A theory of concentric, kink and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers: VI. Asymmetric folding and monoclinal kinking

One of the rules of thumb of structural geology is that drag folds, or minor asymmetric folds, reflect the sense of layer-parallel shear during folding of an area. According to this rule, right-lateral, layer-parallel shear is accompanied by clockwise rotation of marker surfaces and left-lateral by counterclockwise rotation. By using this rule of thumb, one is supposed to be able to examine small asymmetric folds in an outcrop and to infer the direction of axes of major folds relative to the position of the outcrop. Such inferences, however, can be misleading. Theoretical and experimental analyses of elastic multilayers show that symmetric sinusoidal folds first develop in the multilayers, if the rheological and dimensional properties favor the development of sinusoidal folds rather than kink folds, and that the folded layers will then behave much as passive markers during layerparallel shear and thus will follow the rule of thumb of drag folding. The analyses indicate, however, that multilayers whose properties favor the development of kink folds can produce monoclinal kink folds with a sense of asymmetry opposite to that predicted by the rule of thumb. Therefore, the asymmetry of folds can be an ambiguous indicator of the sense of shear.The reason for the ambiguity is that asymmetry is a result of two processes that can produce diametrically opposed results. The deformation of foliation surfaces and axial planes in a passive manner is the pure or end-member form of one process. The result of the passive deformation of fold forms is the drag fold in which the steepness of limbs and the tilt of axial planes relative to nonfolded layering are in accord with the rule of thumb.The end-member form of a second process, however, produces the opposite geometric relationships. This process involves yielding and buckling instabilities of layers with contact strength and can result in monoclinal kink bands. Right-lateral, layer-parallel shear stress produces left-lateral monoclinal kink bands and left-lateral shear stress produces right-lateral monoclinal kink bands. Actual folds do not behave as either of these ideal end members, and it is for this reason that the interpretation of the sense of layer-parallel shear stress relative to the asymmetry of folds can be ambiguous.Kink folding of a multilayer with contact strength theoretically is a result of both buckling and yielding instabilities. The theory indicates that inclination of the direction of maximum compression to layering favors either left-lateral or right-lateral kinking, and that one can predict conditions under which monoclinal kink bands will develop in elastic or elastic—plastic layers. Further, the first criterion of kink and sinusoidal folding developed in Part IV remains valid if we replace the contact shear strength with the difference between the shear strength and the initial layer-parallel shear stress.Kink folds theoretically can initiate only in layers inclined at angles less than to the direction of maximum compression. Here φ is the angle of internal friction of contacts. For higher angles of layering, slippage is stable so that the result is layer-parallel slippage rather than kink folding.The theory also provides estimates of locking angles of kink bands relative to the direction of maximum compression. The maximum locking angle between layering in a nondilating kink band and the direction of maximum compression is . The theory indicates that the inclination of the boundaries of kink bands is determined by many factors, including the contact strength between layers, the ratio of principal stresses, the thickening or thinning of layers, that is, the dilitation, within the kink band, and the orientation of the principal stresses relative to layering. If there is no dilitation within the kink band, the minimum inclination of the boundaries of the band is to the direction of maximum compression, or to the direction of nonfolded layers. Here α is the angle between the direction of maximum compression and the nonfolded layers. It is positive if clockwise.Analysis of processes in terminal regions of propagating kink bands in multilayers with frictional contact strength indicates that an essential process is dilitation, which decreases the normal stress, thereby allowing slippage and buckling even though slopes of layers are low there.     

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.     

