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FOLDING
Folding is a concept that embraces all geologic processes by which surfaces in rocks become curved during deformation. Since folds are permanent deformation structures with no or little loss of cohesion of the folded layer, folding refers to the essentially slow, ductile behaviour of relatively soft and/or hot rocks. Beyond the descriptive, anatomical classifications, much of the early geologic work on folding processes focused on the deformation of stratified sediments. Different folding mechanisms combine a few basic processes involving the geometrical (layer thickness and spacing) and physical (viscosity, viscosity contrast, anisotropy) properties of the rocks. This lecture deals with some consideration on genetic, mechanical aspects concerning the development of folds. The important point to note is that stress alone is insufficient to cause folding: A planar surface must first exist to define the fold shape, and the orientation of this planar marker with respect to the stress direction controls in many ways the attitude of the resulting fold.
Folding processes
Most models of fold formation ignore body forces and the effect of the material enclosing the layers (both are treated as viscous fluids), which in practice has a very important role in determining or modifying the fold geometry. Flat layers may become curved in several ways.
Rotation
Obviously, folding rotates parts of the layers. The continuous change in orientation of the stiff layers with respect to the shortening direction first produces a marked decrease in compressive resistance of the rock mass. This is a form of bulk weakening, which accompanies the progressive modification of the internal geometry of the rock mass without any change in material properties (e.g. viscosity); it is accordingly termed structural softening. Structural softening is followed by an increase in compressive resistance, structural hardening. Softening and hardening are the conditions for onset, growth and decay of instabilities. Evidently, any other mechanism of strain softening that may be present (e.g. a change in the effective material properties caused by processes such as microfracturing, pressure solution, etc...) will also affect the stability of the system.
Lengthening
Fold hinges most often mark the site where the folds began to amplify. This is usually over a short distance so that any fold axis plunges towards the two extremities of the hinge segment. Further limb rotation and subsequent fold growth (amplification) accompanies lengthwise, not necessarily symmetrical migration of the hinge tips.
Mechanical role of layers: Active / passive folding
Any deformation involves displacement of material points, for example particles. A passive particle has no interaction with its neighbours; it only moves. An active particle interacts with its neighbours and its displacement is affected by that of neighbouring particles. In geology, the compositional layering has mechanical properties influencing the strain pattern and the folding process. For example, boundaries between layers with contrasting strengths (viscosity) may slip or localise shear deformation, hence guiding the way curvatures develop. Folding is active when deformation takes place at the layer scale and the strength difference between layers directly affects the deformation pattern. Conversely, passive folding takes place at the grain scale while layers have no significant competence contrasts and so do not generate any stress acting across and/or parallel to the layer boundaries; layering serves merely as a geometrical strain marker. Passive folds grow during heterogeneous flow and their shape reflects the pattern of heterogeneous deformation. Passive folding is experimented by folding a stack of modelling-paste layers (think also of oil traces or scum on slowly flowing water). Active folding is experimented by bending a pile of cards that glide one upon the other.
Folding mechanisms
Folds can result from layer-parallel compression, uneven loading oblique to perpendicular to layers, or from amplification of surface irregularities during deformation flow.
Bending
Bending involves forces applied and acting at high angles to layers that may or may not have competence contrasts. A layer subjected to bending is like a notebook supported at the ends and loaded in the middle. The notebook bends downward when the load is placed in the middle.
Mathematically, modelling the bending of layers works well for small folds, but becomes inexact for large segments of the lithosphere because of inelastic behaviour.
Meso-scale bending
Drape folds and forced folds may occur on outcrop scale around local objects.
Buckling
Buckling is a well-known active mechanism for the development of rounded folds in a competent layer (i.e. a layer with low rate of ductile flow) enclosed in an incompetent (with high rate of ductile flow) medium of sufficient viscosity contrast. Gently pushing the two extremities of a paper sheet on a table towards each other reproduces this folding mechanism. When the force is small the sheet remains flat. As the force is slowly increased, it suddenly becomes curved. This rapid change from a flat to a curved (buckled) form at a particular force is due to the development of a mechanical instability.
Similarly, geological buckling involves the flexural instability of a stiff interface, layer or stack of layers under lateral, i.e. layer parallel compression. The presence of layers of contrasting competence produces a mechanical anisotropy , which is essential for buckling. The strong layer(s) fold(s) while the weaker matrix fills in gaps. Such conditions are usual geological situations and thus explain why buckle folds are very common in the Earth’s crust. Natural buckling systems can be sub-divided into several groups, namely folds formed on: (i) a single interface; (ii) two interfaces, which define a single layer in a matrix; (iii) several layers; and (iv) a mineral fabric such as an earlier cleavage.
