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Philosophical Magazine,Vol. 86, Nos. 21–22, 21 July–1 August 2006, 3393–3407

Finite amplitude folding of single layers:elastica, bifurcation and structural softening

S. M. SCHMALHOLZ*y

Physics of Geological Processes, University of Oslo,PO Box 1048 Blindern, 0316 Oslo, Norway

(Received 28 October 2004; in final form 25 April 2005)

The amplification of viscous single-layer folds, from infinitesimal amplitudes upto finite amplitudes and large strains, is investigated analytically. Analyticalsolutions for finite amplitude folding of viscous layers valid for large viscositycontrasts and for post-buckling of elastic columns are shown to be identical. Thefailure of the classical, exponential amplification solution for folding is quantifiedusing a nonlinear amplification equation similar to the Landau equation. Theevolution of fold amplitude–strain for single layers with different initialamplitudes and viscosity contrasts essentially depends on a single parameterrather than three parameters as commonly assumed (strain, initial amplitudeand viscosity contrast). This single parameter is constructed by scaling thestrain with the crossover strain, which is the specific value of strain at whichthe linear solutions fail. Scaling the strain with the crossover strain yields acollapse of all amplitude evolution paths for different initial amplitudes andviscosity contrasts onto a single amplification path. Analytical solutions for theevolution of the layer-parallel deviatoric stress within the layer during foldingare presented showing a decrease of the layer-parallel deviatoric stresswith increasing fold amplitude. All stress–amplitude evolution paths fordifferent initial amplitudes and viscosity contrasts can be collapsed onto asingle stress–amplitude evolution path, if the amplitude is scaled by thecrossover amplitude. The decrease in stress is proportional to a decrease ineffective viscosity of the layer during folding. This decrease in effective viscosityrepresents structural softening, because the true, Newtonian viscosity of the layerremains constant.

1. Introduction

Single-layer folds are an impressive and common manifestation of rock deformation(figure 1). Most single-layer folds are considered the result of a mechanical instabil-ity. This instability is generated if a flat, competent layer is compressed parallel toits interfaces. Usually, the layer is embedded in a mechanically weaker material(referred to here as matrix). The development of folds due to such a mechanicalinstability is termed folding in this study. Folding of single layers has been studiedextensively in the past decades and folding studies have been performed using

*Email: [email protected] at: Geological Insitute, ETH Zentrum, CH-8092 Zurich.

Philosophical Magazine

ISSN 1478–6435 print/ISSN 1478–6443 online # 2006 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/14786430500197785

analytical [1–6], numerical [7–10] and experimental methods [ 4, 11–14]. Analyticalstudies generally provide the best insight into the mechanics of folding, because theyyield a mathematical relationship between the involved physical parameters. Anexample of such a relationship is the dominant wavelength formula for Newtonianlayers, which predicts the fastest growing fold wavelength as a function of theviscosity contrasts between layer and matrix [1]. However, analytical solutions areonly derivable for setups—which are significantly simplified compared to the naturalfolding situation—by considering for example homogeneous materials or two-dimensional pure shear shortening only. In addition, the equations describingsingle-layer folding are in principle nonlinear and are usually linearized to makethe derivation of analytical solutions feasible. Hence, most analytical results arestrictly valid only for the initial stages of folding where fold amplitudes areinfinitesimal [5, 15]. In other words, most analytical solutions are valid only forfold shapes where fold amplitudes are so small that the folds are hardly observable.Therefore, the initial stages of the fold amplification are understood much betterthan the entire folding process involving finite amplitudes as observed in nature(figure 1). This is true despite the fact that single-layer folding can nowadaysbe easily simulated numerically, because the numerical simulations only providespecific numbers for specific input parameters, but do not reveal the underlyingmechanics of the folding process.

Figure 1. Two natural single-layer fold trains. Top picture is taken in the area aroundVale Figueiras, SW Portugal, and bottom picture is taken at Geoparken Oslo, Norway.

