Tectonophysics 340 (2001) 195 – 213
www.elsevier.com/locate/tecto
Strain and competence contrast estimation from fold shape
Stefan M. Schmalholz *, Yuri Yu. Podladchikov
Geologisches Institut, ETH Zentrum, 8092 Zürich, Switzerland
Received 14 February 2000; accepted 21 February 2001
Abstract
A new method to estimate strain and competence contrast from natural fold shapes is developed and verified by analogue
and numerical experiments. Strain is estimated relative to the nucleation amplitude, AN, which is the fold amplitude when the
amplification velocities caused by kinematic layer thickening and dynamic folding are identical. AN is defined as the initial
amplitude corresponding to zero strain because folding at amplitudes smaller than AN is dominantly by kinematic layer
thickening. For amplitudes larger than AN, estimates of strain and competence contrast are contoured in thickness-to-wavelength
(H/l) and amplitude-to-wavelength (A/l) space. These quantities can be measured for any observed fold shape. Contour maps
are constructed using existing linear theories of folding, a new nonlinear theory of folding and numerical simulations, all for
single-layer folding. The method represents a significant improvement to the arc length method. The strain estimation method is
applied to folds in viscous (Newtonian), power-law (non-Newtonian) and viscoelastic layers. Also, strain partitioning in fold
trains is investigated. Strain partitioning refers to the difference in strain accommodated by individual folds in the fold train and
by the whole fold train. Fold trains within layers exhibiting viscous and viscoelastic rheology show different characteristic strain
partitioning patterns. Strain partitioning patterns of natural fold trains can be used to assess the rheological behaviour during
fold initiation. D 2001 Elsevier Science B.V. All rights reserved.
Keywords: Strain estimation; Folding; Fold shape; Competence contrast; Strain partitioning; Fold trains
1. Introduction
A major geological question concerning folding is
how much strain (or shortening) is associated with
observed fold shapes. In this study, strain is defined as
the difference between the deformed and the initial
length of a line element normalized by its initial
length. Information of strain is relevant to palinspastic
*
Corresponding author. Now at Geomodelling Solutions
GmbH, Binzstrasse 18, 8045 Zürich, Switzerland. Fax: +41-1455-6390.
E-mail address: stefan.schmalholz@geomodsol.com
(S.M. Schmalholz).
reconstructions of folded regions such as mountain
belts. Although numerous studies have investigated
the mechanics of folding using analytical techniques
(e.g., Biot, 1961; Johnson and Fletcher, 1994; Hunt et
al., 1996), analogue experiments (e.g., Currie et al.,
1962; Ramberg, 1963; Hudleston, 1973; Abbassi and
Mancktelow, 1992) and numerical simulations (e.g.,
Dieterich, 1970; Cobbold, 1977; Zhang et al., 1996;
Mancktelow, 1999; Schmalholz et al., 2001), there is
no satisfactory method to estimate strain from natural
fold shapes. One reason is that the existing analytical
folding solutions are only valid for small limb dips
(10 to 15, e.g., Fletcher and Sherwin, 1978),
whereas most natural folds exhibit much larger limb
0040-1951/01/$ - see front matter D 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 0 4 0 - 1 9 5 1 ( 0 1 ) 0 0 1 5 1 - 2
196
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
dips. An intuitively attractive estimation method is the
so-called arc length method (e.g., Dahlstrom, 1969).
This method assumes that the observed fold arc length
corresponds to the initial fold wavelength (Fig. 1).
Hence, strain can be estimated from the difference
between the observed arc length and fold wavelength.
However, the arc length method results in large errors
if it is applied to folds that developed in settings where
the competence contrast between the folded layer and
its matrix is small. These errors are a result of the
relatively large component of homogeneous layer
thickening (Sherwin and Chapple, 1968). This study
applies a new analytical solution for folding valid up
to large limb dips (Schmalholz and Podladchikov,
2000) and numerical simulations of folding to develop
a strain estimation method. This method reliably
estimates strain and has the additional advantage that
it enables estimation of the competence contrast.
The unknown material properties and initial geometry of natural folds are the major obstacles to strain
estimation. An essentially flat layer with infinitesimal
fold amplitudes may accommodate several hundreds
of percent strain by homogeneous layer thickening
without developing observable fold limb dips. A layer
that appears unfolded in the field may have accommodated substantial layer-parallel strain. With increasing limb dip the strain that is accommodated by layer
thickening decreases, and that accommodated by
folding increases. In this study, we define a nucleation
amplitude, AN (Table 1), as the initial (or reference)
amplitude for strain estimation in order to circumvent
the obstacle of unknown initial geometry. Only the
strain that is accommodated while amplitudes are
larger than AN is, therefore, estimated. The strain that
is accommodated while amplitudes are smaller than
AN is ignored because this strain is not recorded
through the fold shape.
The variation of fold shape with strain depends on
material properties. This was shown by analogue and
numerical experiments where the variation of geometrical parameters has been recorded as a function of
strain. Manifestations of material dependence are the
alteration of the ratios of (i) amplitude to initial
amplitude (Chapple, 1968; Hudleston and Stephansson, 1973; Abbassi and Mancktelow, 1992), (ii) arc
length to initial arc length (Hudleston, 1973; Johnson
and Fletcher, 1994), (iii) amplitude to thickness
(Hudleston, 1973), (iv) amplitude to wavelength (Currie et al., 1962) and (v) wavelength to thickness (Lan
and Hudleston, 1995a). One specific fold shape,
corresponding to different material properties, may
theoretically correspond to different strains. Without
knowledge of the material properties, a unique strain
estimate is not possible from a single geometrical
parameter that quantifies the fold shape. To avoid this
obstacle, we employ two observable geometrical
parameters simultaneously to characterize folds. Thus,
during progressive folding, strain is contoured in the
thickness-to-wavelength (H/l) and the amplitude-towavelength (A/l) space. These ratios can be measured
for any observed fold shape. Strain contours as a
function of A/l and H/l provide a ‘‘strain contour
map’’, which uniquely defines strain for a continuous
range of competence contrasts. The idea of this study
is to use two observable geometrical ratios to constrain two unknowns: the competence contrast and the
strain that is accommodated by the folds after they
exceed the nucleation amplitude.
Fig. 1. Major geometrical parameters of folds are the arc length (Larc), wavelength (l), amplitude (A) and thickness (H ). In the initial folding
stages, where amplitudes are very small, the arc length has approximately the same length than the wavelength.
