A theory of concentric, kink and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers: IV. Development of sinusoidal and kink folds in multilayers confined by rigid boundaries

This part concerns folding of elastic multilayers subjected to principal initial stresses parallel or normal to layering and to confinement by stiff or rigid boundaries. Both sinusoidal and reverse-kink folds can be produced in multilayers subjected to these conditions, depending primarily upon the conditions of contacts between layers. The initial fold pattern is always sinusoidal under these ideal conditions, but subsequent growth of the initial folds can change the pattern. For example, if contacts between layers cannot resist shear stress or if soft elastic interbeds provide uniform resistance to shear between stiff layers, sinusoidal folds of the Biot wavelength grow most rapidly with increased shortening. Further, the Biot waves become unstable as the folds grow and are transformed into concentric-like folds and finally into chevron folds. Comparison of results of the elementary and the linearized theories of elastic folding indicates that the elementary theory can accurately predict the Biot wavelength if the multilayers contain at least ten layers and if either the soft interbeds are at most about one-fifth as stiff as the stiff layers, or there is zero contact shear strength between layers.Multilayers subjected to the same conditions of loading and confinement as discussed above, can develop kink folds also. The kink fold can be explained in terms of a theory based on three assumptions: each stiff layer folds into the same form; kinking is a buckling phenomenon, and shear stress is required to overcome contact shear strength between layers and to produce slippage locally. The theory indicates that kink forms will tend to develop in multilayers with low but finite contact shear strength relative to the average shear modulus of the multilayer. Also, the larger the initial slopes and number of layers with contact shear strength, the more is the tendency for kink folds rather than sinusoidal folds to develop. The theoretical displacement form of a layer in a kink band is the superposition of a full sine wave, with a wavelength equal to the width of the kink band, and of a linear displacement profile. The resultant form resembles a one-half sine curve but it is significantly different from this curve. The width of the kink band may be greater or less than the Biot wavelength of sinusoidal folding in the multilayer, depending upon the magnitude of the contact shear strength relative to the average shear modulus. For example, in multilayers of homogeneous layers with contact strength, the Biot wavelength is zero so that the width of the kink band in such materials is always greater than the Biot wavelength. In general, the higher the contact strength, the narrower the kink band; for simple frictional contacts, the widths of kink bands decrease with increasing confinement normal to layers. Widths of kink bands theoretically depend upon a host of parameters — initial amplitude of Biot waves, number of layers, shear strength of contacts between layers, and thickness and modulus ratios of stiff-to-soft layers — therefore, widths of kink bands probably cannot be used readily to estimate properties of rocks containing kink bands. All these theoretical predictions are consistent with observations of natural and experimental kink folds of the reverse variety.Chevron folding and kink folding can be distinctly different phenomena according to the theory. Chevron folds typically form at cores of concentric-like folds; they rarely form at intersections of kink bands. In either case, they are similar folds that develop at a late stage in the folding process. Kink folds are more nearly akin to concentric-like folds than to chevron folds because kink folds form early, commonly before the sinusoidal folds are visible. Whereas concentric-like folds develop in response to higher-order effects near boundaries of a multilayer, kink folds typically initiate in response to higher-order shear, as at inflection points near mid-depth in low-amplitude, sinusoidal fold patterns. Chevron folding and kink folding are similar in elastic multilayers in that elastic “yielding” at hinges can produce rather sharp, angular forms.     

