The shortening and extension of mechanically layered ductile rock generates folds and pinch-and-swell structures (also referred to as necks or continuous boudins), which result from mechanical instabilities termed folding and necking, respectively. Folding and necking are the preferred deformation modes in layered rock because the corresponding mechanical work involved is less than that associated with a homogeneous deformation. The effective viscosity of a layered rock decreases during folding and necking, even when all material parameters remain constant. This mechanical softening due to viscosity decrease is solely the result of fold and pinch-and-swell structure development and is hence termed structural softening (or geometric weakening). Folding and necking occur over the whole range of geological scales, from microscopic up to the size of lithospheric plates. Lithospheric folding and necking are evidence for significant deformation of continental plates, which contradicts the rigid-plate paradigm of plate tectonics. We review here some theoretical and experimental results on folding and necking, including the lithospheric scale, together with a short historical overview of research on folding and necking. We focus on theoretical studies and analytical solutions that provide the best insight into the fundamental parameters controlling folding and necking, although they invariably involve simplifications. To first order, the two essential parameters to quantify folding and necking are the dominant wavelength and the corresponding maximal amplification rate. This review also includes a short overview of experimental studies, a discussion of recent developments involving mainly numerical models, a presentation of some practical applications of theoretical results, and a summary of similarities and differences between folding and necking.

One of the principal objectives of theoretical research in any department of knowledge is to find the point of view from which the subject appears in its greatest simplicity. (J. Willard Gibbs, 1880, acceptance letter of Rumford Prize)

About 200 years ago Sir James Hall made his famous analogue experiments on layer-parallel shortening of linen and woollen cloth layers with vertical confining pressure provided by “a door (which happened to be off the hinges)” loaded with weights (Hall, 1815; Fig. 1). During shortening the layers deflected laterally (orthogonal to the shortening direction), which is a process termed buckling (mainly used for elastic material) or folding (mainly used for viscous material; Fig. 2a, b). James Hall was probably the first to show that natural folds in rock are the result of a horizontal compression. However, some of the first scientific observations on folds were already made more than 100 years earlier, by Marsili and Scheuchzer in 1705, in the European Alps around Lake Uri in Switzerland (see Hantke, 1961, Ellenberger, 1995, Vaccari, 2004; Fig. 3a). If a competent layer (i.e. one having a higher mechanical strength or greater resistance to deformation) is extended, rather than shortened, it will not usually deflect but it will either break (brittle or fracture boudinage) or it locally thins in a ductile manner (necking; Fig. 2c, d). The term boudinage goes back to Lohest (1909) but necking and pinch-and-swell structure were probably first described by Ramsay in 1866 (Ramsay, 1866; Cloos, 1947; Lloyd et al., 1982; Fig. 3b). We use necking here as a general term to refer to local thinning due to the inherent mechanical instability of a competent layer under extension and pinch-and-swell to refer to the periodic structure that develops from repetition of necking zones along the layer. The finite strain deformation geometry of a competent layer is thus fundamentally different for layer-parallel shortening and extension (Fig. 4), although the initial stages of both folding and necking instabilities can be mathematically explained with the same theory (e.g. Smith, 1975, 1977). The mechanical processes controlling the different behaviour of competent layers under compression and extension are the main focus of this review.

Original sketch of James Hall's (1815) folding experiment.

Natural single-layer folds

Numerical simulations of folding and necking resulting from
layer-parallel shortening and extension, respectively, of the same competent
layer with the same initial lateral thickness variation (initial
perturbation). The model was initially 20 times wider than the initial layer
thickness and the medium above and below the layer was 5 times thicker than
the layer. The initial perturbation was introduced as a reduction in the
layer thickness by 10 %, with the width of this perturbation equal to the
layer thickness. The layer interface at the two edges of the perturbation has
been smoothed to avoid numerical inaccuracies. The geometries are the result
of finite element simulations (performed for this study) with a reference
viscosity ratio of 100, a power-law exponent of the layer of 10 and of the
embedding medium of 3. The colours indicate the square root of the second
invariant of the stress tensor

Multilayer folds

Folding and necking are processes that result from instabilities in elastic, plastic and viscous material caused by layer-parallel compression and extension, respectively, of mechanically competent layers. In this review, the overall deformation behaviour is assumed to be ductile and continuous so that fracturing does not play any controlling role. However, after some amount of ductile necking a layer often fractures around the necked region, which is a process termed ductile fracture (e.g. Dieter, 1986), and necking can act as a ductile precursor to brittle boudinage. In nature, brittle structures can also act as precursors that localise ductile shearing (Segall and Simpson, 1986; Mancktelow and Pennacchioni, 2005; Pennacchioni and Mancktelow, 2007) and hence may trigger the formation of a pinch-and-swell structure (Gardner et al., 2015).

The structures resulting from folding and necking can have a wide variety of different geometries, especially in multilayers (Ramberg, 1955; Ramsay and Huber, 1987; Price and Cosgrove, 1990; Goscombe et al., 2004; Fig. 5). As noted by Ramsay and Huber (1987), “folds are perhaps the most common tectonic structure developed in deformed rocks” and a thorough understanding of folding is therefore essential to understand the deformation of typically layered or foliated rocks. Folds can also be generated by passive flow or bending but here we focus only on folds resulting from mechanical instability due to layer shortening. Pinch-and-swell structure seems to be less frequent in nature than folds and this review will provide potential reasons to explain this observation. Folding and necking (and also brittle boudinage) can occur simultaneously, with necks and/or boudins commonly forming in the limbs of folds (Ramberg, 1959; Fig. 5c). In nature, folding and necking are always three-dimensional (3-D) processes (Fig. 6), but most theoretical results are based on 2-D models.

Outcrops showing the three-dimensional (3-D) geometry of folds.

Lithospheric folding and necking.

Folding and necking are important tectonic processes because they occur over the whole range of geological scales, from microscopic dimensions up to the size of lithospheric plates. Lithospheric folding and necking in tectonic plates contradict the paradigm of plate tectonics sensu stricto, which states that tectonic plates are rigid and deformation only occurs at plate boundaries. Lithospheric folding can occur on a length scale of several thousands of kilometres, such as in Central Asia (Burov et al., 1993; Fig. 7a). Lithospheric necking is also important, for example, in the formation of rift basins (Zuber and Parmentier, 1986) and of magma-poor passive continental margins, because many of these margins are characterised by so-called necking domains in which the crustal thickness decreases from normal crustal thickness (30–35 km) to ca. 5–10 km (Sutra et al., 2013). Lithospheric necking can also generate a crustal-scale pinch-and-swell structure (Fletcher and Hallet, 1983; Gueguen et al., 1997; Fig. 7b) and is a first-order process during slab detachment (Lister et al., 2008; Schmalholz, 2011; Duretz et al., 2012; Bercovici et al., 2015).

The literature on the mechanics of folding and necking is vast because these processes (i) occur from millimetre to kilometre scale, (ii) were modelled using a variety of constitutive equations such as elastic, plastic, viscous, viscoelastic or viscoelastoplastic, (iii) can be driven by imposed boundary displacements, velocities or stresses, or by gravity, (iv) were studied for different bulk deformation geometries such as pure shear, simple shear or wrenching, (v) were studied for single- or multilayer configurations, (vi) were studied for isotropic and anisotropic materials, (vii) were studied in 2-D and 3-D and (viii) were studied using analytical solutions, laboratory (analogue) experiments, or numerical simulations. We present here only a small selection of studies and results. Further information on the mechanics of folding and necking in rock can be found in textbooks (Price and Cosgrove, 1990; Johnson and Fletcher, 1994; Pollard and Fletcher, 2005) and in other review articles (Hudleston and Treagus, 2010; Cloetingh and Burov, 2011).

We focus on studies that investigated particular mechanical aspects of folding and necking and on studies that applied the results to geological observations and problems. Particular geological questions concerning folding and necking are, for example, (i) which parameters control the observed geometry of folds and pinch-and-swell structures, (ii) how much shortening or extension is required to generate the observed finite (high) amplitude folds and pinch-and-swell structures, (iii) does folding and necking change the overall (effective) strength of a rock unit, and (iv) how much force or stress is required to generate observed folds and necks, particularly on the lithospheric scale?

Theoretical studies and analytical solutions invariably involve simplifications but can provide the best insight into the fundamental parameters controlling a mechanical process. The aim of such studies is thus to determine the (hopefully) small number of (non-dimensionalised) controlling parameters and to investigate their specific influence.

