Theory of Elasticity
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10-1 CHAPTER 10 DEFORMATION 10.1 Stress-Strain Diagrams And
EN380 Naval Materials Science and Engineering Course Notes, U.S. Naval Academy CHAPTER 10 DEFORMATION 10.1 Stress-Strain Diagrams and Material Behavior 10.2 Material Characteristics 10.3 Elastic-Plastic Response of Metals 10.4 True stress and strain measures 10.5 Yielding of a Ductile Metal under a General Stress State - Mises Yield Condition. 10.6 Maximum shear stress condition 10.7 Creep Consider the bar in figure 1 subjected to a simple tension loading F. Figure 1: Bar in Tension Engineering Stress () is the quotient of load (F) and area (A). The units of stress are normally pounds per square inch (psi). = F A where: is the stress (psi) F is the force that is loading the object (lb) A is the cross sectional area of the object (in2) When stress is applied to a material, the material will deform. Elongation is defined as the difference between loaded and unloaded length ∆푙 = L - Lo where: ∆푙 is the elongation (ft) L is the loaded length of the cable (ft) Lo is the unloaded (original) length of the cable (ft) 10-1 EN380 Naval Materials Science and Engineering Course Notes, U.S. Naval Academy Strain is the concept used to compare the elongation of a material to its original, undeformed length. Strain () is the quotient of elongation (e) and original length (L0). Engineering Strain has no units but is often given the units of in/in or ft/ft. ∆푙 휀 = 퐿 where: is the strain in the cable (ft/ft) ∆푙 is the elongation (ft) Lo is the unloaded (original) length of the cable (ft) Example Find the strain in a 75 foot cable experiencing an elongation of one inch. -
Paleoproterozoic Tectonic Evolution of the Trans-North China Orogen: Toward a Comprehensive Model
Paleoproterozoic tectonic evolution of the Trans-North China Orogen: toward a comprehensive model. Pierre Trap, Michel Faure, Wei Lin, Nicole Le Breton, Patrick Monié To cite this version: Pierre Trap, Michel Faure, Wei Lin, Nicole Le Breton, Patrick Monié. Paleoproterozoic tectonic evolution of the Trans-North China Orogen: toward a comprehensive model.. Precambrian Research, Elsevier, 2012, 222-223, pp.191-211. 10.1016/j.precamres.2011.09.008. insu-00628119 HAL Id: insu-00628119 https://hal-insu.archives-ouvertes.fr/insu-00628119 Submitted on 2 Jan 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. Paleoproterozoic tectonic evolution of the Trans-North China Orogen: Toward a comprehensive model Pierre Trapa, Michel Faureb, Wei Linc, Nicole Le Bretonb, Patrick Moniéd UMR-CNRS 6249 Chrono-Environnement, Université de Franche-Comté, 16 route de Gray a 25030 Besançon Cedex, France Institut des Sciences de la Terre d‟Orléans, CNRS, Université d‟Orléans (UMR 6113), b 45071 Orléans Cedex 2, France State Key Laboratory of Lithosphere Evolution, Institute of Geology and Geophysics, c Chinese Academy of Sciences, Beijing 100029, China Géosciences Montpellier, UMR CNRS 5243, Université Montpellier II, 34095 Montpellier d Cedex 5, France Abstract In this contribution we present a reconstruction of the overall lithotectonic architecture, from inner zones to external ones, of the Paleoproterozoic Trans-North China Orogen, within the North China Craton. -
Dairy Syncline Mine Project Record of Decision
United States Department of Agriculture U.S. Forest Service Caribou-Targhee National Forest Final Record of Decision Dairy Syncline Mine Project, Caribou County, Idaho April 2020 Forest Service – Caribou-Targhee National Forest – April 2020 In accordance with Federal civil rights law and U.S. Department of Agriculture (USDA) civil rights regulations and policies, the USDA, its Agencies, offices, and employees, and institutions participating in or administering USDA programs are prohibited from discriminating based on race, color, national origin, religion, sex, gender identity (including gender expression), sexual orientation, disability, age, marital status, family/parental status, income derived from a public assistance program, political beliefs, or reprisal or retaliation for prior civil rights activity, in any program or activity conducted or funded by USDA (not all bases apply to all programs). Remedies and complaint filing deadlines vary by program or incident. Persons with disabilities who require alternative means of communication for program information (e.g., Braille, large print, audiotape, American Sign Language, etc.) should contact the responsible Agency or USDA’s TARGET Center at (202) 720-2600 (voice and TTY) or contact USDA through the Federal Relay Service at (800) 877-8339. Additionally, program information may be made available in languages other than English. To file a program discrimination complaint, complete the USDA Program Discrimination Complaint Form, AD- 3027, found online at http://www.ascr.usda.gov/complaint_filing_cust.html and at any USDA office or write a letter addressed to USDA and provide in the letter all of the information requested in the form. To request a copy of the complaint form, call (866) 632-9992. -