3394 S. M. Schmalholz

Despite simplifying assumptions, analytical solutions describe the essential and

fundamental mechanical processes of folding and are, furthermore, able to provide

appropriate dimensionless numbers characterizing the folding process [16]. In this

study, single-layer folding of a Newtonian viscous layer embedded within a viscous

matrix under pure shear shortening is studied analytically in two dimensions. Layer

and matrix are treated as isotropic materials and effects of gravity are not consid-

ered. This study focuses solely on analytical solutions describing the nonlinear stages

of single-layer folding, particularly, the finite amplification of viscous layers and thestress evolution within the layer. Furthermore, the analysis focuses on the amplifica-

tion of folds, which exhibit the dominant wavelength and the wavelength selection

process [1, 2] is not considered. The main goals of this study are (i) to establish a

mathematical link between analytical solutions for elastic post-buckling and viscous

finite amplitude folding, (ii) to introduce suitable scaling parameters that eliminate

the dependence of the amplitude–strain and stress–amplitude evolution on the two

parameters initial amplitude and viscosity contrast and (iii) to present an analytical

equation describing the decrease in layer-parallel deviatoric stress during finite

amplitude folding.

2. Elastica and finite amplitude solution

2.1. Elastica

The nonlinear equation describing the deflection of an elastic column under

compression is [17, 18]

@2�

@s2¼ �

F

EIsin � ð1Þ

where �, s, F, E and I are the deflection angle, the incremental arc length, the

compressive load, Young’s modulus and the moment of inertia, respectively(figure 2b). A solution for equation (1) can be derived, which relates the slope of

the deflected column to the arc length coordinate, and is known as the Elastica [19].

Mathematical approximations of the Elastica make it possible to derive an analytical

relationship between the horizontal column displacement, u, and the column

amplitude, A, and is given by (see [17], their pages 39 to 41 and our figure 2b)

u ¼ A2 p2

4ð�0=2Þð2Þ

where �0/2 is the initial length of the column (note; dividing �0 by 2 is done in order touse the same parameter �0 for both the Elastica and the following folding solutions).

Substituting u¼ �0/2� �/2 (with �/2 being the chord of the deflected column) in

equation (2) yields the relationship

A

�¼

1

p�0�

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�

�

�0

sð3Þ

Finite amplitude folding of single layers 3395

The above equation describes the finite amplification, quantified by the parameter

A/�, of an elastic column with increasing strain, or shortening, in nondimensional

form, where strain is quantified by the parameter �0/� which is also often termed

stretch.

2.2. Finite amplitude solution (FAS)

A Newtonian layer is embedded in a Newtonian matrix with smaller viscosity

(figure 2a). Pure shear deformation with the shortening direction parallel to the

initially flat layer interfaces is applied. Linear stability analysis assumes that

geometrical perturbations superposed on the flat layer grow exponentially with

time, t [1, 2]. The evolution equation for the amplitude of the dominant wavelength

can be written as

@A

@t¼ ð1þ �ÞeA ð4Þ

L0~λ0

A0<<λ0

L

A

λ

θA

λ/2

(a) Folding of a single viscous layerHydrodynamic stability problem

(b) Buckling of an elastic columnStability of structures problem(Elastica)

λ0/2

ds

Initial situation

Current situation, pure shear deformation

Layer

Matrix (viscous)

Figure 2. (a) Setup for viscous single-layer folding under pure shear. L0, L, A0, A, �0and � are initial arc length, current arc length, initial amplitude, current amplitude, initialwavelength and current wavelength, respectively. The amplifying fold wavelength correspondsto the dominant wavelength. (b) Setup for post-buckling analysis of a compressed elasticcolumn. � and ds are the deflection angle and the incremental arc length, respectively.


where e and � are the absolute value of the background shortening strain rate andthe maximal value of the dimensionless growth rate, respectively. For Newtonianlayers embedded in Newtonian matrix the maximal value of the growth rate, derivedby the thin-plate method, is [1]

� ¼4

3

�L

�M

� �2=3

ð5Þ

where �L and �M are the Newtonian viscosities of the layer and the matrix,respectively. The factor (1þ �) in equation (4) is the sum of the contributionsfrom kinematic pure shear shortening (i.e. 1) and the result of the linear stabilityanalysis for folding (i.e. �). Analytical solutions for the growth rate, based onthin-plate and perturbation methods, have been derived for Newtonian [1],non-Newtonian (power-law; [2, 20] elastic [1] and viscoelastic rheologies [21]. Forthe treatment of the nonlinear analysis it is convenient to transform time derivativesinto strain derivatives and to use amplitudes, which are scaled by their correspondingwavelength. This yields

@AA

@"¼ ð2þ �ÞAA, using

@

@t¼

@

@"

@"