A theory of concentric, kink and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers: IV. Development of sinusoidal and kink folds in multilayers confined by rigid boundaries

This part concerns folding of elastic multilayers subjected to principal initial stresses parallel or normal to layering and to confinement by stiff or rigid boundaries. Both sinusoidal and reverse-kink folds can be produced in multilayers subjected to these conditions, depending primarily upon the conditions of contacts between layers. The initial fold pattern is always sinusoidal under these ideal conditions, but subsequent growth of the initial folds can change the pattern. For example, if contacts between layers cannot resist shear stress or if soft elastic interbeds provide uniform resistance to shear between stiff layers, sinusoidal folds of the Biot wavelength grow most rapidly with increased shortening. Further, the Biot waves become unstable as the folds grow and are transformed into concentric-like folds and finally into chevron folds. Comparison of results of the elementary and the linearized theories of elastic folding indicates that the elementary theory can accurately predict the Biot wavelength if the multilayers contain at least ten layers and if either the soft interbeds are at most about one-fifth as stiff as the stiff layers, or there is zero contact shear strength between layers.Multilayers subjected to the same conditions of loading and confinement as discussed above, can develop kink folds also. The kink fold can be explained in terms of a theory based on three assumptions: each stiff layer folds into the same form; kinking is a buckling phenomenon, and shear stress is required to overcome contact shear strength between layers and to produce slippage locally. The theory indicates that kink forms will tend to develop in multilayers with low but finite contact shear strength relative to the average shear modulus of the multilayer. Also, the larger the initial slopes and number of layers with contact shear strength, the more is the tendency for kink folds rather than sinusoidal folds to develop. The theoretical displacement form of a layer in a kink band is the superposition of a full sine wave, with a wavelength equal to the width of the kink band, and of a linear displacement profile. The resultant form resembles a one-half sine curve but it is significantly different from this curve. The width of the kink band may be greater or less than the Biot wavelength of sinusoidal folding in the multilayer, depending upon the magnitude of the contact shear strength relative to the average shear modulus. For example, in multilayers of homogeneous layers with contact strength, the Biot wavelength is zero so that the width of the kink band in such materials is always greater than the Biot wavelength. In general, the higher the contact strength, the narrower the kink band; for simple frictional contacts, the widths of kink bands decrease with increasing confinement normal to layers. Widths of kink bands theoretically depend upon a host of parameters — initial amplitude of Biot waves, number of layers, shear strength of contacts between layers, and thickness and modulus ratios of stiff-to-soft layers — therefore, widths of kink bands probably cannot be used readily to estimate properties of rocks containing kink bands. All these theoretical predictions are consistent with observations of natural and experimental kink folds of the reverse variety.Chevron folding and kink folding can be distinctly different phenomena according to the theory. Chevron folds typically form at cores of concentric-like folds; they rarely form at intersections of kink bands. In either case, they are similar folds that develop at a late stage in the folding process. Kink folds are more nearly akin to concentric-like folds than to chevron folds because kink folds form early, commonly before the sinusoidal folds are visible. Whereas concentric-like folds develop in response to higher-order effects near boundaries of a multilayer, kink folds typically initiate in response to higher-order shear, as at inflection points near mid-depth in low-amplitude, sinusoidal fold patterns. Chevron folding and kink folding are similar in elastic multilayers in that elastic “yielding” at hinges can produce rather sharp, angular forms.     

Impact of mechanical anisotropy and power-law rheology on single layer folding

The infinitesimal and finite stages of folding in nonlinear viscous material with a layer-parallel anisotropy were investigated using numerical and analytical methods. Anisotropy was found to have a first-order effect on growth rate and wavelength selection, and these effects are already important for anisotropy values (normal viscosity/shear viscosity) < 10. The effect of anisotropy must therefore be considered when deducing viscosity contrasts from wavelength to thickness ratios of natural folds. Growth rates of single layer folds were found to increase and subsequently decrease during progressive deformation. This is due to interference between the single layer folds and chevron folds that form in the matrix as a result of instability caused by the anisotropic material behaviour. The wavelength of the chevron folds in the matrix is determined by the wavelength of the folded single layer, which can explain the high wavelength to thickness ratios that are sometimes found in multilayer sequences. Numerical models including anisotropic material properties allow the behaviour of multilayer sequences to be investigated without the need for resolution on the scale of individual layers. This is particularly important for large-scale models of layered lithosphere.     

