DOCSLIB.ORG

Principal Stresses

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Principal Stresses Chapter 4 Principal Stresses Anisotropic stress causes tectonic deformations. Tectonic stress is defined in an early part of this chapter, and natural examples of tectonic stresses are also presented. 4.1 Principal stresses and principal stress axes Because of symmetry (Eq. (3.32)), the stress tensor r has real eigenvalues and mutually perpendicu- lar eigenvectors. The eigenvalues are called the principal stresses of the stress. The principal stresses are numbered conventionally in descending order of magnitude, σ1 ≥ σ2 ≥ σ3, so that they are the maximum, intermediate and minimum principal stresses. Note that these principal stresses indicate the magnitudes of compressional stress. On the other hand, the three quantities S1 ≥ S2 ≥ S3 are the principal stresses of S, so that the quantities indicate the magnitudes of tensile stress. The orientations defined by the eigenvectors are called the principal axes of stress or simply stress axes, and the orientation corresponding to the principal stress, e.g., σ1 is called the σ1-axis (Fig. 4.1). Principal planes of stress are the planes parallel to two of the stress axes, or perpendicular to one of the stress axes. Deformation is driven by the anisotropic state of stress with a large difference of the principal stresses. Accordingly, the differential stress ΔS = S1 − S3, Δσ = σ1 − σ3 is defined to indicate the difference1. Taking the axes of the rectangular Cartesian coordinates O-123 parallel to the stress axes, a stress tensor is expressed by a diagonal matrix. Therefore, Cauchy’s stress (Eq. (3.10)) formula is ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ t1(n) σ1 00 n1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ t(n) = t2(n) = 0 σ2 0 n2 . t3(n) 00σ3 n3 1ΔS is sometimes called the stress difference [2]. However, we keep this term for another quantity (Eq. (11.19)). 101 102 CHAPTER 4. PRINCIPAL STRESSES Figure 4.1: Stress ellipsoid and principal stress axes. White lines indicate the principal planes of stress. ∴ t1(n) = σ1n1,t2(n) = σ2n2,t3(n) = σ3n3. (4.1) The planes perpendicular to the stress axes have unit normals, n = (100)T,(010)T,(001)T. Hence, the traction vectors upon the planes are ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ σ1 0 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ t(n) = 0 , σ2 , 0 . 0 0 σ3 Therefore, the principal stresses equal the normal stresses working on the principal stress planes. Shear stress vanishes on the planes. The normal and shear stresses are written in this coordinates as 2 2 2 σN = t(n) · n = σ1n + σ2n + σ3n (4.2) 1 2 3 1/2 = | |2 −| |2 1/2 = 2 2 + 2 2 + 2 − 2 + 2 + 2 2 σS t(n) σN σ1n1 σ2n2 σ3n3 σ1n1 σ2n2 σ3n3 . (4.3) If there is a plane on which traction vanishes, the state of stress is said to be plane stress (§3.4). The stress tensor corresponding to this state has a null principal stress. If σ3 = 0, traction on the 2 + 2 + 2 = σ1σ2-plane vanishes. Combining Eq. (4.1) and n1 n2 n3 1, we have t (n) 2 t (n) 2 t (n) 2 1 + 2 + 3 = 1. (4.4) σ1 σ2 σ3 This is an ellipsoid equation whose principal axes coincide with the stress axes and radii equals |σ1|, |σ2| and |σ3| (Fig. 4.1). This is called Lame’s´ stress ellipsoid. Regarding t(n) as a position vector with its initial point at the coordinate origin, the end point of the vector is on the ellipsoid, provided that 0 <σ3. The ellipsoid becomes a sphere for the hydrostatic state of stress and its radius equals the hydrostatic stress. The stress ellipsoid is given uniquely from the stress tensor regardless of the orientation of coordinate axes, although Eq. (4.1) was derived for a specific orientation of the axes. The basic invariants (§C.6) of the stress tensor r are σI, σII and σIII. Those of the tensor S are SI, SII and SIII. These quantities designate the “absolute values” of the stress tensor, independent from the orientation of rectangular Cartesian coordinates. We can define variables with the same 4.2. TECTONIC STRESS 103 independence from the basic invariants. Among them is the stress ratio σ − σ Φ= 2 3 . (4.5) σ1 − σ3 It is seen that the ratio is in the range of 0 ≤ Φ ≤ 1. When the state of stress is hydrostatic, the stress ratio is not defined. For anisotropic stress states, the stress ratio designates the shape of the stress elliosoid. If Φ=0, σ3 = σ2 <σ1, then, the ellipsoid is prolate. Φ=1 corresponds to an oblate ellipsoid, σ3 <σ2 = σ1. The stress ellipsoid has axial symmetry for the extreme cases Φ=0 and 1. In contrast, intermediate Φ indicates triaxial stresses σ3 = σ2 = σ1. In this case, the stress ellipsoid and stress tensor have orthorhombic symmetry. 