Impact of Mechanical Anisotropy and Power-Law Rheology on Single Layer Folding ⁎ Thomas Kocher , Stefan M

Tectonophysics 421 (2006) 71–87 www.elsevier.com/locate/tecto

Impact of mechanical anisotropy and power-law rheology 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 2006 Available online 22 June 2006

Abstract

The infinitesimal and finite stages of folding in nonlinear viscous material with a layer-parallel anisotropy were investigated using 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 must therefore be considered when deducing viscosity contrasts from wavelength to thickness ratios of natural folds. Growth rates of single layer folds were found to increase and subsequently decrease during progressive deformation. This is due to interference between the single layer folds and chevron folds that form in the matrix as a result of instability caused by the anisotropic material behaviour. The wavelength of the chevron folds in the matrix is determined by the wavelength of the folded single layer, which can explain the high wavelength to thickness ratios that are sometimes found in multilayer sequences. Numerical models including anisotropic material properties allow the behaviour of multilayer sequences to be investigated without the need for resolution on the scale 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 significantly influence fold geometry and growth rate, what effects might be recognizable in natural folds, and 1.1. Folding and mechanical anisotropy in natural how the interplay between anisotropy and other rocks rheological parameters, such as nonlinear (power-law) viscosity, might modify fold development. This paper Folds develop in layered and schistose rocks for uses numerical and analytical methods to investigate the which the material properties would be intuitively influence of layer-parallel anisotropy on single layer expected to vary with direction (i.e. they are anisotrop- buckle folding, which for isotropic materials is perhaps ic). However, it is not immediately clear what degree of the most studied case example (see the literature review anisotropic rheological behaviour is necessary to in Section 1.3). Individual mineral grains possess a crystalline structure and most of them are inherently anisotropic, ⁎ Corresponding author. E-mail addresses: [email protected] (T. Kocher), i.e. some or all mechanical properties vary with [email protected] (S.M. Schmalholz), direction (e.g. Linker et al., 1984). However, whether [email protected] (N.S. Mancktelow). a specific volume of rock – as an aggregate of minerals

