Localised folding in general deformations

TECTO-125622; No of Pages 16 Tectonophysics xxx (2012) xxx–xxx

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Localised folding in general deformations Alison Ord a,⁎, Bruce Hobbs a, b a b

Centre for Exploration Targeting, School of Earth and Environment, The University of Western Australia, Perth, 6009, Australia CSIRO, Perth, Western Australia, Perth 6102, Australia

a r t i c l e

i n f o

Article history: Received 12 March 2012 Received in revised form 12 September 2012 Accepted 16 September 2012 Available online xxxx Keywords: Localised folding Biot theory Chaotic folding Swift–Hohenberg equation Crenulations Geometrical nonlinearity

a b s t r a c t One control on the buckling of a layer (or layers) embedded in a weaker matrix is the reaction force exerted by the deforming matrix on the layer. If the system is linear and this force is a linear function of the layer deflection, as for linear elastic and viscous materials, the resulting buckles can be sinusoidal or periodic. However if the system is geometrically nonlinear, as in general non-coaxial deformations, or the matrix material is nonlinear, as for nonlinear elastic, non-Newtonian viscous and plastic materials, the buckling response may be localised so that individual packets of folds form; the resulting fold profile is not sinusoidal. These folds are called localised folds. Most natural folds are localised. One view is that irregularity derives solely from initial geometrical perturbations. We explore a different view where the irregular geometry results from a softening material or geometrical nonlinearity without initial perturbations. Localised folds form in a fundamentally different way than the Biot wavelength selection process; the concept of a dominant wavelength does not exist. Folds grow and collapse sequentially rather than grow simultaneously. We discuss the formation of localised folds with recent considerations of constitutive behaviour at geological strain rates for general three-dimensional deformations. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The theory of buckling of a layer, or layers, embedded in another material has been studied in geology since the classical work of Hall (1815) although the concepts involved clearly go back nearly 100 years earlier to the Bernoulli family and to Coulomb and Euler. The folds produced by Hall in layers of cloth (Fig. 1a) are localised in the sense that they are not sinusoidal and occur in localised packets. Such irregularity has been widely recognised in the geological literature (Fig. 1, Abbassi and Mancktelow, 1990; Biot et al., 1961; Biot, 1965; Fletcher and Sherwin, 1978; Hudleston and Treagus, 2010; Johnson and Fletcher, 1994; Kocher et al., 2006; Mancktelow, 2001; Mancktelow, 1999; Price and Cosgrove, 1990; Ramberg, 1959; Schmalholz, 2006; Zhang et al., 1996; Zhang et al., 2000). In fact most experimentally produced folds are localised including, as a selection, those produced by Biot et al. (1961), Ghosh (1966), Hudleston (1973), Manz and Wickham (1978), Ramberg (1959), Ramsay (1967, Figs. 3–51, 7–29, 7–35), and Watkinson (1976). The present paper is concerned with such localised folding behaviour. Most studies of fold irregularity in the geological literature derive from the classical work of Biot in the period 1937–1984, some of which are brought together in Biot (1965). That body of work has had a profound impact on the way in which geologists view the folding process but one should appreciate that it is based on small deflections of thin, inextensible layers and linear stability ⁎ Corresponding author. E-mail address: [email protected] (A. Ord).

analyses employing linear constitutive relations and geometries. The results are strictly valid for infinitesimal deflections of layers embedded in linear materials such as linear (Hookean) elastic and/or Newtonian viscous materials undergoing coaxial deformations with no shear stress parallel to the layer, but there is a tendency in the geological literature to extend the results of the linear theory to describe finite non-coaxial deformations of thick, extensible layers with nonlinear matrix materials and nonlinear geometries (see Hudleston and Treagus, 2010). This paper is concerned with nonlinear aspects of buckling theory. In Hobbs and Ord (2012) we show that one would expect from the linearity of the classical Biot-problem that fold systems developed during coaxial deformations at finite strains would be sinusoidal or perhaps non-sinusoidal but still periodic. That paper also shows that the introduction of nonlinear constitutive relations in the form of nonlinear elasticity or strain-rate viscosity weakening behaviour leads to localised folding. The accompanying paper by Schmalholz and Schmid (2012) extends the examples to multilayered power-law viscous materials. Individual natural fold systems show a range of wavelength to thickness ratios ranging from about 2 to somewhere in the range of 15–20 (Table 1 in Hudleston and Treagus, 2010) and generally they lack periodicity (Fig. 1). This observation is commonly interpreted as an outcome from the Biot theory but representing a relatively flat dispersion relation (Biot et al., 1961) so that the wavelength amplification process is not very efficient at selecting a dominant wavelength and is influenced by the statistics of initial geometrical perturbations. It has been well documented that initial geometrical imperfections of sufficiently large wavelength are capable of inducing an irregular buckling response in linear materials undergoing coaxial deformations with no need to appeal to

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A. Ord, B. Hobbs / Tectonophysics xxx (2012) xxx–xxx

Fig. 1. Examples of localised folds. (a) Experimentally produced folds in layers of cloth (Hall, 1815). Model is ca. 1 m across. (b) Folds in fine grained slate, Cornwall. UK. Photo: Tim Dodson. (c) Folds. Cap de Creus. Spain. (d) Folds from Harvey's Retreat, Kangaroo Island, Australia.

nonlinear material or geometrical behaviour of any kind (Johnson and Fletcher, 1994; Mancktelow, 1999, 2001; Schmalholz, 2006; Zhang et al., 1996, 2000). This issue was considered in some depth by Biot (1965) and Biot et al. (1961) and an analytical solution to this problem is given for finite deformations by Muhlhaus et al. (1994, pp 228–231) for Newtonian viscous materials in coaxial deformation histories with constant velocity boundary conditions. That analysis shows that initial perturbations with a wavelength much smaller than the Biot wavelength will grow slowly compared to initial imperfections with wavelengths close to or larger than the Biot wavelength. Individual large wavelength imperfections result in localised fold packets. The issue is: Are all irregularities in natural fold systems derived from initial imperfections or are there other ways of inducing such irregularity? The influence of initial perturbations has been the overwhelming emphasis in the geological literature to the exclusion of considerations of softening nonlinear behaviour as it was for many years in the mechanics literature (Augusti, 1964; Budiansky and Hutchinson, 1964; Koiter, 1963; Thompson and Hunt, 1973; Ziegler, 1956). However it is widely recognised in the more recent mechanics literature that other mechanisms of inducing an irregular or localised buckling response involve the development of some form of softening nonlinearity in the geometry or constitutive behaviour of the materials involved and that this process is distinct from the Biot model (Tvergaard and Needleman, 1980; Whiting and Hunt, 1997). In this paper we explore situations where the fold irregularity arises as a natural consequence of the nonlinearity of perfect systems. The word perfect is meant to imply that there are no imposed imperfections in the system that result in localisation of deformation. If initial imperfections are present they result in a decrease of the load bearing capacity of the system and may control the site and rate

of growth of any localised response. However the presence of imperfections is not a necessary condition for the development of localised folding. In many systems, although the material behaviour remains linear, geometrical nonlinearities can arise, initially or at finite deformations, that induce unstable behaviour (Cross and Greenside, 2009; Hunt and Hammond, in press; Ortiz and Repetto, 1999) and so one could also ask: In the absence of suitable initial perturbations does nonlinear behaviour emerge in buckling systems at large deflections even in systems that are linear at small deflections? We have discussed this in some detail for coaxial deformation histories (Hobbs and Ord, 2012) and although some workers including Muhlhaus (1993), Schmalholz (2006), and Schmalholz and Podladchikov (2000) have reported broadening of the dispersion relationship and bifurcation behaviour at finite deflections of linear viscous materials it appears that no nonlinearity arises that leads to localisation. There is a possibility that amongst the many variables and assumptions inherent in the small deflection, thin layer, coaxial deformation theories there remains an aspect that induces non-sinusoidal behaviour in the absence of initial perturbations once thick layers and/or large deflections are taken into account but to date no such effects have been documented (Hobbs and Ord, 2012). We note however that all such work published to date neglects shear stresses parallel to the folding layer; the work in this paper shows that inclusion of such shear stresses for thick layer theories introduces what are essentially geometrical nonlinearities that lead to localisation in otherwise linear systems. One fundamental aspect of the Biot theory is that in restricting the discussion to linear materials, the force exerted on the folding layer by the embedding matrix must be a linear function of the deflection of the layer (for elastic materials) or, for viscous materials, a linear