4.2 Tectonic stress The lithostatic state of stress causes isotropic contraction for a homogeneous material. Deviation from this state results in strain. Tectonic stress is generally defined as this deviation. The lithostatic state of stress is used as the reference. Other states of stress can also be taken as the reference (§7.2). The average σ + σ + σ trace r σ S + S + S trace S S σ ≡ 1 2 3 = = I or S ≡ 1 2 3 = = I (4.6) 0 3 3 3 0 3 3 3 is said to be mean stress. For lithostatic stresses, mean stress coincides with pressure. Accordingly, it is convenient to define deviatoric stress s = r − σ0I (4.7) as the deviation from the lithostatic state of stress. This is the deviatoric tensor introduced in Section C.6. Let us define the symbol T as T = S − S0I. (4.8) It is obvious that a deviatoric tensor is symmetric, so that the tensor has its principal orientations parallel to the stress axes (§C.5). If T1, T2 and T3 are the principal deviatoric stresses, they are related to the principal stresses as Ti = Si − S0. (4.9) Combining Eqs. (4.6), we have 2S − S − S 2S − S − S 2S − S − S T = 1 2 3 ,T= 2 3 1 ,T= 3 1 2 . (4.10) 1 3 2 3 3 3 Using the stress ratio, these are rewritten as 2 − Φ 2Φ − 1 Φ+1 T = ΔS, T = ΔS, T = − ΔS. (4.11) 1 3 2 3 3 3 104 CHAPTER 4. PRINCIPAL STRESSES Figure 4.2: Diagram for the explanation of the Hubbert-Rubey model. Hubbert-Rubey model In the previous chapter, we considered only the lithostatic state of stress. Our first application of the deviatoric stress tensor is the statics of a thrust sheet. There are thrust sheets that travel horizontally for more than 100 km. The Hubbert-Rubey model explains why such long-distance displacement is possible or what it implies. The very low friction of the fault surface is the key [86, 87]. Consider a thrust sheet that has a thickness of t and is advancing in the x direction (Fig. 4.2). For simplicity, the topographic surface coincides with the top of this sheet. We pay attention to a part of the sheet with length L in the x direction and unit length in the y direction. The tectonic forces pushing at the rear and front of this portion are τxxt in the +x direction and −(1 +Δ)τxxt in the −x direction. Therefore, the net force to push the portion of the sheet in the +x direction is −Δτxxt. Frictional resistance is the one significant force for the sheet. The resistance per unit basal area is σzx, and the law of friction gives σzx = μfσzz = μfρgt, where μf is the coefficient of friction and ρ is the mean density of the sheet. Therefore, a portion feels the resistance, −μfρgLt, in the +x direction. The motion of the sheet is so slow that inertia is negligible. Hence, all forces acting on the portion is balanced: −Δτxxt = −μfρgLt. Therefore, we obtain Δτxx = ρμfgL. In order to evaluate the coefficient of friction, we substitute Δτxx = 100 MPa as the representative magnitude of tectonic stresses and ρ = 2.75 × 103 kg m−3 in this equation. We consider a thrust sheet that travels hundreds of kilometers so that we assume L = 100 km. The result is that μf = 0.036. The friction at the base of those giant thrust sheets is very low compared to the coefficient of the friction of ordinary rocks. Pore fluid lowers the friction (§6.5). Octahedral shear stress The deviatoric stress tensor has the characteristic equation shown in Eq. (C.48), and its first basic invariant is TI = trace T = T1 + T2 + T3 = 0. (4.12) 4.2. TECTONIC STRESS 105 Figure 4.3: Basic invariants of deviatoric stress tensor versus stress ratio. The second basic invariant satisfies 1 1 T = T : T = T T II 2 2 ij ij i,j = 1 2 + 2 + 2 + 2 + 2 + 2 T11 T22 T33 T12 T23 T31 (4.13) 2 = 1 − 2 + − 2 + − 2 + 2 + 2 + 2 (S11 S22) (S22 S33) (S33 S11) S12 S23 S31 6 1 = (S − S )2 + (S − S )2 + (S − S )2 ≥ 0. 6 1 2 2 3 3 1 It is obvious in the last line that τII ≥ 0. TII is related to the principal deviatoric stresses as 1 1 T = T 2 + T 2 + T 2 = (T + T + T )2 − 2T T − 2T T − 2T T II 2 1 2 3 2 1 2 3 1 2 2 3 3 1 = −T1T2 − T2T3 − T3T1. (4.14) The transformation at the second equal sign is based on Eq. (4.12). Substituting Eq. (4.11), we obtain Φ2 − Φ+1 T = (ΔS)2. (4.15) II 3 1 Δ 2 Φ= 1 1 Δ 2 Φ= Therefore, TII ranges from 4 ( S) at 2 to 3 ( S) at 0 and 1.
