Determination of finite strain from passive initially planar markers deflected in ductile rocks undergoing simultaneous superposition of heterogeneous strains

Tecronophysics,

120 (1985) 119-131

Elsevier Science Publishers

119

B.V.. Amsterdam

- Printed

in The Netherlands

DETERMINATION OF FINITE STRAIN FROM PASSIVE INITIALLY PLANAR MARKERS DEFLECTED IN DUCTILE ROCKS UNDERGOING SIMULTANEOUS

JACQUES

SUPERPOSITION

OF HETEROGENEOUS

STRAINS

INGLES

lnstliut Universitaire de Technologie, Dkparrement de GCnie Civil, 50 A, chemin des Maraichers, 31062, Toulouw Cedex (France) and Laboratoire de Tectonophysique, Universitk P. Sabatier, Toulouse (France) (Received

October

30, 1984; revised version

accepted

February

12, 1985)

ABSTRACT

Ingles,

J., 1985. Determination

rocks undergoing From an empirical

we study

passive

planar

initially

from passive

superposition

initially

of heterogeneous

model of strain in ductile heterogeneous

bulk pure shear, sometimes

of finite strain

simultaneous

the influence

markers

the instantaneous

and

strain,

of the parameters describe

markers

deflected

shear zones simultaneously

for determining

in ductile

Tectonophysics, 120: 119-131.

of the deformation

a method

from these deflected

planar strains.

undergoing

a

on the shape changes

of

the finite

strain,

and

even

markers.

INTRODUCTION

Shear zones with deformed fields (Ramsay and Graham,

walls represent the most general type of displacement 1970; Coward, 1976; Cobbold, 1977). These strain

patterns may result from the superposition of several strain fields geologically synchronous or not (Ramsay, 1980). In a study of the reorientation of rigid inclusions

by simultaneous

combination

of pure shear and homogeneous,

steady in

time simple shear, Ghosh and Ramberg (1976) have shown that the rotation passive marker line is the same as that of an infinitely long rigid inclusion. developed

an empirical

model of strain

in ductile

zones simultaneously

of a We

undergoing

bulk pure shear and heterogeneous simple shear and showed that continuity deformation at the boundary of the sheared rock requires that the shear strain

of be

non-steady in time (Ingles, 1983). This study allowed us to determine the kinematics of the deformation (field of rates of displacement, particle path) and the finite strain. In this paper we study the influence of the parameters of the deformation on the shape changes of passive initially planar markers and describe a method for determining the finite strain, and even sometimes the instantaneous strain, from these deflected markers.

0040-1951/85/$03.30

0 1985 Elsevier Science Publishers

B.V.

120 MODEL

The principle

of the method

by which particle

given in detail in an earlier pubhcation The hypothesis

made

and the main

here in order to provide We study undergoing principal zone) strain

plane

strain steady

rates;

results

path and finite strain

are found

1983) and will not be repeated

from this earlier

is

here.

work are summarized

continuity.

a bulk strain

(Ingles,

of a ductile

and

uniform

shear

pure

P, > 0 for contraction

and a heterogeneous (Fig. la):

simple

incompressible

rock

simultaneously

with axes x y (e, = -P,.

and &, < 0 for extension

shear parallel

are the

of the shear

to the x axis. The rate of shear

+ = ‘I&(Y)

(1) is constant in a central zone of homogeneous strain (g(y) = 1 for 0 G y G h,). decreases finearly outside this (k, Q,Y G h) and vanishes at the limit of the shear zone (y >, h); the shear zone is symmetrical semi-thickness is taken as unit of length.

relative

to the x axis and its initial

Continuity of deformation at the boundary of the shear zone leads to a relation between i, and the rate of translation ;ir of the non-sheared zone showing that the in time, shear strain is generally non-steady ‘/ = zi-,g(.4 exp(A/k, where k = (7 + h,,)/2,

(2) h,,

being

the initial

semi-thickness

of the homogeneously

\\ a------------~____..s

Fig. 1. Deformed of y; (b) deformed

\\

shape of an initially shape.

