ELASTICITY AND CONSTITUTION OF THE EARTH'S INTERIOR*
|
|
- Jonas Augustus Franklin
- 5 years ago
- Views:
Transcription
1 JOURNAL OF GEOPHYSICAL RESEARCH VOLUME 57, NO. 2 JUNE, 1952 ELASTICITY AND CONSTITUTION OF THE EARTH'S INTERIOR* BY FRANCIS BIRCtt Harvard University, Cambridge, Massachusetts (Received January 18, 1952) ABSTRACT
2 The one-dimensional Earth according to Jeffreys (1939a) and Gutenberg (1948, 1951) 14 VELOCITY, KM/$EC 12 1 Wave Velocities Iooo 2ooo ooo 4ooo sooo DEPTH, KM Depth (km) 4 5 6
3 Quick aside: radial vs. equal area vs. equal volume 11.6 Equal Volume Equal Area 11.6 lower mantle (% area corresponds to % volume)
4 14 Vp 12 1 Wave Velocities 8 6 Vs Williamson-Adams V - 4/3 V -- K g/p - - (OP/Op dpf p g(r) dr/ h(r appropriate for: 4 hydrostatic homogeneous 2 adiabatic layer Depth (km)
5 Build-a-Planet Part II PRESSURE looo IO G CM/SEO 2 GRAVITY... '"'"" 5OO 5 DENSITY I DEPTH, IN 13 KM I F'IG. 2.--DENSITY, PRESSURE, AND ACCELERATION WITHIN THE EARTH (AFTER BULLEN)
6 V bulk (m/s) Outer Core Inner Core density (g/cm 3 ) 4 2 Lower Mantle Pressure (GPa)
7 Constant parameters K=45 GPa ρ=5 g/cc 1 1 V bulk (m/s) 8 6 Lower Mantle 8 6 density (g/cm 3 ) Pressure (GPa)
8 Linear Increase of density, constant K K=45 GPa 1 1 ρ=4.2 g/cc dρ/dp=.1 V bulk (m/s) 8 6 Lower Mantle 8 6 density (g/cm 3 ) Pressure (GPa)
9 K=-V dp/dv V~1/ρ ρ=4.2 g/cc Hooke s law Earth 1 1 dρ/dp= V bulk (m/s) 8 6 Lower Mantle 8 6 density (g/cm 3 ) K: Bulk Modulus (GPa) Pressure (GPa) Pressure (GPa)
10 K=-V dp/dv Definition of K= -VdP/dV V~1/ρ dρ/dp-polynomial 1 1 K: Bulk Modulus (GPa) fit (4th order) V bulk (m/s) Lower Mantle density (g/cm 3 ) Pressure (GPa) Pressure (GPa)
11 Introduction to Elasticity 1. Deep Earth context 2. Stress, strain and elastic tensors 3. Elasticity and symmetry/anisotropy 4. Thermodynamics of elasticity 5. Beyond Hooke: stress-strain equations of state on board 6. Introduction to lattice dynamics 7. Thermoelasticity 8. Experimental technique 9. Mysteries of Elasticity a. Composite behavior b. Softening: -phase transformations -with iron and volatiles c. Frequency dependence
12 Why mineral physicists like salt 2P 1.5 : 5 1. LITHIUM SODIUM P()TASSIUM RUBIDIUM CESIUM ?/?. 2. FIG. 3--COMPRESSION OF THE ALKALI METALS
13 Birch-Murnaghan Equation of state [ (V ) 7/3 ( V ) 5/3 ] P(V, T ) = 3 2 K T { V V [ 1 3 (V 4 (4 K T ) V ) 2/3 1]}
14 Mie-Gruneisen-Debye Equation of state P(V, T ) = P(V, T ) + P th (V, T ) P th (V, T ) = γ(v ){E th(θ, T ) E th (θ, T )} V ( ) 3 T θ/t t 3 E th (θ, T ) = 9nRT θ (e t 1) dt γ(v ) = γ ( V V ) q θ(v ) = θ exp { [γ γ(v )] q }
15 14 Adiabatic temp gradient 14 dt/dp = (ot/op)s = Ta/pC V bulk (m/s) Outer Core Birch s thermal Williamson-Adams Inner Core density (g/cm 3 ) Lower Mantle ( -.o /g) = - g2 ( - c /g) dr combination o y = Ks/pC and τ ~ deviation from adiabaticity 15 2 Pressure (GPa)
16 Elasticity of a mechanical composite I
17 Elasticity of a mechanical composite II Reuss Bounds: constant stress boundary condition Voigt Bounds: constant strain boundary condition Polycrystalline elasticity constraints = rotations
18 Introduction to Elasticity 1. Deep Earth context 2. Stress, strain and elastic tensors 3. Elasticity and symmetry/anisotropy 4. Thermodynamics of elasticity 5. Beyond Hooke: stress-strain equations of state (Introduction to lattice dynamics) (Thermoelasticity) 8. Experimental Techniques 9. Mysteries of Elasticity a. Composite behavior b. Softening: -phase transformations -with iron and volatiles c. Frequency dependence