@t¼

@

@"e ð6Þ

with " being the logarithmic strain, log(�0/�), and AA ¼ A=�. The factor (1þ �) is

replaced by (2þ �) because the evolution equation (6) is now written for AA insteadof A and the wavelength shortens kinematically while the amplitude increaseskinematically due to the background pure shear. Schmalholz and Podladchikov [5]derived a nonlinear finite amplitude equation

@AA

@"¼ ð2þ �ÞAA

1

1þNð7Þ

where the nonlinear factor N is

N ¼2p2�AA2

ð3p2 þ 9ÞAA4 þ ðp2 þ 6ÞAA2 þ 1ð8Þ

The nonlinear term N is derived assuming that the average strain rate within thefolding layer is not the constant pure shear shortening strain rate, e, but is a functionof the fold arc length, L (see section ‘‘Structural softening’’ and figure 2a). An explicitanalytical solution for equation (7) could not be found, but an implicit solution canbe written as

�0�

¼L

�

�0L0

� ��=ð2þ�ÞAA

AA0

!1=ð2þ�Þ

ð9Þ

where L0 is the initial fold arc length. L can be approximated by

L ¼ �ð1þ aAA2Þ, a ¼

p2

1þ 3AA2ð10Þ


The ratio of arc length to wavelength, L/� only depends on the ratio of amplitude towavelength, AA. Substituting equation (10) into equation (9) provides an equationthat describes the fold amplification (i.e. A/�) with increasing strain (i.e. �0/�) depen-dent on initial conditions (i.e. A0/�0) and material properties (i.e. � or alternatively�L/�M). A Taylor expansion of equation (9) assuming that the growth rate goesto infinity provides an approximate solution valid for large viscositycontrasts (see equation (5)). The Taylor expansion is performed by introducing theparameter �i¼ 1/� into equation (9) and expanding equation (9) around �i¼ 0.If the factor a in equation (10) is set to p2 (which provides an accurate descriptionof the fold arc length up to a limb dip of about 45�) and if AA ¼ A=�0 is usedin equation (10), then the approximation can be written as

A

�¼

1

p�0�

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�

�

�0

sð11Þ

which is identical to equation (3). Therefore, folding of an embedded Newtonianlayer with large viscosity contrast is described by an identical amplitude–strainrelationship than post-buckling of an elastic column.

The linear solution predicting exponential growth (equation (6)) breaks downat specific values for the amplitude and strain (figure 3a). At these values theamplification changes from an exponential growth to a layer length controlledgrowth [5] and the amplification rate (i.e. incremental increase in amplitude perincremental increase in strain) is more or less identical to the amplification rateof the Elastica (figure 3a). The amplification for different values of �, which corre-sponds to different values of the viscosity contrast, is different during the exponentialgrowth, but similar during the layer length controlled growth (figure 3b). The ampli-fication predicted by the FAS is verified by finite element simulations for single-layerfolding solving the two-dimensional Stokes equations for incompressible fluids [22](figure 3c) and has been verified with numerical solutions based on a spectral-finitedifference method [5]. Folding with growth rates larger than 10 corresponds toviscosity contrasts larger than around 25, and all amplification curves for foldingwith growth rates larger than 10 occur within a relatively narrow zone within thespace log(�0/�)� log(A/�) (figure 3d).

3. Bifurcation and scaling

A Taylor expansion of the FAS equation (7) using equation (8) for smallamplitudes and truncating terms higher than order three in the amplitude yields

@AA

@"¼ aAA� abAA3, with a ¼ 2þ � and b ¼ 2p2�: ð12Þ

This equation has the form of a Landau equation [23] or the equation describinga pitchfork bifurcation [24]. Dropping the cubic term from equation (12)yields the classical solution of the linear stability analysis, which predicts an


exponential growth. In the case of a folding instability, the growth rate � is always

larger than zero. Consequently, the factor a times b in front of the cubic term

in equation (12) is always larger than zero and the cubic term is substracted from

the linear term. This indicates a supercritical bifurcation where the amplification

is moderated, or damped, with increasing amplitude. An analytical solution for

equation (12) can be written as

AA ¼AA0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

bAA20 þ ð1� bAA2

0Þ expð�2a"Þq : ð13Þ

0 0.2 0.4 0.6 0.8 1 1.2−8

−6

−4

−2

0

2

log(λ0 / λ)

log(

A /

λ)

(a)

EXP 100FAS 100Elastica

0 0.2 0.4 0.6 0.8 1 1.2−8

−6

−4

−2

0

2

log( λ0 / λ )

log(

A /

λ)