A range of fault slip styles on progressively misoriented planes during flexural-slip folding,Cape Fold Belt,South Africa

Flexural slip folds are distinctive of mixed continuous-discontinuous deformation in the upper crust, as folding is accommodated by continuous bending of layers and localized, discontinuous slip along layer interfaces. The mechanism of localized, layer-parallel slip and the stress and fluid pressure conditions at which flexural slip occurs are therefore distinctive of shear localization during distributed deformation. In the Prince Albert Formation mudstone sequence of the Karoo Basin, the foreland basin to the Cape Fold Belt, folds are well developed and associated with incrementally developed bedding-parallel quartz veins with slickenfibers oriented perpendicular to fold hinge lines, locally cross-cutting axial planar cleavage, and showing hanging wall motion toward the fold hinge. Bedding-parallel slickenfiber-coated veins dip at angles from 18° to 83°, implying that late increments of bedding-parallel shear occurred along unfavorably oriented planes. The local presence of tensile veins, in mutually cross-cutting relationship with bedding-parallel, slickenfiber-coated veins, indicate local fluid pressures in excess of the least compressive stress.Slickenfiber vein microstructures include a range of quartz morphologies, dominantly blocky to elongate-blocky, but in places euhedral to subhedral; the veins are commonly laminated, with layers of quartz separated by bedding-parallel slip surfaces characterized by a quartz-phyllosilicate cataclasite. Crack-seal bands imply incremental slickenfiber growth, in increments from tens of micrometers to a few millimeters, in some places, whereas other vein layers lack evidence for incremental growth and likely formed in single slip events. Single slip events, however, also involved quartz growth into open space, and are inferred to have formed by stick-slip faulting. Overall, therefore, flexural slip in this location involved bedding-parallel faulting, along progressively misoriented weak planes, with a range of slip increments.     

A theory of concentric, kink and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers: V. asymmetric folding in interbedded chert and shale of the franciscan complex, san francisco bay area, california

The origin of tight, asymmetric, kink-like or chevron-like folds in interbedded shales and radiolarian cherts of the Franciscan Complex in the San Francisco Bay area has been somewhat of a mystery for many years. Stephenson Ellen provided many clues as to the origin and indicated that the folds became asymmetric as a result of layer-parallel shear. He believed that the original folds were conjugate kink folds.As a result of reexamination of most of the folds studied by Ellen, of experimentation with elastic multilayers and of the theories developed in Parts III and IV of this series of papers, we believe that the original folds were mostly chevron rather than kink folds. Thus, we suggest that the folds formed by a combination of layer-parallel shortening and layer-parallel shear when the rocks were soft and pore pressures were high.Several lines of evidence suggest that typical folds in the Franciscan are asymmetric chevron folds. A combination of theory of finite simple shear and of experimentation with elastic multilayers indicates that the tight folds of the Franciscan could have been produced by smaller angles of simple shear if the original folds were typical chevron folds rather than typical kink folds. Several field observations, including thickening of shales but not of cherts in hinges of folds and lack of deformation of radiolaria in the cherts, indicate that the cherts were soft and the shales very soft at the time of folding. The pore-water pressures in the shales probably were high. Such conditions theoretically favor concentric-like and chevron folding, not kink folding. Finally, most of the asymmetric folds in a quarry exposure can be reconstructed geometrically as typical chevron folds but not as typical kink folds subjected to simple shear.     

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.     

Folding and faulting of strain-hardening sedimentary rocks

The question of whether single- or multi-layers of sedimentary rocks will fault or fold when subjected to layer-parallel shortening is investigated by means of the theory of elastic-plastic, strain-hardening materials, which should closely describe the properties of sedimentary rocks at high levels in the Earth's crust. The most attractive feature of the theory is that folding and faulting, intimately related in nature, are different responses of the same idealized material to different conditions.When single-layers of sedimentary rock behave much as strain-hardening materials they are unlikely to fold, rather they tend to fault, because contrasts in elasticity and strength properties of sedimentary rocks are low. Amplifications of folds in such materials are negligible whether contacts between layer and media are bonded or free to slip for single layers of dolomite, limestone, sandstone, or siltstone in media of shale. Multilayers of these same rocks fault rather than fold if contacts are bonded, but they fold readily if contacts between layers are frictionless, or have low yield strengths, for example due to high pore-water pressure. Faults may accompany the folds, occurring where compression is increased in cores of folds. Where there is predominant reverse faulting in sedimentary sequences, there probably were few structural units.     