Interface buckling
Interface buckling occurs on many scales, from regional scale as the unconformity between the Mesozoic cover and older basement in the Alps, to small-scale examples observable in the field. In experiments of shortening two-layer analogue models, buckling of the interface starts as symmetric sinusoidal deflections that change their geometry as they amplify into alternating cusps (points formed by two intersecting arcs) and lobes (rounded bends). Cuspate-lobate folds are characteristic of interfaces between media of strongly differing viscosity. The cusps always point into the stronger of the two materials.
Thus, in outcrops exhibiting cuspate-lobate forms, it is possible to know at a glance the layer that was the stiffest at the time of folding. Analytical solutions indicate that cuspate-lobate folds may evolve from the kinematic amplification of sinusoidal folds subjected to shortening greater than about 10%. Analytical work has also shown that folding of the interface between two linear viscous materials is
not a mechanically instability; mechanical instabilities can develop only in materials with power-law viscous flow laws. However, the growth of cuspate-lobate folds is enhanced in materials with non- linear viscosity.
Single layer buckling
Buckled single layers are common in nature, for example, an isolated sandstone or limestone bed in a thick shale or marl sequence or a vein of igneous rock intruded into an unlayered matrix. The buckled layer maintains its thickness throughout, thus producing a parallel, concentric fold. Experimental buckle folds are usually symmetric.
Multilayer buckling
A multilayer is a package of different layers, which is the most common situation in geology: a sedimentary succession is often a more or less regular alternation between two or three rock types (e.g. sandstones and shales in turbidites). The alternating layers have variable thickness and competence. Theoretical and experimental studies have shown that the behaviour of a multilayer depends upon a number of factors, e.g. the number and thickness of competent layers, the spacing between the competent layers, the competence contrast among the layers and the competence of the medium confining the multilayer.
contact strain extends away from the layer into the matrix. For a viscous matrix, the disturbance in the zone of contact strain has died down to approximately 1% of its maximum value at a distance of about one wavelength from, and on either side of the folded layer.
Influence of discontinuities: Flexural-slip and flexural flow:
A multilayer can be a pile of competent layers separated by surfaces of discontinuity or alternating layers of highly contrasting competence. The mechanical consequence is that the competent layers on either side of the surface of discontinuity or of a weak layer may easily slide relative to each other. This shear “decoupling” of layers allows a fold to accommodate a greater flexure than if the stack deforms as a single layer. Flexural-slip describes discrete faulting, usually coincident with bedding planes and accompanying folding. A classical simulation is to bend a book or pile of paper sheets; increasing bending about the fold axis is accommodated by increasing slip between the pages of the book or sheets of the pile. The the thickness of individual sheets does not change, meaning that each sheet makes a parallel fold (i.e. layer surfaces remain parallel). Slip is an important part of folding because layer-parallel stresses increase with increasing rotation of the limbs.
When the shear stress exceeds shear resistance of weak layers or layer boundaries, the strong layers in the limbs slip over each other towards and usually perpendicular to the hinges, which are fixed
from layer to layer. Therefore, slickensides and fibrous mineral growth or other movement indicators showing reverse dip-slip on bedding planes within fold limbs are common criteria for flexural slip. Slip values are greatest at inflexion points, on the limbs, and decrease to zero at the fold hinge. The amount of displacement increases as folds tighten and also depends on the spacing of slip planes. Note that hingeward slip implies opposite movement directions from one limb to the next, yet consistency is maintained from anticline to syncline. Structural variations include ramp faults connecting separate layer-parallel faults and duplex contained within layer-parallel floor and roof faults. The temperature and pressure at which flexural- slip folding occurs are generally low. Flexural flow describes bedding-parallel shear homogeneously distributed within the ductile layer being folded between stiffer layers. Like for flexural slip, bedding-parallel shear in limbs is opposite across the axial plane. The strain pattern due to hingeward shear tends to develop thickened hinges between thinned limbs, i.e. flexural-flow folds are mostly similar. Flexural-flow is sometimes applied to the weak layers that take up bedding-parallel motion within larger parallel folds, generally under low metamorphic grade. In this case, the stiff, active layers tend to keep their thickness throughout the deformation to produce and control the overall shape of concentric and/or parallel folds while the incompetent layers undergo flexural flow. In order to maintain similarity from bed to bed, ductile material moves out of the limbs into the hinges. Natural examples of such similar folds show intense foliation in the fold limbs, which dies away from limbs towards hinge zones. The intensity of shear strain depends on fold shape and position within the fold, with shear strain equal to limb dip in radians.