– will be isotropic or anisotropic depends on a number of additional factors, such as average grain orientation, the spatial distribution of grains, and the scale of observation. On a small scale, where individual grains can be distinguished, properties will generally be anisotropic due to the crystalline structure of the component minerals, but on a larger scale, the behaviour is determined by the average orientation of the constituent grains. A monomineralic aggregate may be isotropic with respect to its mechanical properties if the individual, inherently anisotropic grains are randomly oriented, but anisotropic if a lattice preferred or shape preferred orientation of the minerals exists. A poly- mineralic rock can have bulk anisotropic behaviour not only due to the orientation of individual grains, but also due to the arrangement of the different components. A two-component aggregate of two cubic minerals will be roughly isotropic if the components are randomly distributed. However, an arrangement of the same two components into layers will cause the system to be anisotropic with respect to deformation (Biot, 1965a; Johnson and Fletcher, 1994). The anisotropy of such a system is not an inherent property of the individual components, but a property of the system as a whole, which we will refer to as structural anisotropy in the following. Fig. 1. (a) Turbiditic sequence of finely laminated siltstones and thick Natural rocks are commonly layered, due to homogeneous sandstones from the Markan fold belt, Iran. (b) The sedimentary bedding or metamorphic segregation (e.g. finely layered parts can be approximated by a homogeneous, but Dewers and Ortoleva, 1990). However, the arrangement anisotropic rheology, whereas the sandstone beds are resolved and modelled as homogeneous, isotropic material of higher viscosity. of the components into compositional layers is only one of a range of possible structures that can lead to structural anisotropy. Other possible configurations in a competent single layer within an anisotropic matrix two-component system, such as elliptical or square- fills the gap between the two end-member systems shaped domains in an otherwise homogeneous matrix, represented by (1) single layer folds in an isotropic have been investigated recently by Mandal et al. (2000), matrix, and (2) a multilayer system made up of Treagus (2003) and Fletcher (2004). In natural rocks, individually isotropic materials arranged in regular mechanical anisotropy will usually be determined by a layers of similar thickness. The strength of this approach combination of both the inherent anisotropy of the is that it allows material properties determined on length component minerals (e.g. phyllosilicates), and the scales that differ by many orders of magnitude to be effects of structural anisotropy (e.g. alternating quartz/ accurately modelled, without the need to maintain feldspar-rich and mica-rich layering in a gneiss). resolution on the scale of the fine layering or of the In this paper we investigate the process of buckling individual minerals (e.g. with a finite element grid). instability of single layer folds in anisotropic, nonlinear The two end-member systems mentioned above have viscous materials (i.e. the possible effects of elasticity been extensively investigated in the past. In the and plasticity in natural rocks are excluded). This following we examine how anisotropy and nonlinearity, situation, where a single (isotropic or anisotropic) and in particular the combination of these two effects, competent layer is embedded in a less competent may alter the conclusions that were drawn from anisotropic matrix, or in a multilayer sequence featuring investigations considering the individual effects in layer thicknesses much smaller than that of the single isolation. The current study restricts itself to the case competent layer, can be frequently found in nature, e.g of a pre-existing anisotropy with a constant magnitude in turbiditic sequences with strongly varying layer and fixed orientation relative to material points, such as thickness (Fig. 1). Modelling such a system as a found in banded or schistose metamorphic rocks or T. Kocher et al. / Tectonophysics 421 (2006) 71–87 73 layered sediments. It neglects any changes in magnitude limited to a certain value to guarantee that the ratio or orientation of the anisotropy relative to material between the smallest and largest viscosity in the whole points during the deformation process, for example due model domain does not exceed 106. Viscosity differ- to the progressive development of a lattice preferred ences that are too large would prevent proper conver- orientation, mineral reactions induced by differential gence of the numerical code. stresses (Milke et al., 2004), or a newly formed axial The shear and normal viscosity are fixed to a local plane cleavage. coordinate system parallel and perpendicular to the The effects of anisotropic flow properties of both plane of anisotropy (i.e. foliation or bedding), which layer and matrix on the growth rates and the wavelength rotates as the material is deformed. The formulation selection of buckling folds are investigated first. The used in the FLASH code accounts for this reorientation study is then extended to consider finite deformation, by tracking the vector normal to the anisotropy plane analysing the amplification behaviour of folds in an (the so-called ‘director’), and rotating the material anisotropic matrix to establish whether the results properties according to the change in the director obtained from the growth rate spectra can be extrapo- orientation. The finite element formulation of the lated to higher strain. A comparison of finite fold shapes constitutive law as well as the director formulation are and strain rate fields for isotropic and anisotropic equivalent to the one presented by Mühlhaus et al. rheologies reveals some important differences, which (2002a,b), derived from a more general formulation of have implications for (1) the estimation of viscosity the material matrix reorientation. A concise description contrasts from finite fold shapes of single layer folds of the derivation is given in the Appendix. (Talbot, 1999), and (2) the understanding of deformation in multilayer systems in which individual layer 1.3. Previous work thicknesses vary over orders of magnitude (e.g. Fig. 1). Depending on the scale of observation, a multilay- 1.2. Numerical methods ered system can either be resolved into individual layers or be described as an effectively homogeneous, but The numerical experiments in this study were anisotropic, material. As was shown by Bayly (1964) performed using the finite element code FLASH and Biot (1965a), a sequence of layers of different (Kocher, 2006). This code solves the Stokes equations isotropic materials shows a bulk anisotropic behaviour for an incompressible, anisotropic nonlinear viscous with respect to pure and simple shear deformation. This fluid. Nine-node quadrilateral elements with quadratic bulk behaviour can be described mathematically by shape functions are used to discretise velocity, whereas assigning average normal and shear viscosities, μN and pressure is interpolated linearly using three degrees of μS, to the multilayer sequence. The thickness ratios of freedom per element (Cuvelier et al., 1986). The bulk the individual layers are not considered in this anisotropic viscosity in 2D requires a shear and a normal formulation, which means that the bending resistance viscosity to describe the fluid behaviour (e.g Biot, (Timoshenko and Woinowsky-Krieger, 1959; Turcotte 1965a). Thus power-law rheology is described by and Schubert, 2002) of the individual layers is not included in the mathematical description. Such a ðÞ11 ðÞ11 leff ¼ ref e n ; leff ¼ ref e n ; ð Þ homogeneous, anisotropic viscous material develops N cN II S cS II 1 internal instability under compression (Biot, 1965a; eff eff where μN and μS are the effective normal and shear Cobbold et al., 1971). The absence of microphysical ref ref viscosities, cN and cS constants describing the properties in the model leads to infinitesimally small viscosity at the reference deformation state (i.e. when wavelengths for the developing buckle folds. The use of ϵ˙II =1), and n the stress exponent (Ranalli, 1995), which this specific formulation of anisotropy therefore implies was assumed to be equal for both the shear and the that the thickness of the individual layers in the normal viscosity. The second invariant of the strain rate anisotropic material is small compared to the thickness tensor ϵ˙II is calculated as of the competent layer that is resolved in the model. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Taking into account internal bending moments requires e ¼ e2 þ e2 : ð Þ II xx xy 2 the introduction of an additional parameter describing an internal length scale of the anisotropic material (Biot, This formulation can lead to extremely high values of 1965b; Latham, 1985; Mühlhaus et al., 2002a). viscosity if the magnitude of strain rate approaches zero The bulk anisotropic formulation (without micro- at some point in the model. The viscosity is therefore structural effects) has frequently been used to 74 T. Kocher et al. / Tectonophysics 421 (2006) 71–87 approximate the deformation of multilayer systems, for to flexural flow were high (δ>50) and led the authors to example by Casey and Huggenberger (1985), who conclude that anisotropy was unlikely to be an important implemented this formulation in a finite element code to factor determining the strain patterns in natural single examine chevron-type folds. More theoretical studies on layer folds. Nevertheless, a comparison of fold shapes the deformation of anisotropic material were presented and strain patterns within competent layers may provide by Cobbold (1976) and Weijermars (1992),who better constraints on the rheological behaviour at the discussed the reorientation of the anisotropy planes time of deformation (Lan and Hudleston, 1996). during progressive deformation. They pointed out that Williams (1980) investigated fold shapes and strain the instantaneous stretching and shortening axes will no pattern within a periodic multilayer system as a function longer necessarily be parallel to the principal stress axes, of the viscosity contrast between the two layers. He which has important effects on finite structure formation presented a model to estimate the lock-up angle of (Kocher and Mancktelow, in press). chevron folds, and concluded that chevron folds in a Folds are typical deformation structures in layered multilayer sequence represent the fold shape requiring rocks and buckle folding of linear and nonlinear viscous the least energy dissipation for their formation. layers has been extensively studied (e.g Biot, 1957, 1965b; Ramberg, 1961, 1963, 1964; Chapple, 1968; 1.4. Degree of anisotropy in natural rocks Fletcher, 1974, 1977; Smith, 1975, 1977; Schmalholz and Podladchikov, 1999; Mancktelow, 2001, and many Of the different geometrical setups investigated by others). Fletcher (1974, 1977) and Smith (1975, 1977) Treagus (2003), the multilayer system composed of two independently derived equivalent analytical solutions isotropic materials of equal thickness with different describing the growth rate spectra of small harmonic viscosities proved to be the most anisotropic one. perturbations of the layer interface at the onset of Treagus showed that the maximum anisotropy in this deformation. Fletcher (1974) discussed the possibility of case is expressed as adapting this analytical solution to the case of an ð : Tl þ : Þ2 anisotropic matrix. His approach relies on an approx- d ¼ 0 5 c 0 5 ; ð Þ max l 3 imation introduced by Biot (1965a), who suggested that c a viscous anisotropic half space can be described by a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ μ μ single viscosity l ¼ l dl . However, this where c = strong / weak is the viscosity contrast of the Normal Shear δ μ μ analytical solution does not account for matrix defor- two materials, and = Normal / Shear the bulk viscous mation effects that could become important during anisotropy factor. Eq. (3) can be approximated by progressive deformation. l c Many workers have investigated structural features dmaxc þ 0:5 ð4Þ 4 associated with multilayered rocks, such as kink bands or chevron folds (e.g. Ramberg, 1964, 1970a,b; even at small viscosity contrasts, with the relative error Ramberg and Strömgård, 1971; Ramsay, 1974; Ram- compared to Eq. (3) smaller than 1% for μc ≥9. berg and Johnson, 1976; Wadee et al., 2004), by Although, as outlined above, there is a considerable resolving the individual layer thickness in their models. amount of published work on theoretical aspects of the Cobbold et al. (1971) suggested that kink bands and deformation of anisotropic materials, there is unfortu- chevron folds were only end-members in a continuous nately only little experimental or field data constraining series of different kinds of deformation, leading to the magnitude of the anisotropy factor δ in natural rocks. structures such as box folds or boudin-like structures. Bayly (1970) considered a bi-laminate of isotropic and From analogue experiments employing finely-layered anisotropic rocks and proposed that δ was larger than plasticine separated by graphite powder, Cobbold et al. 12.5 for mica-rich phyllites, and around 2 for mica-poor (1971) proposed that the concept of internal instability phyllites. Combining these results with laboratory could indeed be applied to materials that are on average experiments on the anisotropy of a wax-aluminum homogeneous but anisotropic. flake mixture led him to conclude that anisotropy factors Several papers have also considered strain patterns of 25 or more do occur in natural rocks. Shea and and the style of deformation associated with folding of Kronenberg (1993) performed experiments on schistose an anisotropic layer. Hudleston et al. (1996) investigated rocks for different compression directions and described the strain patterns in isotropic and anisotropic parallel the anisotropy as the ratio of the yield strength parallel to single layer folds. The anisotropy values required to and at 45° to the foliation. They found that the yield change the folding kinematics of the layer from bending strength varies little with orientation of the foliation, T. Kocher et al. / Tectonophysics 421 (2006) 71–87 75 with an average anisotropy value of ca. 2. However, analytical growth rate spectra for power-law material. because the yield mode was not the same for all samples, Fig. 2a and b show the respective growth rate spectra for the results are difficult to compare and must be treated a power-law layer in a Newtonian matrix and vice versa. with caution. Fig. 2c and d show the comparable spectra for an In order to calculate bulk anisotropy factor values for anisotropic Newtonian layer in an isotropic Newtonian multilayer systems using Eq. (4), quantitative data on matrix, and for an isotropic Newtonian layer in an natural viscosity contrasts are required. Different anisotropic Newtonian matrix. The viscosity contrast methods have been applied in the past to determine between the normal viscosities of the layer and matrix this contrast (Talbot, 1999). Hara and Shimamoto was 50 in every case. The solid lines in all figures are the (1984) for example found viscosity contrasts varying appropriate analytical solutions after Fletcher (1974). from 23 to 136 for quartz veins in various types of For the case of an anisotropic competent layer (Fig. 2c), schist, and Cruikshank and Johnson (1993) mention no analytical solution is available. viscosity contrasts of up to 100 between sandstones and The results in Fig. 2a–d demonstrate the strong shales. Applying Eq. (4), a factor of δ=10–20 would effects of anisotropy on the growth rates and dominant then be a conservative first-order estimate for the degree wavelengths of the developing folds, and show that they of anisotropy that can be expected in multilayers. are of the same order of magnitude as effects due to However, a review of experimental rock deformation nonlinear rheology. An anisotropic competent layer in data (Talbot, 1999, Fig. 5) indicates that values of which the shear viscosity is lower than the normal viscosity contrast found in natural settings may be viscosity exhibits slower growth rates and a slightly anywhere between 1 and several orders of magnitude, longer, but less pronounced, dominant wavelength than depending on a number of factors, such as confining the corresponding Newtonian fold (Fig. 2c). On the pressure, water content, grain size, differential stress, other hand, an anisotropic Newtonian matrix surround- and temperature. These viscosity contrasts would imply ing an isotropic Newtonian layer results in a strong much higher anisotropy values than those suggested by increase in the growth rate and a larger dominant the studies of Bayly (1970), Hara and Shimamoto wavelength (Fig. 2d). For δ=10, the growth rate is more (1984) and Cruikshank and Johnson (1993). A wide than doubled and the dominant wavelength increased range of different anisotropy values δ have been applied by 50% compared to the isotropic Newtonian case. in past studies on anisotropy, e.g. up to 50 by Lan and The Biot-approximationpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the anisotropic matrix by lMatrix ¼ l dl Hudleston (1996) and 100 by Weijermars (1992). Normal Shear closely fits the results, show- Compared to the few values of δ directly derived from ing a discrepancy between the analytical approxima- natural examples and experimental data, these values tion and the numerical results of ≤5%, with a appear to be of the correct order, although higher tendency to increase for higher degress of anisotropy anisotropy values cannot be excluded. (Fig. 2d). The approximation of the matrix viscosity cannot be applied in the same way to the layer, as the 2. Growth rate spectra of single layer folds in dashed line inpFig.ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c shows. The analytical solution lLayer ¼ l dl anisotropic and power-law material with Normal Shear for an anisotropy factor of δ=50 in the layer clearly does not reproduce the Layer-parallel pure shear shortening of a single corresponding numerical results. strong layer embedded in a weaker matrix leads to Fig. 2a and b demonstrate that the linearised buckling instability and folding of the competent layer, analytical solution of Fletcher (1974) for power-law if an initial perturbation of the layer interface is present. material matches the nonlinear finite element results to At the onset of folding, when the amplitude is small high accuracy. The growth rates in power-law materials compared to the layer thickness, fold development can are always higher than in Newtonian materials, be described by analytical solutions (e.g. Fletcher, irrespective of whether the layer or the matrix is 1974), which provide growth rates for sinusoidal initial nonlinear. However, the dominant wavelength may perturbations as a function of wavelength. The wave- increase or decrease depending on which of the two length for which the growth rate is a maximum is called domains behaves nonlinearly. Anisotropic behaviour in the dominant wavelength (Biot, 1961). In order to either layer or matrix has opposite effects on the growth investigate the effect of mechanical anisotropy on the rate (Fig. 2c,d), while the dominant wavelength is growth rates of single layer folds, separate spectra were always larger for anisotropic material compared to calculated numerically for both anisotropic matrix and isotropic material, irrespective of whether it is the matrix anisotropic layer, and compared to numerical and or the layer that is anisotropic. 76 T. Kocher et al. / Tectonophysics 421 (2006) 71–87