Please cite this article as: Ord, A., Hobbs, B., Localised folding in general deformations, Tectonophysics (2012), http://dx.doi.org/10.1016/ j.tecto.2012.09.020

A. Ord, B. Hobbs / Tectonophysics xxx (2012) xxx–xxx Table 1 Symbols used in the text with units. Quantity

Description

Units, typical values

A A(t)

Temperature dependent material constant Function that measures the amplification of a layer deflection with time Bending rigidity of the plate A function which measures the reaction of the embedding medium to deflection External loadings tangential to the plate Changes in activation enthalpy associated with backward and forward jumps of dislocation kinks Change in activation enthalpy at 0 K Deformed thickness of a layer Wave number Wave-number of nth mode Boltzmann constant Length of a dislocation Length of plate Undeformed, deformed layer Resultant external moments Stress resultants in the von Karman equation Stress exponent in the viscous power law A number Parameters Non-dimensional axial load External load per unit area Matrix reaction force for the nth mode Time Volume change Width of dislocation kink Deflection of a layer Deflection, initial deflection, deflection rate of nth mode in a layer Position coordinate measured along length of layer Coordinates of a material point in the deformed, undeformed state Function of the axial force parallel to a layer as well as of the constitutive behaviour of a layer Function of the constitutive behaviour of a layer Angle between layer and X Shear strain rate, reference shear strain rate Wavelength Wavelength of nth mode Shear modulus, viscosity of embedding material Initial or reference shear stress, shear stress in deformed state Peierls stress at 0 K Critical stress separating thermally activated glide from viscous creep Amplification rate Amplification rate of the nth mode Axial stress; stress components Dilation angle Parameters that describe the growth of a localisation; all are a function of time A parameter

m s−1 m

D F(w,t) fx , fy ΔH−, ΔH+ ΔH0 h k kn K L L L; L′ mx, my Nx, Ny, Nxy N n p, q P q qn t ΔV w⁎ w wn, wn0, w_ n x X, Y α β φ γ_ ,γ_ 0 λ λn μ,η τ0, τ τ˜ τ^ ω ωn σ; σij Ψ ς1, ς2, ς3, ς4, ϑ1, ϑ2 ξ

Pa m3 m−3 s−1, m−3 Pa J J m m−1 m−1 J K−1 m m Pa m Pa m Dimensionless Dimensionless Dimensionless Dimensionless Pa Pa s m3 m m m m m m−2 s−1, m−2

Degrees s−1 m m Pa, Pa s Pa Pa Pa s−1 s−1 Pa Degrees

m−2 s

3

demonstrated (Hobbs and Ord, 2012) although the inclusion of shear stresses parallel to the buckling layers within the classical Biot theory does introduce geometrical nonlinearities and is worthy of consideration. (ii) Nonlinear behaviour of the embedding matrix is responsible for localisation as has been well documented in elastic, plastic and viscous materials by Tvergaard and Needleman (1980) and in a large number of papers by Budd and Peletier (2000), Budd et al. (1999, 2001), Hunt (2006), Hunt and Wadee (1991), and Hunt et al. (1989, 1996a,1996b, 1997b, 2000). See also Hobbs and Ord (2012) and Schmalholz and Schmid (2012). (iii) Geometrical nonlinearities induced by boundary conditions, in particular non-coaxial deformations, or system behaviour may be responsible for localisation. Examples are presented by Hunt et al. (1989), Whiting and Hunt (1997) and Hunt and Hammond (in press). We explore the last two possibilities in this paper. We first discuss the concept of localised folding (Section 2) and draw out the distinctions between Biot folding and localised folding. We then (Section 3) visit recent developments in modelling the constitutive behaviour of materials deforming by dislocation motion and point out (following in particular Cordier et al., 2012; and Kocks, 1987) that the stress exponent, N, commonly used in power-law viscous materials may have larger values under geological strain rates than those observed experimentally. This introduces the possibility that rocks behave more like rate insensitive plastic materials than the rate sensitive viscous materials commonly assumed from laboratory deformation experiments. This possibility of plastic behaviour is worthy of considerable study and has been considered recently by Schmalholz and Maeder (2012) with respect to boudin development. In particular the volume changes associated with metamorphic reactions suggest a non-associative plastic response (Hobbs et al., 1990; Vermeer and de Borst, 1984) and hence the formation of shear bands in the embedding material during folding. We also show that these volume changes can be responsible for chevron-style folding in some multilayer materials. The treatment is then extended to numerical examples (Section 4) involving nonlinear reaction forces exerted by the embedding material, in particular those arising from rate insensitive plastic materials. We emphasise the influence of boundary conditions in inducing localised behaviour in constitutively linear materials as it arises in coaxial deformations to large deflections, deformations involving shortening parallel to the layer and simultaneous shearing normal to the layer and in simple shearing deformation histories. As expected from a large amount of work on nonlinear elastic materials (Hunt, 2006) the result is a rich set of non-Biot localised and chaotic responses both in the folding layers and in the embedding material in the form of crenulations and of axial plane structures. The notation used in this paper is summarised in Table 1. 2. Governing equations for folding in general deformations

function of the rate of deflection (Biot, 1937, 1965; Hunt et al., 1996b; Whiting and Hunt, 1997). One purpose of this paper is to consider nonlinear reaction forces exerted by the embedding material on the folding layer as would arise if the embedding material is nonlinear elastic, power-law viscous or plastic. However the main emphasis is to consider geometrical nonlinearities arising from general, noncoaxial deformations. Thus, departures from sinusoidal folding can conceivably have three origins other than the growth of initial imperfections: (i) The strict Biot theory when extended to include large deflections and thick plates may introduce nonlinearities that cause the system to localise or the dispersion curve may broaden with increased strain as proposed by Muhlhaus (1993) and Schmalholz and Podladchikov (2000). To date no such effects have been

Consider a single thin layer of linear elastic, viscous or visco-elastic material embedded in a matrix of a weaker material. The embedding material can exert forces both normal and tangential to the layer as it deforms. Biot (1965) showed that if this system is shortened coaxially then the initial flat (but with small perturbations) configuration of the layer is unstable and folds develop provided the contrast in mechanical properties between the layer and matrix is large enough. This problem is linear with respect to constitutive behaviour and also geometrically. Our approach in this paper is to understand the simplest of geometrical situations before proceeding to more complicated multilayered geometries or geometries involving initial imperfections. Our main goal is to investigate the influence of nonlinear behaviour, both in the matrix and arising from the boundary conditions, on the development of folds. We also want to understand the influence of any initial

Please cite this article as: Ord, A., Hobbs, B., Localised folding in general deformations, Tectonophysics (2012), http://dx.doi.org/10.1016/ j.tecto.2012.09.020

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geometrical irregularities on the form of the folds that develop with a nonlinear matrix. Again, in order to advance to this level of complication we need first to understand the situation in well prescribed, simple systems with no initial geometrical imperfections. In the literature on folding of layered materials two different types of deformation responses are identified. We call these two kinds of behaviour Biot folding and localised folding. The first originates from the work of Biot (1937, 1965) where thin layers of a material with linear constitutive behaviour (elastic or Newtonian viscous or combinations of the two) are inter-layered with a material (which we call the matrix or embedding medium) with different linear constitutive behaviours and shortened coaxially. In these models the force parallel to the initial orientation of the layers remains constant so that at least for small deflections, w, the shear stresses parallel to the layers can be neglected and the whole system remains linear in both material and geometrical sense. No softening behaviour arising from constitutive or geometrical origins or from boundary conditions is considered. The result is that the governing equation, for small deflections (by small is meant limb dips of less than ca. 10°, see Hobbs and Ord, 2012), is: 4