(c)

FAS 100FEM 100

0 0.2 0.4 0.6 0.8 1 1.2−8

−6

−4

−2

0

2

log(λ0 / λ)

log(

A /

λ)

(b)

KinematicFAS 25FAS 50FAS 100FAS 500Elastica

0 0.2 0.4 0.6 0.8 1 1.2−8

−6

−4

−2

0

2

log( λ0 / λ )

log(

A /

λ)

No folding solutions

Growth rates larger than 10

Growth rates smaller than 10

Diagram includes viscous and elastic rheologies

(d)

Figure 3. Finite amplification of singe-layer folds. The parameter log (A/�) is plotted versuslog(�0/�). The ratio A/� represents fold amplification and the ratio �0/� represents strain.(a) The finite amplitude solution (labelled FAS 100) deviates from the exponential solutionderived by the linear theory (labelled EXP 100) at a certain value of A/� and �0/�. Viscositycontrast is 100. For larger amplitudes and strains the fold amplification rate is close to theamplification rate of a compressed elastic column (labelled Elastica). (b) Amplification fordifferent values of the viscosity contrast (numbers in legend). The amplification during expo-nential growth differs for different viscosity contrasts, but is similar for larger strains andamplitudes in the regime of layer length controlled growth. The line labelled ‘‘Kinematic’’shows kinematic amplification for a viscosity contrast of one where the amplification is dueto pure shear flow only. (c) The FAS is verified by a numerical finite element simulation(FEM) for a viscosity contrast of 100. (d) The amplification for growth rates larger than 10(i.e. viscosity contrasts larger than 25) occurs within a relatively narrow zone in the spacelog(A/�)� log(�0/�).


Equation (13) provides a better approximation of the FAS than the exponential

solution (figure 4). Equation (13) further predicts that AA asymptotically approaches

a certain value, and for values of AA larger than the asymptotic limit, equation (13)

also fails to correctly predict the fold amplification (figure 4). The asymptotic limit of

equation (13) for AA can be used as an approximation of the amplitude at which both

the exponential solution and the solution of equation (13) break down and fail to

correctly predict the FAS (figure 4). The asymptotic limit for large strain can be

derived by either setting the left-hand side of equation (12) to zero and solving for AA,

or simply by setting the term multiplied by the exponent in equation (13) to zero,

which both yields

AA ¼ �1

pffiffiffiffiffiffi2�

p ¼ AC ð14Þ

The plus or minus sign indicates the symmetry of the amplification and the sign is

controlled by the sign of the initial amplitude. The asymptotic limit in equation (14)

is termed the crossover amplitude, AC, and is chosen as the value of the amplitude

at which the exponential solution and the linear theory breaks down (figure 4).

Note, that AC can also be derived from a Taylor expansion of the nonlinear term

N (equation (8)) for small amplitudes and solving for the value of AA, which

provides N¼ 1. If N is of order one then the nonlinear contribution cannot be

neglected and the linear solution is invalid [5]. The fact that AC depends on 1/�indicates that increasing viscosity contrasts cause smaller values of AC. In other

words, the larger the viscosity contrast and the stronger the folding instability,

the smaller the amplitude range for which the linear theory, predicting exponential

Crossover

amplitude

Crossover

strain

Figure 4. Definition of crossover amplitude and crossover strain. Results are calculated fora viscosity contrast of 100. The finite amplitude solution (FAS) deviates from the exponential(EXP) solution at certain values of A/� and �0/�. For small amplitudes the FAS can beapproximated by an equation of Landau (LAU) type. The asymptotic limit of the LAU isdefined as the crossover amplitude, and the strain between any initial amplitude and thecrossover amplitude, calculated by the exponential solution, is defined as crossover strain.


growth, is valid. If the initial ratio of A0/�0 is smaller than AC, then the linear theory

and the exponential solution can be applied. The strain accommodated during fold-

ing from an initial value of A0/�0 up to AC is according to the exponential solution

given by

"C ¼1

2þ �log

AC

AA0

!¼

1

2þ �log

1

pffiffiffiffiffiffi2�

pAA0

!ð15Þ

where "C is termed the crossover strain (figure 4). The value of "C is a maximum

estimate for the strain that is accommodated during the linear stages of folding

exhibiting an exponential amplification.The amplification predicted by the FAS depends on three parameters, which are

strain, the ratio of initial amplitude to wavelength (A0/�0) and the viscosity contrast