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
Table 1
List of symbols used in the text
a
ad
add
ae
ade
A
A0
AC
AN
D
e
en
earc
ė
G
H
H0
l
l0
ldd
lde
leff
Larc
Larc0
ml
mm
n
P
R
Reff
vceff
Vdyn
Vkin
general folding growth rate
growth rate for ductile layers
dominant growth rate for ductile layers
growth rate for elastic layers
dominant growth rate for elastic layers
fold amplitude
initial fold amplitude
pffiffiffiffiffiffi
crossover amplitude: AC ¼ 1=ðp 2aÞ
nucleation amplitude: AN = 1/(2a)
flexural rigidity of the layer
engineering strain: e =(l0 l)/l0
natural strain: en = ln(l0/l)
strain that is estimated by the
arc length method
pure shear background strain rate
shear modulus of the layer
thickness of the layer
initial thickness of the layer
fold wavelength
initial fold wavelength
dominant wavelength for ductile layers
dominant wavelength for elastic layers
effective dominant wavelength
fold arc length
initial fold arc length
viscosity of the layer
viscosity of the matrix
power-law exponent of the layer
layer-parallel stress
dominant wavelength ratio:
R = ldd/lde or alternatively R = ade/add
effective dominant wavelength ratio:
Reff = n max(1,R)
effective viscosity contrast
dynamic folding velocity
kinematic folding velocity
The aim of this paper is to provide a practical tool
for field geologists, which enables strain and competence contrast estimation from fold shape. We also
show all steps of the mathematical derivation of our
strain estimation method to present the assumptions
and limitations of our method. The study starts from
deriving the nucleation amplitude (the reference
amplitude for strain estimation). In the next section
we derive an amplitude, designated the crossover
amplitude, at which the existing theories of folding
break down. Both sections, quantifying the initial
stages of the folding instability, are potentially useful
for strain corrections if the initial fold amplitude is
known. In the succeeding section, new analytical
197
relationships for the nonlinear folding stages (Schmalholz and Podladchikov, 2000) are used to choose the
geometrical parameters for the strain estimation
method and to evaluate its range of applicability.
Next, numerical simulations are used to construct
the new method of strain and competence contrasts
estimates. All results are derived for ductile (viscous
and power-law) as well as for viscoelastic layers to
avoid a restriction of the strain estimation method to a
certain layer rheology. Otherwise, this rheology needs
to be known by the geologist for the observed folds,
which is unlikely in the majority of the cases. The new
method is then verified by numerical and analogue
experiments and applied to constrain the strain partitioning in natural and experimental fold trains.
2. The nucleation amplitude for ductile and viscoelastic layers
A layer subjected to layer-parallel shortening
accommodates strain by both layer thickening and
rotation. The deformation process, during which considerable rotation of fold limbs takes place, is termed
folding or buckling (e.g., Ramsay and Huber, 1987;
Price and Cosgrove, 1990). Layer thickening and
folding can occur simultaneously, but it is possible
to discriminate between the velocity fields of the two
mechanisms (e.g., Biot, 1965; Johnson and Fletcher,
1994). If the layer and the matrix exhibit the same
material properties, the amplification velocity of the
top layer boundary is controlled exclusively by the
kinematic velocity, which arises due to layer thickening (e.g., Biot, 1965; Johnson and Fletcher, 1994):
Vkin ¼ e_
H
:
2
ð1Þ
The kinematic velocity is a linear function of the
˙
pure shear background strain rate e_ and the layer
thickness H (Fig. 1, for symbols see Table 1). In pure
shear, the vertical ( y) distance of the top layer
boundary from the origin of the coordinate system
(point of zero velocity) is half of the layer thickness H
plus the amplitude A. The contribution of the amplitude to that distance can be neglected to calculate the
kinematic velocity, because in the initial stages (i.e.,
198
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
very small limb dips) the amplitude is much smaller
than the thickness of the layer (Fig. 1).
If the material properties of the layer are different
than those of the matrix, then the dynamic velocity is
non-zero because of the instability of the layer. This
component of the velocity is (e.g., Biot, 1965; Johnson and Fletcher, 1994):
Vdyn ¼ Aa_e:
ð2Þ
The dynamic velocity is a linear function of the
fold amplitude A, a dimensionless growth rate a and
the strain rate.
For ductile (viscous and power-law) layers embedded in a viscous matrix the thin-plate theory (e.g.,
Timoshenko and Woinowsky-Krieger, 1959) provides
the dimensionless growth rate (e.g., Fletcher, 1974):
!
1 pH0 2 n l0 mm
ad ¼ n=
:
ð3Þ
þ
3 l0
2p H0 ml
Here ml, mm, H0, l0 and n are the effective viscosity
of the layer, the viscosity of the matrix, the initial
layer thickness, the initial wavelength of a sinusoidal
perturbation and the power-law exponent of the layer,
respectively. The subscript ‘‘d’’ indicates that the
growth rate corresponds to a ductile layer. The growth
rate has a maximum as a function of wavelength,
which is obtained by setting the derivative of Eq. (3)
with respect to l0 to zero and solving for l0. This
maximum corresponds to the dominant wavelength
(Fletcher, 1974; Smith, 1977):
ldd ¼ 2pH0 ð6nÞ1=3 ðml =mm Þ1=3 :
ð4Þ
The second subscript ‘‘d’’ indicates that the
wavelength is the dominant one. It is assumed that
the dominant wavelength is selected and preserved
during the initial stages of folding because perturbations with this wavelength grow exponentially faster
than all other perturbations. Substituting Eq. (4) into
Eq. (3) gives the dominant growth rate for ductile
layers:
add ¼
pffiffiffi
2=3
4 n ml
:
3 mm
ð5Þ
In Eq. (5), the growth rate is explicitly dependent
on material properties. Alternatively, add can be
expressed through ldd and H0 using Eq. (4). This
makes the dominant growth rate explicitly dependent
on geometric parameters:
add
n
¼ 2
p
ldd
H0
2
:
ð6Þ
In Fig. 2, the kinematic velocity and several
dynamic velocities, for different material properties
Fig. 2. Growth of kinematic and dynamic velocities during shortening. (A) The dynamic velocities (solid lines) increase with increasing
viscosity contrast (vc). The natural strain, en, at which the dynamic velocity equals the kinematic velocity depends on the initial amplitude to
thickness ratio (here A0/H0 = 0.02). (B) For smaller A0/H0 ratios (0.01) than in (A), more strain is necessary for the dynamic velocities to exceed
the kinematic velocities. The intersection between dynamic and kinematic velocities defines the nucleation amplitude.
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
(viscous rheology, n = 1) and initial amplitudes, are
plotted versus the natural strain, which is defined by:
en ¼ lnðl0 =lÞ;
ð7Þ
where l is the current fold wavelength. The dynamic
velocity grows faster than the kinematic velocity with
increasing natural strain, and exceeds the kinematic velocity after a certain amount of natural strain. This
amount of natural strain depends on the material properties and the initial amplitude. The amplitude, at
which the dynamic velocity is equal to the kinematic
velocity, is termed nucleation amplitude AN.
AN is derived by equating Eqs. (1) and (2) and
solving the result for A/H:
AN ¼
A
1
¼
;
H
2a
ð8Þ
which is dimensionless (the amplitude is measured in
units of the layer thickness). Characterizing the growth rate by the value obtained for the dominant wavelength yields:
p2 1
AN ¼
2 n
H0
ldd
2
:
ð9Þ
For elastic (subscript ‘‘e’’) layers embedded in a
viscous matrix, the thin-plate theory provides the
dimensionless growth rate (Turcotte and Schubert,
1982):
1 2p
ae ¼ 4mm e_ l
!