_

----k

___._.._.__.... _. . . . . . . -.~_..._. ‘\

linear passive marker

through

\

the ductile shear zone: (a) variation

121

strained

part of the shear zone, p = P,t characterizes

central

We can define through

a bulk index of non-coaxiality

the shear zone and varying

local and instantaneous shear zone;

s differs

shearing

as ir

from the kinematical

(ir

of the strain,

during

index of non coaxiality,

used by some authors (Hudleston and Hooke, positive

the intensity

of the bulk

t is the time.

pure shear,

r = i-,//Q\,

the progressive s = +/6,,

vorticity

number

constant

deformation

varying

and a

as Jo through

of Truesdell

the

(1953) and

for steady flows in zones of simultaneous superposition 1980; Means et al., 1984); r and s have the sign of e, for

> 0) and range from zero for pure shear to infinity

for simple

shear. For a steady particle

flow of the bulk

rock outside

the shear zone the equation

of the

path is:

x=x,exp(p)+rf(y,)[exp(p)-ll/k Y =.yn exp(-p)

(3)

where (x0, yO) are the initial

coordinates

of the point

considered

and:

f(J),,) = exp(L$y)$ The finite strain (strain trajectories, the tensor of deformation gradients: 1 --ax

ax \

ax,

ah

--ay

ay

aAxo

j

/ =

finite

strains)

is determined

from

\ A

B

c

E

aye \

Here C=O,

principal

/

A=Er=exp(p)and:

B=rg(y)[exp(~)-II/k

(4)

Because of the bulk direction is given by:

pure

shear,

the shear

displacement

parallel

to the shear

)i,

x=Y

J Bdy, 0

=

J

(54

O'B exp( p)dy

from (4) it follows: “v=r[exp(p)-llJ(y,)/k and the total shear displacement X

(5b) across the zone (f( yO) = k for y 2 h) is:

r=r[ev(p)-ll This value can also be obtained

DEFORMATION

OF PASSIVE INITIALLY

(5c) from the eqn. (3) of the particle PLANAR

path.

MARKERS

We study in fact the shape changes of passive initially linear markers which are the traces of the original planes measured in the XZ profile of the shear zone.

122

The equation direction

of a straight

line L,

making

an initial

angle

(Ye with the shear

(Fig. lb) is: where no = cot a0

x0 = no.Yo + 60 When deflected, the particle

the equation

of the strained

curve L is determined

path and the angle (Yof its tangent

from eqn. (3) of

with the shear-direction

is given by

(see Appendix): n=cot

a=n,exp(2p)+rg(y)exp(p)[exp(p)-l]/k

(6)

Out of the shear zone (y > h), g(y) = 0 and L is a straight ay3 with the shear-direction

line making

an angle

(Fig. lb):

n 3 = cot 01~= no exp(2p) This

orientation

Ramberg,

(7)

is that

of a passive

marker

line

in pure

shear

(Ghosh

and

1976, p. 9).

In the central part homogeneously strained (0 < y < II,), g(y) = 1 and L is also a straight line making an angle (Y, with the shear-direction (Fig. lb): n,=cot

a,=n3+rexp(p)[exp(p)-l]/k

In the heterogeneous parabola,

shear zone (h,
its tangent

makes an angle

varies linearly

(Y with the shear-direction

and so L is a

(Fig. lb) given by

(6). Where

a set of sub-parallel

planes

crosses

a heterogeneous

shear

zone,

the

deformation sets up folding (Ramsay, 1967). The axis of the fold is controlled by the line of intersection of the initial planar structure and the XZ profile of the zone. Its axial plane is parallel to the zone and its trace in the plane is located at a position yrn where L exhibits a tangent normal to the shear direction, that is to say, where n = 0 and for (see Appendix): Y, =

exd -d +

(1- hiI)no 2+xdd-l)

This may happen only if h, Q y,,, Q h. From (8) and for a positive shearing and y, 2 h, implies

eQ(p) a

(a,

> 0), Y,,, < h implies

kn0

(9)

+xp(d -11

This relation

allows us to study

tion on the conditions

of existence

the influence

of the parameters

and on the location

the fold both when the shear zone is contracted Influence

that 71/2 < (Y()< V,

that (see Appendix):

changes

of the deforma-

of the axial plane of

(P, > 0) or extended

(e, < 0).

of p = 6, t

From (9) it follows that y,,, exists if 1p 1 >, 1pi 1 with: p,=ln

r I r+kn,

(see Appendix) I

(10)

123

Fig. 2. Warping of initially linear passive marker with increasing p (numbered) during la) contraction perpendicular to the shear zone or (b) extension perpendicular tq the shear zone.