Load more

Recommended publications
  • Strike and Dip Refer to the Orientation Or Attitude of a Geologic Feature. The
    Name__________________________________ 89.325 – Geology for Engineers Faults, Folds, Outcrop Patterns and Geologic Maps I. Properties of Earth Materials When rocks are subjected to differential stress the resulting build-up in strain can cause deformation. Depending on the material properties the result can either be elastic deformation which can ultimately lead to the breaking of the rock material (faults) or ductile deformation which can lead to the development of folds. In this exercise we will look at the various types of deformation and how geologists use geologic maps to understand this deformation. II. Strike and Dip Strike and dip refer to the orientation or attitude of a geologic feature. The strike line of a bed, fault, or other planar feature, is a line representing the intersection of that feature with a horizontal plane. On a geologic map, this is represented with a short straight line segment oriented parallel to the strike line. Strike (or strike angle) can be given as either a quadrant compass bearing of the strike line (N25°E for example) or in terms of east or west of true north or south, a single three digit number representing the azimuth, where the lower number is usually given (where the example of N25°E would simply be 025), or the azimuth number followed by the degree sign (example of N25°E would be 025°). The dip gives the steepest angle of descent of a tilted bed or feature relative to a horizontal plane, and is given by the number (0°-90°) as well as a letter (N, S, E, W) with rough direction in which the bed is dipping.
    [Show full text]
  • Part 3: Normal Faults and Extensional Tectonics
    12.113 Structural Geology Part 3: Normal faults and extensional tectonics Fall 2005 Contents 1 Reading assignment 1 2 Growth strata 1 3 Models of extensional faults 2 3.1 Listric faults . 2 3.2 Planar, rotating fault arrays . 2 3.3 Stratigraphic signature of normal faults and extension . 2 3.4 Core complexes . 6 4 Slides 7 1 Reading assignment Read Chapter 5. 2 Growth strata Although not particular to normal faults, relative uplift and subsidence on either side of a surface breaking fault leads to predictable patterns of erosion and sedi­ mentation. Sediments will fill the available space created by slip on a fault. Not only do the characteristic patterns of stratal thickening or thinning tell you about the 1 Figure 1: Model for a simple, planar fault style of faulting, but by dating the sediments, you can tell the age of the fault (since sediments were deposited during faulting) as well as the slip rates on the fault. 3 Models of extensional faults The simplest model of a normal fault is a planar fault that does not change its dip with depth. Such a fault does not accommodate much extension. (Figure 1) 3.1 Listric faults A listric fault is a fault which shallows with depth. Compared to a simple planar model, such a fault accommodates a considerably greater amount of extension for the same amount of slip. Characteristics of listric faults are that, in order to maintain geometric compatibility, beds in the hanging wall have to rotate and dip towards the fault. Commonly, listric faults involve a number of en echelon faults that sole into a low­angle master detachment.
