Strain distribution across an experimental single-layer fold

Experiments have been carried out to study the effects of progressive deformation on the shape of folds and the variation in two-dimensional strains on cross-sections of singlelayer folds in a less competent matrix, in a pure-shear plane-strain deformation box with no volume change. The layer shortening continues after buckling has set in, leading to thickening of the fold hinge and with progressive buckling the layer elongates. During the layer elongation stage of folding the hinges continue to thicken, whereas the limbs thin out. Concentric folds are a combination of Class 1a type in the outer arc which gradually change to Class Ib type and then to Class 3 folds of Ramsay (1967) in the inner arc. Tangential longitudinal strains and shearing strains predominate in the fold-hinge zone and in the fold limbs of the buckling layer, respectively. Initially, uniform layer-flattening strains perpendicular to the layering develop which become extensive strains in the outer fold arc and compressive strains in the inner fold arc with progressive buckling. In the outer fold arc the extensive strains are distributed laterally over a wider zone and are of a lower magnitude than the compressive strains which are restricted to a narrow zone in the inner fold arc. The neutral surface first appears when the initial layer-flattening strains are removed due to extensive strains on the outer arc and with progressive buckling migrates towards the inner fold arc and extends laterally on the outer fold arc.     

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.     

The Migif–Hafafit gneissic complex of the Egyptian Eastern Desert: fold interference patterns involving multiply deformed sheath folds

The Wadi Hafafit Complex (WHC) is an arcuate belt of orthogneisses, migmatites and other high-grade metamorphic rocks, which marks the boundary between the Central Eastern and the South Eastern Deserts of Egypt. In the WHC, gneissic meta-gabbro outlines macroscopic fold interference patterns characterized by elliptical to irregular culminations cored by gneissic meta-tonalite to meta-trondhjemite. The five main culminations of the WHC have previously been labeled A (most northerly), B, C, D and E (most southerly). A detailed structural investigation of B, C, D and E reveals that these structures are a result of the interference of four macroscopic fold phases, the first three of which may represent a single deformation event. The first folding involved sheath-like fold nappes, which were transported to the N or NW, assisted by translation on gently dipping mylonite zones. The regional gneissosity and mineral extension lineations formed during this folding event. The fold nappes were deformed by mainly open upright small macroscopic and mesocopic folds with approximately NE-trending hinges. As a probable continuation of the latter folding, the sheaths were buckled into large macroscopic folds and monoclines with the same NE-trends. The fourth macroscopic folding resulted from shortening along the NE–SW direction, producing mainly NW–SE-trending upright gently plunging folds. Gravitative uplift is disputed as a component of the deformation history of the WHC. The peculiarities of the fold interference pattern result from the interesting behaviour of sheath folds during their refolding.     

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.     

Side-stepping of axial surface traces in superposed folding

During the refolding of an early non-isoclinal fold (say,F ₁) we may find an offset or side-stepping of the axial surfaces of the later folds (say,F ₂). The offsets can be seen in both type 2 and type 3 interference patterns. An analysis of the shear fold model shows that there is a maximum limit for the magnitude of side-stepping. The side-stepping is larger for larger interlimb angles ofF ₁. It decreases with progressive tightening ofF ₂. By recognizing such side-stepping we can predict on which side the F₁ hinge should lie even if the hinge is unexposed or lies outside the domain of observation. The general rule for the sense of side-stepping is the same for shear folds, flexural slip folds and buckling folds. However, the side-stepping in buckling folds should be used with caution, sinceF ₂ folds on buckled single-layers may show an offset whose sense is opposite to that predicted by the general rule.     

A technique for the analysis of refold structures

When folds of multiple episodes interact, complex interference patterns or refolds result. Simple patterns may be directly interpreted by the structural geologist, but more complex ones are often ambiguous and undecipherable. A technique to analyze many refolds is presented which does not directly require any field measurements of structural data. Instead it requires a series of parallel views through the refold, as can be obtained in mines or from a slabbed rock sample. Individual folds are identified on the level maps and hinge trends and plunges are calculated. Stereonet plots and cross-sections yield the fold orientations and shapes. Assumptions inherent to this technique are (1) the first folds were cylindrical and (2) the second folding motions were approximately those of similar (shear) type folds.The above technique was applied to a small-scale example from the Kootenay Arc, Washington. Analysis showed nearly coaxial folds of complex en échelon, non-cylindrical and non-similar type characterizing both generations. This example demonstrates that the technique is applicable to refolds that do not meet the idealized assumptions about folds. This technique should prove valuable to geologists working in other refold areas and is particularly applicable to mining problems for which the necessary three-dimensional data are available.     

