JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, B03407, doi:10.1029/2003JB002421, 2004
Folding of a finite length power law layer
Daniel W. Schmid1 and Yuri Y. Podladchikov
Geologisches Institut, ETH Zentrum, Zürich, Switzerland
Fernando O. Marques
Departamento Geologia and Centro de Geofisica Universidade de Lisboa, Faculdade Ciências, Universidade de Lisboa,
Lisbon, Portugal
Received 27 January 2003; revised 17 September 2003; accepted 22 December 2003; published 16 March 2004.
[1] Folding of an isolated finite length power law layer embedded in a Newtonian viscous
matrix is investigated and compared to conventional folding experiments where the layer is
of infinite length or in direct contact with lateral boundaries. The approach employed is a
combination of the complex potential method for the basic state and the thin plate
approximation for the linear stability analysis and is verified by finite element models. The
resulting theory reveals that the aspect ratio of a layer has a first-order influence on the
development of folds. The aspect ratio competes with the effective viscosity contrast
for dominant influence on the folding process. If the aspect ratio is substantially larger
than the effective viscosity contrast, the conventional theories are applicable. In other
situations, where the aspect ratio is smaller than the effective viscosity contrast, substantial
corrections must be taken into account, which lead to a new folding mode that is mainly
characterized by decreasing growth rates with increasing effective viscosity contrast
(relative to the far-field shortening rate). This new folding mode helps explain the absence of
large wavelength to thickness ratio folds in nature, which may be due to the limitations of
INDEX TERMS: 3210 Mathematical
aspect ratios rather than large effective viscosity contrasts.
Geophysics: Modeling; 5120 Physical Properties of Rocks: Plasticity, diffusion, and creep; 8005 Structural
Geology: Folds and folding; 8020 Structural Geology: Mechanics; KEYWORDS: folding, fold, inclusion, clast,
aspect ratio
Citation: Schmid, D. W., Y. Y. Podladchikov, and F. O. Marques (2004), Folding of a finite length power law layer, J. Geophys.
Res., 109, B03407, doi:10.1029/2003JB002421.
1. Introduction
[2] Folding is a primary mechanism by which layered
rocks accommodate shortening. Thus folds contain important
information about the deformation history and can potentially
be used to decipher the kinematic history and mechanical
behavior of rocks within a certain outcrop or region. A sound
understanding of the mechanics of folding instabilities is
required to achieve this. Principles of fold mechanics have
been well developed (for summary, see Price and Cosgrove
[1990], Johnson [1977], and Johnson and Fletcher [1994]);
however, previous work on folding assumes either infinitely
long layers or rigid lubricated walls that directly apply
boundary conditions onto the layer ends. This may not be
the case in nature, where folds are always observed in layers
of finite length and are often isolated from the far-field
shortening that drives the folding instability (Figure 1).
Questions then arise as to how the aspect ratio of a layer
and the relative isolation from the far-field forces affect fold
development.
[3] We address these questions with a modification of
previous works. We assume that a finite length layer initially
has an elliptical shape. Employing Muskhelishvili’s [1953]
complex potential method, we determine the mechanics of
this basic state, which we then combine with the linear
stability analysis based on the thin plate approximation. To
overcome the strong simplifications imposed by the analytical methods, we use a finite element model that allows us to
verify the results and illustrates the folding of finite length
layers up to large strains. The configuration studied is a
power law layer embedded in a weaker Newtonian matrix
subjected to plane strain, pure shear far-field flow in two
dimensions with all materials assumed to be incompressible.
Our analysis shows that the layer aspect ratio has a firstorder influence on the development of folds and governs
new folding modes. The conventional theories remain
applicable if the aspect ratio is substantially larger than the
effective viscosity contrast. For all other cases, the folding
indicated by the finite length layer theory predicts significantly reduced growth rates that may explain why large
wavelength to thickness ratios are not observed in nature.
1
Now at Physics of Geological Processes, University of Oslo, Oslo,
Norway.
2. Basic State Analysis
Copyright 2004 by the American Geophysical Union.
0148-0227/04/2003JB002421$09.00
[4] In order to study the folding instability it is necessary
to understand the stable configuration, i.e., the basic state.
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2003]. We only give the expressions relevant for the
following thin plate analysis here. The horizontal stress
within the layer is
sxx ¼ 2
mm ða þ 1Þðamm þ mm þ 2aml Þ
e_ ;
a2 mm þ mm þ 2aml
ð1Þ
and the horizontal strain rate within the layer is
e_ xx ¼
Figure 1. Extremity of a folded quartz vein, Almograve,
Portugal.