1 Useful Definitions Or Concepts
ETH Zurich Institute for Mechanical Systems Department of Mechanical and Process Engineering Center of Mechanics Winter 06/07 Nonlinear Continuum Mechanics Exercise 9 Prof. Dr. Sanjay Govindjee 1 Useful Definitions or Concepts 1.1 Elastic constitutive laws One general type of elastic material model is the one called Cauchy elastic material, which depend on only the current local deformation of the material and has no dependence on past history. This type is often to general and a slightly more restrictive model called Green elastic materials which are derivable from a scalar function (strain energy density function) are often used in mechanics. • Cauchy elastic S = Sˆ(deformation measure) ∂Ψ • Green elastic S = ∂deformation measure) The definition of Green elastic materials coincide with that of hyperelasticity. From hyperelasticity and thermody- namic consistency, one obtains, ∂Ψ P = (1) ∂F or ∂Ψ S = 2 . (2) ∂C Objectivity requires that Ψˆ depend only the stretches or for example on the Right Cauchy deformation tensor, Ψ(ˆ C) . (3) If the material is isotropic, then the Ψ can also be a function of the Left Cauchy deformation tensor, Ψ(ˆ b) (4) or equivalently, Ψ(ˆ I1(b),I2(b),I3(b)) (5) Ψ(ˆ I1(C),I2(C),I3(C)) . (6) 1 ETH Zurich Institute for Mechanical Systems Department of Mechanical and Process Engineering Center of Mechanics Winter 06/07 Nonlinear Continuum Mechanics Exercise 9 Prof. Dr. Sanjay Govindjee 2 Applications Problem: Consider the Neo-Hookean model, 2 Ψ = c1(I1 − 3) + c2(J − 1) + c3lnJ. (7) What should c1, c2, c3 be such that this model is consistent with linear elasticity? Express ci (i = 1, 2, 3) in terms of the Lame constants. -
Crack Tip Elements and the J Integral
EN234: Computational methods in Structural and Solid Mechanics Homework 3: Crack tip elements and the J-integral Due Wed Oct 7, 2015 School of Engineering Brown University The purpose of this homework is to help understand how to handle element interpolation functions and integration schemes in more detail, as well as to explore some applications of FEA to fracture mechanics. In this homework you will solve a simple linear elastic fracture mechanics problem. You might find it helpful to review some of the basic ideas and terminology associated with linear elastic fracture mechanics here (in particular, recall the definitions of stress intensity factor and the nature of crack-tip fields in elastic solids). Also check the relations between energy release rate and stress intensities, and the background on the J integral here. 1. One of the challenges in using finite elements to solve a problem with cracks is that the stress field at a crack tip is singular. Standard finite element interpolation functions are designed so that stresses remain finite a everywhere in the element. Various types of special b c ‘crack tip’ elements have been designed that 3L/4 incorporate the singularity. One way to produce a L/4 singularity (the method used in ABAQUS) is to mesh L the region just near the crack tip with 8 noded elements, with a special arrangement of nodal points: (i) Three of the nodes (nodes 1,4 and 8 in the figure) are connected together, and (ii) the mid-side nodes 2 and 7 are moved to the quarter-point location on the element side. -
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 ............ -
GEOLOGIC FRAMEWORK, TECTONIC EVOLUTION, and DISPLACEMENT HISTORY of the ALEXANDER TERRANE Georgee
TECTONICS, VOL. 6, NO. 2, PAGES 151-173, APRIL 1987 GEOLOGIC FRAMEWORK, TECTONIC EVOLUTION, AND DISPLACEMENT HISTORY OF THE ALEXANDER TERRANE GeorgeE. Gehrels1 and Jason B. Saleeby Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena Abstract. The Alexander terrane consists of Devonian (Klakas orogeny). The second phase is upper Proterozoic(?)