Some of the first mathematical studies on bending of beams were performed by
Galilei (1638), who studied the strength of beams under beam-orthogonal
loading (Fig. 8a). A beam is a 2-D layer that is much longer than thick and a
number of simplifications can therefore be made for the geometrical
description of the bending. A first beam theory was developed in the 18th
century with major contributions from Euler and Bernoulli and is hence often
termed the Euler–Bernoulli beam theory (e.g. Timoshenko, 1953; Szabo, 1987).
It is assumed that the central (neutral) line in the beam is neither extended
nor shortened and that the inner side of the beam is shortened while the
outer side is extended; i.e. there are both layer-parallel extensional
and compressional strains in the beam due to bending. A major result of the
Euler–Bernoulli beam theory is that it can relate the layer-parallel strain,

Frequently used symbols and their consistent meaning throughout the text.

Sketch illustrating the Euler–Bernoulli beam theory and the related thin-plate approximations.

Using the above expressions for

Equation (3) describes the balance of moments acting on a compressed beam
that can deflect freely because the beam is not embedded in another material
and gravitational stresses arising due to the deflection are also not
considered. If there is an additional stress,

Multilayer folds showing that the size of individual folds (quantified by distance between neighbouring hinges of the same type) is related to their layer thickness and that folds become systematically smaller as their layer thickness becomes thinner. Carbonates with silicate-rich layers belonging to the Jurassic El Quemado formation. The sample was found by Stéphane Leresche around the Mount Fitz Roy, southern Patagonia, and the photo was made by Yoann Jaquet.

Equation (5) is a 1-D equation that can be used to study 2-D folding
(buckling) because of the assumption that the horizontal strain can be
expressed by the second spatial derivative of the vertical deflection,

Sander (1911) observed that, for multilayers within a quartz phyllite, the
size of individual folds is related to their layer thickness and that folds
become systematically smaller as their layer thickness becomes thinner (e.g.
Fig. 10). He termed this observation the “rule of fold size” (in original
German:

Configuration for analytical folding

Based on Biot's correspondence principle (1954, 1956, 1961), the thin-plate
equation for elastic material can be easily applied to viscous and
viscoelastic materials (Biot, 1957). For incompressible viscous material, the
horizontal total stress to calculate the moment,

Dimensionless amplification rate,

For comparison with a viscous layer, the dominant wavelength for an elastic
layer embedded in a linear viscous medium is

As discussed in some detail by Biot (1961), the selectivity of amplification,
and therefore the tendency to develop a clear sinusoidal form with a
wavelength approximating that of the dominant wavelength, depends on the
relative bandwidth of the amplification rate curve, which he defined as the
wavelength difference at half the maximum amplification rate

A different approach to derive

The stream function approach was later used by Fletcher (1974) to perform a
hydrodynamic stability analysis for single-layer folding of a power-law
viscous layer. Detailed mathematical descriptions of the stream function
approach in combination with a hydrodynamic stability analysis are given in
Fletcher (1974, 1977), Smith (1975, 1977), Johnson and Fletcher (1994) and
Pollard and Fletcher (2005). We term this approach here stability analysis,
but this approach has also been termed perturbation method or thick-plate
analysis. We do not use the term thick-plate analysis to avoid confusion with
the thick- or shear-deformable plate theory, which is an elaboration of the
thin-plate equation that considers also shearing within the layer (e.g. Wang
et al., 2000). As will be shown later, the same stability analysis can also
be applied to study necking (Smith, 1975, 1977). The stability analysis
assumes that geometrical perturbations are superposed on a flat layer that is
shortening and thickening by pure shear (the basic state, Fig. 11). In a
stability analysis, the initial deflection,

Fletcher (1974) also derived a solution for

Dimensionless amplification rate,

Contours of the ratio of dominant wavelength to layer thickness
(thinner lines) and dimensionless amplification rate (growth rate; thicker
lines) for folding

For power-law viscous flow, the approximate formula (for

Sketch illustrating the linearisation of power-law flow law
(black solid line indicates the exact flow law,

Stability analysis is performed with linear equations and hence the non-linear
power-law flow law must be linearised. This is done by assuming that every
quantity, such as strain rate (e.g.

A note on nomenclature: for linear viscous fluids (

Stability analysis is of great practical importance in nearly all branches of
mechanics because it shows whether mathematically and mechanically correct
solutions are actually possible (or stable) in nature. For example, pure
shear shortening and thickening of a perfectly straight (rectangular) and
competent viscous layer embedded in a less viscous medium, which takes place
without folding, is a mathematically and mechanically correct solution.
However, this solution is not stable because any small geometrical
perturbations, which are always present in natural materials, amplify with
faster velocities than the corresponding pure shear velocities and cause
folding. Homogeneous pure shear deformation of a competent viscous layer
therefore does not occur in nature, although the mathematical solution is
correct. Stability analysis is thus essential to determine whether correct
mechanical solutions are applicable to natural processes. For pure shear
shortening of a layer of thickness

The derivation of the theoretical dominant wavelength and corresponding amplification rates for linear and power-law viscous materials are only strictly valid for infinitesimal amplitudes but have been verified by both laboratory deformation experiments and numerical simulations at small fold amplitudes (e.g. Biot et al., 1961; Mancktelow and Abbassi, 1992; Schmalholz, 2006). The further development of fold geometries and the “selection” and “locking” of a fold wavelength have been discussed and analysed in a number of studies (Sherwin and Chapple, 1968; Fletcher, 1974; Fletcher and Sherwin, 1978; Hudleston and Treagus, 2010). However, a detailed discussion of their results is beyond the scope of this review.

The theories outlined above are valid for single layers with the imposed
layer-parallel displacement, velocity or load directly applied at the ends of
the layer. However, folded veins or dikes are not infinitely long and
therefore a layer-parallel load is usually not directly applied at their
ends. Schmid et al. (2004) considered folding of power-law viscous layers
with a finite length, essentially corresponding to isolated inclusions
removed from the lateral boundaries and embedded in a more extensive linear
viscous medium. They considered the layers as ellipses with large aspect
ratios (i.e. length to thickness ratio). If the viscous medium is shortened
in a direction parallel to the long axis of the ellipse, the stresses in the
surrounding viscous medium will cause deformation and folding of the isolated,
elongate ellipse. Finite-length single-layer folding is controlled by the
dimensionless ratio

As can be seen from Eq. (9) or from a time integration of Eq. (16), both the
standard thin-plate approach and the stability analysis give an exponential
growth of the fold amplitude with time, whereby the amplification rate is
constant. This exponential growth is the result obtained if the mathematical
analysis is carried out to first-order in slope. The solution in Eq. (12)
provides the amplification rate for all possible wavelengths (or Fourier
components) and with this solution the evolution of fold shapes can be
calculated analytically for certain initial layer geometries, such as an
initial isolated bell-shaped geometry (Biot et al., 1961). The analytical
treatment is possible because the bell-shaped geometry can be represented as
an infinite cosine series by a known Fourier integral expression

Evolution of fold geometry for an initial geometrical bell-shaped
perturbation calculated with the analytical solution of Biot et al. (1961;
Eq. 27; red line), the LAF solution of Adamuszek et al. (2013; magenta line)
and a numerical finite element simulation (black layer). The parameters for
the bell-shaped perturbation are

Approximate finite amplitude solutions for folding (

As noted above, there is always a passive or kinematic component of
layer-parallel shortening and thickening that dominates at very small
amplitudes before “explosive” folding manifests itself due to the
exponential amplification rate. Indeed, as also presented above, in the limit
of a perfectly planar layer, folds never develop regardless of the viscosity
ratio and the layer will simply shorten and thicken. The result of this
effect, which is not specifically considered in the thin-plate or stability
analysis solutions above, is that the non-dimensional wavenumber,

It is of course expected that the initial exponential amplification with a
constant rate of Eq. (16) must eventually break down, because fold amplitudes
cannot grow exponentially at the same rate forever during shortening; the
initial dynamic rate (

For the non-sinusoidal, bell-shaped initial perturbation of Fig. 16, with a
viscosity ratio of 75, comparison with the numerical model indicates that the
approximate solution of Eq. (27) is very good up to a maximum limb dip of ca.
25