19 Introduction to Elasticity 1. Deep Earth context 2. Stress, strain and elastic tensors 3. Elasticity and symmetry/anisotropy 4. Thermodynamics of elasticity 5. Beyond Hooke: stress-strain equations of state (Introduction to lattice dynamics) (Thermoelasticity) 8. Experimental techniques 9. Mysteries of Elasticity a. Composite behavior b. Softening: -phase transformations (Landau) -with iron/volatiles? c. Frequency dependence
20 Determining Elastic Properties 1. Directly measure density as a function of pressure, temperature 2. Measure wavespeeds directly (ultrasonic techniques) 3. Probe lattice dynamics (Brillouin spectroscopy and Nuclear Inelastic X-ray spectroscopy) 3b. Calculate lattice dynamics 4. Shockwave techniques: Determine internal energy directly at high P,T
21 High Pressure Diamond Anvil Cell
22 X-ray beam experimental procedure I 2θ 2θ Laser heating Unknown Phase stability Density Elastic properties Standard as a function of temperature
23 High Pressure before about 1 years ago, no in situ high P,T. All was quenched from high Temperature. High Temperature Characterize Sample Synchrotron added in situ It has added two major discoveries, and a host of minor embarassments Synchrotron X-ray diffraction
24 Results for FeO (Crowhurst et al., 28) Fig. 1. Acoustic wave velocities as a function of pressure for propagation in the (1) plane of singlecrystal (Mg.94,Fe.6 )O in an argon pressure-transmitting medium. Circles indicate present data acquired by impulsive stimulated scattering. Uncertainties are given by ± 2s, where s is the formal SE. Squares indicate data obtained by Jackson et al. via Brillouin scattering(19). Lines are calculated velocities based on linear extrapolations of the elastic moduli obtained by Jackson et al.(19). (Top) Body wave velocities. Solid circles and open circles are data for propagation along[11]and[1],respectively.(bottom) Velocities of the wave that propagates at the interface between the sample and the pressure-transmitting medium. The interfacial wave has no dependence on direction under these conditions. ncemag.org SCIENCE VOL JANUARY
25 Fig. 3. (A)Bulkmodulusversuspressure.Circlesindicatepresentdatacalculatedonthebasisofthec ij shown in Fig. 2. Squares are values calculated on the basis of the data of Jackson et al.(19). The solid line is based on a thermodynamic description of the HS-to-LS transition with parameters scaled from those that fit the compression data of Lin et al. (11). The dashed line is the average of separate equations of state for the HS and LS phases [see (B)]. Error bars indicate two SEs obtained from fits to the measured velocities. (B) Experimental data of Lin et al. (11) showingdensityof(mg.83,fe.17 )O versus pressure. The dashed lines represent equations-of-state (fourth-order Eulerian finite-strain) fits (22) to the HS (low pressure) and LS phases. The thick solid line is a fit to the data based on Eq. 1. The thin solid line is the first-principles theoretical prediction of Tsuchiya et al.(15). Results for FeO (Crowhurst et Fig. 2. Measured pressure dependence of the elastic moduli c ij and anisotropy factor A =(c 11 c 12 )/2 c 44.Circlesindicatethepresentdata,andsquaresaredataofJacksonet al.(19). Lines are linear fits to the latter data. Error bars indicate two SEs obtained from fits to the measured velocities. al., 28) A B
26 Elasticity of hydrous phases Structure and elasticity of serpentine at high-pressure Mainak Mookherjee a,, Lars Stixrude b
