folds exist between folds of different

generations. In progressively older fold phases, the amplitude/wavelength

ratio of folds is progressively greater, and the geometric forms of all

folds are closer to the 'similar' (class 2) fold form.

A conspicuous feature of folds in all three areas is the variable

nature of their geometric form. This is most apparent in the synthesised

data. The distributions of intercept values (on a t as 2/co/plot) for

folds in layers of different composition, or for folds of different

generations, have high values of variance and show considerable mutual

overlap. These same features are observed in the distributions of 'shape'

and 'amplitude' (of a harmonic analysis) of the inner and outer arcs of

folds in competent layers, and are also observed in the 'shape' and

'amplitude' distributions for folds of different generations.

Both in the Culm sediments and the Moine granulites, the relative

competence of layers of different composition (deduced from the fold

338

geometry) is distinct and consistent. In the metamorphic rocks of Monar,

competence decreases with an increase in the biotite content, and appears

to increase with increasing grain size. In the Culm sediments, sandstone

is much more competent than slate, and competence again appears to increase

with the grain size. In the Sambuco gneisses, however, competence vaxies

with slight changes in composition and texture of the rock. The main

compositional difference between the Sambuco gneisses and the Moine

granulites is the presence of hornblende in the former. Some property of

this mineral seems to be the major factor in determining the local change

in relative competence of the gneiss layers at Sambuco. In the meta—

morphic rocks of both Monar and Sambuco it is estimated that the maximum

'viscosity' contrast between layers of different composition is in the

order of 10:1 to 15:1.

In all instances the competent layers take on a geometric fold form

of class 1C, and the incompetent layers a variable fold form, but

predominantly one of fold class 3.

Variations of t'a2 with cos2a are nearly linear in all the measured

folds. The 'mean' fold shape of folds of different generations in each

region is close to a 'similar' (class 2) fold form. All the distributions

of intercept values (on a t'a2/cos2a graph) for all folds of any one

generation in each area are unimodal. The one exception is found in the F2

folds of Boscastle where the distribution is bimodal.

The most distinct differences between fold morphology in the three

regions is brought out in the harmonic analysis of single folded surfaces.

Most of the folds in the Monar granulites and the Sambuco gneisses match

the ideal fold forms (fig. 2.14) of 'shape' categories B, C, D & E,

indicating fold shapes varying between sinusoidal and forms very rounded

in the hinge regions. In contrast, the folds at Boscastle (of both

generations), having more angular hinges and straighter limbs, match

339

*shape' categories D, E and F. In all instances, both the inner and outer

arcs of folds in the most competent layers have the highest values of the

ratio b3/b1

(i.e. the folds are more rounded in the hinge regions). The

lower the competence contrast across a folded surface, the lower the value

of the ratio b3/b1 (and the more closely do the fold shapes approach a

sinusoidal form).

The higher the ratio of amplitude/wavelength of folds, the more

rounded are the folds in the hinge ,regions (with the exception of the Fl

folds of Boscastle).

In all three regions buckling is considered to have been the primary

process responsible for fold initiation. Continued fold development

is thoucht to have taken place by processes of both buckling and flattening.

The flattening either followed or accompanied the buckling. Where several

generations of folds are present in one region, the folds of older

generations appear to have undergone more flattening than younger

generation folds. The F2 folds of Boscastle show very little effects

of flattening. They also differ from all the other folds analysed in

that they show geometric features of both buckling and kinking. This is

attributed to the rock-mass having properties of both a rheological

contrast between the layers (of sandstone and slate), and a direction

of weakness and potential slip parallel both to the layering and the Fl

slaty cleavage.

In summary, the general pattern of fold geometry in the rocks of

these three regions is very much the same, and is consistent with a hypot-

hesis of fold development involving buckling in a multilayer (with layers

of different 'viscosity'), modified by varying amounts of flattening.

The folds in the Culm rocks of Boscastle, and particularly the F2 folds

there, do show a number of features not observed in folds in the other

regions. These features were probably produced by kinking. This

340

difference between the folds in the Culm and folds in the other regions

reflects the different mechanical state of the rocks, and the differences

in external conditions (of temperature and pressure)that existed at the

time of fold development .

