Dynamics of the Mantle and Lithosphere ETH Zürich Continuum Mechanics in Geodynamics: Equation cheat sheet
|
|
- Arnold Ward
- 5 years ago
- Views:
Transcription
1 Dynamics of the Mantle and Lithosphere ETH Zürich Continuum Mechanics in Geodynamics: Equation cheat sheet or all equations you will probably ever need Definitions 1. Coordinate system. x,y,z or x 1,x 2,x define points in D space. 2. Field variable. T x,y,z - temperature field - temperature varying in space. Indexed variables. v i,i = 1,2, implies v 1,v 2,v, i.e. three functions. 4. Repeated indices rule also called Einstein summation convention,i = 1,2,,implies i=1 = v Tensor = indexed variable + the rule of transformation to another coordinate system. 6. Useful tensor - Kronecker delta, in D: δ i j = Traction = a force per unit area acting on a plane a vector. 8. Traction sign convention. Compression is negative in mechanics, but positive in geology. 9. Mean stress/strain: σ = σ ii / = trσ/, ε = ε ii / = θ also called dilatation 10. Deviatoric stress/strain: σ = σ i j σδ i j, ε = ε i j εδ i j Stress tensor a matrix, two indexed variables, tensor of rank two: σ i j = σ i j = σ11 σ 12 2D σ 21 σ 22 σ 11 σ 12 σ 1 σ 21 σ 22 σ 2 D σ 1 σ 2 σ 2. Meaning of the elements: each row are components of the traction vectors acting on the coordinate planes. diagonal elements are normal stresses. off diagonal elements are shear stresses.. Special properties: Symmetric for homogeneous material.
2 4. What for? It is a magic tool: if you multiply the stress tensor treated as a simple matrix by a unit vector, n j, which is normal to a certain plane, you will get the traction vector on this plane Cauchy s formula: 5. How do you get it? Usually by solving the equilibrium equations T ni = σ i j n j = 6. NB: The number of equilibrium equations is less than the number of unknown stress tensor components. Strain and strain rate tensors a matrix, two indexed variables, tensor of rank two. 2. Meaning of the elements: diagonal elements are elongation relative changes of length in coordinate axes directions. off diagonal elements are shears deviations from 90 of the angles between lines coinciding with the coordinate axes directions before deformation.. Special properties: symmetric j=1 σ i j n j 4. What for? It is a measure of the deformation. It will be used in the rheological relationships. 5. How do you get it? via velocity/displacements: ε i j = 1 2 = vi + v j j v v2 2 + v v 2 + v v v v 2 12 v1 1 v2 2 2 v 1 6. NB: The number of velocity components is smaller than the number of strain rate components. Rheology, Stress-strain relationships A functional relationship between second rank tensors: incompressible viscous: σ i j = Pδ i j + 2η ε i j elastic: σ i j = λε kk δ i j + 2µε i j Maxwell visco-elastic for deviators: ε i j = σ i j 2G + σ i j 2η 2
3 2. Meaning of the elements: λ,η,µ - are parameters material properties. What for? In the equilibrium equations: first substitute stress via strainrate, than strainrate via displacement velocities, which results in a closed system of equations, meaning that the number of equations is equal to the number of unknowns velocities or displacement. 4. How to find? measure rheology in the lab. 5. NB. There are three major classes of rheologies: Reversible elastic rheology at small stresses and strains. Rate dependent creep - irreversible. Examples are Newtonian viscous or power-law rheology, which is usually thermally activated, pressure independent. Intermediate stress levels. Rate independent instantaneous ultimate yielding at large stresses. Frequently pressure sensitive, temperature independent. Also called: plastic or frictional brittle behavior. General continuum mechanics recipe: How to derive a closed system of equations. 1. Conservation laws Conservation of mass: ρ t + ρv i = 0 Conservation of momentum equilibrium force balance vi ρ where g i is the gravitational acceleration vector. Conservation of energy: j = σ i j j + ρg i E t + v E j + q i = ρq j where E is energy, q i the energy flux vector, and Q an energy source heat production. 2. Thermodynamic relationships Equations of state 1: the caloric equation where c p is heat capacity, and T is temperature. E = c p ρt Equations of state 2: relationships for the isotropic parts of the stress/strain tensors ρ = f T,P where P is pressure note: ρ = ρ 0 ε kk ;P = σ = σ kk /.
