Peter J. Hudleston

Fold morphology and some geometrical implications of theories of fold development

Abstract The geometric properties of folds are considered to be of prime importance in fold classification and in the analysis of natural folds. Methods of precise description of fold geometry are necessary for natural fold analysis, and they enable a link between theoretical and experimental work and natural folds to be made. Various methods of analysing data for the spatial attitude of folds are briefly mentioned, and a detailed examination of methods of representing the geometric form of folds in profile section is made in two parts, one dealing with the forms of single layers and the other with the shapes of individual surfaces. In both cases existing methods of geometrical analysis are critically appraised and many are found to be impracticable. Folded layer geometry can most usefully be represented by two parameters that are both functions of apparent dip. These are thickness, t , and a new parameter, φ, which derives from and is used in conjunction with dip isogons. A refinement of geometrical fold classification is made in terms of these parameters, their interrelationship is examined and their relative merits are considered. In the case of single folded surfaces, a simple application of harmonic analysis provides a useful means of describing the fold shape. The most basic and suitable segment of a folded surface for analysis is a “quarter-wavelength” unit between adjacent hinge and inflexion points. Such a choice of unit leads to a harmonic series consisting only of the odd terms of a sine series. The first few harmonic coefficients are sensitive parameters of fold shape, most information being contained in the first two coefficients, b 1 and b 3 . Plots of b 3 against b 1 and log b n against log n are useful means of representing and comparing coefficients for different folds. Many natural folds seem to be closely matched in shape by a member of an ideal series of mathematical forms, and this leads to a rapid visual method of harmonic analysis that involves no measurements. This has proved most useful in studies of the comparative morphology of folded rocks. Theories of fold development are discussed with particular reference to the development of folds by buckling in isolated competent layers embedded in a less competent matrix. The geometric properties of folds predicted by the theories are examined in terms of the descriptive parameters discussed in the first part of the paper. Harmonic analysis is used to describe the progressive changes in fold shape predicted by recent theory for the buckling of a layer starting as a low-amplitude sine wave. The geometric properties of passive folds are briefly considered, and the problem of “similar” folds is discussed. The geometric forms taken up by two idealised models of a buckled layer are calculated and compared. The clearest distinction between the models is brought out in the isogon patterns or by the thickness variations with dip. The effect of compression in modifying the geometry of folds is considered. Plots of thickness or φ against apparent dip for parallel folds that have been uniformly “flattened” can be so adjusted as to give straight line relationships. Most natural fold shapes are closely represented by straight lines on such adjusted graphs, and the slope or intercept of the best-fit straight line to natural fold data is an empirical parameter of folded layer shape. A simulated effect of simultaneous buckling and flattening of a layer is described that predicts relationships between thickness and dip that have been observed in natural and experimentally produced folds.

An analysis of “Single-layer” folds developed experimentally in viscous media

Abstract Experiments have been performed to study the development of folds in single viscous layers embedded in a less viscous matrix, shortened parallel to the layering under conditions of plane strain and pure shear. This is thought to best simulate the development of small folds in isolated competent rock layers under conditions of regional metamorphism. Biot has suggested that distinct folds will not form by buckling at viscosity contrasts of less than 100:1. Effective viscosity ratios of layer to matrix of between 10 and 100 to 1 were used in the experiments, and folds were observed to develop by buckling in all cases. Analysis of arc length and limb dip with progressive deformation shows that there is always a stage of initial layer-parallel shortening during which folds appear, that gives way to a stage of development where the arc length of the folds changes only slightly for further increase in deformation. The amount of layer-parallel shortening that takes place increases with decreasing viscosity contrast, but the transition from a stage of layer shortening to one of nearly constant arc length seems to take place when the folds have mean limb dips of 10 – 20°, irrespective of viscosity contrast. No relative thickening in the hinges or thinning in the limbs was observed in any of these experiments. Analyses of wavelength/thickness ratios, amplitude, fold shape and layer-parallel shortening show that folding in the experiments is best accounted for in the early stages by the theory of Sherwin and Chapple, and in the later stages by the mathematical models of Chappie. Experiments made at very low viscosity contrasts with folds initially present in the competent layer whose wavelengths were considerably larger than the predicted dominant wavelength, lead to the development of folds with thickening in the hinges and thinning in the limbs. These folds had geometries that can be simulated by a process of simultaneous buckling and flattening. The results of the experiments may be used as an aid in the interpretation of natural fold geometry, and estimates of viscosity contrast and amount of deformation can be made where suitable folds exist.