8.2 SUICARY AND CONCLUSIONS

The principal concern of this thesis has been with fold morphology.

Attempts have been made to link theoretical and experimental work to

natural fold studies by means of accurate geometrical fold analysis.

The existing methods of geometrical fold analysis have been critically

reviewed; many were found to be impracticable. Two new analytical

techniques have been developed, one involving the use of dip isogons (and

concerned with folded layer geometry), and the other based on harmonic

analysis of single folded surfaces. A simple and rapid method of visual

harmonic analysis has been developed that should prove useful to the

field geologist.

Theories of fold development have been discussed with particular

emphasis on the predictions they make for fold shape and the development

of fold shape in an isolated competent layer embedded in a less competent

matrix.

A series of buckling experiments on single layers of a viscous

material embedded in a less viscous medium at low viscosity contrasts

have been described. The shape and progressive changes in shape of the

experimentally developed folds have been analysed and shown to be

consistent with buckling theory. The most important outcome of this work,

not predicted by the theory, was the observation that layer shortening

ceased when the developing folds attained limb dips of about 15°. This

appeared to be a geometrical effect and, in the range of viscosity

contrasts used, to be indelpendent of the value of the viscosity ratio.

341

Three detailed analyses of minor folds in fold belts of different

ages have been described. Systematic differences in fold geometry have

been shown to exist between folds of different generations, and have also

been related to differences in layer composition. The geometry of all

these folds was shown to be consistent with that of folds formed by

buckling and modified by flattening during or after the buckling. The

folds in the Culm sediments differ from the folds in the other regions

in their angular style, and in showing geometrical features of both buckling

and kinking. Estimates of the maximum 'viscosity' contrast in the rocks

of Sambuco and Monar, made from a study of isolated buckle folds, gave

values of 10:1 to 15:1. Estimates of shortening in the profile planes

of the folds were made in the same study. In the F2 folds of Monar, a

value of the ratioiXiA2 was estimated at 10, and in the F2 folds of

Sambuco, the estimated value of this ratio was 5.

In the one instance (Monar F2 folds) where the directions of the

principal axes of the finite bulk strain could be determined, the maximum

extension axis was found to lie at a small angle to the fold axes.

342

ACKNOWLEDGEMENTS

I should like to thank Prof. J. G. Ramsay for suggesting the topic

of research and for supervision during the course of the work.

Many other people have given advice and assistance at one time or

another and I gratefully acknowledge their help.

Drs. N. Gay and N. Price assisted in the design of the shear box

used in the experiments. Dr. D. Elliott, Mr. T. Sibbald and Mr. C. Stabler

were particularly helpful in discussing certain aspects of the work.

Mrs. L. Meadows did most of the typing and Mrs. J. Date helped with

the duplicating work.

The research was carried out under the tenure of a. University of

London Postgraduate Studentship which is gratefully acknowledged.

343

REFERENCES

Agterberg, P.P. 1967. Computer Techniques in Geology.

Earth-Sci. Rev., Vol. 3, pp. 47-77.

Ashwin, D. P. 1957. The Structure and Sedimentation of the Culm

Sediments between Boscastle & Bideford, North Devon. Unpublished

Ph.D. Thesis, Univ. of London.

Barber, N. F. 1966. Fourier Methods in Geophysics. in S.K. Runcorn (ed.),

Methods and Techniques in Geophysics, Vol. 2. pp. 123-204. Inter-

science Publ., New York.

Bell, R. T., and J. B. Currie. 1964. Photoelastic Experiments related

to Structural Geology. Proc. Geol. Ass. Canada, Vol. 15, pp. 33-51.

Beloussov, V. V. 1960. Toctonophysical Investigations. Geol. Soc. Am.

Bull., Vol. 71, pp. 1255-1270.

Biot, M. A. 1957. Folding Instability of a Layered Viscoelastic Medium

under Compression. Proc. Roy. Soc. (London), Ser. A, Vol. 242, pp.

444-454. Biot, M. A. 1959. On the Instability and Folding Deformation of a Layered

Viscoelnqtic Medium in Compression. J. Appl. Mechanics, Ser. E,

Vol. 26, pp. 393-400.

