Philosophical Magazine,
Vol. 86, Nos. 21–22, 21 July–1 August 2006, 3409–3423
Fold amplification rates and dominant wavelength
selection in multilayer stacks
D. W. SCHMID* and YU. Y. PODLADCHIKOV
Physics of Geological Processes, University of Oslo, Oslo, Norway
(Received 21 September 2005; in final form 22 September 2005)
A combination of thin- and thick-plate theories, and finite element models is used
to systematically analyze folding in multilayer stacks. We show that if the
interlayer spacing is large, individual layers fold as single layers, if the spacing is
small the entire stack folds as one effective single layer. In between, a third folding
mode exists that is characterised by a dominant wavelength that scales with n1=3 ,
irrespective of total number of layers, n. The maximum growth rates in the true
multilayer-folding mode are higher than the corresponding single layer growth
rates, increase with n and are bounded by a saturation value that is directly
proportional to the viscosity contrast. This growth rate saturation as well as the
applicability of the true multilayer-folding mode with respect to interlayer spacing
can be explained by the normal and inverse contact strain theory. The true
multilayer-folding mode is expected to be the most frequent mode in nature,
because it exhibits the highest growth rates and has a relatively large applicability
range with respect to interlayer spacing. The increased growth rates in multilayer
folding are especially important for systems where the corresponding single layer
values are not sufficient to drive the folding instability, such as folding in
low-viscosity contrast layers and detachment folding.
1. Introduction
Multilayer systems with large number of layers (>10) have regularly been modelled
analytically and numerically by introducing effective properties of the multilayer
stack [1–5] and exploiting symmetries of the expected fold pattern formation [6].
Folds in tightly stacked multilayers exhibit a large variety of shapes and tend to
comprise kinks and chevrons, angular folds with straight limbs and sharp hinges
(figure 1). Such sharp features tend to be associated with elasto-plastic material
properties and consequently systems with this rheology are thoroughly studied [7, 8].
However, the use of the sharpness of the emerging fold patterns as a rheology
indicator must be pursued carefully, since Cobbold et al. [4] pointed out: ‘‘The idea,
for example, that kink-bands develop in brittle environments only is erroneous’’.
Indeed, analogue models with fluid-like bilaminates such as different waxes and
plasticine [9] exhibit the entire range of fold shapes that occur in natural multilayers,
while previous numerical models employing simple fluids tend to produce gentle,
sinusoidal folds [10].
*Corresponding author. Email: schmid@fys.uio.no
Philosophical Magazine
ISSN 1478–6435 print/ISSN 1478–6443 online ß 2006 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/14786430500380175
3410
D. W. Schmid and Yu. Y. Podladchikov
Figure 1. Multilayer folding in an accretionary wedge, Makran, Iran (photo courtesy
S. Schmalholz). The range of layer thicknesses and strength in this turbiditic sequence
produces a broad range of fold shapes ranging from angular chevron folds to open,
gentle folds.
Our approach here is to explicitly resolve the individual members of the analyzed
multilayer stack as normal, isotropic continua, and therefore let the effective
anisotropy and the geometry of the system evolve with strain. We focus on the
systematic analysis of the most simple multilayer case: only two different materials,
competent layers and weaker matrix, embedded in two infinite matrix half-spaces
and subjected to layer parallel pure shear, figure 2. What are the dominant
wavelengths and maximum growth rates for such a system? How do they depend on
layer-interlayer spacing, material property contrast, and number of layers? In order
to understand the complexity of the emerging system we restrict the rheology of the
materials used to the simplest linear viscous (Newtonian) fluids, governed by Stokes
approximation for slowly creeping flows. The method used to study this system is
a combination of thin-plate and thick-plate analytical theories for the initial stages,
complemented by finite element models for large strains.
2. System configuration
We study a simple multilayer stack in two dimensions, figure 2, which is completely
described by the following variables: the number of layers, n, the thickness of the
3411
Fold amplification rates and dominant wavelength selection
Eff. Single Layer
???