A finite-element study of fold propagation in a viscous layer

A finite-element model of a viscous layer contained in a viscous matrix and undergoing layer-parallel compression is used to examine the hypothesis that a long chain of folds, as found in real rocks, can originate from one initial perturbation to the layer geometry. This hypothesis is tested by determining the velocity with which a perturbation spreads along layers of various viscosities.An insight is gained into the roles played by local strain and local layer strength in the folding mechanism. The results show that for layers with viscosity ratios comparable with those of real rocks it is impossible for long chains of folds to originate from one perturbation. The authors conclude that rock layers contain many initial perturbations and folding originates at all perturbation sites simultaneously. The growth of such folds depends on the amplitude and shape of the initial perturbation and on subsequent interference between folds.     

Rheological Properties of Fractal Deformation in Multilayer Folds

Abstract: The fractal dimensions of folds are related to layer thickness and viscosity of the multilayer. This paper discusses how the thickness, viscosity, and anisotropic degree affect the rheological deformation of fractal folds in multilayers. The number of layers, their thicknesses, viscosities, and anisotropic degree of multilayers cooperate to affect the rheological deformation of folds, which is not controlled by a single rheological factor. A greater anisotropic degree of multilayers is favorable to develop the more complex and disharmonious fractal folds.     

A numerical investigation and geological discussion of the relationship between folding,kinking and faulting

The layer-parallel compression of a regular bilaminate consisting of layers with materials described by an incompressible power-law elastic model is considered. The average mechanical properties of this idealised multilayer are then represented by those of an equivalent anisotropic continuum with internal resistance to bending. Changes in material properties that accompany uniform finite shortening are accounted for. Interpretation of the internal instability analysis for such a continuum, introduced in the companion paper involves the use of a spectrum which at a given level of strain, scans all directions within the continuum for relative susceptibility to a heterogeneous simple shearing instability.Estimates of nonlinear material properties from reported experiments on the behaviour of various rocks in the time-independent deformation regime, and geometric parameters such as the volume fraction of each material and the number of confined layers are considered. The shapes of the resulting spectra may be used to predict natural conditions that will favour the initiation of repetitive buckle folds or more localized disturbances such as kink-bands and faults. Results suggest that for typical properties of sedimentary multilayers, kinking is strongly favoured over repetitive buckling where the weaker material occupies only a very small volume fraction of the multilayer. The effect of significant imperfections leading to slippage between layers is discussed.Finally, a simple classification of structure genesis is proposed in which the mechanical relationships between apparently diverse structures is illustrated.     

The determination of ‘buckling rotation’ and its application to theoretical and experimental models of buckle folds

The quantity ‘buckling rotation’ is defined, for buckle folds, as the total rotation of a fold limb minus the rotation that would occur due to pure shear if no competence contrast existed. Using existing models (theoretical and experimental) of buckle-fold development, the quantity ‘buckling rotation’ has been calculated for successive small increments of strain and plotted against strain or limb dip. The resulting curves are skewed and bell-shaped, indicating an initial sharp increase in buckling rotation early in fold development followed by a gentle, asymptotic decrease. The curve height and position are dependent on the competence contrast and, in multilayer systems, on the ratio of competent to incompetent layer thickness. The initial sharp increase in buckling rotation corresponds to the period of most active layer-parallel shortening during fold development.     

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.     

Monoclinal bending of strata over laccolithic intrusions

Sedimentary strata on top of some laccolithic intrusions are nearly horizontal and little deformed, but are bent into steeply dipping monoclinal flexures over the peripheries of these intrusions. This form of bending is not explained by previous theories of laccolithic intrusion, which predict either horizontal undeformed strata over the center and faulted strata around the periphery, or strata bent continuously into a dome. However, a slight generalization of these theories accomodates the observed form and contains the previous forms as special cases. A critical assumption is that the strength of contacts within a multilayered overburden is overcome locally by layer-parallel shear. If this strength is less than the strength of the layers themselves, then layers over the center remain bonded together and display negligible bending, whereas layers over the periphery slip over one another and are readily bent into a monoclinal flexure.     