The basic state has the same boundary conditions, material
properties, and geometrical configuration as the folding
model, but no perturbation that could grow is present. In
our analysis the basic state is represented by an elliptical
inclusion (cylinder) subjected to a far-field pure shear
shortening parallel to the inclusion long axis (Figure 2).
Such an elliptical inclusion, as well as the ellipsoid in three
dimensions, has the exceptional property that all stress and
strain rate components within the inclusion are constant
with respect to space and are completely described by single
values [e.g., Eshelby, 1957] under homogenous boundary
conditions such as pure shear far-field flow. Constant values
inside the inclusion simplify the analysis of finite length
layer folding as it allows for the combination of the finite
length layer basic state with the conventional folding
analysis that relies on constant values.
[5] We realize that this combination results in a deviation
between the geometry of the basic state and the thin plate
approximation that we employ in the linear stability analysis. Our thin plate approximation requires a basic state with
a straight layer of constant thickness that is much longer
than thick. This is not the case for elliptical finite length
layers. However, over the majority of the layer length, the
layer-matrix interface slope and thickness variations due to
the finiteness of the layer will be negligible compared to the
interface slope caused by the relatively frequent, most
unstable interface perturbations, whose growth is analyzed
here. Exceptions are the tips of the layer where thickness
and interface slope due to the finite nature of the layer
change rapidly and likely cause end effects. These end
effects, however, should not affect the rest of the layer.
Johnson and Fletcher [1994] have shown that small slope
and thickness variations only affect the third-order analysis
and we therefore conclude that the errors introduced by
combining a slightly different basic state with our thin plate
approximation are negligible for most of the investigated
cases where the fold wavelength is shorter than the layer
length (see comparison with finite element models).
2.1. Basic State of a Newtonian Finite Length Layer
[6] We have recently used Muskhelishvili’s [1953] complex potential method to derive all closed form solutions
for deformable elliptical inclusions in two-dimensional,
plane strain, general shear [Schmid and Podladchikov,
ða þ 1Þ2 mm
e_ ;
ða2 þ 1Þmm þ 2aml
ð2Þ
where a is the aspect ratio of the layer (Figure 2), ml is the
viscosity of the layer, mm is the viscosity of the matrix, and e_
is the far-field strain rate. These expressions for sxx and e_ xx
exhibit familiar limits. For infinitely long layers (a ! 1)
the layer horizontal stress is
sxx ¼ 4ml e_ 2mm e_ :
ð3Þ
With the additional assumption that the viscosity of the
layer is significantly higher than that of the matrix, we can
omit the second term to get sxx = 4mle_ , which is the total
horizontal stress value usually used in viscous folding
theory [Biot et al., 1961]. On the other hand, if we assume a
very high viscosity (ml ! 1) for finite length layers,
equation (1) yields
sxx ¼ 2mm ð1 þ aÞ_e;
ð4Þ
which is identical to the result Mandal et al. [2001] obtained
based on Jeffery’s [1922] theory.
[7] For infinitely long layers the expression for e_ xx yields
the far-field value, e_ , as the horizontal strain rate in the layer.
This is equivalent to applying the boundary conditions
through a rigid, lubricated boundary directly onto the layer
ends. The infinitely rigid inclusion is not deformable and
consequently equation (2) yields e_ xx = 0 for ml ! 1.
[8] To simplify the analysis, it is practical to reduce the
complexity of the derived expressions for sxx and e_ xx
Figure 2. Illustration of a finite length layer subject to
horizontal (x parallel) compression, where h and l are the
layer thickness and length, respectively. Note that a typical
real layer is likely to have a larger aspect ratio than shown
here.
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through approximate forms. Introducing the dimensionless
parameter Da
Da ¼
ml 1
;
mm a
is the effective viscosity in the Da 1 limit. The Da 1
limit yields
me ¼ mne
_
ð5Þ
1
e_
1 þ 2Da
am 1n
m
:
2
ð11Þ
Following the differences in the geometry of the basic state,
the straight and the curved overbar symbols distinguish the
infinite and finite layer cases, respectively.
we can approximate sxx and e_ xx as
sxx 4ml
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ð6Þ
3. Linear Stability Analysis
e_ xx 1
e_ :
1 þ 2Da
ð7Þ
The error introduced by the Da approximations is negligible,
i.e., smaller than 10% under the condition that the layer
aspect ratio is larger than 20. Parameter Da is useful since it
readily allows for analyzing the competing effects of the
aspect ratio, a, versus the viscosity contrast, ml/mm. If Da 1, the layer has a much larger aspect ratio than viscosity
contrast with respect to the matrix and all the classical
values of folding analysis are recovered (as can be seen
from equations (6) and (7)) and shown in the following.