Testing models for obliquely plunging lineations in transpression: a natural example and theoretical discussion

Theory predicts that stretching lineations in an ideal vertical transpressional zone should be either vertical or horizontal. Many field descriptions of transpressional zones, however, indicate a range of lineation orientations between these extremes. Several theoretical models have been developed to explain such departures from expected lineation orientation, and we discuss these in the context of a field example from the Archean Superior Province in the North American craton. Existing models are insufficient to explain obliquely plunging lineations in this example because: (1) obliquely plunging lineations cannot be accounted for by shear zone boundary effects imposed by a no-slip condition, (2) foliations and lineations vary independently, (3) the vorticity-normal section is subhorizontal, limiting possibilities for inclined simple shear, (4) high vorticity is needed for finite strains and lineations to match previously proposed triclinic models, but vorticity is relatively low, and (5) juxtaposed east and west plunging lineations are unlikely in the previously proposed triclinic models. Because existing theoretical models are not applicable to our field example, we contemplate a new model to explain obliquely plunging lineations within quasi homogeneous transpression.

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

Abstract 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 = μ 1 μ 2 ) 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 μ 1 μ 2 = 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 μ 1 μ 2 = 10 , and in experiments for μ 1 μ 2 = 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.

Information from fold shapes

Abstract Folds contain information about the deformation rocks have undergone and the condition of the rocks during deformation. How much of this can we decipher? The geometrical characteristics of folds and the strain distribution within them are perhaps the key to unlocking this information. If the folded layers were originally planar, they reflect inhomogeneous deformation, but without additional originally planar or linear markers of other orientations, we cannot determine the state of strain. The strain in a parallel fold, for example, can be accommodated either by flexural slip or by tangential longitudinal strain, two quite different but geologically realistic strain distributions among an infinite number of possible ones. There are some constraints imposed by fold shapes on possible strain distributions. Fold asymmetry reflects, in most circumstances, sense of shear strain parallel to the general orientation of the folded layering. Problems may arise in shear zones with large strains and in foliated rocks in which kink bands develop at small strains. Information on the orientations of principal stresses cannot be obtained from folds, unless the strain is small and the mechanism of folding understood, as for kink bands, or unless the bulk flow is of constant vorticity, which is difficult to demonstrate. The distribution of measured values of wavelength/thickness in a population of single-layer folds, together with measures of strain and estimates of amplification, can be used to estimate the viscosity ratio of the stiff layer to its matrix and the degree of non-linearity in the flow law, if effective power-law flow is assumed. Information on the rheological state of rocks at the time of folding can also be obtained from the pattern of curvature variation in individual single-layer folds, as demonstrated by the use of computer simulations of folding. Where applied, such methods indicate that the slow natural flow of rock involved in folding is mostly consistent with non-linear power-law rheology, as expected from the results of experimental rock deformation involving crystal-plastic deformation mechanisms.

STRAIN COMPATIBILITY AND SHEAR ZONES : IS THERE A PROBLEM?

In analyzing deformation in rocks, it is important to ensure that the solutions obtained satisfy strain compatibility. This creates a challenge to understanding patterns of strain associated with shear zones, in which measured strain may appear incompatible with the strain in the shear zone walls. Flattening strains are common in natural shear zones with locally straight and parallel boundaries: to satisfy compatibility conditions such strains require volume loss across the shear zone or deviations from plane strain, with or without discontinuities between the shear zone and the wall rock. In the case of shear zones for which there is no evidence of volume loss or discontinuities along the shear zone walls, problems of strain compatibility may be resolved if individual shear zones are linked together in an appropriate fashion. Shear zones commonly occur in anastomosing arrays, and simple configurations of such arrays and the strains associated with them are examined. It is shown that local transpression with strain compatibility can be accounted for in this way. Quite complex local strain patterns can develop in simple arrays.

Similar Folds, Recumbent Folds, and Gravity Tectonics in Ice and Rocks

Under steady state flow in glaciers, a banding will form and tend towards parallelism with the flow lines in basal regions of high shear. A simple mathematical model shows that departures from the steady state cause a reorientation of the flow lines such that they depart from parallelism with the banding. If a suitable bedrock ridge is present, the banding will subsequently become deformed into subsimilar folds, whose axial surfaces are very nearly parallel to the direction of maximum extension of the associated strain. The cleavage pattern, geometry, and orientation of many folds in rocks suggest a similar origin during gravitationally-induced flow away from an orogenic welt. In some cases buckling may be a secondary response, resulting in non-similar folds.