A theory of concentric, kink and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers: V. asymmetric folding in interbedded chert and shale of the franciscan complex, san francisco bay area, california

The origin of tight, asymmetric, kink-like or chevron-like folds in interbedded shales and radiolarian cherts of the Franciscan Complex in the San Francisco Bay area has been somewhat of a mystery for many years. Stephenson Ellen provided many clues as to the origin and indicated that the folds became asymmetric as a result of layer-parallel shear. He believed that the original folds were conjugate kink folds.As a result of reexamination of most of the folds studied by Ellen, of experimentation with elastic multilayers and of the theories developed in Parts III and IV of this series of papers, we believe that the original folds were mostly chevron rather than kink folds. Thus, we suggest that the folds formed by a combination of layer-parallel shortening and layer-parallel shear when the rocks were soft and pore pressures were high.Several lines of evidence suggest that typical folds in the Franciscan are asymmetric chevron folds. A combination of theory of finite simple shear and of experimentation with elastic multilayers indicates that the tight folds of the Franciscan could have been produced by smaller angles of simple shear if the original folds were typical chevron folds rather than typical kink folds. Several field observations, including thickening of shales but not of cherts in hinges of folds and lack of deformation of radiolaria in the cherts, indicate that the cherts were soft and the shales very soft at the time of folding. The pore-water pressures in the shales probably were high. Such conditions theoretically favor concentric-like and chevron folding, not kink folding. Finally, most of the asymmetric folds in a quarry exposure can be reconstructed geometrically as typical chevron folds but not as typical kink folds subjected to simple shear.     

Opposite senses of fold asymmetry

If the orientation of the principal compressive stress is oblique to layering, viscous multilayers fold in response to the layer-parallel shortening and develop asymmetric interfaces in response to the layer-parallel shear. A theoretical analysis of folding of viscous multilayers with different slip laws at layer contacts shows that the sense of asymmetry of folds is determined largely by the behavior of the layer contacts and the sense of layer-parallel shear during folding.

For a given sense of layer-parallel shear, the sense of asymmetry of folds can be reversed by changing only the behavior of the layer contacts. If the slip rate is linearly proportional to the shear stress at layer contacts, the resistance to slip is the same everywhere along interfaces, and the folds develop the sense of asymmetry of drag folds. If the slip rate is a nonlinear function of the shear stress at layer contacts, however, the resistance to slip varies with position along interfaces, and folds develop the sense of asymmetry of monoclinal kink folds.

For a given variable resistance to slip at layer contacts, the sense of asymmetry depends on the sense and magnitude of the layer-parallel shear and on the thickness-to-wavelength ratio of the multilayer.

For finite multilayers with variable resistance to slip at contacts, an increase in the layer-parallel shear stress decreases the dominant wavelength and increases the amplification factor for the initial perturbation of the interface.

The multilayer consists of linear viscous layers and is confined by thick, viscous media. Resistance to slip at layer contacts is modeled theoretically by a powerlaw relationship between rate of slip and contact shear stress. The equations, derived to 2nd order in the slopes of the interfaces, describe the growth of asymmetric folds from initial, symmetric perturbations.     

A theory of concentric, kink, and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers: I. Introduction

Most folds in stratified rock are similar in form to ideal kink, concentric or chevron folds, in which there are discontinuities in slope or curvature of bedding planes. In this respect most folds appear to be closely related to faults, traces of which can be considered to be lines across which there are discontinuities of displacement of layers. Further, the close association of reverse faults and folds or monoclinal flexures seems to indicate that theories of faulting and folding should be closely related.The theory of characteristics is a mathematical tool with which we can obtain insights into processes involving discontinuities. Theoretical characteristic lines are directions across which certain variables might be discontinuous and they are directions along which discontinuities propagate. The theory has been widely applied in plasticity theory and in fluid mechanics and theoretical studies of faulting have suggested that faults are analogous to the lines of discontinuity predicted by plasticity theory. Elasticity and viscosity theories, on which theories of folding have been founded, exclude the existence of characteristic lines in the materials unless the equilibrium equations, rheological properties or strains are nonlinear. However, all folding theories are nonlinear to some extent and the theories can be modified so that they predict lines of discontinuity for some conditions of loading and deformation.Theories of folding will be developed in subsequent papers of this series in order to predict conditions under which characteristic lines can exist in multilayered materials and in order to determine the conditions that must be satisfied across and along the characteristic lines. The theory should help us to recognize lines of apparent discontinuity in natural and experimental folds and study of these lines should provide further understanding of mechanisms of folding.Experimental studies of folding of a wide variety of materials, including alternating layers of rubber and gelatin, modeling clay and grease or graphite, and potter's clay and rubber or cardboard, suggest that the patterns of folding in these materials begin with sinusoidal forms, transform into concentric or kink forms and then into chevron forms as the multilayers are shortened axially. A suitable theory of folding of multilayers should account for these observations.     