Hence an infinitely long layer and the conventional
configuration where rigid, lubricated walls impose the
boundary conditions directly onto the layer ends are
identical. If Da 1, the aspect ratio of the finite length
layer is smaller than the viscosity contrast and expressions
are obtained that govern new folding modes.
2.2. Basic State of a Power Law Finite Length Layer
[9] Laboratory measurements show that ductile rocks
almost never behave as Newtonian materials, but exhibit
nonlinear, power law behavior. In order to take the
corresponding effects into account, we introduce a power
law material layer from which the Newtonian case can
always be deduced as an end-member. The above basic
state analysis remains valid, however, the viscosity of the
layer must be substituted by an effective viscosity, me, which
is a function of the strain rate [Fletcher, 1974; Smith, 1975].
For the following thin plate analysis the effective viscosity
of the layer can be expressed as [Fletcher and Hallet, 1983;
Schmalholz et al., 2002]
me ¼
B ð
e_ xx
2
0
1ð1n1Þ
ð1n1Þ
1
BB
1
Þ B
C
¼ @
;
¼
e_ A
e_
2 1 þ 2Da
2 1 þ 2 me
mm a
ð8Þ
1
n1
where B is a material constant, and n is the power law
exponent. For n > 1 this implicit expression can be
approximated through
me ¼ me
m
1þ 2 e
mm a
n1 !
;
ð9Þ
[10] Using the thin plate approximation, folding of
a power law viscous layer in a Newtonian matrix is
determined by (see Schmalholz et al. [2002] for detailed
derivation)
me h3 @ 5 w
@2w
þ sxx h 2 þ qm ¼ 0:
4
3n @x @t
@x
Here, qm is the vertical component of the stress (resistance)
exerted by the matrix onto the top and bottom layer
boundaries and w is the deflection of the layer.
[11] The general solution form of w can be expressed as
w ¼ Aðt Þ sinðkxÞ
B ð1n1Þ
e_
2
qm ¼ 4mm k
@w
:
@t
ð14Þ
Substituting equations (14) and (13) into equation (12)
results in an ordinary differential equation for A(t) with the
following solution:
3hkn
Aðt Þ ¼ Að0Þ exp
t
;
s
xx
me h3 k 3 þ 12mm n
ð15Þ
where A(0) is the initial amplitude of the perturbation.
Equation (15) shows the well-known result that initially the
amplification of the perturbations is exponential with time
(see Schmalholz and Podladchikov [2000] for finite
amplitude behavior). The part of the exponent in front of t
is termed growth rate, a:
a¼
3hkn
sxx :
me h3 k 3 þ 12mm n
ð16Þ
Since all initial perturbations, independent of the wavelength, are exponentially amplified, it is necessary to
determine the wavelength that exhibits the maximum
growth rate:
rffiffiffiffiffiffiffiffiffiffi
me
l ¼ 2p 3
h:
6nmm
ð10Þ
ð13Þ
where A(t) is the amplitude of the sinusoidal perturbation
with time, and k is the wave number, related to the
wavelength l through k = 2p/l. We choose qm to represent
two viscous half-spaces [Biot, 1961]:
where
me ¼
ð12Þ
ð17Þ
Substituting equation (17) into equation (16) and using the
general expression for sxx, equation (6), we obtain the
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Table 1. Summary of Folding Modesa
Case
Growth Rate
2=3
General
Dominant Wavelength
1=3
1 me
2p 6n
h
mm
1=3
1 me
2p 6n
h
mm
a 1=3 2 me n=3
2p 12n
h
am
4n me
e_
3 mm
2=3
4n me
e_
3 mm
2an2=3 2 me n=3
e_
3
a mm
1
1þ2Da
Large aspect ratio classical limit Da 1
Small, finite aspect ratio Da 1
Condition
1=3
1 me
2p 6n
<a
mm
1=3 1 me
a > max mme ; 2p 6n
m
m
m
a 1=3 2 me n=3
2p 12n
<
a < mme
am
m
m
m
1
Parameters are a = l/h, Da = me/mm/a, me = (B/2)e_ ðn1Þ , and me = ml for n = 1, and me = me[1 + (2 me/mm/a)n1] for n > 1. The conditions are imposed
by the Da parameter and by the assumption that at least one dominant wavelength must fit onto the layer.