Fracture termination and step-over at bedding interfaces due to frictional slip and interface opening

Three types of fracture intersection with bedding contacts have been investigated within numerical experiments: fracture transection through bed contacts, termination (abutment) at contacts and step-over of fractures at bedding contacts. To evaluate the mechanisms responsible for different fracture intersections with bed contacts, the numerical experiments explored deformation associated with end-member conditions of sliding-only interfaces and opening-only interfaces. A third suite of models explored the combined influence of both sliding and opening, as a fracture approached the interface. In contrast to our initial supposition that interface sliding promotes fracture termination, the sliding-only interfaces encouraged propagation of fractures straight through the modeled interface. In contrast, the opening-only interfaces yielded either fracture termination or initiation of a new fracture near the ends of the open interface segment (several centimeters from parent fracture in these models). These results suggest that local interface opening near the tip of approaching fractures, rather than sliding, is responsible for fracture termination and step-over at bedding contacts. Combined sliding and opening yielded fracture termination in models with weak interfaces (μ=0; c=0 MPa; T=0 MPa) and either fracture step-over or termination at moderate-strength interfaces (μ=0.65; c=3.25 MPa; T=5 MPa). Fracture termination occurs at moderate-strength interfaces when the stresses along the interface are not great enough to initiate a new step-over fracture. Fracture termination is more likely under conditions of shallower burial depth, lower layer-parallel effective tension and fluid-driven fracture propagation rather than remote layer-parallel tension. Furthermore, thicker beds and greater layer-parallel effective tension may produce greater distances of fracture step-over than thinner beds and more compressive layers. These results may assist in the prediction of subsurface fracture networks and associated fluid flow paths.     

A theory of concentric, kink and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers: VII. development of folds within Huasna Syncline, San Luis Obispo County, California

Folds in the Huasna area of the southern Coast Ranges of California provide an opportunity to study different fold forms and to estimate dimensional and relative rheological properties of rocks at the time of folding. Plunging, concentric-like and chevron-like folds with wavelengths ranging from about 0.1 to 1 km are clearly visible in natural exposures at the south end of the Huasna syncline, which has a wavelength of 12–16 km. Examination of two fresh roadcut exposures in the Miocene Monterey Formation suggests that folding within part of the Monterey was accommodated primarily by layer-parallel slip between structural layers with thicknesses ranging from 30 to 43 m, even though lithologic layers range from a few mm to a few dm in thickness. This part of the Monterey is folded into a series of concentric-like folds, with chevron-like folds at their cores and with a ratio of wavelength to total thickness of layers of about . Theoretical analysis of multilayers, comprised of identical, elastic or elastic—plastic layers with frictionless contacts, indicates that the effective, or weighted-average thickness of structural layers corresponding with an ratio of 0.42 is about 41 m. Thus, the theoretical predictions are roughly in agreement with available data concerning these folds.Thicknesses of structural units in other folds of this area are inadequately known to closely check theoretical predictions, but most of the data are consistent with predictions. An exception is the Huasna syncline which has a larger wavelength than we would predict. There are several likely explanations for this discrepancy. Layers in the underlying Franciscan complex may have taken part in the folding, making our estimates of total thickness too small. The basement rocks may have been much softer, relative to the overlying sedimentary rocks, than we assumed. The Huasna syncline could be partly a result of gravitational instability of relatively low density, Miocene siliceous and porcelaneous shales, overlain by relatively high density, Pliocene sandstones.The Huasna syncline and some of the smaller folds in the Miocene rocks are doubly in the northwest—southeast direction. Further, the maximum compression was approximately normal to the traces of the large faults in this part of California.     