Experimental Deformation of Synthetic Magnetite-Bearing Calcite Sandstones' Effects on Remanence, Bulk Magnetic Properties, and Magnetic Anisotropy

We have quantified effects of experimental deformation on the magnetic properties of a set of synthetic “calcite sandstone” samples containing magnetite. The deformation was carried out in a microcomputer-controlled apparatus that adjusted the applied differential stress as needed to maintain a constant strain rate of 10−5 s−1. Most samples were deformed under dry conditions, but a few were deformed with a pore fluid present; the samples deformed under dry conditions required substantially higher differential stresses. Macroscopically ductile shortening strains of up to 25% produced the following irreversible changes in magnetic properties: (1) increased bulk coercivity, remanence coercivity, and mean anhysteretic remanence susceptibility; (2) decreased mean low-field susceptibility; (3) decreases in the component of remanence parallel to shortening; (4) smaller decreases for most samples in the component normal to shortening, resulting in a net “rotation” of the remanence away from the shortening axis; (5) larger decreases in the normal component in a few samples, resulting in a net “rotation” of the remanence towards the shortening axis; (6) increased magnetic anisotropy; and (7) increased “deformation” of initial magnetic ellipsoids. A comparison of data for samples deformed under dry and wet conditions (higher and lower differential stresses, respectively) indicates that remanence reorientation and susceptibility anisotropy are controlled primarily by bulk strain (i.e., rotation and displacement of particles), whereas coercivity and anhysteretic anisotropy are controlled dominantly by microstrain or intragranular stress.

Recumbent folding in the base of the Barnes Ice Cap, Baffin Island, Northwest Territories, Canada

Recumbent folds exposed in an ice cliff at the southeast side of the Barnes Ice Cap occur in banded ice and have hinges subparallel to the glacier margin. The folds appear similar in shape and attitude to others expressed on the glacier surface as a series of irregular lenses of white ice that are elongate parallel to the margin and are surrounded by blue ice. Such lenses are all around the margin of the south dome of the ice cap. Both sets of folds are thought to have a common origin. Fold geometry and fabric studies suggest that the ice is behaving homogeneously on the scale of the folds and that the banding is essentially passive. Flow considerations indicate that banding or foliation will tend to become parallel to the particle paths near the glacier base and toward the margin under steady-state conditions. However, departures from the steady state in the form of minor advances or retreats may change the flow pattern sufficiently for the particle paths to depart from parallelism with the banding, which may then become passively deformed and eventually folded. For this to occur, the bedrock surface must be appropriately irregular. A simple mathematical model describes this process and successfully accounts for the geometrical features of the folds observed. This theory is consistent with earlier observations of the Barnes Ice Cap, which suggest that there have been fluctuations in the position of the ice-cap margin in the last few centuries.

Numerical modeling of fold initiation at thrust ramps

Abstract The movement of the hangingwall over the footwall at thrust ramps produces a variety of structures found commonly at different scales in fold-and-thrust belts. The nature of the structure depends on the relative rigidity of the hangingwall and footwall, friction along the fault, fault ‘dip’, fault displacement, and the boundary conditions of the deformation. Structures include fault-bend style folds, fault-propagation style folds and wedge folds. We investigate the initial stages of development of such structures using the finite-difference code FLAC. The rock layers are represented as continua with elastic-plastic Mohr-Coulomb constitutive relations, and the fault and bounding bedding planes are assigned normal and shear stiffnesses and coefficients of friction. Under conditions with all layers compressed, antisymmetric fault-propagation style folds develop in both the hangingwall and footwall. With rigid footwall and restricted far-field slip in the hangingwall, a single fault-propagation style fold develops. With far-field displacement of the hangingwall allowed, broader antisymmetric wedge folds develop if hangingwall and footwall are deformable, and a single fault-bend style fold develops if the footwall is rigid. All structures become accentuated with increasing slip on the fault. Where both hangingwall and footwall are deformable, the deformation reduces the ramp angle and tends to minimize distortion of the rock adjacent to the fault. Fault-propagation style folds, paired wedge folds and fault-bend style folds are common in nature, and small-scale examples can be found in various stages of development. Continued slip on thrust faults may lead to the mature structures commonly seen in fold-and-thrust belts.