Geometrical classification of multilayered folds

A new classification scheme based on the degree of fluctuation in the geometry of different layers of a multilayered fold is suggested. The classification scheme uses the degree of fluctuation in geometry in terms of the standard deviation (σ_n) of the thickness parameters t_α′ (orthogonal thickness parameter) and T_α′ (axial plane parallel thickness parameter) for n number of layers, and dip angle α. The degree of fluctuation in the geometry of a multilayered fold can be represented by σ_n(t_α′) or σ_n(T_α′) versus α plots on a Cartesian plane. In the proposed classification scheme, multilayered folds have been divided into two broad categories, namely `isodeviatoric' and `anisodeviatoric'. Isodeviatoric folds have a constant fluctuation in the geometry of different layers recorded in terms of σ_n(t_α′) or σ_n(T_α′) for α>10°. A special type of isodeviatoric fold is recognised as `analogous fold' in which each layer exhibits identical geometry [σ<sub>n</sub>(t<sub>α</sub>′) or σ<sub>n</sub>(T<sub>α</sub>′)=0]. Plots of isodeviatoric folds lie parallel to the abscissa (α) and those of analogous folds lie along the abscissa in the σ<sub>n</sub>(t<sub>α</sub>′) or σ<sub>n</sub>(T<sub>α</sub>′) (ordinate) versus α (abscissa) diagram. Analogous folds have been divided into ten varieties (1A1, 1A2, 1A3, 1B, 1C, 2, 3A, 3B, 3C and composite-analogous types). The anisodeviatoric folds do not exhibit constant fluctuation (deviation) in the geometry of different constitutive layers. Such folds have been subdivided into `peri-analogous', `sub-analogous', `sub-non-analogous', `non-analogous' and `strongly non-analogous' types. This classification scheme is applied to folds developed in low-grade metasedimentary rocks of the Mahakoshal Group and low- to medium-grade rocks of the Chhotanagpur Granite Gneiss Complex in central India.     

The geometrical classification of noncylindrical folds

The majority of naturally occurring folds are noncylindrical if definitions are strictly applied. A new classification of noncylindrical folds using a triangular plot and based on measurements of interlimb angle and hinge angle is proposed. The end-members of the triangular plot are planes, cylindrical isoclines, and isoclinal domes. An infinite range of cylindrical and noncylindrical plane fold shapes may be represented. Noncylindrical nonplane folds may be represented on the plot using proportional circles to signify the degree of non-planarity. The triangular diagram is used to classify large-scale folds from north Norway and their origin is discussed.     

Statistical classification of macroscopic folds as cylindrical, circular conical, or elliptical conical

A folded surface can be represented by the orientation of normals to the surface measured at several locations. When plotted on the unit sphere, the pattern of normals determines the type of fold. Poles from a cylindrical fold give a great circle on the unit sphere, whereas poles of a circular conical fold give a small circle, and poles from an elliptical conical fold give the projection of an ellipse onto the surface of the sphere. Several statistical tests that appear in the literature for classifying folds are discussed and compared. All but one of the tests use quantities obtained from an iterative least-squares procedure that fits the appropriate curve on the sphere. The classification procedure is illustrated with folds from the Canadian Rocky Mountains and uses for examples a cylindrical fold and a circular conical fold from the Smoky River coal field near Grande Cache, Alberta, and an elliptical conical fold near Jasper, Alberta. This methodology has resulted in new coal reserves in the Grande Cache area.This paper was presented at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.     

Petrological evidence for the development of refolded folds during a single deformational event

The Pelona Schist, which forms the lower plate of the Vincent thrust in the San Gabriel Mountains of southern California, has undergone a complex history of folding. The youngest folds in the schist (style 2 folds) range in shape from open to tight and fold both compositional layering and schistosity. These are superposed upon isoclinal folds with axial-plane schistosity (style 1 folds) that, in turn, overprint older isoclinal folds (also called style 1 folds). Samples from the hinges of style 2 folds contain two generations of muscovite. Muscovites of the older generation are parallel to the folded (style 1) schistosity. The newer muscovites recrystallized during and/or after style 2 folding. Microprobe analysis indicates that the two generations of muscovite are very similar in composition, although the new muscovites tend to have slightly higher paragonite and celadonite contents than the old muscovites. From the gross similarity of the two groups of muscovite, it is concluded that the style 1 and style 2 folds were produced during a single progressive deformation. The slightly higher paragonite and celadonite contents of the new muscovites are thought to indicate that both pressure and temperature were increasing during the deformation. This is consistent with the deformation being due to underthrusting of the Pelona Schist beneath the upper plate of the Vincent thrust.     