Biot, M.A. 1961. Theory of Folding of Stratifies,'. Viscoelastic Media

and Its Implications in Tectonics and Orogenesis. Geol. Soc. Am,

Bull., Vol. 72, pp. 1595-1620.

Biot, M. A., H. Ode, and W.L. Roever. 1961. Experimental Verification

of the Theory of Folding c.7 Stratified Viscoelastic Media. Geol.

Soc. Am. Bull., Vol. 72, pp. 1621-1630.

Biot, M.A. 1964. Theory of Viscous Buckling of a Confined Multilayered

Structure. Geol. Soc. Lm. Bull., Vol. 75, pp. 563-568.

344

Biot, M.A. 1965a. Mechanics of Incremental Deformation. John Wiley &

Sons, Inc., New York, 504 pp.

Biot, M.A. 1965b. Further Development of the Theory of Internal Buckling

of Multilayers. Geol. Soc. Am. Bull., Vol. 76, pp. 833-840.

Biot, M.A. 1965c. Theory of Similar Folding of the First and Second Kind.

Geol. Soc. Am. Bull., Vol. 76, pp. 251-258.

Breddin, H. 1957. Tektonische Fossil and Gesteinsdeformation im Gebiet

von St. Goarhausen. Decheniana, Vol. 110, pp. 289-350.

Breddin, H., and H. Furtak. 1962. Zur Geometrie asymmetrischer Falten.

Geol. Mitt. Aachen, Vol. 3, pp. 197-219.

Busk, H.G. 1929. Earth Flexures. Cambridge Univ. Press, London 106 pp.

Campbell, J.D. 1951. Some Aspects of Rock Folding by Shear Deformation.

Am. J. Sci., Vol. 249, pp. 625-639.

Campbell, J.D. 1958. En Echelon Folding. Econ. Geol., Vol. 53, PP.448-472.

Carey, S.W. 1954. The Rheid Concept in Geotectonics. J. Geol. Soc. Austr.

Vol. 1, pp. 67-117.

Chadwick, B. 1968. Deformation and Metamorphism in the Lukmanier Region,

Central Switzerland. Geol. Soc. Am. Bull., Vol. 79, pp.1123-1150.

Chapple, W.M. 1964. A Mathematical Study of Finite Amplitude Rock-Folding.

Unpublished Ph.D. Thesis. California Institute of Technology.

Chapple, W.M. 1968. A Mathematical Theory of Finite-Amplitude Folding.


Cloos, E. 1947. Colite Deformation in the South Mountain Fold, Maryland.


Cruden, D.M. 1968. Methods of Calculating the Axes of Cylindrical Folds:

Areview. Geol. Soc. Am. Bull., Vol. 79, PP. 143-148.

Currie, J.B., H.W. Patnode and R.P.Trump. 1962. Development of Folds

in Sedimentary Strata. Geol. Soc. Am. Bull., Vol. 73, pp. 655-674. Dearman, W.R. and E.C. Freshney. 1966. Repeated Folding at Boscastle,

North Cornwall, England. Proc. Geol. Assoc., Vol. 77, pp. 199-215.

345

de Sitter, L.U. 1964. Structural Geology. 2nd. Ed. McGraw-Hill Book Co.,

Yew York, 551 pp.

Dewey, J.F. 1965. Nature and Origin of Kink-Bands. Tectonophysics,

Vol. 1, pp. 459-494. Donath, F.A. 1963. Fundamental Problems in Dynamic Structural Geology.

in T.W. Donnelly (ea.), The Earth Sciences. pp. 83-103. Univ. of

Chicago Press, Chicago, Ill.

Donath, F.A. and R.B. Parker. 1964. Folds and Folding. Geol. Soc. Am.

Bull., Vol. 75, PP. 45-62.

Elliott, D. 1965. The Quantitative Mapping of Directional Minor

Structures. J. Geol., Vol. 73, pp. 865-880.

Elliott, D. 1968. Interpretation of Fold Geometry from Isogonic Maps.

J. Geol. Vol. 76, pp. 171-190.

Fleuty, M.J. 1964. The Description of Folds. Proc. Geol. Assoc., Vol.

75, 140. 461-492. Flinn, D. 1958. On Tests of Significance of Preferred Orientation in

Three Dimensional Fabric Diagrams. J. Geol., Vol. 66, pp. 526-539.