2
2p
D
PH0 ;
l
ð10Þ
where D and P are the flexural rigidity of the layer and
the layer-parallel membrane stress (averaged over layer thickness), respectively. For incompressible materials, the flexural rigidity is:
D¼
1
GH03 ;
3
ð11Þ
where G is the shear modulus of the layer. As in the
ductile case, the growth rate function yields a domi-
199
nant wavelength for elastic layers (e.g., Biot, 1961;
Turcotte and Schubert, 1982):
pffiffiffiffiffiffiffiffiffi
ð12Þ
lde ¼ 2pH0 G=P:
The corresponding dominant growth rate for elastic
layers is then:
ade ¼
p P H0
:
3 mm e_ lde
ð13Þ
To obtain ‘‘effective’’ parameters valid for ductile
and viscoelastic layers, we now employ linear viscoelastic folding theory (Schmalholz and Podladchikov,
1999, 2001) to express the elastic dominant wavelength and growth rate. To this end, it is assumed that
finite amplitude folding occurs in lithospheric regions
dominated
by ductile behaviour (i.e., Deborah num˙
bers mle_ /G < 1 (e.g., Poliakov et al., 1993; Schmalholz
and Podladchikov, 1999)). Consequently, the membrane stress P is expected to become
limited by the
˙
viscous membrane stress 4mle_ (e.g., Turcotte and
Schubert, 1982) at
˙ very small strain. Therefore, in
this study P = 4mle_ is assumed. Folded layers that
exhibit growth rates derived for elastic layers are, in
this study, viscoelastic because the membrane stress is
assumed to be viscous (Schmalholz and Podladchikov, 2000). The dominant wavelength ratio R introduced by Schmalholz and Podladchikov (1999) is
defined as:
1=3 1=2
ldd
1 ml
P
R¼
¼
;
6n mm
G
lde
ldd
:
lde ¼
ð14Þ
R
The dominant wavelength ratio R relates the dominant growth rates through the proportionality:
ade ¼ add R:
ð15Þ
For layers exhibiting a viscoelastic Maxwell rheology (i.e., elastic and viscous element connected in
series), the folding mode (ductile or elastic) that
exhibits the largest growth rate will tend to dominate
in nature. Therefore, the R parameter is controlling if
a compressed viscoelastic (Maxwell model) layer
folds ductilely (R < 1) or elastically (R >1) (Schmal-
200
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
holz and Podladchikov, 1999, 2001). Comparison of
this criterion with the relationship between the dominant wavelengths (Eq. (14)) shows that the faster
mode of folding is characterized by shorter dominant
wavelength.
Eq. (5) establishes a relation between the viscosity
contrast and add. To preserve this relationship in the
context of viscoelastic folding, an effective viscosity
contrast is defined by the condition that the dominant
growth rates are given by expressions identical to Eq.
(5). Solving Eq. (5) (n = 1) for the viscosity contrast
and substituting add and ade yields then an ‘‘effective’’ viscosity contrast:
( 3=2 pffiffiffi m
)
3
l
R<1
4 add ¼ n mm ;
pffiffiffiffiffiffiffiffi
vceff ¼ 3 3=2 3
3=2
¼ nR3 mml ; R > 1
4 ade ¼ 4 ðadd RÞ
m
pffiffiffi
m
¼ nðmaxð1; RÞÞ3=2 l :
mm
(16)
In a similar way an effective dominant wavelength
can be introduced that preserves the relation between
Fig. 3. (A) The alteration of the nucleation amplitude AN versus leff (Eq. (17)). AN is the amplitude at which the kinematic velocity equals the
dynamic velocity. AN increases strongly for decreasing leff. (B) The alteration of the crossover amplitude AC versus leff. AC is the amplitude at
which the linear theories break down. AC increases strongly for decreasing leff. (C) The continuous folding process can be separated in three
stages: (i) kinematic layer thickening (A < AN), (ii) exponential growth of the amplitude (AN < A < AC), and (iii) layer length controlled growth of
the amplitude (AC < A). (D) The maximum strain that can be accommodated between AN and AC is 40% for a viscosity contrast of 2. For a
viscosity contrast of 100, the maximum strain reduces to 13%. The strain decreases strongly if viscosity contrasts or Reff are increased, where
Reff = n max(1,R).
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
the dominant wavelength and the growth rate given in
Eq. (6). The effective dominant wavelength can be
written:
(
leff ¼
ldd
H0
ldd
H0
pffiffiffi
n;
pffiffiffipffiffiffi
n R;
R < 1ðductile foldingÞ
R > 1ðelastic bucklingÞ
ldd pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n maxð1; RÞ:
H0
¼
)
(17)
The effective dominant wavelength yields a general growth rate valid for ductile and elastic layers:
a¼
leff
p
2
:
ð18Þ
This growth rate can be substituted into Eq. (8) to
provide a general nucleation amplitude for ductile and
viscoelastic layers:
AN ¼
1
1 p 2
¼
:
2a 2 leff
ð19Þ
Decreasing values of leff cause larger nucleation
amplitudes because the growth rates decrease with
decreasing values of leff and this requires larger
amplitudes to maintain the same dynamic velocity
(Eq. (2), Fig. 3A). The formation of large amplitude
folds having large arc length to thickness ratios
requires fast initial growth rates, i.e., a >> 1 and
leff >>1, which result in small nucleation amplitude
values (Eq. (19), Fig. 3A). Therefore, the nucleation
amplitude is likely to be smaller than the real initial
amplitude for observable folds of practical interest, a
conclusion of primary importance for the following
discussion.
3. The crossover amplitude for ductile and viscoelastic layers
The classical linear theory of folding described
above becomes invalid at a certain amplitude, because
the assumptions that allow linearization are strictly
valid only for infinitesimal amplitudes (or very small
limb dips) (e.g., Chapple, 1968; Schmalholz and
201
Podladchikov, 2000). The linear theory cannot provide its own limits and, therefore, a new nonlinear
finite amplitude theory (Schmalholz and Podladchikov, 2000) is used to establish the limits of the linear
theory. The nonlinear theory estimates the crossover
amplitude at which the linear theory breaks down as
(cf., Schmalholz and Podladchikov, 2000):
AC ¼
A 1 1
1
pffiffiffiffiffiffi ¼ pffiffiffi
:
l p 2a
2leff
ð20Þ
The crossover amplitude is a dimensionless number that measures the amplitude in units of the current
wavelength. Amplitudes smaller than AC grow exponentially (as predicted by the classical, linear solutions), whereas amplitudes larger than AC grow slower
than exponential, being controlled by the layer’s
resistance to stretching (Fig. 3B). This growth is
designated as layer length controlled growth (Schmalholz and Podladchikov, 2000). To view nucleation
(Eq. (19)) and crossover amplitudes on a single plot,
AN is multiplied by the initial dominant thickness to
wavelength ratio. Thus, AN is also normalized by the
wavelength. In Fig. 3C, the nucleation and crossover
amplitudes are plotted versus the dominant wavelength to thickness ratio for viscous rheologies. This
phase diagram distinguishes three stages of viscous
folding, which are kinematic thickening, exponential
growth and layer length controlled growth (Fig. 3B).
