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.     

The transition from upright to recumbent folding in the Variscan fold belt of southwest England: a model based on the kinematics of simple shear

A transition from upright folds, at high structural levels, to recumbent folds at depth is described from the Variscan fold belt in southwest England. The folds tighten and cleavage intensifies progressively as the axial plane dip decreases. A simple shear model is developed in which the shortening of a multilayer and its folding produces initially upright open folds which tighten as they rotate during increasing shear strain. The model predicts the observed relationship between interlimb angle and axial plane dip and is used to discuss the development of the structure of north Cornwall.     

3D numerical modeling of forward folding and reverse unfolding of a viscous single-layer: Implications for the formation of folds and fold patterns

The main aims of this study are to show (i) that non-cylindrical three-dimensional (3D) fold shapes and patterns can form during a single, unidirectional shortening event and (ii) that numerical reverse modeling of 3D folding is a feasible method to reconstruct the formation of 3D buckle-folds. 3D viscous (Newtonian) single-layer folding is numerically simulated with the finite element method to investigate the formation of fold shapes during one shortening event. An initially flat layer rests on a matrix with smaller viscosity and is shortened in one direction parallel to the layering. Forward modeling with different initial geometrical perturbations on the flat layer and different lateral boundary conditions generates non-cylindrical 3D fold shapes and patterns. The simulations show that, in reality, the initial layer geometry and the boundary conditions strongly control the final fold geometry. Fold geometries produced from the forward folding models are used as initial setting in numerical reverse folding models with parameters identical to those of forward models. These reverse models accurately reconstruct the initial geometry of forward models with also only one extension event parallel to the previous shortening direction. The starting geometry of the forward models is inaccurately reconstructed by the reverse models if a significantly different viscosity ratio than in the forward models is used. This work demonstrates that reverse modeling has a high potential for reconstructing the deformation history of folded regions and rheological constraints such as viscosity ratio. Reverse models may be applied to natural 3D fold shapes and patterns in order to determine if they formed (i) during a single or multiple deformation events and (ii) as active buckle-folds with a viscosity ratio 1 or as passive, kinematic folds without buckling. This approach may find much application to fold interference patterns, in particular.     

Calcite fabrics around folds as indicators of deformation history

In the Morcles nappe (lowermost Helvetic nappe in western Switzerland) two phases of folding have been established. In this article the relationship between the calcite fabrics present in folded limestones and the folding history is analysed. Calcite fabrics around the first- and second-phase folds are related to the second phase of deformation. The following fabric patterns have been found. (1) The fabric geometry around a second-phase fold from the most internal part of the inverted limb of the nappe (locality Saillon) can be related to an overall simple shear deformation sequence, the sense of shear being related to the last advance of the Morcles nappe over the underlying autochthonous sedimentary cover of the Aiguilles Rouges massif. (2) In more external parts of the inverted limb (locality Petit Pré) the fabrics around a second-phase fold are interpreted as indicating a change in the deformation history from simple shear to a strain regime which can account for shortening along the first-phase cleavage by the formation of buckle folds. This change in the local strain regime could be related to a ‘locking’ of the frontal folds of the nappe during the last overthrust shear movements. (3) In the normal limb of the nappe the fabrics around a late second-phase kink fold (locality Neimia) are earlier than the folding event and appear to be rotated passively by the fold. (4) The fabrics around a first-phase fold (locality La Routia) are later than the folding event and overprint the fold.     

Folding of a detachment and fault – Modified detachment folding along a lateral ramp,southwestern Montana,USA

The inversion of the Middle Proterozoic Belt sedimentary basin during Late Cretaceous thrusting in Montana produced a large eastwardly-convex salient, the southern boundary of which is a 200 km-long oblique to lateral ramp subtended by a detachment between the Belt rocks and Archean basement. A 10 km-long lateral ramp segment exposes the upper levels of the detachment where hanging wall Belt rocks have moved out over the Paleozoic and Mesozoic section. The hanging wall structure consists of a train of high amplitude, faulted, asymmetrical detachment folds. Initial west-east shortening produced layer parallel shortening fabrics and dominantly strike slip faulting followed by symmetrical detachment folding. “Lock-up” of movement on the detachment surface produced regional simple shear and caused the detachment folds to become asymmetrical and faulted. Folding of the detachment surface after lock-up modified the easternmost detachment folds further into a southeast-verging, overturned fold pair with a ramp-related fault along the base of the stretched mutual limb.     

