Models of strain variation in nappes and thrust sheets: A review

Tectonophyrics,

201

88 (1982) 201-233

Elsevier Scientific

MODELS

~blis~ng

Company,

OF STRAIN

Amsterdam-Punted

VARIATION

in The Netherlands

IN NAPPES

AND THRUST

SHEETS:

A REVIEW

DAVID

J. SANDERSON

Department

of Geology,Queens Unioersiiy, Berfast (United Kingdom)

(Sub~tted

July 3, 1981; revised version received

December

14. 1981)

ABSTRACT

Sanderson,

J., 1982. Models of strain variation

(Editor),

Strain within nappes rectangular simple

shear components

resting

operating

for the models and solutions direction

of shear.

characteristic shortening

Simple

transport

parallel

considers

shear

of gravitational

Differential

on a thrust

using simple boundary

to the thrust.

Deformation

can be modelled

case where

with a>

of nappes

gives rise to additional

movements. prolate

reversals fabrics

strains

a stretch

in the transport

(a > 1) and oblate strains

a brief consideration

in the shear direction

a=l.

tensors

pure and

are determined

with a stretch Recumbent

1. and thrust

wrench-type

nappe

Finally

An original

to various

(a) in the

fold nappes,

sheets with layer parallel

I.

in this type of model are found and used to interpret

produce

gradient

simple shear combined

arise as a special

strains

By including

conditions.

to be subject

strains.

heterogeneous

zones

coltapse,

with (Y<

sheets: a review. In: G.D. Williams

plane is considered

found for the principal

model

and folding

and thrust

and thrust sheets can be modelled

prism of material

A tw~dimensio~al

in nappes

Belts. Tectonophysics, 88: 201-233.

Strain within Thrust

of bending during

side-wall direction

shears. Solutions ramps

for the principal

and steep zones parallel

these differential

transport

to

models

(a < 1).

strains

produced

the strain history.

in a sheet passing

These predict

over a ramp

possible crenulation

indicates

of early thrust

at ramps.

INTRODUCTION

Mapping of geological structures often reveals patterns of variation in their orientation which, traditionally, have been attributed to refolding. Many studies, however, have demonstrated changing structural patterns in relation to variation in finite strain. This is clearly illustrated by Ramsay and Graham (1970) in their study of shear zones, where variations in the orientation and intensity of cleavage represent variations in finite strain. This work has formed the basis for many studies of structural patterns at various scales. The models presented in this paper have been developed from the simple shear model of Ramsay and Graham, which occurs as a special case. 0040- 195 I /82/oooO-oooO/SO2.75

6 1982 Elsevier Scientific

Publishing

Company

202

The main attempts

impetus

to the development

to interpret

maps

Caledonides

(Sanderson

(Sanderson,

1979: Rattey and Sanderson.

to simple model.

strain

of the models

of the orientations

in this paper

of fabrics

et al., 1980) and in the Variscides

models

one must

There is, in general.

and

of southwestern

1982). In attempting

analyse

carefully

no way to invert

arose from

folds in the Irish

both

England

to relate field studies the field data

field data into a unique

and

strain

the

model.

especially in the absence of quantitative strain data. The best one can hope to achieve is a model which is consistent with observation. and which explains and predicts

the

important

property

of models can act as an important

In this paper, apptication. examine

relationships

it is the models

The approach

in some

impetus

I wish to examine.

is to set up models,

them over a wide range of parameters.

calculated

and, where possible.

direct

application.

model

predictions

situation.

This

rather

(1976) and Ramsay

than

discuss

follows

that

rather

predictive

than the details

of their and to

for the finite strains

in algebraic briefly.

the application

by Ramsay

The

with the simplest.

Solutions

are discussed fully

fashion.

to field study.

starting

these are presented

Some field examples approach

coherent

and

mainly

to a particular

Graham

(1970).

(1980) for simple shear zones. A major contribution

and provides

THRUST

SHEETS.

a good example

NAPPES

AND

of the use to which strain

SHEAR

ZONES:

A TWO-DIMENSIONAL

for

to illustrate field

Coward

to models of

strain in thrust sheets was made by Coward and Kim (1981 f with application Moine Thrust zone. This work introduces the “differential transport model” paper

are

form. suitable

models

to the of this

can be put.

STRAIN

MODEL

Simple shear xws Ramsay

and Graham

shear zones affecting

(1970) have analysed

undeformed

the displacement

rock. If the zones are bounded

and strain

in ideal

by parallel

planes

of infinite extent, or approximate this condition. they show that the deformation within the zone must consist of simple shear parallel to the bounding planes and/or volume change. The deformation gradient tensor (or “deformation matrix”), D, is given by: 1 D=o i0

0 Y 10 0 1-l-A I

(1)

where y is the shear strain and A the dilation. vector u(x,y, z) to v’(x’, y’. I’) such that:

D expresses

the transformation

v’=Dv The terminology

of a

(2) used for tensors

more fully by Means (1976). In a study of shear zones,

follows

thrust

that of Malvern

sheets and nappes

(1969) and is discussed

it is convenient

to choose

203

Fig. I. Diagram of thrust sheet or nappe showing reference frame used in this paper. Shaded plane is thrust plane fx.y )* with x parallel to direction of transport.

reference axes with sume physicaf significance and which lead to simpfe sp~~jfi~a~i~n of vectors and tensors. I have chosen: x-paratlel to the shear direction or “direction of tectonic transport”. z-normal to the zone boundary or thrust plane; thus xy is the thrust plane+ (see Fig. 1). The choice of this reference frame corresponds.to that used by Coward (1976) and Ramsay (1980) but differs in axis labelling from that of Ramsay and Graham (1970), Cobbold (1977) and others. If there is no volume change (A = 0) within the zone then I) corresponds to a simple shear. The principal strains (qt, q2) and their orientations can be specified by simple equations derived from Ramsay (1980) with minor modi~~ation~ 77% = (y’f4

+ 1)“’ i- yJ2 =expfsinhh’

712=(y2/4+ R=q,/nz= tan 28’ = 2/y

y/2]

1)1’2--Y/2=expI-sinh-‘y/2} ((y2/4fl)“*+y/2)*=exp{2sinh-‘y/2}

(3af f3b) (Jc) (Jd)

where 6’ is the angle between the max. principal strain and the x-axis (Fig. 1). With the use of modern electronic calculators the hyperbolic expressions above allow rapid calculation of principal strains.

The shear strain may vary for different planes across the shear zone, giving rise to heterogeneous simple shear. In eq. 1, y must be replaced by some function of z, i.e., y =f( 2). Heterogeneous simple shear has been recognized in many natural zones of strain variation which are bounded by undeformed rock. The zones vary from microscopic to megascopie scales (e.g. Ramsay and Graham, 1970; Coward et al., 1973; Escher and Watterson, 1974; Beach, 1974; Coward, 1976, 198Oa; Stephenson, 1976, etc.). It is interes~n~ and I believe significant, that most of these examples involve either small-scale shear zones or larger zones of wrench-type o~entatio~ (i.e.