The strain that is accommodated between AN and
AC during folding can be calculated using the linear
theory. Classical, linear analytical theories assume an
exponential growth of the fold amplitude such that
(e.g., Biot, 1965; Johnson and Fletcher, 1994):
A ¼ A0 expðaen Þ;
ð21Þ
where A0 is the initial amplitude of the fold. To
calculate the strain accommodated between AN and
AC, A in Eq. (21) is replaced by AC and A0 is replaced
by AN. The strain is then:
1
AC ldd
ln
a
AN H0
2 pffiffiffi 2 2 l
p
¼
ln pffiffiffiffiffiffiffi eff
:
leff
Reff p2
en ¼
ð22Þ
202
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
This strain is small for folding parameters of
practical interest (Fig. 3D), and from now on we
express strain by the so-called engineering strain, e,
which is related to the natural strain en (Eq. (7)) by:
e¼
l0 l
l
1
:
¼1
¼1
l0
l0
expðen Þ
ð23Þ
The overall strain accommodated by most natural
folds is dominated by the layer length controlled
growth stage of folding because the strain accommodated between AN and AC is small (Fig. 3C and D).
The linear analytical solutions for fold amplification
(e.g., Biot, 1961; Johnson and Fletcher, 1994) are,
therefore, unsuitable to derive a strain estimation
method and we have to use a new nonlinear solution
valid for finite amplitude folding (Schmalholz and
Podladchikov, 2000).
4. The strain contour map
In this section, a strain contour map is constructed
that allows strain and competence contrast estimation
from fold shapes without knowledge of material
properties and initial geometries. The strains are
estimated with respect to the nucleation amplitude AN.
Numerical simulations of viscous and viscoelastic
single-layer folding were performed to provide a basis
for the analytical treatment using a combined spectral/
finite-difference method (Schmalholz et al., 2001)
(Fig. 4). These simulations establish known or real
strains that can be compared to estimated strains.
Applying the arc length method assumes that
observed fold arc lengths (Larc, cf. Fig. 1) correspond
to initial fold wavelengths. Therefore, the ‘‘arc length
strain’’ can be calculated by:
earc ¼
Larc l
:
Larc
ð24Þ
As expected, the arc length strains estimated for
viscous folds that developed in settings where viscosity contrasts are small exhibit the largest deviations
from the real strain (Fig. 5). This deviation is caused
by the large component of homogeneous layer thickening (Sherwin and Chapple, 1968). Layers with large
growth rates (either large viscosity contrast or R>1)
Fig. 4. Numerically simulated single-layer fold shapes for three different material properties. The three folds initially exhibited a sinusoidal fold
shape corresponding to the theoretical dominant wavelength to thickness ratios. Increasing viscosity contrasts cause larger wavelength to
thickness ratios and faster amplitude growth. Numbers above folds indicate engineering strains in percent.
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
203
in Eq. (26) is identical to that for the arc length
method (Eq. (24)). The finite amplitude solution
provides the arc length method in the limit of very
large growth rates.
Schmalholz and Podladchikov (2000) showed that
the arc length of a folded layer with initial sinusoidal
shape can be approximated by:
Larc
p2 ðA=lÞ2
¼1þ
:
l
1 þ 3ðA=lÞ2
Fig. 5. Accuracy of the arc length method. The arc length method is
applied to numerically simulated fold shapes and strain estimates
obtained by the arc length method are compared with real strains
known from numerical simulations. The dotted line for ‘‘chevron
type’’ folding corresponds to folding with pure limb rotation where
the initial arc length is unaltered during folding. The arc length
method gets more inaccurate with decreasing viscosity contrast
(vc). The arc length method is accurate for viscoelastic layers with
R = 2 and different ratios of membrane stress to shear modulus
( P/G = PG) because folds with large growth rates tend to develop
‘‘chevron type’’ fold shapes.
show a ‘‘chevron type’’ fold evolution and the arc
length method is accurate because for folding of
layers with large growth rates, the limbs are rigid
and exclusively rotate around the fold hinge (Fig. 5;
for a geometrical description see Price and Cosgrove,
1990).
The new finite amplitude solution derived by
Schmalholz and Podladchikov (2000) is given by:
en ¼ ln
Larc l0
Larc0 l
a=ð2þaÞ
þln
A l0
A0 l
1=ð2þaÞ
; (25)
where Larc0 is the initial arc length of the fold, for
which a first order Taylor expansion of Eq. (25) about
a 1 gives:
en ¼ ln
Larc l0
Larc0 l
ð26Þ
Assuming that l0/Larc0 1, and employing Eq.
(23) to relate natural and engineering strains, the strain
ð27Þ
Substituting Eq. (27) into Eq. (26), replacing the
natural strain by the engineering strain (Eq. (23)),
assuming that l0/Larc0 1 and solving the resulting
equation for A/l yields:
pffiffiffi
e
A
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
l
p2 ðp2 þ 3Þe
ð28Þ
Eq. (28) gives the finite amplitude evolution with
increasing strain for folded layers exhibiting large
growth rates. The amplitude depends on the square
root of the strain. Note that this dependence is similar
to the relations between the amplitude and the strain
(or the axial load) that is known in the engineering
literature as post-buckling (e.g., Bazant and Cedolin,
1991), where axially compressed, elastic beams are
investigated.
After substitution of Eqs. (27) and (6) (for n = 1),
the finite amplitude solution for strain (Eq. (25))
depends on two geometric ratios: the amplitude to
wavelength ratio A/l and the initial wavelength to
thickness ratio ldd/H0. Therefore, the observable
ratios A/l and l/H are chosen as coordinates for the
construction of a strain contour map. Observed wavelength to thickness ratios are usually < 10 (e.g.,
Johnson and Fletcher, 1994; Sherwin and Chapple,
1968). Consequently, H/l is employed to represent
small wavelength to thickness ratios on linear coordinate axes (Fig. 6). The finite amplitude approximation (Eq. (28)) is valid for small H/l values (which
correspond to large ldd/H0 values and large growth
rates). Therefore, the strain increase with A/l can be
estimated assuming H/l 0 with Eq. (28). However,
this approximation (Eq. (28)) must be corrected for
increasing H/l (or decreasing ldd/H0). Importantly,
results of the finite amplitude solution (Eq. (25)) show
204
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
Fig. 6. The strain contour map. The measured ratios A/l and H/l of any observed fold shape allow strain and competence contrast estimation
using the strain contour map. Lines marked with numbers in percent are contour lines of the corresponding strain in percent.
that all strain contours exhibit a more or less constant,
negative slope of A/l with increasing H/l (Fig. 6).