Adamuszek et al. (2013) presented a comprehensive finite amplitude solution
for both single sinusoidal folds and for multiple waveforms represented by a
Fourier series, such as the case for the bell-shaped perturbation (their
Appendix B). They combined the results of stability analysis with the two
previously proposed corrections to the original linearised theories outlined
above. Their large amplitude folding (LAF) model includes both the effect of
homogeneous shortening and thickening on the non-dimensional wavenumber and
the need to consider the shortening rate of the arc length, rather than the
shortening rate imposed at the boundaries. The LAF model is described by a
coupled system of ordinary differential equations involving time derivatives,
so that the time evolution has to be calculated numerically. The LAF solution
allows for quite exact prediction of fold geometries up to moderate limb dips
of

The improvement in fit resulting from these additional corrections can be
seen for the bell-shaped perturbation example in Fig. 16. As noted
previously, the simple solution of Eq. (27) breaks down at limb dips of

The finite amplitude solutions outlined above are based effectively on an
elaboration of the linear thin-plate approach. It is also possible to further
elaborate the linear first-order stability analysis in order to obtain
results that are more accurate for finite, but still small amplitudes, by
considering sinusoidal terms up to third-order (Fletcher, 1979; Johnson and
Fletcher, 1994). This higher-order stability analysis provides results which
are valid up to moderate limb dips of

One particular result of the finite amplitude solution is that the
layer-parallel load,

Structural softening during folding (black line) and necking (red line) of the
simulations shown in Fig. 4. The effective
viscosity is calculated by the ratio of the area-averaged square root of the
second invariant of the deviatoric stress tensor,

Implicit in developing the analytical solutions above, as well as in many analogue
and numerical models, is that there are basically three stages of fold
development. As is obvious from Eq. (16), the amplification velocity

An important, and still controversial question, is the magnitude of the
effective viscosity ratio implicit in observed natural single-layer folds
(e.g. Hobbs et al., 2008; Schmid et al., 2010). Using Eq. (16) for the
amplification of the dominant wavelength with the maximal amplification rate,
integrating it in time with

As a concise summary, the simpler solutions for the dominant wavelength and corresponding maximal amplification rate derived above are listed in Table 2.

Approximate solutions for dominant wavelength and maximal amplification rates for folding and necking.

Result of a numerical simulation (performed for this study) of
linear viscous multilayer folding for a viscosity ratio of

Many natural folds seem to have a cylindrical structure; i.e. their lateral extend along the fold axis is considerably larger than their wavelength. For such cylindrical fold shapes the above discussed 2-D solutions are applicable. However, other natural folds are clearly not cylindrical (such as dome and basin fold shapes) or have been formed by two consecutive events of deformation. For such fold shapes, 3-D models are required. Three-dimensional viscous folding has also been studied analytically with both the thin-plate (Ghosh, 1970) and the stability analysis for linear viscous (Fletcher, 1991) and power-law viscous fluids (Fletcher, 1995; Mühlhaus et al., 1998). The finite amplitude solution in Eq. (29) has been elaborated for 3-D folding to take into account the ratio of the two orthogonal, layer-parallel shortening (or extension) rates and was verified with 3-D numerical simulations (Kaus and Schmalholz, 2006).

Multilayer folds (Figs. 5, 10) are more frequent in nature than single-layer folds. The mechanics of elastic multilayer buckling was already discussed in Smoluchowski (1909) and we provide here only a short review. Viscous multilayer folding is also discussed in detail in Johnson and Fletcher (1994) and a review of multilayer folding can be found in Hudleston and Treagus (2010).

The thin-plate approach shows that if a multilayer is composed of

Impact of spacing of competent layers (

In a series of articles, Biot developed a theory of internal buckling and
multilayer folding (Biot, 1964a, b, 1965a, b). In his theory of internal
buckling, he considered multilayers as a confined anisotropic or stratified
medium under compressive stress (Biot, 1964a, 1965a). This simplifies the
mathematical analysis, because the individual deformation of each competent
and incompetent layer is neglected. For a confined multilayer, the top and
bottom boundaries remain straight so that the top and bottom layers in the
multilayer cannot fold. Application of this internal buckling theory to
confined viscous multilayers with competent and incompetent layers of equal
thickness,

Biot (1965b) further showed that in a homogeneous anisotropic (orthotropic) medium two essential types of internal instability can occur, corresponding to either folding or localised kink-band formation. Following Biot's theory of internal instability, Cobbold et al. (1971) performed analytical analyses and laboratory experiments for compression of confined multilayers representing a homogeneous, anisotropic material and could validate the application of the analytical results for an anisotropic medium to true confined multilayers consisting of competent and incompetent layers. They further concluded that (i) the instability due to multilayer compression is mainly governed by the degree of anisotropy rather than rheological properties such as elasticity or viscosity, (ii) the form of the internal structure depends on the angle between layer orientation (or anisotropy orientation) and compression direction, (iii) sinusoidal folds (developing for low degrees of anisotropy) and conjugate kink bands (developing for high degree of anisotropy) are end-members of a range of fold structures that can develop in anisotropic material, (iv) chevron folds (i.e. folds with long straight limbs and short angular hinges; e.g. Paterson and White, 1966) can form by either convergence of conjugate kink bands or by progressive straightening of limbs during amplification of initially sinusoidal folds and (v) the scale of an internal structure in an anisotropic (but statistically homogeneous) rock depends on the scale of the elements that cause the anisotropy.

Johnson and Fletcher (1994) presented solutions based on stability analysis for selective amplification during low-amplitude folding for many examples of multilayers with different configurations of competent and weak layers and embedding medium. They showed that the strength of the folding instability generally increases with (i) the number of competent layers involved, (ii) increasing viscosity ratios between the competent and weak layers and (iii) larger thickness of the weak medium embedding the multilayer (see also Ramberg, 1962, his Fig. 12). They further show that the amplification of multilayer folds with all layers having free slip interfaces is significantly stronger than for multilayers with all layers having no slip (bonded) interfaces (Johnson and Fletcher, 1994). The difference in interface condition (free slip vs. bonded) is much more significant for folding of a multilayer than for folding of a single layer (Johnson and Fletcher, 1994).

A third-order stability analysis of viscous multilayer folding for small but finite amplitudes was performed by Johnson and Pfaff (1989) to study fold shapes in multilayers. They distinguished three end-member forms in multilayers: parallel, constrained and similar folds. Parallel (concentric) folds develop in multilayers confined by a soft medium, whereas constrained (internal) forms develop in multilayers confined by a very competent medium. Similar (chevron) forms develop if wavelengths are short relative to the thickness of the multilayer.

Schmid and Podladchikov (2006) applied the stability analysis to show that
folding of embedded multilayers can occur in essentially three modes: as an
effective single layer, as a true multilayer or as real single-layer folding
(Schmid and Podlachikov, 2006). They consider a multilayer embedded in a weak
medium, so that the top and bottom layer in the multilayer can fold, which
differs from the configuration of internal folding in confined multilayers
considered by Biot (1965a) (see above). The multilayer is made of competent layers
with equal thickness,

A particular problem of multilayer folding is the mechanism of formation of
asymmetric parasitic folds (

Some approximate solutions for the dominant wavelength and maximal amplification rates for multilayers are summarised in Table 2.

Lithospheric folding is of general geodynamic importance because it demonstrates that large regions of the interior of tectonic plates are deformable. Internal deformation of tectonic plates contradicts a basic principle of plate tectonics sensu stricto, which states that tectonic plates are essentially rigid and deformation should only occur at plate boundaries. Continental lithospheric folding and necking can be considered as a specific and important component of continental tectonics.

Probably the first large-scale folding analysis was performed by
Smoluchowski (1909), who considered an elastic beam and applied Eq. (5).
Smoluchowski (1909) equated the load term

During shortening of the crust and lithosphere, rock layers can deform by either distributed ductile creep (e.g. thickening or folding) or localised brittle faulting and plastic shearing (thrusting), corresponding to bulk viscous or brittle-plastic flow, respectively. Several studies have therefore investigated the parameters that control whether shortening of rock layers is more likely to take place by folding or faulting (e.g. Johnson, 1980; Erickson, 1996; Simpson, 2009; Yamato et al., 2011). The question of “folding vs. faulting” is of particular interest for fold-and-thrust belts (e.g. Jura, Zagros or Appalachians). For example, Johnson (1980) applied the stability analysis described above to elasto-plastic, strain-hardening layers and showed that during shortening of such layers folding is more likely than faulting for (i) multilayers and (ii) frictionless layer contacts. In many analytical studies involving plastic deformation, the plastic layer is characterised either by a representative, constant yield stress or by a large power-law stress exponent, which mimics a constant von Mises stress (e.g. Martinod and Davy, 1992; Martinod and Molnar, 1995). However, the pressure sensitive yield stress of rocks is usually represented by a Mohr–Coulomb failure criterion, which is difficult to treat with analytical methods. Hence, numerical simulations have been applied to quantify the parameters that cause folding or faulting in fold-and-thrust belts (e.g. Simpson, 2009; Yamato et al., 2011). For example, Yamato et al. (2011) numerically calculated amplification rate vs. fold wavelength curves for sedimentary sequences. Based on these curves, they determined the folding–faulting boundary for the sedimentary sequences and applied the results to the Zagros fold belt. In this review, we focus on ductile folding studies and do not discuss in further detail the folding vs. faulting issue.