Fig. 2. Growth rate spectra of a single layer fold for 4 different setups: (a) isotropic power-law layer, isotropic Newtonian matrix (b) isotropic Newtonian layer, isotropic power-law matrix (c) anisotropic Newtonian layer, isotropic Newtonian matrix (d) isotropic Newtonian layer, anisotropic Newtonian matrix. Symbols are numerical results, solidpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lines are analytical values according to Fletcher (1974). The dashed line in (c) shows the l ¼ l dl δ analytical solution of Fletcher (1974), with Layer Normal Shear for L =50 (corresponding to the numerical results marked by the triangles). The dotted line represents the same reference run in all 4 plots (isotropic Newtonian layer and matrix, viscosity contrast=50). n is the stress exponent, and δ the anisotropy factor as defined in the text.

Laboratory experiments as well as data inferred stress exponents nLayer/nMatrix (Fig. 3a). The two effects from studies on lithospheric scales indicate that the are essentially additive and can result in very high deformation of natural rocks is accurately described by growth rates. In contrast, the dominant wavelength nonlinear constitutive laws, except for very low stress depends on the ratio of the stress exponents, but for levels and temperatures (e.g. Borch and Green, 1987; equal exponents n will be dominated by the layer effect Turcotte and Schubert, 2002; Freed and Burgmann, (Fig. 3a). If both layer and matrix are anisotropic, the 2004). If neither the matrix nor the layer is of dominant wavelength of the fold will be larger, Newtonian viscosity, then a combination of matrix because anisotropy in both the layer and matrix causes and layer effects will determine the growth rate and an increase in the dominant wavelength. Whether the wavelength of the developing folds. A combination of growth rate actually increases or decreases depends on effects can also be assumed if both layer and matrix the ratio of the anisotropy coefficients δLayer/δMatrix. are anisotropic. A power-law rheology of both matrix Yet Fig. 3b shows that if matrix and layer possess the and layer leads to higher growth rates compared to the same degree of anisotropy, the matrix effects strongly Newtonian isotropic case irrespective of the ratio of dominate the growth rate spectrum, causing a marked T. Kocher et al. / Tectonophysics 421 (2006) 71–87 77

Fig. 3. Fold growth rate spectra for (a) isotropic power-law layer and matrix with equal stress exponents n (b) Newtonian anisotropic material with equal 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 the different degrees of anisotropy. The initial perturba- isotropic Newtonian case. tion wavelength corresponded to the dominant A comparison with the analytical solution of Fletcher wavelength of the isotropic Newtonian matrix and (1974), which only accounts for the anisotropic matrix layer, as given by the analytical solution of Fletcher (δ=50, solid line in Fig. 3b), demonstrates that the effect (1974). The viscosity contrast between the normal of the anisotropic layer cannot be neglected. However, viscosity of the layer and the normal viscosity of the based on natural observations, one can argue that the matrix was fixed at 20 for all runs, and the maximum matrix is usually the domain with stronger anisotropy initial perturbation amplitude set to 2% of the layer than the layer (Hudleston et al., 1996). An example is thickness. shown in Fig. 1, where the behaviour of the massive Fig. 4 shows the normalized amplitude and the sandstone beds can be supposed to be close to isotropic, corresponding total growth rates, plotted against the whereas the fine layering in the surrounding sandstone- bulk stretch (X/X0) of the domain, for folds in shale unit implies a strong anisotropy. It can therefore be anisotropic (Fig. 4a,b) and power-law materials (Fig. assumed that the anisotropy value δ in the layer will 4c,d). The faster amplification of the folds in an rarely be higher than in the matrix. Based on the anisotropic matrix is readily shown in Fig. 4a, which observation that the influence of the matrix dominates was expected from the higher values of initial growth the growth rate diagram for equal anisotropy effects of rates established in the previous section. The higher the both matrix and layer (Fig. 3b), and the fact that the degree of anisotropy, the faster the fold amplification. matrix of a fold is often the more anisotropic domain in For example, an anisotropy factor of δ=5 causes the natural examples, the subsequent experiments focus on amplitude to be roughly twice as large at a stretch of the effects of an anisotropic matrix, assuming the layer 0.85, compared to isotropic material. Note that this to be isotropic. anisotropy value is rather small relative to the possible range in natural rocks, as discussed in Section 1.4. A 3. Finite amplification behaviour of single layer comparison of the growth rates for the different folds in an anisotropic matrix amplification curves (Fig. 4b) shows that the initial growth rates for each curve are not only higher than for In order to investigate progressive fold amplifica- the case of an isotropic matrix, but that the growth rates tion and growth rate evolution in an anisotropic actually increase during progressive deformation, reach- matrix, growth rates were calculated for a Newtonian ing a value that is almost twice as large as the initial isotropic layer embedded in a Newtonian matrix with value for an anisotropy factor of 10 in the matrix. With 78 T. Kocher et al. / Tectonophysics 421 (2006) 71–87

