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.     

Volumetric strains in neutral surface folding

Neutral surface folding is a significant contributor to fold development. This mechanism produces contrasting strains in the inner and outer arcs of the folded layer that arise from principal stress orientations that are approximately parallel and perpendicular to the layer. We demonstrate that such stress patterns imply significant gradients of mean stress across the folding layer, being more tensional on the outer arc and more compressive in the inner arc. This could pump fluids towards the outer arc during folding and result in heterogeneous volume changes. We conclude that the neutral surface folding model should be adapted to accommodate volumetric strains, in order to explain dilatational structures (e.g. open fractures, veining) on the extrados and volume-loss structures (e.g. pressure solution seams, stylolitic cleavages) on the intrados. This dilatation has economic implications as it allows prediction of sites of mineralization and zones of secondary permeability in fold-related hydrocarbon traps.     

Single-layer folding in marble and limestone: An experimental study

Folding experiments have been carried out on single-layers of Carrara marble and Solnhofen limestone at a confining pressure of 275 bars, temperature of 400°C, and strain rates of 5.5×10⁻⁷ to 8.2×10⁻⁷. The marble and limestone layers were embedded in a rock-salt matrix and in a matrix of a mixture of 60% fine-grained halite and 40% fine-grained calcite, respectively, and deformed to different percentages of bulk shortening. Aspect ratios of the layers varied between 11.25 and 15. The stress-strain relationship reveals that strain increased with a very small increment in compressive stresses, once folding was initiated.With progressive deformation the bulk strain is compensated by folding along one fold hinge. The resulting folds are concentric and a combination of class 1a, 1b and 3 type. The changes in the arc length, layer thickness, limb dip and wavelength with progressive folding in marble layers, are discussed.The microstructure and texture of the folded marble and limestone layers have been investigated optically and by means of an X-ray texture goniometer. The inner fold arc exhibits a strong preferred orientation, whereas in the outer fold are the preferred orientation is poorly developed. Differences in the fabric in medium-grained marble and fine-grained limestone layers have been attributed to the difference in mechanism of deformation.     

Strain analysis in folds (Infrahelvetic complex,Central Alps)

Strain analysis based on initially uniformly oriented elliptical particles in an oolitic limestone (Blegi oolite) was used to study the homogeneity of the state of strain on various scales, kinematics of folding and deformation mechanisms. A computer (reduced means) method for strain analysis is presented which is based on deforming a population of ellipses with shape and orientation properties of measured undeformed ooids. The strain values obtained with this method are within an accuracy of about 10% (in terms of axial ratios) and are in good agreement with the ones obtained with existing graphical methods. The state of strain is homogeneous on the scale of a thin section, handspecimen and outcrop, provided that regions around relatively strong fossils and regions of marked variations in lithology are avoided. Whole rock strains and strains as indicated by ooids alone are similar. Strain patterns in folds in limestones embedded in sandstones, shales and marl are compatible with bending accompanied simultaneously with a shortening perpendicular to the axial surface. The shortening may be attributed to the shear strains related to fold asymmetry and overthrusting. Strains on the outer arcs of a competent dolomite layer compare well with theoretical and experimental fold models; strain patterns include complex contact strains and change along the fold hinge line across a transverse fault which was active during the folding process. Strains parallel to the hinge line are more or less uniform but do not necessarily represent a plane strain state. Volume change took place during deformation. It was accomplished by pressure solution processes, the pressure solved material being partly redeposited. Pressure solution accounts for only a relatively small fraction of the bulk finite strain and was accompanied by plastic flow. Intracrystalline deformation together with grain boundary sliding and/or grain boundary migration went hand in hand with recrystallization (noteably grain growth).     

Deformation in the hinge region of a chevron fold,Valley and Ridge Province,central Pennsylvania

The hinge region of an asymmetrical chevron fold in sandstone, taken from the Tuscarora Formation of central Pennsylvania, U.S.A., was studied in detail in an attempt to account for the strain that produced the fold shape. The fold hinge consists of a medium-grained quartz arenite and was deformed predominantly by brittle fracturing and minor amounts of pressure solution and intracrystalline strain. These fractures include: (1) faults, either minor offsets or major limb thrusts, (2) solitary well-healed quartz veins and (3) fibrous quartz veins which are the result of repeated fracturing and healing of grains. The fractures formed during folding as they are observed to cross-cut the authigenic cement. Deformation lamellae and in a few cases, pressure solution, occurred contemporaneously with folding. The fibrous veins appear to have formed as a result of stretching of one limb: they cross-cut all other structures. Based upon the spatial relationships between the deformation features, we believe that a neutral surface was present during folding, separating zones of compression and extension along the inner and outer arcs, respectively. Using the strain data from the major faults, the fold can be restored back to an interlimb angle of 157°; however, the extension required for such an angle along the outer arc is much more than was actually measured. This disparity between observed and required deformation suggests that the rest of the folding strain may be attributed to minor faulting, isolated severe pressure solution and to slight grain movements; we were not able to recognize the latter. We propose that a single episode of deformation produced the chevron fold causing the brittle deformation after the sandstone had been lithified. This brittle deformation was accomplished by faulting together with the translation of individual sandstone blocks which do not contain significant internal deformation.     