Biot (1961) also derived the dominant wavelength for a viscous layer floating
on an inviscid medium in the field of gravity, which can be written as

Lithospheric folding has also been studied with the stability analysis
(Zuber, 1987; Martinod and Davy, 1992; Burov et al., 1993; Martinod and
Molnar, 1995), which yields more accurate (but also more complicated)
solutions for the amplification rate without changing the first-order results
and conclusions of studies based on the thin-plate approach. The solution in
Eq. (35) is strictly valid only for infinitesimal amplitudes and

Martinod and Molnar (1995) performed a stability analysis in which they
considered a power-law viscous rheology and Mohr–Coulomb plastic yield
strength of the Indian oceanic lithosphere. They argued that the oceanic
lithosphere is overlain by unconsolidated sediments with average density of
2300 kg m

A frequently applied model for viscous deformation of the continental
lithosphere is the thin viscous sheet model (England and McKenzie, 1982).
This model assumes that lithospheric folding is negligible and that the
lithosphere deforms by homogeneous, kinematic thickening. Thin-sheet models
consequently assume that vertical velocities due to folding are less than
vertical velocities due to kinematic thickening and hence they assume
amplification rates for folding

In most branches of science, proposed analytical solutions can be tested by direct observation or experiment. However, in the case of folding and necking of rock layers as considered here, this is impossible due to the long times and large forces, pressures, temperatures, and (often) length scales involved. By necessity, the development of such structures to large amplitude can only be studied by analogue and, more recently, numerical modelling. Initially these analogue models were only qualitative but progressively became more quantitative with the application of correct scaling laws (Hubbert, 1937) and the use of materials more rheologically similar to rocks but deformable at lower stresses, temperatures and pressures. Only a limited personal selection of the many published studies can be presented here but, in keeping with the overall theme of the review, we particularly highlight those studies that either specifically constrained analytical solutions or attempted to extend them to higher amplitudes typical of natural geometries.

The pioneer in analogue modelling of folding was Hall (1815), who “conceived that two opposite extremities of each bed being made to approach, the intervening substance, could only dispose of itself in a succession of folds, which might assume considerable regularity, and would consist of a set of parallel curves, alternatively convex and concave towards the centre of the earth”. To test this premise he carried out his now famous experiments using layers of cloth to demonstrate that the folds he observed in nature could develop by shortening of horizontally layered rocks by application of a horizontal force (Fig. 1). The experiments were entirely qualitative but established the basic principle. Since then, a large number of analogue experimental studies have investigated the influence of material contrast (e.g. viscosity ratios), constitutive equations (elastic, linear and power-law viscous, plastic and different combinations), material anisotropy and initial perturbation geometry on the initiation and development of single- and multilayer folds and boudins. Only a limited selection is presented here as examples.

In a companion paper to Biot (1961), Biot et al. (1961) presented a series of
analogue models aiming to provide experimental verification of the analytical
results for folding of stratified viscoelastic media (Biot, 1957, 1961).
These experiments considered layer-parallel shortening of both elastic and
viscous layers embedded in a viscous matrix. Biot's thin-plate theory is for
a layer of infinite length and predicts an amplification rate as a function
of normalised wavelength (or wavenumber) given by Eq. (12) with a dominant
wavelength, corresponding to the maximum amplification rate, given by
Eq. (13). In an analogue model, a layer of infinite length is unattainable
and an initial infinitesimal amplitude perturbation spectrum of perfect
random white noise (all wavelength components present and with equal
amplitude) is also unrealistic. A novel alternative approach proposed by Biot
et al. (1961) was to consider the amplification of an initial isolated
bell-shaped perturbation. This can be represented as an infinite cosine
series by a known Fourier integral expression given in Eq. (26) and the
evolving fold geometry with time (strain) can be calculated with Eq. (27).
Biot et al. (1961) used this approach in a numerical evaluation of the time
history of folding and sideways propagation away from an isolated
perturbation but not in their analogue models. In these models, they used
thin plates of aluminium or cellulose acetate butyrate (elastic layers) or
roofing tar (viscous layers) in a corn syrup viscous matrix, without any
prescribed initial perturbation, to establish a good correspondence with
theory – at least for the very high viscosity ratios (

Ramberg and Stephansson (1964) performed laboratory experiments on folding of
a viscous plate (made from molten mixtures of colophony and diethyl
phthalate) floating on an aqueous solution of potassium iodide to verify the
dominant wavelength for folding under gravity given in Eq. (34). They showed
that the value of

Ghosh (1966) studied single-layer folding under simple shear, using combinations of modelling clay, putty and wax. He noted that the fold axis developed parallel to the major axis of the strain ellipse on the surface of the layer (i.e. perpendicular to the principal component of shortening within the layer), which, for generally oblique layering, is not necessarily parallel to a principal axis of the applied bulk strain. He also noted that the single layer folds are, at least initially, generally symmetric despite the simple shear boundary conditions. This is consistent with the later, general observation of Lister and Williams (1983) that single layer buckle folds are good examples of coaxial spinning deformation (their Fig. 4) and agrees with results of numerical models (Viola and Mancktelow, 2005; Llorens et al., 2013a). Ghosh (1968) also did analogue experiments on multilayer folding to develop rough qualitative constraints on the transition from conjugate to chevron to concentric folds. Currie et al. (1962) had previously also qualitatively investigated single- and multilayer folding in elastic materials using photoelastic gelatin. With this experimental technique they could not only investigate the influence of layer thickness and ratios in elastic properties on fold wavelength but also analyse the stress trajectories in the layer and matrix during folding. Their experiments provided an excellent visual representation of the zone of contact strain around a folding layer and the consequent development of disharmonic or harmonic folding depending on the spacing between layers (their Plate 2).

Hudleston (1973) performed experiments to study the development of single-layer folds with shortening parallel to the layer. The material used for both layer and matrix were mixtures of ethyl cellulose in benzyl alcohol, which, at the low concentrations used in his experiments, is effectively linear viscous. The viscosity ratios considered were between 10 and 100 and thus much lower than those used by Biot et al. (1961). One of the aims of the experiments was to establish that folding to finite amplitude with such low ratios, and correspondingly short wavelength to thickness ratios, was possible, in contrast to what was implied in the original papers of Biot (1961) and Biot et al. (1961). In these experiments, Hudleston (1973) also specifically investigated layer-parallel shortening and thickening and the transition to rapid (explosive) fold amplification, as well as making harmonic analyses of the experimental fold shapes.

Cobbold (1975a) carried out analogue experiments to study the sideways propagation of folds away from an initial isolated perturbation in a single layer undergoing layer-parallel shortening, using a pure-shear deformation rig (Cobbold, 1975b; Cobbold and Knowles, 1976). Materials used were well-calibrated paraffin waxes of different melting points, with power-law stress exponents of ca. 2.6 and an effective viscosity ratio between layer and matrix of ca. 10. Conceptually this was an experimental investigation of the process considered theoretically and numerically by Biot et al. (1961) with an initial isolated bell-shaped perturbation, but for power-law viscous materials and a much lower (and more realistic) viscosity ratio. However, Cobbold (1975a) used a cylindrical form for the initial perturbation, rather than a bell-shape with the known Fourier integral representation of Eq. (26), and did not consider the propagation in terms of amplification of Fourier spectral components. Instead, he introduced the important concept of the perturbation flow lines (Passchier et al., 2005) to qualitatively investigate the sideways spread of the folding instability.

Gairola (1978) made single-layer fold experiments with plasticene layers embedded in putty to investigate the effects of progressive deformation on fold shape and particularly on the internal strain within the layer and on the varying position of the neutral surface. He found that the appearance of the neutral surface depends on the “ductility contrast” between the layer and matrix, and the amount of strain. A neutral surface may not appear at all if the contrast between layer and matrix is very small, due to the strong component of layer-parallel shortening, which agrees with recent results of numerical simulations (Frehner, 2011). These experimental results can also explain why thin-plate results become more inaccurate for smaller viscosity ratios, because the thin-plate results are based on the assumption of a neutral line in the centre of the folding layer.