(�L/�M, due to the dependence on �). In figures 5a and 5c the evolution of A/�is plotted for different values of A0/�0 and �L/�M versus �0/�, which represents

1 1.2 1.4 1.6−0.4

−0.2

0

0.2

0.4

λ0 / λ

A /

λ

(a) Before data collapse

0.8 1 1.2 1.4 1.6−0.4

−0.2

0

0.2

0.4

( λ0 / λ ) / εC

A /

λ

(b) After data collapse

A0/λ0=1e−6

A0/λ0=1e−4

A0/λ0=1e−2

A0/λ0=1e−6

A0/λ0=1e−4

A0/λ0=1e−2

1 1.2 1.4 1.6−0.4

−0.2

0

0.2

0.4

λ0 / λ

A /

λ

(c) Before data collapse

0.8 1 1.2 1.4 1.6−0.4

−0.2

0

0.2

0.4

( λ0 / λ ) / εC

A /

λ

(d) After data collapse

µL/µM= 50

µL/µM=100

µL/µM=500

µL/µM= 50

µL/µM=100

µL/µM=500

A0/λ0=1e−4A0/λ0=1e−4

µL/µM = 100 µL/µM = 100

Figure 5. Data collapse for the amplification with different initial amplitudes and viscositycontrasts. (a) The ratio A/� for a viscosity contrast of 100 is plotted versus �0/� for threedifferent ratios of A0/�0. (b) The same amplification as in figure (a) is now plotted versus theratio (�0/�)/eC. eC is the crossover strain defined in equation (15). Scaling the ratio �0/� by eCprovides a data collapse and all A/� versus (�0/�)/eC curves fall together onto a single line.(c) The ratio A/� is plotted versus �0/� for three different viscosity contrasts. (d) The sameamplification as in figure (c) is plotted versus the ratio (�0/�)/eC. Similar to figure (b), scalingby eC provides a data collapse and all A/� versus (�0/�)/eC curves fall onto nearly the samecurve.


progressive strain. The crossover strain, defined in equation (15), also depends on

both A0/�0 and �L/�M (due to its dependence on �). In figures 5b and 5d the same

A/� versus �0/� (corresponding to amplification–strain) evolutions as in figure 5a

and 5c are plotted, but the parameter �0/� is divided by the corresponding value

of the crossover strain, "C. This scaling of the parameter �0/� with "C collapses

the amplification–strain curves for different values of A0/�0 and �L/�M onto

essentially a single amplification–strain curve. The crossover strain is the appro-

priate parameter to scale the parameter �0/� in the FAS to eliminate the

dependence of the amplification–strain evolution on the parameters A0/�0 and

�L/�M.The observed success of scaling the parameter �0/� by "C makes it possible to

construct a single, universal amplification–strain curve, which is valid for

all reasonable values of A0/�0 and �L/�M. The presumably typical values of

A0/�0¼ 3e�3 and �L/�M¼ 75 are chosen to construct the universal

amplification–strain curve (figure 6a). The value A0/�0¼ 3e�3 corresponds to a

limb dip of 1 degree. This universal amplification–strain curve is generated by sub-

stituting the values of A0/�0¼ 3e�3 and �L/�M¼ 75 into equations (9) and (10) and

dividing the parameter �0/� with the value of "C, which is calculated from equations

(5) and (15) for the values of A0/�0¼ 3e�3 and �L/�M¼ 75. All other amplification–

strain curves for different values of A0/�0 and �L/�M can now be constructed by

multiplying the parameter (�0/�)/"C of the universal curve with the value of "Ccorresponding to the desired values of A0/�0 and �L/�M. For example, in figure