4 . Rheological relationships for deviators ε i j = R σ i j, σ i j 4. Energy flux vector vs. temperature gradient q i = k T where k is the thermal conductivity. Summary: The general system of equations for a continuum media in the gravity field. vi ρ E ρ t + ρv i = 0 1 i E x j x j = σ i j j + ρg i 2 + q i = ρq E = c p ρt 4 ρ = f T,P 5 ε i j = R σ i j, σ i j 6 q i = k T 7 where ρ is density, v i velocity, g i gravitational acceleration vector, E energy, q i heat flux vector, Q an energy source heat production, e.g. by radioactive elements, c is heat capacity, T temperature, P pressure and k thermal conductivity. Known functions, tensors and coefficients: g i,c p, f..,ρ 0,R...,k Unknown functions: ρ,v i, σ, σ i j,q i,e,t. The number of unknowns is thus equal to the number of equations. Example: The Stokes system of equations for slowly moving incompressible linear viscous Newtonian continuum materials. ρ 0 c p T T = x j σ i j = ρ 0 g i = 0 9 j k T + ρ 0 Q 10 ε i j = σ i j 2η 11 σ i j = Pδ i j + σ i j 12 Known functions, tensors and coefficients: g i,c p,q,ρ 0,η,k Unknown functions: v i,p, σ i j,σ i j,t. The number of unknowns is thus equal to the number of equations. 4
5 2D version, spelled out Choice of coordinate system and new notation for 2D: σxx σ g i = 0, g,x i = x,z,v i = v x,v z,σ i j = xz σ zx σ zz Note that σ zx = σ xz. The 2D Stokes system of equations the basis for basically every mantle convection/lithospheric deformation code: v x + v z = 0 1 σ xx + σ xz = 0 14 σ xz + σ zz ρg = 0 15 T ρ 0 c p t + v T x + v z σ xx = P + 2η v x σ zz = P + 2η v z vx σ xz = η + v z 2 T = k T 2 T Known: g,q,c p,ρ 0,η,k. Unknown: v x,v z,p,σ xx,σ xz,σ zz,t. Number of equations? ρ 0 Q 19 5
Chapter 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 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 informationDynamics of Glaciers
Dynamics of Glaciers McCarthy Summer School 01 Andy Aschwanden Arctic Region Supercomputing Center University of Alaska Fairbanks, USA June 01 Note: This script is largely based on the Physics of Glaciers
More informationContinuum mechanism: Stress and strain
Continuum mechanics deals with the relation between forces (stress, σ) and deformation (strain, ε), or deformation rate (strain rate, ε). Solid materials, rigid, usually deform elastically, that is the
More informationElements of Continuum Elasticity. David M. Parks Mechanics and Materials II February 25, 2004
Elements of Continuum Elasticity David M. Parks Mechanics and Materials II 2.002 February 25, 2004 Solid Mechanics in 3 Dimensions: stress/equilibrium, strain/displacement, and intro to linear elastic
More informationRock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth
Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth References: Turcotte and Schubert, Geodynamics, Sections 2.1,-2.4, 2.7, 3.1-3.8, 6.1, 6.2, 6.8, 7.1-7.4. Jaeger and Cook, Fundamentals of
More informationNumerical Modelling in Geosciences. Lecture 6 Deformation
Numerical Modelling in Geosciences Lecture 6 Deformation Tensor Second-rank tensor stress ), strain ), strain rate ) Invariants quantities independent of the coordinate system): - First invariant trace:!!