Therefore, A/l in Eq. (28) is replaced by A/l C1H/
l, where C1 is the average slope of a strain contour.
The finite amplitude solution is only accurate for
viscosity contrasts >50 due to usage of thin-plate
assumptions (Schmalholz and Podladchikov, 2000).
This is confirmed by numerical results of viscous
single-layer folding with small viscosity contrasts.
These numerical results indicate a change in strain
contour slope, if H/l values are larger than these H/l
values, which belong to the folding line for a viscosity
contrast of around 50 (Fig. 6). A folding line is the
characteristic line containing all points that are
defined at different strains through the measured A/l
and H/l values for a fixed viscosity contrast (Fig. 6).
The initial point of such a line corresponds to the
initial dominant wavelength and the nucleation amplitude. Numerical results show that the second slope has
an approximately constant value, designated C2, for
all strain contours. The folding line for a viscosity
contrast of 50 defines the boundary, at which the
strain contour slope changes from C1 to C2. The slope
C1 is applied for strain contours above the folding line
for a viscosity contrast of 50. This folding line was
approximated by least squares as:
A
H
0:22 þ 2:43
l
l
ð29Þ
The regression equation permits calculation of the
value of H/l (termed H50), at which the break in slope
of a strain contour occurs for a given A/l. The strain
contour map (Fig. 6) is then constructed by solving
Eq. (28) for strain and correcting A/l by the slopes C1
and C2. This yields:
e¼
p2 Z 2
1 þ Z 2 ð3 þ p2 Þ
with
Z¼
A=l þ C1 H=l;
A=l þ C2 H=l þ ðC1 C2 ÞH50 ;
if H=l= < H50
:
if H=l > H50
H50 ¼ ðA=l þ 0:22Þ=2:43; C1 ¼ 0:8; C2 ¼ 0:4
ð30Þ
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
Fig. 7. Verification of the strain estimation method by numerical
simulations. Real strains from viscous folds with different viscosity
contrast (vc) are plotted versus strains estimated by the strain
contour map (Fig. 6). Estimated strains are close to real strains
which justifies usage of only two constant slopes to approximate the
change of A/l with varying H/l along a fixed strain contour (see
Fig. 6).
The slope C1 of the strain contours above the
folding line for a viscosity contrast of 50 is approximated by least square fit using the finite amplitude
solution. Alternatively, C1 can be derived analytically
by the derivative of A/l with respect to H/l by
applying the rules of implicit differentiation to the
finite amplitude solution presented in Eq. (25). The
slope C2 of the strain contours below this folding line
is approximated by least square fit using three numer-
205
ical simulations for viscosity contrasts of 10, 25 and
50.
Numerically calculated folding lines for viscosity
contrasts of 10, 25, 50 and 250 are plotted within the
strain contour map (Fig. 6). These folding lines permit
estimation of the viscosity contrast for viscous folding
or, for unknown rheology, the ‘‘effective’’ viscosity
contrast.
In Fig. 7, the real strain is plotted versus the strain
estimated using the strain contour map for several
numerical simulations. The accuracy of the strain
predictions justifies the usage of the constant slopes
C1 and C2, and the usage of the approximated folding
line for a viscosity contrast of 50 (Eq. (29)) as the
boundary between the two domains of different strain
contour slopes.
5. Verification of the strain contour map by
numerical and analogue experiments
To verify the strain estimation method, two numerical simulations were performed with our spectral/
finite-difference code for folding of layers with initial
random perturbations (Schmalholz et al., 2001). The
initial amplitude to thickness ratio was 0.02 and the
amplitudes exhibited an uncorrelated (‘‘white noise’’)
distribution. One layer was pure viscous with a
viscosity contrast of 100 and the other layer was
viscoelastic (linear viscoelastic Maxwell material;
e.g., Turcotte and Schubert, 1982; Findley et al.,
1989; Shames and Cozzarelli, 1997) with a viscosity
Fig. 8. Numerical simulations of fold train evolution for viscous and viscoelastic rheologies. The initial random perturbations corresponded to an
uncorrelated (‘‘white noise’’) amplitude distribution.
206
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
Fig. 9. Verification of the strain estimation method by numerical
simulations (Fig. 8). Each fold limb within the fold trains was
treated as a fold limb of a fold that is symmetric to the fold axial
plane. The measured ratios A/l and H/l were used to estimate
strains using the strain contour map. These estimates are compared
with real strains of each fold limb, and good coincidence is
obtained.
contrast of 2500 and R = 2 (Fig. 8). The method is
applied to every individual fold limb, which is treated
as half of a fold that is symmetric with respect to the
fold axial plane. For the purpose of strain estimation,
the wavelength is measured as two times the horizontal distance between two neighboring fold hinges.
This distance does not correspond to a wavelength
in the strict mathematical sense (cf., Fletcher and
Sherwin, 1978). However, the good correlation (Fig.
9) between the real strains versus strains estimated by
Eq. (30) shows that (i) the use of the horizontal
distance between neighboring hinges as wavelength
is acceptable and (ii) the strain estimation method is
applicable to folds with natural, asymmetric shape.
Furthermore, the strain estimation method was
applied to results of analogue and numerical experiments performed by other authors. Perturbation C in
Fig. 9 of Abbassi and Mancktelow (1992) was considered. The real strain of the fold, which developed in
the middle part of the shortened layer, was determined
using the experimentally deformed grid. The layer at
2.1% shortening was considered as initial geometry.
The shapes of the middle fold at 8.6%, 12.9% and
22.5% bulk strain were used to measure the ratios A/l
and H/l. The estimated strains using our method
compare well to the real strains determined from the
analogue experiment (Fig. 10). A case illustrated by
Hudleston (1973) (his Fig. 3B), consists of two folds
within a layer (viscosity contrast = 24) that was shortened by a bulk deformation of 6 (around 59%). In this
case, the estimated strains are about 10% less than the
real strains, a deviation that is attributed to the
unknown initial geometry and the small viscosity
contrast of the experimental configuration. However,
this discrepancy is small in comparison to the error
from the arc length method that yields a strain of
about 30%. In Lan and Hudleston (1995b), their Fig. 1
was considered, which shows two ductile folds produced by finite element simulations. The two ductile
folds exhibited a power-law exponent n = 1 and
n = 10, and were shortened 40%. Lan and Hudleston
used an initial amplitude to thickness ratio of 0.1 (A0/
H0 = 0.1) for a viscosity contrast of 100. This initial
amplitude is larger than the corresponding nucleation
amplitude (AN). However, a comparison between real
strain and strain estimated by our method is possible,
if the real strain is corrected. The strain that would
Fig. 10. Verification of the strain estimation method by analogue
(Abbassi and Mancktelow, 1992; Hudleston, 1973) and numerical
(Lan and Hudleston, 1995a,b; Mancktelow, 1999) experiments of
other authors. The solid line represents a perfect fit. Estimated
strains correspond well with real strains except for the analogue
experiments of Hudleston (1973). The reason is the low viscosity
contrast of 24 and the unknown initial geometry. However, the arc
length method estimates for this case a strain of around 30%, which
is considerably more inaccurate than our strain estimation method.
