impact of mechanical anisotropy and power-law rheology on single

(2006) 71–87www.elsevier.com/locate/tecto

Tectonophysics 421

Impact of mechanical anisotropy and power-lawrheology on single layer folding

Thomas Kocher ⁎, Stefan M. Schmalholz, Neil S. Mancktelow

Geological Institute, ETH Zurich, CH-8092 Zurich, Switzerland

Received 30 November 2005; received in revised form 20 March 2006; accepted 13 April 2006Available online 22 June 2006

Abstract

The infinitesimal and finite stages of folding in nonlinear viscous material with a layer-parallel anisotropy were investigatedusing numerical and analytical methods. Anisotropy was found to have a first-order effect on growth rate and wavelength selection,and these effects are already important for anisotropy values (normal viscosity/shear viscosity) <10. The effect of anisotropy musttherefore be considered when deducing viscosity contrasts from wavelength to thickness ratios of natural folds. Growth rates ofsingle layer folds were found to increase and subsequently decrease during progressive deformation. This is due to interferencebetween the single layer folds and chevron folds that form in the matrix as a result of instability caused by the anisotropic materialbehaviour. The wavelength of the chevron folds in the matrix is determined by the wavelength of the folded single layer, which canexplain the high wavelength to thickness ratios that are sometimes found in multilayer sequences. Numerical models includinganisotropic material properties allow the behaviour of multilayer sequences to be investigated without the need for resolution on thescale of individual layers. This is particularly important for large-scale models of layered lithosphere.© 2006 Elsevier B.V. All rights reserved.

Keywords: Anisotropy; Single layer folding; Buckling; Structural softening

1. Introduction

1.1. Folding and mechanical anisotropy in naturalrocks

Folds develop in layered and schistose rocks forwhich the material properties would be intuitivelyexpected to vary with direction (i.e. they are anisotrop-ic). However, it is not immediately clear what degree ofanisotropic rheological behaviour is necessary to

⁎ Corresponding author.E-mail addresses: [email protected] (T. Kocher),

[email protected] (S.M. Schmalholz),[email protected] (N.S. Mancktelow).

0040-1951/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.tecto.2006.04.014

significantly influence fold geometry and growth rate,what effects might be recognizable in natural folds, andhow the interplay between anisotropy and otherrheological parameters, such as nonlinear (power-law)viscosity, might modify fold development. This paperuses numerical and analytical methods to investigate theinfluence of layer-parallel anisotropy on single layerbuckle folding, which for isotropic materials is perhapsthe most studied case example (see the literature reviewin Section 1.3).

Individual mineral grains possess a crystallinestructure and most of them are inherently anisotropic,i.e. some or all mechanical properties vary withdirection (e.g. Linker et al., 1984). However, whethera specific volume of rock – as an aggregate of minerals

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Fig. 1. (a) Turbiditic sequence of finely laminated siltstones and thickhomogeneous sandstones from the Markan fold belt, Iran. (b) Thefinely layered parts can be approximated by a homogeneous, butanisotropic rheology, whereas the sandstone beds are resolved andmodelled as homogeneous, isotropic material of higher viscosity.

72 T. Kocher et al. / Tectonophysics 421 (2006) 71–87

– will be isotropic or anisotropic depends on a numberof additional factors, such as average grain orientation,the spatial distribution of grains, and the scale ofobservation. On a small scale, where individual grainscan be distinguished, properties will generally beanisotropic due to the crystalline structure of thecomponent minerals, but on a larger scale, the behaviouris determined by the average orientation of theconstituent grains. A monomineralic aggregate may beisotropic with respect to its mechanical properties if theindividual, inherently anisotropic grains are randomlyoriented, but anisotropic if a lattice preferred or shapepreferred orientation of the minerals exists. A poly-mineralic rock can have bulk anisotropic behaviour notonly due to the orientation of individual grains, but alsodue to the arrangement of the different components. Atwo-component aggregate of two cubic minerals will beroughly isotropic if the components are randomlydistributed. However, an arrangement of the same twocomponents into layers will cause the system to beanisotropic with respect to deformation (Biot, 1965a;Johnson and Fletcher, 1994). The anisotropy of such asystem is not an inherent property of the individualcomponents, but a property of the system as a whole,which we will refer to as structural anisotropy in thefollowing.

Natural rocks are commonly layered, due tosedimentary bedding or metamorphic segregation (e.g.Dewers and Ortoleva, 1990). However, the arrangementof the components into compositional layers is only oneof a range of possible structures that can lead tostructural anisotropy. Other possible configurations in atwo-component system, such as elliptical or square-shaped domains in an otherwise homogeneous matrix,have been investigated recently by Mandal et al. (2000),Treagus (2003) and Fletcher (2004). In natural rocks,mechanical anisotropy will usually be determined by acombination of both the inherent anisotropy of thecomponent minerals (e.g. phyllosilicates), and theeffects of structural anisotropy (e.g. alternating quartz/feldspar-rich and mica-rich layering in a gneiss).

In this paper we investigate the process of bucklinginstability of single layer folds in anisotropic, nonlinearviscous materials (i.e. the possible effects of elasticityand plasticity in natural rocks are excluded). Thissituation, where a single (isotropic or anisotropic)competent layer is embedded in a less competentanisotropic matrix, or in a multilayer sequence featuringlayer thicknesses much smaller than that of the singlecompetent layer, can be frequently found in nature, e.gin turbiditic sequences with strongly varying layerthickness (Fig. 1). Modelling such a system as a

competent single layer within an anisotropic matrixfills the gap between the two end-member systemsrepresented by (1) single layer folds in an isotropicmatrix, and (2) a multilayer system made up ofindividually isotropic materials arranged in regularlayers of similar thickness. The strength of this approachis that it allows material properties determined on lengthscales that differ by many orders of magnitude to beaccurately modelled, without the need to maintainresolution on the scale of the fine layering or of theindividual minerals (e.g. with a finite element grid).