27 Single crystal elastic anisotropy (Inner core anisotropy?)
28 Waves propagating through anisotropic media Slide borrowed from Ronald D. Kriz Engineering Science and Mechanics Virginia Polytechnic Institute and State University Blacksburg, Virginia 2461
Elasticity, the fourth-rank tensor defining the strain of crystalline
Elasticity of MgO and a primary pressure scale to 55 GPa Chang-Sheng Zha*, Ho-kwang Mao, and Russell J. Hemley Geophysical Laboratory and Center for High Pressure Research, Carnegie Institution of Washington,
More informationChemical Composition of the Lower Mantle: Constraints from Elasticity. Motohiko Murakami Tohoku University
Chemical Composition of the Lower Mantle: Constraints from Elasticity Motohiko Murakami Tohoku University Acknowledgements Jay D. Bass (University of Illinois) Stanislav V. Sinogeikin (University of Illinois)
More informationInterpreting Geophysical Data for Mantle Dynamics. Wendy Panero University of Michigan
Interpreting Geophysical Data for Mantle Dynamics Wendy Panero University of Michigan Chemical Constraints on Density Distribution Atomic Fraction 1.0 0.8 0.6 0.4 opx cpx C2/c garnet il olivine wadsleyite
More informationSimultaneous sound velocity and density measurements of NaCl at high temperatures and pressures: Application as a primary pressure standard
American Mineralogist, Volume 97, pages 1670 1675, 2012 Simultaneous sound velocity and density measurements of NaCl at high temperatures and pressures: Application as a primary pressure standard Masanori
More informationPressure Volume Temperature Equation of State
Pressure Volume Temperature Equation of State S.-H. Dan Shim ( ) Acknowledgement: NSF-CSEDI, NSF-FESD, NSF-EAR, NASA-NExSS, Keck Equations relating state variables (pressure, temperature, volume, or energy).
More informationFirst-principles thermoelasticity of bcc iron under pressure
First-principles thermoelasticity of bcc iron under pressure Xianwei Sha and R. E. Cohen Carnegie Institution of Washington, 5251 Broad Branch Road, NW, Washington, D.C. 20015, USA Received 17 May 2006;
More informationGS388 Handout: Radial density distribution via the Adams-Williamson equation 1
GS388 Handout: Radial density distribution via the Adams-Williamson equation 1 TABLE OF CONTENTS ADIABATIC COMPRESSION: THE ADAMS WILLIAMSON EQUATION...1 EFFECT OF NON-ADIABATIC TEMPERATURE GRADIENT...3
More informationThermal equation of state of (Mg 0.9 Fe 0.1 ) 2 SiO 4 olivine
Physics of the Earth and Planetary Interiors 157 (2006) 188 195 Thermal equation of state of (Mg 0.9 Fe 0.1 ) 2 SiO 4 olivine Wei Liu, Baosheng Li Mineral Physics Institute, Stony Brook University, Stony
More informationElasticity of single crystal and polycrystalline MgSiO 3 perovskite by Brillouin spectroscopy
GEOPHYSICAL RESEARCH LETTERS, VOL. 31, L06620, doi:10.1029/2004gl019559, 2004 Elasticity of single crystal and polycrystalline MgSiO 3 perovskite by Brillouin spectroscopy Stanislav V. Sinogeikin Department
More informationStrain-related Tensorial Properties: Elasticity, Piezoelectricity and Photoelasticity
Strain-related Tensorial Properties: Elasticity, Piezoelectricity and Photoelasticity Torino, Italy, September 4-9, 2016 Alessandro Erba Dipartimento di Chimica, Università di Torino (Italy) alessandro.erba@unito.it
More informationINTRODUCTION TO THE PHYSICS OF THE EARTH S INTERIOR
INTRODUCTION TO THE PHYSICS OF THE EARTH S INTERIOR SECOND EDITION JEAN-PAULPOIRIER Institut de Physique du Globe de Paris PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building,
More informationElasticity of single-crystal aragonite by Brillouin spectroscopy
Phys Chem Minerals (2005) 32: 97 102 DOI 10.1007/s00269-005-0454-y ORIGINAL PAPERS Lin-gun Liu Æ Chien-chih Chen Æ Chung-Cherng Lin Yi-jong Yang Elasticity of single-crystal aragonite by Brillouin spectroscopy
More informationUnified analyses for P-V-T equation of state of MgO: A solution for pressure-scale problems in high P-T experiments
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi:10.1029/2008jb005813, 2009 Unified analyses for P-V-T equation of state of MgO: A solution for pressure-scale problems in high P-T experiments Yoshinori
More informationEART162: PLANETARY INTERIORS
EART162: PLANETARY INTERIORS Francis Nimmo Last Week Global gravity variations arise due to MoI difference (J 2 ) We can also determine C, the moment of inertia, either by observation (precession) or by
More informationEOS-FIT V6.0 R.J. ANGEL
EOS-FIT V6. R.J. AGEL Crystallography Laboratory, Dept. Geological Sciences, Virginia Tech, Blacksburg, VA46, USA http://www.geol.vt.edu/profs/rja/ ITRODUCTIO EosFit started as a program to fit equations
More information101 year anniversary of Max Theodor Felix von Laue s first application of x-rays to crystallography!