204

involving

mainly

few exceptions

horizontal (Escher

displacement

and Watterson.

shear has not proved very successful The exceptions belts.

on sub-vertical 1974: Coward,

in the interpretation

tend to be in deep-crustal

Is there a reason

shear planes).

With only a

1980a) heterogeneous of thrust-type

environments

rather

for this? Do we need to modify

simple

deformation.

than shallow

the heterogeneous

thrustsimple

shear model?

Continuity

ucr0.u strain domain bo~n~uries

There is a fundamental difference between most thrust sheets and simple shear zones. The latter are characterized by continuity of material across the shear zone boundary, undeformed being

thus bedding, dykes, veins, etc. can be mapped continuously from rock into the shear zones. The reason for this continuity lies in there

no strain

maintained outside

in planes

between

parallel

to the shear plane,

the strain ellipsoid

within

the zone and the undeformed

may be sphere

(Fig. 2).

Using eq. 2 we may put z = 0 to decompose deformation

I

and thus continuity

‘B

in the xy-plane.

is an identity

representing

the

matrix:

i+,i

21

We obtain

This minor

D to a 2 X 2 minor

the same identity

minor

regardless

of the values of y or 1 + A and this

applies to the undeformed state where y = 0 and (1 + A) = I. Continuity across the zone boundary (xy ) is not unique to simple shear zones and any two deformation

gradient

continuity

This concept

condition.

deformation deformation

~~=[

:,

by Cobbold gradient

i,

:I]

tensors

(1977)

in which the xy minors

has been and

developed

Schwerdtner

tensors have strain continuity

and~~=ly

tZ

~~]

are equal satisfy this

in other

(1982).

In

forms of banded general

any

two

across the x): plane if:

(4)

Heterogeneous simple shear is a simple form of deformation which is compatible with the undeformed state (i.e. an identity matrix) and is apparently easily achieved in nature. Nappes and thrusts sheets are not, however, subject to the constraint of continuity of stlain across the thrust plane. The thrust itself represents a plane across which there is a discontinuity of displacement of material; indeed faults may be viewed as of displacement or velocity discontinuities (Odit, 1960). Using the terminology Means (1976, p. 221) the thrust may form an incoherent strain domain boundary.

205

undeformed

deformed

Fig. 2. Diagram

to show continuity

between

undeformed

material

and that deformed

by simple

shear.

Shear strain y = tan 4.

Two-dimensional thrust sheet/nappe model To develop a more realistic model we can remove the constraint for continuity with the undeformed state, and hence allow strain in planes parallel to the thrust. To keep the model simple at this stage we will allow a stretch (q,) along the x-axis (i.e. parallel to the shear direction), but none parallel to y (i.e. 9, = 1). This leads to a two-dimensional model in which the intermediate principal stretch, q2 = q, = 1. Two such models are compared with a simple shear zone in Fig. 3. We can write the deformation. gradient tensor for such models as: 77, D=O !0

0 VZY 10 0 rlr 1

Fig. 3. A. Simple shear zone, note continuity

with undeformed

B. Combination

of simple shear (y = 1) and stretch

C. Combination

of simple shear ( y = 1) and stretch ( nx =0.7).

B and

C represent

continuous

simple

with unsheared.

thrust

sheet

models

rocks outside.

(VJ, = 2) parallel with discontinuity

y = 1.

to shear direction. at thrust

piane,

but sheared

part

206

or more simply in two-dimensions:

where rl,Xand q, are the stretches parallel to x and z, respectively, and y .= tan 4, $ being the angle through which a line originally normal to the thrust plane is sheared (Fig. 3). If the deformation within the thrust sheet is banded (i.e. varies in ;-direction only) then ?I, must be constant through the sheet to maintain strain continuity. We can consider the deformation in Fig. 3 as due to the superposition of a simple shear on a pure shear thus:

This superposition implies a pure shear followed by a simple shear and is chosen to produce a simple interpretation of y, but does not necessarily imply a particular strain history (see later). We may further factorize D to isolate a dilational component in (5a), thus:.

If there is no dilation f A = 0) we obtain:

which is an upper triangular matrix discussed at length in geological terms by Matthews et al. (1971) and used to interpret shear zones by Coward (1976). The parameter, LY,is the stretch in the x-axis, and, since we are assuming plane strain with no volume change, a! --t is the stretch in I. This is a pure shear with a strain ratio, R = (Y’.y is the shear strain of the x-axis; remembering x is a reference axis and not the maximum principal strain axis, X (Fig. 4). Matthews et al. (1971, 1974) derive expressions for the orientation and axial ratios

Fig. 4. Diagram x-axis.

to define parameters

X and Z are principal

used in two-dimensional

strain axes.

thrust

sheet models. a is stretch

paraliei

to

of the strain ellipse in terms of (Yand y, which with minor simplification are:

(7’4 From eqs. 7a, b or their graphical representation (Fig. 5) the axial ratios and orientation of the principal strains in the xz-plane can be calculated. This plane corresponds to the xz plane of the strain ellipsoid. Graphs similar to Fig. 5 have appeared in previous publications, but some care is needed to avoid confusion in their use. Matthews et al. (1971) plot (Y’and y as axes with loci of 9’ (their #) and R, otherwise their graph yields the same results as Fig. 5. Coward (1976, fig, 3) gives an incorrect version of my Fig. 5, later correctly produced in Coward (1980b, fig. 7). Coward and Kim (1981, fig. 9) produce a graphical solution of the deformation gradient tensor: A’/2

[0

Y

1

1

to analyse the strain within the thrust plane (x+v), hence their hi12 = ru’j2in Fig. 5. (This analysis will be discussed more fully later.) If we now allow the shear strain, y, to change through the thrust sheet to represent heterogen~us shear strain, then we substitute y =f(z) in eq. 6. For the strain gradients to remain a function of z only, i.e. banded deformation, a: must be constant throughout the thrust sheet. In the simple models shown in Fig. 3 only two parts of the thrust sheet are represented, an upper part with y = 0 and a lower part with y = 1. In Fig. 3B (Y= 2, representing lengthening along x, whereas in Fig. 3C cu= 0.7, representing shortening along x. Since y can vary continuously through the sheet asf(t) we can model more realistic shear zones which, since a: = constant, will have strain fields plotting along Ioci of a in Fig. 5. The practical problems in modelling strain in thrust sheets involve selection of suitable values of a and y. For the thrust sheets whose upper boundary is the earth’s surface then y = 0 at the top of the sheet, and would be expected to increase downward due to drag on the thrust plane. The form of y =f(z) is generally unknown. For viscous flow high y-values will occur within a boundary layer at the base of the sheet. The growth of the boundary layer will depend on the nature of the flow (Newtonian or non-Newtonian), its velocity and the effective viscosity; the boundary layer growing more rapidly with decreasing Reynolds number. Tectonic flow with low Reynolds number should be characterized by large boundary layers but this may be reduced by slip on the thrust plane. These and related dynamical aspects of defo~at~on within nappes have been the subject of some theoretical and scale model analyses by Ramberg (1977, 1980).

208

so

80

70

60 6‘ 50.