More information1 Exercise: Linear, incompressible Stokes flow with FE
Figure 1: Pressure and velocity solution for a sinking, fluid slab impinging on viscosity contrast problem. 1 Exercise: Linear, incompressible Stokes flow with FE Reading Hughes (2000), sec. 4.2-4.4 Dabrowski
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationLecture 8: Tissue Mechanics
Computational Biology Group (CoBi), D-BSSE, ETHZ Lecture 8: Tissue Mechanics Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2015/16 7. Mai 2016 2 / 57 Contents 1 Introduction to Elastic Materials
More informationRheology of Soft Materials. Rheology
Τ Thomas G. Mason Department of Chemistry and Biochemistry Department of Physics and Astronomy California NanoSystems Institute Τ γ 26 by Thomas G. Mason All rights reserved. γ (t) τ (t) γ τ Δt 2π t γ
More informationGetting started: CFD notation
PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =
More informationContinuum Mechanics. Continuum Mechanics and Constitutive Equations
Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform
More informationChapter 2: Fluid Dynamics Review
7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading
More informationBasic Equations of Elasticity
A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ
More informationDETAILS ABOUT THE TECHNIQUE. We use a global mantle convection model (Bunge et al., 1997) in conjunction with a
DETAILS ABOUT THE TECHNIQUE We use a global mantle convection model (Bunge et al., 1997) in conjunction with a global model of the lithosphere (Kong and Bird, 1995) to compute plate motions consistent
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 informationMathematical Background
CHAPTER ONE Mathematical Background This book assumes a background in the fundamentals of solid mechanics and the mechanical behavior of materials, including elasticity, plasticity, and friction. A previous
More informationCH.9. CONSTITUTIVE EQUATIONS IN FLUIDS. Multimedia Course on Continuum Mechanics
CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS Multimedia Course on Continuum Mechanics Overview Introduction Fluid Mechanics What is a Fluid? Pressure and Pascal s Law Constitutive Equations in Fluids Fluid Models
More informationSubduction II Fundamentals of Mantle Dynamics
Subduction II Fundamentals of Mantle Dynamics Thorsten W Becker University of Southern California Short course at Universita di Roma TRE April 18 20, 2011 Rheology Elasticity vs. viscous deformation η
More information12. Stresses and Strains
12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)
More informationSEISMOLOGY I. Laurea Magistralis in Physics of the Earth and of the Environment. Elasticity. Fabio ROMANELLI
SEISMOLOGY I Laurea Magistralis in Physics of the Earth and of the Environment Elasticity Fabio ROMANELLI Dept. Earth Sciences Università degli studi di Trieste romanel@dst.units.it 1 Elasticity and Seismic
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139
MASSACHUSETTS NSTTUTE OF TECHNOLOGY DEPARTMENT OF MATERALS SCENCE AND ENGNEERNG CAMBRDGE, MASSACHUSETTS 39 3. MECHANCAL PROPERTES OF MATERALS PROBLEM SET SOLUTONS Reading Ashby, M.F., 98, Tensors: Notes
More informationThe Navier-Stokes Equations
s University of New Hampshire February 22, 202 and equations describe the non-relativistic time evolution of mass and momentum in fluid substances. mass density field: ρ = ρ(t, x, y, z) velocity field:
More informationCONSERVATION OF ENERGY FOR ACONTINUUM
Chapter 6 CONSERVATION OF ENERGY FOR ACONTINUUM Figure 6.1: 6.1 Conservation of Energ In order to define conservation of energ, we will follow a derivation similar to those in previous chapters, using
More informationStress equilibrium in southern California from Maxwell stress function models fit to both earthquake data and a quasi-static dynamic simulation
Stress equilibrium in southern California from Maxwell stress function models fit to both earthquake data and a quasi-static dynamic simulation Peter Bird Dept. of Earth, Planetary, and Space Sciences
More informationq v = - K h = kg/ν units of velocity Darcy's Law: K = kρg/µ HYDRAULIC CONDUCTIVITY, K Proportionality constant in Darcy's Law
Darcy's Law: q v - K h HYDRAULIC CONDUCTIVITY, K m/s K kρg/µ kg/ν units of velocity Proportionality constant in Darcy's Law Property of both fluid and medium see D&S, p. 62 HYDRAULIC POTENTIAL (Φ): Φ g
More information20. Rheology & Linear Elasticity
I Main Topics A Rheology: Macroscopic deformation behavior B Linear elasticity for homogeneous isotropic materials 10/29/18 GG303 1 Viscous (fluid) Behavior http://manoa.hawaii.edu/graduate/content/slide-lava
More informationChapter 1 Fluid Characteristics
Chapter 1 Fluid Characteristics 1.1 Introduction 1.1.1 Phases Solid increasing increasing spacing and intermolecular liquid latitude of cohesive Fluid gas (vapor) molecular force plasma motion 1.1.2 Fluidity
More information3D Elasticity Theory
3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.
More informationTensor Transformations and the Maximum Shear Stress. (Draft 1, 1/28/07)
Tensor Transformations and the Maximum Shear Stress (Draft 1, 1/28/07) Introduction The order of a tensor is the number of subscripts it has. For each subscript it is multiplied by a direction cosine array
More informationNumerical Heat and Mass Transfer
Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 15-Convective Heat Transfer Fausto Arpino f.arpino@unicas.it Introduction In conduction problems the convection entered the analysis
More information- Marine Hydrodynamics. Lecture 4. Knowns Equations # Unknowns # (conservation of mass) (conservation of momentum)
2.20 - Marine Hydrodynamics, Spring 2005 Lecture 4 2.20 - Marine Hydrodynamics Lecture 4 Introduction Governing Equations so far: Knowns Equations # Unknowns # density ρ( x, t) Continuity 1 velocities
More informationChapter 9: Differential Analysis of Fluid Flow
of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known
More informationPhysics of Continuous media
Physics of Continuous media Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2012 October 26, 2012 Deformations of continuous media If a body is deformed, we say that the point which originally had
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 informationGame Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit
More informationAE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.
AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says
More information1 Stress and Strain. Introduction
1 Stress and Strain Introduction This book is concerned with the mechanical behavior of materials. The term mechanical behavior refers to the response of materials to forces. Under load, a material may
More informationChapter 9: Differential Analysis
9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationIntroduction to Fluid Mechanics
Introduction to Fluid Mechanics Tien-Tsan Shieh April 16, 2009 What is a Fluid? The key distinction between a fluid and a solid lies in the mode of resistance to change of shape. The fluid, unlike the
More informationUseful Formulae ( )
Appendix A Useful Formulae (985-989-993-) 34 Jeremić et al. A.. CHAPTER SUMMARY AND HIGHLIGHTS page: 35 of 536 A. Chapter Summary and Highlights A. Stress and Strain This section reviews small deformation
More informationMHA042 - Material mechanics: Duggafrågor
MHA042 - Material mechanics: Duggafrågor 1) For a static uniaxial bar problem at isothermal (Θ const.) conditions, state principle of energy conservation (first law of thermodynamics). On the basis of
More informationMECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso
MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium
More informationGlacier Dynamics. Glaciers 617. Andy Aschwanden. Geophysical Institute University of Alaska Fairbanks, USA. October 2011
Glacier Dynamics Glaciers 617 Andy Aschwanden Geophysical Institute University of Alaska Fairbanks, USA October 2011 1 / 81 The tradition of glacier studies that we inherit draws upon two great legacies
More informationLecture 3: 1. Lecture 3.
Lecture 3: 1 Lecture 3. Lecture 3: 2 Plan for today Summary of the key points of the last lecture. Review of vector and tensor products : the dot product (or inner product ) and the cross product (or vector
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 informationMechanics PhD Preliminary Spring 2017
Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n
More informationExercise solutions: concepts from chapter 10
Reading: Fundamentals of Structural Geology, Ch 0 ) The flow of magma with a viscosity as great as 0 0 Pa s, let alone that of rock with a viscosity of 0 0 Pa s, is difficult to comprehend because our
More information1 Hooke s law, stiffness, and compliance
Non-quilibrium Continuum Physics TA session #5 TA: Yohai Bar Sinai 3.04.206 Linear elasticity I This TA session is the first of three at least, maybe more) in which we ll dive deep deep into linear elasticity
More informationChapter 2: Basic Governing Equations
-1 Reynolds Transport Theorem (RTT) - Continuity Equation -3 The Linear Momentum Equation -4 The First Law of Thermodynamics -5 General Equation in Conservative Form -6 General Equation in Non-Conservative
More informationComputational Astrophysics
Computational Astrophysics Lecture 1: Introduction to numerical methods Lecture 2:The SPH formulation Lecture 3: Construction of SPH smoothing functions Lecture 4: SPH for general dynamic flow Lecture
More informationConstitutive Equations
Constitutive quations David Roylance Department of Materials Science and ngineering Massachusetts Institute of Technology Cambridge, MA 0239 October 4, 2000 Introduction The modules on kinematics (Module
More informationSurface force on a volume element.
STRESS and STRAIN Reading: Section. of Stein and Wysession. In this section, we will see how Newton s second law and Generalized Hooke s law can be used to characterize the response of continuous medium
More informationRheology: What is it?