Fig. 4. Amplitude and growth rate development for an isotropic Newtonian layer embedded in an anisotropic Newtonian matrix (a, b), and for an isotropic 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 initially horizontal, and only the competent layer interface is perturbed with a sinusoidal waveform of maximum 2% amplitude of the layer thickness. The wavelength is equal to the dominant wavelength for the isotropic Newtonian case of μc =20 according to the formula of Fletcher (1974). The amplification 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 growth Newtonian case, as predicted by the analytical so- rate is higher, but is reached after a smaller amount of lutions, but no increase in growth rate occurs during shortening. progressive deformation. Comparing these results to the amplification and The observed increase in amplification rate of folds growth rate history of power-law folds (Fig. 4c,d), it in anisotropic matrix are not predicted by the analytical can be seen that even a stress exponent of n=5 in both theories (Fletcher, 1974; Johnson and Fletcher, 1994). matrix and layer is not high enough to make up for the However, they can be explained and understood by effects of an anisotropy factor of 10 in the matrix, even looking at the deformation processes in the matrix itself, though the effects of both domains being nonlinear are which will be considered below in the general additive (Fig. 3a). The amplification of a fold in discussion. power-law material always lags behind that in an anisotropic matrix for the parameter combinations 4. The effect of matrix anisotropy and nonlinearity chosen in our experiments. The growth rate curves on finite fold shape and matrix deformation for folds in a power-law matrix look fundamentally different from those of folds in anisotropic material. As established above, initial infinitesimal growth rates The initial growth rates are higher than for the and amplification during progressive fold development T. Kocher et al. / Tectonophysics 421 (2006) 71–87 79 show considerable differences depending upon the layer. Amplitude and wavelength of the chevron folds rheology of the matrix (Newtonian, power-law or are almost constant through the whole model domain, anisotropic). These effects presumably also have an with vertical axial planes parallel to the axial planes of influence on the finite fold shape of the competent folds in the competent layer. layers, and on the strain fields in the surrounding If the effects of an anisotropic and power-law matrix matrix. To investigate these influence, a Newtonian are now combined (δ=6, n=3; Fig. 5d), the folds isotropic single layer with the same initial random amplify even faster, and are well developed at 40% perturbation was deformed in a Newtonian, power-law, shortening. Chevron folds with the same wavelength Newtonian-anisotropic and power-law anisotropic ma- and amplitude as the competent layer do occur, but trix. The finite folds at 15%, 25% and 40% shortening develop only in the close vicinity of the layer. Away are shown in Fig. 5. The viscosity ratio between the from the competent layer, the amplitude of the chevron normal viscosity of the layer and the normal viscosity of folds is attenuated due to the power-law properties of the the matrix is 50 in all runs. A pure shear background material. While the hinge regions of the matrix folds are deformation field with periodic vertical velocities at the angular in Newtonian anisotropic rheology, they are lateral boundaries and a free surface at the top were folded on a very small scale (in fact the scale of applied. The matrix-layer interface was perturbed using numerical resolution) in a power-law anisotropic a red noise random signal (biased towards longer material. However, the fold limbs are relatively straight wavelengths) with a maximum amplitude of 5% of the and correspond to the limbs in the competent layer in layer thickness. The lines in the matrix in Fig. 5 are both rheologies. In contrast to the Newtonian isotropic passive markers parallel to the plane of anisotropy, with matrix, the deformation in power-law anisotropic an irregular initial vertical spacing which increases away material starts with an early and simultaneous formation from the layer. of conjugate kink bands (Fig. 5d, at 15% shortening), A first look at the folds that develop in the four which broaden and merge during progressive deforma- different matrix rheologies reveals major differences. In tion. The kink bands originate from the limbs of the a Newtonian isotropic matrix (Fig. 5a), the amplitudes developing fold in the competent layer. The angular of the folds are small, and the perturbation induced in folds that develop in power-law anisotropic matrix are the matrix by the competent layer quickly decays with the product of merging of two kink bands, and do not increasing distance away from the embedded competent develop directly as in the Newtonian anisotropic matrix layer (Ramsay, 1967). If the same Newtonian layer (compare the regions in the two boxes in Fig. 5d). The buckles in a matrix with stress exponent of n=3 (Fig. small scale matrix folding occurs where the kink bands 5b), the growth rates are higher from the beginning, as have not yet merged, leaving the plane of anisotropy predicted by the analytical solution. This results in between two conjugate kink bands in an orientation stronger amplification of the folds for the same amount parallel to the maximum shortening direction. The of bulk shortening compared to the case of a Newtonian matrix folds develop a characteristic box shape in some matrix. Since the fold amplifies faster, less layer places. thickening is acquired compared to Fig. 5a, but again The observed differences in the finite geometry are the matrix itself is only deformed in a zone of contact associated with highly different strain rate patterns in the strain in close proximity to the competent layer. In fact, matrix for different rheologies, which are illustrated in there is no marked difference in the overall structural Fig. 6. The strain rate maxima in Newtonian and power- pattern between the two cases, except for the slightly law material are concentrated in comparatively narrow larger amplitude of the power-law folds. bands on either side of the competent layer (Fig. 6a,b). However, if the matrix is Newtonian but anisotropic For a Newtonian matrix, the contour lines of the strain (δ=6), the finite fold shape looks very different from the rates are arranged symmetrically around the normal to isotropic examples (both Newtonian and power-law; the competent layer, whereas in power-law material a Fig. 5c). Not only are the amplitudes of the folds in the tendency exists to form shear zones inclined at 45° to the competent layers larger than in the Newtonian or power- layer (along the directions of highest shear stress), law case, but the matrix is also strongly deformed, which link to form a weakly interacting network of showing chevron-type folds of the same wavelength as zones of high strain rate (Fig. 6b). However, in both the competent layer. Hinge collapse features can be Newtonian and power-law materials, the magnitude of identified in the matrix in areas close to the competent the strain rate tensor quickly decays further away from layer due to local geometrical constraints, whereas the competent layer, and at some distance from the layer angular chevron folds develop further away from the deviates only slightly from the background value of one. 80 T. Kocher et al. / Tectonophysics 421 (2006) 71–87