Single Layer
s
h
HB
Base
Figure 2. System studied in this paper. If the individual layers of the n-layered multilayer
stack are far apart (h/s large) the layers do not interact and behave as single layers; if they are
very close (h/s small) they act as an effective single layer. The question addressed here is what
happens in the intermediate h/s range. Another parameter studied is the influence of a free slip
base at a distance HB from the lowermost member of the stack.
individual strong layer, h, the thickness of the interlayer spacing, s, the distance
between the lowermost layer and a free slip base, HB , the viscosity of the strong
layers, l , and the viscosity of the matrix and the interlayer material, m . Assuming
incompressibility and slow, low Reynolds number flow the system is governed by the
momentum balance
ij, j ¼ 0
ð1Þ
vi, i ¼ 0
ð2Þ
and the mass conservation
where ij is the total stress and vi the velocity vector [11]. The rheology is
ij ¼ 2"_ij pij
"_ij ¼ 12 vi, j þ vj,i
ð3Þ
ð4Þ
where p is the pressure, ij the Kronecker delta, and "_ij the strain rate. The system is
driven by box-normal pure shear velocities with constant strain rate "_BG . Hence the
lateral walls move with v1 ¼ "_BG x1 , and the top and bottom with v2 ¼ "_BG x2 .
3412
D. W. Schmid and Yu. Y. Podladchikov
The top and bottom boundaries have a free slip condition and the vertical velocities
at the lateral walls are periodic. While the top boundary is always far away from the
multilayer stack, the bottom boundary can be brought close to the stack to study
the effect of detachment folding [12] on multilayers.
3. Methods
3.1. Thin-plate analysis
Thin-plate analysis [13, 14] is the most elegant method to study single layer folding in
that it represents the entire system in one equilibrium equation. In the case of a single
viscous layer in a matrix
l h3 @5 w
@2 w
þ
h
þ qm ¼ 0
xx
@x2
3 @x4 @t
ð5Þ
where w is the deflection of the layer, t is the time, xx ¼ 4l "_BG the horizontal stress
within the layer, and qm is the resistance of the matrix to the folding. The general
solution form for the deflection is
w ¼ AðtÞ sinðkxÞ
ð6Þ
where AðtÞ is the amplitude of a sinusoidal perturbation with time, and k is the wave
number, related to the wavelength through k ¼ 2=. If we assume that the layer
is embedded in two infinite (low) viscous half spaces the matrix resistance term
becomes [14]
qm ¼ 4m k
@w
@t
ð7Þ
Solving equation (5) for AðtÞ results in
AðtÞ ¼ Að0Þ exp
12hkl "_BG
t
l h3 k3 þ 12m
ð8Þ
which shows the well known result that the initial amplification is exponential with
time, with a growth rate _
_ ¼
12hkl
"_BG
l h3 k3 þ 12m
The wavelength which growth fastest, called dominant wavelength, is
l 1=3
B ¼ 2
h
6m
and the corresponding maximum growth rate
2=3
4 l
_ B ¼ 3
"_BG
m
ð9Þ
ð10Þ
ð11Þ
These results were originally derived by Biot [14], hence the subscript B, and are used
throughout this paper as the single layer reference case. If there is more than one
Fold amplification rates and dominant wavelength selection
3413
layer the stack may be approximated as one effective single layer. Ignoring the
matrix material in between the competent layers we have to adjust the effective
thickness of the system and account for the increase in the in plane force, which
changes equation (5) to
l nh3 @5 w
@2 w
þ xx nh 2 þ qm ¼ 0
4
@x
3 @x @t
ð12Þ
Mathematically this is identical to Biot’s [14] treatment of an isolated multilayer
stack, however, he argued that the matrix resistance term is reduced by a factor n.
The resulting expressions for the dominant wavelength and maximum growth
rate are
nl 1=3
¼ 2
h
ð13Þ
6m
nl 2=3
_ ¼ 43
ð14Þ
"_BG
m
The simplifying assumptions underlying the thin-plate approximation, especially that
shear deformations are not considered, limit its applicability to multilayer systems,
where shear between and inside the individual layers is to be expected.