分形褶皱复杂性的流变因素研究

根据Ramberg的纵弯褶皱粘性力学实验,在褶皱形态的分形分析基础上,利用分形理论和褶皱的流变学理论导出了褶皱的分数维(D)与岩层厚度(h)和粘度(μ)间的关系式,并探讨了褶皱复杂性对褶皱分数维的影响,从中获得有关复杂褶皱的流变学信息。影响分形褶皱复杂程度的因素很多,主要因素包括岩层的厚度和粘度。因此,对褶皱的分形测量和岩层厚度及粘度的分析,可以定量分析分形褶皱形成的流变机理。这一研究是褶皱的非线性流变学理论研究的一个尝试。     

The development of asymmetrical folds in a cross-laminated siltstone

A group of folds in alternating pelites and cross-laminated siltstones is described. An interpretation of the finite strain state, in the competent silt layers, is proposed on the basis of an analysis of the angle between cross-lamination and the principal surface of accumulation. Strain magnitudes are greatest in the fold hinge where domains of layer parallel shortening and layer parallel extension are separated by a neutral surface. Strain magnitudes in the fold limbs are small and are largely related to the development of the asymmetry of the folds. In the incompetent pelitic layers, strain in the fold limbs has a large, layer parallel shear component. Deformation in the pelites is accompanied by, and presumably partially achieved by, migration of quartz from areas where there is a tendency for volume to decrease, to areas where it is tending to increase. This process involves local increases in volume of more than 50%.A kinematic model is proposed for development of the folds. It involves early development of small symmetrical folds followed by their modification to asymmetrical, parasitic structures on the limbs of later folds. In the late stages of folding, continued shortening perpendicular to the axial surface orientation is achieved by development of a conjugate crenulation cleavage.     

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.     

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.     

Strain and competence contrast estimation from fold shape

A new method to estimate strain and competence contrast from natural fold shapes is developed and verified by analogue and numerical experiments. Strain is estimated relative to the nucleation amplitude, A_N, which is the fold amplitude when the amplification velocities caused by kinematic layer thickening and dynamic folding are identical. A_N is defined as the initial amplitude corresponding to zero strain because folding at amplitudes smaller than A_N is dominantly by kinematic layer thickening. For amplitudes larger than A_N, estimates of strain and competence contrast are contoured in thickness-to-wavelength (H/λ) and amplitude-to-wavelength (A/λ) space. These quantities can be measured for any observed fold shape. Contour maps are constructed using existing linear theories of folding, a new nonlinear theory of folding and numerical simulations, all for single-layer folding. The method represents a significant improvement to the arc length method. The strain estimation method is applied to folds in viscous (Newtonian), power-law (non-Newtonian) and viscoelastic layers. Also, strain partitioning in fold trains is investigated. Strain partitioning refers to the difference in strain accommodated by individual folds in the fold train and by the whole fold train. Fold trains within layers exhibiting viscous and viscoelastic rheology show different characteristic strain partitioning patterns. Strain partitioning patterns of natural fold trains can be used to assess the rheological behaviour during fold initiation.     

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.     