低级变质沉积岩区压溶作用在褶皱变形中的意义

拉卡兰褶皱带中,发育于Ballarat-Bandigo冲断带中的低级变质砂、泥岩的宏观构造以间离劈理和人字形褶皱为特征,而且劈理在褶皱中呈扇形发育。劈理和褶皱的几何关系分析显示:劈理和褶皱的形成为压溶作用、压扁作用、弯曲作用和被动旋转共同作用的结果,而褶皱砂、泥岩中变形构造则以与压溶作用和再沉淀过程有关的显微构造为其典型特征。Fry法进行的全岩应变测量显示,褶皱砂岩的内部应变相当低(X/Z=1.40—1.83),褶皱应变格局给出变形机制的信息包括:缩短过程中的压扁作用和压溶作用、褶皱过程中由弯滑导致的层平行剪应变、以及褶皱后期发育阶段内弧区强烈的压溶作用。宏观构造、显散构造以及应变特征多方面信息证明:低级变质的沉积岩在褶皱变形过程中,压溶作用为一重要的变形机制。应变分解显示在30％—50％的总地壳水平缩短量下,弯曲导致的缩短最为14％—36％,压扁导致的缩短量为3％—14％,压溶导致的缩短量为8％—26％,而且压溶作用主要发生在褶皱内弧区。     

A theory of stress and strain variations in viscous layers, and its geological implications

A system of contrasting viscous layers in oblique deformation is considered. A method of deriving the attitudes and relative values of the principal deviatoric stresses in a layer of particular attitude and viscosity, is presented. For most oblique layers the method is numerical and graphical, but some algebraic solutions are given. Examples demonstrate that stresses refract from layer to layer, and vary in size. The results are more useful to the geologist as infinitesimal strains. It is shown that strain is not homogeneous in layered viscous systems, and strong variations in orientation, size and shape of strain ellipsoids may occur. Some tentative geological implications are suggested.     

Strain patterns and shortening in a folded thrust sheet: An example from the southern appalachians

Total strain patterns estimated across the Pulaski thrust sheet of the southwest Virginia Appalachians show an approximately homogeneous, plane strain deformation associated with folding and distortion above a subsurface décollement. Estimated strains are low (1.2 < < 2.0) with a subvertical extension. Chlorite fibers in pressure fringes on framboidal pyrite indicate that non-rotational deformation produced weak cleavage and pencil structure in mudrock. Variations in shape of pencils and fiber lengths in pressure fringes define highest strains in fold hinges and adjacent to contraction faults. Fabric transitions, delineated by distribution and intensity of cleavage, pencil structure and bedding fissility across the thrust sheet are strain dependent. Balanced cross-sections suggest 35% horizontal shortening due to regional folding and faulting within the Pulaski sheet. Strain integration techniques give 17–35% horizontal shortening associated with cleavage formation. Removal of this strain indicates that cleavage was superposed on open to tight, class-3 folds. Pre-existing thickness variations and anomalous low strains in tight folds require early folding accomodated by intergranular deformation (perhaps controlled grainboundary sliding). Suppression of cleavage formation and penetrative strain was possibly due to higher pore fluid pressure in the early stages of thrust sheet deformation. Observed variations in bedding-cleavage angle and low cleavage fans are compatible with this deformation sequence.     

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.     

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.     

Cleavage-fold relationships and their implications for transected folds: an example from southwest Virginia,U.S.A.

A non-coaxial deformation involving pre-folding initiation of cleavage perpendicular to bedding is proposed to explain non-axial planar cleavage associated with mesoscopic folds in part of the Appalachian foreland thrust-belt of southwest Virginia. Folds are gently plunging, asymmetric, upright to slightly inclined, sinusoidal forms with non-axial fanning cleavage. They show extreme local variations in type and degree of transection and the consistency of transection direction. These relations are further complicated by hinge migration.Cleavage-fan angles, bedding-cleavage angles and δ transection values appear influenced by fold tightness, and in part by fold flattening strain. Fold flattening increments are considered simultaneous with folding. Axial surface traces, and not cleavage traces, coincide with the principal extension direction in fold profiles. Geometric modelling of cleavage fanning and bedding-cleavage angle variations for various theoretical folding modes suggest that folding in limestone and sandstone layers was by tangential longitudinal strain. Significant shape modification and change in bedding-cleavage relations occurred after limb dips of 40 and 50° were attained in limestone and sandstone respectively. Mud-rock class 1C folds with convergent cleavage fans show features transitional between buckling and flexural flow. Initiation of ‘cleavage’ fabrics during layer-parallel shortening prior to significant folding may be important for cleavage evolution in some deformed rocks.     