    [Show full text]
  • Kinematics of the Northern Walker Lane: an Incipient Transform Fault Along the Pacific–North American Plate Boundary
    Kinematics of the northern Walker Lane: An incipient transform fault along the Paci®c±North American plate boundary James E. Faulds Christopher D. Henry Nevada Bureau of Mines and Geology, MS 178, University of Nevada, Reno, Nevada 89557, USA Nicholas H. Hinz ABSTRACT GEOLOGIC SETTING In the western Great Basin of North America, a system of dextral faults accommodates As western North America has evolved 15%±25% of the Paci®c±North American plate motion. The northern Walker Lane in from a convergent to a transform margin in northwest Nevada and northeast California occupies the northern terminus of this system. the past 30 m.y., the northern Walker Lane has This young evolving part of the plate boundary offers insight into how strike-slip fault undergone widespread volcanism and tecto- systems develop and may re¯ect the birth of a transform fault. A belt of overlapping, left- nism. Tertiary volcanic strata include 31±23 stepping dextral faults dominates the northern Walker Lane. Offset segments of a W- Ma ash-¯ow tuffs associated with the south- trending Oligocene paleovalley suggest ;20±30 km of cumulative dextral slip beginning ward-migrating ``ignimbrite ¯are up,'' 22±5 ca. 9±3 Ma. The inferred long-term slip rate of ;2±10 mm/yr is compatible with global Ma calc-alkaline intermediate-composition positioning system observations of the current strain ®eld. We interpret the left-stepping rocks related to the ancestral Cascade arc, and faults as macroscopic Riedel shears developing above a nascent lithospheric-scale trans- 13 Ma to present bimodal rocks linked to Ba- form fault.
    [Show full text]
  • Late Mesozoic Compressional to Extensional Tectonics in The
    Late Mesozoic compressional to extensional tectonics in the Yiwulüshan massif, NE China and its bearing on the evolution of the Yinshan-Yanshan orogenic belt: Part I: Structural analyses and geochronological constraints Wei Lin, Michel Faure, Yan Chen, Wenbin Ji, Fei Wang, Lin Wu, Nicolas Charles, Jun Wang, Qingchen Wang To cite this version: Wei Lin, Michel Faure, Yan Chen, Wenbin Ji, Fei Wang, et al.. Late Mesozoic compressional to extensional tectonics in the Yiwulüshan massif, NE China and its bearing on the evolution of the Yinshan-Yanshan orogenic belt: Part I: Structural analyses and geochronological constraints. Gond- wana Research, Elsevier, 2013, 23 (1), pp.54-77. 10.1016/j.gr.2012.02.013. insu-00681290 HAL Id: insu-00681290 https://hal-insu.archives-ouvertes.fr/insu-00681290 Submitted on 21 Aug 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Late Mesozoic compressional to extensional tectonics in the Yiwulüshan massif, NE China and its bearing on the evolution of the Yinshan–Yanshan orogenic belt: Part I: Structural analyses and geochronological constraints Wei Lina Michel Faureb Yan Chenb Wenbin Jia Fei Wanga Lin Wua Nicolas Charlesb Jun Wanga Qingchen Wanga a State Key Laboratory of Lithospheric Evolution, Institute of Geology and Geophysics, Chinese Academy of Sciences, P.O.
    [Show full text]
  • Faults and Joints
    133 JOINTS Joints (also termed extensional fractures) are planes of separation on which no or undetectable shear displacement has taken place. The two walls of the resulting tiny opening typically remain in tight (matching) contact. Joints may result from regional tectonics (i.e. the compressive stresses in front of a mountain belt), folding (due to curvature of bedding), faulting, or internal stress release during uplift or cooling. They often form under high fluid pressure (i.e. low effective stress), perpendicular to the smallest principal stress. The aperture of a joint is the space between its two walls measured perpendicularly to the mean plane. Apertures can be open (resulting in permeability enhancement) or occluded by mineral cement (resulting in permeability reduction). A joint with a large aperture (> few mm) is a fissure. The mechanical layer thickness of the deforming rock controls joint growth. If present in sufficient number, open joints may provide adequate porosity and permeability such that an otherwise impermeable rock may become a productive fractured reservoir. In quarrying, the largest block size depends on joint frequency; abundant fractures are desirable for quarrying crushed rock and gravel. Joint sets and systems Joints are ubiquitous features of rock exposures and often form families of straight to curviplanar fractures typically perpendicular to the layer boundaries in sedimentary rocks. A set is a group of joints with similar orientation and morphology. Several sets usually occur at the same place with no apparent interaction, giving exposures a blocky or fragmented appearance. Two or more sets of joints present together in an exposure compose a joint system.