See the text for a more detailed description of the comparison.
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
have been accommodated during folding from AN up
to A0/H0 = 0.1 is calculated using Eq. (21). AN is
determined by Eqs. (4) and (9). The calculations yield
that a strain of around 6% for n = 1 and around 4% for
n = 10 would have been accommodated between AN
and A0/H0 = 0.1. The viscous fold (n = 1) accommodated around 46% strain and the ductile fold (n = 10)
around 44% strain, if AN is considered as initial
amplitude. The estimated strains are close to the real
strains (Fig. 10). Finally, the strain estimation method
was applied to finite element simulations of singlelayer folding performed by Mancktelow (1999). The
deformed layer in his Fig. 12b at a logarithmic strain
of 1.112 (around 67% strain) was considered. For
each individual fold, the ratios A/l and H/l were
measured, and the estimated strains are again plotted
versus the real strain of 67%.
207
The application of the newly developed strain
estimation method to analogue and numerically produced fold shapes demonstrates that the method gives
reliable strain estimates and can be used to estimate
strains from natural fold shapes.
6. Strain partitioning in fold trains
In the following the term strain partitioning refers
to the difference in strain that is accommodated by the
whole fold train (bulk strain) and the individual fold
limbs within the fold train (individual strain). To
investigate strain partitioning we employ the numerical simulations of fold trains presented in Fig. 8. The
bulk strain of the total layer is plotted versus individual strains that are accommodated by individual folds
Fig. 11. Strain partitioning in numerically simulated viscous and viscoelastic fold trains. The individual strains of individual folds within the fold
train are plotted versus the layer bulk strain. If all individual strains would lie on the dotted line, no strain partitioning is present because all
individual folds accommodate the same strain than the fold train. The viscous fold train shows slightly increasing strain partitioning, whereas the
viscoelastic fold train shows a strong increase in strain partitioning already at the beginning of shortening.
208
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
Fig. 12. (A) Individual strains of folds within the numerically simulated viscous and viscoelastic fold trains (see Fig. 8) at a bulk strain of
13% are plotted versus the wavelength to thickness ratios (l/H) of the corresponding folds. The viscous fold train shows little strain
partitioning and little variation in the l/H ratios. The viscoelastic fold train shows strong strain partitioning where larger l/H ratios correspond
to smaller individual strains. (B) The same individual strains are plotted versus the restored initial wavelength to thickness ratios (l0/H0). The
l/H ratios were restored using Eq. (33).
(Fig. 11). Each fold limb is considered to be the limb
of a fold that is symmetric with respect to the fold
axial plane. Fig. 11 shows that the strain partitioning
in the viscoelastic layer is much stronger than in the
pure viscous layer.
Individual strains at a layer bulk strain of 13%
are plotted versus the corresponding wavelength to
thickness ratio (l/H) of individual folds (Fig. 12A). In
the viscous case, all folds show more or less the same
strain and similar l/H ratios. In contrast, individual
folds in the viscoelastic layer accommodate very
different amounts of strain and l/H ratios vary
strongly. The initial l/H ratio (l0/H0) can be restored
if it is assumed that the area of the fold does not
change during folding and that the initial arc length is
approximately equal to the initial wavelength. The
initial fold area Larc0H is then equal to the current fold
area LarcH and the equation for the conservation of
area (or mass) is:
l0 H0 ¼ Larc H:
ð31Þ
Under pure shear shortening, the initial wavelength
is related to the current wavelength through the
equation:
l0 ¼ lexpðen Þ:
ð32Þ
Substituting H0 = l0/(l0/H0) into Eq. (31) and
rearranging yields:
l0
l expð2en Þ
:
¼
H0 H Larc =l
ð33Þ
The ratio of Larc/l can be expressed through the
ratio of A/l (see Eq. (27)) and the natural strain can be
replaced by the engineering strain using Eq. (23).
Therefore, l0/H0 can be calculated through l/H, A/l
and the strain. In Fig. 11B, the average restored values
of l0/H0 for individual folds in the viscous layer is
14, and comparable to the theoretical dominant
wavelength (Eq. (4)) of 16 for a viscosity contrast
of 100.
The strain estimation method is used to investigate
strain partitioning within natural fold trains from
Ramsay and Huber (1987) (Fig. 13A) and Weiss
(1972) (Fig. 13B). The measured data, the estimated
strains and the restored initial wavelength to thickness
ratios are presented in Tables 2 and 3. The fold train of
Fig. 13A (Table 2) has approximately constant strain
of 70% for all individual folds, whereas in the fold
train of Fig. 13B (Table 3) strains vary from 35% to
62%. In Fig. 14A, the estimated strains of individual
folds within three different fold trains are plotted
versus the restored ratio of l0/H0. The three fold
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
trains are the train shown in Fig. 13A, the train
produced by the numerical simulation of Mancktelow
(1999) (the same fold train used for the verification of
our method) and the train generated by our numerical
simulation for viscous rheology (Fig. 11). All three
fold trains show little strain partitioning. In Fig. 14B,
the estimated strains of individual folds within two
209
other fold trains are plotted versus the restored ratio of
l0/H0. The fold trains are the train shown in Fig. 13B
and the train with viscoelastic rheology produced by
our numerical simulation presented in Fig. 11. The
strain partitioning within these fold trains exhibits the
same characteristic relation to the restored ratio of l0/
H0, namely that increasing values of l0/H0 correspond
to decreasing strains.
7. Discussion and conclusion
Fig. 13. Natural examples of fold trains. (A) Folded layer (thin
section, picture from Ramsay and Huber, 1987) where five fold
limbs were used to estimate strain (see Table 1). (B) Folded layer
(picture from Weiss, 1972) where six fold limbs were used to
estimate strain (see Table 2).
It this paper, a new method is proposed for the
strain and competence contrast estimates as a function
of two easily measurable geometrical parameters: the
thickness to wavelength (H/l) ratio and the amplitude
to wavelength (A/l) ratio (Fig. 6, Eq. (30)). Analytical
solutions for folding of ductile and viscoelastic layers
are combined using linear viscoelastic folding theory
(Schmalholz and Podladchikov, 1999, 2001) to enable
the applicability of our method to natural folds without knowing the deformation mechanism. Our method
reliably estimates strain from observed fold shapes
and is a considerable improvement to the arc length
method.
The major shortcoming of the method is that the
nucleation amplitude AN must be defined as initial
amplitude corresponding to zero strain. If the natural
(or real) initial amplitude is greater or less than AN, the
method over- or underestimates, respectively, the
strain. For purpose of comparison, the value of AN
for a viscosity contrast of 50 provides a reasonable
reference value to define small or large initial amplitude. AN for a viscosity contrast of 50 is 0.03. For
viscosity contrasts < 50 thin-plate approximations are
inaccurate, because layer thickening influences the
deformation of the layer. In general, smaller AN leads
to larger growth rates (cf. Eq. (8)). In the initial
folding stages large growth rates cause exponential
amplitude growth within small amounts of strain.