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.     

Evolution of nonperiodic forms in geological folds

An elastic layer resting on a viscoelastic (Maxwell) foundation is used to model the initiation and subsequent development of a geological fold. This study focuses on the effect of nonlinear terms, of both geometric and material origin, in the formulation of the governing differential equation. The result, during the initial elastic phase of deformation, is to admit the infinite variety of shapes familiar to the field of localized buckling. Their continued effect during the evolutionary phase allows the development of nonperiodic forms hitherto ignored by linear stability analyses. Numerical results are presented for two viscoelastic models subjected to loading conditions of constant end displacement and constant rate of end displacement.     

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。     

Kinematic interpretation of folds in alpine-type peridotites

From a general understanding of the flow mechanisms in alpine-type peridotites, it is possible to describe without ambiguity the general flow regime and its directions in a massif. This result provides the means for an investigation of the origin of the folding in pyroxenitic layers independent of any preconceived theory on folding.The folds are usually isoclinal and of the flexural-flow type as demonstrated by petrofabric studies in hinges. Their axes are always parallel or subparallel to a mineral lineation which in turn is parallel or close to the orientation of the fabric elements defining the flow line. Their axial plane, which usually coincides with the foliation, is parallel to or close to the flow plane. This conclusion, also supported by paragenetic observations, shows that the folds were formed or transposed during the plastic flow responsible for the development of structures (foliation and lineation), textures and preferred mineral orientations. In the case of the Lanzo Massif and a few other Iherzolite massifs, the flow occurred during the intrusion from the mantle. The mapping in Lanzo yields evidence of a large-scale U-shaped fold with a remarkable pattern of mesoscopic folds attached to it: the tight isoclinal folds are restricted to the limbs of the largescale structure, and the open folds locally refolding former isoclinal ones to the hinge area where the angle between the folded pyroxenitic layering and the axial-plane foliation is large. Stereograms of the field structures in this hinge area clearly illustrate the geometric relations mentioned above.This folding, characterized by its axis and axial plane respectively close to the flow line and flow plane, can be explained either by rotation towards the flow line of non-cylindrical-fold axes or by direct formation in a non-plane flow when the flow line is initially contained in the layering or close to it. In this respect, the folding may bring information on the minor flow component, complementary to that given on the major flow component by considering the textures and fabrics. Finally this folding is shown to be ubiquitous in plastically deformed peridotites. It is proposed that these conclusions be extended to other domains submitted to intense non-plane flow.     

Parallel, similar and constrained folds

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

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

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

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

Finite-element folds of similar geometry

Model folds of similar geometry have been produced by using the finite-element method and the constitutive relations of a layer of wet quartzite embedded in a marble matrix with an initially sinusoidal configuration and a 10° limb dip. The power law for steady-state flow of Yule Marble (Heard and Raleigh, 1972) is used for the matrix and our new law for Canyon Creek quartzite deformed in the presence of water is used for the layer. The equiv- alent viscosity of the wet quartzite is highly temperature-sensitive, giving rise to a strong temperature dependence of the quartzite: marble viscosity ratio which, at a strain rate of 10⁻¹⁴/sec, drops from 543 at 200° to 0.13 at 800°C. At 375°C (η_q/η_m = 10), concentric folds develop at all strains to 80% natural shortening and stress, finite strain and viscosity distributions are somewhat similar to those found previously. Raising the temperature to 550° C (η_q/η_m = 1), at any stage of prior amplification, causes the folds to flatten with increasing strain, accompanied by attenuation of limbs and thickening of hinges, leading to folds with similar geometries and isoclinal folds at extreme strains. The effects are more pronounced at higher temperatures and at 700° C (η_q/η_m = 0.3) limb attenuation is so severe as to give rise to unrealistic geometries. At temperatures below about 600° C (η_q/η_m = 2), similar folds do not form. It thus appears as if a viscosity contrast near unity is required to produce similar folds in rocks, under the conditions simulated and different temperature dependencies of viscosities of materials in layered sequences is one important means of reducing viscosity contrasts.     

