Arvid M. Johnson

Development of faults as zones of deformation bands and as slip surfaces in sandstone

Three forms of fault are recognized in Entrada and Navajo Sandstones in the San Rafael Desert, southeastern Utah; deformation bands, zones of deformation bands, and slip surfaces. Small faults occur asdeformation bands, about one millimeter thick, in which pores collapse and sand grains fracture, and along which there are shear displacements on the order of a few millimeters or centimeters. Two or more deformation bands adjacent to each other, which share the same average strike and dip, form azone of deformation bands. A zone becomes thicker by addition of new bands, side by side. Displacement across a zone is the sum of displacements on each individual band. The thickest zones are about 0.5 m and total displacement across a thick zone rarely exceeds 30 cm. Finally,slip surfaces, which are through-going surfaces of discontinuity in displacement, form at either edge of zones of highly concentrated deformation bands. In contrast with individual deformation bands and zones of deformation bands, slip surfaces accommodate large displacements, on the order of several meters in the San Rafael Desert.The sequence of development is from individual deformation bands, to zones, to slip surfaces, and each type of faulting apparently is controlled by somewhat different processes. The formation of zones apparently involves strain hardening, whereas the formation of slip surfaces probably involves strain softening of crushed sandstone.

Mechanics of growth of some laccolithic intrusions in the Henry mountains, Utah, I: Field observations, Gilbert's model, physical properties and flow of the magma

The shapes of sills and laccolithic intrusions and associated host rock deformation were studied at several locations on the flanks of the Henry Mountains. Diorite sills range from 0.5 to 10 m in thickness, are less than 1 km2 in areal extent, and have blunt terminations. The laccolithic intrusions range from 10 to 200 m in thickness, and from 1 to 3 km2 in areal extent. The host rock, principally sandstone and shale, is deformed along closely spaced cataclastic shear planes. This deformation is concentrated at contacts, especially near sill terminations and over laccolith peripheries. The diorite contains plagioclase phenocrysts which are usually sheared in a thin zone adjacent to each contact. Field observations suggest that sills are the forerunners of laccolithic intrusions which form only after magma has spread far enough laterally (greater than about 1 km2 in the Henry Mountains) to gain leverage to bend the overburden upward. Further injection of magma results in laccolithic peripheries or terminations with one of three distinct cross-sectional forms: (1) blunt termination of the diorite accompanied by bending and minor faulting of the host rock; (2) termination at a peripheral diorite dike cutting upward across the host rock; or (3) abrupt termination of the diorite against a nearly vertical fault zone. In order to study some of the processes of sill and laccolith intrusion, mechanical models for the driving pressure, physical properties, and flow behavior of the diorite magma are derived and discussed. A static driving pressure (equal to the difference between total magma pressure and lithostatic pressure) of up to 700 bar is estimated. The rheological behavior of the magma in the Henry Mountains is unknown. However, flow behavior is calculated assuming three of the more common models for fluids: Newtonian viscous, pseudoplastic, and Bingham. Suspended crystals probably contributed to the finite strength of the magma (estimated to be at least 103 dyn/cm2 for the Henry Mountains magma) which enables it to support dense zenoliths and also fixes maximum limits on the lengths of sills or dikes. Pressure in magma flowing along tabular intrusions of uniform thickness drops linearly in the flow direction for all three rheological materials. Thickening of tabular intrusions tends to make the pressure drop less rapidly, but pressure drops more rapidly in the tapered region near a termination. Pressure distributions under these and other conditions are derived in order to use them in the models of host rock deformation presented in Part II.

Mechanics of growth of some laccolithic intrusions in the Henry mountains, Utah, II: Bending and failure of overburden layers and sill formation