6b the amplification–strain curve for A0/�0¼ 1e�3 and �L/�M¼ 50 is plotted

using the original equations (9) and (10), and additionally by multiplying the

1 1.1 1.2 1.3 1.4 1.5 1.6 17. 1.80

0.05

0.1

0.15

0.2

0.25

λ0 / λ

A /

λ

(b) Verification of scaling relation

A0/λ0 = 1e−3

µL/µM = 50

0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

0.05

0.1

0.15

0.2

0.25

0.3

(λ0 / λ) / eC

A /

λ

(a) Universal amplification curve

A0/λ0 = 3e−3

µL/µM = 75

true FASscaled FAS

Figure 6. (a) The universal amplification–strain curve is constructed using the values for A0/�0 and �L/�M displayed in the graph. The strain (i.e. �0/�) is scaled by eC, which is calculatedfor the displayed values of A0/�0 and �L/�M from equation (15). (b) The universal curveshown in (a) makes it possible to construct all amplification–strain curves for different valuesof A0/�0 and �L/�M by simply multiplying the horizontal coordinate of the universal curvewith the crossover strain, which corresponds to the desired values of A0/�0 and �L/�M. Here,the values are A0/�0¼ 1e�3 and �L/�M¼ 50. The curve labelled ‘true FAS’ is the true solutionfor finite amplitude folding governed by equations (9) and (10), whereas the curve labelled‘scaled FAS’ is constructed by multiplying the horizontal coordinate of the universal curve infigure 6a with the value of eC for A0/�0¼ 1e�3 and �L/�M¼ 50.


parameter (�0/�)/"C of the universal amplification–strain curve from figure 6a withthe value of "C corresponding to values of A0/�0¼ 1e�3 and �L/�M¼ 50.

4. Structural softening

For pure shear, the background shortening strain rate, e, is given by [3]

e ¼ �1

�

@�

@t: ð16Þ

The deviatoric layer stress in the direction of shortening, �, is then �¼ 2�Le.However, the FAS assumes that the layer parallel strain rate during folding, eL, isgiven by

eL ¼ �1

L

@L

@tð17Þ

and, consequently, the deviatoric layer-parallel stress �L, is given by �L¼ 2�LeL.The viscosity of the layer is then �L¼ �L/(2eL). However, if one focuses on thepure shear deformation and ignores the internal folding within the deformeddomain, then the effective viscosity for pure shear at the model boundary at thelocation of the layer is given by �eff¼ �L/(2e) [25]. Using equation (10) for the arclength and substituting it into equation (17), using equations (7) and (8) andthe relation @�=@" ¼ ��, and replacing time derivatives with strain derivatives(see equation (6)) provides after some algebraic manipulation

eLe¼

�L�¼

3ð3þ p2ÞAA4þ ð6� 3p2ÞAA2

þ 1

3ð3þ p2ÞAA4 þ ð2�p4 þ 6þ p2ÞAA2 þ 1

!� 1� ð4þ 2�Þp2AA2: ð18Þ

The right term in equation (18) is a Taylor expansion of the correct, middle term

around AA ¼ 0. Equation (18) can be slightly modified to provide an equation

which differs less than 10% from equation (18) for values of AA less than 0.05

eLe�

3ð3þ p2ÞAA4þ ð6þ p2ÞAA2

þ 1

3ð3þ p2ÞAA4 þ ð2�p2 þ 6þ p2ÞAA2 þ 1

!� 1� 2�p2AA2: ð19Þ

The only reason for considering this modified expression is that its Taylor expan-sion is simpler and more compact than the Taylor expansion in equation (18). TheTaylor expansion of equation (19) around AA ¼ 0 (right term of above equation)shows that the ratio eL/e is always decreasing for values of � and AA larger thanzero. The ratio eL/e is identical to the stress ratio �L/�. Furthermore, the ratio eL/eis equal to the ratio �L/(2�Le) and hence to the ratio �eff/�L.

Just as for the amplification, A/�, the evolution of the parameter �L/� dependson the parameters A0/�0 and �L/�M (due to dependence on �). To plot �L/� versus�0/�, the values of �0/� are calculated from equation (9) for given values of AA,which are then substituted into equation (18) to provide the ratio �L/� (figure 7).The decrease of the ratio �L/� is more similar for different values of �L/�M,


if �L/� is plotted versus A/� rather than versus �0/� (figure 7a and 7c). An essentiallysingle line describing the evolution of �L/� with progressive values of A/� fordifferent values of �L/�M is obtained, if A/� is scaled by AC (figure 7d). Scaling�0/� by eC shows that the maximum rate of decrease of �L/� occurs at (�0/�)/eC¼ 1(figure 7b).