Schedule Rheology basics Viscous, elastic and plastic Creep processes Flow laws Yielding mechanisms Deformation maps Yield strength envelopes Constraints on the rheology from the laboratory, geology, geophysics
More informationDifferential relations for fluid flow
Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow
More informationINDEX 363. Cartesian coordinates 19,20,42, 67, 83 Cartesian tensors 84, 87, 226
INDEX 363 A Absolute differentiation 120 Absolute scalar field 43 Absolute tensor 45,46,47,48 Acceleration 121, 190, 192 Action integral 198 Addition of systems 6, 51 Addition of tensors 6, 51 Adherence
More informationStress, Strain, and Viscosity. San Andreas Fault Palmdale
Stress, Strain, and Viscosity San Andreas Fault Palmdale Solids and Liquids Solid Behavior: Liquid Behavior: - elastic - fluid - rebound - no rebound - retain original shape - shape changes - small deformations
More informationNavier-Stokes Equation: Principle of Conservation of Momentum
Navier-tokes Equation: Principle of Conservation of Momentum R. hankar ubramanian Department of Chemical and Biomolecular Engineering Clarkson University Newton formulated the principle of conservation
More informationIntroduction to Seismology Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Stress and Strain
More informationChapter 5. The Differential Forms of the Fundamental Laws
Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations
More informationSoft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies
Soft-Body Physics Soft Bodies Realistic objects are not purely rigid. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Deformed
More informationNotes 4: Differential Form of the Conservation Equations
Low Speed Aerodynamics Notes 4: Differential Form of the Conservation Equations Deriving Conservation Equations From the Laws of Physics Physical Laws Fluids, being matter, must obey the laws of Physics.
More information1 2D Stokes equations on a staggered grid using primitive variables
Figure 1: Staggered grid definition. Properties such as viscosity and density inside a control volume (gray) are assumed to be constant. Moreover, a constant grid spacing in x and -direction is assumed.
More informationLarge time-step numerical modelling of the flow of Maxwell materials
Large time-step numerical modelling of the flow of Maxwell materials R.C. Bailey Geology and Physics Depts., University of Toronto, Toronto, Ontario M5S 1A7, Canada August 1, 2003 Maxwell viscoelastic
More informationDynamics of Ice Sheets and Glaciers
Dynamics of Ice Sheets and Glaciers Ralf Greve Institute of Low Temperature Science Hokkaido University Lecture Notes Sapporo 2004/2005 Literature Ice dynamics Paterson, W. S. B. 1994. The Physics of
More informationStress, Strain, Mohr s Circle
Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected
More informationCHAPTER 8 ENTROPY GENERATION AND TRANSPORT
CHAPTER 8 ENTROPY GENERATION AND TRANSPORT 8.1 CONVECTIVE FORM OF THE GIBBS EQUATION In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?
More informationMechanics of solids and fluids -Introduction to continuum mechanics
Mechanics of solids and fluids -Introduction to continuum mechanics by Magnus Ekh August 12, 2016 Introduction to continuum mechanics 1 Tensors............................. 3 1.1 Index notation 1.2 Vectors
More informationChapter 1. Governing Equations of GFD. 1.1 Mass continuity
Chapter 1 Governing Equations of GFD The fluid dynamical governing equations consist of an equation for mass continuity, one for the momentum budget, and one or more additional equations to account for
More informationLoading σ Stress. Strain
hapter 2 Material Non-linearity In this chapter an overview of material non-linearity with regard to solid mechanics is presented. Initially, a general description of the constitutive relationships associated
More informationFinite Element Method in Geotechnical Engineering
Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps
More informationRheology III. Ideal materials Laboratory tests Power-law creep The strength of the lithosphere The role of micromechanical defects in power-law creep
Rheology III Ideal materials Laboratory tests Power-law creep The strength of the lithosphere The role of micromechanical defects in power-law creep Ideal materials fall into one of the following categories:
More informationChapter 6: Momentum Analysis
6-1 Introduction 6-2Newton s Law and Conservation of Momentum 6-3 Choosing a Control Volume 6-4 Forces Acting on a Control Volume 6-5Linear Momentum Equation 6-6 Angular Momentum 6-7 The Second Law of
More informationGeodynamics Lecture 5 Basics of elasticity
Geodynamics Lecture 5 Basics of elasticity Lecturer: David Whipp david.whipp@helsinki.fi! 16.9.2014 Geodynamics www.helsinki.fi/yliopisto 1 Goals of this lecture Introduce linear elasticity! Look at the
More informationContents. I Introduction 1. Preface. xiii
Contents Preface xiii I Introduction 1 1 Continuous matter 3 1.1 Molecules................................ 4 1.2 The continuum approximation.................... 6 1.3 Newtonian mechanics.........................
More informationContinuum Mechanics and Theory of Materials
Peter Haupt Continuum Mechanics and Theory of Materials Translated from German by Joan A. Kurth Second Edition With 91 Figures, Springer Contents Introduction 1 1 Kinematics 7 1. 1 Material Bodies / 7
More informationContinuum Mechanics Fundamentals
Continuum Mechanics Fundamentals James R. Rice, notes for ES 220, 12 November 2009; corrections 9 December 2009 Conserved Quantities Let a conseved quantity have amount F per unit volume. Examples are
More informationOCR Physics Specification A - H156/H556
OCR Physics Specification A - H156/H556 Module 3: Forces and Motion You should be able to demonstrate and show your understanding of: 3.1 Motion Displacement, instantaneous speed, average speed, velocity
More informationOther state variables include the temperature, θ, and the entropy, S, which are defined below.