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). T. Kocher et al. / Tectonophysics 421 (2006) 71–87 81

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, for the same runs as shown in Fig. 5a–d, at 15% bulk shortening: (a) isotropic Newtonian matrix (b) isotropic power-law matrix (c) anisotropic Newtonian 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 the 5. Structural softening: rheological control on strain rate field for an anisotropic matrix (Fig. 6c). In integrated strength profiles this case, the deformation is no longer concentrated around the competent layer, but extends far out into the The preceding section addressed the geometric and matrix. These far-ranging areas of high strain rate reflect kinematic differences that result from folding of a the developing internal instability in the matrix and the Newtonian isotropic layer embedded in matrix materials onset of chevron fold formation. The strain-rate pattern of different rheologies. The major differences in fold looks different again for the case of an anisotropic development and finite structure (Figs. 4 and 5) suggest power-law matrix (Fig. 6d). The power-law rheology that there might also be significant differences in the influence the deformation in two ways: (1) localisation dynamics of fold formation in layer and matrix. takes places where high strain rates occur, which causes As was argued by Casey and Butler (2004), the onset the bands of high strain rates observed in the case of of a buckling instability leads to a significant stress Newtonian anisotropic matrix to become narrower, and decrease of the bulk rock volume. To assess the (2) the bands are no longer perpendicular to the layer. influence of matrix rheology on the magnitude of this The resulting structures are well-developed kink bands structural softening (i.e. the decrease in stress during dipping at 70°–80° after 15% shortening. The maximum shortening under constant strain rate, Schmalholz et al., strain rate magnitudes are the highest of all four 2005), the horizontal stresses were integrated over a rheologies. length of ten times the layer thickness along the lateral 82 T. Kocher et al. / Tectonophysics 421 (2006) 71–87