5. Discussion

The standard method to investigate folding instabilities analytically is the pertur-bation, or thick-plate, method [3]. This method starts from the complete set ofthe equations of motion and analyses the motion of an infinitesimal, geometricalperturbation, which is superposed on a basic, stable flow field. A first-orderanalysis using the perturbation method provides an exponential growth of the

1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

λ0 / λ

τ L /

τ

(a) Before data collapse

0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

( λ0 / λ ) / εC

τ L /

τ

(b) After data collapse

0.001 0.01 0.1 0.50

0.2

0.4

0.6

0.8

1

A / λ

τ L /

τ

(c) Before data collapse

0.1 0.5 1 50

0.2

0.4

0.6

0.8

1

( A / λ ) / AC

τ L /

τ

(d) After data collapse

µL/µM= 50

µL/µM= 100

µL/µM= 500

µL/µM= 50

µL/µM= 100

µL/µM= 500

µL/µM= 50

µL/µM= 100

µL/µM= 500

µL/µM= 50

µL/µM= 100

µL/µM= 500

FEM 100

Figure 7. Structural softening. �L¼ 2�LeL and �¼ 2�Le, where �L, e and eL are the viscosityof the layer, the background pure shear shortening strain rate and the strain rate within thefolding layer in a direction parallel to the layer interfaces, respectively. (a) The ratio �L/� isplotted versus �0/� for three different viscosity contrasts. (b) The same plot as in figure 7a, butwith �0/� scaled by the crossover strain, eC. The maximal stress drop (i.e. largest slope of thecurve �L/� versus (�0/�)/eC occurs for (�0/�)/eC¼ 1. (c) The ratio �L/� is plotted versus A/� forthree different viscosity contrasts. (d) The same plot as in figure 7c, but with A/� scaled by thecrossover amplitude, AC. Scaling of A/� with AC provides a data collapse and all �L/� versus(A/�)/AC curves collapse more or less onto the same line. The crosses are results of a numericalfinite element simulation for a viscosity contrast of 100.


fold amplitude. A third-order perturbation analysis showed that the fold amplifica-tion slows with increasing fold amplitude [3]. However, as Johnson and Fletcher [3]mentioned, the third-order analysis is only applicable up to maximum limb slopes ofless than 30� and the algebra for third- and higher-order analysis becomes so com-plex that the perturbation method essentially becomes infeasible. This shows thata feasible analytical solution valid for finite amplitudes cannot be derived bya straightforward extension of the first-order theories to higher-order theories.Therefore, the FAS captures the essential mechanics of the folding process, whichis a progressive decrease of the layer-parallel strain rate compared to the backgroundshortening strain rate during folding (generating structural softening). This essentialprocess is directly implemented into the amplitude evolution equation derived by thefirst-order perturbation analysis, circumventing an infeasible, nonlinear treatment ofthe equations of motion. The FAS was specifically designed to provide a simple,analytical prediction of the fold amplitude growth with progressive strain. The FASis not suitable to predict correctly the fold shape or the internal shear deformationof the folding layer for large strains.

In this study, the analytical result of the thin-plate theory for the maximumgrowth rate (equation (5)) has been applied, because of its compactness andsimplicity. For a direct comparison of the FAS with numerical results, the valueof the maximum growth rate should be calculated using the perturbation method.

This study considers viscous, Newtonian layers embedded in viscous material.Growth rates and dominant wavelengths for more complex rheologies, such asnon-Newtonian (power-law) or viscoelastic rheologies, have been derived by linearstability analysis [2, 20, 26]. Implementing the maximal growth rates correspondingto the dominant wavelengths valid for these more complex rheologies into the FASequation (7) will provide a good approximation for the finite amplitude evolution,if power-law and viscoelastic effects do not modify the layer-parallel stresssignificantly (e.g., an elastic loading phase for viscoelastic materials active over asignificant range during the amplification).

The decrease in deviatoric stress during single-layer folding also has implicationsfor the state of stress within the lithosphere during shortening. Folding of thecrust-mantle boundary during lithospheric shortening with a constant rate canreduce the stress and consequently the effective viscosity of the lithosphere by upto one order of magnitude [25].

6. Conclusions

This study establishes a mathematical link between single-layer folding of viscouslayers described by the theory of hydrodynamic stability and post-buckling ofan elastic column described by the theory of stability of structures. An embeddedviscous single-layer (for large viscosity contrasts) amplifies identical to an elasticcolumn with progressive shortening strain.