褶皱复杂程度的分形分类

本文根据褶皱的白相似性特征,引用分形概念,分析褶皱的分形分布特性,并利用MichaelBamsley（1986）的分形插值理论首次模拟褶皱的复杂形态,主要利用插值点（x、y）和褶皱扰动系数d模拟各种褶皱,在此基础上,提出褶皱复杂程度的分形分类方案。这种新的分类方案即褶皱形态的传统分类名称附缀分形分类名称,解决了传统分类方法不能区分褶皱复杂性的难题。这种方法使褶皱的形态分析达到较精确的定量描述和分类,分形分类方法是对褶雏形态分类的一个补充。     

Noncylindrical flexural slip folds in nature and experiment

Some naturally formed folds in North Cornwall, England, show the following geometrical features:

1. (a) each fold is noncylindrical;

2. (b) the profile shape varies along the hinge-line (chevron-shaped at culmination, rounded at terminations);

3. (c) hinge-lines and axial surfaces of some folds curve strongly in certain restricted areas. Micro-structures indicate that the folds formed by geometric bending and flexural slip.

The progressive development of folds like these has been simulated by subjecting multilayered plasticine models to layer-parallel compression in coaxial stages. This technique allows folds to be observed and measured after each deformation stage. The folds initiate at irregularities in the layering or stress-field. As each fold amplifies, it acquires a characteristic three-dimensional shape: generally the profile is chevron-shaped at the point of maximum amplitude and is rounded at the terminations. Growth of a fold also involves a lengthening of the hinge-line by propagation of the terminations (e.g. into areas of previously unfolded layering). Individual folds also tend to trigger the development of new folds at either side, eventually forming a fold complex with regular wavelength. Propagating fold complexes may interfere by processes of linking and blocking that are not strictly the same as interference effects in other wavelike processes. The interference of fold complexes is described with reference to layer surfaces and crosssections. Causes of noncylindricity are examined. Geometrical aspects of naturally formed folds are analyzed in the light of the experimental findings.     

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.     

Orogen-transverse tectonic window in the Eastern Himalayan fold belt: A superposed buckling model

The Eastern Lesser Himalayan fold-thrust belt is punctuated by a row of orogen-transverse domal tectonic windows. To evaluate their origin, a variety of thrust-stack models have been proposed, assuming that the crustal shortening occurred dominantly by brittle deformations. However, the Rangit Window (RW) in the Darjeeling-Sikkim Himalaya (DSH) shows unequivocal structural imprints of ductile deformations of multiple episodes. Based on new structural maps, coupled with outcrop-scale field observations, we recognize at least four major episodes of folding in the litho-tectonic units of DSH. The last episode has produced regionally orogen-transverse upright folds (F₄), the interference of which with the third-generation (F₃) orogen-parallel folds has shaped the large-scale structural patterns in DSH. We propose a new genetic model for the RW, invoking the mechanics of superposed buckling in the mechanically stratified litho-tectonic systems. We substantiate this superposed buckling model with results obtained from analogue experiments. The model explains contrasting F₃–F₄ interferences in the Lesser Himalayan Sequence (LHS). The lower-order (terrain-scale) folds have undergone superposed buckling in Mode 1, producing large-scale domes and basins, whereas the RW occurs as a relatively higher-order dome nested in the first-order Tista Dome. The Gondwana and the Proterozoic rocks within the RW underwent superposed buckling in Modes 3 and 4, leading to Type 2 fold interferences, as evident from their structural patterns.     

