Labao Lan

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.

Rheological controls on the shapes of single-layer folds

Abstract Information about rheology can potentially be gained from analyzing the shapes of folds in isolated buckled layers. We employ two-dimensional finite element models of incompressible flow in power-law viscous fluids to investigate this. We first show that the shape of the initial perturbation has relatively little effect on the final shape of the folds when the buckling instability is high. We find that the most significant factor affecting the shapes of single-layer folds is the stress exponent, n L , in the flow law of the layer. The hinges of outer arcs become sharper as n L increases and the limbs become relatively longer and straighter. These differences can be expressed quantitatively by defining a curvature index, ki , which has a value of 0 for a fold formed by circular arcs and 1 for a chevron fold. Results show a dependence of ki on n L that is well-defined for L / h > 10, with ki increasing with n L . Data for experimentally-produced folds are consistent with the numerical results. Limb dip, ki and L/h can all be readily measured on natural folds and provide a basis for comparing the shapes of natural and computer-simulated folds. The data for small folds in siltstone layers in shales in the central Appalachians are consistent with highly non-linear flow of the stiff siltstone layers during buckling.

Flexural flow folding: Does it occur in nature?

Flexural flow produces a continuous deformation equivalent to that of a flexed deck of cards of vanishing card thickness. It is assumed to be a common process by which parallel folds are formed in nature in competent, anisotropic rock layers. The strain pattern produced by flexural flow contrasts strongly with that produced in parallel folds formed by the bending or buckling of isotropic competent layers, but it has not been documented in nature. Two basic questions are thus: (1) If anisotropy is sufficiently large, will flexural flow develop, and (2) if so, are competent rocks sufficiently anisotropic for them to respond to deformation by flexural flow? A measure of planar anisotropy, A , is given by the ratio of viscosity in shortening to viscosity in shear. Finite-element modeling indicates that if the value of this ratio is greater than about 50, the strain pattern in a buckle fold in a competent layer will be almost indistinguishable from one of flexural flow. Such a high anisotropy, however, is unlikely in natural competent layers. The most texturally anisotropic rocks tend to be those rich in phyllosilicates, and these usually behave in an incompetent fashion. Common competent rocks, such as those rich in quartz and feldspar, are rarely significantly anisotropic. A strong preferred crystallographic orientation as a result of crystal-plastic flow can produce anisotropic layer behavior, but A is unlikely to exceed ∼10, which is insufficiently great for folding by flexural flow. We thus conclude that flexural flow is unlikely in single competent layers. The aggregate response to deformation, however, of composite layers of alternating competent and incompetent rocks may be highly anisotropic and on average correspond to flexural flow.

Pure shear deformation of square objects, and applications to geological strain analysis

Abstract Geological objects that deform differently from the rock matrix, such as pebbles or other clasts, are unlikely to have been originally circular or elliptical in section, and must therefore be expected to deform heterogeneously and change shape irregularly. We investigate this process with finite element models of pure shear deformation of square objects in three orientations, square, skew and diagonal, in a matrix with different viscosity. Modelling shows that ‘squares’ deform irregularly, with competent objects becoming barrel shaped and ‘fish mouthed’ (cf. boudins), whereas incompetent objects become bone shaped or elongate lobes. The object aspect ratios ( R O ) are less different from the bulk strain ratio ( R S ) than for equivalent circular objects. In contrast, diagonal squares deform almost homogeneously into ‘rhombs’, with aspect ratios closer to those for circles. Asymmetrically oriented ‘skew squares’ behave intermediately, developing skew flag and hooked shapes according to competence contrasts, that might be misdiagnosed as shear criteria. All these square objects (and circles in theory), show almost linear strain paths of object versus bulk ( R −1), with slope related to viscosity ratio, object shape and orientation. Linear relationships are also found for concavity/convexity shape factors for ‘squares’. The results have implications for strain analysis and competence contrasts in rocks.

Finite-element models of buckle folds in non-linear materials

Abstract Buckle-fold development in single layers has been simulated by means of a two-dimensional finite element model incorporating incompressible power-law viscous fluids. We have produced folds with the stiff layer initially parallel to and oblique to the bulk maximum shortening direction and investigated how fold growth rates and geometry vary as functions of viscosity ratio and power-law exponent. Results for single-layer buckling and the related instabilities of boudinage, mullions and inverse folding closely match theoretical predictions in terms of initial growth rates, which depend strongly on the power-law exponent. The results further show that the fold shape is sensitive to changes in the power-law exponent of the stiff layer. The most striking feature is that the fold hinges become sharper and the limbs become relatively longer and straighter as the exponent increases. Asymmetry of folding can arise in bulk pure shear. Models of asymmetric folds with inclinations of the stiff layer of up to 30° have been made. The results indicate that initial growth rates are independent of the inclination of the stiff layer. Results of this work suggest that a number of geometrical properties of folds may be useful in providing information on the rheological behavior of rocks deformed under very slow natural conditions.

Rock rheology and sharpness of folds in single layers

Abstract Many folds in rocks display angular profiles with sharp hinges and straight limbs. Previous studies by many authors have demonstrated that such fold shapes in multilayers result from an intrinsic anisotropy possessed by layered or foliated rocks. Our results of two-dimensional finite-element modeling show that folds with sharp hinges and straight limbs may also develop in isolated competent layers under suitable rheological conditions. Two basic material properties that affect fold shape are non-linearity and anisotropy. Viscous-plastic flow, power-law flow (strain-rate softening), and strain softening are three types of non-linear behavior. All lead to folds with sharp hinges and straight limbs. Anisotropy has a similar influence on fold geometry. Angularity of folds in isolated competent layers increases with increasing non-linearity of the layer or increasing anisotropy of the layer, and a quantitative relationship between fold angularity and degree of non-linearity or anisotropy may be established. Virtually identical angular fold shapes may be produced by either non-linear or anisotropic layer behavior. The strain distribution associated with these shapes is very different, however. For non-linear behavior, strain is focused in the fold hinges and minimal on the limbs, and for anisotropic behavior the reverse is the case, with shear in the limbs being dominant. These differences suggest that it should be possible to distinguish non-linear from anisotropic rheological behavior using layer shape and strain pattern.