Influence of anisotropy
A bedding-parallel anisotropy is an intrinsic property of multilayers. Theoretical and experimental work on homogeneous, anisotropic multilayers shows that there is a range of fold shapes that can form under anisotropy-parallel compression. The type of fold is determined by the mechanical anisotropy. Symmetric, sinusoidal folds in multilayers with weak anisotropy give way to folds with gently diverging axial planes and ultimately to box folds in multilayers with high anisotropy. Fold shapes propagate away from the folded layer much farther into an anisotropic than in an isotropic matrix.
Shear folding
Differential slip along closely spaced planes or simple shear on closely spaced shear zones parallel to the axial surface and oblique to the folded layer produces ideally similar folds. This passive mechanism is called shear or slip folding.
Kinking – angular folding: Effect of mechanical anisotropy
Kinks have straight limbs between sharp to angular hinges whose axial planes define the kink band boundaries more simply termed kink planes. Short limbs define the kink bands. Kink bands occur in strongly anisotropic rock where the anisotropy is either beds with a finite thickness or foliation
with very thin layers. Their particular geometry is controlled by the rotation through an angle α of a
set of thin layers within the kink bands. Ideally, kinking involves no internal strain in the layers, only rotation around the kink hinges. Therefore, flexural slip in the limbs is inherently linked to kinking to insure the continuity of layers across the kink band boundaries.
The formation of kink bands is predicted by theoretical analysis of the viscous deformation of materials with a strong planar anisotropy. The models differ in the way the kink grows and in the geometry of the deformation, which is specified with two angles: βi between the kink plane and the within-kink layers and βe between the kink plane and layers out of the kink band. There are two main mechanisms:
- Model 1: kink band boundary migration (also termed mobile hinge).
- Model 2: kink bands as shear zones (fixed hinge) Kink band boundary migration
The two kink band boundaries migrate away from a central nucleation line into the undeformed material. In this case angles βi and βe remain constant in the widening kink bands.
Kink band shear zones
The two kink band boundaries mark the fixed boundaries of a shear zone at the onset of kink band development. In this case, the kinked segment maintains a constant length during shear-induced rotation. If βi > βe , dilation must take place between the kink-band layers. Rotation larger than βi = βe causes layer thinning, which might be a blocking factor.
Experiments revealed that kink bands commonly occur in conjugate sets with opposing asymmetry when the maximum compressive stress is (sub-) parallel to a pre-existing planar anisotropy. However, they do not develop along planes of high shear strain, which indicates that they are not true shear zones. Still, by analogy with faults, the apparent and relative displacement of long-limb layers across the kink band defines three sorts of kink bands:
- Normal kink bands in which there is a volume decrease in the kink band.
- Reverse kink bands in which there is a volume increase in the kink band.
- Neutral kinks in which the volume remains constant.
- The deformation has only involved a plane strain.
Controls of fold wavelength
One considers that a single viscous layer embedded in a matrix of lower viscosity is very thin compared to the fold wavelength (the so-called thin-plate theory). Buckling produces a fold system that has a symmetric, periodic, sinusoidal shape. The analysis deals with the nucleation, i.e. the investigation is limited to very small amplitude, first buckle folds resulting from infinitesimal deformation.
Theory
In mathematical treatment, if a laterally compressed layer is perfect, then it simply thickens during shortening without folding. An imperfection is required to induce buckle folding. This initial imperfection might be present in the layer prior to the imposition of compressive stress or may be a local instability that develops while compression is applied. Technically, it is simulated with one or several superposed low-amplitude sinusoidal functions that describe the layer boundaries. The theory first assumes that the medium that confines the layer resists layer-perpendicular deflection. Then, the most-stable shape is the one that needs least amount of layer-parallel stress, i.e. the least elastic strain energy in both the layer and the surrounding material to emerge spontaneously. Results indicate that, although all of the primary irregularities might start to grow, only one sinusoidal, regular fold train with one particular wavelength grows preferentially as deformation proceeds. This most stable, selected and amplified sinusoidal response is the dominant wavelength.
