GG611 Structural Geology Sec1on Steve Martel POST 805
|
|
- Angelina Merritt
- 5 years ago
- Views:
Transcription
1 GG611 Structural Geology Sec1on Steve Martel POST 805 Lecture 5 Mechanics Stress, Strain, and Rheology 11/6/16 GG611 1 Stresses Control How Rock Fractures hkp://hvo.wr.usgs.gov/kilauea/update/images.html 11/6/16 GG
2 Predicted Coulomb Stress Changes Caused by 1992 Landers Earthquake, California From King et al., 1994 (Fig. 11) Coulomb stress change caused by the Landers rupture. The ley-lateral ML=6.5 Big Bear rupture occurred along doked line 3 hr 26 min ayer the Landers main shock. The Coulomb stress increase at the future Big Bear epicenter is bars. hkp://earthquake.usgs.gov/research/modeling/papers/landers.php 11/6/16 GG611 3 I. Stress at a point II. Strain at a point III. Rheology Outline IV. Applica1ons: Displacement and Stress Fields V. Appendix: Eigenvectors, eigenvalues, and principal stresses 11/6/16 GG
3 Stress at a Point Stresses refer to balanced internal "forces (per unit area)". They differ from force vectors, which, if unbalanced, cause accelera1ons "On -in conven1on": The stress component σ ij acts on the plane normal to the i- direc1on and acts in the j- direc1on 1 Normal stresses: i=j 2 Shear stresses: i j 11/6/16 GG611 5 Dimensions of stress: force/unit area Conven1on for stresses Tension is posi1ve Compression is nega1ve Stress at a Point Follows from on-in conven1on Consistent with most mechanics books Counter to most geology books 11/6/16 GG
4 σ ij = σ xx σ yx σ xy σ yy Stress at a Point 2-D 4 components σ xx σ xy σ xz σ ij = σ yx σ yy σ yz σ zx σ zy σ zz 3-D 9 components For rota1onal equilibrium, σ xy = σ yx, σ xz = σ zx, σ yz = σ zy ; stress matrix is symmetric In nature, the state of stress can (and usually does) vary from point to point 11/6/16 GG611 7 Stress at a Point Analogy with vectors The components of a vector vary with the reference frame, even though the vector does not For certain reference frame orienta1ons, some vector/ tensor components are zero The non-zero components are meaningful and illumina1ng in a reference frame where some components are zero 11/6/16 GG
5 Principal Stresses Have magnitudes and orienta1ons Principal stresses act on planes which feel no shear stress Principal stresses are normal stresses Principal stresses act on perpendicular planes owing to symmetry of stress tensor The maximum, intermediate, and minimum principal stresses are usually designated σ 1, σ 2, and σ 3, respec1vely Designated by a single subscript 11/6/16 GG611 9 Principal Stresses F Principal stresses represent the stress state most simply G σ ij = σ σ 2 2-D 2 components H σ σ ij = 0 σ σ 3 3-D 3 components * If σ 1 = σ 2 = σ 3, the state of stress is called isotropic. This occurs beneath a s1ll body of water. 11/6/16 GG
6 Applica1on: Ver1cal Strike-slip Faults 11/6/16 GG Applica1on: Dip-slip Faults 11/6/16 GG
7 Strain at a point: Basic Concepts Normal strain (ε): change in rela1ve line length Shear strain (γ): change in angle between originally perpendicular lines Volumetric strain (Δ): change in rela1ve volume Based on rates of change of displacement Strains are dimensionless 11/6/16 GG Finite strain at a point Chain rule relates difference in ini1al posi1ons (dx and dy) of neighboring points to difference in displacements (dx and dy ) At a point, displacement deriva1ves are constants Matrix rela1ng difference in displacement [du] to difference in ini1al posi1on [dx] contains constants Unit circles (spheres) deform to ellipses (ellipsoids) du x = u x x dx + u x y dy du y = u y x dx + u y y dy u x u y = u x x u y x u x y u y y dx dy [J u ] du [ ] = [ J u ][ dx] 11/6/16 GG
8 Infinitesimal strain at a point In infinitesimal strain, displacement deriva1ves are small rela1ve to 1 The infinitesimal strain matrix contains normal strains (on main diagonal) and shear strains (offdiagonal terms) The infinitesimal strain matrix is symmetric Infinitesimal principal strains are perpendicular ε ij = ε xx ε yx ε xy ε yy u x x = 1 u x 2 y + u y x ε 1 0 ε ij = 0 ε 2 1 u x 2 y + u y x u y y 11/6/16 GG Rheology The rela1onship between the flow or deforma1on of a material and the loads causing the flow or deforma1on Typically relates stress to strain, or stress to strain rate In reality, rheology is a complicated func1ons of pressure, temperature, fluid content, etc. We generally use simple rheologic models 11/6/16 GG