Simple shear of deformable square objects

Abstract Finite element models of square objects in a contrasting matrix in simple shear show that the objects deform to a variety of shapes. For a range of viscosity contrasts, we catalogue the changing shapes and orientations of objects in progressive simple shear. At moderate simple shear ( γ =1.5), the shapes are virtually indistinguishable from those in equivalent pure shear models with the same bulk strain ( R S =4), examined in a previous study. In theory, differences would be expected, especially for very stiff objects or at very large strain. In all our simple shear models, relatively competent square objects become asymmetric barrel shapes with concave shortened edges, similar to some types of boudin. Incompetent objects develop shapes surprisingly similar to mica fish described in mylonites.

Non-ellipsoidal inclusions as geological strain markers and competence indicators

Abstract Geological objects that do not deform homogeneously with their matrix can be considered as inclusions with viscosity contrast. Such inclusions are generally treated as initially spherical or ellipsoidal. Theory shows that ellipsoidal inclusions deform homogeneously, so they maintain an ellipsoidal shape, regardless of the viscosity difference. However, non-ellipsoidal inclusions deform inhomogeneously, so will become irregular in shape. Geological objects such as porphyroblasts, porphyroclasts and sedimentary clasts are likely to be of this kind, with initially rectilinear, prismatic or superelliptical section shapes. We present two-dimensional finite-element models of deformed square inclusions, in pure shear (parallel or diagonal to the square), as a preliminary investigation of the deformation of non-ellipsoidal inclusions with viscosity contrast. Competent inclusions develop marked barrel shapes with horn-like corners, as described for natural ductile boudins, or slightly wavy rhombs. Incompetent inclusions develop ‘dumb-bell’ or bone shapes, with a surprising degree of bulging of the shortened edges, or rhomb to sheath shapes. The results lead to speculation for inclusions in the circle to square shape range, and for asymmetric orientations. Anticipated shapes range from asymmetric barrels, lemons or flags for competent inclusions, to ribbon or fish shapes for incompetent inclusions. We conclude that shapes of inclusions and clasts provide an important new type of strain marker and competence criterion.

The effects of rheology on the strain distribution in single layer buckle folds

Abstract Folds in rock provide opportunities for establishing how fabric and deformation mechanisms vary with local variations in strain and strain rate, and strain distribution in folds should, in principle, provide information on rheological conditions during folding. To investigate this, we use finite element models to simulate buckling in single-layer folds in incompressible, power-law materials in plane strain. Numerical results show that the pattern of strain variation in buckle folds is sensitive to variations in the power-law exponent, n L , of the stiff layer. For a given amplitude, wavelength/thickness, and ratio of viscosities, m , of layer to matrix, strain and strain gradient along the axial trace increase more rapidly away from the neutral surface (on both sides) for power-law ( n L > 1) materials than for Newtonian ( n L = 1) materials. Buckling strain is superimposed on early uniform layer-parallel shortening, which becomes greater as initial amplitude decreases and as n L and m decrease. The effect of this superposition can best be described by reference to the infinitesimal neural surface (INS) and finite neutral surface (FNS), which vary in position with degree of overall shortening. The INS and FNS move from the outer arc towards the inner arc during buckling, the latter following the former. Thus, at any stage of folding, the layer can be divided into three zones with different coaxial, non-linear strain histories. Fabric in natural folds is expected to reflect such unsteady flow conditions.

Rheological Information from Geological Structures

The contrast in rheological properties between layers of different composition or texture, and between stiff inclusions and their matrix, gives rise to perturbations in flow that result in structures. Theory and modeling allow us to understand the conditions necessary for such structures to form and, conversely, we can use the form of the structures to infer possible rheological conditions for the rocks during natural deformation. We review here several structures and their use as indicators of rheological behavior, based on theory, numerical and experimental models, and observations on naturl structures.Theory predicts that a dominant wavelength/thickness (Ld/h) exists for both folding and boudinage that depends on the ratio of viscosities of layer to matrix, the homogeneous shortening undergone by the layer, and the exponent,n, in a flow law of power-law type. The measurement of averageL/h, and of the shortening within the layers allows an estimate of the power-law exponent of the stiff layer to be made. Also, numerical modeling shows that fold hinges become sharper asn increases and the limbs become relatively longer and straighter. The dynamic growth of pinch and swell instabilities only overcomes the kinematic decay in nonlinear flow, thus the existence of pinch and swell is by itself evidence of nonlinear behavior.Strain rate and strain, increase more rapidly away from the neutral surface for a layer of power-law rheology (withn>1) than for a layer of Newtonian rheology. Thus, strain gradient across a fold hinge, at fixed amplitude or limb dip, increases with increasingn.Porphyroclasts in mylonites develop characteristic rims of recrystallized grains that are drawn out into trails of σ or ° shape by the perturbed flow of the, material around the clast. Experimental evidence suggests that σ shapes occur if flow is linear, whereas nonlinear flow may give rise to the δ shape.The inference of rheological behavior from structures is complementary to the determination of rheological properties of rocks in the laboratory. What data there is suggest that constitutive relations for rocks undergoing ductile deformation in which many structures develop are highly nonlinear. There is general qualitative agreement between flow laws inferred on the basis of experimental results and those inferred from observation of structural characteristics.