Neurath and Smith (1982) performed folding experiments with wax models, measured the effective viscosities and power-law exponents for the wax models, and compared the experimentally determined amplifications rates with the corresponding theoretical rates. They showed that theoretical and measured amplification rates more or less agreed with the theoretical rate as derived by Smith (1975, 1977, 1979) and the equivalent results of Fletcher (1974, 1977).

Abbassi and Mancktelow (1990) investigated the influence of initial
perturbation shape on fold shape, establishing that markedly asymmetric
folds, even with overturned limbs, could develop by amplification of a small
initially asymmetric irregularity, despite the fact that the imposed boundary
condition was layer-parallel shortening in a pure shear deformation rig
(Mancktelow, 1988a). Abbassi and Mancktelow (1992) and Mancktelow and
Abbassi (1992) employed the isolated bell-shaped perturbation technique
originally developed by Biot et al. (1961) directly in analogue experiments,
both to investigate the effects of perturbation geometry on fold shape and
lateral propagation (Cobbold, 1975a) and to experimentally determine fold
amplification rates. Instead of calculating a numerical forward model for a
specific amplification rate curve as done by Biot et al. (1961), they
reversed the approach and used the changing shape of an initial bell-shaped
perturbation with known initial values of

Marques and Podladchikov (2009) placed a thin layer of either plasticine or polyethylene between viscous polydimethylsiloxane (PDMS; Dow Corning SGM36) below and Fontainebleau quartz sand above. The PDMS represents the ductile part of the lithosphere, the quartz sand the brittle parts and the thin layer of either plasticine or polyethylene the thin elastic core, which is easily flexed but unstretchable/unbreakable. Their results show that a very thin, elastic layer between an overlying brittle and underlying viscous medium produces folding as the dominant deformation mechanism during shortening, and not brittle faulting or viscous homogeneous thickening.

Recently, Marques and Mandal (2016) have made experiments to investigate the buckling and post-buckling behaviour of an elastic single layer (cellophane, plasticine, or polyethylene film) in a linear viscous medium (PDMS silicone putty). The experiments were performed in two stages: a first stage of buckling by layer-parallel shortening at different rates and a second stage of buckling relaxation with fixed lateral boundaries. They found major contradictions between their experimental results and both the analytical results of Biot (1961) for the buckling phase and with the analytical solutions and conclusions of Sridhar et al. (2002) for the buckling relaxation stage. Their results have still to be explained by theoretical models.

Analogue experiments on single- and multilayer folding have generally
investigated a geometry where the principal bulk shortening direction is
within the layer and the principal extension axis is perpendicular to the
layer. Experiments with oblique layers are technically difficult because the
layer ends tend to slide along the boundaries. Oblique loading of the ends
also introduces unavoidable additional perturbation components, because the
planar boundary is no longer parallel to the axial plane of the developing
folds. Grujic and Mancktelow (1995) carried out pure and simple shear
analogue experiments, where the intermediate axis was perpendicular to the
layer (i.e. both the principal shortening and extension directions were
within the layer). Models were generally constructed of power-law (

Davy and Cobbold (1991) modelled the lithosphere as two, three or four layers: brittle crust (quartz sand), ductile crust (silicone), brittle mantle (quartz sand) and ductile asthenosphere (sugar solution). Variation in the rheology with depth (e.g. temperature dependence of viscosity) was not considered in the simplified model but the potential effects of erosion were. They investigated the interplay between buckling and lithospheric thickening, showing that thickening style is mainly dependent on mantle behaviour, as well as demonstrating the effect of low degrees of coupling, when the brittle crust can detach and buckle independently of mantle layers.

Martinod and Davy (1994) modelled the development of periodic instabilities in continental and oceanic lithosphere under compression. The lithosphere was modelled as a stack of alternating brittle (quartz sand) and ductile (silicone putty) layers. As with Davy and Cobbold (1991), there was no vertical variation within the layers themselves. For small strain, the deformation modes mainly depend on the spatial distribution of the brittle layers and the amplitude of buckling is an exponential function of horizontal strain, as would be expected for folding (Eq. 9).

The terms “boudin” and “boudinage” were first introduced by Lohest et al. (1908) and Lohest (1909) as a descriptive term for sausage-like structures (hence “boudin”, which is a French word for blood sausage) that they observed in the High-Ardenne Slate Belt, which were developed in psammitic layers embedded within a more pelitic matrix. However, recent studies now interpret these classic “boudins” of Lohest and co-authors to be in fact “mullions” (Urai et al., 2001; Kenis et al., 2002, 2004), developed due to layer shortening. “Pinch and swell” was already used by Matson (1905) as a purely descriptive term for the geometry of peridotite dykes from near Ithaca, New York, but without a sketch and with the implication that this was an original intrusive rather than tectonic structure. A short but relatively comprehensive summary of early literature on boudinage is given by Cloos (1947). By this time, there were already descriptions in the literature of more ductile pinch-and-swell structures (e.g. Ramsay 1866; Harker 1889; Walls 1937), but Cloos (1947) concentrated more on examples involving fracture and interpreted the initial fractures as tension joints normal to the direction of extension. However, he notes that “the barrel shape of the classical boudins is somewhat puzzling but seems to be a function of incipient flowage in the competent layer”. Fracture development producing rectangular or barrel-shaped boudins is promoted by the dynamic (or tectonic) underpressure inherently developed in an extending competent layer, in contrast to the overpressure developed if the layer is shortened (Mancktelow, 1993, 2008). This under- or overpressure is associated with corresponding refraction in the principal stress axes in the more competent layer (Mancktelow, 1993), so that extensional fractures are nearly perpendicular to layering, as typically reported for brittle boudins (e.g. Cloos, 1947). As discussed by Rast (1956), the difference in behaviour between barrel-shaped and lozenge-shaped boudinage directly reflects the mechanical response of the layer: if the layer is effectively elasto-plastic (i.e. “brittle”) it develops extensional fractures (joints) and rectangular or barrel-shaped boudins; if viscous flow dominates (at least initially), mechanical instability will lead to necking and the development of pinch-and-swell or lozenge shapes.

We focus here on studies investigating ductile necking instability. Many studies on boudinage consider brittle boudinage or study deformation with an initial configuration where the competent layer is already broken or already includes weak layers separating the layer. Such studies are useful to investigate the kinematic evolution of boudins but yield no insight into the necking instability. An extensive review of boudinage and necking is also provided in Price and Cosgrove (1990).

Galilei (1638) performed one of the first experiments to test the tensile
strength of columns (Fig. 8) and the first mathematical study of necking was
probably by Considère (1885) (see also Dieter, 1986, in his Sect. 8-3).
Assuming a homogeneous layer with thickness

The analysis above, which is based on the early work of Considère,
assumes that the flow stress is only dependent on strain. A similar analysis
can be done for a material that is both strain and strain-rate sensitive. The
strain-rate sensitivity is described by a standard, strain-rate hardening
power-law viscous flow law; i.e.

Smith (1975, 1977) applied the stability analysis to both folding and necking
of linear and power-law viscous layers embedded in a linear and power-law
viscous medium. He showed that the dominant wavelength solution for folding
and necking is identical (for the same material parameters), but that the
corresponding amplification rates for folding and necking are different
(Fig. 14). The maximal amplification rate for necking, namely the maximum
from Eq. (18) for

Neurath and Smith (1982) also performed necking experiments with wax models
in addition to the folding experiments. For necking the measured
amplification rates where significantly higher (a factor of 2–3) than the
theoretical ones (Neurath and Smith, 1982). They suggested that the
discrepancy could be due to some kind of strain softening by which the power-law exponent would increase with increasing strain. They show analytically
that strain softening can be described with an effective power-law stress
exponent

A simple 1-D analytical solution for the evolution of thinning during necking
of an incompressible power-law layer can be found by assuming that the layer
is free (no embedding medium) and that plane sections in the layer remain
plane during necking (Emerman and Turcotte, 1984; Schmalholz et al., 2008).
The extension rate parallel to the layer can then be expressed by a change of
the layer thickness, that is

Comparison with numerical simulations shows that the above simple analytical
solution provides reasonably accurate results for the evolution of thinning
(

Some approximate solutions for the dominant wavelength and maximal amplification rate for necking are listed in Table 2.