finite stages of single layer folding. The effects of anisotropy on wavelength selection and growth rates are of similar order to those of power-law viscosity, for n and δ in the range of natural values. Anisotropy is therefore a rock property that cannot be simply ignored, even if its values are low (i.e. δ<10). This is especially true for the deduction of physical rock parameters, such as viscosity contrasts, from wavelength to thickness ratios measured on natural folds (see Talbot (1999) for a summary). Anisotropy adds another unknown to the system, in addition to the difficulties introduced by the lack of constraint on the initial perturbation shape and amplitude in natural examples (e.g. Mancktelow, 2001). As demonstrated here, anisotropy is a first-order effect directly influencing the growth rate of folds and the σ Fig. 7. Plot of the integrated horizontal normal stress xx,normalized amount of layer-parallel shortening acquired during σ0 against the initial stress value xx, showing the stress drop due to structural deformation. Without better constraints on the degree softening in the four experiments in Fig. 5. Anisotropy, although only weak (δ=6), strongly influences the strength of the bulk rock. of anisotropy in the rock at the time of formation, for example from laboratory studies, an accurate estimation of the viscosity contrast is very difficult, if not boundaries of the four models in Fig. 5. The resulting impossible. integrated stresses, normalized over the initial stress Our results also establish that a competent aniso- value, for a Newtonian layer embedded in a Newtonian- tropic layer exhibits lower growth rates than a isotropic, power-law isotropic, Newtonian-anisotropic comparable isotropic layer (Fig. 2c). As the shear and power-law anisotropic matrix are plotted in Fig. 7. viscosity of the layer decreases, the kinematics within Note that because strain rates were kept constant in the the competent layer changes from layer-parallel short- numerical experiments, the horizontal normal stress σxx ening/stretching, or ‘folding of the first kind’,to‘folding is directly proportional to the effective bulk viscosity of of the second kind’, i.e. flexural flow (Biot, 1965b). Biot the material. showed that folding of the second kind occurs if The bulk strength decreases with increasing short- rffiffiffiffiffi l l ening (or increasing fold amplification) for all four >> : s ; ð Þ l 0 2 l 5 matrix rheologies. Weakening is smallest for a New- s n tonian rheology, where no feedback between the strain where μ is the viscosity of the matrix, and μn and μs are rate and the material properties exists. The power-law the normal and shear viscosity of the layer. Normalizing matrix rheology does provide this feedback mechanism, the viscosities in the inequality (5) against the matrix leading to a somewhat larger strength reduction of the viscosity μ and reformulating gives domain. In a Newtonian anisotropic matrix, the developing matrix instability leads to weakening that l >> : l3: ð Þ n 0 04 6 is even more pronounced than for a power-law matrix. s However, the strongest effects of weakening are This inequality demonstrates that the layer deforma- recorded if the layer is embedded in an anisotropic tion for an anisotropy of δ=50 and a normal viscosity power-law matrix. The formation of kink bands and contrast of 10 (Fig. 2c) is dominated by flexural flow folding of the competent layer leads to a reduction in the whereas, in the isotropic material, layer-parallel short- strength of the material to approximately 20% of its ening/stretching dominates. A further reduction of the initial value after 15% shortening. At higher strains, a shear viscosity would cause the competent layer to reduction in the strength of the material of up to an order buckle internally, and the matrix would then essentially of magnitude is possible. act as a rigid confinement (Biot, 1965b). The inequality (6) provides a quantitative explanation for the high 6. Discussion anisotropy values required by Hudleston et al. (1996) to obtain flexural flow: a lower normal viscosity contrast The numerical experiments have demonstrated the would allow flexural flow to become more important strong influence of anisotropy on both the initial and the at lower anisotropy values. However, because the T. Kocher et al. / Tectonophysics 421 (2006) 71–87 83 normalized shear viscosity scales in a cubic manner in surrounding anisotropic matrix may be quite common Eq. (6), it must be in the range of the matrix viscosity in nature and explain the very high wavelength to for flexural flow to be important in the competent thickness ratios (up to 50) of some chevron folds layer, even if the normal viscosity of the layer is high. (Ramsay, 1974). In a nonlinear anisotropic material, It follows that, although the competent layers usually kink bands and small-scale folding of the matrix occurs, show less evidence for mechanical anisotropy (Hudle- rather than chevron folds on the scale of the competent ston et al., 1996), a careful assessment of the degree layer. The initial kink band orientation is in good of anisotropy in both layer and matrix is necessary to agreement with the analogue model results of Wadee et understand the kinematics of folding, particularly at al. (2004). In our experiments, the formation of kink low normal viscosity contrast. bands and small-scale folds were found to represent end- The observed strong variations in the fold growth rate members of a range of possible deformation styles: high in anisotropic Newtonian rock have not been previously anisotropy values favour the development of small scale documented. They can be explained by considering the folding, whereas high stress exponents promote the deformation in the matrix itself. In isotropic matrix formation of kink bands. Nevertheless, both processes material, the area around a single folded layer influenced can occur simultaneously. A comparison with the work by the perturbation flow field of that layer is roughly of Jiang et al. (2004) shows that the two end-members equal to one wavelength of the fold (e.g. Ramsay, 1967, can also be observed in numerical experiments for fig. 7–82). For an anisotropic matrix, this is no longer anisotropic elasto-plastic material. However, a major the case. Due to the internal instability that develops in difference is that, at intermediate anisotropy values, the the matrix (Biot, 1965a), the anisotropy planes start to small scale folding is restricted to the hinge regions of buckle with the same wavelength and a similar the kink bands in our experiments, whereas small scale sinusoidal perturbation velocity field as the competent folding occurs within the kink bands in elasto-plastic layer. The perturbation spreads through the matrix and materials (Jiang et al., 2004). The origin of this the growth rate of the single layer increases while the difference is not yet understood. perturbation extends into the material. The larger the The chevron folds on the scale of the competent layer anisotropy factor of the matrix, the faster the perturba- that occur in a nonlinear anisotropic matrix partly form tion propagates, and the stronger the increase in growth by direct evolution from the sinusoidal perturbation near rate. During this spreading period, the competent layer the competent layer, and partly by broadening and and the matrix deform almost ‘in phase’, i.e. with the merging of conjugate kink bands further away from the same perturbation velocity field, leading to a reduction layer (compare the areas in the boxes in Fig. 5d, at 25% in vertical stresses which oppose folding. However, and 40% shortening). The formation of chevron folds by deformation of the matrix with a sinusoidal shape is not kink band intersection was described geometrically by the mechanically preferred one for an anisotropic Ramsay and Huber (1987), but the dynamic transition material (Williams, 1980). The anisotropic matrix from kink bands to chevrons can only be demonstrated would rather form angular or chevron folds. The by numerical or analogue models (Wadee et al., 2004). transition from sinusoidal to chevron geometry takes Our results confirm a suggestion by Paterson and Weiss place at low limb dips, as was demonstrated by Fletcher (1966) that kink bands can form as an intermediate step and Pollard (1999), and this chevron fold development in chevron fold formation for a nonlinear rheology. A soon starts to interfere with, and hamper the growth of, second conclusion from this observation is that chevron the single layer fold. As a result, the growth rate of the fold geometry can be the result of different deformation single layer fold quickly decays, even to below values processes in different rheologies, with quite different for the corresponding isotropic case. kinematics leading to similar final fold geometry. The examples presented in Fig. 5 of finite folds in In our experiments, nonlinear rheology was a matrix material with different rheologies have shown necessity for the formation of kink bands. This is in that – starting from exactly the same geometrical setup agreement with most analogue experimental work – a variety of very different-looking structures can result published on kink band formation (e.g Cobbold et al., for different combinations of anisotropic and linear or 1971), where plasticine, which has a strongly nonlinear nonlinear mechanical properties. The very regular rheology, was used as the analogue material. However, chevron folds that develop in Newtonian isotropic we cannot exclude the possibility that a nonlinear shear matrix are of the same wavelength as the folds in the viscosity (but a linear normal viscosity) would be competent layer. This ‘overprinting’ of the wavelength sufficient to form kink bands, as was suggested by of a more competent layer on the folds in the Ramberg and Johnson (1976). 84 T. Kocher et al. / Tectonophysics 421 (2006) 71–87