40

30

20-

10-

O3

2

I

4

5

6

78910

20

30

40

fro

100

R

Fig. 5. Plot of strain ratio, R. against lines and loci of constant

a-values,

I will now comment

angle, B’. between

dashed

X-strain

lines of constant

on some examples

axis and shear direction

shear strain,

of nappes

(x-axis).

Solid

y.

and thrust

sheets displaying

a

wide range of (Yand y parameters. Examples

with a * I, i.e. simple shear zones

Heterogeneous increasing

simple

shear zones have variable

y the maximum

principal

stretch

y, but LY= t throughout.

( TJ~) will rotate

towards

With

the shear

direction from an initial 45’ position and the strain ratio increases (eqs. 3a-d or Fig. 5, cr = 1 line). Ramsay and Graham ( 1970) describe many small-scale examples of this type of shear zone and larger-scale, wrench-type zones of this type are common (Coward et al., 1973; Beach, 1974; Stephenson, 1976, etc.). Thrust-type shear zones (i.e. those with shallow dipping shear planes) appear to be much less common than wrench-type zones. One very good example is described by Escher and Watterson (1974) and Escher et al. (1975) from the Nagssugtoquidian belt in Greenland. The shear plane dips ZOO-40’ NW, the shear being to the SE. Below the zone two dyke swarms with different o~entations, can be traced into the Nagssugt~uidian belt and from their reorientation and change in thickness Escher

et al. (1975) calculate a shear strain, y - 6. It is significant that continuity of the dykes can be demonstrated from undeformed rocks into those deformed within the belt, thus demonstrating that (Y= 1 along the boundary. Other examples of flat lying shear zones have been described from the Precambrian of southern Africa by Coward (1980a), but these are associated, at least locally, with shortening and thickening of the sheet (a! ( 1).

If the thrust sheet or nappe retains an approximately banded deformation then, from the continuity condition, the (Yvalue throughout should be fairly constant. If a < 1 then the sheet is shortened in the direction of shear (x-axis). Where y = 0 in the upper parts of the sheet the maximum principal stretch will be subvertical (i.e. 5’ = 900). As y increases, 5’ will rotate towards the shear plane and this will be accompanied by a rapid increase in the strain ratio (Fig. 5). Thus thrust sheets and nappes exhibiting layer-parallel shortening will develop upright folds and cleavage near the surface, which will flatten towards the base (Fig. 6B). Tightening of folds and intensification of cleavage will accompany this shallowing. Rotation of structures through angles much greater than 45” is possible (cf. simple shear zones), but large shear strains are required to produce sub-parallelism of cleavage and the shear plane. Examples of this type of strain field are common in thrust sheets and high-level nappes. The transition from upright to recumbent folding and accompanying strain increase in Millook nappe in northern Cornwall, England (Sanderson, 1979; Rattey and Sanderson, 1982) displays the features characteristic of this type of nappe.

Fig. 6. Patterns

of maximum

sheet with layer parallel

stretch

shortening

trajectories

(cleavage?)

for (A) simple shear zones (a=

(fy < 1); (C) thrust sheet with layer-parallel

lengthening

I): (B) thrust ((Y> 1).

210

Mitra

and

thrusting

Elliott

(1980)

in the southern

can be followed

downward

can also be demonstrated Ridge

Anticline,

have

mapped

Appalachians. to become

asymptotic

on a regional

which

Mitra

and

the cleavage

in an extensive

A steep cleavage,

to the thrust

scale by the cleavage

Elliott

relate

area

high in the thrust plane.

This pattern

fan across

to the eastward

of

sheet.

the Blue

descent

of the

decollement exposing higher levels in the thrust complex and hence steeper cleavage in the east. Much layer-parallel shortening is evident from the folding in the Valley and Ridge province, Harris

and Milici.

Coward parts

some of which is localized

at ramps

in the thrust

planes

(e.g.

1977).

and Kim (1981) report

of the Moine

Thrust

Erriboll.

Similar shortening

explain

fabric

and

strain

layer-parallel

complex;

and thickening variation

shortening

particularly across

and folding

in front

has been suggested shear

zones

in various

of the thrust by Coward

in southern

zone

at

( 1980a) to

Rhodesia

and

Botswana. Large amounts of layer-parallel shortening producing upright folds and cleavage is seen in many high-level fold belts. This produces a serious “space problem” if the structures

extend

by considerable

to depth in the crust, since the shortening crustal

thickening

and isostatic

uplift.

must be accommodated

Coward

and Siddans

have recently suggested a “thin skinned” solution to this problem slate belt of North Wales. They consider the layer-parallel shortening to a large high-level decollement.

thrust sheet. separated

This solution

resembles

from the main thickness

the relationship

between

(1979)

in the classical to be confined of the crust by a

folding

and decolle-

ment in the Jura and southern Appalachians. Thus many areas of “crustal shortening” may in fact be within thrust sheets, but well above the boundary layer effect and hence show no evidence of the shear strain component of deformation. Such thrust sheets must, however, differ from simple shear zones by having sub-horizontal shortening

components

(i.e. (Y< I).

Examples involving boundary parallel lengthening (a > 1)

With LY> 1, rocks above

the boundary

layer (i.e. y x 0) will have the maximum

principal stretch (qr) sub-parallel to the shear plane and hence flat-lying in a thrust-type zone. As y increases the r],-axis lifts up from the thrust plane, i.e. 6’ increases from zero, passing through some maximum angle and then decreases (Fig. 5). This effect has been pointed out for the strain path when superposing simple shear on rocks previously flattened in the shear plane (Sanderson, 1976), and is the same phenomenon as the cleavage fanning in similar folds discussed by Matthews et al. (197 1). If cleavage tracks the XY-plane of the finite strain ellipsoid then a curious low amplitude sigmoidal curvature to the fabric should develop (Fig. 6C). A weak cleavage is asymptotic to the top of the boundary layer, slightly oblique within it, and becomes stronger and asymptotic to the base. The maximum angle (8’) attainable between cleavage and the thrust plane

211

depends on the value of (Y(Fig. 7). For a: > 2.5,8’ will not exceed 5’, hence these low amplitude sigmoidal patterns may not be mappable in many areas, especially if the thrust sheet is later folded or poorly exposed. Another obvious feature of the strain at high 1yvalues is that the strain ratios do not increase much at low shear strains. For example in Fig. 7, for cx= 2, R only increases from 4 to 5 as the shear strain, y, increases from 0 to 2. Thus moderately large shear strains do not result in zones of anomalously high finite strain. There is no simple correlation between finite strain and the amount of work done in deforming the thrust sheet (see Hsu, 1974, for a general discussion of this phenomenon). Since cleavage remains at a low angle to the thrust plane in this type of thrust sheet or nappe, examples should be common in areas of recumbent folding. Tectonic slides (Bailey, 19 10; Hutton, 1979a) are discontinuities developed during penetrative deformation and represent zones of intensification of regional deformation Hutton (1979b) and Sanderson et al. (1980) discuss the strain at the Horn Head slide zone in the Dalradian rocks of Donegal and consider the strain field to be produced by superposition of simple shear on a regional strain whose n,-axis was originally sub-parallel to the slide. We can consider the regional and slide strains to

c i R

Fig. 7. Enlargement

2

3

4

5

6

of part of Fig. 5 for a>

78910

t

1, showing

RS

I 20

three strains

SO (A, B and C) discussed

-I I . I 100 in text.