-Cambrian through marked by Middle Devonian through Lower Middle(?) Jurassic rocks that underlie much of Permian strata which accumulated in southeastern (SE) Alaska and parts of eastern tectonically stable marine environments. Alaska, western British Columbia, and Devonian and Lower Permian volcanic rocks and southwestern Yukon Territory. A variety of upper Pennsylvanian-Lower Permian syenitic to geologic, paleomagnetic, and paleontologic dioritic intrusive bodies occur locally but do not evidence indicates that these rocks have been appear to represent major magmatic systems. displaced considerable distances from their The third phase is marked by Triassic volcanic sites of origin and were not accreted to western and sedimentary rocks which are interpreted to North America until Late Cretaceous-early have formed in a rift environment. Previous Tertiary time. Our geologic and U-Pb syntheses of the displacement history of the geochronologic studies in southern SE Alaska terrane emphasized apparent similarities with and the work of others to the north indicate rocks in the Sierra-Klamath region and that the terrane evolved through three distinct suggested that the Alexander terrane evolved in tectonic phases. During the initial phase, from proximity to the California continental margin late Proterozoic(?)-Cambrian through Early during Paleozoic time. Our studies indicate, Devonian time, the terrane probably evolved however, that the geologic record of the along a convergent plate margin. -
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. -
Horst Inversion Within a Décollement Zone During Extension Upper Rhine Graben, France Joachim Place, M Diraison, Yves Géraud, Hemin Koyi
Horst Inversion Within a Décollement Zone During Extension Upper Rhine Graben, France Joachim Place, M Diraison, Yves Géraud, Hemin Koyi To cite this version: Joachim Place, M Diraison, Yves Géraud, Hemin Koyi. Horst Inversion Within a Décollement Zone During Extension Upper Rhine Graben, France. Atlas of Structural Geological Interpretation from Seismic Images, 2018. hal-02959693 HAL Id: hal-02959693 https://hal.archives-ouvertes.fr/hal-02959693 Submitted on 7 Oct 2020 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. Horst Inversion Within a Décollement Zone During Extension Upper Rhine Graben, France Joachim Place*1, M. Diraison2, Y. Géraud3, and Hemin A. Koyi4 1 Formerly at Department of Earth Sciences, Uppsala University, Sweden 2 Institut de Physique du Globe de Strasbourg (IPGS), Université de Strasbourg/EOST, Strasbourg, France 3 Université de Lorraine, Vandoeuvre-lès-Nancy, France 4 Department of Earth Sciences, Uppsala University, Sweden * [email protected] The Merkwiller–Pechelbronn oil field of the Upper Rhine Graben has been a target for hydrocarbon exploration for over a century. The occurrence of the hydrocarbons is thought to be related to the noticeably high geothermal gradient of the area. -
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 -
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. -
Basin Inversion and Structural Architecture As Constraints on Fluid Flow and Pb-Zn Mineralisation in the Paleo-Mesoproterozoic S
https://doi.org/10.5194/se-2020-31 Preprint. Discussion started: 6 April 2020 c Author(s) 2020. CC BY 4.0 License. 1 Basin inversion and structural architecture as constraints on fluid 2 flow and Pb-Zn mineralisation in the Paleo-Mesoproterozoic 3 sedimentary sequences of northern Australia 4 5 George M. Gibson, Research School of Earth Sciences, Australian National University, Canberra ACT 6 2601, Australia 7 Sally Edwards, Geological Survey of Queensland, Department of Natural Resources, Mines and Energy, 8 Brisbane, Queensland 4000, Australia 9 Abstract 10 As host to several world-class sediment-hosted Pb-Zn deposits and unknown quantities of conventional and 11 unconventional gas, the variably inverted 1730-1640 Ma Calvert and 1640-1580 Ma Isa superbasins of 12 northern Australia have been the subject of numerous seismic reflection studies with a view to better 13 understanding basin architecture and fluid migration pathways. Strikingly similar structural architecture 14 has been reported from much younger inverted sedimentary basins considered prospective for oil and gas 15 elsewhere in the world. Such similarities suggest that the mineral and petroleum systems in Paleo- 16 Mesoproterozoic northern Australia may have spatially and temporally overlapped consistent with the 17 observation that basinal sequences hosting Pb-Zn mineralisation in northern Australia are bituminous or 18 abnormally enriched in hydrocarbons. This points to the possibility of a common tectonic driver and shared 19 fluid pathways. Sediment-hosted Pb-Zn mineralisation coeval with basin inversion first occurred during the 20 1650-1640 Ma Riversleigh Tectonic Event towards the close of the Calvert Superbasin with further pulses 21 accompanying the 1620-1580 Ma Isa Orogeny which brought about closure of the Isa Superbasin.