Therefore, if a natural layer is characterized by initial
amplitudes >AN, the error in the strain estimate is, in
general, small. Also, further improvements of the
precision of the strain estimates can easily be made
using Eq. (22) to correct for the discrepancy in strain
between the real initial amplitude and the nucleation
amplitude, if the former can be constrained independently.
210
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
Table 2
Measured data, estimated strains and restored l0/H0 ratios from the
fold train in Fig. 13A
Table 3
Measured data, estimated strains and restored l0/H0 ratios from the
fold train in Fig. 13B
Limb
A/l
H/l
e [%]
l0/H0
Limb
A/l
H/l
e [%]
l0/H0
AB
BC
CD
DE
EF
0.70
0.92
0.69
0.61
0.56
0.36
0.26
0.44
0.43
0.51
71
72
72
70
70
11
15
9
10
8
AB
BC
CD
DE
EF
FG
0.22
0.29
0.19
0.34
0.31
0.25
0.07
0.12
0.09
0.33
0.23
0.30
38
50
35
62
58
55
25
20
21
11
14
11
Measured A/l and H/l ratios that lie below the
folding line for a viscosity contrast of 50 indicate
small competence contrast (Fig. 6) and that the natural
initial amplitudes may have been smaller than AN.
Therefore, the strain estimate is a minimum value, and
the real strain may be considerably larger than estimated. However, for folding with small competence
contrast the arc length method is inaccurate (Fig. 5),
and because our method yields more accurate strain
estimates than the arc length method the error is
unlikely to be important. AN is only likely to underestimate natural initial amplitudes for layers with
small competence contrast because the corresponding
AN is relatively large. Under this condition, the strain
contour map allows estimation of the competence
contrast. Also, the strain contour map provides accurate strain estimates for folding of a nonlinear viscous
layer exhibiting large power-law exponents (Fig. 10).
It appears that the observable differences in fold shape
for viscous and strongly power-law rheologies do not
reflect significant differences in strain. This behaviour
suggests that A/l and H/l are in fact the appropriate
coordinates for strain contour maps.
Recently performed numerical simulations of single-layer folding showed, that for both simple and
pure shear conditions, the evolution of the ratio A/l
versus strain is similar. Despite the strong asymmetry
of the folds developed under simple shear, the finite
amplitude solution (Schmalholz and Podladchikov,
2000), used for constructing the strain contour map,
predicted the fold amplification successfully. For the
folds developed under simple shear, the amplitude
was measured as half the vertical distance between the
highest and lowest point of the top layer boundary and
Fig. 14. Two characteristic types of strain partitioning. Estimated strains of individual folds within fold trains are plotted versus restored initial
wavelength to thickness ratios (l0/H0, see Eq. (33)). (A) The fold trains from Fig. 13A (Ramsay and Huber, 1987), from a numerical simulation
of Mancktelow (1999) (see text) and the viscous fold train (Fig. 8) show little strain partitioning and little variation in the restored l0/H0 ratio.
(B) The fold trains from Fig. 13B (Weiss, 1972) and the viscoelastic fold train (Fig. 8) show strong strain partitioning and increasing restored l0/
H0 ratios with decreasing individual strain.
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
Fig. 15. Growth rates for ductile and elastic layers are plotted versus
the wavelength to thickness ratios (l0/H0) of the layers. Growth
rates for ductile layers (Eq. (6)) increase with increasing l0/H0
ratios for different power-law exponents (n). Growth rates for elastic
layers (Eq. (13))
˙ decrease with increasing l0/H0 ratios for different
values of P/(mle).
the wavelength was measured as the horizontal distance between two neighboring, concave upward fold
hinges. These new results indicate that the presented
strain estimation method can also be successfully
applied to asymmetric fold shapes that developed
under simple shear conditions.
There are two different types of strain partitioning
patterns of individual folds within fold trains (Fig. 14):
(i) little variation in strain and restored l0/H0 ratio, and
(ii) strong variation in strain, where restored values of
l0/H0 vary inversely with strain. The two patterns are
observed, respectively, in numerically simulated pure
viscous and viscoelastic (R = 2) fold trains. We speculate that strain partitioning within fold trains is related
to rheology. To justify this speculation, ldd/H0 and lde/
H0 is replaced by l0/H0 within the ductile growth rate
(Eq. (6)) and the elastic growth rate (Eq. (13)),
respectively. These growth rates have different relationships to l0/H0 (Fig. 15). In the ductile case (n = 1, 5
and 10), increasing l0/H0 ratios cause increasing
growth
rate. In contrast, in the elastic case ( P/
˙
mle_ = 50, 250 and 500) increasing l0/H0 ratios cause
decreasing growth rate. Therefore, in ductile layers,
larger wavelengths grow faster, whereas in elastic
layers shorter wavelengths grow faster. In both cases,
the fastest growing wavelength should accommodate
211
the most strain. It is assumed that during compression
the dominant wavelength is selected and locked when
the amplitude of the layer perturbation is approximately equal to AN. Individual fold amplitudes within
a layer need not grow to AN simultaneously. In the pure
viscous case, the dominant wavelength exclusively
depends on the viscosity contrast. This dependence
is unchanged regardless of whether the individual
folds grow simultaneously. Therefore, the development of the same dominant wavelength is expected,
which explains the observed clustering of dominant
wavelengths (Figs. 12 and 14A). However, if the layer
has a viscoelastic or power-law rheology, the situation
may be different. In the viscoelastic cases, as in the
viscous case, an individual fold selects its wavelength
when its fold amplitude is equal to AN, but the
dominant wavelength depends on the layer-parallel
stress. The amplification of an individual fold may
reduce the layer-parallel stress within an entire viscoelastic layer by increasing the fold arc length (Schmalholz and Podladchikov, 2000). A later fold may then
evolve under different stress conditions than the initial
fold. For power-law layers, the dominant wavelength
depends on the effective viscosity contrast. However,
for power-law materials the ‘‘effective’’ viscosity is a
function of the stresses within the layer and matrix
(m = m(s); Fletcher, 1974). The stresses within the
whole fold train may alter due to asynchronous growth
of individual folds. The effective viscosity contrast
may therefore change during shortening of power-law
layers. Consequently, individual folds that have different amplitudes may select wavelengths of different
size.
Also, strong wavelength selectivity, predicted by
the linear theories (e.g., Biot, 1961), decreases strongly
after only a few percent of strain for layers with
relatively large growth rates (i.e., >75) (Schmalholz
and Podladchikov, 2000). This means that the development of a single dominant wavelength within layers
exhibiting large growth rates is improbable. In such
layers wavelengths of different size may develop.
These wavelengths can be shorter or longer than the
dominant wavelength, and are expected to amplify
according to the finite amplitude growth rate spectra
(Schmalholz and Podladchikov, 2000). Consequently,
we speculate that periodic folds with approximately
the theoretical dominant wavelength only develop
when growth rates are small ( < 75). Folds that grow
212
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
rapidly are expected to exhibit heterogeneous wavelength spectra.