A theory of concentric, kink and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers: VII. development of folds within Huasna Syncline, San Luis Obispo County, California

Folds in the Huasna area of the southern Coast Ranges of California provide an opportunity to study different fold forms and to estimate dimensional and relative rheological properties of rocks at the time of folding. Plunging, concentric-like and chevron-like folds with wavelengths ranging from about 0.1 to 1 km are clearly visible in natural exposures at the south end of the Huasna syncline, which has a wavelength of 12–16 km. Examination of two fresh roadcut exposures in the Miocene Monterey Formation suggests that folding within part of the Monterey was accommodated primarily by layer-parallel slip between structural layers with thicknesses ranging from 30 to 43 m, even though lithologic layers range from a few mm to a few dm in thickness. This part of the Monterey is folded into a series of concentric-like folds, with chevron-like folds at their cores and with a ratio of wavelength to total thickness of layers of about . Theoretical analysis of multilayers, comprised of identical, elastic or elastic—plastic layers with frictionless contacts, indicates that the effective, or weighted-average thickness of structural layers corresponding with an ratio of 0.42 is about 41 m. Thus, the theoretical predictions are roughly in agreement with available data concerning these folds.Thicknesses of structural units in other folds of this area are inadequately known to closely check theoretical predictions, but most of the data are consistent with predictions. An exception is the Huasna syncline which has a larger wavelength than we would predict. There are several likely explanations for this discrepancy. Layers in the underlying Franciscan complex may have taken part in the folding, making our estimates of total thickness too small. The basement rocks may have been much softer, relative to the overlying sedimentary rocks, than we assumed. The Huasna syncline could be partly a result of gravitational instability of relatively low density, Miocene siliceous and porcelaneous shales, overlain by relatively high density, Pliocene sandstones.The Huasna syncline and some of the smaller folds in the Miocene rocks are doubly in the northwest—southeast direction. Further, the maximum compression was approximately normal to the traces of the large faults in this part of California.     

A theory of concentric,kink, and sinusoidal folding and of monoclinal flexuring of compressible,elastic multilayers: III. Transition from sinusoidal to concentric-like to chevron folds

A basic, sinusoidal solution to the linearized equations of equilibrium for compressible, elastic materials provides solutions to several problems of folding of multilayers. Theoretical wavelengths are comparable to those predicted by Ramberg, using viscosity theory, and to those predicted by elementary folding theory. The linearized analysis of buckling of a single, stiff, elastic layer, either isolated or within a soft medium, suggests that wavelengths computed with elementary beam theory are remarkably similar to those computed with the linearized theory for wavelength-to-thickness ratios greater than about five. This is half the limit of ten normally assumed for use of the elementary theory.The theory and experiments with deep beams of rubber or gelatin indicate that thick, homogeneous layers folded with short wavelengths assume internal forms strikingly similar to those of the ideal concentric fold. Thus, mechanical layering clearly is not required to produce concentric-like forms.Further, the theory suggests that “arc and cusp” structure, or “pinches”, at edges of deep beams as well as chevron-like forms in single or multiple stiff layers are a result of a peculiar, plastic-like behavior of elastic materials subjected to high normal stresses parallel to layering. In a sense, the elastic material “yields” to form the hinge of the chevron fold, although the strain vanishes if the stresses are released. Accordingly, it may be impossible to distinguish chevron forms produced in elastic-plastic materials, such as cardboard or aluminum and perhaps some rock, from chevron forms produced in purely elastic materials, such as rubber.Analysis of the theory shows that, just as high axial stresses make straight, shortened multilayers the unstable form and sinusoidal waves the stable form, stresses induced by sinusoidal displacements of the multilayer make the sinusoidal waveform unstable and concentric-like waves the stable form. Thus, concentric-like folds appear to be typical of folded multilayers according to our analysis. Further, where the layers have short wavelengths in the cores of the concentric-like folds, the stiff layers “yield” elastically at hinges and straighten in limbs. Thus the concentric-like pattern is replaced by chevron folds as the multilayer is shortened. In this way we can understand the sequence of events from uniform shortening, to sinusoidal folding, to concentric-like folding, to chevron folding in multilayers composed of elastic materials.     