2

∂ w ∂ w þ α 2 þ F ðwÞ ¼ 0 ∂x4 ∂x

ð2:1Þ

where α is a function of the axial force parallel to the layer and of the constitutive behaviour of the layers and F(w) is a function that represents the reaction force on the layer from the matrix as the layer deflects; x is the distance measured along the layer from some reference
2 are point (Fig. 2). In deriving Eq. (2.1) terms of the order of ∂w ∂x neglected (Hunt et al., 1997b; Thompson and Hunt, 1973). Eq. (2.1) is the stationary form of the Swift–Hohenberg equation (Cross and Greenside, 2009; Swift and Hohenberg, 1977): ∂w ∂4 w ∂2 w ¼ 4 þ α 2 þ F ðwÞ: ∂t ∂x ∂x The term stationary refers to situations where ∂w ¼ 0. This equa∂t tion has been studied extensively over the past 35 years and has become one of the archetype equations that describe pattern formation in nonlinear systems (Cross and Greenside, 2009; Cross and Hohenberg, 1993). Notice that Eq. (2.1) has three forms of solutions

Fig. 2. Details of buckling model. Coordinates in the undeformed state are (X, Y). In the deformed state the intrinsic coordinate x is used and measured parallel to the deformed layer. w is the deflection of the layer and F(w) is the reaction force exerted by the embedding medium upon the buckling layer. This reaction force is a linear function of w for linear matrix materials and is nonlinear for other materials such as power-law viscous and plastic materials.

(Hunt et al., in review). If we write w = exp(ξx) as a solution to Eq. (2.1) and F(w) = w then the characteristic equation is 4

2

ξ þ αξ þ 1 ¼ 0: Hunt et al. (in review) point out that the solutions to this equation are of three classes: (i) If α b − 2 solutions are localised and resemble kink bands. (ii) If − 2 b α b 2 solutions are localised and consist of solitary waves. (iii) If α ≥ 2 solutions are periodic and of the form (Eq. (2.2)) below. This delineation into three classes turns out to be important when one considers general deformations. For coaxial deformations α can commonly be identified with, or is a function of, the axial load. This discussion highlights the fact that even for linear constitutive relations (2.1) can lead to localised behaviour depending on the controls on α. Eq. (2.1) has been derived in various ways, including both fluid- and solid-mechanics approaches for the buckling problem, by a large number of authors including Biot (1965), Jaeger (1969), Ramberg (1963), Smith (1977), Thompson and Hunt (1973) and Turcotte and Schubert (1982). In two dimensions the layer deflection, w, is a function of both x and time, t. For the linear problem described above F is always a linear function of w and depends also on the wavelength of the deflection (Biot, 1937). One can readily confirm that if F is linear and for α ≥2 one solution to Eq. (2.1) is: wðx; t Þ ¼ Aðt Þ sinðkxÞ

ð2:2Þ

where k is the wave-number for that deflection and k ¼ 2π λ where λ is the wavelength of the deflection. A(t) is a function that describes the way in which the amplitude evolves with time. Biot (1965) shows that A(t) = w 0 exp(ωt) where w0 is some initial deflection of the layer and ω is the amplification rate of the deflection that arises. Because the problem is linear the system obeys the mathematical Law of Superposition (Boyce and DiPrima, 2005) so that the solution to Eq. (2.1) can be expressed as a Fourier series which is a linear combination of an infinite number of solutions of the form of Eq. (2.2) with different values of wn0 and ωn corresponding to each wave-number, kn ¼ 2πn λ : wðx; t Þ ¼

∞ X

0

wn expðωn t Þ sinðkn xÞ:

ð2:3Þ

n¼1

The linearity of the system, resulting in Eq. (2.3) as a solution to Eq. (2.1), means that each mode, characterised by a particular kn, grows independently of all other modes. Biot (1965) showed that the dispersion curve (the plot of ωn against kn) is bell shaped, skewed towards smaller wave-numbers, and sharply peaked so that one wavelength grows faster than all others; this wavelength is called the dominant wavelength. This wavelength selection process, together with Eq. (2.2), means that the folds that grow according to this theory are always sinusoidal as long as Eq. (2.1) holds with α ≥ 2 and F(w) is linear. We refer to this first type of folding as Biot folding. Fletcher and Sherwin (1978), following Biot et al. (1961), proposed that departures from a sinusoidal form that are commonly observed in natural folds arise from a broad shape for the dispersion curve meaning that the spread of growing wavelengths is large and the wavelength selection process is not very efficient. The issue is considered in some depth by Biot et al. (1961). This means that several terms from Eq. (2.3) appear in the final solution instead of just one (the dominant wavelength) and the fold systems that arise from simple broadening of the dispersion curve are therefore periodic or quasi-periodic and in general are no longer sinusoidal. This follows   since the sum of functions such as sin nπx have a fundamental period λ

Please cite this article as: Ord, A., Hobbs, B., Localised folding in general deformations, Tectonophysics (2012), http://dx.doi.org/10.1016/ j.tecto.2012.09.020

A. Ord, B. Hobbs / Tectonophysics xxx (2012) xxx–xxx 2λ min½n (Boyce and DiPrima, 2005, p 577) where min[n] means the minimum value of n. Quasi-periodicity would correspond to a situation where the quotient of some wave numbers is irrational. Since many natural folds are aperiodic and are commonly localised, simple broadening of the dispersion curve seems an unlikely explanation for the shapes of natural fold systems. This statement is true if any initial imperfections have wavelengths that are small compared to the Biot wavelength. An analytical expression for the growth rates of initial imperfections is given by Muhlhaus et al. (1994) and this is discussed in detail in Hobbs and Ord (2012). Initial imperfections greater in size than the Biot wavelength, tend to grow fastest with a maximum growth rate at the Biot wavelength. Hence if large wavelength imperfections are initially distributed irregularly then irregular folds will grow. This analytical result is true only for Newtonian viscous embedding materials and for coaxial deformations. There is no analytical solution for nonlinear embedding materials but similar effects are to be expected and are confirmed for nonlinear materials by the work of Mancktelow and Abbassi (1992). The second type of folding behaviour identified in the literature follows naturally as a generalisation of Biot folding and arises if the reaction of the matrix to deflection of the layer is no longer linear as occurs if the matrix softens in some manner. F is then no longer a linear function of w and Eq. (2.2) is not a solution to Eq. (2.1). The result is that a linear combination of terms such as Eq. (2.3) can no longer be written as a solution to Eq. (2.1). Thus, in general, a wavelength selection process based on the preferred growth of Fourier components does not operate and the process is quite different to Biot folding. The solutions to Eq. (2.1) are now non-sinusoidal or localised along the layer as shown in Fig. 3 and very sensitive to the magnitude of α in Eq. (2.1) and the form of F; many examples are given in the book devoted to the solutions of Eq. (2.1) by Peletier and Troy (2001). This is known as localised folding; the solutions to Eq. (2.1) are also known as solitary waves. As an example, if F(w) = w − w 2 then Champneys and Toland (1993) show that there are an infinite number of localised solutions to Eq. (2.1). Some of the possible solutions to the stationary nonlinear Swift–Hohenberg equation are shown in Fig. 3. An expression for some of these localised solutions to Eq. (2.1) with F(w) = w − w 3 is given by (Hunt et al., 1996b):

w ¼ ς 1 sechðϑ1 xÞ cosðϑ2 xÞ þ ς 2 sechðϑ1 xÞ tanhðϑ1 xÞ sinðϑ2 xÞ þ ς3 x sechðϑ1 xÞ sinðϑ2 xÞ þ ς 4 x sechðϑ1 xÞ tanhðϑ1 xÞ cosðϑ2 xÞ

ð2:4Þ

Fig. 3. Some of the possible solutions to the stationary nonlinear Swift–Hohenberg equation.