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.     

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.     

Single layer folding in simple shear

Despite the common occurrence of simple shear deformation, laboratory and numerical simulations of folding have so far been almost exclusively in pure shear. Here we present a series of finite-element simulations of single layer folding in simple shear up to high shear strains (γ ≤ 4, and up to 75% shortening of the folding layer). In the simulations we vary the viscosity contrast between layer and its surroundings (25–100), the stress exponent (1 or 3) and the kinematics of deformation (pure- versus simple shear). In simple shear fold trains do not show a clear asymmetry, axial planes form perpendicular to the developing fold train and rotate along with the fold train. Differences in geometries between folds formed in simple and pure shear folds are thus difficult to distinguish visually, with simple shear folds slightly more irregular and with more variable axial plane orientation than in pure shear. Asymmetric refraction of an axial planar cleavage is a clearer indication of folding in simple shear. The main effect of an increase in stress exponent is an increase in effective viscosity contrast, with only a secondary effect on fold geometry. Naturally folded aplite dykes in a granodiorite are found in a shear zone in Roses, NE Spain. Comparison of the folded dykes with our numerical simulations indicates a viscosity contrast of around 25 and a stress exponent of 3. The natural folds confirm that at this moderate viscosity contrast, a significant amount of shortening (20–30%) is achieved by layer thickening instead of folding.     

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.     

Influence of structural position on fracture networks in the Torridon Group,Achnashellach fold and thrust belt,NW Scotland

In fold-and-thrust belts rocks undergo deformation as fold geometries evolve. Deformation may be accommodated by brittle fracturing, which can vary depending on structural position. We use 2D forward modelling and 3D restorations to determine strain distributions throughout folds of the Achnashellach Culmination, Moine Thrust Belt, NW Scotland. Fracture data is taken from the Torridon Group; a thick, coarse grained fluviatile sandstone deposited during the Proterozoic. Modelling infers a correlation between strain and simple curvature; we use simple curvature to infer how structural position and strain control fracture attribute variations in a fold and thrust belt.In high curvature regions, such as forelimbs, fracture intensities are high and fractures are short and oriented parallel to fold hinges. In low curvature regions fractures have variable intensities and are longer. Fracture orientations in these regions are scattered and vary over short distances. These variations do not relate to strain; data suggests lithology may influence fracturing. The strain history of fold structures also influences fracturing; structures with longer deformation histories exhibit consistent fracture attributes due to moderate-high strain during folding, despite present day low curvature. This is in contrast to younger folds with similar curvatures but shorter deformation histories. We suggest in high strain regions fracturing is influenced by structural controls, whereas in low strain regions lithology becomes more important in influencing fracturing.     

褶皱过程中含金石英脉形成的构造机制——以广西龙水金矿石英脉的成因研究为例

本文以广西龙水水政顶倾竖褶皱构造变形分析为基础,根据含金石英脉与非含金石英脉的变形差异,认为作用于岩层上的主剪切应力(σ_s)与主压缩应力(σ_p)随时间(t)的变化是导致岩层褶皱过程中石英脉多样性变形的主因;阐述了上述模型的动态应力场及其动力学过程;提出了构造变形的分解作用、水力压裂作用与裂开-愈合作用是石英脉形成的新构造机制。     

库车坳陷克拉苏-依奇克里克构造带的构造演化

野外露头上的小尺度构造记录着整个褶皱逆冲带运动学演化进程。通过对库车褶皱逆冲带坎亚肯褶皱发育的小尺度构造的特征和相互叠加、切割关系的分析,认为克拉苏-依奇克里克构造带在新生代的构造挤压变形中经历了三个阶段,即平行层面的挤压缩短作用阶段、弯滑与弯流褶皱作用阶段和逆冲断层改造作用阶段。同时,对不同阶段出现的断层滑动数据进行反演和古应力恢复的结果表明,整个变形过程,最大主应力方向σ1(即构造缩短方向)集中分布在NNW—SSE,没有发生大的变化,说明三个阶段的构造变形是在同一稳定应力场下发生的,从而进一步证明所有的变形是同一次构造挤压下的渐进式序列变形,而不是多期构造挤压变形的结果。此次构造挤压变形主要发生在康村沉积期(12～5Ma),对于克依构造带的构造圈闭最终定型起着决定性作用。     

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.