a
general maximum growth rate expression, valid for layers of
all aspect ratios:
1
4n me 2=3
a¼
e_ :
1 þ 2Da 3 mm
ð18Þ
[12] The dominant wavelength and maximum growth rate
expressions for the conventional configuration, where the
layer aspect ratio is larger that the effective viscosity
contrast between layer and matrix, are deduced from the
above equations by analyzing their Da 1 limit. As shown
by Fletcher [1974] and Smith [1975], we obtain
sffiffiffiffiffiffiffiffiffiffi
3 me
l ¼ 2p
h
6nmm
a¼
4n me
3 mm
ð19Þ
2=3
e_ :
ð20Þ
On the other hand, if the viscosity contrast is significantly
larger than the aspect ratio, Da 1, the corresponding
expressions are
[14] However, if the general expressions determining
folding in finite length layers are analyzed versus the
relevant actual values within the layer, then the classical
expressions are recovered. In the case of the dominant
wavelength this is evident since the general expression is
written in terms of the real effective viscosity of the layer.
To obtain the same for the general maximum growth rate we
normalize by the expression for the layer parallel strain rate
(equation (7))
a
¼
e_ xx
4n me
3 mm
2=3
:
ð23Þ
Nevertheless, in order to compare folding in different layers
of finite length it is necessary to analyze fold growth with
respect to the far-field-based values, i.e., e_ and me. For
example, in the case of a Newtonian layer, the fold
amplification rate in a conventional configuration (Da 1)
and a layer where the aspect ratio is small relative to the
viscosity contrast (Da 1) are, if normalized by the actual
shortening rate experienced by the layer, identical. However,
the actual bulk shortening required to develop folds of
identical amplitude is for the Da 1 experiment 2Da times
larger as follows from equation (7).
4.2. Maximum Growth Rate
[15] The most important parameter concerning the development
of folds is the rate at which they grow. If the layer is
ð21Þ
infinitely long or the boundary conditions directly applied to
the layer ends, the growth rate is predicted to increase with
increasing effective viscosity contrast, me/mm. On the other
_ 2=3
1=3
n=3
2=3 _
hand if folds grow in isolated layers of finite length the
1
4n me
9 me
2an
2 me
_
a¼ m
e_ ¼ a
e_ ¼
e_ : growth rate is reduced and actually decreases with further
2
2n mm
3
a mm
2 m ea 3 mm
increase of effective viscosity contrast once the me/mm > a
m
ð22Þ condition is met. In the case of a Newtonian layer the
infinite layer approximation overestimates the actual growth
rate by a factor 1 + 2Da.
[16] Figure 3 illustrates how the fold amplification rate is
4. Analysis
affected by the aspect ratio of the layer. For a given aspect
4.1. Difference of Timescales
ratio the maximum possible growth rate occurs when Da =
[13] The folding mode table (Table 1) reveals that the 1, i.e., when me/mm = a. If me/mm < a, the growth rate is
aspect ratio has a significant influence on the development smaller due to the smaller effective viscosity contrast; if
of folds in layers of finite length. The reason is that the me/mm > a, the growth rate is reduced because the effective
isolation of a finite length layer from the far-field driving viscosity contrast exceeds the aspect ratio and therefore
forces leads to a reduction of the layer parallel compressive emphasizes the isolation of the layer. Consequently, the
stress and strain rate experienced by the layer (see aspect ratio of a layer determines the maximum dominant
equations (6) and (7)). Hence the actual driving force of wavelength (and accordingly effective viscosity contrasts)
fold growth is reduced and in the case of a power law layer that develops into a significant fold for a given bulk strain.
[17] A two-dimensional finite element model for incom(n > 1) the layer is in the large Da limit effectively stiffer
than it would be in the equivalent conventional experiment pressible Newtonian materials was used to verify the
(_me > me, see equation (11)), which slows down the devel- predictions of the analytical theory. Despite the relatively
opment of folds even more.
strong simplifications of the analytical approach the deviasffiffiffiffiffiffiffiffiffiffi
_
a 1=3 2 m n=3
_
3 me
e
l ¼ 2p
h
h ¼ 2p
6nmm
12n
a mm
_
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conventional configuration the effective viscosity is larger
and therefore the dominant wavelength longer.