Matrix folding on the scale of the numerical interplay between large and small scale structures is resolution, which occurs in power-law anisotropic given in Fig. 8. While the thin layers in the matrix have material, could be avoided by introducing bending straight limbs when they correspond to the limbs of the stiffness into the model, e.g. by means of a Cosserat fold in the thick limestone layer, they are strongly theory (Mühlhaus et al., 2002a). This formulation buckled in the hinge region. The matrix properties contains a length parameter that can be interpreted as determine the type of folds that develop in the hinge the thickness of the individual layers of the matrix. In region, but the long, unfolded limbs in the matrix are nature, some length scale in the anisotropic material will due to kink band formation at early deformation stages, always be present, e.g. due to the individual layer causing the layering in these limb regions to be rotated thickness or the width of individual elongate mineral away from the maximum shortening direction. grains (e.g. mica or amphibole). Because bending The structural softening of anisotropic material stiffness was neglected in the current study, no statement during folding is very marked and much stronger than about the wavelength of small-scale matrix folds can be in isotropic Newtonian or power-law rocks. For made. Nevertheless, we do observe that, in power-law structure development that is kinematic-controlled, as anisotropic material, the matrix deformation shows a is frequently argued to be the case in compressional distinct pattern of mainly undeformed, but rotated areas, settings (Tikoff and Wojtal, 1999), this behaviour results and strongly folded areas in the hinge regions of the in strong stress variations of up to an order of competent layer. A possible natural example of this magnitude. However, for structures that are stress-

Fig. 8. Folded multilayer sequence with strongly varying layer thicknesses in a Jurassic limestone/shale sequence of the Cluse du Fier, internal Jura, France. The shaded area at the top left marks a homogeneous, thick limestone layer, and solid black lines highlight the layering. The thin layers in the matrix material are straight on the limbs of the large-scale structure, but strongly folded in the hinge area. The area in which the matrix is strongly folded appears to become wider further away from the competent layer (diverging dashed lines), although this is difficult to establish due to the 3D geometry of the structure and the brittle deformation occurring on the right side of the outcrop. T. Kocher et al. / Tectonophysics 421 (2006) 71–87 85

controlled, such as gravity-driven detachment folding of derived here. The rheological parameters μShear, μNormal sediments on passive continental margins (Sumner et of the fluid describe the rheology in a local coordinate Y Y Y al., 2004), strong variations in strain rates can be system ( n ; s ), where n is a unit vector normal to the Y Y expected to produce very heterogeneous structures. plane of anisotropy, and s is perpendicular to n , Anisotropic material behaviour thus accentuates the parallel to the plane of anisotropy. The constitutive law influence of boundary conditions on the style of in the local coordinate system is: deformation development. RV¼ M4d EV; ð7Þ · 7. Conclusions where Σ′ is the deviatoric stress tensor, E′ is the strain rate tensor, an apostrophe ′ denotes a local quantity (in Y Y This study has demonstrated that mechanical anisot- the ( n ; s ) coordinate system), and M4 is a fourth ropy has a first-order effect on infinitesimal and finite order tensor with all entries equal 0 except: folding of a single layer. Anisotropy of rocks, in ¼ l ; ¼ l ; ¼ l : ð Þ M1111 2 n M2222 2 n M1212 s 8 particular of the matrix, is a parameter that must be taken into account when deducing viscosity contrasts from However, the stresses and strain rates in the finite fold geometry and wavelength to thickness ratios. element formulation are given in a global coordinate Y Y · Anisotropy provides a mechanism to expand the area system ( x ; y ). Therefore, the global strain rates E around a single layer that is influenced by the need to be rotated into the local coordinate system Y Y ( n ; s ) before calculating the stresses from Eq. (7): heterogeneous deformation induced by the single layer folding, to distances much greater than the order of one EV¼ RT d EdR; ð9Þ wavelength typical of isotropic matrix material. The where R is the transformation matrix: internal instability in the anisotropic matrix allows propagation of deformation characteristics (e.g. the fold cosðhÞsinðhÞ R ¼ ; ð10Þ wavelength of the single layer) into the matrix. This is a sinðhÞ cosðhÞ possible explanation for the large wavelength to θ thickness ratios occasionally found in multilayer and the angle between the global x-axis and the local sequences. A nonlinear rheology prevents the direct n-axis. Inserting Eq. (9) into Eq. (7), and rotating the local formation of chevron folds in the matrix, and instead Y Y stresses back into the ( x ; y ) coordinate system gives: favours kink band formation and small scale matrix folding. Chevron fold geometries can result from R ¼ RT dM4dRd EdRT dR: ð11Þ kinematically quite different deformation processes and are not indicative of a specific rheology. The strong This equation can be expanded for the three stress structural weakening observed in anisotropic material components σxx, σyy, σxy of the global deviatoric stress suggests that anisotropic rock properties should be tensor Σ, and rewritten in the following form (collapsing included in larger scale models, especially if the two of the four indices): processes under consideration are stress-controlled. r ¼ Me; ð12Þ σ σ σ σ T ϵ˙ ϵ˙ ϵ˙ ϵ˙ T Acknowledgements where =( xx, yy, xy) and =( xx, yy, xy) are now vectors, and the matrix M can be separated into an SMS thanks Ray Fletcher for the stimulating discus- isotropic and an anisotropic component: sions and helpful comments during their visit at PGP, M ¼ Miso þ Maniso ð13Þ Oslo. TK was supported by the ETH project 0-20998-02. Jean-Pierre Burg is thanked for a review of an earlier 0 1 l version of this manuscript. Comments by two anony- 2 n 00 B C ¼ l ; mous reviewers have helped to improve this paper. Miso @ 02n 0 A 00l 0 s 1 Appendix A. Derivation of the finite element a0 a0 a1 B C formulation for a viscous anisotropic fluid ¼ðl l Þ ; ð Þ Maniso n s @ a0 a0 a1 A 14 a a 0:5 þ a The derivation of the finite element formulation of 1 1 0 the constitutive equation for a transversely anisotropic where a =2n2n2, and a =n n3 −n3n ,withn , n the x 0 1 2 1 1 2 Y1 2 1 2 fluid (Newtonian or power-law) in two dimensions is and y component of the director n . 86 T. Kocher et al. / Tectonophysics 421 (2006) 71–87

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