Formation of small scale folds and faults in northern Ohio

Folds and faults in the Devonian shales of the Berea area, Ohio have been analyzed on a small scale (several meters) to determine their method of formation. A simple elastic model has been developed to account for stress concentration due to lithostatic pressures induced by local relief. This analytical model is then coupled to the beam–column buckling stability theory. The possibility of failure due to faulting is also examined using the Coulomb criterion. Initial results indicate that local stresses due to topography are a probable cause of these structures, though the possible role of regional tectonic effects cannot be eliminated. The state of the local stress field and the possibility of seismic events have important implications for surface and subsurface design problems.     

The geometry and kinematics of flow perturbation folds

Minor folds formed synchronous with ductile deformation in high strain zones can preserve a record of the scale and kinematics of heterogeneous flow. Using structures associated with WNW-directed Caledonian thrusting in N Scotland, we show that localised perturbations in flow resulted in the generation of predominantly cylindrical minor folds with hinges lying at low angles to the transport direction. These define a series of larger-scale fold culminations (reflecting ‘surging flow’) or depressions (reflecting ‘slackening flow’) that are bisected by transport-parallel culmination and depression surfaces. The fold patterns suggest a dominance of layer-normal differential shearing due to gradients in shear strain normal to transport. Culmination surfaces are marked by along-strike reversals in the polarity of structural facing and vergence of minor folds which, contrary to classic fold patterns, define reverse asymmetric relationships. Culmination surfaces separate folding in to clockwise (Z folds) and anticlockwise (S folds) domains relative to the transport lineation. The dip of fold axial planes systematically increases as their strike becomes sub-parallel to transport resulting in a 3D statistical fanning arrangement centred about the transport direction. Thus, mean S- and Z-fold axial planes intersect precisely parallel to the transport lineation and potentially provide a means of determining transport directions in cases where lineations are poorly preserved. Culminations display convergent fold patterns with fold hinges becoming sub-parallel to transport towards the culmination surface and underlying detachment, whilst axial planes define overall concave up listric geometries which are bisected by the culmination surface. Thus, around culminations and depressions there are ordered, scale-independent relationships between transport direction, shear sense, fold facing, vergence and hinge/axial plane orientations. The techniques described here can be applied and used predictively within any kinematically coherent system of ductile flow.     

断层相关褶皱概念模型中的裂缝域

断层相关褶皱对裂缝的控制作用主要体现在岩层的几何形态对裂缝方位的影响和变形对岩石破裂的控制程度。本文在断层相关褶皱概念模型的基础上提出了裂缝域,并以Suppe的平行膝折概念模型为对象,利用几何学方法对模型中区域性裂缝的裂缝域进行分区描述,并提出了断层相关褶皱的曲线概念模型中区域性裂缝的分布规律。以新疆卡普沙良河一处断层转折褶皱剖面上裂缝的实测数据为依据,对模型进行验证,发现测量结果同模型基本一致。该方法为前陆冲断带地区断层相关褶皱对裂缝控制作用的定量预测提供了一种全新的思路。     

Numerical studies of hyperplasticity with single,multiple and a continuous field of yield surfaces

The development of a new constitutive model would normally require a new procedure to be established for derivation of the incremental response. However, for models generated within the framework of hyperplasticity (using single, multiple or continuous yield surfaces), this derivation can be carried out using standard procedures. In this paper we present first the unified incremental response for a model using a single internal variable for general loading conditions. Next, we develop and explore three different numerical techniques for implementation of this procedure. One of the approaches, which seems superior, is extended to the multi‐surface hyperplastic formulation. Finally, to allow integration of continuous hyperplastic models within the same multi‐surface hyperplastic setting, the issue of discretization of the continuous field of yield surfaces is addressed. Copyright © 2003 John Wiley & Sons, Ltd.     