Theoretical studies on small-scale multilayer necking are rare in the geological literature. Most analytical multilayer necking studies have been applied to large-scale necking and lithospheric extension (see below). Most theoretical studies have considered brittle boudinage in order to calculate the stress field in multilayers under extension or to calculate the stress fields for layers with pre-existing vertical fractures, in which case the initial fracturing process itself has not been investigated (e.g. Strömgård, 1973; Mandal et al., 2000).

Cobbold et al. (1971) showed that if the theory of internal instability for folding, as developed by Biot (1957, 1964a), is used for a compression direction orthogonal to the anisotropy orientation, then structures can form that are similar to pinch-and-swell structure (they also used the term internal boudins).

Artemjev and Artyushkov (1971) were probably the first to suggest that rift
systems are caused by crustal thinning due to a necking instability during
lithospheric extension. It was subsequently shown that lithospheric necking
for slow spreading rates (1–3 cm yr

The impact of gravity on necking can be also quantified by an Argand number,
namely the dimensionless ratio of gravitational stress to extensional
stress (Fletcher and Hallet, 1983)

Recent studies on magma-poor rifted margins have identified so-called necking
zones in which the crustal thickness is strongly reduced from a normal
thickness of 30–35 km to about 5–10 km (Peron-Pinvidic and Manatschal,
2009). These necking domains separate the proximal domain from the
hyperextended domains in which the continental crust is strongly thinned
(e.g. Sutra et al., 2013). Recent studies on magma-poor margins indicate that
the continental lithosphere can be significantly extended and necked over
several hundreds of kilometres without a lithospheric breakup, which would
result in the formation of new oceanic crust at a mid ocean ridge (Sutra et
al., 2013). Assuming that pre-rift (initial) geometrical perturbations of
crustal thickness have an amplitude (

Lithospheric extension, rifting and associated sedimentary basin formation in a number of regions worldwide have been attributed to mainly lithospheric necking, such as the region around the Porcupine and Rockall basins in the southern North Atlantic (Mohn et al., 2014; Fig. 7b), the Baikal rift (Artemjev and Artyushkov, 1971) or the western Mediterranean back-arc basin (Gueguen et al., 1997). Furthermore, most kinematic or semi-kinematic (including flexure) models of lithospheric thinning and associated sedimentary basin formation implicitly assume a continuous necking of the lithosphere (McKenzie, 1978; Kooi et al., 1992). Such thinning models are of practical importance for the assessment of hydrocarbon reservoir potential in extensional sedimentary basins (see applications).

Necking has also been suggested to be the controlling process for slab
detachment (Lister et al., 2008; Duretz et al., 2012). Detachment of an
oceanic slab usually occurs when the corresponding ocean is closed and
continental collision has started. The cold and dense oceanic slab is then
hanging more or less vertically in the mantle and is attached to the
overlying, less dense continent. The downward extension is controlled by the
negative buoyancy of the cold slab in the warmer mantle. The analytical
necking solution of Eq. (41) has been applied to show the feasibility of slab
detachment (by using the buoyancy as the driving force

There are fewer experimental studies on the development of viscous
pinch-and-swell necking for several reasons. First, as shown theoretically
by Smith (1975, 1977), Emerman and Turcotte (1984) and Eq. (37) above, the
dynamic growth rate of necking in linear viscous materials is zero. Whereas
for folds the kinematic or passive growth rate due to the homogeneous
component of background strain is

Ramberg (1955) performed compression experiments orthogonal to layering of
layered cakes of putty, plasticene and cheese, with either 1-D or 2-D
compensating extension. The resulting structures are similar to natural
boudinage and pinch-and-swell structure, but such models, like the models of
Hall (1815) on folding, were more illustrative than quantitative. Griggs and
Handin (1960) studied the mechanisms of deep earthquakes and performed
extension experiments with natural rock, but not necessarily scaling length, time
and temperature. Amongst others, they performed experiments with Hasmark and
Luning dolomite and Eureka quartzite layers embedded in Yule marble for
confining pressures of 200 and 500 MPa (2 and 5 kbar) and temperatures of
800

In addition to their experiments on folding in power-law viscous materials, Neurath and Smith (1982) also performed necking experiments. The measured amplification rates where significantly higher (a factor of 2–3) than the theoretical ones and they suggested that the discrepancy could be due to some kind of strain softening, by which the power-law exponent would increase with increasing strain. They showed analytically that strain softening can be described with an effective power-law stress exponent given in Eq. (39). Ghosh (1988) conducted experiments with plaster of Paris resting on a substrate of pitch with equal stretching of the layer in all directions to investigate 2-D chocolate–tablet structure. The study was designed to consider the geometry during progressive development and from the materials chosen could only develop brittle boudins rather than the pinch-and-swell structures considered here. Kidan and Cosgrove (1996) used the same rig employed in earlier folding experiments (Cobbold 1975a, 1975b; Cobbold and Knowles, 1976) to investigate multilayer boudinage, using layers of paraffin wax and plasticine. Their experiments generally developed rectangular boudins due to (sequential) fracturing but internal pinch-and-swell structure in some cases developed on a larger scale, reflecting the overall anisotropy.

More recent experimental studies on boudinage in 2-D or 3-D have used
specially designed rigs (Zulauf et al., 2003) and power-law materials with
high stress exponents, such as plasticine with

Marques et al. (2012) used layers of viscoelastoplastic clay or elastic soft paper in linear viscous PDMS silicone putty to investigate the influence of layer thickness and bulk strain rate on the average boudin width for brittle boudins. Although their natural measurements from south-west Portugal show a clear linear relationship between layer thickness and boudin width, as would be expected from elastic theory, the average boudin width in their experiments shows an exponential dependence on layer width and a power-law dependence on bulk strain rate.

Equations (5) and (8) for elastic and viscous folding, respectively, are linear and the corresponding solutions are periodic; i.e. they can be expressed with trigonometric functions such as cosine or sine. However, most natural fold systems are not strictly periodic but irregular and sometimes localised. Localised folding is characterised by the existence of large amplitudes only over a small region of a shortened layer (after Wadee, 1999). The reason for irregular and localised fold geometries has been controversially discussed in the last decades (e.g. Zhang et al., 1996; Mancktelow, 1999; Schmid et al., 2010; Hobbs and Ord, 2012). There are essentially two fundamental reasons for irregular and localised fold geometries: (1) geometrical and material heterogeneities, and (2) material softening.

Considering the first reason, if linear equations for folding are considered,
then irregular and localised fold geometries can result from (i) an irregular
and localised initial geometry of the layer or (ii) from non-homogeneous
material properties. Concerning (i), in the thin-plate approach one usually
assumes that the layer has initially a constant thickness but that the layer
is initially not perfectly straight; for example, it can have the shape of a
bell-shaped function (Eq. 26; Fig. 16). Stability analysis can consider
initial irregularities either as deviation from a straight layer having
constant thickness or as initial variations in the layer thickness. The
thin-plate approach and the stability analysis can also include the impact of
non-linear rheologies, such as a power-law flow that is strain-rate hardening
(stress increases with increasing strain rate), but these flow laws are in
practice linearised to provide accurate solutions provided fold amplitudes
are small (i.e. limb dips smaller than

Considering now the second reason for localised fold shapes, other types of
non-linearities have also been studied with the thin-plate approach, whereby
the resistance of the material in which the layer is embedded (often termed
the matrix or foundation) is assumed to be non-linear. The linear term for the
matrix resistance in Eq. (5) is usually

Geometrical and material heterogeneities are intuitive reasons for observed irregular fold geometries because natural rock layers are never perfectly straight or homogeneous before folding. Geometrical non-linearities are intrinsic for folding because they arise naturally due to the deviations of the folded layer from the initially straight layer. Linearised equations can predict the fold shapes up to amplitudes for which the final irregularities can already be seen, such as for an initial bell-shaped perturbation (Fig. 16). Non-linearities due to material softening, such as a non-linear matrix resistance, are more difficult to justify, and especially quantify, in a straightforward manner. Non-linear matrix resistance is usually justified by some kind of material strain softening (e.g. Hobbs and Ord, 2012). However, this softening process is usually defined a priori and it is not clear what micromechanical processes actually causes such particular non-linearities related to softening. Typical candidates responsible for softening are, for example, grain size reduction, mineral reactions or fluid-rock interaction. The impact of strain-rate softening on folding has been investigated also with numerical simulations (e.g. Hobbs et al., 2011)

This review focuses on analytical solutions, with some reference to the analogue models that were often used to qualitatively or (semi-)quantitatively test these analytical solutions. However, since the late 1960s more and more numerical studies of folding and necking have been performed. One of the first numerical simulations of folding in a geological context was carried out by Dieterich (1969) and Dieterich and Carter (1969). Stephansson and Berner (1971) already applied the finite element method to various tectonic processes such as folding, deformation of isolated boudins, isostatic adjustment and spreading at the mid-Atlantic ridge.