Ip 1, moves away 1p ) = 1pi 1, y,,, is located at ym = h, and, with increasing towards the boundary of the shear zone. Figure 2 shows the warping of a passive marker line with increasing Ip I: when the zone is contracted (Fig. Za), pi = 0.207 and when the zone is extended (Fig. 2b), pi = - 0.172. For

124

Influence

of r = iT/ti,

From (9), it follows -kn,

r’ = 1 - exp( -p) For

) r I = I ri 1, y,

that y, exists if Ir 1>, ) ri 1 with: (see Appendix)

is located

(11)

at y,,, = h, and with increasing

I

1r 1, moves

away

Q

\

Y ;,

Y

\ \ \

1

\

Fig. 3. Warping

perpendicular

of initially

linear passive

markers

to the shear zone or (b) extension

with increasing

perpendicular

r (numbered)

to the shear zone.

during

(a) contraction

0

-1

Fig. 4. Warping (a) contraction

of initially perpendicular

linear passive markers

1

with several initial orientations

to the shear zone or (b) extension

perpendicular

a0 (numbered)

to the shear zone.

-X

during

126

towards

the boundary

marker

of the shear zone. Figure

line with increasing

(Fig.

of a passive 3a), r, = 1.906

(Fig. 3b), r, = - 3.387.

and when the zone is extended Influence

3 shows the warping

/r / : when the zone is contracted

of a,,

From (9) it follows that JJ_, exists if 7r/2 < CX,,< ~ygiwith: Cot ~Ygi=

For

-f[l

(see Appendix)

-exp(-p)]

cyO= a”,,

_v,,, is located

at ym = h,

(12)

and

with

decreasing

(Ye) moves

away

towards warping

the boundary of the zone that it reaches for 0~~)= n/2. Figure 4 shows the of passive marker lines with several initial orientations (~a: when the zone is and, when the zone is extended (Fig. 4b), contracted (Fig. 4a), CQ,= 144”ll (Y
Using

OF FINITE STRAIN

eqn. (4), the eqn. (6) giving

initially

linear

passive

n=cot

a=n3+Bexp(p),

marker

the orientation

through

the shear zone of the

L. becomes: (13a)

and using (2) n=cot

a=n3+s[exp(p)-l]

(13b)

where n3 = n, exp(2p). When the initial orientation orientation Another zone

(Ye of the marker

line is known,

0~~outside the shear zone gives, from (7) the value of p. practical possibility occurs when the finite strain ellipse outside

is known;

p is determined

from

the

length

of the

cyOcan be determined (A,) ‘I2 = exp( p). The initial orientation Then variation in orientation cy of the deflected marker B (from eqn. 13a) and s (from eqn. 13b) through

compute

B and p, the finite strain

From

axis) can be computed shear

the measure

zone profile

(Ingles,

must

ellipse (lengths

1983). When

be compared

finite strain axis. The total shear displacement

principal

it exists,

the shear semi-axis,

from (7). line can be used of the principal

the schistosity direction

across the zone can be computed

trace on the

of the principal from the variations

of B (eqn. Sa): X -,- =

exp(p)jhBdc.

(where

0

h exp( p) is the unit of length) X

‘=exp(d,-1

to

the shear zone.

and orientations

to the computed

of its

and so, r can be determined

from (5~):

127

Generally,

the parameters

from passive

deflected

of the instantaneous

marker

lines because

for three unknown quantities (g,, ir determined otherwise. then the others deformation

is known

(kinematic,

strain

cannot

be determined

we have two known

and

values

t). If one of these parameters

are computable

formation

only

( p and r ) can be

and the whole story of the

and evolution

of strain patterns).