Flinn, D. 1962. On Folding during Three Dimensional Progressive Deformation.

Quart. J. Geol. Soc., Vol. 118, pp. 385-433.

Flinn, D. 1965. Deformation in Metamorphism. in W.S. Pitcher & G.W. Flinn

(eds.), Controls of Metamorphism. pp. 46-72. Oliver & Boyd, Edinburgh.

Freshney, M.C. McKeown, M. Williams, and W.R. Dearman. 1966.

Structural Observations in the Bude to Tintagel Area of the Coast

of North Cornwall, England. Geol. en Mijnb., Vol. 45, PP. 41-47.

Gay, N.C. 1966. The Deformation of Inhomogeneous Materials and Consequent

Geological Implications. Unpublished Ph.D. Thesis, Univ. of London.

Ghosh, S.K. 1966. Experimental Tests

Strain Ellipsoid in Simple Shear

Vol. 3, pp. 169-185.

Ghosh, S.K. 1967. Experimental Tests

Strain Ellipsoid in Simple Shear

physics, Vol. 4, PP. 341-343.

of Buckling Folds in Relation to

Deformations. Tectonophysics,

of Buckling Folds in Relation to

Deformations: a Reply. Tectono-

346

Ghosh, S.K., and H. Ramberg. 1968. Buckling Experiments on Intersecting

Fold Patterns. Tectonophysics, Vol. 5, pp. 89-105.

Ghosh, S.K. 1968. Experiments of Buckling of Multilayers which permit

Interlayer Gliding. Tectonophysics, Vol. 5, pp. 207-250.

Grovsky, M.V. 1959. The Use of Scale Models in Tectonophysics. Internat.

Geol. Rev., Vol. 1, pp. 31-47.

Harbaugh„ J.W., and F.W. Preston. 1965. Fourier Series Analysis in Geology.

Symposium Computer Applications in Mineral Exploration. Tuscon,

Arizona. V.I. pp. R1-R46.

Hasler, P. 1949. Geologie und Petrographie der Sambuco-Massari-

Gebirgsgruppe zwischen der oberen Valle Leventina und Valle Maggia

im nordlichen Tessin. Schweiz. Min. Petr. Mitt., Vol. 29, pp. 50-150.

Heading, J. 1963. Mathematical Methods in Science and Engineering. Edward

Arnold Ltd., London 628 pp.

Heard, H.C. 1968. Experimental Deformation of Rocks and the Problem of

Extrapolation to Nature. in R.E. Riecker (ed.), Rock Mechanics

Seminar. pp. 439-507. Spec. Report, Terr. Sci. Lab., Air Force

Cambridge Res. Labs., Bed., Massachusetts.

Heim, A. 1921. Geologie der Schweiz. II, Tauchnitz, Leipzig.

Higgins, A.K. 1964. The Structural and Metamorphic Geology of the Area

between Nufenenpass and Basodino, Tessin, South Switzerland.

Unpublished Ph.D. Thesis, Univ. of London.

Hills, E.S. 1963. Elements of Structural Geology. John Wiley & Sons Inc.,

New York, 483 pp.

Hubbert, M.K. 1937. Theory of Scale Models as Applied to the Study of

Geologic Structures. Geol. Soc. Am. Bull., Vol. 48, pp. 1459-1520.

Ickes, E.L. 1923. Similar, Parallel and Neutral Surface Types of Folding.

Econ. Geol., Vol. 18, pp. 575-591.

Johnson, M.R.W. 1956. Conjugate Fold Systems in the Moine Thrust Zone

in the Lochcarron and Coulin Forest Areas of Wester Ross. Geol.

Mag., Vol. 93, ' T.P. 345-350.

347

Johnson, M.R.W. 1965. Torridonian and Moinian. in G.Y. Craig (ed.),

The Geology of Scotland. pp. 79-113. Oliver & Boyd, Edinburgh.

Kelley, J.C. 1968. Least Squares Analysis of Tectonite Fabric Data.


Kreyszig, E. 1967. Advanced Engineering Mathematics. 2nd Ed. John Wiley

& Sons, Inc., New York, 898 pp.