Ultimate locking angles for conjugate and monoclinal kink bands

Honea, Reches and Johnson have introduced a theory of conjugate and monoclinal kinking in rocks. A correction is required in expressions for the ultimate locking angles of kink bands.     

Variscan stress regime rotation: Insights from the analysis of kink folds in the northern margin of the Bohemian Massif,South Poland

We aimed to determine variations in stress regimes during the youngest Variscan deformations in the northern part of the Bohemian Massif. For this purpose, we calculated the orientation of the principal stress and strain axes for kink folds observed in the metamorphic envelope of the Karkonosze Granite, using two methods: 1) the traditional method, incorporating structural diagrams (for conjugate kink folds only), and 2) butterfly diagram analysis. The use of both methods enabled us to determine the stress regime, based not only on conjugate but also on monoclinal kink bands. The obtained results prove that butterfly diagram analysis, when applied to monoclinal kink folds, yields reliable results, especially when calibrated using the internal friction angle (Ф) calculated for the conjugate structures.We identified two generations of kink folds: 1) an older one, developed under sublatitudinal shortening and most probably related to the Early Carboniferous terminal stages of the northwest-directed thrusting of the metamorphic units, and 2) a younger one; produced by north-south Variscan Carboniferous compression, and the emplacement and subsequent doming of the Karkonosze Granite. This is the first study on brittle-ductile structures observed commonly in the metamorphic units of the Bohemian Massif, showing their relation to the granitoid intrusion and complementing the tectonic models that usually omit kink folds.     

A range of fault slip styles on progressively misoriented planes during flexural-slip folding,Cape Fold Belt,South Africa

Flexural slip folds are distinctive of mixed continuous-discontinuous deformation in the upper crust, as folding is accommodated by continuous bending of layers and localized, discontinuous slip along layer interfaces. The mechanism of localized, layer-parallel slip and the stress and fluid pressure conditions at which flexural slip occurs are therefore distinctive of shear localization during distributed deformation. In the Prince Albert Formation mudstone sequence of the Karoo Basin, the foreland basin to the Cape Fold Belt, folds are well developed and associated with incrementally developed bedding-parallel quartz veins with slickenfibers oriented perpendicular to fold hinge lines, locally cross-cutting axial planar cleavage, and showing hanging wall motion toward the fold hinge. Bedding-parallel slickenfiber-coated veins dip at angles from 18° to 83°, implying that late increments of bedding-parallel shear occurred along unfavorably oriented planes. The local presence of tensile veins, in mutually cross-cutting relationship with bedding-parallel, slickenfiber-coated veins, indicate local fluid pressures in excess of the least compressive stress.Slickenfiber vein microstructures include a range of quartz morphologies, dominantly blocky to elongate-blocky, but in places euhedral to subhedral; the veins are commonly laminated, with layers of quartz separated by bedding-parallel slip surfaces characterized by a quartz-phyllosilicate cataclasite. Crack-seal bands imply incremental slickenfiber growth, in increments from tens of micrometers to a few millimeters, in some places, whereas other vein layers lack evidence for incremental growth and likely formed in single slip events. Single slip events, however, also involved quartz growth into open space, and are inferred to have formed by stick-slip faulting. Overall, therefore, flexural slip in this location involved bedding-parallel faulting, along progressively misoriented weak planes, with a range of slip increments.     

Impact of mechanical anisotropy and power-law rheology on single layer folding

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

The geometry and kinematics of a suite of conjugate kink bands, southeastern Australia

A conjugate set of subvertical kink bands is exposed in coastal outcrops of well-foliated Ordovician turbidites near Mystery Bay, Australia. All kink bands with widths exceeding 3 cm have complex internal structures including compound and parasitic kinks, stepped kink boundaries, internal crenulations, variable kink angles and prismatic voids. The kink bands are interpreted to result from rotation of short foliation segments between fixed kink planes with subsequent widening and modification by layer-parallel shear external to the kink band. Layer-parallel shear of both sinistral and dextral sense accompanied kinking and indicates a variable stress system during kink band development.Conjugate kink bands are abundant and are used to estimate bulk strain orientations. In general, the dominant kink set of a conjugate pair is inclined at a lower angle to the external foliation than the weaker set and this angular disparity increases with increasing dominance of one set. These observations are at variance with relationships described from experimental bulk pure shear deformation of anisotropic materials. It is suggested that orthogonal constraints in these experiments restrict layer-parallel shear to within a developing kink band and are, therefore, unlike many natural kink systems. Simple shear experiments can produce structures geometrically similar to natural kink bands.     

A numerical investigation and geological discussion of the relationship between folding,kinking and faulting

The layer-parallel compression of a regular bilaminate consisting of layers with materials described by an incompressible power-law elastic model is considered. The average mechanical properties of this idealised multilayer are then represented by those of an equivalent anisotropic continuum with internal resistance to bending. Changes in material properties that accompany uniform finite shortening are accounted for. Interpretation of the internal instability analysis for such a continuum, introduced in the companion paper involves the use of a spectrum which at a given level of strain, scans all directions within the continuum for relative susceptibility to a heterogeneous simple shearing instability.Estimates of nonlinear material properties from reported experiments on the behaviour of various rocks in the time-independent deformation regime, and geometric parameters such as the volume fraction of each material and the number of confined layers are considered. The shapes of the resulting spectra may be used to predict natural conditions that will favour the initiation of repetitive buckle folds or more localized disturbances such as kink-bands and faults. Results suggest that for typical properties of sedimentary multilayers, kinking is strongly favoured over repetitive buckling where the weaker material occupies only a very small volume fraction of the multilayer. The effect of significant imperfections leading to slippage between layers is discussed.Finally, a simple classification of structure genesis is proposed in which the mechanical relationships between apparently diverse structures is illustrated.     

A finite-element study of fold propagation in a viscous layer

A finite-element model of a viscous layer contained in a viscous matrix and undergoing layer-parallel compression is used to examine the hypothesis that a long chain of folds, as found in real rocks, can originate from one initial perturbation to the layer geometry. This hypothesis is tested by determining the velocity with which a perturbation spreads along layers of various viscosities.An insight is gained into the roles played by local strain and local layer strength in the folding mechanism. The results show that for layers with viscosity ratios comparable with those of real rocks it is impossible for long chains of folds to originate from one perturbation. The authors conclude that rock layers contain many initial perturbations and folding originates at all perturbation sites simultaneously. The growth of such folds depends on the amplitude and shape of the initial perturbation and on subsequent interference between folds.     

Kinematics of fault-related folding in a duplex,lost river range,Idaho, U.S.A.

The Doublespring duplex, located in the Lost River Range of Idaho, is a Sevier age fault-related fold complex in massive limestones of the Upper Mississippian Scott Peak Formation. Folds within the duplex closely resemble fault-bend fold geometrics, with open interlimb angles and low-angle bed cut-offs. Narrow, widely spaced, bedding-parallel shear zones with well-developed pressure solution cleavage alternate with massive, relatively undeformed layers on fold limbs. Shear zones are developed only on the limbs of anticlines, and have similar but unique morphologies in each of three different folds. Incremental strain histories reconstructed from antitaxial fibrous overgrowths and veins within the shear zones constrain the kinematics of folding. Shear zones experienced distributed bedding-parallel simple shear (flexural flow) towards pins near axial surfaces, while adjacent massive layers experienced rotation through an externally fixed extension direction. The absence of footwall synclines and morphological differences in shear zones from adjacent folds suggest that faulting preceded folding. Kinematic histories of folds that have experienced different translational histories are identical, and are not compatible with strain histories predicted from previous kinematic models of fault-bend folding. Shear zone development and fiber growth is instead interpreted to have occurred during low amplitude fixed-hinge buckling in response to initial resistance to translation of the thrust sheet. Fault-bend folding with mobile axial surfaces occurred with translation of the thrust sheets once the initial resistance to translation was overcome and resulted in no penetrative strain.     

Fold propagation in single embedded layers

This paper investigates the folding of single competent layers embedded in a less competent matrix, where the competence contrast is about 10: 1. Folds result from buckling during layer-parallel compression. A geometrical study of natural examples shows that individual folds tend to be grouped into fold complexes.The amplitude varies from a maximum at the centre of a complex to a minimum at each end. Each complex is often centred about a sedimentary lens or nodule which may have triggered the folding and localized the complex. The formation of folds of this kind has been simulated experimentally by deformation of models made from paraffin waxes of known rheological properties. Early in the deformation of a model, buckling starts at a localized site of disturbance, producing only one fold. With further deformation, new folds appear at either side of the initial one. The buckling then propagates along the layering, further folds appearing serially in time and distance. The end result is a complex with many individual folds and a regularly periodic shape.With a competence contrast of 10: 1, the rate of fold propagation is slow, and formation of a periodic complex requires an overall shortening of at least 15%. The shapes of folds formed experimentally are similar to those formed naturally.     

Parasitic folds with wrong vergence: How pre-existing geometrical asymmetries can be inherited during multilayer buckle folding

Parasitic folds are typical structures in geological multilayer folds; they are characterized by a small wavelength and are situated within folds with larger wavelength. Parasitic folds exhibit a characteristic asymmetry (or vergence) reflecting their structural relationship to the larger-scale fold. Here we investigate if a pre-existing geometrical asymmetry (e.g., from sedimentary structures or folds from a previous tectonic event) can be inherited during buckle folding to form parasitic folds with wrong vergence. We conduct 2D finite-element simulations of multilayer folding using Newtonian materials. The applied model setup comprises a thin layer exhibiting the pre-existing geometrical asymmetry sandwiched between two thicker layers, all intercalated with a lower-viscosity matrix and subjected to layer-parallel shortening. When the two outer thick layers buckle and amplify, two processes work against the asymmetry: layer-perpendicular flattening between the two thick layers and the rotational component of flexural flow folding. Both processes promote de-amplification and unfolding of the pre-existing asymmetry. We discuss how the efficiency of de-amplification is controlled by the larger-scale fold amplification and conclude that pre-existing asymmetries that are open and/or exhibit low amplitude are prone to de-amplification and may disappear during buckling of the multilayer system. Large-amplitude and/or tight to isoclinal folds may be inherited and develop type 3 fold interference patterns.     

Stick-slip model for kink band formation in shear zones and faults

Differential shear stresses acting along or adjacent to a non-planar fault surface or shear zone may cause uneven acceleration during slip. Alternatively, at the initiating and closing stages of motion of parts of a stick-slip fault, localised shear stresses may be variable. Stress variation of this nature causes zones of relative compression and tension, especially close to the “stick” zones on the fault. In fissile rocks adjacent to the fault, kink bands form in zones of local relative compression, while stratal extension features such as veins, fractures and extensional crenulations might be expected in the corresponding zones of relative tension. Repetitive motion on the fault should therefore cause the development of a suite of kink bands superimposed on each other and on any complementary extensional structures. Field evidence indicates that the extensional structures are not developed to the same extent as the kinks, perhaps due to ductile flow during layer-parallel extension of phyllosilicate rocks.

The advantages of this model are that it does not require bulk shortening of the shear zone relative to the enclosing less strained rocks, nor does it depend on complex stress orientation changes.     

The development of folds in bedded chert and related rocks in the Malaguide Complex, southern Spain

Folds and folding mechanism in a chert sequence and related rocks of the Maláguide Complex (the uppermost tectonic unit of the Betic Zone) have been investigated. The geometric study shows that folds that developed in the chert sequence are usually angular in shape and asymmetric. Chevron and conjugate folds are common.Folding in bedded chert is explained in terms of a suggested model:

1. (1) Development of folds by kink and conjugate kinking.

2. (2) As the shortening increases, the interlimb angles decrease; in the kink folds this is caused by a reduction of the angle between the layers within the kink and the kink boundaries. There seems to be a relationship between this angle and the asymmetrical thinning-out in the limbs of many folds: the smaller is the angle between the kink boundary and the layers within the kink band, the larger is the reduction of the thickness in these layers.

Single limestone layers embedded in slate deform very probably by a buckling mechanism, implying tangential longitudinal strain and an additional flattening.     

An approach to folding kinematics from the analysis of folded oblique surfaces

Two-dimensional analysis of folded surfaces oblique to the mechanical layering can shed light on the kinematic mechanisms that operated during the development of folds. A new version of the program ‘FoldModeler’, developed in the Mathematica™ environment, is used to obtain the deformed configuration of an initial pattern of oblique surfaces deformed by any combination of the most common kinematic folding mechanisms: flexural flow, tangential longitudinal strain, with or without area change and heterogeneous simple shear. The layer can also undergo any form of homogeneous strain at any moment of the folding process. The outputs of the program provide complete information about the strain distribution in the folded layer that includes graphs of the angle between the oblique surfaces as a function of the inclination of the layering through the fold. These graphs can be very useful to discriminate between the mechanisms that operate in the development of natural folds, and they have been obtained and discussed for the most common combinations of strain patterns. The program is applied to obtain theoretical folds that give a good fit of some natural examples of folded oblique surfaces.     

Centrifuge modelling of deformation of a multi-layered sequence over a ductile substrate: 1. Style and 4D geometry of active cover folds during layer-parallel shortening

Centrifuge analogue modelling illustrates the progressive development of active folds in multilayers upon a ductile substrate during layer-parallel shortening. Models simulate folding of a mechanically stratified sedimentary sequence upon migmatitic gneisses in a large hot orogen, or upon a thick basal evaporite ± shale sequence in deeper levels of fold belts. The absence of a weak low-viscosity and low-density layer at the interface promotes infolding of the cover sequence and ductile substrate, whereas a planar upper surface to the basal ductile substrate is preserved when it is present. Whilst fold style, wavelength, and deformation of the interface with the ductile substrate differ depending on whether a low-viscosity and low-density layer is present at the base of the cover sequence, there is no marked systematic curvature of fold axes as seen in previous sandbox models for fault-bend or fault propagation folding during bulk shortening. Bulk shortening of a layered sequence with relatively thick individual layers above a ductile substrate promotes a regular and upright train of buckle folds, whereas thinner layers promote a more irregular distribution of buckle folds with variable vergence, style, and amplitude. Buckle folds above a ductile substrate progressively develop during bulk shortening from open and upright, to angular and tight, and may further develop into cuspate structures above relatively weak horizons. Relatively thick weak horizons within the layered sequence during bulk shortening interrupt regular fold patterns up structural section and allow out-of-phase folds to develop above and below the weak horizon.