DISCUSSION

This theoretical study is developed from three hypotheses, simultaneous superposition of the deformations, heterogeneous shear with a linear variation and steady flow of the bulk rock outside the shear zone. It is well known that the finite strain patterns superposition

of deformations

depend

upon

resulting

the order

from

the sequential

of superposition

(Ramsay,

1967; Ramberg, 1975). Deformation in natural rocks is admittedly more complex than the one resulting from an homogeneous deformation alone such as simple shear or pure shear. In fact, most of the events of flow of natural ductile rocks result from simultaneous combination of more or less complex heterogeneous deformations. Our model is an advance towards a closer approximation of the natural situations. Most of natural deformations of rocks are heterogeneous. It is obvious that the linear variation of the shear strain assumed in our model is, in fact, a very simplified empirical image of the natural heterogeneity of the shear zones in rocks. This image is realistic

however

in natural

ductile

if we compare shear

Hara et al., 1973; Simpson,

zones

it with the numerous

of various

1983; Williams

thickness

determinations (Ramsay

and

et al., 1984). Moreover,

of the strain Graham,

assuming

1970; a linear

variation, rather than a more complex one, makes easier the mathematical developments but does not fundamentally modify the kinematics of the flow. So, when the natural orientation a linear variation, above.

of the deflected marker line through the finite strain can be determined

the shear zone approximates from the method described

The assumption simplification but

of a steady flow of the bulk rock outside the shear zone is also a may be reasonable provided an average long-term steady flow

exists. It is, for example, the most commonly admitted assumption made for the study of flows of glaciers (Milnes and Hambrey, 1976; Hambrey and Milnes, 1977; Hudleston,

1983) that exhibit

a kinematic

similarity

with the development

of nappes

and thrust sheets (Elliott, 1976; Hudleston, 1977). This hypothesis of steady flow outside the shear zone, leading to a non-steady flow in the shear zone (eqn. 2). is certainly more realistic than the one of a steady flow in the shear zone (Ramberg, 1975) because continuity of deformation in a ductile material implies, in this case. that the flow outside the shear zone be non-steady with a perfectly well-defined rate of translation (eqn. 2).

128 CONCLUSION

Our

theoretical

natural

model

of simultaneous

superposition

of heterogeneous

strains,

image of the natural situations, is however a stage giving a simplified a closer approximation and knowledge of the complex mechanisms of

although towards

flows of rocks (kinematic,

of the parameters

strain patterns).

of the deformation

It allows us to study the influence

on the shape changes

of passive initially

markers and to describe a method for determining the finite sometimes the instantaneous strain, from these deflected markers.

strain,

and

planar even

ACKNOWLEDGEMENTS

The English

author

thanks

two

anonymous

referees

for many

improvements

of the

text and useful suggestions.

APPENDIX

Calculation

of the equation of the deflected marker line

For a steady fiow of the bulk rock outside particle path is:

the shear zone, the equation

of the

.v=yoewh4 and the equation

of the initially

linear

marker

is:

* X() = n,y, + b,

(AZ)

where n, = cot c+ From (Al) and (A2) it follows:

5=

(

no+2 exp(2p)+~f(y~o)[exp(p)-11 I

or:

x=n,exp(2p)~+~f~~~)[exp(p)-1l+h,exp(p) wheref(y,)=uy~+by0+c=aexp(2p)y2+bexp(p)y+c with: 1 a = - 2(1 -h,,)

’

So that the equation x=Ay’+B_y+C

b=

-2a,

c--ah&

of L is: (A3)

129

where: A = _ r exp(2p)]exp(p)

11

-

2k(l - hI,) B = no exp(2p)

+

rexp(p)]exp(p)-11 k(l

- hcl,)

rG, ]exp( P) -

C=b,exp(p)-

=(I -h,)

The angle (Yof the tangent

11

to L with the shear direction

is given by:

n=cota=*=2Ay+B

dy

hence: n=n,exp(2p)+iexp(p)[exp(p)-I)] and because

the initial

l-ly~~O~P)

semi-thickness

of the shear zone is taken

as unit of length,

h exp( p) = 1, so that:

1 -y exdp)

1 -ho,

h-y

=-=&Y(Y) h-h,

then:

+idY) expb)[exp(d -11

n = no exp(2p) The tangent y=y,=

is normal

to the shear direction

no(l-

-&=exp(-p)+

Calculation

(A4) (n = 0) where:

h&>

2r]exp(p)

-11

of the condition of existence

of ym

y, exists if h,
implies

shearing

(kr > 0). y, < h =

that no < 0, thus (7r/2) < (~a < n.