The two end-member systems mentioned above havebeen extensively investigated in the past. In thefollowing we examine how anisotropy and nonlinearity,and in particular the combination of these two effects,may alter the conclusions that were drawn frominvestigations considering the individual effects inisolation. The current study restricts itself to the caseof a pre-existing anisotropy with a constant magnitudeand fixed orientation relative to material points, such asfound in banded or schistose metamorphic rocks or

73T. Kocher et al. / Tectonophysics 421 (2006) 71–87

layered sediments. It neglects any changes in magnitudeor orientation of the anisotropy relative to materialpoints during the deformation process, for example dueto the progressive development of a lattice preferredorientation, mineral reactions induced by differentialstresses (Milke et al., 2004), or a newly formed axialplane cleavage.

The effects of anisotropic flow properties of bothlayer and matrix on the growth rates and the wavelengthselection of buckling folds are investigated first. Thestudy is then extended to consider finite deformation,analysing the amplification behaviour of folds in ananisotropic matrix to establish whether the resultsobtained from the growth rate spectra can be extrapo-lated to higher strain. A comparison of finite fold shapesand strain rate fields for isotropic and anisotropicrheologies reveals some important differences, whichhave implications for (1) the estimation of viscositycontrasts from finite fold shapes of single layer folds(Talbot, 1999), and (2) the understanding of deformationin multilayer systems in which individual layerthicknesses vary over orders of magnitude (e.g. Fig. 1).

1.2. Numerical methods

The numerical experiments in this study wereperformed using the finite element code FLASH(Kocher, 2006). This code solves the Stokes equationsfor an incompressible, anisotropic nonlinear viscousfluid. Nine-node quadrilateral elements with quadraticshape functions are used to discretise velocity, whereaspressure is interpolated linearly using three degrees offreedom per element (Cuvelier et al., 1986). The bulkanisotropic viscosity in 2D requires a shear and a normalviscosity to describe the fluid behaviour (e.g Biot,1965a). Thus power-law rheology is described by

leffN ¼ crefN e� 1n�1ð Þ

II ; leffS ¼ crefS e� 1n�1ð Þ

II ; ð1Þwhere μN

eff and μSeff are the effective normal and shear

viscosities, cNref and cS

ref constants describing theviscosity at the reference deformation state (i.e. whenϵ˙II=1), and n the stress exponent (Ranalli, 1995), whichwas assumed to be equal for both the shear and thenormal viscosity. The second invariant of the strain ratetensor ϵ˙II is calculated as

e�II ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie�2xx þ e�2xy

q: ð2Þ

This formulation can lead to extremely high values ofviscosity if the magnitude of strain rate approaches zeroat some point in the model. The viscosity is therefore

limited to a certain value to guarantee that the ratiobetween the smallest and largest viscosity in the wholemodel domain does not exceed 106. Viscosity differ-ences that are too large would prevent proper conver-gence of the numerical code.

The shear and normal viscosity are fixed to a localcoordinate system parallel and perpendicular to theplane of anisotropy (i.e. foliation or bedding), whichrotates as the material is deformed. The formulationused in the FLASH code accounts for this reorientationby tracking the vector normal to the anisotropy plane(the so-called ‘director’), and rotating the materialproperties according to the change in the directororientation. The finite element formulation of theconstitutive law as well as the director formulation areequivalent to the one presented by Mühlhaus et al.(2002a,b), derived from a more general formulation ofthe material matrix reorientation. A concise descriptionof the derivation is given in the Appendix.

1.3. Previous work

Depending on the scale of observation, a multilay-ered system can either be resolved into individual layersor be described as an effectively homogeneous, butanisotropic, material. As was shown by Bayly (1964)and Biot (1965a), a sequence of layers of differentisotropic materials shows a bulk anisotropic behaviourwith respect to pure and simple shear deformation. Thisbulk behaviour can be described mathematically byassigning average normal and shear viscosities, μN andμS, to the multilayer sequence. The thickness ratios ofthe individual layers are not considered in thisformulation, which means that the bending resistance(Timoshenko and Woinowsky-Krieger, 1959; Turcotteand Schubert, 2002) of the individual layers is notincluded in the mathematical description. Such ahomogeneous, anisotropic viscous material developsinternal instability under compression (Biot, 1965a;Cobbold et al., 1971). The absence of microphysicalproperties in the model leads to infinitesimally smallwavelengths for the developing buckle folds. The use ofthis specific formulation of anisotropy therefore impliesthat the thickness of the individual layers in theanisotropic material is small compared to the thicknessof the competent layer that is resolved in the model.Taking into account internal bending moments requiresthe introduction of an additional parameter describingan internal length scale of the anisotropic material (Biot,1965b; Latham, 1985; Mühlhaus et al., 2002a).

The bulk anisotropic formulation (without micro-structural effects) has frequently been used to


approximate the deformation of multilayer systems, forexample by Casey and Huggenberger (1985), whoimplemented this formulation in a finite element code toexamine chevron-type folds. More theoretical studies onthe deformation of anisotropic material were presentedby Cobbold (1976) and Weijermars (1992), whodiscussed the reorientation of the anisotropy planesduring progressive deformation. They pointed out thatthe instantaneous stretching and shortening axes will nolonger necessarily be parallel to the principal stress axes,which has important effects on finite structure formation(Kocher and Mancktelow, in press).

Folds are typical deformation structures in layeredrocks and buckle folding of linear and nonlinear viscouslayers has been extensively studied (e.g Biot, 1957,1965b; Ramberg, 1961, 1963, 1964; Chapple, 1968;Fletcher, 1974, 1977; Smith, 1975, 1977; Schmalholzand Podladchikov, 1999; Mancktelow, 2001, and manyothers). Fletcher (1974, 1977) and Smith (1975, 1977)independently derived equivalent analytical solutionsdescribing the growth rate spectra of small harmonicperturbations of the layer interface at the onset ofdeformation. Fletcher (1974) discussed the possibility ofadapting this analytical solution to the case of ananisotropic matrix. His approach relies on an approx-imation introduced by Biot (1965a), who suggested thata viscous anisotropic half space can be described by asingle viscosity l ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lNormaldlShearp

. However, thisanalytical solution does not account for matrix defor-mation effects that could become important duringprogressive deformation.