The typical patterns of strain partitioning between
individual folds in fold trains can thus be used to
discriminate deformation mechanisms: (i) folding with
small growth rates (relatively constant wavelengths),
(ii) folding with large growth rates due to viscoelastic
rheology (shorter wavelengths accommodate more
strain), and (iii) folding with large growth rates due
to power-law rheology (longer wavelengths accommodate more strain). Therefore, measured strain partitioning patterns of natural fold trains and our strain
estimation method could be used to assess the deformation mode (ductile or viscoelastic) and the magnitude of the competence contrast during the initiation of
folding.
Acknowledgements
We thank R.C. Fletcher, R.H. Groshong, Jr. and H.
Fossen for helpful reviews. We are grateful to D.W.
Schmid for providing his results of numerical simulations of single-layer folding under simple shear. S.M.
Schmalholz was supported by ETH project No. 0-20499-98.
References
Abbassi, M.R., Mancktelow, N.S., 1992. Single layer buckle folding
in non-linear materials: I. Experimental study of fold development from an isolated initial perturbation. J. Struct. Geol. 14 (1),
85 – 104.
Bazant, Z.P., Cedolin, L., 1991. Stability of structures: elastic, inelastic, fracture, and damage theories. The Oxford Engineering
Science Series. Oxford Univ. Press, New York.
Biot, M.A., 1961. Theory of folding of stratified viscoelastic media
and its implications in tectonics and orogenesis. Geol. Soc. Am.
Bull. 72, 1595 – 1620.
Biot, M.A., 1965. Mechanics of Incremental Deformations. Wiley,
New York.
Chapple, W.M., 1968. A mathematical theory of finite-amplitude
rock-folding. Geol. Soc. Am. Bull. 79, 47 – 68.
Cobbold, P.R., 1977. Finite-element analysis of fold propagation: a
problematic application? Tectonophysics 38, 339 – 358.
Currie, I.B., Patnode, H.W., Trump, R.P., 1962. Development of
folds in sedimentary strata. Geol. Soc. Am. Bull. 73, 655 – 674.
Dahlstrom, C.D.A., 1969. Balanced cross sections. Can. J. Earth
Sci. 6, 743 – 757.
Dieterich, J.H., 1970. Computer experiments on mechanics of finite
amplitude folds. Can. J. Earth Sci. (7), 467 – 476.
Findley, W.N., Lai, J.S., Onaran, K., 1989. Creep and Relaxation of
Nonlinear Viscoelastic Materials. Dover Publications, New York.
Fletcher, R.C., 1974. Wavelength selection in the folding of a single
layer with power-law rheology. Am. J. Sci. 274 (11), 1029 – 1043.
Fletcher, R.C., Sherwin, J., 1978. Arc lengths of single layer folds: a
discussion of the comparison between theory and observation.
Am. J. Sci. 278, 1085 – 1098.
Hudleston, P.J., 1973. An analysis of ‘‘single-layer’’ folds developed experimentally in viscous media. Tectonophysics 16,
189 – 214.
Hudleston, P.J., Stephansson, O., 1973. Layer shortening and foldshape development in the buckling of single layers. Tectonophysics 17, 299 – 321.
Hunt, G., Mühlhaus, H., Hobbs, B., Ord, A., 1996. Localized folding of viscoelastic layers. Geol. Rundsch. 85, 58 – 64.
Johnson, A.M., Fletcher, R.C., 1994. Folding of Viscous Layers.
Columbia Univ. Press, New York.
Lan, L., Hudleston, P.J., 1995a. The effects of rheology on the strain
distribution in single layer buckle folds. J. Struct. Geol. 17 (5),
727 – 738.
Lan, L., Hudleston, P.J., 1995b. A method of estimating the stress
exponent in the flow law for rocks using fold shape. PAGEOPH
145 (3/4), 621 – 635.
Mancktelow, N.S., 1999. Finite-element modelling of single-layer
folding in elasto-viscous materials: the effect of initial perturbation geometry. J. Struct. Geol. 21 (2), 161 – 177.
Poliakov, A., Cundall, P., Podladchikov, Y., Laykhovsky, V., 1993.
An explicit inertial method for the simulation of visco-elastic
flow: an evaluation of elastic effects on diapiric flow in twoand three-layers models. In: Runcorn, D.B.S.a.S.K. (Ed.),
Flow and Creep in the Solar System: Observations, Modeling
and Theory, Proceedings of NATO Advanced Study Institute,
Kluwer, The Netherlands, pp. 175 – 195.
Price, N.J., Cosgrove, J.W., 1990. Analysis of Geological Structures. Cambridge Univ. Press, Cambridge.
Ramberg, H., 1963. Fluid dynamics of viscous buckling applicable
to folding of layered rocks. Bull. Am. Assoc. Pet. Geol. 47 (3),
484 – 505.
Ramsay, J.G., Huber, M.I., 1987. The techniques of modern structural geology. Folds and Fractures, vol. 2. Academic Press,
London, pp. 309 – 700.
Schmalholz, S.M., Podladchikov, Y.Y., 1999. Buckling versus folding: importance of viscoelasticity. Geophys. Res. Lett. 26 (17),
2641 – 2644.
Schmalholz, S.M., Podladchikov, Y.Y., 2000. Finite amplitude folding: transition from exponential to layer length controlled
growth. Earth Planet. Sci. Lett. 181 (4), 619 – 633.
Schmalholz, S.M., Podladchikov, Y.Y., 2001. Viscoelastic folding:
Maxwell versus Kelvin rheology. Geophys. Res. Lett. 28 (9),
1835 – 1838.
Schmalholz, S.M., Podladchikov, Y.Y., Schmid, D.W., 2001. A
spectral/finite-difference method for simulating large deformations of heterogeneous, viscoelastic materials. Geophys. J. Int.
145, 199 – 208.
Shames, I.H., Cozzarelli, F.A., 1997. Elastic and Inelastic Stress
Analysis. Taylor & Francis, Washington, DC.
Sherwin, J., Chapple, W.M., 1968. Wavelengths of single layer
S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213
folds: a comparison between theory and observation. Am. J. Sci.
266, 167 – 179.
Smith, R.B., 1977. Formation of folds, boudinage, and mullions in
non-Newtonian materials. Geol. Soc. Am. Bull. 88 (2), 312 – 320.
Timoshenko, S.P., Woinowsky-Krieger, S., 1959. Theory of Plates
and Shells. McGraw-Hill, Auckland.
213
Turcotte, D.L., Schubert, G., 1982. Geodynamics. Applications of
Continuum Physics to Geological Problems. Wiley, New York.
Weiss, L.E., 1972. The Minor Structures of Deformed Rocks.
Springer-Verlag, Berlin.
Zhang, Y., Hobbs, B.E., Ord, A., 1996. Computer simulation of
single-layer buckling. J. Struct. Geol. 18 (5), 643 – 655.