    [Show full text]
  • Tectonic Features of the Precambrian Belt Basin and Their Influence on Post-Belt Structures
    ... Tectonic Features of the .., Precambrian Belt Basin and Their Influence on Post-Belt Structures GEOLOGICAL SURVEY PROFESSIONAL PAPER 866 · Tectonic Features of the · Precambrian Belt Basin and Their Influence on Post-Belt Structures By JACK E. HARRISON, ALLAN B. GRIGGS, and JOHN D. WELLS GEOLOGICAL SURVEY PROFESSIONAL PAPER X66 U N IT ED STATES G 0 V ERN M EN T P R I NT I N G 0 F F I C E, \VAS H I N G T 0 N 19 7 4 UNITED STATES DEPARTMENT OF THE INTERIOR ROGERS C. B. MORTON, Secretary GEOLOGICAL SURVEY V. E. McKelvey, Director Library of Congress catalog-card No. 74-600111 ) For sale by the Superintendent of Documents, U.S. GO\·ernment Printing Office 'Vashington, D.C. 20402 - Price 65 cents (paper cO\·er) Stock Number 2401-02554 CONTENTS Page Page Abstract................................................. 1 Phanerozoic events-Continued Introduction . 1 Late Mesozoic through early Tertiary-Continued Genesis and filling of the Belt basin . 1 Idaho batholith ................................. 7 Is the Belt basin an aulacogen? . 5 Boulder batholith ............................... 8 Precambrian Z events . 5 Northern Montana disturbed belt ................. 8 Phanerozoic events . 5 Tectonics along the Lewis and Clark line .............. 9 Paleozoic through early Mesozoic . 6 Late Cenozoic block faults ........................... 13 Late Mesozoic through early Tertiary . 6 Conclusions ............................................. 13 Kootenay arc and mobile belt . 6 References cited ......................................... 14 ILLUSTRATIONS Page FIGURES 1-4. Maps: 1. Principal basins of sedimentation along the U.S.-Canadian Cordillera during Precambrian Y time (1,600-800 m.y. ago) ............................................................................................... 2 2. Principal tectonic elements of the Belt basin reentrant as inferred from the sedimentation record ............
    [Show full text]
  • The Role of Pressure Solution Seam and Joint Assemblages In
    THE ROLE OF PRESSURE SOLUTION SEAM AND JOINT ASSEMBLAGES IN THE FORMATION OF STRIKE-SLIP AND THRUST FAULTS IN A COMPRESSIVE TECTONIC SETTING; THE VARISCAN OF SOUTHWESTERN IRELAND Filippo Nenna and Atilla Aydin Department of Geological and Environmental Sciences, Stanford University, Stanford, CA 94305 e-mail: [email protected] scale such as strike-slip faults and thrust-cored folds in Abstract various stages of their development. In this study we focus on the initiation and development of strike-slip The Ross Sandstone in County Clare, Ireland, was faults by shearing of the initial JVs and PSSs and deformed by an approximately north-south compression formation of thrust faults by exploiting weak shale during the end-Carboniferous Variscan orogeny. horizons and the strike-parallel PSSs in the adjacent Orthogonal sets of fundamental structures form the sandstone intervals. initial assemblage; mutually abutting arrays of 170˚ Development of faults from shearing of initial oriented set 1 joints/veins (JVs) and approximately 75˚ fundamental structural elements with either opening or pressure solution seams (PSSs) that formed under the closing modes in a wide range of structural settings has same stress conditions. Orientations of set 2 (splay) JVs been extensively reported. Segall and Pollard (1983), and PSSs suggest a clockwise remote stress rotation of Martel and Pollard (1989) and Martel (1990) have about 35˚ responsible for the contemporaneous described strike-slip faults formed by shearing of shearing of the set 1 arrays. Prominent strike-slip faults thermal fractures in granitic rocks. Myers and Aydin are sub-parallel to set 1 JVs and form by the linkage of (2004) and Flodin and Aydin (2004) reported strike-slip en-echelon segments with broad damage zones faulting formed by shearing of joints formed by an responsible for strike-slip offsets of hundreds of metres.