X- ray Diffrac+on Ge 116 (weeks 9 & 10) March 5, 2013 Jennifer Jackson 101 year anniversary of Max Theodor Felix von Laue s first application of x-rays to crystallography! Done: Radiation Safety Training
More informationDetermining the Elastic Modulus and Hardness of an Ultrathin Film on a Substrate Using Nanoindentation
Determining the Elastic Modulus and Hardness of an Ultrathin Film on a Substrate Using Nanoindentation The Harvard community has made this article openly available. Please share how this access benefits
More informationHydrogen-bonded structure and mechanical chiral response of a silver nanoparticle superlattice
Hydrogen-bonded structure and mechanical chiral response of a silver nanoparticle superlattice Bokwon Yoon 1, W. D. Luedtke 1, Robert N. Barnett 1, Jianping Gao 1, Anil Desireddy 2, Brian E. Conn 2, Terry
More informationIntroduction to Seismology Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 1.510 Introduction to Seismology Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 1.510 Introduction to
More informationPhysics of the Earth and Planetary Interiors
Physics of the Earth and Planetary Interiors 176 (2009) 98 108 Contents lists available at ScienceDirect Physics of the Earth and Planetary Interiors journal homepage: www.elsevier.com/locate/pepi Optimal
More informationBrillouin scattering and X-ray diffraction of San Carlos olivine: direct pressure determination to 32 GPa
ELSEVIER Earth and Planetary Science Letters 159 (1998) 25 33 Brillouin scattering and X-ray diffraction of San Carlos olivine: direct pressure determination to 32 GPa Chang-sheng Zha a, Thomas S. Duffy
More informationComputational support for a pyrolitic lower mantle containing ferric iron
SUPPLEMENTARY INFORMATION DOI: 1.138/NGEO2458 Computational support for a pyrolitic lower mantle containing ferric iron Xianlong Wang, Taku Tsuchiya and Atsushi Hase NATURE GEOSCIENCE www.nature.com/naturegeoscience
More informationThe electronic structure of materials 1
Quantum mechanics 2 - Lecture 9 December 18, 2013 1 An overview 2 Literature Contents 1 An overview 2 Literature Electronic ground state Ground state cohesive energy equilibrium crystal structure phase
More informationThe equation of state of CaSiO perovskite to 108 GPa at 300 K
Ž. Physics of the Earth and Planetary Interiors 120 2000 27 8 www.elsevier.comrlocaterpepi The equation of state of CaSiO perovskite to 108 GPa at 00 K Sang-Heon Shim a,), Thomas S. Duffy a, Guoyin Shen
More informationRaman spectroscopy at high pressure and temperature for the study of Earth's mantle and planetary minerals
Raman spectroscopy at high pressure and temperature for the study of Earth's mantle and planetary minerals Bruno Reynard, Gilles Montagnac, and Hervé Cardon Laboratoire de Géologie de Lyon Coupling HP
More informationEquations of State. Tiziana Boffa Ballaran
Equations o State iziana Boa Ballaran Why EoS? he Earth s interior is divided globally into layers having distinct seismic properties Speed with which body waves travel through the Earth s interior are
More informationElasticity, composition and temperature of the Earth s lower mantle: a reappraisal
Geophys. J. Int. (1998) 14, 291 11 Elasticity, composition and temperature of the Earth s lower mantle: a reappraisal Ian Jackson Research School of Earth Sciences, Australian National University, Canberra
More information16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations
6.2 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive quations Constitutive quations For elastic materials: If the relation is linear: Û σ ij = σ ij (ɛ) = ρ () ɛ ij σ ij =
More informationHydration of Olivine and. Earth s Deep Water Cycle
Hydration of Olivine and Earth s Deep Water Cycle IASPEI October 5, 2005 Olivine Hydration and Earth s Deep Water Cycle J. R. Smyth, (University of Colorado) Dan Frost and Fabrizio Nestola (Bayerisches
More informationCompression of CaTiO 3 and CaGeO 3 perovskites
American Mineralogist, Volume 84, pages 77 81, 1999 Compression of CaTiO 3 and CaGeO 3 perovskites NANCY L. ROSS 1, * AND ROSS J. ANGEL 1 Department of Geological Sciences, University College London, Gower
More informationThermal and compositional structure of the Mantle and Lithosphere
Chapter 1 Thermal and compositional structure of the Mantle and Lithosphere 1.1 Primordial heat of the Earth The most widely accepted planetary formation theory says that the solar system accreted from
More informationEARTHQUAKE WAVES AND THE MECHANICAL PROPERTIES OF THE EARTH'S INTERIOR
EARTHQUAKE WAVES AND THE MECHANCAL PROPERTES OF THE EARTH'S NTEROR By K. E. BULLEN t has come to be realized in the present century that the science of Seismology, in addition to providing information
More informationWe briefly discuss two examples for solving wave propagation type problems with finite differences, the acoustic and the seismic problem.