3.2. Thick-plate analysis
The initial stages of the development of the folding instability in a multilayer system
governed by equations (1) to (4) can be accurately described, without the
simplifications of the thin-plate theory, by the thick-plate analysis [e.g. 15].
Johnson and Fletcher [16] describe in detail the derivation of the thick-plate linear
stability analysis and how an eigenvalue problem must be solved, where the
eigenvalues represent the growth rate of the perturbed interfaces and the eigenvectors
specify the corresponding interface configuration of this fastest growing mode.
Numerically scanning through the wavelength space yields dispersion relations,
similar to equation (9), for which the maximum growth rate and the corresponding
dominant wavelength can be determined. As already pointed out in the introduction,
the thick-plate analysis allows for straightforward modification of the boundary
conditions and therefore the influence of a free slip base may be incorporated.
3.3. Finite element method
While thin- and thick-plate analyses are only valid for the initial stages of the fold
formation [16–18], the finite element method (FEM) allows for studying finite
amplitude folding, free of a priori deformation type assumptions. The code used here
is a personally developed, two-dimensional, implicit FEM code using the seven node
Crouzeix-Raviart triangle [19] to solve the Stokes equations for incompressible,
slowly creeping, viscous materials. The shape functions are continuous (bubble node
enriched), quadratic basis functions for the velocities and discontinuous linear basis
functions for pressure. The incompressibility constraint is taken care of by Uzawa
3414
D. W. Schmid and Yu. Y. Podladchikov
iterations [11]. This mixed method avoids spurious pressures usually appearing due
to the incompressibility constraint [20]. Since viscous fluids do not exhibit any
rheological memory, contrasting for example Maxwellian fluids, re-meshing may be
performed whenever required. Indeed, it is sufficient to move simply the contours
defining the geometry of the system between time steps and re-mesh continuously,
which allows for arbitrary bulk shortening. Since in all presented FEM models
the folds grow from red-noise interface perturbations and relatively long layers
are required in order to grasp some of the complexity of natural multilayer
folds, relatively large resolutions are required and a typical system comprising of
ten competent layers requires approximately one million degrees of freedom to be
solved for.
4. Multilayer stack embedded in viscous half spaces
Systematic thick-plate analysis of a multilayer stack embedded in two viscous half
spaces, figure 3, confirms the anticipation in figure 2: if the individual layers are
either far apart or very close they behave as single layers, either as true single layers
or as one effective single layer, comprising the entire stack. This holds for the
maximum growth rate as well as the dominant wavelength; for large layer spacing
the Biot wavelength serves as a good approximation, for very small spacing B is also
applicable, however, must be divided by the number of layers in order to account
for the thickness of the effective single layer (see top insert in figure 3c where =ðnb Þ
is plotted versus h=s). The reason why especially the (Biot) normalized maximum
growth rate for the l =m ¼ 25 case deviates from unity in the two single layer limits
is that the simplifications of the thin-plate analysis lead to an overestimation of the
growth rates for small viscosity contrasts [21]. The growth rates in the two single
layer limits also exhibit a slight asymmetry in that the effective single layer shows
slightly higher fold amplification rates, which is due to decreased resistance to
interlayer slip in the multilayer stack. Therefore the growth rates of the effective
single layer should be compared to the thick-plate analysis with free slip conditions
at the layer-matrix interface [21].
The most interesting result of the thick-plate folding analysis of the simple
multilayer stack is the existence of a third folding mode, exemplified by the middle
plateau in the =B plots, figures 3a and 3c, which is centred with respect to h=s ¼ 1.