Can flat-ramp-flat fault geometry be inferred from fold shape?: A comparison of kinematic and mechanical folds

The inference of fault geometry from suprajacent fold shape relies on consistent and verified forward models of fault-cored folds, e.g. suites of models with differing fault boundary conditions demonstrate the range of possible folding. Results of kinematic (fault-parallel flow) and mechanical (boundary element method) models are compared to ascertain differences in the way the two methods simulate flexure associated with slip along flat-ramp-flat geometry. These differences are assessed by systematically altering fault parameters in each model and observing subsequent changes in the suprajacent fold shapes. Differences between the kinematic and mechanical fault-fold relationships highlight the differences between the methods. Additionally, a laboratory fold is simulated to determine which method might best predict fault parameters from fold shape. Although kinematic folds do not fully capture the three-dimensional nature of geologic folds, mechanical models have non-unique fold-fault relationships. Predicting fault geometry from fold shape is best accomplished by a combination of the two methods.     

3D fold growth rates

Geological folds are inherently 3D structures; therefore, they also grow in three dimensions. Here, fold growth in all three dimensions is quantified by numerically simulating upright single‐layer folds in 3D Newtonian media. Horizontal uniaxial shortening leads to a buckling instability, which grows from a point‐like initial perturbation in all three dimensions by fold amplification (vertical), fold elongation (parallel to fold axis) and sequential fold growth (parallel to shortening direction) of secondary (and further) folds adjacent to the initial isolated fold. The two lateral directions exhibit similar averaged growth rates, leading to bulk fold structures with aspect ratios in map view close to 1. However, fold elongation is continuous with increasing bulk shortening, while sequential fold growth exhibits jumps whenever a new sequential fold appears and the bulk fold structure therefore suddenly occupies more space. Compared with the two lateral growth directions, fold amplification exhibits a slightly higher growth rate.     

Palaeomagnetic studies of fold development and propagation in the Pembrokeshire old red Sandstone

The geometry of two folds in lower Devonian Old Red Sandstone in Pembrokeshire, deformed in the Hercynian, has been defined at intermediate stages during their evolution by this preliminary palaeomagnetic study. This is possible because remagnetization appears to have occurred in discrete episodes during deformation and the shape can be reconstructed by restoring the fold limbs to an angle at which the remagnetization directions coincide. The plunge of the folds is shown to vary considerably during evolution, one fold having an early easterly plunge which changed to the present westerly plunge, the other fold having an early plunge which was more steeply to the west than its present attitude. Clockwise rotation of the whole of southern Pembrokeshire by 40° towards the end of, or since, the Hercynian deformation, is indicated by the palaeomagnetic data and a structural model for this is proposed.     

Sheath fold development in monoclinic shear zones

We use numerical simulations to investigate the evolution of sheath folds around slip surfaces in simple‐shear‐dominated monoclinic shear zones. A variety of sheath fold shapes develops under general shear, including tubular folds with low aspect ratio eye patterns and tongue‐like structures showing bivergent flanking structures in sections normal to the sheath elongation, which may potentially lead to confusing shear sense interpretations. Not all investigated monoclinic flow end‐members lead to the development of sheath folds sensu stricto (folds with apical angle <90°). The aspect ratio of the eye patterns, R_yz, correlates with the ratio between the principal strain in the Y‐direction and the smaller of the principal strains in the X–Z plane, and thus it could be used in strain analysis.     

Classification of fold interference patterns: a reexamination

A modification of Ramsay's (1967) classification of fold interference patterns is proposed, based upon angular parameters in part different from those used by Ramsay. We use the angle between the axes of the first folds and the pole to the axial planes of the second folds as one of our parameters (gamma). Use of this angle plus beta, one of Ramsay's angles, provides a more natural basis for classification of three-dimensional patterns and in particular avoids several ambiguities in Ramsay's scheme.     

Strain analysis and fold shape in a limestone layer and implications for layer rheology