Excerpt from GEOL557 Numerical Modeling of Earth Systems by Becker and Kaus 2016 1 Wave propagation Figure 1: Finite difference discretization of the 2D acoustic problem. We briefly discuss two examples
More informationLecture 11 - Phonons II - Thermal Prop. Continued
Phonons II - hermal Properties - Continued (Kittel Ch. 5) Low High Outline Anharmonicity Crucial for hermal expansion other changes with pressure temperature Gruneisen Constant hermal Heat ransport Phonon
More information2, from which K = # " 2 % 4 3 $ ) ( )
Introducción a la Geofísica 2010-01 TAREA 6 1) FoG. Calculate the bulk modulus (K), the shear modulus (µ) and Poisson s ratio (ν) for the lower crust, upper mantle and lower mantle, respectively, using
More informationPLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by: [Whitaker, Matthew L.][State University of New York at Stony Brook] On: 28 September 2008 Access details: Access Details: [subscription number 788676199] Publisher Taylor
More informationAnalysis of volume expansion data for periclase, lime, corundum and spinel at high temperatures
Bull. Mater. Sci., ol. 35, No., August, pp. 31 37. c Indian Academy of Sciences. Analysis of volume expansion data for periclase, lime, corundum and spinel at high temperatures BPSINGH, H CHANDRA, R SHYAM
More informationCh 6: Internal Constitution of the Earth
Ch 6: Internal Constitution of the Earth Mantle composition Geological background 88 elements found in the Earth's crust -- of these, only 8 make up 98%: oxygen, silicon, aluminum, iron, calcium, magnesium,
More informationGlobal geophysics and wave propagation
Global geophysics and wave propagation Reading: Fowler p76 83 Remote sensing Geophysical methods Seismology Gravity and bathymetry Magnetics Heat flow Seismology: Directly samples the physical properties
More informationalloys at high pressures. The central peaks at 0 mev correspond to the elastic scattering.
1 2 3 4 5 Supplementary Figure 1 Representative NRIXS spectra of the basaltic glass and Fe-rich alloys at high pressures. The central peaks at 0 mev correspond to the elastic scattering. Open symbols:
More informationPEAT SEISMOLOGY Lecture 9: Anisotropy, attenuation and anelasticity
PEAT8002 - SEISMOLOGY Lecture 9: Anisotropy, attenuation and anelasticity Nick Rawlinson Research School of Earth Sciences Australian National University Anisotropy Introduction Most of the theoretical
More informationMgO was dried at 1100 C overnight, then stoichiometrically combined with 95% enriched 57 Fe 2 O 3
82 Appendix A A.1 Synthesis and Characterization A.1.1 (Mg 0.16 Fe 0.84 )O MgO was dried at 1100 C overnight, then stoichiometrically combined with 95% enriched 57 Fe 2 O 3 (Advanced Materials Techonologies
More informationChapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature
Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte
More informationComputational models of diamond anvil cell compression
UDC 519.6 Computational models of diamond anvil cell compression A. I. Kondrat yev Independent Researcher, 5944 St. Alban Road, Pensacola, Florida 32503, USA Abstract. Diamond anvil cells (DAC) are extensively
More informationOn the study of elastic wave scattering and Rayleigh wave velocity measurement of concrete with steel bar
NDT&E International 33 (2000) 401 407 www.elsevier.com/locate/ndteint On the study of elastic wave scattering and Rayleigh wave velocity measurement of concrete with steel bar T.-T. Wu*, J.-H. Sun, J.-H.
More informationStructure and Dynamics : An Atomic View of Materials
Structure and Dynamics : An Atomic View of Materials MARTIN T. DOVE Department ofearth Sciences University of Cambridge OXFORD UNIVERSITY PRESS Contents 1 Introduction 1 1.1 Observations 1 1.1.1 Microscopic
More informationAn acoustic wave equation for orthorhombic anisotropy
Stanford Exploration Project, Report 98, August 1, 1998, pages 6?? An acoustic wave equation for orthorhombic anisotropy Tariq Alkhalifah 1 keywords: Anisotropy, finite difference, modeling ABSTRACT Using
More informationNumerical Methods in Geophysics. Introduction
: Why numerical methods? simple geometries analytical solutions complex geometries numerical solutions Applications in geophysics seismology geodynamics electromagnetism... in all domains History of computers
More informationElastic properties of minerals and functional materials by Brillouin scattering and laser ultrasonics. Pavel Zinin
Elastic properties of minerals and functional materials by Brillouin scattering and laser ultrasonics Pavel Zinin Scientific and Technological Center of Unique Instrumentation, RAS, Moscow, Russia Motivation
More informationA look into Gassmann s Equation
A look into Gassmann s Equation Nawras Al-Khateb, CHORUS Heavy Oil Consortium, Department of Geoscience, University of Calgary nawras.alkhateb@ucalgary.ca Summary By describing the influence of the pore
More informationIn this section, thermoelasticity is considered. By definition, the constitutive relations for Gradθ. This general case
Section.. Thermoelasticity In this section, thermoelasticity is considered. By definition, the constitutive relations for F, θ, Gradθ. This general case such a material depend only on the set of field
More informationAb initio Predictions of Structural and Thermodynamic Properties of Zr 2 AlC Under High Pressure and High Temperature
CHINESE JOURNAL OF CHEMICAL PHYSICS VOLUME 28, NUMBER 3 JUNE 27, 215 ARTICLE Ab initio Predictions of Structural and Thermodynamic Properties of Zr 2 AlC Under High Pressure and High Temperature Fen Luo
More informationCan seismic anisotropy in Dʺ be used to constrain flow patterns in the lowermost mantle?