Krumbein, W.C. 1966. Classification of Map Surfaces based on the Structure

of Polynomial and Fourier Coefficient Matrices. Geol. Surv. Kansas,

Computer Contrib. No. 7, pp. 12-18.

Loudon, T.V. 1963. The Sedimentation and Structure in the Macduff District

of North Banffshire and Aberdeenshire. Unpublished Ph.D. Thesis,

Univ. of Edinbureh.

Loudon, T.V. 1964. Computer Analysis of Orientation Data in Structural

Geology. Tech. Report Geog. Branch Off. Naval Res., O.N.R. Task No.

389-135, Contr. No. 1228 (26), No. 13, pp. 1-130

McBirney, A.R., and M.G. Best. 1961. Experimental Deformation of Viscous

Layers in Oblique Stress Fields. Geol. Soc. Am. Bull., Vol. 72,

PP. 495-498. Mertie, J.B. 1940. Stratigraphic Measurements in Parallel Folds. Geol.

Soc. Am. Bull., Vol 51, pp. 1107-1134.

Mertie, J.B. 1959. Classification, Delineation and Measurement of

Non—parallel Folds. U.S. Geol. Surv. Profess. Paper 314 B, pp. 91-124.

mukhopadhiai, D. 1964. Caledonian Structure and Metamorphism in Eastern

Part of Glen Strathfarrar, Inverness—shire, Scotland. Unpublished

Ph.D. Thesis, Univ. of London.

MUkhopadhyay, D. 1965a. Effects of Compression on Concentric Folds and

Mechanism of Similar Folding. J. Geol. Soc. India, Vol. 6, pp. 27-41

NUkhopadhyay, D. 1965b. Some Aspects of Coaxial Refolding. Geo. Rund.,

Vol. 55, pp. 819-825.

Norris, D.K. 1963. Shearing Strain in Simple Folds in Layered Media.

Geol. Surv. Canada, Paper 63-2, pp. 26-27.

Osakina„ D.N., M.V. Grovsky, G.V. Vinogradov, and V.P. Pavlov, 1960.

Optical Polarisation Study of Plastic Deformation Processes with

348

the aid of Ethylcellulose Solutions and Gels. Kolloidnyi Zh.,

Vol. 22, pp. 434-441; English Translation: Colloid J. Vol. 22,

PP. 439-445. Paterson, M.S., and L.E. Weiss, 1962. Experimental Folding in Rocks.

Nature, Vol. 195, pp. 1046-1048.

Paterson, M.S. and L.E. Weiss. 1966. Experimental Deformation and Folding

in Phyllite. Geol. Soc. Am. Bull., Vol. 77, pn. 343-374.

Paterson, M.S. and L.E. Weiss. 1968. Folding and Boudinage of Quartz-Rich

Layers in Experimentally Deformed Phyllite. Geol. Soc. Am. Bull.,

Vol. 79, pp. 795-812.

Peach, B.N. and Others. 1913. The Geology of Central Ross-shire. Mem.

Geol. Surv. Scotland.

Phillips, A., and J.G. Byrne, 1968. Structural Analysis by the Construction

of Sections in Highly Deformed Rocks. (Abstract). in "Displacement

within Continents". Joint Symp. of Geol. Soc. Land., and Yorks.

Geol. Soc.

Potter, J.F. 1967. Deformed Micaceous Deposits in the Downtonian of the

Llandeilo Region, South Wales. Proc. Geol. Assoc., Vol. 78, pp.277-288.

Powell, C.M. 1967. Studies in the Geometry of Folding and Its Mechanical

Interpretation. Unpublished Ph.D. Thesis, Univ. of Tasmania.

Preston, F.W., and J.H. Henderson. 1964. Fourier Series Characterisation

of Cyclic Sediments for Stratigraphic Correlation. in D.F. Merriam

(ed.), Symposium on Cyclic Sedimentation. Kansas Geol. Surv. Bull.,

Vol. 2, pp. 415-425.

Price, N.J. 1967. The Initiation and Development of .Asymetrical Buckle

Folds in Non-Metamorphosed Competent Sediments. Tectonophysics,

Vol. 4, pp. 173-201.

Ramberg, H. 1952. The Origin of Metamorphic and Metasomatic Rocks. Univ.