Over a relatively large range of h=s values the dominant wavelength in this
true multilayer mode is near constant. The extent of the plateau increases with
increasing viscosity contrast. Towards the large layer spacing side this can be
understood in terms of Rambergs [22] contact strain theory. The folding instability
causes perturbations on top of the pure shear background component, such as
visualized in figure 4. These perturbations decay exponentially away from the layer,
and Ramberg argued that if two layers are separated by more than the dominant
wavelength they would not affect each other mechanically. B =h for the two presented
viscosity contrast cases are 10.1 and 20.8, which correspond well to the right ends of
the middle plateaus in figures 3a and 3c, respectively. Figures 3a and 3c suggest
that the contact strain theory is also applicable in an inverse sense; if layers are closer
than the inverse of the single layer dominant wavelength, h=s < 1=b , the stack
3415
Fold amplification rates and dominant wavelength selection
(a)
45
µl /µm = 25
(b)
µl /µm = 25
2
n=40
n=40
40
1.8
n=20
1.6
n=10
35
30
25
λ
n=20
λB 20
15
10
40
n=5
1.2
n=3
n=10
n=5
5 n=3
n=2
0
10−2
(c)
α 1.4
αB
n=2
1
10−1
100
h/s
µl /µm = 250
n=40
101
102
0.8
10−2
(d)
λ/(nλB)
10−1
4
100
h/s
µl /µm = 250
101
102
101
102
n=40
35
3.5
n=20
30
3
25
λ
20
λB
λ/(n1/3λB )
n=5
2
15
10
n=10
5
n=5
n=3
n=2
0
n=10
α 2.5
αB
n=20
10−2
n=3
1.5
n=2
1
10−1
100
h/s
101
102
0.5
10−2
10−1
100
h/s
Figure 3. Thick-plate theory based linear stability analysis of multilayer stacks embedded
in two viscous half spaces: dominant wavelength (a, c) and corresponding maximum
growth rates (b, d) for two different viscosity contrasts. All values are normalized by the
corresponding single layer values given by Biot [14]. The two inserts in (c) represent the same
data as plotted in (c), but normalized differently, corroborating the existence and scaling of
three different plateaus of the dominant wavelength in the h/s space.
behaves effectively as a single layer, if 1=b < h=s < b the stack is in a true multilayer
folding mode.
The importance of the true multilayer-folding mode is emphasised by the fact
that its growth is controlled by growth rates that are higher than the corresponding
single layer case, figures 3b and 3d. This reflects the thin-plate based prediction,
equation (14), and corresponding findings by a number of researchers, e.g. Ramberg
[22], Ramsay and Huber [23], and Muhlhaus et al. [2]. However, it is obvious from
figure 3 that there is a saturation of _ with respect to the number of layers n. This is
reflected in figure 5 where layer spacing is set constant, h=s ¼ 1, and the growth rate
is plotted versus the number of layers in the multilayer stack. The thin-plate based
3416
D. W. Schmid and Yu. Y. Podladchikov
Figure 4. Perturbation flow, equals total flow minus pure shear background flow, driven by
a small sinusoidal perturbation on layer with l =m ¼ 100.
h/s=1
102
n2/3
101
µl /µm = 1000
µl /µm = 250
α
αB
µl /µm = 25
100
10−1
100
101
102
103
n
Figure 5. Maximum growth rate at h=s ¼ 1 as function of the number of layers in an isolated
multilayer stack. n2=3 is derived based on the thin-plate theory, equation (14).
Fold amplification rates and dominant wavelength selection
3417
prediction of the growth rate in a multilayer stack, equation (14), over-predicts the
growth rate increase with respect to n substantially and may be used as a limiting
case for large viscosity contrasts and small n. With increasing number of competent
layers the growth rate saturates quite rapidly and for the discussed viscosity contrasts
of 25 and 250 already 9 and 38 layers, respectively, are sufficient to reach 70% of the
saturation growth rate. The saturation growth rate, _ sat , scales directly proportional
to the viscosity contrast and for the investigated range of viscosity contrasts _ sat can
be expressed with less than 10% error as
_ sat l
"_BG
m
ð15Þ
The contact strain theory (and its ‘inverse’ version) may be applied again to explain
the characteristics of the maximum growth rates. If the layer spacing fulfils the
condition 1=b < h=s < b the multilayer stack is in the true multilayer-folding mode
and there is mechanical interaction between the layers. Within this range the
individual layer growth rates contribute to each other and the maximum is reached
for h=s ¼ 1. However, the range over which a single layer can contribute to its
neighbours is a function of the dominant wavelength of the layer, which is why the
growth rate shows a saturation value and large viscosity contrast layers interact over
larger distances.