Can seismic anisotropy in Dʺ be used to constrain flow patterns in the lowermost mantle? Andrew Walker 1, Andy Nowacki 1, Alessandro Forte 2, James Wookey 1 and J.-Michael Kendall 1 1 School of Earth Sciences,
More informationResidual Stress analysis
Residual Stress analysis Informations: strains Macro elastic strain tensor (I kind) Crystal anisotropic strains (II kind) Fe Cu C Macro and micro stresses Applied macro stresses Residual Stress/Strain
More informationPart 5 ACOUSTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA
Part 5 ACOUSTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA Review of Fundamentals displacement-strain relation stress-strain relation balance of momentum (deformation) (constitutive equation) (Newton's Law)
More informationNanoscale Energy Conversion and Information Processing Devices - NanoNice - Photoacoustic response in mesoscopic systems
Nanoscale Energy Conversion and Information Processing Devices - NanoNice - Photoacoustic response in mesoscopic systems Photonics group W. Claeys, S. Dilhair, S. Grauby, JM. Rampnoux, L. Patino Lopez,
More informationDetermination of the hyperfine parameters of iron in aluminous (Mg,Fe)SiO 3 perovskite
Determination of the hyperfine parameters of iron in aluminous (Mg,Fe)SiO 3 perovskite Jennifer M. Jackson Seismological Laboratory, Geological & Planetary Sciences California Institute of Technology VLab
More informationELASTOPLASTICITY THEORY by V. A. Lubarda
ELASTOPLASTICITY THEORY by V. A. Lubarda Contents Preface xiii Part 1. ELEMENTS OF CONTINUUM MECHANICS 1 Chapter 1. TENSOR PRELIMINARIES 3 1.1. Vectors 3 1.2. Second-Order Tensors 4 1.3. Eigenvalues and
More informationTHE ROCK PHYSICS HANDBOOK
THE ROCK PHYSICS HANDBOOK TOOLS FOR SEISMIC ANALYSIS IN POROUS MEDIA Gary Mavko Tapan Mukerji Jack Dvorkin Stanford University Stanford University Stanford University CAMBRIDGE UNIVERSITY PRESS CONTENTS
More information6th NDT in Progress Lamb waves in an anisotropic plate of a single crystal silicon wafer
6th NDT in Progress 2011 International Workshop of NDT Experts, Prague, 10-12 Oct 2011 Lamb waves in an anisotropic plate of a single crystal silicon wafer Young-Kyu PARK 1, Young H. KIM 1 1 Applied Acoustics
More informationSummary. Simple model for kerogen maturity (Carcione, 2000)
Malleswar Yenugu* and De-hua Han, University of Houston, USA Summary The conversion of kerogen to oil/gas will build up overpressure. Overpressure is caused by conversion of solid kerogen to fluid hydrocarbons
More informationLindgren CRYSTAL SYMMETRY AND ELASTIC CONSTANTS MICHAEL WANDZILAK. S.B., Massachusetts Institute of Technology (196'7)
CRYSTAL SYMMETRY AND ELASTIC CONSTANTS by MICHAEL WANDZILAK S.B., Massachusetts Institute of Technology (196'7) Submitted in partial fulfillment of the requirements for the degree of Master of Science
More informationSupplementary Information for. Universal elastic-hardening-driven mechanical instability in α-quartz and quartz. homeotypes under pressure
Supplementary Information for Universal elastic-hardening-driven mechanical instability in α-quartz and quartz homeotypes under pressure Juncai Dong, Hailiang Zhu, and Dongliang Chen * Beijing Synchrotron
More informationPEAT SEISMOLOGY Lecture 3: The elastic wave equation
PEAT8002 - SEISMOLOGY Lecture 3: The elastic wave equation Nick Rawlinson Research School of Earth Sciences Australian National University Equation of motion The equation of motion can be derived by considering
More informationChapter 5 Torsion STRUCTURAL MECHANICS: CE203. Notes are based on Mechanics of Materials: by R. C. Hibbeler, 7th Edition, Pearson