Many workers have investigated structural featuresassociated with multilayered rocks, such as kink bandsor chevron folds (e.g. Ramberg, 1964, 1970a,b;Ramberg and Strömgård, 1971; Ramsay, 1974; Ram-berg and Johnson, 1976; Wadee et al., 2004), byresolving the individual layer thickness in their models.Cobbold et al. (1971) suggested that kink bands andchevron folds were only end-members in a continuousseries of different kinds of deformation, leading tostructures such as box folds or boudin-like structures.From analogue experiments employing finely-layeredplasticine separated by graphite powder, Cobbold et al.(1971) proposed that the concept of internal instabilitycould indeed be applied to materials that are on averagehomogeneous but anisotropic.

Several papers have also considered strain patternsand the style of deformation associated with folding ofan anisotropic layer. Hudleston et al. (1996) investigatedthe strain patterns in isotropic and anisotropic parallelsingle layer folds. The anisotropy values required tochange the folding kinematics of the layer from bending

to flexural flow were high (δ>50) and led the authors toconclude that anisotropy was unlikely to be an importantfactor determining the strain patterns in natural singlelayer folds. Nevertheless, a comparison of fold shapesand strain patterns within competent layers may providebetter constraints on the rheological behaviour at thetime of deformation (Lan and Hudleston, 1996).Williams (1980) investigated fold shapes and strainpattern within a periodic multilayer system as a functionof the viscosity contrast between the two layers. Hepresented a model to estimate the lock-up angle ofchevron folds, and concluded that chevron folds in amultilayer sequence represent the fold shape requiringthe least energy dissipation for their formation.

1.4. Degree of anisotropy in natural rocks

Of the different geometrical setups investigated byTreagus (2003), the multilayer system composed of twoisotropic materials of equal thickness with differentviscosities proved to be the most anisotropic one.Treagus showed that the maximum anisotropy in thiscase is expressed as

dmax ¼ ð0:5Tlc þ 0:5Þ2lc

; ð3Þ

where μc=μstrong /μweak is the viscosity contrast of thetwo materials, and δ=μNormal /μShear the bulk viscousanisotropy factor. Eq. (3) can be approximated by

dmaxclc4þ 0:5 ð4Þ

even at small viscosity contrasts, with the relative errorcompared to Eq. (3) smaller than 1% for μc≥9.

Although, as outlined above, there is a considerableamount of published work on theoretical aspects of thedeformation of anisotropic materials, there is unfortu-nately only little experimental or field data constrainingthe magnitude of the anisotropy factor δ in natural rocks.Bayly (1970) considered a bi-laminate of isotropic andanisotropic rocks and proposed that δ was larger than12.5 for mica-rich phyllites, and around 2 for mica-poorphyllites. Combining these results with laboratoryexperiments on the anisotropy of a wax-aluminumflake mixture led him to conclude that anisotropy factorsof 25 or more do occur in natural rocks. Shea andKronenberg (1993) performed experiments on schistoserocks for different compression directions and describedthe anisotropy as the ratio of the yield strength parallel toand at 45° to the foliation. They found that the yieldstrength varies little with orientation of the foliation,


with an average anisotropy value of ca. 2. However,because the yield mode was not the same for all samples,the results are difficult to compare and must be treatedwith caution.

In order to calculate bulk anisotropy factor values formultilayer systems using Eq. (4), quantitative data onnatural viscosity contrasts are required. Differentmethods have been applied in the past to determinethis contrast (Talbot, 1999). Hara and Shimamoto(1984) for example found viscosity contrasts varyingfrom 23 to 136 for quartz veins in various types ofschist, and Cruikshank and Johnson (1993) mentionviscosity contrasts of up to 100 between sandstones andshales. Applying Eq. (4), a factor of δ=10–20 wouldthen be a conservative first-order estimate for the degreeof anisotropy that can be expected in multilayers.However, a review of experimental rock deformationdata (Talbot, 1999, Fig. 5) indicates that values ofviscosity contrast found in natural settings may beanywhere between 1 and several orders of magnitude,depending on a number of factors, such as confiningpressure, water content, grain size, differential stress,and temperature. These viscosity contrasts would implymuch higher anisotropy values than those suggested bythe studies of Bayly (1970), Hara and Shimamoto(1984) and Cruikshank and Johnson (1993). A widerange of different anisotropy values δ have been appliedin past studies on anisotropy, e.g. up to 50 by Lan andHudleston (1996) and 100 by Weijermars (1992).Compared to the few values of δ directly derived fromnatural examples and experimental data, these valuesappear to be of the correct order, although higheranisotropy values cannot be excluded.

2. Growth rate spectra of single layer folds inanisotropic and power-law material

Layer-parallel pure shear shortening of a singlestrong layer embedded in a weaker matrix leads tobuckling instability and folding of the competent layer,if an initial perturbation of the layer interface is present.At the onset of folding, when the amplitude is smallcompared to the layer thickness, fold development canbe described by analytical solutions (e.g. Fletcher,1974), which provide growth rates for sinusoidal initialperturbations as a function of wavelength. The wave-length for which the growth rate is a maximum is calledthe dominant wavelength (Biot, 1961). In order toinvestigate the effect of mechanical anisotropy on thegrowth rates of single layer folds, separate spectra werecalculated numerically for both anisotropic matrix andanisotropic layer, and compared to numerical and

analytical growth rate spectra for power-law material.Fig. 2a and b show the respective growth rate spectra fora power-law layer in a Newtonian matrix and vice versa.Fig. 2c and d show the comparable spectra for ananisotropic Newtonian layer in an isotropic Newtonianmatrix, and for an isotropic Newtonian layer in ananisotropic Newtonian matrix. The viscosity contrastbetween the normal viscosities of the layer and matrixwas 50 in every case. The solid lines in all figures are theappropriate analytical solutions after Fletcher (1974).For the case of an anisotropic competent layer (Fig. 2c),no analytical solution is available.