Numerical simulations are essential to study folding and necking scenarios for which analytical solutions cannot be derived or for which only approximate analytical solutions exist. Such scenarios are for example (i) the finite amplitude evolution of folding and necking in 2-D and 3-D for which only approximate analytical solutions are available (Chapple, 1968; Kaus and Schmalholz, 2006; Schmalholz, 2006; Schmalholz et al., 2008; Schmid et al., 2008; Grasemann and Schmalholz, 2012; Fernandez and Kaus, 2014; Frehner, 2014; von Tscharner et al., 2016), or (ii) the numerical solution of non-linear folding equations (see Sect. 4.1.; e.g. Hunt et al., 1997; Wadee, 1999). A typical application for numerical simulations is, for example, the study of fold propagation (or serial folding) where folding in a competent layer starts from a localised geometrical perturbation and new folds develop sequentially away from the initial perturbation along the layer (such as shown in Fig. 16). Such fold propagation has been studied in 2-D in single- (e.g. Cobbold, 1977; Mancktelow, 1999; Zhang et al., 2000) and multilayers (Schmalholz and Schmid, 2012) and in 3-D in single layers (Frehner, 2014).

Many numerical simulations of folding consider a layer-parallel compression of the layers and the bulk deformation of the model is close to pure shear. Folding of layers under bulk simple shear has been studied numerically for single layers (Viola and Mancktelow, 2005; Llorens et al., 2013a) and multilayers (Schmalholz and Schmid, 2012; Llorens et al., 2013b). A main result of the simple shear studies is that folding under bulk simple shear does not generate asymmetric fold shapes but more or less symmetric fold shapes similar to the ones generated under bulk pure shear (cf. Lister and Williams, 1983). Also, when layers rotate in a simple-shear zone they can be first shortened until the fold train is more or less orthogonal to the simple-shear zone. Further shear and rotation, however, extends the fold train, which can unfold the layers again (Llorens, et al., 2013b). Laboratory experiments of such single layer folding and unfolding under bulk simple shear have been already performed by Ramberg (1959).

For active folding, a continuous competent layer is actually not required. Adamuszek et al. (2013a) showed that it is sufficient for folding to have competent inclusions (that can be of various sizes) aligned and clustered in a way to form a “layer” of inclusion clusters. If this “layer” of inclusions is embedded in a weaker viscous medium then the layer-parallel shortening also generates folding of the layer consisting of individual inclusions. Adamuszek et al. (2013a) applied their results to a folded sequence of alternating nodular limestone and shale.

Numerical simulations of the extension of power-law viscoplastic (von Mises; Schmalholz and Maeder, 2012) and power-law viscous (Duretz and Schmalholz, 2015) multilayers embedded in weaker power-law viscous medium showed the formation of individual shear zones that crosscut the entire multilayer. The shear zones form after some period of distributed multilayer necking and only occur (i) when the weak inter-layers are power-law viscous and (ii) when the spacing between the competent layers is less than or approximately equal to the thickness of the competent layers. The shear zones crosscutting the entire multilayer form due to the alignment of individual necks in different layers, which is a finite amplitude effect when individual necking zones can form a connected network of weak zones.

The numerical studies mentioned above investigated fundamental mechanical folding and necking processes, but numerical solutions are also useful to study the coupling of folding and necking with other processes such as (i) the generation of heat during folding due to dissipative rock deformation (shear heating) and the related thermal softening caused by thermo-mechanical feed-back with temperature-dependent rock viscosity (Hobbs et al., 2007, 2008; Burg and Schmalholz, 2008), (ii) the conversion of macroscale mechanical work into microscale mechanical work during the reduction and growth of mineral grain size and related softening due to grain size reduction (Peters et al., 2015), (iii) the impact of metamorphic reactions on rock deformation (Hobbs et al., 2010), (iv) coupling of crustal folding or necking with erosion in 2-D (Burg and Podladchikov, 2000; Burov and Poliakov, 2001) and 3-D (Collignon et al., 2015) or (v) the coupling of folding with salt diapirism (Fernandez and Kaus, 2014). A detailed outline of a coupled thermodynamic approach to study rock deformation and the resulting structures is given in the recent textbook by Hobbs and Ord (2014). The impact of shear heating and grain size reduction on lithospheric folding and necking can be significant because both processes cause a mechanical softening of the rock (e.g. Regenauer-Lieb and Yuen, 1998; Regenauer-Lieb et al., 2006). For example, during shortening of the continental lithosphere, shear heating and thermal softening can cause a transition from distributed folding to localised ductile thrusting (Burg and Schmalholz, 2008; Schmalholz et al., 2009; Jaquet et al., 2016).

Numerical simulations are based on a basic set of partial differential equations resulting from the concepts of continuum mechanics. These equations are useful to describe continuous deformation and strain localisation by shear bands (with no loss of velocity continuity). Elaborated numerical algorithms based on continuum mechanics, the so-called extended finite element method or XFEM (Belytschko et al., 2001), are additionally able to model discontinuous fracture, for example due to 3-D folding (Jäger et al., 2008). In geological studies it is more common to apply so-called discrete element methods to study brittle deformation and fracturing. In simple words, these discrete models assume that a material consists of particles that are connected by elastic springs. The force balance is controlled by Newton's law (force equals mass times acceleration), which is an ODE (no spatial derivatives) and not a PDE. A fracture appears when the stress in a spring connecting two particles exceeds the yield strength and the spring connection between the two corresponding particles is then removed. Discrete element modelling has been applied, for example, to study fracturing during detachment folding (Hardy and Finch, 2005) or to study the evolution of brittle boudinage in 2-D and 3-D (Abe and Urai, 2012; Abe et al., 2013).

Recently, Adamuszek et al. (2016) developed the
MATLAB^{©} based software termed Folder, which can be
used to numerically model folding and necking in power-law viscous single-
and multilayers. Folder is freely available under

The stability analysis (Fletcher, 1974; Smith, 1975, 1977) can be used to study the initial, small amplitude stages of both folding and necking. Folding and necking result from the same type of mechanical instability. This instability causes initial geometrical perturbations on the layer interface to amplify with velocities that are faster than the velocities corresponding to the applied bulk deformation (e.g. pure shear). The dominant wavelength for folding and necking is identical for the same material parameters (Fig. 14). Amplification rates for folding and necking increase with increasing viscosity ratio and with increasing power-law stress exponent in both the layer and the embedding medium (Fig. 14).

Folding and necking are processes that can take place in single and multilayers and can also act on all scales. For large-scale folding and necking, gravity decreases the intensity of the folding and necking instabilities. The impact of gravity on folding and necking is usually quantified by some kind of Argand number, which is the ratio of the gravitational stress to the layer-parallel stress driving compression or extension, respectively.

Folding and necking are both associated with structural softening (Fig. 18). The effective viscosity of a rock unit consisting of competent layers embedded in a weaker medium during shortening and extension with a constant bulk rate of deformation can be calculated by the ratio of the area-averaged stress to the bulk rate of deformation. If the shortening and extension would be homogeneous pure shear at a constant rate and the layer would deform by homogeneous thickening and thinning, then the effective viscosity of the layered rock unit would remain constant. However, if folding or necking takes place the effective viscosity decreases during bulk shortening and extension, respectively, because the area-averaged stresses are smaller during folding and necking than during pure shear thickening and thinning. Related to the stress decrease is a decrease in viscous dissipation (i.e. stress times strain rate) and mechanical work rate (i.e. product of boundary stress and velocity integrated over the boundary of the rock unit). The structural softening related to folding and necking hence reduces the mechanical work required to deform the layered rock unit. Therefore, folding and necking are the preferred deformation modes of mechanically layered rock units because folding and necking minimises the work required for the deformation. During structural softening the material properties remain constant and for both linear and power-law viscous material the flow laws are always strain-rate hardening; i.e. the stress increases with increasing strain rate. Hence, structural softening is fundamentally different to material strain softening, where some material property (e.g. cohesion, friction angle or effective viscosity) decreases with progressive strain.

During folding the layer thickness remains more or less constant and the shortening is compensated by a lateral deflection (Fig. 4). During necking the local variation in layer thickness is significant and the extension is compensated by localised thinning of the layer while the central axis of the layer remains more or less straight (Fig. 4).