From (AS), y, > h, = ho, exp( -p)

exd -PI +

(A51

kno

r[exdp)-11

implies:

>O

and thus:

exd -p> a

kno

r[ev(p) - 11

(‘46)

130

of p = 6, t

Influence

From (A6) it follows that _r,,, existsif: (A7a)

When

(r, 6, and p > 0), (A7a) becomes:

the zone is contracted

(A7b)

exp(p)aL

r +Jcn,

and when the zone is extended exp(p)g*

(r, t;_,.and p < 0): (A7c)

0 ), the relations

If p,=ln(*

(A7b) and (A7c) can be summarized

by

/ p

1 a

0

I Ptl.

Influence of r = 3,/P, From (A6) it follows that )m exists if:

If-12 When

r>,---

kn,

(A84

I- exd-d (A8a) becomes:

the zone is contracted, kn,

(A8b)

1 - exp( -PI

and when the zone is extended: kn,

” -

1 -exp(-p)

kno

If ri= -

the relations

1 -expf-p)

(A8b) and (A&c) can be summarized

by 1r 1 a / r, 1.

Influence of a0 Both when the zone is contracted

or extended

it follows from (A7a) that _)J, exists

if: n,, 2 noi =

-

f

[l -

and hence if (a/2)

exp(

-p)]

< CQ6 Cyoiwith noi = cot

cYoi.

REFERENCES Cobbold,

P.R., 1377. Description

perturbation, Coward,

and deformation

and origin bands.

M.P., 1976. Strain within ductile

of banded

deformation

structures.

I. Regionatisation,

Can. J. Earth Sci., 14: 1721-1731. shear zones. Tectonophysics,

34: 181-197.

local

131

Elliott.

D., 1976. The motion

Ghosh,

S.K. and Ramberg,

shear. Tectonophysics, Hambrey, Hara,

Eclogae

Res.. 81: 949-963.

of inclusions

by combination

of pure shear and simple

A.G.,

1977. Structural

geology

of an Alpine

glacier

(Griesgletscher.

Valais,

Geol. Helv., 70: 667-684.

L., Takeda,

K. and Kimura,

Sci. Hiroshima

Univ., 7: l-11.

Hudleston,

sheets. J. Geophys.

34: l-70.

M.J. and Milnes,

Switzerland).

of thrust

H., 1976. Reorientation

P.J., 1977. Similar

T., 1973. Preferred

folds, recumbent

lattice orientation

folds, and gravity

of quartz in shear deformation.

tectonics

J.

in ice and rocks. J. Geol.. 85:

113-122. Hudleston,

P.J., 1983. Strain patterns

Struct. Hudleston,

P.J. and Hooke,

for the development Ingles,

in an ice cap and implications

R.L., 1980. Cumulative

of foliation.

J., 1983. Theoretical W.D., Williams,

a KCl/mica

strain

patterns

Milnes,

in shear zones. J.

A.G.

and

Tectonophysics. Ramberg.

J. Struct.

Hambrey.

in the Barnes

Ice Cap and implications

66: 127-146.

in ductile

J. Struct.

P.F. and Hobbs,

mixture.

deformation

Tectonophysics,

simple shear and bulk shortening. Means,

for strain variations

Geol.. 5: 445-463.

zones simultaneously

undergoing

heterogeneous

Geol., 5: 369-381.

B.E.. 1984. Incremental

deformation

and fabric development

in

Geol., 6: 391-398.

M.J.,

1976.

A method

of estimating

cumulative

strains

in glacier

ice

34: T23-T27.

H.. 1975. Particle paths, displacement

and progressive

strain applicable

to rocks. Tectonophysics.

28: l-37. Ramsay.

J.G., 1967. Folding

Ramsay,

J.G., 1980. Shear zone geometry:

Ramsay.

J.G. and Graham,

Simpson,

C., 1983. Displacement

Struct.

and Fracturing R.H.,

of Rocks. McGraw-Hill, a review. J. Struct.

1970. Strain

variation

and strain patterns

New-York.

Geol., 2: 83-99.

in shear belts. Can. J. Earth.

from naturally

occurring

Sci., 7: 7866813.

shear zone terminations.

J.

Geol.. 5: 497-506.

Truesdell,

C.. 1953. Two measures

Williams,

G.D.,

patterns 177-186.

Chapman,

by progressive

of vorticity.

T.J. and Milton, thrust

faulting

J. Rational N.J.,

Mech. Anal.,

1984. Generation

in the Laksefjord

Nappe.

2: 173-217. and

modification

Finnmark.

of finite

Tectonophysics,

strain 107: