A theory of concentric, kink and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers: VI. Asymmetric folding and monoclinal kinking

One of the rules of thumb of structural geology is that drag folds, or minor asymmetric folds, reflect the sense of layer-parallel shear during folding of an area. According to this rule, right-lateral, layer-parallel shear is accompanied by clockwise rotation of marker surfaces and left-lateral by counterclockwise rotation. By using this rule of thumb, one is supposed to be able to examine small asymmetric folds in an outcrop and to infer the direction of axes of major folds relative to the position of the outcrop. Such inferences, however, can be misleading. Theoretical and experimental analyses of elastic multilayers show that symmetric sinusoidal folds first develop in the multilayers, if the rheological and dimensional properties favor the development of sinusoidal folds rather than kink folds, and that the folded layers will then behave much as passive markers during layerparallel shear and thus will follow the rule of thumb of drag folding. The analyses indicate, however, that multilayers whose properties favor the development of kink folds can produce monoclinal kink folds with a sense of asymmetry opposite to that predicted by the rule of thumb. Therefore, the asymmetry of folds can be an ambiguous indicator of the sense of shear.The reason for the ambiguity is that asymmetry is a result of two processes that can produce diametrically opposed results. The deformation of foliation surfaces and axial planes in a passive manner is the pure or end-member form of one process. The result of the passive deformation of fold forms is the drag fold in which the steepness of limbs and the tilt of axial planes relative to nonfolded layering are in accord with the rule of thumb.The end-member form of a second process, however, produces the opposite geometric relationships. This process involves yielding and buckling instabilities of layers with contact strength and can result in monoclinal kink bands. Right-lateral, layer-parallel shear stress produces left-lateral monoclinal kink bands and left-lateral shear stress produces right-lateral monoclinal kink bands. Actual folds do not behave as either of these ideal end members, and it is for this reason that the interpretation of the sense of layer-parallel shear stress relative to the asymmetry of folds can be ambiguous.Kink folding of a multilayer with contact strength theoretically is a result of both buckling and yielding instabilities. The theory indicates that inclination of the direction of maximum compression to layering favors either left-lateral or right-lateral kinking, and that one can predict conditions under which monoclinal kink bands will develop in elastic or elastic—plastic layers. Further, the first criterion of kink and sinusoidal folding developed in Part IV remains valid if we replace the contact shear strength with the difference between the shear strength and the initial layer-parallel shear stress.Kink folds theoretically can initiate only in layers inclined at angles less than to the direction of maximum compression. Here φ is the angle of internal friction of contacts. For higher angles of layering, slippage is stable so that the result is layer-parallel slippage rather than kink folding.The theory also provides estimates of locking angles of kink bands relative to the direction of maximum compression. The maximum locking angle between layering in a nondilating kink band and the direction of maximum compression is . The theory indicates that the inclination of the boundaries of kink bands is determined by many factors, including the contact strength between layers, the ratio of principal stresses, the thickening or thinning of layers, that is, the dilitation, within the kink band, and the orientation of the principal stresses relative to layering. If there is no dilitation within the kink band, the minimum inclination of the boundaries of the band is to the direction of maximum compression, or to the direction of nonfolded layers. Here α is the angle between the direction of maximum compression and the nonfolded layers. It is positive if clockwise.Analysis of processes in terminal regions of propagating kink bands in multilayers with frictional contact strength indicates that an essential process is dilitation, which decreases the normal stress, thereby allowing slippage and buckling even though slopes of layers are low there.     

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.     

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.     

Parallel, similar and constrained folds

Theoretical analysis of folding of viscous multilayers with free slip or bonding at layer contacts indicates that folds in such multilayers can be described in terms of three end-members:parallel, in which orthogonal thicknesses of layers are largely constant;similar, in which vertical thicknesses of layers and shapes of successive interfaces are essentially constant; andconstrained, in which amplitudes of anticlines and synclines decrease to zero at upper and lower boundaries. Constrained,internal folds form if the multilayer is confined by rigid media; parallel,concentric-like folds form if the multilayer is confined by soft media, provided soft interbeds are sufficiently thin for the stiff layers to fold as an ensemble. Similar,sinusoidal orchevron folds form throughout much of the thickness of a multilayer, for any stiffness of confining media, provided wavelengths of folds are short relative to the thickness of the multilayer or soft interbeds are sufficiently soft and thick for the stiff layers to act independently. The analysis shows that multilayer folds may have the same form regardless of whether the layer contacts are freely slipping or bonded.

The forms of folds in multilayers confined by media with different viscosities above and below depend on the viscosity contrast of the media. For no medium above and a rigid medium below, the forms are concentric-like in the upper part and internal in the lower part of the multilayer. For no medium above and a soft medium below, the folds are concentric-like throughout the multilayer.

The theory indicates that a useful way to analyze forms of folds in rocks or in experiments is in terms of component waveforms, as defined, for example, by Fourier series. The distributions of amplitudes of component waveforms throughout the multilayer appears to be diagnostic, reflecting contrasts in properties of the multilayer and its confining media. Analysis of a large fold in the central Appalachians, Pennsylvania, and of a smaller fold in the Huasna syncline, California, indicates that at least three component waveforms are required to produce the gross forms of those folds.

The theory closely predicts wavelengths and shapes of folds produced in analogous elastic multilayers, indicating that nonlinearities in material behavior, which are inherent in the elastic material but are absent in the viscous material, are less significant than nonlinearities in the boundary conditions, which are the same in elastic and viscous materials.     

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: 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.     

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.     

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.     

The displacement-distance method for contractional kink bands and its application to fold-thrust structures

The displacement-distance (d−x) method can be modified to study the geometry and development of contractional kink bands by dividing displacements into cartesian component vectors. Kink bands are idealised as having constant layer lengths, enabling simple trigonometry to be used to determine the displacement of one wall of the kink band relative to the other wall.

In a consideration of several applications of the d−x method for kink bands, it is shown that displacement is transferred between conjugate and overstepping kink bands in a similar way to displacement transfer between conjugate and overstepping faults and extension fractures. The several different models of kink band formation are shown to each have different displacement characteristics. The d−x method can also be used to study the geometry and evolution of folds related to thrust-propagation and ramps, which are often modelled as having kink band geometries. For instance, the d−x method can be used to show how fault-tip and fault-bend folds cause or accommodate thrust displacement variations, and to estimate displacement rates from the amounts of deformation in different syn-thrust sedimentary layers.     

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.     

Dynamic analysis and two types of kink bands in quartz veins deformed under subgreenschist conditions

In order to clarify deformation mechanisms and behaviours of quartz in a low-temperature regime in the earth's crust, microstructural analyses, particularly on kink bands have been carried out for quartz veins moderately deformed under subgreenschist conditions. Both the dominance of subbasal deformation lamellae and geometry of kink bands suggest that basal (0001) slip was the sole active slip system in the deformed quartz. On a morphological basis, kink bands in the quartz were classified into two types: type I is characterized by conjugate and narrow bands with angular hinge zones, and type II by a wide monoclinal band. Dynamic analyses using deformation lamellae and kink bands have revealed that type I kink bands were formed in grains with basal plane (sub-)parallel to the compression axis, whereas type II kink bands were formed in grains with basal planes inclined to it. Using a numerical model of kinking of elastic multilayers modified after Honea and Johnson (Tectonophysics 30, 197–239, 1976), changes of the level of yielding stress for kinking and the width of kink bands as a function of the angle θ between the slip plane and the compression axis have been examined. The theory predicts that type I kink bands were formed at a higher stress level than type II kink bands, and hence occurrence of type I kink bands suggests that a significant strain hardening occurred in the deformed quartz veins. The theory also well explains the fact that the width of type I kink bands (θ=0 to 10°) is narrower by an order of magnitude than type II kink bands (θ=10 to 80°).     

Some aspects on the origin of fold-type fabrics — theory,experiments and field applications

An applicable interpretation of fabrics should be based mainly on geometrical considerations in order to cover available field data. Therefore a theory on the formation of foldtype fabrics including congruent and concentric flexural-slip folds as well as kink bands in materials subjected to rhombic and different monocline strain types is derived by means of particle motion fields for homogeneous and isotropic bodies. The analysis of experimental results and their comparison with field observations largely confirms the theory and contributes to its improvement. Some trends can be established: With increasing monocline character of the strain type, the probability of concentric and congruent flexural-slip folding is reduced. It is substituted by kink band formation. While the portion of monocline strain is enlarged with depth, rhombic fold symmetry indicates, in the realm of elastic-plastic behaviour, the proximity of the surface of the earth or of a detachment surface. With gradual increase of the rock anisotropy, the development of shear faults, kink folds and, finally, congruent and concentric flexural-slip folds is favoured.     

Some aspects on the origin of fold-type fabrics — theory,experiments and field applications

An applicable interpretation of fabrics should be based mainly on geometrical considerations in order to cover available field data. Therefore a theory on the formation of foldtype fabrics including congruent and concentric flexural-slip folds as well as kink bands in materials subjected to rhombic and different monocline strain types is derived by means of particle motion fields for homogeneous and isotropic bodies. The analysis of experimental results and their comparison with field observations largely confirms the theory and contributes to its improvement. Some trends can be established: With increasing monocline character of the strain type, the probability of concentric and congruent flexural-slip folding is reduced. It is substituted by kink band formation. While the portion of monocline strain is enlarged with depth, rhombic fold symmetry indicates, in the realm of elastic-plastic behaviour, the proximity of the surface of the earth or of a detachment surface. With gradual increase of the rock anisotropy, the development of shear faults, kink folds and, finally, congruent and concentric flexural-slip folds is favoured.     

Geometrical adjustments during rotation of a Jura fold limb

Jura folds do not resemble models of continuously distributed buckling; they are never sinusoidal. Rather, they may be approximated by discrete, externally rotated conjugate kink bands with rounded hinges. Observational details are confusing, and good outcrops are limited; however, several types of geometrical adjustments necessary during growth (particularly external rotation) of a variety of elementary kink models may be plausibly correlated with observed features. The suitable way to the modeling of fold growth and an explanation of observations on different scales seems to be through synthesis of different modes of rotation and adjustment mixed in varying degrees to fit individual folds. A dominant role is played by incompetent beds of variable importance giving rise to several types of disharmonic folding.     

Fold development in the Jura

The term “folding” encompasses a wide range of processes, most of which are poorly understood. Jura folds, though comparatively simple, have developed by a superposition of different types of instabilities both in space and time. They are never periodic and sinusoidal and are more realistically approximated by kink bands with rounded hinges. Thrusting and kinking instabilities had closely similar thresholds, with kinks usually following and deforming thrusts. An analysis of embryonic folds shows that instabilities in the sedimentary cover were initiated primarily at inherited flaws of the basal décollement layer. They thence spread upward, often following stratigraphically higher incompetent layers in secondary décollement and there nucleating secondary instabilities before reaching the surface (disharmonic folding). Embryonic folds thus are usually narrow, emanating from secondary décollement layers that are connected with the basal décollement zone by thrusts nucleated at inherited obstacles. These are eventually overcome, permitting basal décollement to coalesce with kinking instabilities that grow downward from nuclei in higher décollement intervals. In this way folds centered in the basal décollement layer, and consequently of normal width, may be superposed on the narrow embryonic folds. The sequence and importance of the different elements may vary from place to place to result in a vast catalog of fold shapes.     

Finite-element simulation of asymmetric folding

Fold shapes and strain distributions produced in stiff single layers inclined up to 20° to the direction of principal shortening were investigated using finite-element computer models. The finite-element model was formulated for constant-strain quadrilaterals using the constitutive equation for a compressible, linearly viscous fluid. The model of a stiff layer imbedded in a less viscous medium was designed to accommodate 2 Biot wavelengths. Inclinations of the stiff layer to the shortening direction were 0°, 5°, 10°, 15° and 20°. At each inclination folds were produced with viscosity ratios of 17 : 1, 24 : 1 and 42 : 1. Folds were initiated by prescribing symmetric sinusoidal perturbations with limb dips of 2°. Results from models with 0° initial inclinations are similar to results obtained by others. Folds are sinusoidal and symmetric, and strain distributions are symmetrically disposed about axial planes in both the matrix and stiff layer. As layer inclination is increased, these features change. The folds become asymmetric (as measured by the ratio of limb lengths), and the amount of asymmetry increases with inclination. Finite-strain distributions in both the stiff layer and matrix are not symmetrically disposed about the axial plane. Principal strains in the matrix tend to parallel the long limb of the stiff layer, and are “refracted” through the long limb at a larger angle than through the short limb.     

Nucleation and growth of kink bands

The geometric and strain properties of an ideal kink band in a perfectly plastic material with one slip surface are determined and two plausible modes by which a kink band can grow while maintaining these properties are presented. The first mode involves change in the orientation of the boundary surface of the kink band as it broadens. In the second mode the initial orientation of this boundary surface is maintained during broadening.A theoretical model is proposed for kink band formation in three stages: first, in an elastically strained material with a suitably oriented planar anisotropy; a small nucleus is either present or develops, and grows in thickness by one or the other or by both of the growth modes; second, as some critical thickness is reached, this nucleus propagates longitudinally until the excessive elastic distortion surrounding its ends is sufficiently relieved, or until the boundary of the body is reached; and third, the band widens by lateral migration of its boundaries, most probably by mode 2 growth.It is proposed that the direction of longitudinal propagation—and hence the orientation of the kink band with respect to principal normal stress—is decided by the value of a, the kink angle, in the nucleus at the time the critical thickness for propagation is reached. A further proposal is that this value of a is in turn decided by a principle of maximum plastic work: the instantaneous increment in strain or in volume within a nucleus is always the one for which the instantaneous increment in plastic work is a maximum at constant deviatoric stress.Calculations based on this theory for the two growth modes working either separately or together as competing processes yield three models of kink band nucleation in reasonable agreement with published experimental data. The most appealing of these models physically is one in which early mode 1 growth is replaced by mode 2 growth as the increments in mechanical work for the two modes become equal.     

Experimental analysis of folding in simple shear

Six experiments of single-layer folding with simple-shear boundary conditions were completed. Using materials of ethyl cellulose, the viscosity ratio of the stiff layer to matrix ranged from 20 to 100. The experiments were monitored by 10–14 photographs taken at equally spaced time intervals. Strain distributions in both the stiff layer and matrix were calculated from the displacements of over 300 ink dots distributed over the surface of each experiment. Both incremental strain (calculated from the relative displacements of the dots between successive photographs) and accumulating strain were determined on the two-dimensional profile of the materials as they folded.Symmetrical fold wavelengths occur and seem to be controlled by the wavelengths of initial perturbations in the stiff layer. If the Biot wavelength was not present initially, it will not occur in the final waveform. Consequently, in a group of natural folds, the mean value of wavelength/thickness ratios apparently reflects the initial perturbations. The mean value should not be confused with the Biot wavelength and should not be used to calculate viscosity ratios in naturally deformed rocks.Substantial layer thickening occurred only with viscosity ratios of 20. The amount of layer thickening also depends on initial perturbations of the stiff layer. If these perturbations are near the Biot wavelength, they are greatly amplified, the folds grow rapidly and layer thickening is small. If the perturbations are not near the Biot wavelength, amplification is small, the folds grow slowly and layer thickening is much greater.Principal elongations of the accumulated strain in the cores of some of the folds are not symmetrically distributed about axial planes and may cut across the axial plane at angles up to 20°. Strain shadows in the matrix, near the convex side of fold hinges, are also prominent. These triangular-shaped regions of low strain are not symmetrically disposed about fold axial planes, in contrast to strain shadows occurring in folds produced under pure-shear boundary conditions.The rotation of accumulating principal elongations in the stiff layer was calculated at fold inflections. Even though the folds themselves are generally symmetrical, these rotations at opposite fold inflections are not. One fold limb exhibits little rotation of principal elongations during folding while the other has rotations up to 70°. In contrast, folds formed in pure-shear boundary conditions have rotations of principal directions on opposite fold limbs equal in magnitude.     

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.