Chicago Press, Chicago, 317 pp.

Ramberg, H. 1959. Evolution of Ptygmatic Folding. Norsk Geol. Tidsskr.,

Vol. 39, PP. 99-151.

349

Ramberg, H. 1961a. Contact Strain and Folding Instability of a Multi-

layered Body under Compression. Geol. Rund.,Vol. 51, pp. 405-439.

Ramberg, H. 1961b. Relationship between Concentric Longitudinal Strain

and Concentric Shearing Strain during Folding of Homogeneous Sheets

of Rock. Am. J. Sci., Vol. 259, Pp. 382-390.

Ramberg, H. 1963a. Strain Distribution and Geometry of Folds. Bull.,

Geol. Inst. Univ. Uppsala, Vol. 42, pp. 1-20.

Ramberg, H. 1963b. Fluid Dynamics of Viscous Buckling Applicable to

Folding of Layered Rocks. Bull. Am. Assoc. Petrol. Geol., Vol. 47,

PP. 484-515.

Ramberg, H. 1963c. Evolution of Drag Folds.

Geol. Mag., Vol. 100, pp. 97-106.

Ramberg H. 1964a. Selective Buckling of Composite Layers with Contrasted

Rhoological Properties: a Theory for Simultaneous Formation of

Several Orders of Folds. Tectonophysics, Vol. 1, pp. 307-341.

Ramberg, H. 1964b. Note on Model Studies of Folding of Moraines in

Piedmont Glaciers. J. Glaciology, Vol. 5, pp.207- 218.

Ramberg H, 1967. Gravity, Deformation and the Earth's Crust, as Studied

by Centrifuged Models. Academic Press, London 214 PP.

Ramberg, H., and S.K. Ghosh. 1968. Deformation Structures in the Hovin

Group Schists in the Hommelvik Hell Region (Norway). Tectonophysics,

Vol. 6, pp. 311-330.

Ramsay, J.G. 1954. The Stratigraphy, Structure and Metamorphism of the

Lewisian "Inlier" and Moine Rocks of Glen Strathfarrar, Ross-shire.

Unpublished Ph.D. Thesis, Univ. of London.

Ramsay, J.G. 1958. Superimposed Folding at Loch Monar, Inverness-shire

and Ross-shire. Quart. J. Geol. Soc., Vol.113, pp. 271-307.

Ramsay, J.G. 1962a. The Geometry and Mechanics of Formation of "Similar"

Type Folds. J. Geol. Vol. 70, pp. 309-327.

Ramsay, J.G. 1962b. The Geometry of Conjugate Fold Systems. Geol. Mag.,

Vol. 99, pp. 516-526

350

Ramsay, J.G. 1963. Structure and Metamorphism of the Moine and Lewisian

Rocks of the North-west Caledonides. in Johnson & Stewart (eds.),

The British Caledonides. pp. 143-175. Oliver & Boyd Ltd., Edinburgh.

Ramsay, J.G. 1967. Folding and Fracturing of Rocks. McGraw-Hill Book Co.,

New York, 568 pp.

Rast, N. 1964. Morphology and Interpretation of Folds - a Critical Essay.

Geol. J., Vol. 4 pp. 177-188. Reiner, M. 1960. Deformation, Strain and Flow. 2nd. Ea. H.K. Lewis and Co.,

London, 347 PP-

Roberts, J.L. 1966. The Formation of Similar Folds by Inhomogeneous

Plastic Strain, with Reference to the Fourth Phase of Deformation

Affecting the Dalradian Rocks in the Southwest Highlands of Scotland.

J. Geol., Vol. 74, pp. 831-855. Schryver, K. 1966. On the Measurement of the Orientation of Axial Planes

of Minor Folds. J. Geol., Vol. 74, PP. 83-84.

Selwood, E.B. 1961. The Upper Devonian and Lower Carboniferous Stratigraphy

of Boscrstle and Tintagel, Cornwall. Geol. Mag., Vol. 98, pp. 161-167.

Sherwin, J. and W.M. ChapDle, 1968. Wavelengths of Single Layer Folds:

a Comparison between Theory and Observation. Am. J. Sci., Vol. 266,

pp. 167-179.