Figure 6 illustrates the findings of the thin- and thick-plate analysis with the
comparison of two FEM models where the interfaces of a single layer (a) and a
9 layer stack (b) were red-noise perturbed with a maximum amplitude of 1/50 of the
layer thickness. The viscosity contrast is 25 for which B 10 and the perturbation
grows with _ B 10. Such small growth rates are close to be insufficient to drive
proper fold development and figure 6a shows that even at 50% shortening the
amplitude to wavelength ratios are still small and most of the shortening is
accommodated kinematically by layer thickening [cf. 24]. This and the fact that most
natural folds show smaller wavelength to thickness ratios than 10, corresponding to
even smaller viscosity contrasts and growth rates, basically render Newtonian
rheology unfeasible to explain folds in rocks. Other rheologies such as power-law
[21, 25] and visco-elasticity [3, 26] show higher growth rates for small wavelength to
thickness ratios. However, if we simply cut the Newtonian layer into 9 slices and
separate them by a distance equal to the individual layer thickness the thick-plate
analysis predicts substantially larger growth rates that are already 70% of _ sat . Such
a growth rate is larger than the corresponding single layer value for a viscosity
contrast of 100. Indeed, the FEM model shows that folds in the individual slices of
the multilayer stack have at 50% shortening developed large amplitudes, figure 6c.
The fold wavelength to thickness ratio in the individual slice of the stack is larger
than in the real single layer, figure 6a; however, relative to the entire stack thickness,
the dominant wavelength is smaller than B =h. The fold morphologies in the stack
comprise a large variety of shapes, ranging from box folds, over more gentle folds
in the outer parts of the stack to chevrons in the middle. The sharpness of the hinges
and straight limbs are best seen in figure 6c, which does not resemble the fold
morphology one would expect based on the understanding of low viscosity contrast
single layer folding, figure 6a.
3418
D. W. Schmid and Yu. Y. Podladchikov
µl /µm = 25
50% Shortening
(a)
(b)
(c)
Figure 6. Finite element models of folding in an isolated single layer (a) and corresponding
multilayer stack with 9 competent layers (b). The initial interface perturbation is a red-noise
with an amplitude of 1/50 of the individual layer thickness h. (c) is the centre layer of the
multilayer experiment.
Table 1.
Multilayer folding mode table.
Effective single layer
h=s < 1=B
Few layers
n B =h
_ ¼
_ B ¼
Many layers
B =h n
True multilayer
1=B < h=s < B
4 l 2=3
"_BG
3 m
¼ 2
l
6m
1=3
hn
4 nl
3 m
2=3
"_BG
nl 1=3
¼ 2
h
6m
_ ¼
Real single layer
B < h=s
l
"_BG
m
_ B ¼
4 l 2=3
"_BG
3 m
B ¼ 2
l
6m
1=3
h
nl 1=3
h
¼ 2
6m
The question remains why the h=s ¼ 1 value is of such a special importance
irrespective of viscosity contrast. Future work on this subject will have to focus on
symmetry arguments and the transition from shear between the layers in a multilayer
stack to Poiseuille-like flow, which was previously observed in models by Schmalholz
and Podladchikov [26].
The folding in an isolated multilayer stack is summarised in table 1. The most
important finding is the existence of a true multilayer-folding mode, which is
characterized by a n1=3 dependency of the dominant wavelength and increasing
growth rates with increasing number of layers bounded by a saturation value that is
directly proportional to the viscosity contrast. Note that the growth rates for the true
multilayer case are valid for h=s ¼ 1 and must be adjusted for other h=s values
according to figure 3.
Fold amplification rates and dominant wavelength selection
3419
5. Influence of a free slip base
The previous results were obtained under the assumption that two infinite half spaces
are present below and above the multilayer stack. Multilayer folding in nature is
often influenced by the presence of a base over which detachment folding takes place;
an example is the folding in the Jura Mountains, north of the Alps. In such cases it
seems necessary that a weak (low viscous) detachment horizon is present, such as the
Triassic evaporite horizon in the case of the Jura [27]. This effective decoupling
allows for large enough growth rates that lead to finite amplitude folds. Schmalholz
et al. [12] derived an analytical theory that accounts for the presence of a base in the
vicinity of a competent layer subject to lateral compression.