Deformation of host rocks during growth of a laccolithic intrusion is analyzed using the theory of bending a stack of thin elastic plates. The theoretical model suggests that magma spreading laterally in the form of a sill will eventually gain sufficient leverage on the overlying strata to deflect them upward and form a laccolith. The amount of bending increases as the fourth power of the distance the magma spreads, whereas the overburden resists bending as the third power of its effective thickness. Effective thickness is the thickness of a single layer which has the same resistance to bending as a multilayer of similar length and elastic modulus. The effective thickness of overburden in the Henry Mountains is estimated as between 17 and 23 of the actual thickness. The form of bending is similar for Newtonian, pseudoplastic, and Bingham magmas. The magnitude of the bending depends upon the total upward force and its distribution and is not simply related to magma viscosity as has been suggested by several previous investigators. After elastic bending strata should fail over the periphery of an intrusion, the site of maximum bending strain and differential stress predicted by the theory. Field observations described in Part I correlate well with these predictions. Because bending strains are proportional to layer thickness, strata of comparable strength but different thicknesses fail at different stages of laccolith development. This leads to the different cross-sectional forms of laccoliths observed in the field. The effect of host rocks on sill form and growth is analyzed using the elastic solution for an elliptical hole under uniform pressure. The theory suggests that sill thickness increases in proportion to length. The concentration of high stresses near the sill termination should induce permanent deformation and account for the blunt terminations described in Part I. This blunting is most likely to occur in relatively ductile rocks whereas sills simply split brittle rocks and maintain sharp terminations. The driving pressure in sills can be calculated from measurements of length and termination radius of curvature, if the yield strength of the host rocks can be estimated. This driving pressure must be greater than the overburden pressure, but sills apparently do not form or propagate by lifting their overburdens. Instead they propagate by locally deforming the host rock. After spreading over a distance about three times the effective overburden thickness, the overlying layers begin to bend upward significantly. This stage marks the transition from a sill to a laccolithic intrusion.

Analysis of minor fractures associated with joints and faulted joints

Abstract In this paper, we use fracture mechanics to interpret conditions responsible for secondary cracks that adorn joints and faulted joints in the Entrada Sandstone in Arches National Park, U.S.A. Because the joints in most places accommodated shearing offsets of a few mm to perhaps 1 dm, and thus became faulted joints, some of the minor cracks are due to faulting. However, in a few places where the shearing was zero, one can examine minor cracks due solely to interaction of joint segments at the time they formed. We recognize several types of minor cracks associated with subsequent faulting of the joints. One is the kink, a crack that occurs at the termination of a straight joint and whose trend is abruptly different from that of the joint. Kinks are common and should be studied because they contain a great deal of information about conditions during fracturing. The sense of kinking indicates the sense of shear during faulting: a kink that turns clockwise with respect to the direction of the main joint is a result of right-lateral shear, and a kink that turns counterclockwise is a result of left-lateral shear. Furthermore, the kink angle is related to the ratio of the shear stress responsible for the kinking to the normal stress responsible for the opening of the joint. The amount of opening of a joint at the time it faulted or even at the time the joint itself formed can be estimated by measuring the kink angle and the amount of strike-slip at some point along the faulted joint. Other fractures that form near terminations of pre-existing joints in response to shearing along the joint are horsetail fractures. Similar short fractures can occur anywhere along the length of the joints. The primary value in recognizing these fractures is that they indicate the sense of faulting accommodated by the host fracture and the direction of maximum tension. Even where there has been insignificant regional shearing in the Garden Area, the joints can have ornate terminations. Perhaps the simplest is a veer, where the end of one joint segment turns gradually toward a nearby joint segment. The veer is a result of a nearby, shear-stress-free face such as a joint surface. Our greatest difficulty has been explaining long overlap of parallel joint segments, that is, the lack of veer. The only plausible explanation we know is suggested by the research of Cottrell and Rice, that high compression parallel to the joint segments will tend to prevent the joints from turning toward one another. The most interesting and puzzling fractures are stepped joints and associated echelon cracks, in which the slight misalignment of the stepped joints suggests mild left-lateral shear, while the strong misalignment of echelon cracks that continue the traces of the stepped joints suggests strong right-lateral shear. The stepped joints are thought to reflect local left-lateral shearing that acted over an area of several thousand square metres, whereas the stepped echelon cracks reflect local interaction between the tips of nearby joints propagating in different directions.