Figure 3. (a) Far-field normalized growth rate versus farfield-based effective viscosity contrast and (b) normalized
dominant wavelength. The circles at the end of the lines
represent the limit where only one dominant wavelength fits
onto the finite length layer.
tions between analytical and numerical results, which represent the complete solution, are small and the predicted
significant differences to the conventional folding theories
validated. Stronger deviations between analytical and numerical results toward higher Da numbers are expected as
the layer length to wavelength ratio tends toward one and
therefore the described geometrical differences between
basic state and linear stability become considerable.
4.4. Finite Strain Experiments
[19] In order to illustrate the folding characteristics of a
finite length viscous layer up to large deformations and
without the simplification imposed by the analytical techniques, we performed numerical experiments. Three models
with identical far-field strain rates and viscosity contrasts of
100:1 are compared in Figure 4. The initial interface
perturbation of all three experiments was a sinusoid with
the dominant wavelength and initial amplitude of 1/100 of
the maximum layer thickness. The difference is the aspect
ratio, which is varied in order to change the Da number. The
Da number increases from top to bottom, including 0
(boundaries directly connected to the layer), 0.47 and
1.56. For comparison, the initial length of the Da = 0 and
the Da = 0.47 layers was set equal.
[20] At a bulk shortening of 33% in all three experiments,
the fold amplitudes show that the growth rate decreases with
increasing Da number with respect to the bulk strain rate
(see Figure 5). The folds in the Da = 0 configuration have
almost gone through the entire field of active amplification
[Schmalholz and Podladchikov, 2000] and developed significant amplitudes. Already the Da = 0.47 folds have a
considerably lower growth rate and are consequently less
pronounced. This layer ‘‘feels’’ less compression than the
equivalent Da = 0, which can also be seen from the fact that
the Da = 0.47 fold train is longer, although their initial layer
lengths were identical. A further increase in Da leads to an
even slower fold growth and therefore the Da = 1.56 folds
are at this significant bulk shortening still in the initial
stages of fold amplification.
[21] The folds in the Da > 0 experiments are less
developed toward the tips of the layers compared to the
center. This is due to the initial perturbation that was put on
4.3. Dominant Wavelength
[18] The wavelength selection is less affected by the
finiteness of a layer. In the case of a Newtonian layer the
dominant wavelength is entirely insensitive to the layer
aspect ratio. The dominant wavelength of a power law layer
depends on the aspect ratio because the latter affects the
effective viscosity in the large Da limit. Compared to a
Figure 4. Folding of finite length layers. Viscosity
contrast is always 100:1. Bulk shortening in all three
experiments is 33%.
Figure 5. Normalized growth rate versus Da for Newtonian layers with viscosity contrast 100:1. The three dots
are the growth rates measured in the corresponding
experiments (compare Figure 4). The max(Da) point
represents the maximum possible Da for the given viscosity
contrast, which allows for at least one dominant wavelength
on the layer length.
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the entire layer with a fixed wavelength to thickness ratio,
tuned to fit the center of the layer. Because of its elliptical
shape, the thickness of the layer decreases toward the tips.
Consequently, the wavelength to thickness ratio of the
perturbation near the tips is larger than the dominant value,
and hence the folds grow slower. Interestingly, the folded
quartz vein in Figure 1 shows a similar behavior.
[22] The measured growth rates of the finite element
models are all lower than the analytically predicted values.
This is expected as thin plate approximations generally
overestimate the real growth rates [Fletcher, 1977]. The
increase in the difference between real (numerical) and
predicted (analytical) growth rates with increasing Da is
again due to the use of the thin plate approximation. On one
hand, the assumption of negligible shear stresses may not be
exactly correct for the modeled viscosity contrast of 100:1,
and on the other hand, the geometrical deviations between
basic state and thin plate approximation are significant for
the small aspect ratios that were chosen here for ease of
visualization. The errors observed in Figure 3 are substantially smaller due to the larger viscosity contrast and aspect
ratio. However, it is important to note that (1) the predicted
trend of decreasing growth rates with increasing Da is
reproduced and (2) the real growth rates are even smaller
than predicted by the analytical theory.
5. Discussion
[23] We have shown that the aspect ratio of a layer has a
first-order influence on the development of folds. Because
of the isolation from the far-field driving forces, the actual
driving force of the folding instability in a finite length layer
is reduced and therefore the folding growth rate diminished.
The aspect ratio of a layer essentially limits the maximum
wavelength that may develop into significant folds, which
agrees with field observations.