develop simultaneously;

they are part of the same deformation

phase and no specific

strain history was implied by Sanderson et al. The regional strain then represents the strain field away from the boundary layer of the slide, with a = 2. As the slide is approached

y increases,

(1979b) has mapped the slide

the finite strain

cleavage

throughout

following

the cy= 2 locus in Fig. 7. Hutton

( XY plane of defortnation

the zone

and

measured

pebbles

pebbles) whose

as sub-parallel

axial

ratios

to

(X/Z)

increase to about 100 towards the slide. Since pebble outlines are obliterated near the slide the strains may be even higher there. Using Fig. 7, R = 100 and (Y= 2 would be produced bondinage

by y * 20, thus the strong reported

from

this and

LS-fabrics,

other

rotated

slide zones

folds (sheath

(Hutton.

folds) and

1979a) are easily

understood. Tectonic and

slides are common

Scandinavia,

however,

and

structures

in other

a more widespread

belts

in the orthotectonic of recumbent

phenomenon

folds

Caledonides

of Britain

and

There

nappes.

which suggests lengthening

parallel

is,

to the

transport direction of the nappes in parts of the Caledonides and other erogenic belts. Nappes with flat-lying cleavage, sub-parallel to the basal thrust, and a stretching lineation parallel to the transport direction are widely reported from such belts. It is not possible to explain these observations solely in terms of simple shear, since sub-parallelism of cleavage and shear plane requires very large strains. Yet throughout

much

of the

British

and

Scandinavian

Caledonides

strains

are not

particularly high. Shape fabric studies (eg. Borradaile 1973. 1979 in the southwestern Highlands of Scotland) and. more generally. the widespread preservation of detailed sedimentary

structures

attest to this low strain.

Simple shear requires

strain ratios in

excess of 100 to reduce 8’ to less than 5”. Arguments of this sort lead Ramberg (1977) to postulate horizontal spreading and vertical shortening, due to gravitational collapse, as important mechanisms for nappe movement. I do not wish to discuss the dynamical aspects of Ramberg’s models but merely to point out their similarity to strain

models

with LY> 1. Many

of Ramberg’s

scale models

conform

closely

to

banded deformation if one neglects end effects in the collapsing slabs. and. hence. the strain fields within them can be factorized into cy and y components. The presence of the lower boundary layer is obvious in Ramberg’s slip along the base they tend to develop complex “extrusion

models, although for flow” near the base.

Even if one doubts the dynamics of Ramberg’s model in connection with nappe which 1 do not, the reasons for considering such models strongly supports

tectonics,

a > 1 strain Factorizing

in many nappes. struin into (Y mu’ y components

So far we have simply considered using different values of (Yand y to model strain variation in thrust sheets. The collection of strain data and mapping of the orientation of cleavage (B’) can be viewed as a means of verifying, but not necessarily proving, these models. The inverse problem of taking strain data and

213

trying to factorize it to obtain values of a and y is obviously an important procedure. This would allow, not only verification of the model but also, the mapping of (Yand y variation within thrust sheets. In addition a knowledge of y is essential if strains in thrust sheets are to be integrated to yield information on original stratigraphical thickness (Hossack, 1978; Cobbold, 1979) or estimates of ductile displacement across the sheet (cf. Ramsay and Graham, 1970, for simple shear zones). The main problem with any attempt to factorize strain into more easily interpreted components is that one must make certain assumptions. This is very clearly iilustrated by the R,/+type techniques which are widely used to factorize shape fabrics into tectonic and sedimentary components. The basis of the R,/Q method (Ramsay, 1967; Dunnet, 1959, and later developments) is that one assumes an orientation for the tectonic strain component, usually based on the orientation of cleavage and stretching lineation. Any departure of the shape fabric axes from the tectonic strain axes is then used as a basis for factorization. Usually a number of objects are measured and some “‘statistical” estimate of an “optimum” factorization attempted. The results of all factorizations depend on the validity of the assumptions, the ability to make sufficiently accurate measurements, and obtaining sufficiently narrow “confidence limits” for useful estimation of the parameters being calculated. In many circumstances factorization is simply impossible, even if one is confident of the assumptions. Again the R,/# technique illustrates this point, since when the tectonic fabric is parallel to the sedimentary fabric one cannot apply the usual factorization methods (see Hutton, 1979b, for a possible alternative approach). How realistic, therefore, is factorization of cy and y components in a thrust sheet? Let us assume we have good elliptical strain markers, having no ductility contrast with matrix, and a statistically spherical initial shape fabric. The main limit to factorization is the determination of the angle 8’. Figure5 can be used as a simple graphical basis for discussion of factorization. If one knows R and 6’ to some confidence limits, then plotting on Fig. 5 determines the range of possible 01and y values. Let us assume we know R to I 10% and 8’ to *2’, these-are fairly precise limits which are probably rarely attained in strain analysis of natural rock materials. Clearly where the IYand y Ioci are widely spaced, as in the top and left of Fig. 5, reasonably precise factorization is possible. The main problems arise at low 6’ and high R values which correspond to high LYand high y values. Figure 7 shows an enlargement of Fig. 5 in which the B’ axis is stretched to facilitate the use of the diagram. Consider point A, in Fig. 7, with R = 2 and 8’ = 15”, we can estimate a: = 1.25-1.35 and y = 0.5-0.8. With the same accuracy, however, at point B (R = 8, 8’ = 2”) we get fairly precise estimates for OL = 2.5-2.9, but y may vary from 0 to c. 3.3, Further consideration of Fig. 7 will indicate that where cy is high, say > 2.5, then 8’ must be known at confidence limits of a 2O to allow any useful factorization of the Y component. At high vafues of R, where the loci of 1ystart to converge towards

8’ = 0, then cr cannot

be factorized

without

extremely

precise 8’ estimates.

Figure 8 shows two model strains,

both with R r= 16, with a difference in orientation of the X-axis of 3.2”. In one case y = 0. in the other y = 8. If these shape fabrics occurred

in separate

outcrops,

allow such a small angular either, was parallel

measurement

variation

to the thrust plane.

higher. we could not usefully

error and later movement

to be measured,

estimate

would

not

nor would we know which. if limits of y .= 0 I 8 or

Thus with confidence y. nor the ductile

displacements

in the sheet.

The OLvalues also vary, being 4 or 3. This may not seem much. but if one attempts restore the stratigraphical 4 and

3 km respectively.

produce

a considerable

Thus, at high strains possible,

except

under

of a sheet 1 km thick one would obtain

thickness With

a large

difference

length:

to any balanced

thickness

aspect

cross-section

and where 8’ is small, I do not b&eve exceptional

circumstances.

These

to

values of

ratio

this would

(Hossack.