Mechanical models of trishear-like folds

Previous workers have formulated velocity descriptions of the trishear kinematic model of fault propagation-folds, which are inherently non-unique. We present two complete mechanical models of fault-related folding and assess the validity of the assumptions used in the assignment of velocities in the trishear description and to eliminate the untenable situation of an infinite number of possible solutions for velocity fields. The mechanical model of forced-folding, Forced Fold, based on viscous folding theory, is used to derive the velocity fields in an anisotropic sedimentary cover overlying faulted and displaced rigid basement blocks. The solution of displacements around a stress-free fault in an elastic body is used to model fault-arrest folds that form around a fault imbedded in a deformed medium. The results indicate that the velocity fields assumed in the trishear model more closely resemble the velocities derived in the mechanical Forced Fold model than the mechanical model of fault-arrest folding. The Forced Fold model produces a triangular region of concentrated deformation similar to the trishear region assumed in the kinematic models, while the deformation produced by the fault-arrest model is not concentrated within a triangular zone.

A theory of concentric, kink, and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers

Abstract Most folds in stratified rock are similar in form to ideal kink, concentric or chevron folds, in which there are discontinuities in slope or curvature of bedding planes. In this respect most folds appear to be closely related to faults, traces of which can be considered to be lines across which there are discontinuities of displacement of layers. Further, the close association of reverse faults and folds or monoclinal flexures seems to indicate that theories of faulting and folding should be closely related. The theory of characteristics is a mathematical tool with which we can obtain insights into processes involving discontinuities. Theoretical characteristic lines are directions across which certain variables might be discontinuous and they are directions along which discontinuities propagate. The theory has been widely applied in plasticity theory and in fluid mechanics and theoretical studies of faulting have suggested that faults are analogous to the lines of discontinuity predicted by plasticity theory. Elasticity and viscosity theories, on which theories of folding have been founded, exclude the existence of characteristic lines in the materials unless the equilibrium equations, rheological properties or strains are nonlinear. However, all folding theories are nonlinear to some extent and the theories can be modified so that they predict lines of discontinuity for some conditions of loading and deformation. Theories of folding will be developed in subsequent papers of this series in order to predict conditions under which characteristic lines can exist in multilayered materials and in order to determine the conditions that must be satisfied across and along the characteristic lines. The theory should help us to recognize lines of apparent discontinuity in natural and experimental folds and study of these lines should provide further understanding of mechanisms of folding. Experimental studies of folding of a wide variety of materials, including alternating layers of rubber and gelatin, modeling clay and grease or graphite, and potters clay and rubber or cardboard, suggest that the patterns of folding in these materials begin with sinusoidal forms, transform into concentric or kink forms and then into chevron forms as the multilayers are shortened axially. A suitable theory of folding of multilayers should account for these observations.

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

Abstract 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 ± (45°− φ 2 ) 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 ± (90° − φ 2 ) . 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 ± (45° + φ 4 ) to the direction of maximum compression, or ± (45° + φ 4 − α 2 ) 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.

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.

Duplex structures connecting fault segments in Entrada Sandstone

Abstract All stages in the development of a duplex structure—from isolated, stepped fault segments, to segments joined by a single ramp, to segments joined by tens of ramps—are preserved along strike-slip and normal faults in Entrada Sandstone in Arches National Park, Utah. Bedding is either absent or at a high angle to the duplex-like structures in Entrada Sandstone, thus it had no significant role in constraining their geometry. We can reproduce the essential features of a duplex structure along a normal fault with mechanical and kinematic models previously used to simulate duplex structures along thrust faults. However the models do not account for the amount of observed thickening at the step where the structure forms. This suggests that the geometry of duplex-like structures along these strike-slip faults may be a result of interaction between the fault segments.

Rates of seasonal creep of silty clay soil

Summary Profiles of displacement caused by seasonal creep of an expansive silty clay soil were measured at several times of the year with a tiltmeter and flexible pipes installed in two areas near Stanford University in central California. The displacement profiles typically are convex upward and net creep displacement at the ground surface of an undisturbed slope with a gradient of 12 to 14 per cent is about 0.4 in/yr (1 cm/yr) for a layer of soil of five feet (1.5 m) thickness The soil apparently expands and creeps on wetting of the soil and while the soil is at a high moisture content, but the creep rate diminishes with time, as predicted by laboratory tests by Mitchell and Singh. On drying, a point on the surface of a slope contracts roughly along a normal to the slope but directions of horizontal displacements during drying appear to be random and to depend on the location of the point with respect to local shrinkage cracks.