The results in Fig. 2a–d demonstrate the strongeffects of anisotropy on the growth rates and dominantwavelengths of the developing folds, and show that theyare of the same order of magnitude as effects due tononlinear rheology. An anisotropic competent layer inwhich the shear viscosity is lower than the normalviscosity exhibits slower growth rates and a slightlylonger, but less pronounced, dominant wavelength thanthe corresponding Newtonian fold (Fig. 2c). On theother hand, an anisotropic Newtonian matrix surround-ing an isotropic Newtonian layer results in a strongincrease in the growth rate and a larger dominantwavelength (Fig. 2d). For δ=10, the growth rate is morethan doubled and the dominant wavelength increasedby 50% compared to the isotropic Newtonian case.The Biot-approximation of the anisotropic matrix bylMatrix ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lNormaldlShearp

closely fits the results, show-ing a discrepancy between the analytical approxima-tion and the numerical results of ≤5%, with atendency to increase for higher degress of anisotropy(Fig. 2d). The approximation of the matrix viscositycannot be applied in the same way to the layer, as thedashed line in Fig. 2c shows. The analytical solutionwith lLayer ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lNormaldlShearp

for an anisotropy factorof δ=50 in the layer clearly does not reproduce thecorresponding numerical results.

Fig. 2a and b demonstrate that the linearisedanalytical solution of Fletcher (1974) for power-lawmaterial matches the nonlinear finite element results tohigh accuracy. The growth rates in power-law materialsare always higher than in Newtonian materials,irrespective of whether the layer or the matrix isnonlinear. However, the dominant wavelength mayincrease or decrease depending on which of the twodomains behaves nonlinearly. Anisotropic behaviour ineither layer or matrix has opposite effects on the growthrate (Fig. 2c,d), while the dominant wavelength isalways larger for anisotropic material compared toisotropic material, irrespective of whether it is the matrixor the layer that is anisotropic.

Fig. 2. Growth rate spectra of a single layer fold for 4 different setups: (a) isotropic power-law layer, isotropic Newtonian matrix (b) isotropicNewtonian layer, isotropic power-law matrix (c) anisotropic Newtonian layer, isotropic Newtonian matrix (d) isotropic Newtonian layer, anisotropicNewtonian matrix. Symbols are numerical results, solid lines are analytical values according to Fletcher (1974). The dashed line in (c) shows theanalytical solution of Fletcher (1974), with lLayer ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lNormaldlShearp

for δL=50 (corresponding to the numerical results marked by the triangles). Thedotted line represents the same reference run in all 4 plots (isotropic Newtonian layer and matrix, viscosity contrast=50). n is the stress exponent, andδ the anisotropy factor as defined in the text.


Laboratory experiments as well as data inferredfrom studies on lithospheric scales indicate that thedeformation of natural rocks is accurately described bynonlinear constitutive laws, except for very low stresslevels and temperatures (e.g. Borch and Green, 1987;Turcotte and Schubert, 2002; Freed and Burgmann,2004). If neither the matrix nor the layer is ofNewtonian viscosity, then a combination of matrixand layer effects will determine the growth rate andwavelength of the developing folds. A combination ofeffects can also be assumed if both layer and matrixare anisotropic. A power-law rheology of both matrixand layer leads to higher growth rates compared to theNewtonian isotropic case irrespective of the ratio of

stress exponents nLayer/nMatrix (Fig. 3a). The two effectsare essentially additive and can result in very highgrowth rates. In contrast, the dominant wavelengthdepends on the ratio of the stress exponents, but forequal exponents n will be dominated by the layer effect(Fig. 3a). If both layer and matrix are anisotropic, thedominant wavelength of the fold will be larger,because anisotropy in both the layer and matrix causesan increase in the dominant wavelength. Whether thegrowth rate actually increases or decreases depends onthe ratio of the anisotropy coefficients δLayer/δMatrix.Yet Fig. 3b shows that if matrix and layer possess thesame degree of anisotropy, the matrix effects stronglydominate the growth rate spectrum, causing a marked

Fig. 3. Fold growth rate spectra for (a) isotropic power-law layer and matrix with equal stress exponents n (b) Newtonian anisotropic material withequal anisotropy factors δ. Solid lines are analytical solutions according to Fletcher (1974), the dotted line is the same reference run as in Fig. 2a–d.The same normal viscosity contrast (μc=50) is applied.


overall increase in the growth rate compared to theisotropic Newtonian case.

A comparison with the analytical solution of Fletcher(1974), which only accounts for the anisotropic matrix(δ=50, solid line in Fig. 3b), demonstrates that the effectof the anisotropic layer cannot be neglected. However,based on natural observations, one can argue that thematrix is usually the domain with stronger anisotropythan the layer (Hudleston et al., 1996). An example isshown in Fig. 1, where the behaviour of the massivesandstone beds can be supposed to be close to isotropic,whereas the fine layering in the surrounding sandstone-shale unit implies a strong anisotropy. It can therefore beassumed that the anisotropy value δ in the layer willrarely be higher than in the matrix. Based on theobservation that the influence of the matrix dominatesthe growth rate diagram for equal anisotropy effects ofboth matrix and layer (Fig. 3b), and the fact that thematrix of a fold is often the more anisotropic domain innatural examples, the subsequent experiments focus onthe effects of an anisotropic matrix, assuming the layerto be isotropic.

3. Finite amplification behaviour of single layerfolds in an anisotropic matrix

In order to investigate progressive fold amplifica-tion and growth rate evolution in an anisotropicmatrix, growth rates were calculated for a Newtonianisotropic layer embedded in a Newtonian matrix with

different degrees of anisotropy. The initial perturba-tion wavelength corresponded to the dominantwavelength of the isotropic Newtonian matrix andlayer, as given by the analytical solution of Fletcher(1974). The viscosity contrast between the normalviscosity of the layer and the normal viscosity of thematrix was fixed at 20 for all runs, and the maximuminitial perturbation amplitude set to 2% of the layerthickness.