5

and an example of the time evolution of this solution together with the time evolution of the axial force for constant velocity boundary conditions is given in Fig. 4. The values of the parameters ς1,ς2,ς3,ς4, ϑ1,ϑ2 (all of which vary with time) are taken from Fig. 5 of Hunt et al. (1996b). Notice that F(w) = w − w 3/6 is an approximation to F(w) = sinh −1(w) which is a function commonly taken to represent experimentally observed nonlinear elastic response (Whiting and Hunt, 1997). If packets of folds interfere with one another then chaotic folding can arise (Champneys et al., 1999; Hunt et al., 1996b). If processes other than buckling operate and are coupled such as in the development of axial plane structures (Hobbs and Ord, 2012) then the system response is complex. An important distinction between the processes involved in Biot buckling and in localised fold formation is that in the Biot theory all folds form at the same time and all folds with the dominant wavelength grow at the same time at the same rate. In contrast, localised folds form at different times so that there is a sequential development of folds; some folds may form only to collapse later in the deformation history or folds can grow in what were formerly planar regions (Budd and Peletier, 2000; Budd et al., 1999, 2001). The Biot wavelength selection process does not occur. If the processes operating during buckling are described by a set of coupled Swift–Hohenberg equations or by coupled Swift–Hohenberg/reaction–diffusion equations then complex fold systems can form with multiple structures intertwined with each other (Hobbs and Ord, 2012). This presumably is the case in fold systems comprising layers with different nonlinear behaviours or layers buckling with metamorphic reactions coupled to the mechanical behaviour. Although there is a fundamental difference in the processes that operate during the development of Biot and localised fold systems, Biot folding can be (but is not always) the precursor for localised folding. Thus Tvergaard and Needleman (1980) show that for softening plastic materials under constant loading rate, the stress in the layer first rises to a maximum where sinusoidal folds develop; this we identify with Biot folding. Further, Tvergaard and Needleman show that as the stress within the plastic layer decreases due to softening, the system behaviour bifurcates and a localised fold system develops only to be replaced later in the deformation history by a sinusoidal fold system with a wavelength different to that at a peak load. An example where a localised system evolves towards a sinusoidal system is given by Knobloch (2008). Many examples of this sequential development of localised folds are given by Budd and Peletier (2000) and Budd et al. (2001; 1999) and an experimental example is given by Watkinson (1976). This kind of behaviour is explained, in the spirit of Thompson and Hunt (1973), by the system seeking at any particular time the lowest energy configuration. In linear systems there is but one energy minimum and this corresponds to a sinusoidal solution. For nonlinear systems the energy landscape has many or perhaps an infinite number of local minima and the system jumps from one localised solution to another as deformation proceeds (Burke and Knobloch, 2007). The folding response may be fractal (Hunt et al., 1996a). Occasionally the local minimum also corresponds to a new sinusoidal solution. At present the mathematics that enables analytical descriptions of these processes has not been developed (Knobloch, 2008). Although analytical solutions may sometimes be obtained they may not be unique or stable and an infinite number of other solutions may exist. This is particularly true for solutions obtained by some form of linear perturbation analysis (Fletcher, 1974; Smith, 1977). A more recent approach (Burke and Knobloch, 2007; Knobloch, 2008) is to explore the phase space appropriate to the system and search for bifurcations using software such as AUTO (Doedel et al., 2002) or XPPAUT (Ermentrout, 2002). It is notable that computer simulations based on finite element or finite difference approaches commonly do not pick many of the solutions to these nonlinear problems (Hunt et al., 1997a). Examples of localised folding are presented in Section 4.

Please cite this article as: Ord, A., Hobbs, B., Localised folding in general deformations, Tectonophysics (2012), http://dx.doi.org/10.1016/ j.tecto.2012.09.020

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A. Ord, B. Hobbs / Tectonophysics xxx (2012) xxx–xxx

Fig. 4. Some localised solutions to Eq. (2.4) when F(w) is nonlinear. The boundary conditions are constant velocity and the deformation is coaxial shortening. P is the non-dimensional axial load and t is the non-dimensional time. As P decreases the response becomes more localised. These solutions correspond to Eq. (2.4) with values of the various time dependent parameters taken from Hunt et al. (1996b).

The assumptions made here are:

2.1. General non-coaxial deformations In common with most discussions of buckling in the geological literature the above discussion assumes that the deformation of the system is coaxial and that the layer is thin. This means that for small deflections of the embedded layer the shear stress parallel to the layer can be neglected as can any buckling moments arising from such shear stresses. However, most deformation histories in geology are non-coaxial. For finite deflections of a thick layer and for more general deformations of thick layers, shear stresses parallel to the layer are always present, imposed by the embedding matrix, and may be finite in magnitude from the instant deformation begins. This is the situation for instance in a simple shearing deformation of a layer embedded in a weaker matrix (Fig. 5b). As the layer begins to buckle the shear stress, τ, parallel to the layer becomes a function of the deflection of the layer. One way to understand the buckling of a layer in a three dimensional general deformation is to consider the von Karman (1910) equation for the deflection, w, of an elastic plate undergoing large deflections in a general three dimensional deformation. A derivation of this equation is given by Fung (1965, pp 463–470):

(i) (ii) (iii) (iv)

The thickness, h, is much smaller than the length of the plate, L. The deflection of the plate, w, is of the same   order as h.   ∂w The slope is everywhere small, ∂w ≪1;  ∂y ≪1: ∂x Tangential displacements parallel to the plate are infinitesimal. This means that only terms such as ∂w ; ∂w are retained in the ∂x ∂y strain–displacement relations. (v) All strain components, as measured by the Lagrangian Green strain tensor, are small and Hooke's law holds. (vi) The plate is a Kirchoff plate (Hobbs and Ord, 2012) which means that within the plate, tractions on the mid-plane of the plate are negligible and strains vary linearly within the plate thickness.

These assumptions make it clear that any solution to Eq. (2.5) is relevant only to small deflections. In the von Karman Eq. (2.5), Nx, Ny, Nxy (with dimensions Pa m) are the stress resultants:

h

=2

h

=2

h

=2

Nx ¼ ∫ σ xx dz; Nxy ¼ ∫ σ xy dz; N y ¼ ∫ σ yy dz; ∂4 w ∂4 w ∂4 w þ 2 þ ∂x4 ∂x2 ∂y2 ∂y4 " # 2 2 2 1 ∂mx ∂my ∂ w ∂ w ∂ w ∂w ∂w qþ ¼ þ þ Nx 2 þ 2Nxy þ Ny 2 −f x −f y D ∂x ∂y ∂x∂y ∂x ∂y ∂x ∂y ð2:5Þ where (x, y, z) are coordinates relative to the undeformed (flat) plate with z normal to the undeformed plate (Fig. 5a which also shows the sign convention).

−h =2

−h =2

−h =2

q is the external load per unit area (units Pa) applied normal to the mid-surface, mx and my are the resultant external moments (with units Pa m) about the mid-surface arising from shear tractions parallel to the plate surface: mx ¼ hσ zx ; my ¼ hσ zy

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This means that my = fy = 0 and reduces to

7 ∂w ∂y

¼ 0. The von Karman equation

" # ∂4 w 1 ∂mx ∂2 w ∂w qþ ¼ þ Nx 2 −f x : D ∂x ∂x ∂x4 ∂x With the sign conventions given in Fig. 5a in mind and with no shear stresses, the force parallel to the edges of the plate is compressive (so that Nx is negative and we write Nx = − σh where σ is the compressive stress) and the reactive stress normal to the plate is q = − F this reduces to the stationary form of the Swift–Hohenberg equation (cf. Eq. 2.1): 4

2

∂ w σh ∂ w F þ þ ¼ 0: D ∂x2 D ∂x4 With a shear stress parallel to the layer we have in two dimensions: D

  4 2 ∂ w ∂ w ∂mx ∂w þ σh 2 þ F ¼ −f x : 4 ∂x ∂x ∂x ∂x

ð2:6Þ

An equation of this form has also been derived by Hunt et al. (in review) and by Sridhar et al. (2002). One can confirm that Eqs. (2.2)