    [Show full text]
  • Sedimentary Stylolite Networks and Connectivity in Limestone
    Sedimentary stylolite networks and connectivity in Limestone: Large-scale field observations and implications for structure evolution Leehee Laronne Ben-Itzhak, Einat Aharonov, Ziv Karcz, Maor Kaduri, Renaud Toussaint To cite this version: Leehee Laronne Ben-Itzhak, Einat Aharonov, Ziv Karcz, Maor Kaduri, Renaud Toussaint. Sed- imentary stylolite networks and connectivity in Limestone: Large-scale field observations and implications for structure evolution. Journal of Structural Geology, Elsevier, 2014, pp.online first. <10.1016/j.jsg.2014.02.010>. <hal-00961075v2> HAL Id: hal-00961075 https://hal.archives-ouvertes.fr/hal-00961075v2 Submitted on 19 Mar 2014 HAL is a multi-disciplinary open access L'archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destin´eeau d´ep^otet `ala diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publi´esou non, lished or not. The documents may come from ´emanant des ´etablissements d'enseignement et de teaching and research institutions in France or recherche fran¸caisou ´etrangers,des laboratoires abroad, or from public or private research centers. publics ou priv´es. 1 2 Sedimentary stylolite networks and connectivity in 3 Limestone: Large-scale field observations and 4 implications for structure evolution 5 6 Laronne Ben-Itzhak L.1, Aharonov E.1, Karcz Z.2,*, 7 Kaduri M.1,** and Toussaint R.3,4 8 9 1 Institute of Earth Sciences, The Hebrew University, Jerusalem, 91904, Israel 10 2 ExxonMobil Upstream Research Company, Houston TX, 77027, U.S.A 11 3 Institut de Physique du Globe de Strasbourg, University of Strasbourg/EOST, CNRS, 5 rue 12 Descartes, F-67084 Strasbourg Cedex, France.
    [Show full text]
  • Chapter 8 Large Strains
    Chapter 8 Large Strains Introduction Most geological deformation, whether distorted fossils or fold and thrust belt shortening, accrues over a long period of time and can no longer be analyzed with the assumptions of infinitesimal strain. Fortunately, these large, or finite strains have the same starting point that infinitesimal strain does: the deformation and dis- placement gradient tensors. However, we must clearly distinguish between gradi- ents in position or displacement with respect to the initial (material) or to the final (spatial) state and several assumptions from the last Chapter — small angles, addi- tion of successive phases or steps in the deformation — no longer hold. Finite strain can get complicated very quickly with many different tensors to worry about. Most of our emphasis here will be on the practical measurement of finite strain rather than the details of the theory but we do have to review a few basic concepts first, so that we can appreciate the differences between finite and infinitesimal strain. Some of these differences have a profound impact on how we analyze de- formation. CHAPTER 8 FINITE STRAIN Comparison to Infinitesimal Strain A Plethora of Finite Strain Tensors There are lots of finite strain tensors and they come in pairs: one referenced to the initial state and the other referenced to the final state. The derivation of these tensors is usually based on Figure 7.3 and is tedious but straightforward; we will skip the derivation here but you can see it in Allmendinger et al. (2012) or any good continuum mechanics text. The first tensor is the Lagrangian strain tensor: 1 ⎡ ∂ui ∂u j ∂uk ∂uk ⎤ 1 " Lij = ⎢ + + ⎥ = ⎣⎡Eij + E ji + EkiEkj ⎦⎤ (8.1) 2 ⎣∂ X j ∂ Xi ∂ Xi ∂ X j ⎦ 2 where Eij is the displacement gradient tensor from the last Chapter.