[24] Natural folds show a preference for small wavelength
to thickness ratios (<10), as reported by Sherwin and
Chapple [1968]. Such small values imply that Biot’s theory
of viscous folding [Biot, 1961] is not applicable because the
measured wavelength to thickness ratios require such small
viscosity contrasts that the corresponding growth rates
would be too small to develop folds. For ductile materials,
theoretical investigations such as layer parallel shortening
[Sherwin and Chapple, 1968] and the introduction of power
law materials [Fletcher, 1974; Smith, 1975], were thus
developed to explain small wavelength to thickness ratios.
These theories successfully explain how small wavelength
to thickness ratios are possible, but not how the growth of
large wavelength to thickness ratio folds is suppressed.
Laboratory measurements show that the effective viscosity
of rocks varies many orders of magnitudes as a function of
conditions and compositions [e.g., Carter and Tsenn, 1987].
Accordingly it seems likely that effective viscosity contrasts
between layer and matrix in nature span a considerable
range. Hence, even if power law rheology is taken into
account, we would expect the frequent observation of large
(
10) wavelength to thickness ratio folds, which is not the
case (Talbot [1999] reports one value substantially larger
than 10). Therefore the growth of large wavelength to
thickness ratio folds seems suppressed, which is a characteristic feature of finite length layer folding.
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[25] Arguably, natural layers admissible for fold amplification are likely to have aspect rations <1000. For example,
veins often have aspect ratios in the range of 100– 500 [e.g.,
Vermilye and Scholz, 1995]. Even if a layer has an aspect
ratio that nominally exceeds this range it is unlikely that the
effective aspect ratio, relevant for folding, is substantially
larger. The effective aspect ratio is determined by the
longest segment of a layer along which the deviations from
the perfect plane are smaller than half of the layer thickness.
Larger amplitudes of initial perturbation move folding out
of the exponential amplification where dominant folding
frequencies are expected to develop, into the large-amplitude mode characterized by weak wavelength selectivity
and kinematic dominance of heterogeneities [Schmalholz
and Podladchikov, 2000]. For example, a 1-km-long 1-mthick layer should not deviate more than half a meter from a
perfect plane, anywhere along its 1 km length in order to
maintain an effective aspect ratio of 1000.
[26] If we assume maximum effective aspect ratios of
approximately 500, the applicability of conventional folding
theories is limited to effective viscosity ratios substantially
smaller than 500 (Da 1) (see Figure 3). The growth of folds
in layers with larger effective layer-matrix viscosity contrast
will be substantially reduced according to the folding of finite
length theory and as verified by the two-dimensional finite
element models. Therefore the lack of natural fold trains
indicating large viscosity contrasts is due to the limited
(effective) aspect ratios in nature rather than due to absence
of large effective viscosity contrasts. The reason why dominant wavelengths in the range of 10 – 100 are rare is
inconclusive.
6. Conclusions
[27] Taking the layer aspect ratio into account, we have
developed an analytical theory for the folding of a finite
length, power law layer embedded in a Newtonian matrix. It
has been shown that the aspect ratio is a key parameter
controlling the development of folds and that its importance
is governed by the relative values of aspect ratio and
effective viscosity contrast. If the aspect ratio is substantially larger than the effective viscosity contrast then the
importance of the aspect ratio is negligible and the conventional expressions, which describe an end-member case in
the folding of finite length layer theory, are applicable. For
all other cases, new expressions have been derived and our
key finding is that the fold growth rates are substantially
reduced relative to the bulk shortening rate and that the
growth of large wavelength to thickness ratio folds is
suppressed. Therefore, in order to interpret the geological
history recorded in folds, the layer aspect ratio must be
considered as a parameter of first-order importance.
[28] Acknowledgments. This research was supported by the ETH
Zurich, grant TH 0-20650-99. We wish to thank Stefan Schmalholz, JeanPierre Burg, Boris Kaus, and Hilary Paul for helpful and inspiring discussions. We would like to acknowledge the thorough reviews by Wen Jeng
Huang, Arvid M. Johnson, Ray Fletcher, and the comments by Associate
Editor Leonid Germanovitch, all of which helped to improve this manuscript.
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F. O. Marques, Departament Geologia, Facutas Ciências, Universidade de
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fc.ul.pt)
Y. Y. Podladchikov, Geologisches Institut, Sonneggstr. 5, ETH Zentrum,
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D. W. Schmid, Physics of Geological Processes, University of Oslo, Pb
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