Early formed regional antiforms and synforms that fold younger matrix schistosities: their effect on sites of mineral growth

In the metamorphic cores of many orogenic belts, large macroscopic folds in compositional layering also appear to fold one or more pervasive matrix foliations. The latter geometry suggests the folds formed relatively late in the tectonic history, after foliation development. However, microstructural analysis of four examples of such folds suggests this is not the case. The folds formed relatively early in the orogenic history and are the end product of multiple, near orthogonal, overprinting bulk shortening events. Once large macroscopic folds initiate, they may tighten further during successive periods of sub-parallel shortening, folding or reactivation of foliations that develop during intervening periods of near orthogonal shortening. Reactivation of the compositional layering defining the fold limbs causes foliation to be rotated into parallelism with the limbs.Multiple periods of porphyroblast growth accompanied the multiple phases of deformation that postdated the initial development of these folds. Some of these phases of deformation were attended by the development of large numbers of same asymmetry spiral-shaped inclusion trails in porphyroblasts on one limb of the fold and not the other, or larger numbers of opposite asymmetry spirals on the other limb, or similar numbers of the same asymmetry spirals on both limbs. Significantly, the largest disparity in numbers from limb to limb occurred for the first of these cases. For all four regional folds examined, the structural relationships that accompanied these large disparities were identical. In each case the shear sense operating on steeply dipping foliations was opposite to that required to originally develop the fold. Reactivation of the folded compositional layering was not possible for this shear sense. This favoured the development of sites of approximately coaxial shortening early during the deformation history, enhancing microfracture and promoting the growth of porphyroblasts on this limb in comparision to the other. These distributions of inclusion trail geometries from limb to limb cannot be explained by porphyroblast rotation, or folding of pre-existing rotated porphyroblasts within a shear zone, but can be explained by development of the inclusion trails synchronous with successive sub-vertical and sub-horizontal foliations.     

Centrifuge modelling of deformation of a multi-layered sequence over a ductile substrate: 1. Style and 4D geometry of active cover folds during layer-parallel shortening

Centrifuge analogue modelling illustrates the progressive development of active folds in multilayers upon a ductile substrate during layer-parallel shortening. Models simulate folding of a mechanically stratified sedimentary sequence upon migmatitic gneisses in a large hot orogen, or upon a thick basal evaporite ± shale sequence in deeper levels of fold belts. The absence of a weak low-viscosity and low-density layer at the interface promotes infolding of the cover sequence and ductile substrate, whereas a planar upper surface to the basal ductile substrate is preserved when it is present. Whilst fold style, wavelength, and deformation of the interface with the ductile substrate differ depending on whether a low-viscosity and low-density layer is present at the base of the cover sequence, there is no marked systematic curvature of fold axes as seen in previous sandbox models for fault-bend or fault propagation folding during bulk shortening. Bulk shortening of a layered sequence with relatively thick individual layers above a ductile substrate promotes a regular and upright train of buckle folds, whereas thinner layers promote a more irregular distribution of buckle folds with variable vergence, style, and amplitude. Buckle folds above a ductile substrate progressively develop during bulk shortening from open and upright, to angular and tight, and may further develop into cuspate structures above relatively weak horizons. Relatively thick weak horizons within the layered sequence during bulk shortening interrupt regular fold patterns up structural section and allow out-of-phase folds to develop above and below the weak horizon.     

Development of conjugate shear bands during bulk simple shearing

In rocks possessing a strong planar fabric, shear bands of constant shear sense and oriented at an oblique angle to the foliation are considered by many authors to be characteristic of a non-coaxial bulk deformation history, whereas conjugate shear bands are considered to indicate coaxial shortening. However, in two areas where bulk deformation history appears to be non-coaxial (Cap Corse, Corsica and Ile de Groix, Brittany), conjugate shear bands are observed. In order to investigate this problem, experiments were performed by bulk simple shearing using Plasticine as a rock analogue. When slip between layers of the model is permitted, shear bands of normal-fault geometry form with both the same and opposite shear sense as the bulk simple shearing at approximately the same angle with the layering (40°) irrespective of layer orientation in the undeformed state (for initial orientations of 50, 30 and 15°). Shear bands are initially formed within individual layers and may propagate across layer interfaces when further movement along these is inhibited. The existence of conjugate shear bands in Corsica and Ile de Groix is therefore not incompatible with a model of bulk simple shearing for these two regions. In field studies, one should perhaps exercise care in using shear bands to determine the kind of motion or the sense of bulk shearing.     

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.