We present in figure 7 a systematic thick-plate analysis of multilayer folding over
a base, where the same systems as in figure 3 are examined, but a base is closely
underlying the stack with the base to stack distance equal to 1/10th of the combined
thickness of all competent layers in the stack. If only one layer is present, n ¼ 1, the
growth rates and the dominant wavelength are greatly reduced compared to the Biot
equivalents, _ B and B , as shown by Ramberg [28] and Schmalholz et al. [12].
In terms of the maximum growth rate, figures 7b and 7d, the thick-plate analysis
shows again the existence of two single layer domains: for small h=s ratios the stack
folds in the detachment folding regime [12] with its corresponding small growth rates
and small wavelength to thickness ratios, for large h=s most of the individual layers
are beyond the reach of the influence of the base and the eigenvalue analysis
essentially gives _ B and B . In between these two single layer domains again a true
multilayer-folding mode exists, which is characterized by increased growth rates
resulting from the mechanical interaction of the individual folding instabilities. For
h=s ¼ 1 and l =m ¼ 25 already five layers are sufficient to yield growth rates that
are approximately equal to the corresponding _ B . If more layers are present even
higher growth rates result, however, as for the case of an isolated multilayer stack,
the vertical interaction length between individual layers is limited depending on the
dominant wavelength and therefore the maximum possible growth reach the
saturation value _ sat given in equation (15), cf. figure 8a. Figure 8a also illustrates
that the highest increase in _ occurs when the system goes from one to two layers,
and that the thin-plate based prediction of the growth rate increase proportional to
n2=3 is representative for small to medium number of layers in the stack; in case of
l =m ¼ 250 for 2 n 100. The dominant wavelength plots versus the interlayer
spacing, figures 7a and 7c, show only two plateaus instead of three in the case of the
isolated stack, cf. figure 3. The large h=s value plateau represents individual single
layers that do not interact with each other or the base, the small h=s value plateau,
best seen in the =ðnB Þ insert in figure 7c, represents an effective single layer that is
approximated by the theory given by Schmalholz et al. [12]. In the true multilayer
folding mode domain these two end-member cases are matched and no simple
description of the exact characteristics is given. This is exemplified in figure 8b, where
the influence of the number of layers on the dominant wavelength is studied.
It is obvious that the thin-plate based expression for the dominant wavelength in
a multilayer stack, equation (13), does not provide a good approximation, contrasting the case of an isolated multilayer stack where it represents perfect agreement.
3420
(a)
D. W. Schmid and Yu. Y. Podladchikov
µl /µm = 25
4.5
4
(b)
HB/(nh) = 0.1
n=40
1.8
µl /µm = 25
HB/(nh) = 0.1
n=40
1.6
n=20
λ
λB
3.5
1.4
3
1.2
n=10
n=5
1
α
αB 0.8
2.5
n=20
2
n=3
n=2
0.6
1.5
n=10
0.4
1
n=5
0.5 n=3
n=2
n=1
0
10−2
(c)
0.2
n=1
10−1
100
101
h/s
µl /µm = 250 HB/(nh) = 0.1
3
0
10−2
102
(d)
2.5
10−1
100
101
h/s
µl /µm = 250 HB/(nh) = 0.1
n=40
λ/(nλB)
2.5
102
2
n=20
2
n=10
1.5
n=40
λ
1.5
λB
α
αB
n=5
n=3
n=2
1
1 n=20
n=10
0.5
0.5 n=5
n=3
n=2
n=1
0
10−2
10−1
100
h/s
101
102
0
10−2
n=1
10−1
100
101
102
h/s
Figure 7. Thick-plate theory based linear stability analysis of multilayer stacks close to an
underlying base. With the exception of HB =ðnhÞ ¼ 0:1 the configuration is identical to figure 3.