Singh, R.P. 1967. Experimental Tests of Buckling Folds in Relation to

Strain Ellipsoid in Simple Shear Deformations: a Discussion.

Tectonophysics, Vol. 5, pp. 341-343.

Speight, J.G. 1965. Meander Spectra of the Augabunga River. J. Hydrol.,

Vol. 3, pp.1-15.

Stabler, C.L. 1968. Simplified Fourier Analysis of Fold Shapes.

Tectonophysics, Vol. 6, pp. 343-350.

Stauffer, M.R. 1966. An EMpirical-Statistical Study of Three-Dimensional

Fabric Diagrams as Used in Structural Analysis. Can. J. Earth Sci.,

Vol. 3, PP. 473-498.

Stone, R.O., and J. Dugundji. 1965. A study of Microrelief- Its Mapping,

Classification and Quantification by Means of a Fourier Analysis.

Eng. Geol., Vol. 1, pp. 89-187.

351

Timoshenko, S. 1960. Strength of Materials. Part 1. D. van Nostrand Co. Inc.,

Princeton, New Jersey, 442 pp.

Turner, F.J., and L.E. Weiss. 1963. Structural Analysis of Metamorphic

Tectonites. McGraw-Hill Book Co., New York, 545 pp.

van Hise, C.R. 1896. Principles of North American Pre-Cambrian Geology.

U.S. Geol. Surv. 16th Ann. Rept., part 1, pp. 571-843.

Watson, G.S. 1966. The Statistics of Orientation Data. J. Geol., Vol. 74,

pp. 786-797.

Wenk, E. 1962. Plagioclas als Indexmineral in den Zentralalpen. Schweiz.

Min. Petrogr. Mitt., Vol. 42, pp. 139-152.

Whitten, E.H.T., and J.J. Thomas. 1965. Geometry of Folds Portrayed by

Contoured by Maps: a New Method for Representing the Geographical

Variability of Folds. Geol. Soc. Am. Spec. Paper No . 87, p.186.

Whitten, E.H.T. 1966a. Structural Geology of Folded Rocks ▪ Rand McNally

& Co., Chicago, 663 pp.

Whitten, E.H.T. 1966b. Sequential Multivariate Regression Methods and

Scalars in the Study of Fold-Geometry Variability. J. Geol., Vol. 74,

PP• 744-763. Williams, E. 1965. The Deformation of Competent Granular Layers in

Folding. Am. J. Sci., Vol. 263, pp. 229-237.

Williams, E. 1967. Notes on the Determination of Shortening by Fle.Xure

Folding Modified by Flattening. Papers & Proc. Roy. Soc. Tasmania,

Vol. 101, pP. 37-41.

Wilson, G. 1952. rtYgmatic Structures and Their Formation. Geol. Mag.,

Vol. 89, pp. 1-21.

Wilson, G. 1967. The Geometry of Cylindrical and Conical Folds. Proc.

Geol. Assoc., Vol. 78, pp. 179-210.

Wynne-Edwards, H.R. 1963. Flow Folding. Am. J. Sci., Vol 261. pp. 793-814.

Zwart, H.J. 1964. The Development of Successive Structures in the

Devonian and Carboniferous of Devon and Cornwall. Geol. en Mijnb.

Vol. 43, PP. 516-526.

352

APPENDIX

PROGRAMME FOR THE CALCULATION OF HARMONIC COEFFICIENTS.

A listing is given of a programme to calculate the harmonic co-

efficients bn of a periodic function f(x) represented by discrete values

of f(x) in the period 2n. The IBM 'Share' subroutine FORIT is used to

calculate the values of the harmonic coefficients and is incorporated

in the programme. This is written in FORTRAN IV to run on an IBM 7094

computer under the IBSYS monitor.

The equations on which the calculations are based are given in

section 2.5. 2N+1 values of f(x) are taken over the range 0 - W, at

intervals of W/2N+1 (see figs. 2.9 & 2.10). For a 'quarter wavelength

unit' of a folded profile (see fig. 2.10) N+1 values of f(x) are measured

in the range 0 - WA (the origin is the first point taken). Data for

the programme are the value of N9 the number of coefficients M to be

computed (M less than or equal to N), the value of W/4 and N values

of f(x) in the range 0 - W/4 in increasing order of size (excluding

the value at the origin).