Fig. 5. A layer embedded in another weaker material undergoing a general deformation. (a) Stresses on the plate in a general three dimensional deformation showing sign convention. The stresses σzz, σzx, σzy are induced by deformation of the embedding medium. (b) Variation of the shear stress on a layer, L, embedded in a weaker matrix, as folds nucleate during a simple shearing deformation in two dimensions. Undeformed state is referred to coordinates (X, Y). Coordinate axes in deformed state, L′ , are (x, y), layer thickness in deformed state is h. w is the layer deflection normal to its initial length. The shear stress on the layer in the undeformed state is τ0 and τ in the deformed state. The layer is rotated through the angle φ during deformation. The inset shows the initial conditions comprising a layer embedded in a weaker matrix.

fx, fy are the external loadings (with units Pa) tangential to the plate: f x ¼ σ zx ; f y ¼ σ zy and D (with units Pa m 3) is the bending rigidity of the plate and is related to the elastic moduli and thickness of the plate. In the above, body forces are neglected and the stress components, σij, are components of the Kirchhoff stress tensor (Fung, 1965). The von Karman equation can be extended to power-law viscous materials (Muhlhaus et al., 1998; Sridhar et al., 2002). The nonlinearity of the von Karman equation (Von Karman, 1940; Von Karman and Biot, 1940) means that even after 100 years, solutions to the equation are few and far between. An example is Muhlhaus et al. (1998) who solve the equation for a power-law viscous plate embedded in a Newtonian viscous material. A difficulty is that the deflection is always a function of both x and y even for plane deformations. This means that the resulting buckling pattern in three dimensions always involves curved fold axes as is ubiquitous in natural examples. We want to examine a two dimensional situation and an approach is to take a slice through a three dimensional model where we make the assumptions that the buckle axes parallel to y are locally straight and parallel to y and there is no imposed shear loading parallel to y.

Fig. 6. Dependence of power-law exponent N on stress. (a) Proposed relationship by Kocks (1987) between shear strain rate and shear stress. τ^ defines the boundary between a region where dislocation motion is controlled by thermally activated kink migration so that the behaviour is rate insensitive and a region where dislocation motion is dominated by viscous glide and the behaviour is power-law viscous. (b) Summary of relationships. At low stresses the behaviour is relatively rate insensitive with large values of N. Within the experimentally accessible region values of N are in the range of 1 to about 5. At high stresses the dislocation velocity saturates near the velocity of sound and the behaviour is again characterised by large values of N (motivated by Cordier et al., 2012).

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and (2.3) are no longer solutions to Eq. (2.6) even if F(w) is a linear function of w. We discuss below a simple shearing deformation of a thick layer where a shear stress is present parallel to the layer, imposed by the shearing embedding material, from the start of the deformation. First we clarify what is meant by small deflections. The approximation that is made in all of the previous discussion
and 2 in the derivation of Eq. (2.1) by all authors is that sin2 φ≈ ∂w ≪1. For φ=10o, ∂x 2 o 2 sin φ=0.03 and for φ=20 , sin φ=0.1 so we take 10° as perhaps the value of φ where nonlinearities due to shear stresses become important. This approximation is the same as that needed for Eq. (2.1) to be a reasonable approximation for the general equation of buckling (Hobbs and Ord, 2012; Thompson and Hunt, 1973). The value of φ=15° proposed by Chapple (1968) from different arguments is quite reasonable. For values of φ say greater than about 10° nonlinear behaviour is expected to dominate even for Newtonian viscous materials and localised solutions are expected. 2.2. Simple shearing deformations If a layer undergoing a simple shearing deformation with an initial shear stress τ0 undergoes a small deflection, w, so that it is rotated through an angle φ (Fig. 5b), then the shear stress parallel to the deflected where x is measured parallel to layer is τ=τ0 cos 2φ and using sinφ ¼ ∂w ∂x 
2  and, the layer in the deformed state we arrive at τ ¼ τ 0 1− ∂w ∂x neglecting terms in

D


2 ∂w ∂x

, Eq. (2.6) becomes

" # ∂4 w ∂2 w ∂w ∂2 w þ Ph þ F ¼ τ þ 1 : −2h 0 ∂x ∂x4 ∂x2 ∂x2

ð2:7Þ

Eq. (2.2) is clearly not a solution to Eq. (2.7) even if F(w) is a linear function of w and localised solutions are expected even at small deflections. In the examples presented in Section 4, the F(w) term in Eq. (2.7) represents any constitutive nonlinearity whereas the term on the right hand side of Eq. (2.7) represents a geometrical nonlinearity. Thus localised solutions to the folding problem are expected in general non-coaxial deformations and for large deflections in coaxial deformations of thick layers even for linear elastic and viscous materials where F(w) is linear. Such localised behaviour is well documented in simple shearing experiments by Ghosh (1966), Manz and Wickham (1978) and Ramberg (1959) and by the general deformation modelled by Watkinson (1976) although it is unlikely that the materials used in those experiments exhibited linear constitutive behaviour. 3. Constitutive relations for geological materials It is commonly assumed that the mechanical behaviour of rocks can be extrapolated from those established under laboratory conditions to the strain rates and stresses operating under geological deformation conditions although the pitfalls in doing so have been discussed by Paterson (1987, 2001) and Poirier (1985). This means that the power-law viscous behaviour of many geological materials established in the laboratory at strain rates of 10 −4 s −1 to perhaps 10 −8 s −1 is routinely extrapolated and used to model mechanical behaviour at strain rates of 10 −9 s −1 to 10 −15 s −1. Kocks (1987) and Kocks et al. (1975) pointed out that in metals the relationship between stress and strain rate can be divided into two regimes (Fig. 6a). First there exists a high stress regime (where the stress is larger than a critical value, τ^ , which is temperature dependent) where the relationship is of the typical power-law form: γ_ ¼ γ_ 0



τ τ0

N

ð3:1Þ

γ_ is the plastic strain rate, τ is the shear stress and N is a number which for minerals is typically between 1 and about 5; γ_ 0 is the strain rate when the shear stress is τ0 and is temperature dependent. This regime Kocks calls the viscous glide regime. Secondly, if τbτ^ , dislocation motion is controlled by thermally activated drift and is essentially rate insensitive. This is typical of behaviour at low stresses and low strain rates. The material behaviour can be described by a power law of the form (3.1) but N is now very large compared to 5 and in the limit N → ∞ so that the material is perfectly plastic although still thermally activated so that γ_ 0 is a function of temperature. This regime is the thermally activated regime. An assumption that powerlaw mechanical behaviour established at laboratory accessible strain rates can be used at slow geological strain rates means that one assumes that N is independent of strain rate whereas the Kocks argument proposes that N is a function of strain rate and becomes large at low stresses (and hence small strain rates). The above proposition of Kocks (1987) is impossible to confirm for rocks in the laboratory since the strain rates required are experimentally inaccessible. However there is no theoretical justification for Eq. (3.1) except for N = 1 and N = 3 so that extrapolation to geological strain rates is more a matter of optimism and convenience than being based on theoretical analysis. Recently however a number of studies including those of Cordier and coworkers (Amodeo et al., 2011; Carrez et al., 2009; Cordier et al., 2012), Monnet et al. (2004), Naamane et al. (2010), Nabarro (2003), and Tang et al. (1998) have modelled the relationship between dislocation velocity and stress for various materials at various temperatures and pressures using molecular and dislocation dynamics and compared the simulated results with experimental observations. Both the viscous glide and thermally activated regimes of Kocks (1987) are recognised. The above work proposes that Eq. (3.1) should be replaced by γ_ ¼ A

−

þ L ΔH ðτÞ−ΔH ðτÞ sinh KT w ðτÞ2

ð3:2Þ

where A is a temperature dependent material constant that depends on dislocation geometry and kink structure, L is the dislocation length, w* is the width of dislocation kinks and is stress dependent, ΔH − and ΔH + are the changes in activation enthalpies associated with dislocation–obstacle interactions for backward and forward jumps of the dislocation kinks, T is the absolute temperature and K is the Boltzmann constant. Notice that ΔH is a function of stress. An expression for ΔH, given by Kocks et al. (1975), is p q  τ ΔH ðτÞ ¼ ΔH0 1− τ˜