Fig. 4 shows the normalized amplitude and thecorresponding total growth rates, plotted against thebulk stretch (X/X0) of the domain, for folds inanisotropic (Fig. 4a,b) and power-law materials (Fig.4c,d). The faster amplification of the folds in ananisotropic matrix is readily shown in Fig. 4a, whichwas expected from the higher values of initial growthrates established in the previous section. The higher thedegree of anisotropy, the faster the fold amplification.For example, an anisotropy factor of δ=5 causes theamplitude to be roughly twice as large at a stretch of0.85, compared to isotropic material. Note that thisanisotropy value is rather small relative to the possiblerange in natural rocks, as discussed in Section 1.4. Acomparison of the growth rates for the differentamplification curves (Fig. 4b) shows that the initialgrowth rates for each curve are not only higher than forthe case of an isotropic matrix, but that the growth ratesactually increase during progressive deformation, reach-ing a value that is almost twice as large as the initialvalue for an anisotropy factor of 10 in the matrix. With

Fig. 4. Amplitude and growth rate development for an isotropic Newtonian layer embedded in an anisotropic Newtonian matrix (a, b), and for anisotropic power-law layer and matrix (c, d). μc is the viscosity contrast of the normal viscosities, δ the anisotropy factor. The circular symbols at X /X0=1 in (b) and (d) are the analytical values of the growth rate at infinitesimal amplitude according to Fletcher (1974). The anisotropy is initiallyhorizontal, and only the competent layer interface is perturbed with a sinusoidal waveform of maximum 2% amplitude of the layer thickness. Thewavelength is equal to the dominant wavelength for the isotropic Newtonian case of μc=20 according to the formula of Fletcher (1974). Theamplification of a fold in isotropic material (solid line) can be analytically described by the solution of Schmalholz and Podladchikov (2000).


increasing anisotropy, the value of the maximum growthrate is higher, but is reached after a smaller amount ofshortening.

Comparing these results to the amplification andgrowth rate history of power-law folds (Fig. 4c,d), itcan be seen that even a stress exponent of n=5 in bothmatrix and layer is not high enough to make up for theeffects of an anisotropy factor of 10 in the matrix, eventhough the effects of both domains being nonlinear areadditive (Fig. 3a). The amplification of a fold inpower-law material always lags behind that in ananisotropic matrix for the parameter combinationschosen in our experiments. The growth rate curvesfor folds in a power-law matrix look fundamentallydifferent from those of folds in anisotropic material.The initial growth rates are higher than for the

Newtonian case, as predicted by the analytical so-lutions, but no increase in growth rate occurs duringprogressive deformation.

The observed increase in amplification rate of foldsin anisotropic matrix are not predicted by the analyticaltheories (Fletcher, 1974; Johnson and Fletcher, 1994).However, they can be explained and understood bylooking at the deformation processes in the matrix itself,which will be considered below in the generaldiscussion.

4. The effect of matrix anisotropy and nonlinearityon finite fold shape and matrix deformation

As established above, initial infinitesimal growth ratesand amplification during progressive fold development


show considerable differences depending upon therheology of the matrix (Newtonian, power-law oranisotropic). These effects presumably also have aninfluence on the finite fold shape of the competentlayers, and on the strain fields in the surroundingmatrix. To investigate these influence, a Newtonianisotropic single layer with the same initial randomperturbation was deformed in a Newtonian, power-law,Newtonian-anisotropic and power-law anisotropic ma-trix. The finite folds at 15%, 25% and 40% shorteningare shown in Fig. 5. The viscosity ratio between thenormal viscosity of the layer and the normal viscosity ofthe matrix is 50 in all runs. A pure shear backgrounddeformation field with periodic vertical velocities at thelateral boundaries and a free surface at the top wereapplied. The matrix-layer interface was perturbed usinga red noise random signal (biased towards longerwavelengths) with a maximum amplitude of 5% of thelayer thickness. The lines in the matrix in Fig. 5 arepassive markers parallel to the plane of anisotropy, withan irregular initial vertical spacing which increases awayfrom the layer.

A first look at the folds that develop in the fourdifferent matrix rheologies reveals major differences. Ina Newtonian isotropic matrix (Fig. 5a), the amplitudesof the folds are small, and the perturbation induced inthe matrix by the competent layer quickly decays withincreasing distance away from the embedded competentlayer (Ramsay, 1967). If the same Newtonian layerbuckles in a matrix with stress exponent of n=3 (Fig.5b), the growth rates are higher from the beginning, aspredicted by the analytical solution. This results instronger amplification of the folds for the same amountof bulk shortening compared to the case of a Newtonianmatrix. Since the fold amplifies faster, less layerthickening is acquired compared to Fig. 5a, but againthe matrix itself is only deformed in a zone of contactstrain in close proximity to the competent layer. In fact,there is no marked difference in the overall structuralpattern between the two cases, except for the slightlylarger amplitude of the power-law folds.

However, if the matrix is Newtonian but anisotropic(δ=6), the finite fold shape looks very different from theisotropic examples (both Newtonian and power-law;Fig. 5c). Not only are the amplitudes of the folds in thecompetent layers larger than in the Newtonian or power-law case, but the matrix is also strongly deformed,showing chevron-type folds of the same wavelength asthe competent layer. Hinge collapse features can beidentified in the matrix in areas close to the competentlayer due to local geometrical constraints, whereasangular chevron folds develop further away from the

layer. Amplitude and wavelength of the chevron foldsare almost constant through the whole model domain,with vertical axial planes parallel to the axial planes offolds in the competent layer.

If the effects of an anisotropic and power-law matrixare now combined (δ=6, n=3; Fig. 5d), the foldsamplify even faster, and are well developed at 40%shortening. Chevron folds with the same wavelengthand amplitude as the competent layer do occur, butdevelop only in the close vicinity of the layer. Awayfrom the competent layer, the amplitude of the chevronfolds is attenuated due to the power-law properties of thematerial. While the hinge regions of the matrix folds areangular in Newtonian anisotropic rheology, they arefolded on a very small scale (in fact the scale ofnumerical resolution) in a power-law anisotropicmaterial. However, the fold limbs are relatively straightand correspond to the limbs in the competent layer inboth rheologies. In contrast to the Newtonian isotropicmatrix, the deformation in power-law anisotropicmaterial starts with an early and simultaneous formationof conjugate kink bands (Fig. 5d, at 15% shortening),which broaden and merge during progressive deforma-tion. The kink bands originate from the limbs of thedeveloping fold in the competent layer. The angularfolds that develop in power-law anisotropic matrix arethe product of merging of two kink bands, and do notdevelop directly as in the Newtonian anisotropic matrix(compare the regions in the two boxes in Fig. 5d). Thesmall scale matrix folding occurs where the kink bandshave not yet merged, leaving the plane of anisotropybetween two conjugate kink bands in an orientationparallel to the maximum shortening direction. Thematrix folds develop a characteristic box shape in someplaces.