Folds are developed in thin limestone layers within slates of the McKay Formation exposed to the east of the Rocky Mountain Trench, British Columbia, Canada. They possess geometrical characteristics expected of development by buckling. Strain in the profile plane of a selected fold is similar to that predicted by tangential longitudinal strain, except that magnitudes are too low for the observed curvature. This is attributed to inhomogeneity of strain on the scale of measurement, largely because of pressure solution. Material removed by pressure solution from the inner arc of the fold appears to form veins perpendicular to the hinge, a direction of tectonic stretching. Bedding-parallel stylolites developed diagenetically prior to tectonism.Layer-parallel shortening during the initiation of buckling was less than 20%, and probably less than 10%. The mean arclength/thickness ratio is 6.5 and 7.1, with a dispersion of 0.48 and 0.37 for local and regional populations of 29 and 212 folds, respectively. Application of buckling theory to this data suggests that folding followed a non-linear flow law. The viscosity contrast between limestone and slate would be higher and the power law exponent lower, if initial irregularities in the layers were in the form of a constant amplitude spectrum rather than one of white roughness. The data do not allow a choice of initial amplitude spectrum to be made, nor do they closely constrain estimates of n the power law exponent and viscosity contrast.Deformation in the limestone layers was accommodated by intracrystalline flow (twin gliding), pressure solution, and extensional veining (the last two linked by diffusive mass transfer). The first two dominated deformation in the profile plane of the fold and the last, in association with fracturing, allowed for extension parallel to the hinge. Experimental and theoretical considerations suggest that deformation by a combination of these processes should be non-linear. The non-linear flow law deduced from buckling analysis is consistent with expectations based on observations of active deformation mechanisms.     

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.     

中下部地壳拆离断层带演化中的褶皱作用:以辽南变质核杂岩为例

作为变质核杂岩构造的重要组成部分,拆离断层带内广泛发育的褶皱构造与其寄主岩石一样记录了中下地壳拆离作用过程。选取辽南变质核杂岩金州拆离断层带内褶皱构造作为研究对象,基于叶理与褶皱构造关系分析,划分了褶皱期次与阶段性;通过形态组构分析、结晶学组构分析及石英古温度计等技术方法的应用,初步分析了拆离断层内褶皱的形成机制,为辽南地区拆离作用过程提供约束。根据褶皱形成与拆离作用的时间关系,将拆离带内褶皱分为拆离前褶皱、拆离同期褶皱和拆离后褶皱;拆离作用同期的褶皱按时间早晚分为早期(a1)阶段、中期(a2)阶段、晚期(a3)阶段。不同阶段褶皱的野外形态、叶理与褶皱关系等方面的差异,以及形态组构与结晶学组构的特征,为判断和恢复褶皱的形成机制提供了佐证,揭示出拆离断层带褶皱是在纵弯压扁和顺层流变的共同作用下递进剪切变形的产物。在拆离作用过程中, a1阶段和a2阶段褶皱以纵弯、压扁褶皱作用为主,a3阶段褶皱以弯滑作用为主。褶皱作用记录了拆离断层一定温度范围内(主要集中在380~500 ℃)的变形特征,拆离作用从早期到晚期的演化整体处于相对稳定的应变状态下。对金州拆离断层带而言,在区域NW-SE向伸展过程中,还伴随着NE-SW向微弱的收缩。     

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.     

Strain and rotation in a multilayered fold

Strain was estimated in a fold of Cambrian interlayered siltstones and pelites by determining the preferred orientation of chlorite grains with an X-ray goniometer. Strains so obtained and the postulate that continuity be preserved allowed unfolding of the fold and the determination of rigid body rotations that accompanied the strain. Petrologic investigation showed no sign of major differential volume changes in the siltstones, and this in conjunction with measured strains led to the conclusion that one of the silty layers making up the fold was not, originally, a bed of uniform thickness but a lenticular body, probably representing a single ripple on a ripple-marked tidal flat.Unfolding by piecemeal fitting of unstrained domains shows that none of the principal axes of strain lie consistently parallel to or at right angles to the fold axis. Rock material was displaced with components orthogonal to the profile plane as well as parallel to it. Strain due to compaction during an early history of increasing sediment overburden cannot be separated from strain during tectonic deformation. Its influence is most clearly seen in differential volume change between siltstones and pelites. Additional differential volume changes within pelite beds could have occurred at any time of the deformational history.A comparison of the orientation of strain and rotation axes in the two limbs of the fold, and also comparison of the same orientations in Eulerian coordinates (Cartesian coordinates in the observed fold) and in Lagrangean coordinates (Cartesian in the unfolded fold) make it probable that episodes of relatively uniform strain both preceded and followed the buckling episode that produced the sharp hinge in the competent silt-stone. The siltstone may have been less indurated and thus no more competent than the pelite during early deformation.