Structural pattern and strain history of a superposed fold system in the Precambrian of Central Rajasthan,India. II. Strain history

Analysis of shapes of folds, together with other structures such as axial plane foliation boudinage, mullions and cross joints, show that the F₁ folds in the ‘main Raialo syncline’ were formed by buckling, and were subsequently modified by flattening normal to the axial planes and lengthening along the axis. The apparent buckle shortening of the F₁ folds generally ranges between 70 and 80%. The folds were formed by simple shear (giving place to pure shear at certain stages) in an almost north-south direction on subhorizontal beds. Progressive deformation in the later stage of F₁ folding resulted in gentle upright folding of F₁ axial planes on F₁′ axes slightly oblique to F₁. The F₂ folds, whose average shortening ranges from 20 to 30%, were also formed by buckling caused by horizontal compression in a nearly northwest-southeast direction. This folding was preceded and followed in some instances by homogeneous strain, as deduced fro mthe shapes of the F₂ folds and the nature of variation of the F₁ lineations. The F₃ conjugate structures developed when the maximum compressive strain was vertical and the intermediate compressive strain northwest-southeast, almost normal to the subvertical F₂ axial planes. The increase in the amplitude of the F₂ folds in the last phase of F₂ folding in certain zones resulted in an excess of vertical load, which dissipated with the formation of the F₃ structures. In the last stage of movement (F₄) the maximum compressive strain became horizontal along the strike of F₂ axial planes, whereas the minimum compressive strain was normal to them. The F₄ structures, therefore, point to a longitudinal shortening with reference to large scale F₂ folding.     

Early formed regional antiforms and synforms that fold younger matrix schistosities: their effect on sites of mineral growth

In the metamorphic cores of many orogenic belts, large macroscopic folds in compositional layering also appear to fold one or more pervasive matrix foliations. The latter geometry suggests the folds formed relatively late in the tectonic history, after foliation development. However, microstructural analysis of four examples of such folds suggests this is not the case. The folds formed relatively early in the orogenic history and are the end product of multiple, near orthogonal, overprinting bulk shortening events. Once large macroscopic folds initiate, they may tighten further during successive periods of sub-parallel shortening, folding or reactivation of foliations that develop during intervening periods of near orthogonal shortening. Reactivation of the compositional layering defining the fold limbs causes foliation to be rotated into parallelism with the limbs.Multiple periods of porphyroblast growth accompanied the multiple phases of deformation that postdated the initial development of these folds. Some of these phases of deformation were attended by the development of large numbers of same asymmetry spiral-shaped inclusion trails in porphyroblasts on one limb of the fold and not the other, or larger numbers of opposite asymmetry spirals on the other limb, or similar numbers of the same asymmetry spirals on both limbs. Significantly, the largest disparity in numbers from limb to limb occurred for the first of these cases. For all four regional folds examined, the structural relationships that accompanied these large disparities were identical. In each case the shear sense operating on steeply dipping foliations was opposite to that required to originally develop the fold. Reactivation of the folded compositional layering was not possible for this shear sense. This favoured the development of sites of approximately coaxial shortening early during the deformation history, enhancing microfracture and promoting the growth of porphyroblasts on this limb in comparision to the other. These distributions of inclusion trail geometries from limb to limb cannot be explained by porphyroblast rotation, or folding of pre-existing rotated porphyroblasts within a shear zone, but can be explained by development of the inclusion trails synchronous with successive sub-vertical and sub-horizontal foliations.     

Faulting and fracturing in the Jurassic carbonate ramp folds of southern Rif ridges (Northern Morocco)

In the South Rifian ridges (SRR), the dominated structures correspond to the faulted anticline characteristic of a foreland orogeny context, front of the Rif Alpine belt. These anticlines correspond to thrust propagation folds. Geometric model of these structures shows that the normal faults have controlled the Mesozoic sedimentation during extensive episodes and participated in determining areas of thrusting during Miocene compressional phases. However, the normal fault strike which is relative to the direction of the shortening determined the geometry of diverse folds whether into the frontal ramps, lateral, or oblique. In the meantime, the systematic fracturing study in the Jurassic limestone beds, in different parts of the folds with axes oriented E-W, NW-SE, and NE-SW, permits to propose a relative fracturing chronology and tries investigating the relationship between folding and fracturing. The three main fracture families, oblique, transversal, and axial, appear simultaneously during the amplification of the fold. The simple shear in the limb contributes the latest to the folding reactivation and the density of the intensification of these microfractures. Likewise, given the important downslope fold limb dip of the ramp propagation folds, theoretically the shear intensity is more important, and micro-fractures are more important in the downslope fold limb, thus the uphill one.     