The observed differences in the finite geometry areassociated with highly different strain rate patterns in thematrix for different rheologies, which are illustrated inFig. 6. The strain rate maxima in Newtonian and power-law material are concentrated in comparatively narrowbands on either side of the competent layer (Fig. 6a,b).For a Newtonian matrix, the contour lines of the strainrates are arranged symmetrically around the normal tothe competent layer, whereas in power-law material atendency exists to form shear zones inclined at 45° to thelayer (along the directions of highest shear stress),which link to form a weakly interacting network ofzones of high strain rate (Fig. 6b). However, in bothNewtonian and power-law materials, the magnitude ofthe strain rate tensor quickly decays further away fromthe competent layer, and at some distance from the layerdeviates only slightly from the background value of one.

Fig. 5. Finite fold shapes of an isotropic Newtonian layer embedded in (a) isotropic Newtonian matrix (b) isotropic power-law matrix (n=3) (c)anisotropic Newtonian matrix (δ=6) (d) anisotropic power-law matrix (n=3, δ=6), at 15%, 25% and 40% shortening. The same initial perturbation(with a maximum amplitude of 5% of the layer thickness) was used in all four experiments (red noise, biased towards longer wavelengths).


Fig. 6. Contour plots of the second invariant of the strain rate tensor (as defined in Eq. (2)), normalized against the background strain rate invariant, forthe same runs as shown in Fig. 5a–d, at 15% bulk shortening: (a) isotropic Newtonian matrix (b) isotropic power-law matrix (c) anisotropicNewtonian matrix (d) anisotropic power-law matrix. The high strain rate lobes in the Newtonian material (a) are oriented perpendicular to the layer,whereas in a power-law matrix (b) they are alligned at ± 45° to the maximum compression direction, following the planes of highest shear stress.


This picture is very different when looking at thestrain rate field for an anisotropic matrix (Fig. 6c). Inthis case, the deformation is no longer concentratedaround the competent layer, but extends far out into thematrix. These far-ranging areas of high strain rate reflectthe developing internal instability in the matrix and theonset of chevron fold formation. The strain-rate patternlooks different again for the case of an anisotropicpower-law matrix (Fig. 6d). The power-law rheologyinfluence the deformation in two ways: (1) localisationtakes places where high strain rates occur, which causesthe bands of high strain rates observed in the case ofNewtonian anisotropic matrix to become narrower, and(2) the bands are no longer perpendicular to the layer.The resulting structures are well-developed kink bandsdipping at 70°–80° after 15% shortening. The maximumstrain rate magnitudes are the highest of all fourrheologies.

5. Structural softening: rheological control onintegrated strength profiles

The preceding section addressed the geometric andkinematic differences that result from folding of aNewtonian isotropic layer embedded in matrix materialsof different rheologies. The major differences in folddevelopment and finite structure (Figs. 4 and 5) suggestthat there might also be significant differences in thedynamics of fold formation in layer and matrix.

As was argued by Casey and Butler (2004), the onsetof a buckling instability leads to a significant stressdecrease of the bulk rock volume. To assess theinfluence of matrix rheology on the magnitude of thisstructural softening (i.e. the decrease in stress duringshortening under constant strain rate, Schmalholz et al.,2005), the horizontal stresses were integrated over alength of ten times the layer thickness along the lateral

Fig. 7. Plot of the integrated horizontal normal stress σxx, normalizedagainst the initial stress value σxx

0 , showing the stress drop due to structuralsoftening in the four experiments in Fig. 5. Anisotropy, although onlyweak (δ=6), strongly influences the strength of the bulk rock.


boundaries of the four models in Fig. 5. The resultingintegrated stresses, normalized over the initial stressvalue, for a Newtonian layer embedded in a Newtonian-isotropic, power-law isotropic, Newtonian-anisotropicand power-law anisotropic matrix are plotted in Fig. 7.Note that because strain rates were kept constant in thenumerical experiments, the horizontal normal stress σxx

is directly proportional to the effective bulk viscosity ofthe material.

The bulk strength decreases with increasing short-ening (or increasing fold amplification) for all fourmatrix rheologies. Weakening is smallest for a New-tonian rheology, where no feedback between the strainrate and the material properties exists. The power-lawmatrix rheology does provide this feedback mechanism,leading to a somewhat larger strength reduction of thedomain. In a Newtonian anisotropic matrix, thedeveloping matrix instability leads to weakening thatis even more pronounced than for a power-law matrix.However, the strongest effects of weakening arerecorded if the layer is embedded in an anisotropicpower-law matrix. The formation of kink bands andfolding of the competent layer leads to a reduction in thestrength of the material to approximately 20% of itsinitial value after 15% shortening. At higher strains, areduction in the strength of the material of up to an orderof magnitude is possible.

6. Discussion

The numerical experiments have demonstrated thestrong influence of anisotropy on both the initial and the

finite stages of single layer folding. The effects ofanisotropy on wavelength selection and growth rates areof similar order to those of power-law viscosity, for nand δ in the range of natural values. Anisotropy istherefore a rock property that cannot be simply ignored,even if its values are low (i.e. δ<10). This is especiallytrue for the deduction of physical rock parameters, suchas viscosity contrasts, from wavelength to thicknessratios measured on natural folds (see Talbot (1999) for asummary). Anisotropy adds another unknown to thesystem, in addition to the difficulties introduced by thelack of constraint on the initial perturbation shape andamplitude in natural examples (e.g. Mancktelow, 2001).As demonstrated here, anisotropy is a first-order effectdirectly influencing the growth rate of folds and theamount of layer-parallel shortening acquired duringdeformation. Without better constraints on the degreeof anisotropy in the rock at the time of formation, forexample from laboratory studies, an accurate estima-tion of the viscosity contrast is very difficult, if notimpossible.