ð3:3Þ

where ΔH0 is the activation enthalpy at 0 K, τ˜ is the Peierls stress at 0 K and p, q are parameters such that 0 b p ≤ 1 and 1 ≤ q ≤ 2. Values of p and q are obtained from experiments and/or from computer simulations but Kocks et al. (1975) suggest that p = 0.5 and q = 1.5 are representative values. Cordier et al. (2012) have explored the behaviour of MgO at various pressures, strain rates and temperatures and show that under geological conditions MgO is likely to behave as a material with large values of N to the extent that it may resemble more a perfectly plastic, rate insensitive material than a power-law material with relatively small values of N. At the other extreme of behaviour at high stresses and/or high strain rates dislocation velocity is limited by the velocity of sound in the crystal so that the constitutive behaviour at high strain rates is again approximated by a power law with large values of N (Fig. 6b). This discussion indicates that there is some value in exploring the behaviour of folding systems for matrix values of N in the range 5 ≤ N ≤ ∞. In particular for plastic constitutive behaviour the volume

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changes resulting from metamorphic reactions introduce the possibility of non-associative plastic constitutive behaviour and hence the development of localised deformation in the matrix. Such behaviour is otherwise not permitted for matrix constitutive behaviour of the form (3.1) with small values of N. The term plastic is reserved for materials that possess a yield surface (Hobbs et al., 1990). For stress states within the yield surface the material is elastic and plastic states occur on the yield surface. The shape of the yield surface depends on temperature, strain and strain-rate and the yield stress may follow Eq. (3.1). Following Cordier et al. (2012) we consider a yield surface that is rate independent. We also only consider isothermal situations with no hardening or softening of the yield stress. Our material has a Tresca yield criterion (Jaeger, 1969) but the same results are obtained for a Von Mises yield criterion. In addition, plastic volumetric changes can be represented using another surface, the potential surface (Vermeer and de Borst, 1984), which is parallel to the yield surface when there is zero dilation. To represent volume changes a quantity called the dilation angle, ψ, is defined and given by sin ψ = (plastic dilation rate)/(plastic shear rate) . For the perfect plastic material used here the condition ψ = 0 is known as associated plasticity whereas the condition ψ > 0 is known as non-associated plasticity. We represent metamorphic reactions that involve a positive plastic volume change by perfect non-associative plasticity here. The materials used also have positive or negative elastic dilations depending on the value of the elastic bulk modulus and the deviatoric stress state.

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4. Some examples 4.1. Localised folding examples for coaxial deformation histories Biot (1937) showed that for a coaxial deformation history the response of a linear elastic material to the buckling behaviour of a layer is proportional to the deflection of the layer and to the wave-number of the buckle. For a series of n superposed buckles with wave-numbers kn ¼ 2nπ λ , where λ is a wavelength, the reaction force, qn, of the matrix for the nth mode is qn ¼ −4μkn wn

ð4:1Þ

where μ is the shear modulus of the matrix and wn is the deflection associated with the nth mode. The resulting buckling system is linear and the response is a single sinusoidal fold system (Biot, 1965). A similar argument follows for a Newtonian linear viscous matrix so that _n qn ¼ −4ηkn w

ð4:2Þ

_ n is the deflection rate where η is the viscosity of the matrix and w associated with the nth mode (Hobbs and Ord, 2012; Muhlhaus et al., 1998). Again, if one neglects the shear stress parallel to the layer, the resulting buckling system is a single sinusoidal fold system (Muhlhaus et al., 1998). However if the matrix is composed of

Fig. 7. Single layers of Newtonian viscous material embedded in power-law viscous materials with N indicated. (a–d) The models have been shortened parallel to the initial orientation of the layer. (e–f) The models have been shortened parallel to the initial orientation of the layer and sheared sinistrally through about 10°. Bulk shortening in each case is (a) 51% (b) 45% (c) 31% (d) 31% (e) 45% (f) 34% (g) 40% and (h) 45%. The initial viscosity ratio between layer and embedding material is 100 in each case. Localised folding develops in all cases including the constitutively linear case (N=1).

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material with a nonlinear constitutive relation then the response of the matrix material to deflections of the layer is no longer a linear function of the deflection or deflection rate and Eqs. (4.1), (4.2) cannot be used. Moreover, there is no guarantee that individual modes will grow independently of each other so that treatment of individual superimposed deflections in terms of a Fourier series is not possible in general (Hunt et al., 1997b). For most cases the response is now a localised fold system although in some cases such as the nucleation of the fold system, where the displacements are very small and the system can be taken to be linear, the embryonic fold system may be sinusoidal and can be described by a treatment identical to that of Biot (1965). However, as we have indicated in Section 2, if shear stresses parallel to the buckling layer contribute to a buckling moment then a geometrical nonlinearity arises that may induce localised solutions for linear materials. These behaviours are illustrated in Fig. 7 where a layer of Maxwell material (N = 1) is embedded in materials with values of N = 1, 2, 5, 10 and no initial perturbations are included in the model. The computational code used is the finite difference code, FLAC2D (ITASCA, 2008). Experience from other workers (for instance, Hunt et al., 1997a) suggests that such a code is not capable of detecting all of the possible solutions to nonlinear problems as are posed here. For all values of N localised folds form and even for N = 1 the folds are localised. All of these folds nucleate as aperiodic structures so that the Biot solution for N = 1 and the Fletcher–Smith solutions for N > 1 do not hold at fold nucleation (Fletcher, 1974; Smith, 1977). The important control is exerted by the value of τ^ , the critical stress separating thermally activated glide from viscous creep, as defined by Kocks et al. (1975). If τ^ is small (as would be the situation corresponding to high temperatures) then the fold growth rate is small whereas if τ^ is large (corresponding to lower temperatures) then the growth rate is larger (Figs. 8 and 9).

4.2. Localised folding examples for simple shearing deformation histories Fig. 10 shows the development of single layer localised folding in (sinistral) simple shearing deformations. The results are similar to those reported by Llorens et al. (in press) except that initial perturbations are included in that study whereas we explore systems with no added initial perturbations. The matrix materials are power-law viscous for Fig. 10(a, b, c, d) with N = 1, 2, 5 and 10 respectively. In Fig. 10(e, f) the matrix material is purely plastic with dilation angles of 0° and 12° respectively. The undeformed state for all of these is as shown in Fig. 11(a) so that the layer is in the incremental shortening field for all examples shown here. There are no initial perturbations included in these models. Since there is a shear stress on the layer at the instant of fold nucleation, all of these situations show the development of localised folding including the case for N = 1. Also shown are contours of the instantaneous shear strain; this is a measure of the instantaneous buckling growth rate and this increases gradually from N = 2 to N = 10 and to the plastic case with no dilation. The plastic matrix with no dilation shows contours of instantaneous shear strain localising weakly along the shearing direction whilst the plastic matrix with ψ = 12° shows strong localisation approximately parallel to the shearing direction. Details of this localisation are shown in Fig. 11. 4.3. Influence of volume changes associated with mineral reactions Metamorphic reactions are invariably associated with volume changes either directly through the ΔV of the reaction and/or through the release or production of volatiles such as H2O. If such volume changes occur during plastic deformation the dilation is coupled directly to the deformation and the constitutive behaviour is said to be non-associative (Hobbs and Ord, 1989;

Fig. 8. Fold behaviour as a function of the magnitude of τ^ . (a) If τ^ is small, corresponding to high temperatures then fold amplification is small. (b) If τ^ is large, corresponding to low temperatures, fold amplification is high.