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.     

Layer shortening and fold-shape development in the buckling of single layers

The progressive development of folds by buckling in single isolated viscous layers compressed parallel to the layering and embedded in a less viscous host is examined in several ways; by use of experiments, an analogue model to simulate simultaneous buckling and flattening and by an application of finite-element analysis.The appearance of folds with a characteristic wavelength in an initially flat layer occurs in the experiments for viscosity ratios (μ_layer/μ_host = μ₁/μ₂) of between 11 and 100; progressive fold development after the initial folds have appeared is similar in the experiments and in the finite-element models. Except for the finite-element model for μ₁/μ₂ = 1,000 layer-parallel shortening occurs in the early stages of folding and a stage is reached where little further changes in arc length occur. The amount of layer-parallel shortening increases with decreasing viscosity contrast, and becomes relatively unimportant after the folds have attained limb dips of about 15°–25°.Thickness variations with dip are only significant here for the finite-element model with μ₁/μ₂ = 10, and in experiments for μ₁/μ₂ = 5 where the layer is initially in the form of a moderate-amplitude sine wave. The variations range from a parallel to a near-similar fold geometry, and in general depend on the viscosity contrast, the degree of shortening and the initial wavelength/thickness ratio. They are very similar to the variations predicted by the analogue model of combined buckling and flattening. The difference between the thickness/dip variations in a fold produced by buckling at low viscosity contrast and one produced by flattening a parallel fold is marked at high limb dips and very slight at low limb dips.Many natural folds in isolated rock layers or veins show thickness/dip relationships expected for a flattened parallel fold, and some show relationships expected for buckling at low viscosity contrasts. Studies of the wavelength/thickness ratios in natural folds have suggested that competence contrast is often low. Many folds in isolated rock layers or veins whose geometry may vary between parallel and almost similar, and may be indistinguishable from those of flattened parallel folds, have probably developed by a process of buckling at low viscosity contrasts.     

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.     

Kinematics of compressional fold development in convergent wrench terranes

Kinematic models are presented for compressional fold development in wrench and convergent wrench terranes that relate fold shortening, axial rotation, and axial extension. Fold shortening may be derived from final fold geometry. Existing fold geometry and axial orientation, two readily measurable quantities, provide the data needed to determine the relative components of shearing and convergence within the fold system. Analyses utilizing these kinematic models indicate that folds developed in sedimentary rocks in the wrench borderlands of both the Rineonada and San Andreas wrench faults in central California are the product of strongly convergent wrenching. The axes of these folds have been rotated no more than a few degrees during the course of their development. In contrast, folds developed in the Alpine Schists along the Alpine fault in New Zealand and in Pleistocene sediments along the southern limit of the San Andreas fault suggest an almost pure wrench setting and large (>25 °) axial rotations.

Significant axial extension is inherent in wrench-related compressional folds. This axial extension is commonly manifest in the form of normal and strike-slip faults that are internal to the folds and trend at high angles to the fold axes. The relative amount of axial extension diminishes as the degree of convergence increases. This axial extension, and the associated extensional features, can be a diagnostic indication of the influence of wrenching.     

纵弯叠加褶皱地区应力场研究-以川东北地区为例

川东北地区主体构造为北西向大巴山弧形构造带和北东-北东东向川东弧形褶皱带,发育早期北东向纵弯褶皱和晚期北西向纵弯褶皱,两者构成明显的纵-纵复合叠加,形成典型的限制褶皱、横跨褶皱、斜跨褶皱和移褶等。早、晚两期褶皱和共轭“X”节理均反映出早期应力场为北西-南东向水平挤压,晚期应力场为北东-南西向挤压。     

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.