Our results also establish that a competent aniso-tropic layer exhibits lower growth rates than acomparable isotropic layer (Fig. 2c). As the shearviscosity of the layer decreases, the kinematics withinthe competent layer changes from layer-parallel short-ening/stretching, or ‘folding of the first kind’, to ‘foldingof the second kind’, i.e. flexural flow (Biot, 1965b). Biotshowed that folding of the second kind occurs if

lls

>> 0:2ffiffiffiffiffilsln

r; ð5Þ

where μ is the viscosity of the matrix, and μn and μs arethe normal and shear viscosity of the layer. Normalizingthe viscosities in the inequality (5) against the matrixviscosity μ and reformulating gives

ln >> 0:04l3s : ð6ÞThis inequality demonstrates that the layer deforma-

tion for an anisotropy of δ=50 and a normal viscositycontrast of 10 (Fig. 2c) is dominated by flexural flowwhereas, in the isotropic material, layer-parallel short-ening/stretching dominates. A further reduction of theshear viscosity would cause the competent layer tobuckle internally, and the matrix would then essentiallyact as a rigid confinement (Biot, 1965b). The inequality(6) provides a quantitative explanation for the highanisotropy values required by Hudleston et al. (1996) toobtain flexural flow: a lower normal viscosity contrastwould allow flexural flow to become more importantat lower anisotropy values. However, because the


normalized shear viscosity scales in a cubic manner inEq. (6), it must be in the range of the matrix viscosityfor flexural flow to be important in the competentlayer, even if the normal viscosity of the layer is high.It follows that, although the competent layers usuallyshow less evidence for mechanical anisotropy (Hudle-ston et al., 1996), a careful assessment of the degreeof anisotropy in both layer and matrix is necessary tounderstand the kinematics of folding, particularly atlow normal viscosity contrast.

The observed strong variations in the fold growth ratein anisotropic Newtonian rock have not been previouslydocumented. They can be explained by considering thedeformation in the matrix itself. In isotropic matrixmaterial, the area around a single folded layer influencedby the perturbation flow field of that layer is roughlyequal to one wavelength of the fold (e.g. Ramsay, 1967,fig. 7–82). For an anisotropic matrix, this is no longerthe case. Due to the internal instability that develops inthe matrix (Biot, 1965a), the anisotropy planes start tobuckle with the same wavelength and a similarsinusoidal perturbation velocity field as the competentlayer. The perturbation spreads through the matrix andthe growth rate of the single layer increases while theperturbation extends into the material. The larger theanisotropy factor of the matrix, the faster the perturba-tion propagates, and the stronger the increase in growthrate. During this spreading period, the competent layerand the matrix deform almost ‘in phase’, i.e. with thesame perturbation velocity field, leading to a reductionin vertical stresses which oppose folding. However,deformation of the matrix with a sinusoidal shape is notthe mechanically preferred one for an anisotropicmaterial (Williams, 1980). The anisotropic matrixwould rather form angular or chevron folds. Thetransition from sinusoidal to chevron geometry takesplace at low limb dips, as was demonstrated by Fletcherand Pollard (1999), and this chevron fold developmentsoon starts to interfere with, and hamper the growth of,the single layer fold. As a result, the growth rate of thesingle layer fold quickly decays, even to below valuesfor the corresponding isotropic case.

The examples presented in Fig. 5 of finite folds inmatrix material with different rheologies have shownthat – starting from exactly the same geometrical setup– a variety of very different-looking structures can resultfor different combinations of anisotropic and linear ornonlinear mechanical properties. The very regularchevron folds that develop in Newtonian isotropicmatrix are of the same wavelength as the folds in thecompetent layer. This ‘overprinting’ of the wavelengthof a more competent layer on the folds in the

surrounding anisotropic matrix may be quite commonin nature and explain the very high wavelength tothickness ratios (up to 50) of some chevron folds(Ramsay, 1974). In a nonlinear anisotropic material,kink bands and small-scale folding of the matrix occurs,rather than chevron folds on the scale of the competentlayer. The initial kink band orientation is in goodagreement with the analogue model results of Wadee etal. (2004). In our experiments, the formation of kinkbands and small-scale folds were found to represent end-members of a range of possible deformation styles: highanisotropy values favour the development of small scalefolding, whereas high stress exponents promote theformation of kink bands. Nevertheless, both processescan occur simultaneously. A comparison with the workof Jiang et al. (2004) shows that the two end-memberscan also be observed in numerical experiments foranisotropic elasto-plastic material. However, a majordifference is that, at intermediate anisotropy values, thesmall scale folding is restricted to the hinge regions ofthe kink bands in our experiments, whereas small scalefolding occurs within the kink bands in elasto-plasticmaterials (Jiang et al., 2004). The origin of thisdifference is not yet understood.

The chevron folds on the scale of the competent layerthat occur in a nonlinear anisotropic matrix partly formby direct evolution from the sinusoidal perturbation nearthe competent layer, and partly by broadening andmerging of conjugate kink bands further away from thelayer (compare the areas in the boxes in Fig. 5d, at 25%and 40% shortening). The formation of chevron folds bykink band intersection was described geometrically byRamsay and Huber (1987), but the dynamic transitionfrom kink bands to chevrons can only be demonstratedby numerical or analogue models (Wadee et al., 2004).Our results confirm a suggestion by Paterson and Weiss(1966) that kink bands can form as an intermediate stepin chevron fold formation for a nonlinear rheology. Asecond conclusion from this observation is that chevronfold geometry can be the result of different deformationprocesses in different rheologies, with quite differentkinematics leading to similar final fold geometry.

In our experiments, nonlinear rheology was anecessity for the formation of kink bands. This is inagreement with most analogue experimental workpublished on kink band formation (e.g Cobbold et al.,1971), where plasticine, which has a strongly nonlinearrheology, was used as the analogue material. However,we cannot exclude the possibility that a nonlinear shearviscosity (but a linear normal viscosity) would besufficient to form kink bands, as was suggested byRamberg and Johnson (1976).