The insert in (b) shows the eigenvectors, i.e. the amplitude configuration, of the fastest
growing mode for l =m ¼ 25, n ¼ 5, and h=s ¼ 1. The insert in (c) represents the data plotted
in c), but normalised differently in order to demonstrate the existence of two dominant
wavelength plateaus in the h=s space.
Finite amplitude folding over a base is illustrated with the FEM model in figure 9
at 60% shortening and the system configuration l =m ¼ 25, h=s ¼ 1,
HB =ðnhÞ ¼ 0:1, and n ¼ 5. The amplitude configuration of the fastest growing
mode for this system is predicted by the eigenvector plot insert in figure 7b, which
shows that there should be a rapid increase in amplitude with distance to the base,
which is confirmed by the FEM model. The small growth rate for this system,
approximately equal to an isolated l =m ¼ 25 single layer system, requires large
strains to develop significant amplitudes, the folds also appear more isolated and less
angular than in the equivalent case of a multilayer stack embedded in viscous half
spaces, figure 6.
3421
Fold amplification rates and dominant wavelength selection
(a)
(b)
h/s=1 HB /(nh)=0.1
101
102
µl /µm = 250
n2/3
µl /µm = 25
100
µl /µm = 250
101
α
αB
λ
λB
10−1
10−2
100
h/s=1 HB /(nh)=0.1
µl /µm = 25
n1/3
100
101
102
103
10−1
100
n
101
102
103
n
Figure 8. Influence of the number of layers n at constant h=s ¼ 1 and HB =ðnhÞ ¼ 0:1 on the
maximum growth rate (a) and the dominant wavelength (b). n2=3 and n1=3 are the
corresponding values derived with the thin-plate theory, equations (13) and (14), respectively.
µl /µm = 25 60% Shortening
Base
Figure 9. Finite element model illustrating the influence of a base on the development of the
folding instability. The model configuration corresponds to the insert in figure 7b, the initial
interface perturbation is red-noise with amplitude h=50.
6. Conclusions
Using a combination of thin- and thick-plate analysis and finite element models, we
have shown that folding in isolated multilayer systems is characterized by three
different modes: single layer, effective single layer, and the true multilayer-folding
mode. These modes are summarized in the multilayer-folding mode table (table 1).
The true multilayer-folding mode is expected to be the most prominent folding mode
in nature because it occurs over a wide range of layer spacing and exhibits faster
growth rates than the corresponding single layer cases. The limits of the validity of
the true multilayer-folding mode in terms of the interlayer spacing are given by
1=B < h=s < B , which is equivalent to the contact strain theory by Ramberg [22]
3422
D. W. Schmid and Yu. Y. Podladchikov
and its inverse. The contact strain theory also limits the vertical interaction of
individual layers with respect to the maximum growth rate, which leads to
saturation, described by _ sat ="_BG ¼ l =m . The dominant wavelength in the true
multilayer-folding mode scales with n1=3 and therefore does not saturate with respect
to n. Hence, folding in the true multilayer mode always feels the total number of
layers in the stack in terms of the dominant wavelength, but only a fraction of n with
respect to the maximum growth rate. The ratio of dominant wavelength to stack
thickness, =ðnhÞ, in isolated multilayer stacks is always smaller than the
corresponding single layer Biot equivalent, B =h. However, relative to the thickness
of the individual layer in the stack this ratio is larger, i.e., > B .
FEM models confirm that the stacking of layers increases the growth rate and
even small viscosity contrast multilayer stacks with a small number of layers develop
significant folds within 50% shortening. The fold geometries in such models
comprise box and chevron folds, which are typical features of elasto-plastic
rheologies. Finally, increased growth rates in multilayer stacks are especially
important in cases where the corresponding single layer growth rates are low, such as
low viscosity contrast folding or detachment folding. In the latter case a few layers
can be sufficient to compensate for the growth rate decreasing effect of a base.
Acknowledgments
We would like to thank Stefan Schmalholz for many stimulating discussions on the
subject of multilayer folding over many years. Part of this research was funded by
a grant of the ETH Zurich: TH 0-20650-99.
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