In folding, a particular wavelength can be selected and “locked in” if the
fold arc length does not vary significantly during fold amplification, which
is the case for large viscosity ratios (

Shortening of viscous single- and multilayers generates folds. Different multilayer configurations and flow laws can generate a wide variety of fold shapes but the preferred deformation mode will always be folding. Extension of power-law viscous single layers can generate necking, for which there is essentially no simple shear deformation within the layer around the necking zone (i.e. plane sections remain plane). In contrast, during extension of power-law viscous, embedded multilayers individual shear zones can form, which crosscut the entire multilayer and hence generate a significant simple shear deformation within the multilayer. Therefore, the general deformation mode for shortening competent layers is independent of the single- or multilayer configuration, whereas for extension of competent layers the deformation mode can be dependent on this configuration.

The maximal amplification rates for folding and necking for the same material
parameters are significantly different (Fig. 14). Amplification rates for
folding are larger than the ones for necking for the same material
parameters. While folding occurs for linear and power-law viscous rheologies,
necking only occurs for power-law viscous rheologies. Since the amplification
rates for necking are smaller than the ones for folding, significant necking
(

The yield stress for brittle fracture is typically described by a Mohr–Coulomb failure criterion, which is based on parameters determined by Byerlee (1978; Byerlee's law). These yield stresses are usually 4 times larger during compression than during extension (Sibson, 1974), due to the implicit development of dynamic over- and underpressure, respectively (Mancktelow, 1993, 2008). Hence, layers under layer-parallel compression can deform viscously up to much larger stresses than layers under layer-parallel extension before fracture occurs. The available stress range for folding without fracturing is therefore much larger than the stress range for necking without fracturing.

The range of material parameters and of flow stresses for significant necking is significantly smaller than the corresponding range for folding, which may be the main reason why pinch-and-swell structure is less frequent in nature than folding, and also why brittle boudinage is more frequent than pinch-and-swell structures.

The main direct applications of analytical and numerical solutions for
folding are the estimation of (i) the bulk shortening that was necessary to
generate an observed fold and (ii) the viscosity ratio during the formation
of the observed fold (Sherwin and Chapple, 1968; Talbot, 1999; Hudleston and
Treagus, 2010). Natural rock viscosities are commonly estimated using
laboratory-derived flow laws but the extrapolation from laboratory
(10^{©}
based software, the fold geometry toolbox, which determines automatically the
values of

Fold geometries can also be used to estimate the dominant folding mechanism. Schmalholz et al. (2002) distinguished three types of folding mechanism depending on the controlling material parameters: (i) matrix-controlled folding (controlled by viscosity ratio between layer and matrix), (ii) detachment folding (controlled by the thickness of the weak layer below a strong layer) and (iii) gravity folding (controlled by the ratio of gravity to viscous stress, namely the Argand number, Eq. 34). They presented a diagram that allows estimation of the dominant folding mechanism from the fold geometry alone.

Numerical simulations of necking have shown that during necking initially straight and vertical lines remain vertical and straight (Schmalholz et al., 2008). This feature justifies the application of thermo-kinematic models to lithospheric necking and the associated formation of sedimentary basins (e.g. McKenzie, 1978; Kooi et al., 1992). These thermo-kinematic models subdivide the lithosphere laterally into a series of vertical columns, whose independent thinning is quantified by thinning factors. Such models have been applied to reconstruct the thermo-tectonic history of extensional sedimentary basins, which is useful to evaluate the potential of hydrocarbon reservoirs. Two-dimensional thermo-kinematic models of lithospheric extension are significantly faster to compute than 2-D thermo-mechanical models and can hence be used efficiently in combination with automated inversion or optimisation methods (Poplavskii et al., 2001; White and Bellingham, 2002; Ruepke et al., 2008).

The mathematical solutions for folding and necking have also been used to assess the deformation style of the outer shell of the moons of Jupiter. Dombard and McKinnon (2001) investigated the grooved terrain of Ganymede and argued that the regular structural periodicity found in this grooved terrain could be the result of an extensional necking instability. Dombard and McKinnon (2006) also argued that topographic undulation, with a ca. 25 km wavelength, observed on Jupiter's icy moon Europe could be due to contractional folding.

Significant progress has been made in understanding and quantifying the mechanical processes of folding and necking since the pioneering folding experiments of Hall and the pinch-and-swell observations of Ramsay (1866). The geometry and mechanical evolution of many fold trains can be explained by the dominant wavelength theory of Biot (1957) and Ramberg (1962) and its elaboration to power-law viscous rheology by Fletcher (1974) and Smith (1977). Folding and necking in viscous layers are the result of a hydrodynamic instability. Folding and necking are the preferred deformation modes because they minimise the mechanical work required to shorten or extend mechanically layered rock on all scales. The most important quantities to analyse folding and necking are the dominant wavelength and the corresponding maximal amplification rate. The two quantities allow the estimation of fundamental parameters relevant for folding and necking, such as the effective viscosity ratio or the Argand number, and also allow an evaluation of whether folding or necking instabilities are sufficient to generate observable fold or pinch-and-swell structures.

Folding and necking instabilities should in principle always be active when
ductile, layered rocks are shortened or extended on all scales. However,
observable folds and necks (pinch-and-swell structure) are usually only
generated when the dimensionless amplification rate

Folds are more frequent in nature than pinch-and-swell structure because folding can occur in layered rock that deform according to viscous and power-law viscous flow laws while necking only occurs in rock with power-law viscous (or other non-linear) behaviour. For the same material parameters the amplification rates for folding are also larger than the ones for necking. Furthermore, stresses during folding (compression) can be significantly larger than stresses during necking (extension) before the rock fails by fracture. Hence, brittle boudinage is more frequently observed than continuous necking.

Despite the vast literature on folding and necking there are still many open questions and challenges. For example, 3-D finite amplitude folding and necking in power-law viscous multilayers have not been investigated analytically and numerically in detail. Particular future challenges are to quantify the coupling of folding and necking with other processes acting during rock deformation, such as fracturing, shear heating, grain-size evolution, fluid flow and metamorphic reactions. The concept of continuum mechanics can provide the system of equations that describes these coupled processes and numerical algorithms will be able to solve these equations. However, these equations and related numerical simulations will include many parameters and one of the biggest challenges may be to determine the controlling parameters (e.g. via dimensional analysis) and to make the coupled thermodynamic processes comprehensible. In that sense, one of the main objectives for future research on folding and necking is encapsulated in the famous statement of J. Willard Gibbs quoted at the beginning of this review.

This is a review article and hence most results and figures in this article
have been taken or modified from already published articles, which are cited
in this review. The topographic data used in Fig. 7a were obtained with the
online tool Geocontext-Profiler
(

Mathematical folding studies either use the thin-plate equation or the stability analysis, which is based on a stream function solution for the full 2-D force balance equations. We show here how the thin-plate equation can be derived from the full 2-D force balance equations, based on the derivation of a general extended thin-sheet equation by Medvedev and Podladchikov (1999a, b). The thin-plate equation for folding is essentially derived by vertical integration and approximation of the 2-D force balance equations.

The force balance equations in 2-D without gravity are

The thin-plate approach of Biot (1961) assumes that only vertical tractions
act on the layer boundaries, that horizontal tractions are negligible and
that

An essential step in the derivation of the dominant wavelength solution for
folding was the derivation of a correct term for the resistance of the
viscous embedding medium, which depends not only on the amplitude,

The result

The value of

We thank Peter Hudleston and Boris Kaus for their reviews, and Dani Schmid, Marta Adamuszek and Marcin Dabrowski for their comment, editor Susanne Buiter for her invitation to contribute this review to Solid Earth and for her comments, and Ray Fletcher for his informal remarks. We are grateful to Yuri Podladchikov for many lessons, discussions and constructive criticism over the last 20 years. Stefan Schmalholz thanks Sergei Medvedev for his valuable help in improving Appendix A. We also greatly appreciate many fruitful and stimulating discussions on folding and necking with Ray Fletcher, Dani Schmid, Sergei Medvedev, Marcel Frehner, Marcin Dabrowski, Marta Adamuszek and Jean-Pierre Burg over the years. Neil Mancktelow thanks John Ramsay for firing an interest in the study of folds and boudins in the field and in the application of analogue and numerical models to better understand their development. Edited by: S. Buiter Reviewed by: P. Hudleston and B. J. P. Kaus