Volcanic rocks of the witwatersrand triad,south Africa. I: Description,classification and geochemical stratigraphy

The Witwatersrand triad contains thick volcanic sequences confined largely to the Dominion Group at the base and the Ventersdorp Supergroup at the top. These volcanic sequences are of late-Archaean to early-Proterozoic age and are amongst the oldest supracrustal volcanic sequences erupted onto the Archaean Kaapvaal craton. The volcanic rocks have suffered low-grade greenschist facies metamorphism but primary textures and, in some samples, primary mineralogies are well preserved. Major and trace element analyses of a large number of samples of volcanic rocks from boreholes in the Klerksdorp area, Western Transvaal, indicate that the chemical compositions of the lavas are largely unmodified by secondary processes and can be used for classification, stratigraphic studies and petrogenetic interpretation. The volcanics of the Dominion Group are a bimodal tholeiitic basalt-rhyolite association with considerable compositional variation from basalt to tholeiitic andesite within the basaltic suite. The Ventersdorp Supergroup is a weakly developed bimodal tholeiitic basalt-dacite suite. Basic rocks in the Klipriviersberg Group at the base and in the overlying Platberg Group are largely basaltic, whereas those of the Allanridge Formation at the top are largely tholeiitic andesites. Within the Klipriviersberg Group the more primitive samples occur at the top of the sequence. Phenocryst-rich porphyries of the Makwassie Formation in the Platberg Group are largely of dacitic composition and have textural features suggestive of an ash flow origin. A distinct geochemical stratigraphy is present within the volcanic sequences and application of discriminant function analysis and an empirical discrimination grid based on ratios of immobile elements P, Ti, and Zr allows for unknown samples to be placed within their correct stratigraphic unit.     

Stress-induced viscosity changes and the existence of dominant wavelengths in folds

The theory of folding in stratified media presented by Biot and Ramberg has been extended by considering a more general type of material response. The model consists of a viscous layer embedded in a less viscous medium, compressed parallel to the layering. A transition from Newtonian to non-Newtonian behavior is considered and an approximate solution to the governing equation is discussed. The results give the effect of local, stress-induced changes in the viscosity on the profile of the fold. The results indicate that as the transition to non-Newtonian behavior proceeds, the wavelength selection process described by Biot and Ramberg breaks down; the wavelength of the fold which develops probably will not be the “dominant” wavelength defined by Biot.     

Magnetic fabric in folds of the easternmost Rheno-Hercynian Zone

The magnetic fabric in folds was investigated in the easternmost Rheno-Hercynian Zone, the Nízký Jeseník Mts. In their eastern areas, the rocks show signs of only weak achimetamorphism and very gentle ductile deformation; SE vergent buckle folds of long wavelength are developed whose magnetic fabric can be easily unfolded geometrically. In the central areas, spaced cleavage and NW vergent buckle folds can be found; the folds can be unfolded mostly only partially. In the western areas, NW vergent cleavage folds and very well developed slaty cleavage occur. The magnetic fabric in the folds is homogeneous, the folds cannot be unfolded at all. The cleavage is transformed into metamorphic schistosity at the western border of the area.     

Successive development of plane noncylindrical folds in progressive deformation

In the history of superposed deformations of the iron formations at the western border of the Kolar Gold Field in S India, an important event was the successive growth of broadly coaxial plane noncylindrical folds in course of a progressive deformation concomitant with development of ductile mesoscopic shear zones. The noncylindrical folds were initiated as active folds by the creation of a buckling instability at successive stages on newly developed foliation surfaces. The nucleation of noncylindrical folds and the subsequent axial-plane folding of the tightened mature folds are explained by the mechanical inhomogeneity of the rocks and the heterogeneous character of strain. The correlation between increasing tightness and increasing noncylindricity of the folds indicates that the initial curvatures of hinge lines were accentuated by an extension parallel to the subhorizontal stretching lineation. From the patterns of deformed lineations over folds of varying tightnesses, it is concluded that the passive accentuation of hinge-line curvatures was mostly achieved when the folds had already become isoclinal or very tight.