Matrix folding on the scale of the numericalresolution, which occurs in power-law anisotropicmaterial, could be avoided by introducing bendingstiffness into the model, e.g. by means of a Cosserattheory (Mühlhaus et al., 2002a). This formulationcontains a length parameter that can be interpreted asthe thickness of the individual layers of the matrix. Innature, some length scale in the anisotropic material willalways be present, e.g. due to the individual layerthickness or the width of individual elongate mineralgrains (e.g. mica or amphibole). Because bendingstiffness was neglected in the current study, no statementabout the wavelength of small-scale matrix folds can bemade. Nevertheless, we do observe that, in power-lawanisotropic material, the matrix deformation shows adistinct pattern of mainly undeformed, but rotated areas,and strongly folded areas in the hinge regions of thecompetent layer. A possible natural example of this

Fig. 8. Folded multilayer sequence with strongly varying layer thicknesses inFrance. The shaded area at the top left marks a homogeneous, thick limestonematrix material are straight on the limbs of the large-scale structure, but strofolded appears to become wider further away from the competent layer (divegeometry of the structure and the brittle deformation occurring on the right

interplay between large and small scale structures isgiven in Fig. 8. While the thin layers in the matrix havestraight limbs when they correspond to the limbs of thefold in the thick limestone layer, they are stronglybuckled in the hinge region. The matrix propertiesdetermine the type of folds that develop in the hingeregion, but the long, unfolded limbs in the matrix aredue to kink band formation at early deformation stages,causing the layering in these limb regions to be rotatedaway from the maximum shortening direction.

The structural softening of anisotropic materialduring folding is very marked and much stronger thanin isotropic Newtonian or power-law rocks. Forstructure development that is kinematic-controlled, asis frequently argued to be the case in compressionalsettings (Tikoff and Wojtal, 1999), this behaviour resultsin strong stress variations of up to an order ofmagnitude. However, for structures that are stress-

a Jurassic limestone/shale sequence of the Cluse du Fier, internal Jura,layer, and solid black lines highlight the layering. The thin layers in thengly folded in the hinge area. The area in which the matrix is stronglyrging dashed lines), although this is difficult to establish due to the 3Dside of the outcrop.


controlled, such as gravity-driven detachment folding ofsediments on passive continental margins (Sumner etal., 2004), strong variations in strain rates can beexpected to produce very heterogeneous structures.Anisotropic material behaviour thus accentuates theinfluence of boundary conditions on the style ofdeformation development.

7. Conclusions

This study has demonstrated that mechanical anisot-ropy has a first-order effect on infinitesimal and finitefolding of a single layer. Anisotropy of rocks, inparticular of the matrix, is a parameter that must be takeninto account when deducing viscosity contrasts fromfold geometry and wavelength to thickness ratios.Anisotropy provides a mechanism to expand the areaaround a single layer that is influenced by theheterogeneous deformation induced by the single layerfolding, to distances much greater than the order of onewavelength typical of isotropic matrix material. Theinternal instability in the anisotropic matrix allowspropagation of deformation characteristics (e.g. the foldwavelength of the single layer) into the matrix. This is apossible explanation for the large wavelength tothickness ratios occasionally found in multilayersequences. A nonlinear rheology prevents the directformation of chevron folds in the matrix, and insteadfavours kink band formation and small scale matrixfolding. Chevron fold geometries can result fromkinematically quite different deformation processesand are not indicative of a specific rheology. The strongstructural weakening observed in anisotropic materialsuggests that anisotropic rock properties should beincluded in larger scale models, especially if theprocesses under consideration are stress-controlled.

Acknowledgements

SMS thanks Ray Fletcher for the stimulating discus-sions and helpful comments during their visit at PGP,Oslo. TKwas supported by the ETH project 0-20998-02.Jean-Pierre Burg is thanked for a review of an earlierversion of this manuscript. Comments by two anony-mous reviewers have helped to improve this paper.

Appendix A. Derivation of the finite elementformulation for a viscous anisotropic fluid

The derivation of the finite element formulation ofthe constitutive equation for a transversely anisotropicfluid (Newtonian or power-law) in two dimensions is

derived here. The rheological parameters μShear, μNormal

of the fluid describe the rheology in a local coordinatesystem (Yn ;Ys ), where Yn is a unit vector normal to theplane of anisotropy, and Ys is perpendicular to Yn ,parallel to the plane of anisotropy. The constitutive lawin the local coordinate system is:

R V¼ M4d E� V; ð7Þwhere Σ′ is the deviatoric stress tensor, E

·′ is the strain

rate tensor, an apostrophe ′ denotes a local quantity (inthe (Yn ;Ys ) coordinate system), and M4 is a fourthorder tensor with all entries equal 0 except:

M1111 ¼ 2ln; M2222 ¼ 2ln; M1212 ¼ ls: ð8ÞHowever, the stresses and strain rates in the finite

element formulation are given in a global coordinatesystem (Yx ;Yy ). Therefore, the global strain rates E

·

need to be rotated into the local coordinate system(Yn ;Ys ) before calculating the stresses from Eq. (7):

E� V¼ RT d E�dR; ð9Þwhere R is the transformation matrix:

R ¼ cosðhÞ �sinðhÞsinðhÞ cosðhÞ

� �; ð10Þ

and θ the angle between the global x-axis and the localn-axis.

Inserting Eq. (9) into Eq. (7), and rotating the localstresses back into the (Yx ;Yy ) coordinate system gives:

R ¼ RT dM4dRd E�dRT dR: ð11ÞThis equation can be expanded for the three stress

components σxx, σyy, σxy of the global deviatoric stresstensorΣ, and rewritten in the following form (collapsingtwo of the four indices):

r ¼ Me�; ð12Þwhere σ=(σxx, σyy, σxy)

T and ϵ˙=(ϵ˙xx, ϵ˙yy, ϵ˙xy)T are now

vectors, and the matrix M can be separated into anisotropic and an anisotropic component:

M ¼ Miso þManiso ð13Þ

Miso ¼2ln 0 0

0 2ln 0

0 0 ls

0B@

1CA;

Maniso ¼ ðln � lsÞ�a0 a0 �a1a0 �a0 a1�a1 a1 �0:5þ a0

0B@

1CA; ð14Þ

where a0=2n12n2

2, and a1=n1n23−n13n2, with n1, n2 the x

and y component of the director Yn .


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impact of mechanical anisotropy and power-law rheology on single

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