Dominant wavelength
Two key factors control the dominant wavelength:
- the layer thickness;
- the viscosity ratio (the strength contrast) between layer and matrix, which both are treated as Newtonian viscous materials.
For a single competent layer of thickness h and viscosity μL embedded in a weaker matrix of infinite
thickness and viscosity μ (^) M. Internal forces (^) Fint (resistance of the competent layer) and external
forces Fext (matrix resistance) act together against the development of a fold with first wavelength Wi : 2 3 L x int (^2) d x
2 h e F 3W e
π μ
M d x ext x
W e F e
μ
π
The model is shortened at a rate ε^ x =dex dt by the amount e (^) x in the x-direction. The dominant
wavelength ( Biot-Ramberg analysis ) is the wavelength with the smallest total force
( Ftot^ =^ Fint^ +^ Fext). The following equation expresses the initial dominant wavelength^ Wd:
( ) 1 3 Wd = 2 hπ μ (^) L 6 μ (^) M (1)
This relatively simple relationship has been experimentally and numerically verified and is applicable only to small-amplitude folds. Equation (1) clearly states that:
- The wavelength is independent of both the amounts of compressive load and the strain rate.
- The wavelength is directly proportional to the thickness (^) h of the competent layer; thus, different wavelengths will arise in different layers of variable thickness, in all of which the shortening strain is constant. Thicker layers produce longer wavelengths. Variable intensities of fold development do not indicate variable intensities of deformation.
- The wavelength depends only on the cube root of the layer-to-matrix viscosity ratio.
The impact of the strength ratio on the wavelength/thickness ratio (hence fold style) can be visualised by rearranging equation (1):
( ) 1 3 Wd h = 2 π μL 6 μM
Note that in this equation the ratio (^) μL μ (^) M is merely the ratio of viscosity between the layer and its
embedding material. A related feature is that as μL approaches μ (^) M, the dominant wavelength
approaches a value of 3.46 h.
Equation (1) can be reorganized as:
( )
3 μL μM =0.024 Wd h
indicating that the viscosity contrast can be approximated from wavelength and thickness measurements.
A 0 is the amplitude of the initial, sinusoidal perturbation. Its presence in this equation implies that
the amplitude of the initial perturbation influences the final geometry of the waveform. Finite amplitudes of folds may reflect the original variations in amplitude of existing irregularities such as ripple marks as much as the competence contrast and the imposed bulk shortening.
P A is the amplification factor , which gives the amplification rate. It integrates the sum (^) ( A (^) k +Ad)
but the total expression, that also includes viscosity contrast and layer thickness, is quite complex. If amplification is exponential, it should enter an “explosive” mode when the incremental amplification should increase enormously with respect to shortening for high amplification factors. Estimates suggest that this should occur when the amplification factor of the dominant wavelength is approximately 1000 and for high values of viscosity contrast. Complex expressions avoid this problem.
History of buckling
A contrast in competence among the associated layers is essential for buckling. The development of a buckle fold is an unstable process conveniently divided into four stages:
- Layer-parallel shortening.
- Nucleation of the buckling instability.
- Amplification of the buckle-fold.
- Locking up and shortening in pure shear. 1) Incubation: Initial homogeneous shortening
In experimental buckling, compressed layers do not produce folds for the first 20% or so of shortening. Instead, the individual layers increase their thickness essentially to compensate homogeneous, layer-parallel shortening. The amount of homogeneous, elastic and inelastic strain before buckling begins is a function of the strain rate and of the relative mechanical properties of the layers undergoing buckling. The layer thickness remains constant; therefore, there is no shear strain within and parallel to the shortened layers.
2) Nucleation
Buckle initiation is difficult and generally requires some form of perturbation on the initially shortened/thickened layer. Nucleation involves rotation of the layering at selected sites where there are inherent (e.g. initial bedding variation) or generated (e.g. local fluctuation in applied boundary stress) heterogeneity in the deformation. A certain wavelength of perturbation is selectively amplified. This amplification builds the buckle folds; the selected wavelength is related to the mechanical character of the stiff layers.