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Fig. 9. Fold behaviour as a function of τ^ for coaxial deformation of a layer. (a, b) N = 5; (c, d) N = 10. For low values of τ^ (a, c) low intensity shears develop in the matrix but fold growth is slow. At high values of τ^ (b, d) periodic deflections of the layer develop at low amounts of shortening as revealed by the distribution of instantaneous shear strains. At higher amounts of shortening these periodic deflections collapse into localised distributions of instantaneous shear strain and fold localisation. Keys show minimum and maximum magnitudes of contours of shear strain increment for a, b and c, and full range for d.

Vermeer and de Borst, 1984). Such materials are prone to localisation if the potential surface representing the deformation arising from dilation is not parallel to the plastic yield surface. In Figs. 10(f) and 11 we present the results of simple shearing experiments for plastic materials with a Tresca yield surface and a dilation given by sin ψ = 0.2 where ψ is the dilation angle so that in this case ψ = 12°. One can see that localised deformation zones develop at an angle of ψ/4 to the shear planes as predicted by non-associative plasticity theory (Hobbs et al., 1990; Vermeer and de Borst, 1984). These localised shearing zones comprise packets of crenulations as shown in Fig. 11(b), (d). In these models (and Figs. 12, 13), the yield stress of the plastic layers is 107 MPa and the viscosity of the viscous layers is 1021 Pa s. In Fig. 12 we show the response to a simple shearing deformation of multilayer stacks of alternating elastic–plastic and Maxwell materials with different thickness ratios for the layers. The initial undeformed state is as shown in Fig. 11. In Fig. 12(a, c, e) there is no plastic volume

change whilst in Fig. 12 (b, d, f), ψ = 12°. Localised structures develop in all cases but for situations with no plastic dilation the fold hinges tend to be rounded; for those involving plastic volume change the hinges are sharper and the folds resemble more chevron folds. Fold axial planes initiate normal to the principal axis of the deformation rate and rotate towards the shearing plane as deformation proceeds. 4.4. Influence of imperfections We have not explored the influence of initial imperfections in a systematic manner as has Mancktelow (2001) for Maxwell materials. In Fig. 13 we show the influence of increasing the amplitude of initial perturbations from zero to 0.08 of the layer thickness, with N =1, and for a plastic matrix with a coaxial deformation. Such changes in amplitude make little difference to the shapes of the localised folds but do have a very significant influence on the load bearing capacity of the system

Fig. 10. Development of single layer folds for different matrix materials; simple shearing deformation. The plots are of instantaneous shear strain with contour interval of 0.2. All bulk shear strains are identical. Single layer is a linear Maxwell material. The strength difference between layer and matrix is 100 in all cases. For the matrix materials: (a) N = 1. Max. contour is 1.0. (b) N = 2. Max. contour is 0.8. (c) N = 5. Max. contour is 1.75. (d) N = 10. Max. contour is 4.2. (e) Perfectly plastic, no volume change. Max. contour is 5.6. (f) Perfectly plastic, dilation rate is 0.2 × shear strain rate. Max. contour is 2.0. The undeformed state for each of these models is as shown in Fig. 11(a,c).

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Fig. 11. Localised crenulation zones in plastic matrix with volume change. Simple shearing deformation. (a), (c) Undeformed and deformed states. (b), (d) Zoom into part of deformed model showing localised zones of crenulations forming at 3° to the shearing plane; this angle is that expected from the specified dilation (Vermeer and de Borst, 1984). The folded layer is a Maxwell material as described in the text. The evenly spaced markers are purely passive and exist only to highlight the local deformation. The difference between (a), (b) and (c), (d) is only in the thickness of the folded layer.

and on the rate at which folds grow. This is compatible with the observations of Mancktelow (2001). Thus as one would expect, initial imperfections reduce the load bearing capacity as is widely demonstrated in the mechanics literature (Thompson and Hunt, 1973). An important point to note is that for the system with no perturbations the axial force rises as the folds nucleate and then remains constant for a substantial period of buckle growth. For systems with perturbations the axial force decreases very soon after folds form. 5. Discussion The structural patterns in deformed rocks in the form of folds and associated foliations and lineations show an endless range of diversity and complexity. One goal of Structural Geology is to understand the origin of this diversity and if possible use such diversity to say something about the constitutive behaviour and deformation conditions that lead to what we observe in the field. The approaches developed for fold development in the geological literature have largely been linear and a review of some of this work from a linear point of view is given by Hudleston and Treagus (2010). A linear system is one where the output of the system is proportional to the input. Examples are linear viscous systems where the strain rate is proportional to the imposed stress. Linear systems obey the mathematical Law of Superposition which says that if two or more functions are solutions to an equation that describes the evolution of the system then any linear combination of these solutions is also a solution. Thus methods involving Fourier series are important in solving linear problems. The evolution of folds has been treated as a linear problem in most of what has been written in the structural geology literature. This means

that for the most part only small deflections of a thin layer (or layers) are considered in a coaxial deformation. This means that moments arising from shear stresses parallel to the layer can be neglected. An additional assumption is that the force exerted on the layer by the deforming matrix is a linear function of the layer deflection as is the case for linear elastic and viscous materials (Biot, 1937, 1965). In situations where the embedding material is nonlinear, as in power-law viscous materials, the constitutive behaviour has been linearised (Fletcher, 1974; Smith, 1977) so that the reaction force at the instant folds nucleate is approximately linear. This is a perfectly valid mathematical procedure but it obscures the fact that subsequent growth of the fold is governed by a nonlinear matrix response. This means that although these solutions are valid for the instant of fold nucleation (when folds do not actually exist) they cannot describe the subsequent evolution of the fold shape. The issue is that in linearising the problem the response of the system naturally becomes linear even though the system response is intrinsically nonlinear. Fold evolution continues until the assumptions in the linearising procedure are no longer valid. At this stage the neglected nonlinear terms may become dominant leading perhaps to localised behaviour and what is known as nonlinear saturation (Cross and Greenside, 2009) where initial exponential growth rates decrease as nonlinear terms compete with and eventually overpower the initially dominant linear terms. This overall linear approach to the folding problem results in the well known Ramberg–Biot Eq. (2.1) which has sinusoidal or periodic solutions (for α≥2) depending on the spread of the dispersion curve (Biot et al., 1961; Biot, 1965). Commonly natural folds are non-periodic and the linear approach then demands that any departure from periodicity is due to initial imperfections in the system. Thus the rich

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Fig. 12. Influence of plastic volume change on geometry of folds. (a) Plastic layers 5× thickness of thin Newtonian viscous layers, ψ = 0°. (b) Plastic layers 5× thickness of thin Newtonian viscous layers, ψ= 12°. (c) Newtonian viscous layers 5× thickness of thin plastic layers, ψ= 0°. (d) Newtonian viscous layers 5× thickness of thin plastic layers, ψ = 12°. (e) Zoom into model. Both plastic and Newtonian viscous the same thickness, ψ= 0°. (f) Zoom into model. Both plastic and Newtonian viscous are of the same thickness, ψ = 12°.

diversity of fold geometries seen in natural examples is attributed in the linear (Hudleston and Treagus, 2010) approach solely to the influence of initial imperfections. Another approach to understanding fold geometry, the one adopted in this paper, is to expand upon the observation that in almost all cases the buckling problem is nonlinear. Thus even for the classical coaxial deformation of a layer the problem becomes nonlinear once the shear stresses on the layer become significant (corresponding to a limb dip of perhaps 10°). If the imposed deformation is non-coaxial, the shear stress parallel to the layer cannot be neglected even for initial infinitesimal deflections. Hence for thick layers the buckling moment due to this layer parallel shear stress contributes to the fold development just as the compressive stress parallel to the layer contributes in the coaxial case. This buckling moment due to shear stresses is a function of the layer deflection so that Eq. (2.1) is replaced by Eq. (2.6) even for a linear embedding material. Thus a Fourier series such as Eq. (2.3) is no longer a solution to the governing equations describing fold nucleation and evolution and more complicated wave-trains in the form of localised solutions arise. The lack of a Fourier series as a solution to the governing equations also means that a wavelength selection process as defined by (Biot, 1965; Hunt et al., 1997a) no longer applies and the concept of a dominant wavelength is no longer present. Even for small non-coaxial deformations of thin layers, if the embedding material is nonlinear then Eq. (2.2) is no longer a solution to Eq. (2.1) and localised solutions result. Thus in all situations, except for coaxial deformations of linear materials with layers thin enough that the moments arising from layer parallel shear stresses can be neglected, localised solutions describe the evolution of the buckling layers with no need to appeal to initial imperfections to explain departures from strictly sinusoidal or periodic geometries. We attribute the rich diversity and complexity