    [Show full text]
  • Gy403 Structural Geology Kinematic Analysis Kinematics
    GY403 STRUCTURAL GEOLOGY KINEMATIC ANALYSIS KINEMATICS • Translation- described by a vector quantity • Rotation- described by: • Axis of rotation point • Magnitude of rotation (degrees) • Sense of rotation (reference frame; clockwise or anticlockwise) • Dilation- volume change • Loss of volume = negative dilation • Increase of volume = positive dilation • Distortion- change in original shape RIGID VS. NON-RIGID BODY DEFORMATION • Rigid Body Deformation • Translation: fault slip • Rotation: rotational fault • Non-rigid Body Deformation • Dilation: burial of sediment/rock • Distortion: ductile deformation (permanent shape change) TRANSLATION EXAMPLES • Slip along a planar fault • 360 meters left lateral slip • 50 meters normal dip slip • Classification: normal left-lateral slip fault 30 Net Slip Vector X(S) 40 70 N 50m dip slip X(N) 360m strike slip 30 40 0 100m ROTATIONAL FAULT • Fault slip is described by an axis of rotation • Rotation is anticlockwise as viewed from the south fault block • Amount of rotation is 50 degrees Axis W E 50 FAULT SEPARATION VS. SLIP • Fault separation: the apparent slip as viewed on a planar outcrop. • Fault slip: must be measured with net slip vector using a linear feature offset by the fault. 70 40 150m D U 40 STRAIN ELLIPSOID X • A three-dimensional ellipsoid that describes the magnitude of dilational and distortional strain. • Assume a perfect sphere before deformation. • Three mutually perpendicular axes X, Y, and Z. • X is maximum stretch (S ) and Z is minimum stretch (S ). X Z Y Z • There are unique directions
    [Show full text]
  • THE GROWTH of SHEEP MOUNTAIN ANTICLINE: COMPARISON of FIELD DATA and NUMERICAL MODELS Nicolas Bellahsen and Patricia E
    THE GROWTH OF SHEEP MOUNTAIN ANTICLINE: COMPARISON OF FIELD DATA AND NUMERICAL MODELS Nicolas Bellahsen and Patricia E. Fiore Department of Geological and Environmental Sciences, Stanford University, Stanford, CA 94305 e-mail: [email protected] be explained by this deformed basement cover interface Abstract and does not require that the underlying fault to be listric. In his kinematic model of a basement involved We study the vertical, compression parallel joint compressive structure, Narr (1994) assumes that the set that formed at Sheep Mountain Anticline during the basement can undergo significant deformations. Casas early Laramide orogeny, prior to the associated folding et al. (2003), in their analysis of field data, show that a event. Field data indicate that this joint set has a basement thrust sheet can undergo a significant heterogeneous distribution over the fold. It is much less penetrative deformation, as it passes over a flat-ramp numerous in the forelimb than in the hinge and geometry (fault-bend fold). Bump (2003) also discussed backlimb, and in fact is absent in many of the forelimb how, in several cases, the basement rocks must be field measurement sites. Using 3D elastic numerical deformed by the fault-propagation fold process. models, we show that early slip along an underlying It is noteworthy that basement deformation often is thrust fault would have locally perturbed the neglected in kinematic (Erslev, 1991; McConnell, surrounding stress field, inducing a compression that 1994), analogue (Sanford, 1959; Friedman et al., 1980), would inhibit joint formation above the fault tip. and numerical models. This can be attributed partially Relating the absence of joints in the forelimb to this to the fact that an understanding of how internal stress perturbation, we are able to constrain the deformation is delocalized in the basement is lacking.
    [Show full text]
  • Pan-African Orogeny 1
    Encyclopedia 0f Geology (2004), vol. 1, Elsevier, Amsterdam AFRICA/Pan-African Orogeny 1 Contents Pan-African Orogeny North African Phanerozoic Rift Valley Within the Pan-African domains, two broad types of Pan-African Orogeny orogenic or mobile belts can be distinguished. One type consists predominantly of Neoproterozoic supracrustal and magmatic assemblages, many of juvenile (mantle- A Kröner, Universität Mainz, Mainz, Germany R J Stern, University of Texas-Dallas, Richardson derived) origin, with structural and metamorphic his- TX, USA tories that are similar to those in Phanerozoic collision and accretion belts. These belts expose upper to middle O 2005, Elsevier Ltd. All Rights Reserved. crustal levels and contain diagnostic features such as ophiolites, subduction- or collision-related granitoids, lntroduction island-arc or passive continental margin assemblages as well as exotic terranes that permit reconstruction of The term 'Pan-African' was coined by WQ Kennedy in their evolution in Phanerozoic-style plate tectonic scen- 1964 on the basis of an assessment of available Rb-Sr arios. Such belts include the Arabian-Nubian shield of and K-Ar ages in Africa. The Pan-African was inter- Arabia and north-east Africa (Figure 2), the Damara- preted as a tectono-thermal event, some 500 Ma ago, Kaoko-Gariep Belt and Lufilian Arc of south-central during which a number of mobile belts formed, sur- and south-western Africa, the West Congo Belt of rounding older cratons. The concept was then extended Angola and Congo Republic, the Trans-Sahara Belt of to the Gondwana continents (Figure 1) although West Africa, and the Rokelide and Mauretanian belts regional names were proposed such as Brasiliano along the western Part of the West African Craton for South America, Adelaidean for Australia, and (Figure 1).
    [Show full text]