3) Amplification
Amplification is the progressive vertical growth of the fold. Theoretical studies of folding and the resulting concept of dominant wavelength are normally only valid for the first increment (i.e. nucleation) of buckling, when fold amplitude is so small that it is practically invisible. Once buckling has been initiated, shortening can continue by rotation of the limbs. Buckling becomes progressively easier, and the dip of limbs increases rapidly compared to the rate of bulk shortening. Buckling is at that stage a structural softening process, i.e. the layer resistance against shortening diminishes with progressive strain (amplification) while the material properties remain constant. The buckle-fold is amplified at a rate that depends on the ductility contrast between the stiff and soft layers (faster amplification for larger contrast). A 15° dip for the limbs is about the limit in amplitude for which the dominant wavelength analysis expressed by (1) becomes inoperative. The weak matrix layers continue to shorten homogeneously while the fold amplifies. Progressive shortening of the system is thus composed of two parts: one part is directly associated with the bending of the layers to form folds, the other part consists of an additional strain at each point with a
5bis) refolding
Alternatively continued compression may cause the buckled layer to buckle again. The layer has then a new effective thickness nearly equal to the height of the flattened buckles and so, in accordance with the buckling theory, will buckle with a larger wavelength. Refolding under changing stress directions can lead to complex structures (interference patterns).
Multilayer
Complicated mathematical expressions are required to describe the behaviour of a multilayer sequence because they must include all variables, notably the spacing of stiff layers and the degree of cohesive strength between layers within the sequence. Therefore, experimental deformation of multilayered models has been crucial to identify some of the physical factors that control the shapes of folds. Models consisting of layers of different thicknesses and mechanical properties are complex systems that show specific behaviours: At stage (2), the buckling instability is related to the mechanical character and location of the thickest, stiffest layers within the sequence. At stage (3), while the stiffest, thickest layers buckle as single units, the multilayer sequence as a whole will undergo flexural-slip folding. The nature and degree of development of minor structures in the relatively soft layers will depend on the local strain environments created during the folding of stiffer layers.
Viscous rheology and folding
People who used numerical methods to examine how folds grow usually assumed the materials to be ideally viscous.
Stress distribution
Their result shows a complex relationship between folding and stress orientation.
- In the hinge zones, the maximum compressive stress is parallel to the layer on the concave sides of folds where layer-parallel shortening occurs, and it is roughly perpendicular to the layer on the convex sides where layer-parallel elongation occurs.
- In the limbs, the maximum compressive stress tends to rotate with the limbs until limb dips become steep, at which point it returns toward its original orientation and tends to be at high angle to the bedding.
The magnitudes of the stress also vary across the fold and throughout the course of the deformation. These changes reflect the fact that the competent layer bears a large proportion of the force applied to the system when the layer is parallel to the shortening direction, but its strengthening effect decreases as the limbs rotate to higher angles. Influence of viscosity contrast
Numerical modelling demonstrated that initial, homogeneous layer shortening absorbs much of the bulk deformation and folding becomes a less important process where the viscosity contrast between layer and matrix is low.
Single-layer buckle folds
It has been shown that any sort of shape can develop. Because of the temperature dependence of the rheologies, an increase in temperature changes the mechanical behaviour of the system, thereby affecting the geometry of the folds that develop. The shape of the fold can vary from class 1B through class 1C to nearly class 2 depending on the viscosity ratio, the amount of shortening and the wavelength thickness ratio.
- Where the competent layer is much stiffer than the matrix (^) ( μ (^) L μ (^) M> (^50) ), the amplification
rate of buckling is very fast and the competent layer deflects vigorously into the lower-competent surrounding material. Folds with a large wavelength compared to the thickness of the competent layer first develop, whereby the length of the competent layer is not or little changed. During further deformation limbs rotate up to more than 90°. Large wavelength, rounded forms are produced, such as ptygmatic folds.
- Where competence contrast is low (^) ( μ (^) L μ (^) M< (^10) ), the amplification rate is slow. Then folding
is unlikely to develop. Instead, most of the deformation will consist of layer shortening and thickening partly expressed by low-amplitude, short wavelength folds on the boundaries of the competent layer. With further shortening these folds take alternating round and sharp shapes. These are cuspate-lobate folds. The deflection of softer rock into more competent rock produces the cusps that point into the stiffer rock. In three dimensions a linear fold mullion structure forms parallel to the fold axes.
The differences in behaviour however, are minor. Accordingly, the models provide useful insight into the geometry of natural folds, which also indicates that assuming a linear viscous rheology is probably a reasonable first-order approximation. The question arises as to what difference initial layer shortening will make to the dominant wavelengths predicted by equation (1). Workers who have taken layer shortening into account show