1979).

strain

exceptional

factorization

is

circuf~stances

are clearly illustrated by attempts to factorize strain in the Moine Thrust sheets and. since they allow some interesting conclusions to be drawn. I shall discuss them briefly. Within the Cambrian rocks of the Moine Thrust zone. worm burrows ~~~#~jt~z~~.s). known locally as “pipes”, provide strain markers which in many ways are ideal for factorization In undeformed rocks the pipes are normal to bedding and circular in cross-section, hence iay~r-parallel strain ratios and shear strains are easily measured. They were first used by McLeish (197 1) in an essentially two-dimens~ona1 strain study.

He demonstrated

the heterogeneous

nature

of the strain,

with high strains

in

the mylonite zones. A further study of mylonitic pipe rock by Wilkinson et al. ( 1975) favoured a simple shear deformation with y = 12.0 and 9.0 obtained for two samples.

This corresponds

to strain

ratios of X/Z*

100. In a more wide ranging

Fig. S. Plot of two strain ellipses with axial ratios R = 15. The straight indicate

separation

of two bedding

y = 0, a = 4. 3. Ellipse produced

planes

originally

lines above and below each ellipse

the same distance

by 7 = 8. cs = 3. For further

discussion

apart. see text.

A. Ellipse produced

by

215

study of deformed “pipes”, Coward and Kim (1981) factorized strain into layerparallel stretch and shear strain. They point out that the thrusts generally parallel bedding in “flats” and that refraction of cleavage and pipes is also in zones parallel to bedding. Thus the banded nature of the deformation is sub-parallel to bedding and hence (Y/V factorization should be possible. Coward and Kim’s analysis is three-dimensional and will be discussed again later, but the main two-dimensional conclusion of their work is to indicate variable layer-parallel shortening of up to 50% (i.e. (Y= 0.5) in some areas. Near the Moine Thrust at Glencoul, however, the “pipes” are elongate, with ratios of c. 5: 1, parallel to the WNW-ESE transport direction (i.e. (Y= 5). Thus there is considerable variation in (Yas well as y in the Moine Thrust zone. One wider implication of these results, and the application of the thrust strain model, concerns the role of irrotational and rotational strains in the mylonite zones. Johnson (1967) proposed an essentially irrotational strain within the Moine mylonites, whereas Wilkinson et al. (1975) argued for rotational simple shear. In essence this debate reflects the more widespread discussion of the role of rotational versus irrotational strain in natural rock deformation. Using the thrust strain model I think we can go some way to resolving this problem. Both McLeish (197 1) and Wilkinson et al. (1975) point out that the pipes form small angles, c. 2’ with bedding in the mylonites. McLeish shows clearly that an irrotational strain requires bedding to be initially at high angles to X, or that a simple shear must have a shear plane initially at a moderate angle to bedding. The latter inte~retation was restated by Wilkinson et al. Neither of these interpretations are consistent with Coward and Kim’s evidence for thrusting and banded deformation sub-parallel to bedding. Let us consider (Y= 5, as suggested by the elliptical sections of the pipes in bedding, and a shear strain of y = tan (88”) = 30, which is necessary to rotate the pipes to within 2O of a bedding-parallel shear plane. We can plot these parameters on Fig. 7. point C, and conclude that X should he at about ! ’ to the shear plane (bedding) and R 2: 61. These strain values are in general agreement with those of McLeish and Wilkinson et al., but now consistent with a more realistic model of bedding parallel shear in the thrust sheet.’ Incidently this example again illustrates that very high shear strains accompanied by a: > 1 are possible without producing extreme axial ratios (R). A shear strain of y = 30 in simple shear (a = 1) gives R = 902 (cf. R = 61 above). Finally I should point out that this factorization of mylonite strains would not have been possible with elliptical markers, since one simply cannot measure mean long axis orientations to sufficient accuracy in the field. The analysis does, however, depend on the assumption of bedding-parallel shear and thrusting which must be established from other observations.

So far I have discussed only the finite strain in the thrust model, i.e., the strain field. One could envisage the strain at any point in the thrust sheet as following

21h

many different markers

histories

(Elliott,

In this section

or strain

paths.

Unless

1972). one can say little about I wish to distinguish

clearly

one has suitable

incremental

the strain path from the strain

between

the factorization

of strain

strain field. and

the true strain path. In factorizing

the deformation

gradient

tensor (eys. 5a, b), I chose to premultiply

a pure shear by a simple shear. This was done, quite deliberately, useful,

and potentially

“mathematical cation

measurable.

convenience”

of tensors.

parameters

in order to produce

CYand y. We must not confuse

with the process of superposing

strains

this

by premultipli-

A strain history of pure shear followed by simple shear is different

from one of simple shear followed

by pure shear. given by:

Equation 8 is different from ey. 6: if (Yand y are the same in both then the resulting tensors are different. Two different deformation gradient tensors produce the same finite strain if all their elements are equal. (Here I include the rotational as well as distortional components of strain.) Thus for the same deformation gradients:

and hence, equating

eiements: ( 1Oa)

lx1 = Ly?_ and

(10t.J)

Ly; ‘Y, = cw,y, Substituting

1Oa in lob:

(Y, = C2y,

(where a, = LYE = a)

t 1Oc)

From eq. 10~. one can see that for the same pure shear (a) we need to use a different simple shear ( yZ ) acting prior to pure shear. than that ( yr ) acting after pure shear. to produce the same finite strain. The use of eq. 6 in preference to eq. 8 simply allows

one to define y more simply, since, in Fig. 4, the use of eq. 8 would give: CL ‘y= tan+ Coward ( 1980b) and Coward and Kim (198 1) have compared the (r and y for both sequences. Clearly if one knows that the sequence of deformation occurred in a given order, say pure shear followed by simple shear, then one can easily model the strain history. In a nappe or thrust sheet there is rarely any such simple sequence. The nappe

models of Ramberg

(1977, 1979) indicate

shear strains. To consider the simultaneous development

the interdependence

of stretch

and

of a and y strains we need to work

217

with

the strain-rate

Ramberg, tensors

tensors,

1975). To obtain

rather

than

the combined

the deformation velocity

gradients

(Hsu,

1967;

field we must sum the strain-rate

for pure and simple shear:

(11) (simple

(pure shear)

shear)

Following Ramberg (1975) the deformation and simple shear rates, is given by:

ILexdit)

D=

& {exp(lt)

gradient

tensor, assuming

constant

pure

- exp(-it)}

exp( --it)

0

exp( it)

y/l

sinh( it)

0

exp( -rr)

1

(12)

Selecting values of i (the rate of change of extension) and p (the shear strain rate) allows calculation of D and hence the finite strain path. The ratio y/i in eq. 12 can take values from 0 (pure shear) to infinity

(simple shear). The ratio will vary through

a thrust sheet being low in the upper parts and high in the boundary layer near the base. The simplest possible model would use I as a time-like parameter and have p increase downward through the model. A very useful approach to understanding the kinematics and dynamics of thrust sheets would be to analyse these strain rates in scale models, eg, using grids, and compare them with data from incremental strain indicators A

in natural

examples.