STRUCTURAL MECHANICS: CE203 Chapter 5 Torsion Notes are based on Mechanics of Materials: by R. C. Hibbeler, 7th Edition, Pearson Dr B. Achour & Dr Eng. K. El-kashif Civil Engineering Department, University
More informationGarnet yield strength at high pressures and implications for upper mantle and transition zone rheology
Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi:10.1029/2007jb004931, 2007 Garnet yield strength at high pressures and implications for upper mantle and transition zone rheology
More informationRadiation pattern in homogeneous and transversely isotropic attenuating media
Radiation pattern in homogeneous and transversely isotropic attenuating media Satish Sinha*, Sergey Abaseyev** and Evgeni Chesnokov** *Rajiv Gandhi Institute of Petroleum Technology, Rae Bareli, UP 229010
More informationP314 Anisotropic Elastic Modelling for Organic Shales
P314 Anisotropic Elastic Modelling for Organic Shales X. Wu* (British Geological Survey), M. Chapman (British Geological Survey), X.Y. Li (British Geological Survey) & H. Dai (British Geological Survey)
More information3D Stress Tensors. 3D Stress Tensors, Eigenvalues and Rotations
3D Stress Tensors 3D Stress Tensors, Eigenvalues and Rotations Recall that we can think of an n x n matrix Mij as a transformation matrix that transforms a vector xi to give a new vector yj (first index
More informationPhysics and Chemistry of the Earth and Terrestrial Planets
MIT OpenCourseWare http://ocw.mit.edu 12.002 Physics and Chemistry of the Earth and Terrestrial Planets Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationPressure Volume Temperature (P-V-T) Relationships and Thermo elastic Properties of Geophysical Minerals
Pressure Volume Temperature (P-V-T) Relationships and Thermo elastic Properties of Geophysical Minerals A PROPOSAL FOR Ph.D PROGRAMME BY MONIKA PANWAR UNDER THE SUPERVISION OF DR SANJAY PANWAR ASSISTANT
More informationUnderstand basic stress-strain response of engineering materials.
Module 3 Constitutive quations Learning Objectives Understand basic stress-strain response of engineering materials. Quantify the linear elastic stress-strain response in terms of tensorial quantities
More informationGrüneisen parameters and isothermal equations of state
American Mineralogist, Volume 85, pages xxx xxx, Grüneisen parameters and isothermal equations of state L. VOČADLO,, * J.P. POIRER, AND G.D. PRICE Department of Geological Sciences, University College
More informationElasticity of Plagioclase Feldspars
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Elasticity of Plagioclase Feldspars J. Michael Brown Department of Earth and Space Sciences Box 35-1310 University of Washington Seattle, WA 98195
More informationEffect of Water on the Sound Velocities of Ringwoodite in the Transition Zone
Effect of Water on the Sound Velocities of Ringwoodite in the Transition Zone Steven D. Jacobsen Department of Geological Sciences, Northwestern University, Evanston, IL 60208 Joseph R. Smyth Department
More informationSynchrotron facilities and the study of the Earth s deep interior
INSTITUTE OF PHYSICS PUBLISHING Rep. Prog. Phys. 68 (2005) 1811 1859 REPORTS ON PROGRESS IN PHYSICS doi:10.1088/0034-4885/68/8/r03 Synchrotron facilities and the study of the Earth s deep interior Thomas
More informationLecture #8 Non-linear phononics
Lecture #8 Non-linear phononics Dr. Ari Salmi www.helsinki.fi/yliopisto 10.4.2018 1 Last lecture High pressure phononics can give insight into phase transitions in materials SASER can be used to generate
More informationNDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.
CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo
More informationOptical Imaging Chapter 5 Light Scattering
Optical Imaging Chapter 5 Light Scattering Gabriel Popescu University of Illinois at Urbana-Champaign Beckman Institute Quantitative Light Imaging Laboratory http://light.ece.uiuc.edu Principles of Optical
More informationMohr's Circle and Earth Stress (The Elastic Earth)
Lect. 1 - Mohr s Circle and Earth Stress 6 Mohr's Circle and Earth Stress (The Elastic Earth) In the equations that we derived for Mohr s circle, we measured the angle, θ, as the angle between σ 1 and
More informationSound Attenuation at High Temperatures in Pt
Vol. 109 006) ACTA PHYSICA POLONICA A No. Sound Attenuation at High Temperatures in Pt R.K. Singh and K.K. Pandey H.C.P.G. College, Varanasi-1001, U.P., India Received October 4, 005) Ultrasonic attenuation
More informationFile name: Supplementary Information Description: Supplementary Figures, Supplementary Tables and Supplementary References
File name: Supplementary Information Description: Supplementary Figures, Supplementary Tables and Supplementary References File name: Supplementary Movie 1 Description: The movie shows compression behaviour
More informationChapter 3. Load and Stress Analysis
Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3
More informationMechanics of Biomaterials
Mechanics of Biomaterials Lecture 7 Presented by Andrian Sue AMME498/998 Semester, 206 The University of Sydney Slide Mechanics Models The University of Sydney Slide 2 Last Week Using motion to find forces
More informationHydrogeophysics - Seismics
Hydrogeophysics - Seismics Matthias Zillmer EOST-ULP p. 1 Table of contents SH polarized shear waves: Seismic source Case study: porosity of an aquifer Seismic velocities for porous media: The Frenkel-Biot-Gassmann
More informationEquation of state of (Mg 0.8,Fe 0.2 ) 2 SiO 4 ringwoodite from synchrotron X-ray diffraction up to 20 GPa and 1700 K
Eur. J. Mineral. 2006, 18, 523-528 Equation of state of (Mg 0.8,Fe 0.2 ) 2 SiO 4 ringwoodite from synchrotron X-ray diffraction up to 20 GPa and 1700 K MASANORI MATSUI 1, *, TOMOO KATSURA 2, AKIRA KUWATA
More informationElastic Properties of Polycrystalline Solid Helium
JLowTempPhys(2010)160:5 11 DOI 10.1007/s10909-010-0173-8 Elastic Properties of Polycrystalline Solid Helium Humphrey J. Maris Sebastien Balibar Received: 29 March 2010 / Accepted: 16 April 2010 / Published
More informationSpin crossovers in the Earth mantle. Spin crossovers in the Earth mantle
Spin crossovers in the Earth mantle Spin crossovers in the Earth mantle Renata M. Wentzcovitch Dept. of Chemical Engineering and Materials Science Minnesota Supercomputing Institute Collaborators Han Hsu
More informationElements of Rock Mechanics
Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider
More informationBorehole Geophysics. Acoustic logging measurements
Acoustic logging measurements - Review of basic physics background - Concept of P- and S-wave measurements and logging tools - Tube waves - Seismic imaging - Synthetic seismograms - Field application examples
More informationMacroscopic theory Rock as 'elastic continuum'
Elasticity and Seismic Waves Macroscopic theory Rock as 'elastic continuum' Elastic body is deformed in response to stress Two types of deformation: Change in volume and shape Equations of motion Wave
More informationCase Study: Residual Stress Measurement
Case Study: Residual Stress Measurement Life Prediction/Prognostics 15 Alternating Stress [MPa] 1 5 1 2 service load natural life time with opposite residual stress intact (no residual stress) increased
More informationdoi: /nature09940
LETTER doi:10.1038/nature09940 Spin crossover and iron-rich silicate melt in the Earth s deep mantle Ryuichi Nomura 1,2, Haruka Ozawa 1,3, Shigehiko Tateno 1, Kei Hirose 1,3, John Hernlund 4, Shunsuke
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Standard Solids and Fracture Fluids: Mechanical, Chemical Effects Effective Stress Dilatancy Hardening and Stability Mead, 1925
More informationTextures in experimentally deformed olivine aggregates: the effects of added water and melt.
Textures in experimentally deformed olivine aggregates: the effects of added water and melt. F. Heidelbach 1, a, B. Holtzman 2, b, S. Hier-Majumder 2, c and D. Kohlstedt 2, d 1 Bayerisches Geoinstitut,
More informationPEAT SEISMOLOGY Lecture 2: Continuum mechanics
PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a
More informationStructural Analysis I Chapter 4 - Torsion TORSION
ORSION orsional stress results from the action of torsional or twisting moments acting about the longitudinal axis of a shaft. he effect of the application of a torsional moment, combined with appropriate
More informationComplete set of elastic constants of -quartz at high pressure: A first-principles study
Complete set of elastic constants of -quartz at high pressure: A first-principles study Hajime Kimizuka, 1,2, * Shigenobu Ogata, 1,3 Ju Li, 4 and Yoji Shibutani 1,3 1 Department of Mechanical Engineering,
More informationSupplementary Table 1. Parameters for estimating minimum thermal conductivity in MoS2
Supplementary Table 1. Parameters for estimating minimum thermal conductivity in MoS2 crystal. The three polarizations (TL1 TL2 and TA) are named following the isoenergydecomposition process described
More informationMotivation. Confined acoustics phonons. Modification of phonon lifetimes Antisymmetric Bulk. Symmetric. 10 nm
Motivation Confined acoustics phonons Modification of phonon lifetimes 0 0 Symmetric Antisymmetric Bulk 0 nm A. Balandin et al, PRB 58(998) 544 Effect of native oxide on dispersion relation Heat transport
More information