of fold shapes observed in natural examples to the nonlinearity of the system. Initial imperfections have an influence in controlling where folds might nucleate and certainly control the load bearing capacity of the system and how fast buckles grow but they are not essential in controlling fold system profiles and do not influence the dynamics of fold evolution. There may be no simple analytical solution to the nonlinear forms of Eq. (2.1) and an indefinite number of solutions commonly exist (Champneys and Toland, 1993; Peletier and Troy, 2001). The situation is made more frustrating by the observation that calculations of buckling behaviour based on finite element or finite difference methods are not generally capable of revealing the diversity of possible solutions (Hunt et al., 1997a). The lack of analytical and finite element solutions to nonlinear problems is emphasised by Burke and Knobloch (2007) and Knobloch (2008). The issue is that for nonlinear problems the energy landscape can be endlessly variable and the system seeks local minima in this landscape only to move to a neighbouring minimum as deformation proceeds. For linear systems only one energy minimum exists and this corresponds to a sinusoidal solution. The theoretical developments to date suggest that for many nonlinear systems the energy landscape comprises two or more intertwined surfaces. There are two ways in which a system can move from one local minimum to another. One way is to move smoothly from one local minimum in a single surface to another over the intervening energy barrier. The other is to jump from a maximum in one surface to a minimum in an adjacent surface. This pattern of evolution is referred to as snakes and ladders (Burke and Knobloch, 2007). To date no work on this kind of behaviour has been published with respect to buckling of visco-elastic layers but it is a profitable line of future research. Studies using this approach for simple elastic systems (Hunt and Hammond, in press) offer important insights into the nonlinear folding problem. For the present

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Fig. 13. Influence of amplitude of initial perturbations on fold evolution. Embedding material is plastic with properties given in the text. In the left column are plots of force at the end of a layer against the magnitude of layer deflection at the second fold hinge to the right in the right hand column. Top row: no initial imperfections 34% shortening. Middle row: initial amplitude of 0.005 of the layer thickness. 36% shortening. Bottom row: initial amplitude of 0.08 of the layer thickness. 27% shortening.

such problems need to be analysed using software that tracks the evolution of nonlinear systems and identifies bifurcations in behaviour (Doedel et al., 2002; Ermentrout, 2002). The suggestion made by the work of Cordier et al. (2012) and Kocks (1987) that materials may behave as rate insensitive plastic materials at low strain rates rather than power-law viscous materials with relatively low values of N offers interesting opportunities to explore the influence of plastic behaviour on buckling behaviour. The brief exploration into this behaviour in this paper indicates, as expected, that plasticity results in localised buckling behaviour. Indications that many geological materials may behave as plastic materials rather than as power-law viscous materials with relatively small values of N are given by the conclusion (Schmalholz and Maeder, 2012) that in strain-rate hardening materials high values of N are required to produce boudinage and that boudins develop readily in perfectly plastic materials (N = ∞). Equally some discussions of fold development (Aerden et al., 2010) indicate that plastic behaviour is more applicable than viscous behaviour with low values of N. However the introduction of non-associative plasticity, as could arise from volume changes associated with metamorphic reactions, produces localised structures, in the form of localised zones of crenulations, in the matrix together with localised folds in the embedded layer. These localised zones of crenulations form (and remain)

parallel to the plane of localisation predicted by non-associated plasticity theory so that they are close to the shearing plane in a simple shearing deformation. The axial planes of folds produced in simple shearing deformations initiate normal to the principal axis of deformation rate at 45° to the shearing plane and rotate towards the shearing plane during deformation. Thus at high shearing strains the localised zones of crenulations are very close to axial plane for the folds but vergence patterns given by the crenulations may not be reflected in changes of fold limb orientation, a feature common in some rocks (Bell et al., 2003). At relatively low shear strains these zones of crenulations transect the axial planes. The introduction of plastic dilation into the evolution of buckles in multilayers leads to a sharpening of the fold profiles so that the rounded hinges formed for zero dilation now more resemble chevron folds. This is in keeping with the observations of Hunt and Hammond (in press) where the formation of kink like structures is related to volume changes associated with particle interactions. For zero plastic volume change no work is done against the mean stress in changing the thickness between adjacent layers and the resultant fold shapes are rounded since there is no energy advantage to be obtained for the volume change to be either localised or distributed. If plastic volume changes are present then the system attempts to minimise the work done by these volume changes against the mean stress by keeping the limbs as straight as possible

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and concentrating as much of the deformation as possible at the hinges. This would seem to be one explanation for the widespread development of chevron folds in natural multilayers. Finally there is the interesting observation in Fig. 7(a) that in a coaxial deformation of a layer with a Newtonian matrix material and no imperfections, the system immediately develops localised folds instead of the expected sinusoidal solution. Investigation shows that the deflections of the layer are localised at the very first signs of deflection of the layer. Our tentative explanation of this result is that for the perfect system the situation is that described by Riks (1998) where the system attempts to reach the critical buckling load of the perfect system but that the phenomenon of clustered bifurcations is present so that near to the perfect critical load a number of other possible buckling paths are available due to the intrinsic nonlinearity of the governing Eq. (2.6) and the phenomenon of mode swapping takes place where a number of possible symmetrical and asymmetrical buckling paths are available. The introduction of initial random, small geometrical perturbations means that the system never reaches the perfect critical load and picks the solution for infinitesimal deflections of this system where the behaviour is essentially linear; this corresponds to the Biot sinusoidal solution. If the amplitude of initial perturbations is increased even further then the system becomes unstable and localised folds again form governed by the perturbations. Clearly further work is needed on this topic. 6. Conclusions (i) Periodic solutions exist for the buckling of a single layer embedded in a weaker matrix only for coaxial shortening deformations of a thin layer with a constitutively linear matrix. The term thin means that buckling moments that arise from shear stresses parallel to the layer can be neglected. This is the classical Biot solution for linear elastic and linear viscous materials. (ii) For coaxial deformations and for general non-coaxial deformations of thick layers, in which the layer is shortened, bending moments arising from shear stresses parallel to the layer cannot be neglected (as in the Biot case) and the resulting buckling behaviour is localised in the sense that localised packets of folds form. This is true for both linear and nonlinear matrix materials. (iii) If the matrix is nonlinear (nonlinear elastic, power-law viscous or plastic) localised folds always form independently of whether the deformation is coaxial or not. (iv) The mechanism of formation of localised folds differs from the Biot mechanism in that fold packets develop sequentially rather than a single wave-train growing simultaneously everywhere; no wavelength selection process operates and there is no dominant wavelength. (v) Fold localisation is an intrinsic outcome of deforming systems with nonlinear geometries and constitutive relations. Localisation occurs independently of the existence of initial irregularities. Initial irregularities may control the position of localised folds but not whether they form or not. Initial irregularities decrease the load bearing capacity of the buckling system and increase the probability that a fold will nucleate and grow. (vi) The possibility that rate independent plasticity characterises the constitutive behaviour of deforming geological materials needs greater attention. In particular the development of non-associative plasticity arising from volume changes during syn-deformational metamorphic reactions can induce localised zones of crenulations. These zones transect the axial planes of the developing folds but can resemble axial plane structures at high strains. However the vergence of these crenulations is independent of position relative to the fold limbs. Such structures are common in deformed rocks. (vii) The volume changes associated with metamorphic reactions, in conjunction with plastic responses, can lead to the development

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of chevron-style fold systems. The requirement to minimise the work done during plastic dilation means that folds become angular and more sharply defined than when volume changes are accommodated by distributed dissipation arising from volume change.

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