THREE-DIMENSIONAL

SHEETS AND

MODEL

OF

DIFFERENTIAL

TRANSPORT

WITHIN

THRUST

NAPPES

The previous section dealt with a two-dimensional model of a thrust and considered the deformation in the profile or xz-plane. Such models may be applied to thrust

sheets

constant

and

nappes

displacement

with

along

relatively

strike.

simple

In this section

geometry

and

I will consider

subject

to fairly

a three-dimen-

sional model which allows differential transport of different parts of the structure. In the model material particles are considered to be displaced in the x-direction, but the amount of displacement may change along the y-axis (Fig. 9). This displacement gradient in the xy plane gives rise to a wrench-type shear strain, with xz as the shear plane and the x-axis as the shear direction, in addition to the thrust-type shear discussed

in the two-dimensional

The deformation components:

1:

i”

:::‘I=[;

gradient

model. tensor

I” (wrench)

;].[a

can be considered

; (thrust)

r].

I 0 a

to be composed

01

(Y-’ 0

(pure shear)

1

of three

(13)

Fig. 9. Diagram

to simw reference

t/, and U, represent

two different

axes and definitian displacements

of &

parallel

for differential

transport

model of thrusting.

to .x-axis which give rise to the shear component

yw = tan I/+.

The order of premuItiplication of these tensors is again chosen for .‘mathematical convenience” and does not imply a strain history, The parameters yw. yr and Q have a simple specification with this factorization (Fig. 9): Yw = tanr//,

(140)

yT = tan \i/r

(14h)

a =rt,

(1467)

The subscripts W and T denote wrench and thrust geometry of the simple shear components. The order of multipIication of the simple shears does not affect the factorization. since: [s

I”

~i;[::