Chapter 3 Thermodynamics In order to complete the formulation we need to express the stress tensor T and the heat-flux vector q in terms of other variables. These expressions are known as constitutive
More informationThe finite difference code (fully staggered grid) includes a marker-in-cell Lagrangian marker
GSA Data Repository 2018289 Ruh et al., 2018, Shale-related minibasins atop a massive olistostrome in an active accretionary wedge setting: Two-dimensional numerical modeling applied to the Iranian Makran:
More information2. Conservation Equations for Turbulent Flows
2. Conservation Equations for Turbulent Flows Coverage of this section: Review of Tensor Notation Review of Navier-Stokes Equations for Incompressible and Compressible Flows Reynolds & Favre Averaging
More informationAdvanced Course in Theoretical Glaciology
Advanced Course in Theoretical Glaciology Ralf Greve Institute of Low Temperature Science Hokkaido University Lecture Notes Sapporo 2015 These lecture notes are largely based on the textbook Greve, R.
More informations ij = 2ηD ij σ 11 +σ 22 +σ 33 σ 22 +σ 33 +σ 11
Plasticity http://imechanicaorg/node/17162 Z Suo NONLINEAR VISCOSITY Linear isotropic incompressible viscous fluid A fluid deforms in a homogeneous state with stress σ ij and rate of deformation The mean
More informationMAE4700/5700 Finite Element Analysis for Mechanical and Aerospace Design
MAE4700/5700 Finite Element Analsis for Mechanical and Aerospace Design Cornell Universit, Fall 2009 Nicholas Zabaras Materials Process Design and Control Laborator Sible School of Mechanical and Aerospace
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 information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More informationThermal-Mechanical Behavior of Oceanic Transform Faults
Presented at the COMSOL Conference 2008 Boston Thermal-Mechanical Behavior of Oceanic Transform Faults COMSOL Conference - Boston, Massachusetts October 2008 Emily C. Roland - MIT/WHOI Joint Program Mark
More informationPROCESS SYSTEMS ENGINEERING Dr.-Ing. Richard Hanke-Rauschenbach
Otto-von-Guerice University Magdeburg PROCESS SYSTEMS ENGINEERING Dr.-Ing. Richard Hane-Rauschenbach Project wor No. 1, Winter term 2011/2012 Sample Solution Delivery of the project handout: Wednesday,
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 informationContinuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms
Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive
More informationEKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009)
EKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009) Dr. Mohamad Hekarl Uzir-chhekarl@eng.usm.my School of Chemical Engineering Engineering Campus, Universiti
More information202 Index. failure, 26 field equation, 122 force, 1
Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic
More informationCourse Syllabus: Continuum Mechanics - ME 212A
Course Syllabus: Continuum Mechanics - ME 212A Division Course Number Course Title Academic Semester Physical Science and Engineering Division ME 212A Continuum Mechanics Fall Academic Year 2017/2018 Semester
More informationRobust characteristics method for modelling multiphase visco-elasto-plastic thermo-mechanical problems
Physics of the Earth and Planetary Interiors 163 (2007) 83 105 Robust characteristics method for modelling multiphase visco-elasto-plastic thermo-mechanical problems Taras V. Gerya a David A. Yuen b a
More informationIntroduction to Geology Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 12.001 Introduction to Geology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. RHEOLOGICAL MODELS Rheology
More informationBasic Theorems in Dynamic Elasticity
Basic Theorems in Dynamic Elasticity 1. Stress-Strain relationships 2. Equation of motion 3. Uniqueness and reciprocity theorems 4. Elastodynamic Green s function 5. Representation theorems Víctor M. CRUZ-ATIENZA
More informationRheology and the Lithosphere
Rheology and the Lithosphere Processes in Structural Geology & Tectonics Ben van der Pluijm WW Norton+Authors, unless noted otherwise 3/8/2017 16:51 We Discuss Rheology and the Lithosphere What is rheology?
More informationChapter 3 Stress, Strain, Virtual Power and Conservation Principles
Chapter 3 Stress, Strain, irtual Power and Conservation Principles 1 Introduction Stress and strain are key concepts in the analytical characterization of the mechanical state of a solid body. While stress
More information