The failure of the linear folding theory and the corresponding exponentialsolution can be described mathematically by a nonlinear amplitude equation similarto a Landau equation or the equation describing a pitchfork bifurcation. The asymp-totic limit for large strain of this nonlinear amplitude equation quantifies the


amplitude, termed crossover amplitude, at which the linear theory becomes invalid.The crossover amplitude corresponds to a crossover strain, which is accumulatedbetween any initial amplitude and the crossover amplitude. Layers exhibitinginitial amplitudes larger than the crossover amplitude never grow exponentiallyand linear theory is not applicable. Scaling by the crossover strain collapsesall amplitude–strain curves for different initial amplitudes and viscosity contrastonto a single amplitude–strain curve (figure 6). Similarly, scaling by thecrossover amplitude collapses all stress–amplitude curves for different initialamplitudes and viscosity contrast onto a single stress–amplitude curve. Scalingby the crossover amplitude and the crossover strain eliminates the dependence ofthe finite amplitude solution on the two parameters initial amplitude and viscositycontrast.

During the development of a singe-layer fold under shortening with constantrate, the averaged deviatoric layer-parallel stress within the layer decreases.The stress decrease represents a decrease in the effective viscosity of the layer, definedby the ratio of stress to twice pure shear strain rate, and represents structuralsoftening, because the true Newtonian viscosity of the layer remains constant.

Acknowledgements

I thank Yuri Podladchikov for the inspiring discussions and the revelation of variousmathematical tricks, and James Connolly for improving the English of themanuscript. Reviews by L. Moresi and an anonymous reviewer are gratefullyacknowledged. This study was supported by a centre of excellence grant to PGPfrom the Norwegian Research Council.

References

[1] M.A. Biot, Geol. Soc. America Bull. 72 1595 (1961).[2] R.C. Fletcher, Am. Jour. Sci. 274 1029 (1974).[3] A.M. Johnson and R.C. Fletcher, Folding of Viscous Layers (Columbia University Press,

New York, 1994).[4] H. Ramberg, Bull Am. Ass. Petr. Geol. 47 484 (1963).[5] S.M. Schmalholz and Y.Y. Podladchikov, Earth Planet. Sci. Lett. 181 617 (2000).[6] R.B. Smith, Bull. Seismol. Soc. Am. 86 1601 (1975).[7] J.H. Dieterich and N.L. Carter, Am. J. Sci. 267 129 (1969).[8] N.S. Mancktelow, J. Struct. Geol. 21 161 (1999).[9] S.M. Schmalholz, Y.Y. Podladchikov and D.W. Schmid, Geophy. J. Int. 145 199 (2001).[10] Y. Zhang, B.E. Hobbs, A. Ord and H.B. Muhlhaus, J. Struc. Geol. 18 643 (1996).[11] M.R. Abbassi and N.S. Mancktelow, J. Struc. Geol. 14 85 (1992).[12] M.A. Biot, H. Ode and W.L. Roever, Geol. Soc. Am. Bull. 72 1621 (1961).[13] P.R. Cobbold, Tectonophysics 27 333 (1975).[14] P.J. Hudleston, Tectonophysics 16 189 (1973).[15] W.M. Chapple, Geol. Soc. Am. Bull. 79 47 (1968).[16] S.M. Schmalholz, Y.Y. Podladchikov and J.-P. Burg, J. Geopys. Res. 107 Doi 10.1029/

2001JB000355 (2002).[17] Z.P. Bazant and L. Cedolin, in Stability of Structures: Elastic, Inelastic, Fracture, and

Damage Theories, The Oxford Engineering Science Series (Oxford University Press,New York, 1991).


[18] M. Sathyamoorthy, Nonlinear Analysis of Structures, CRCMechanical Engineering Series(CRC Press, Boca Raton, Boston, 1998).

[19] L. Euler, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes(Marcum-Michaelem Bousquet & Socios., Lausannne & Geneve, 1744).

[20] R.B. Smith, Geol. Soc. Am. Bull. 88 312 (1977).[21] S.M. Schmalholz and Y.Y. Podladchikov, Geophys. Res. Lett. 26 2641 (1999).[22] C. Cuvelier, A. Segal and A.A. van Steenhoven, Finite Element Methods and the

Navier-Stokes Equations (D. Reidel Publishing Company, Dordrecht, 1986).[23] P.G. Drazin and W.H. Ried, in Hydrodynamic Stability (Cambridge University Press,

Cambridge, 1981).[24] J. Guckenheimer and P. Holmes, Nonliner oscillations, dynamical systems, and bifurca-

tions of vector fields, in Applied Mathematical Sciences, Vol. 42 (Springer, New York,1983).

[25] S.M. Schmalholz, Y.Y. Podladchikov and B. Jamtveit, Terra Nova 17 66 (2005).[26] S.M. Schmalholz and Y.Y. Podladchikov, Geophys. Res. Lett. 28 1835 (2001).


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