r

8jii

;

~~~1~

f

l‘i./::

I”

91

(15)

The order of multiplication of the pure shear by the combined simple shears is subject to the same considerations as discussed for the two-dimensional model. The order in eq. 13 being chosen to give the simplest factorization. A model of this type has been used by Coward and Kim (1981) to explain and analyse the deformation within bedding planes in the Moine Thrust zone. Rattey (1980) and Rattey and Sanderson (1982) have also used a model of this type with (Y= 1, to determine incremental strains and to predict fold axis orientations within thrust sheets in southwestern Cornwall. I shah develop the strain model in two stages; initially with cy=I I and then consider the affect of ty + 1. With (Y= 1 aIgebraic solution for principal strains of the deformation gradient tensor in eq. 13 is possible. Since the model involves differential displacement along the thrust plane, I shall call it simply the “differential transport model” of thrusting. differential

transport model, with a = I

With LY= I, the deformation gradient tensor simplifies to eq. 15. We can consider the deformation within the sheet as operating on two orthogonal slip planes, parallel

219

to the xy-piane (thrust plane) and the xz-plane, with a common slip axis, parallel to x. Slip of this form gives rise to volume constant plane strain, with q2 = f, r13= s; 1 and K = ln( q, /q,)/ln( q2/q3) = 1. The strain ratio, R = TJ,/vi is given by: R=f{A&(A2-4)“2)

@a)

with: A==2+y$f-y$

Wb)

These results are derived in the appendix. The strain ratios for different values of yr and yw are plotted in Fig. IO, which illustrates one interesting feature of these results. The strain ratios are fairly intensitive to changes in the smallest of the two shear components. For example, a thrust sheet with yr = 1.5 has R = 4 if yw I= 0, R = 4.27 if yw = 0.5 and R = 5.05 if yw = 1. Thus even at fairly low strains, the finite strain within the sheet does not change much for increasing yw, provided yw a yr, and vice versa. Figure 11 shows the orientation of the principal strain axes. The orientation of the Y-axis is controlled simply by the ratio, yw/yr: 8, = arctan CYW%)

07a)

and always lies in the yz plane of the thrust sheet. The X-strain axis rotates towards the shear direction (x-axis), such that the angle between the two, 8, is given by: 8, = + arctan[2/(

& -i- Y+)‘~~]

0.5

( 17b)

3

45

Fig. IO. Loci of strain ratios (R) plotted on graph of yT verses yw, n = 1. Tltiti sloping lines represent equal

yT/yw.

Fig. I 1. Equal area stereogram y*=fY$+y+/)‘;~. great

showing orientation

see text. Note position

circles for constant

yT /uw.

of principal

strains

of Y-axis fixed by yT,/yw

The thrust

reference

for varying yw and y.r. with a = 1. ratio (see ey. 17a). X and 2 lie on

axes are indicated

and strain

axes plotted

for a

sin&r-al yw shear.

Obviously

if either yw or yr = 0, the deformation

is simple shear and ey. 17b reduces

to eq. 3d. Using these solutions

for the strain

can model and interpret to vertically

through

various

in thrust sheets with differential

changes

along strike, i.e. in y-direction,

the sheet. This model

is directly

applicable

transport

we

in addition

to the strain

at

side-wall ramps oriented parallel to the xz-plane of the thrust. Such ramps have been widely recognized in the southern Appalachians (Harris and Milici, 1977) and increasingly in other areas of the thrust tectonics. Many of the wrench faults reported from thrust complexes would now be regarded as side-wall ramps. At some level in the thrust sheet the yr component can be regarded as constant, thus as the side-wall ramp is approached drag produces an increasing yw component, and hence increasing yw/yT ratio. With increasing yw, the XY-plane (cieavage) steepens and its strike rotates towards the transport direction (x-axis). In Fig. 12 I have modelied the effect of increasing yw from 0 to 4, on a constant yr = 1.5. The X-axis “100ps” towards parallelism with the transport direction and is never oriented more than 15”

221

Fig. 12. Equal area stereogram “looping”

from

showing

yT = 1.5. Note steepening

constant

of X-axis. For further

the xz-plane.

(cleavage)

for yw increasing

from 0 to 4 at

and swing of strike of XY towards

rotation

x with increasing

y,,, and slight

discussion

Higher

of XY-plane

see text.

yr values

produce

even less deviation

of X from

the

xz-plane. Simulation

of a number

of such side-wall

ramp models allows certain

conclusions

to be drawn: (1) Cleavage side-wall

(XY-plane)

will

steepen

and

its strike

will rotate

towards

the

ramp.

(2) The “fold axis” of the cleavage (i.e. p intersection trend of the thrust and parallel (3) The stretching

lineation

to the side-wall

in Fig. 12) is normal

to the

ramp.

(X) rotates slightly out of the xz-plane,

but this swing

will not be great and in most cases X makes only a slight angle with the side-wall ramp. Larger swings in X occur where yr is small. An example:

the “‘strike swings” and “‘steep zones” of north Mayo

The three features observed in the model can be clearly observed in the D, strain field of the Caledonian deformation in north Mayo, Ireland (Fig. 13). Sanderson et al. (1980) have described part of this strain field in terms of a southerly increasing dextral,

wrench-type,

shear zone superposed

on the gently inclined

nappe pile. Using

Fig. 13. Map of main D, Caledonian

fabric elements

of Achill Island, the other east of Belmullet zones are noted. The general

the orientation data simple

found

nappe

transport

of the cleavage

outside

in northern

is to the west. (After Sanderson

and stretching

the steep zone, Sanderson

shear to produce

Mayo. Note two steep zones one south

(B on map). Sense of wrench type shears responsible

a model of the strain

hneation,

for shear

et al.. 1980.)

together

with shape fabric

et al. superposed

an E-W

dextral

field. This analysis

produced

a close

correspondence between observed and model orientations of fabrics and symmetry of shape fabrics. The analysis depended, however, on assuming a particular strain history of nappe emplacement followed by wrench-type shear; a sequence which, although probable, was not known with certainty. In the model proposed in this paper, the strain field produced does not depend on the order of superposition of thrust and wrench-type shears (from eq. 15). Thus the strain field in northern Mayo can be interpreted in terms developed by Sanderson et al. (1980), but without the necessity of assuming a sequence of strain. Another steep zone runs eastward from Beimullet (Fig. 13) and may represent a compliment~y sinistraI-tie steep zone, but insufficient fabric data exist to analyse

223

this at present. In general terms, we can interpret the strain field in northern Mayo as involving a series of nappes advancing westward at different rates. Within these nappes differential movement gives rise to wrench-type shear zones parallel to the direction of tectonic transport. These zones modify the strain field producing steep zones with sub-vertical cleavage and strong L or LS fabrics. We must examine the model further before obtaining an explanation for the L-fabrics.

We now consider the addiiion of a pure shear component (eq. 13) involving a stretch, 01, in the direction of tectonic transport. This more general model, with cy+ 1, produces a triaxial strain, i.e. K+ 1. Hence the treatment in the appendix does not yield X, ==1, and the resulting cubic equation cannot be factorized to yield a quadratic. Since the algebraic solution of the cubic equation is complicated, I have chosen to solve the tensor DDT by numerical methods, yw

ff-‘yr

1

0

0

cc2

(181 I

Iterative solutions of the eigenv~u~ and eig~nv~tors of DDT can be found using standard computer subroutines. The calculations for this paper were carried out using a general purpose package for handling 3 X 3 matrices written in BASIC for an APPLE II microcomputer. The.eigenproblem is calculated by the arbitrary vector method, Figure 14 shows some results produced for different cy values by increasing the shear strains such that yw = yT, some other solutions for y-r = constant are shown in Fig 15. The main conclusion drawn from many solutions involving different . , combmatlons of yw, yr and a is that the a value has an important influence on the s~rnrnet~ of the resulting strain ellipsoid. This may be summarized as follows: cy= 1 produces plane strain ( K = 1) cyC 1 produces oblate strain ( K K 1) ar > 1 produces prolate strain (lilr

1)

Stereograms of the loci of principal strain axes indicate similar general features to the a: = 1 case with steepening and strike rotation occurring in zones of high yw. There are two specific features of the strain field that I wish to emphasize at this stage. Where a: ( 1) the XY-plane will be upright in the upper parts of nappes and thrust sheets (i.e. where yr is low), hence any increase in yw approaching a side-wall ramp will produce a large strike swing about a steeply inclined axis. In effect the graph in Fig. 5 may be used to predict the angle between cleavage and the transport direction

Fig. 14. Flinn diagram

showing

when a = 1. Loci of a -4,

solutions

to deformation

2, 0.5 and 0.2 constructed

to vafues of yw = y,.. Note prolate

strains

by using yw as the shear strain.

when a z

gradient

for constant

tensor of differential yw/y-r

transport

= 1. The numbered

model

points

refer

I, ablate when a < 1.

This large strike swing would be very similar

to the

passive rotation of cleavage into a later wrench-type shear zone. Such zones should be fairly obvious and may be mistakenly attributed to a sep&rate, later deformation phase. Similar zones of high yw occurring where yr is large, as in the boundary at the base of a nappe, would be characterized by steepening of the XY-planes. very different

“drag effects”

Where a > 1, the strains and

of the XY-plane wiil become prolate

are possible

layer Thus

in (r =CI thrust sheets.

in zones of differential

displacement

hence

nappes, become

these may be characterized by L-fabrics. In the higher levels of such where yr --f O? prolate shape fabrics will develop fairly rapidly and may K = co. At this point “axis swapping” occurs with a transition from

flat-lying XY-planes, through L-fabrics, to steep XY-planes with the X-axis remaining sub-horizontal throughout. This involves a swapping of Y and Z-axis and is similar to the strain paths discussed by Sanderson (1976, case 3s). As discussed previously, it is in the recumbent fold belts, possibly produced by gravity collapse, that ey> 1 may be expected. Thus the relationship between linear fabrics, side-wall ramps and steep zones trending parallel to the transport direction should typify such belts as the orthotectonic Cafedonides of Britain and Scan-