The comments on subroutine FORIT are part of the original 'Share'

programme.

C

FOURIER ANALYSIS OF FOLD PROFILES

PROGRA LI IN

C SCALE = THE VALUE OF W/4 effirt-T-=zieVr--1;15-2

C FNT = VALUES OF f- ( X )

R-OF-DATA-SET-S- HA =FoLLiztvi-----_Eixa.v=pievrasFTefarikslEsTro

CON V A •

—At4E-THF=sEcOND_viaEur-o----E-isrr=•

FLI-14LECMEA;==ANAWitEFIES=00 0-/-r._E_,S DTMENSTON-A-(-401-0-81-4-01-vF NT-(-1-001-vAF (1-001

PEAC4-5=*-99=NUMBER

KOUts/T= WRITE (6.202)KOU

"ZI CALF

WRTTE-10204 ) N • M • SLALL

NMID = N+1

READ (5,102) (FNT(I), I=2,NMID ) •- MI:71 1 011 " Ll DMIVIVAIWA Vag • ft g '.M1 2 :

C MAKES THE PERIOD W EQUAL TO 2P1e

15 FNT(I) = SCALE * FNT(I) *MB: , — z • zf,..."1== Ai I I GE---n-7--PTZ2ffARE-7APRANGFA

C IN CORRECT ORDER IN THE RANGE 0 - PI

K = I +J

IF (NMID - (I+J+1)) 42043.41

443 J = J +

K = NMID -J Z!==i7-3

IF (I•EQ•NMID) GO TO 65

42 J = —1

J = J+2

AFNT(I) = FNT(K)

6b DO 62 1=2,NMID if M• W_ES

C 2N+1 VALUES OF FNT IN THE RANGE 0 - 2P1 APE GENERATED • '1' - -1== = I CI

DO 25 I=1,N

25 FNT(J) = - FNT(I + 1) -

IF (IER •GT• 0) GO TO 12

at)--E-01411tietET=4E14tiaallE4ZROINtftet

..=••==3/41.1112••1*/1Z=s11111. m401=13 It1U= .61'.:4

121HWIFH DATA SE' NUMBER 9 I4) wt

E.NU

MAX = M + 1

CKOUN MBERT—STOP

t4-1 f2 WRITE(692U3) IER KOUNT

etaMieM 99 FORMAT (I6)

V-0,---FORMA=1,6A=ECTHEFawIL) 102 FORMAT (12F6e398X) A4.1a r.. m. - n..•=u.ta 202 FORMAT (///29H RESULTS FOR DATA SE, NUMBER 914 //)

_-4ttBRoLET4tsriapnET C

C IER=0 NO ERROR C- RtA ERE

C IER=2 M LESS-THAN 0

C N MUST-BE-GREATER THAN UR LOUAL-TO-M

ER

e. • --.=i

C

USES

•

• •

0

•

:V= " = = - 2= pr.=. 03 Erg •

C JOHN-W1LEY-ANO-S

TER e4. 0 •

0 •

• 0

DEXTNG,M.HROUGHTHENvERROCEDUREBIHASBEEAEHE = 2 • '

C-- mocrI-F-1-ED-Tcy-sTmPL---rFY-T HE COMPUT rONS •

SUBROUTINE FORITTFNTITWICIAIB,IL.R, =.' a.;12 - - •

C C_ ____cHec uter-teRRo

C

20 IF(M) 30. 40+40

RETURN

0 IER=1

C

C 45_

COEF=24,0/1-20*AWT-101

SS-r-IN(CONST) Cl COS (CONS C=1.0

J=1 MX' — .7,==

70 U2=0.0

I=2*N+1

C FORM FOURIER COEFFICIENTS RECURSIVELY

7b U0=ENT(I)+2.0*c*u1-U2

80.80.75 2 I= 3.WE:=,- 'ark-Me:4 .z.":""

B(J)=COEF*5*u1

90 O=C1*C-5I*S

S=c 1 *S +Si *C C.—

..)== J+-1

ro-o—A-r-r crib __T=URN

END

the morphology and development of folds a

Documents