225

Fig. 15, Flinn diagram, continuous lines indicate loci of finite strains with yT =O and I, dashed lines indicate yw values. Lines constructed for a = 1.5 in an attempt to model the strain field of the Keem conglomerate. Strain data plotted as stars with tie lines indicating different analyses of same outcrop. With the exception of one outcrop in a flat-lying slide zone, all data lie between yT =O and 1.

dinavia. The zones of prolate strain in the Bygdin conglomerate, elongate parallel to the regional stretching direction (Hossack, 1968) are the sort of fabrics to be expected in zone of differential displacement in nappes of this type. In northern Mayo it is significant that prolate strain characterizes the steep zones. On Achill Island (Fig. 13), the Keem conglomerate can be traced for a short distance southward towards the steep zone and the shape fabric becomes increasingly prolate. Figure 15 shows a plot of the shape fabric data from this conglomerate, together with loci representing the model strain fields for 01I= 1.5 and various yr values. A more detailed study of this strain field will be presented elsewhere, but the close correspondence between observed and model fields supports the previous discussion of the fabric patterns (see also Sanderson et al., 1980). STRAIN AND STRAIN HISTORY AT RAMPS

So far we have considered thrust sheets moving on planar surfaces, but it is well known that many thrusts have sharp steps or ramps which generally allow the thrust

226 to cut up-section.

as it is translated sheet passing

Rich (1934) showed over them. Figure

sheet is first bent upwards the ramp. It is this bending I shall discuss probably

folding

of the sheet

16 shows a very simple representation

of a thrust

over a ramp. Neglecting

the “toe” of the sheet, it can be seen that the

as it enters the ramp and then downwards which produces

two very simple

more realistic,

that these ramps produce

in order

models

as it passes out

the folding over the ramp. In this section of this bending,

to demonstrate

the second

some features

of which

of the strain.

is

and

particularly the strain history, within the sheet. Various dynamical models of ramping have been proposed (Wiftschko. 1979: Berger and Johnson, 1980). hut my sole concern

here is with simple kinematic/stra~n

models.

Bending fold mode/ Figure 17 illustrates one way in which material could flow over a ramp. The vertical thickness of the sheet is kept constant and bending achieved by vertical shear. Using

The shear strains are of opposite the convention of dextral shear

Fig. 16. Diagrammatic is indicated

representation

by the line segment

of bending

parallel

sense on entering as positive, there

of thrust sheet during

to bedding.

and leaving the ramp. is a positive shear on

ramping.

The sequence

of bending

227

Fig. 17. “Bending (right)

model”

and leaving

(left)

for a ramp (see text). A. Finite s&rain state. B. Increments the ramp.

C. Possible

patterns

of cleavage

developed

of shear on entering due to bending

strain

reversals.

entering

the ramp and a negative

the shear strains y=

is related

shear on leaving

it (Fig. 17B). The magnitude

of

simply to the ramp angle, 8:

tan6

(19)

Material originally on the lower flat will suffer equal and opposite shears on passing through the ramp to the upper flat, with no nett finite strain. Material originally in the ramp

suffers

only the negative

shear as it collapses

occupying the ramp at the end of thrusting finite strain is distributed as in Fig. 17A.

to form the toe. Material

suffers only the positive

shear. Thus the

From eq. 19, the bending strain will be fairly small for low ramp angles. Using a typical value of 6 = 20°, the resulting shear strain is y = 0.36, which is equivalent to a strain ratio, R = 1.4. These bending strains may be superposed on strains already developed by drag on the thrust plane.

228

Flexurul flow

Figure flexural

model

18 shows another flow parallel

remaining

constant.

model for ramp folding

to bedding,

This is probably

sheet at high structural

in which the deformation

with the orthogonal

thickness

of the thrust

is by sheet

a more realistic

levels. If we consider

response of a layered thrust the toe to be undeformed, then the

shear strain due to each bend is given by: y = 2 tan 6/2

(20)

The sense of incremental

shear is shown in Fig. 18B. As material passes from the lower to upper flat it will have a finite strain distribution as shown in Fig. 18A. The strains in the flats are y, and that in the ramp is 2y, y being given from eq. 20. Again there is a reversal of incremental strains

strain on entering

from 2-r to y. The average finite strains

incremental

strain

:

0

the upper flat to reduce the finite

are again fairly low for typical

ramp

Q

Fig. 1X. “Flexural flow” model for a ramp. A. Finite strain state (assuming no deformation in “toe”). B. Incremental strains, read from right to left, on passing from lower flat. to ramp, to upper flat. C. Pattern of cleavage developed due to bending strain reversals (see text).

229

angles

of S = 20°, which would

may be in addition

produce

to any generated

most of the shear strain

y = 0.35 and R = 1.4. Again

may occur in incompetent

these may be considerably

these strains

in the sheet due to drag. In a multilayered layers and

sheet

hence the strains

in

higher than the average.

Strain history in simple ramp models Although the average finite strains developed around they may represent important localized strain variation. common first bent discussion

to both models,

is the necessity

a ramp are generally small, A more interesting feature,

for shear reversals

to occur as material

is

one way and then the other as it passes over the ramp. I will restrict of this strain reversal to the more realistic, flexural flow model.

Consider the strain history.of material travelling from the lower to upper flat. In the lower flat, material will have suffered a shear in the direction of tectonic transport (sinistral or negative in Fig. 18). This, combined with any drag on the thrust plane, would produce a cleavage dipping to the right. The cleavage may be intensified in incompetent units. On entering the ramp an additional negative shear strain is added. This would be expected to further intensify the cleavage, which would steepen somewhat due to the external rotation of bedding being greater than the internal rotation of cleavage due to strain. On leaving the ramp, the layer suffers a positive (dextral) shear, opposite to the previous shearing and sense of transport. This reversal of shear strain would reduce the finite strain in the sheet. If the fabric in the sheet simply reverse (positive)

reflected

the finite strain

shear, however,

as to cause it to be shortened. cause

the pre-existing

cleavages

cleavage

will operate

it too would decrease on an existing

(S, in Fig.

18C) to be crenulated, applied

produces a similar pattern of cleavages (Fig. 17C). It is suggested, on the basis of these models, that crenulation

producing

to the bending of the main

may occur when sheets pass over ramps. An important

The

in such a way

I think it much more likely that this shortening

in the upper flat. The same type of argument

sheet cleavage

in intensity.

cleavage

would two model thrust

consideration

is the magnitude of the strain components developed, and these depend on the ramp angle 6. With both models average shear strains of co.35 are predicted for typical ramp angles of 20’. Since Ramsay and Graham (1970) report schistosity at nearly 45“ to simple shear zones in undeformed rock, it is clearly possible for recognizable fabrics to form at low shear strains. In my experience y = 0.35 is just about the limit of detection for crenulation of a slaty cleavage, as it represents about 16% shortening. The S, cleavage in Figs. 17C and 18C is, however, in a very favourable orientation for crenulation being at a small angle to the minimum incremental stretch. If the reverse shear was localized in incompetent bands it may be more easily recognised. A search for this “SZ” cleavage in multilayered through ramps with S > 20’ might confirm its existence.

lithologies

having passed

230

CONCLUSIONS

The models discussed in this paper were developed in response to problems encountered in mapping fabrics. folds and finite strain in the field. In the course of the development of the models many new relationships were discovered, The models now need to be tested against further field data. One encouraging feature of this approach has been the jndep~ndent Formuiat~on and application of the d~ff~reI~tja~ transport model by Coward and Kim (1981). These workers have shown very clearly how, with careful study of strain markers, useful mapping and ~nte~retation of strain in thrust sheets is possible. Further development of the models is possible. An e~an~ination of convergent and divergent flow which would provide stretch components in the.~-direction wouid be an obvious area for further study. APPENDIX To find the principaI strains from a deformation gradient tensor. i?, one simply forms the tensor DB’r. This is a symmet~cal tensor which has eigenvalues equal to the principal quadratic elongations (X, = ?!I i = 1,3) and eigenvectors parallel to the principal axes. BD’ is the inverse of the Cauchy deformation tensor (Malvern, 1969) and is sometimes referred to as the Finger tensor (D. Mainprice, pers. commun.~ 1980). if:

Solving the characteristic equation: (D@--hC)=O we

get:

(tiy~‘.cy:-x){l-x)‘-(y:.~y?2_)(1-X)=0

(A.11

Hence h = 1 is a solution to &hecharacteristic equation and thus an eigenvalue of DD r. This demonstrates that one principal quadratic elongation is unity and, for constant volume deformation. this must be X2 = I, hence the deformation is plane.strain. Dividing through eq. A, 1 to eliminate (I- A) produces a quadratic equation which simplifies to: V-(2+y$+y”+)h+I=o This has solutions: h=;(A++-4)‘“7 where: A=2+&+.Yf

231

(16a,b) in main text.

These are equations The eigenvectors

can be found

by substituting

each eigenvalue

into:

(DDT--XI)e=O and solving the resulting eigenvector, yr-plane.

set of homogeneous

e, (0, yw, or),

equations

corresponding

to find the eigenvector,

The plunge of the Y-axis in the yr-plane

1 yields the

e. Using X, =

to the Y-axis of the finite strain

ellipsoid

and lying in the

is given by: (A.2)

8, =arctan(~w/~r) The eigenvector,

e,, corresponding

to the X-axis obviously

lies in the plane normal

to Y or ez and makes

an angle 8, with the x-axis: f?, = jarctan[2/( Equations

& + v$)“‘]

(A.3)

A.2 and A.3 correspond

to eqs. l7a, b in the main text.

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