9 Rheology: Viscous (fluid) behavior Shear strain rate is propor1onal to shear stress hkp://manoa.hawaii.edu/graduate/content/slide-lava 11/6/16 GG Rheology: Duc1le (plas1c) behavior No strain un1l stress reaches a cri1cal level hkp:// hkp://hvo.wr.usgs.gov/kilauea/update/images.html 11/6/16 GG hkp://upload.wikimedia.org/wikipedia/commons/8/89/ropy_pahoehoe.jpg 9
10 11/6/16 Rheology: BriKle behavior (fracture) Deforma1on is discon1nuous 11/6/16 GG hkp://upload.wikimedia.org/wikipedia/commons/8/89/ropy_pahoehoe.jpg Rheology: Linear elas1c behavior Deforma1on reverses when stress is relieved Stress and infinitesimal strain are linearly related Principal strains and principal stresses are parallel in isotropic materials [σ] = [C][ε] hkps://thegeosphere.pbworks.com/w/page/ /sumatra hkp:// data/assets/image/0006/3021/seismic_hammer.jpg 11/6/16 GG
11 Applica1ons: Displacement and Stress Fields Displacement and stresses over a region can be obtained by solving boundary value problems Requirements Body geometry Boundary condi1ons (e.g., stress components or displacements ac1ng on a boundary surface) Rheology Governing equa1on(s) for equilibrium and compa1bility General solu1on Specific solu1on that honors boundary condi1ons 11/6/16 GG Geologic Problem: Radia1ng Dikes Shiprock, New Mexico Aerial view showing radial dikes Both images from hkp://en.wikipedia.org/wiki/shiprock#images 11/6/16 GG
12 Displacement Field Around a Pressurized Hole in an Elas1c Plate Geometry and boundary condi>ons Displacement field For u 0 > 0 ( u r u 0 ) = ( a r) 11/6/16 GG Displacement and Stress Fields Around a Pressurized Hole Displacement field Stress field σ 2 σ 1 Foru 0 > 0 :( u r u 0 ) = ( a r) For P < 0 : ( σ 1 P) = a r ( ) 2 ( σ 2 P) = ( a r) 2 11/6/16 GG
13 Dikes Open in Direc1on of Most Tensile Stress, Propagate in Plane Normal to Most Tensile Stress Most tensile stress hkp://hvo.wr.usgs.gov/kilauea/update/images.html 11/6/16 GG Modeled vs. Measured Displacement Fields Around a Fault Model for Fric1onless, 2D Fault in an Elas1c Plate Model Displacement field GPS Displacements, Landers /6/16 GG
14 Modeled vs. Measured Co-seismic Displacement, Landers Earthquake Co-seismic displacement field due to the 1992, Landers EQ" Co-seismic deforma1on can be modeled using elas1c disloca1on theory (based on Massonnet et al., 1993) 11/6/16 GG Stress Fields Around a Fric1onless, 2D Model Fault in an Elas1c Plate vs. Observa1ons Model stress field: Most tensile stress & stress trajectories Tail cracks at end of lek-lateral strike-slip fault Note loca1on and orienta1on of tail cracks 11/6/16 GG
15 Appendix: Eigenvectors, eigenvalues and principal stresses 11/6/16 GG I Stress! vector! (trac1on) τ = lim F / A A 0 Trac1on Vector on a Plane Trac1on vectors can be added as vectors A trac1on vector can be resolved into normal (τ n ) and shear (τ s ) components A normal trac1on (τ n ) acts perpendicular to a plane A shear trac1on (τ s ) acts parallel to a plane Local reference frame n-axis is normal to plane s-axis is parallel to plane 11/6/16 GG
16 Cauchy s Formula Transforms stress state at a point to the trac1on ac1ng on a plane with normal n Transforms normal vector n to the trac1on vector τ τ j = n i σ ij Trac1on component that acts in the j-direc1on Dimensionless weigh1ng factor (cosine between the n- and i- direc1ons;) Expansion (2D) τ x = n x σ xx + n y σ yx τ y = n y σ xy + n y σ yy Stress component that acts on a plane with its normal in the j-direc1on, and that acts in the j-direc1on 11/6/16 GG Cauchy s Formula E Deriva1on Contribu1ons to τ x 1 τ x = w ( 1) σ xx + w ( 2) σ yx Based on a force balance Note that all contribu1ons must act in x-direc1on 2 F x = A n A x A n F x + A x A y A n F x A y 3 τ x = n x σ xx + n y σ yx Similarly 4 τ y = n x σ xy + n y σ yy n x = cosθ nx = cosθ x n y = cosθ ny = cosθ y 11/6/16 GG
17 Principal Stresses III Eigenvectors and eigenvalues A τ has a magnitude and direc1on B τ from Cauchy s formula C τ x τ y τ x τ y σ xx σ xy = τ n x n y = σ xx σ yx σ xy σ yy σ yx σ yy n x n y Let λ = τ = λ n x n y n x n y n is unit normal to plane A n The form of (C ) is [A][X]=λ[X], and [σ] is symmetric 11/6/16 GG Principal Stresses 1 [σ][x]=τ[x] 2 This is an eigenvalue problem (e.g., [A][X]=λ[X]) A [σ] is a stress tensor (represented as a square matrix) B τ is a scalar C [X] is a vector D [X], [σ][x], and τ[x] all point in the same direc1on 3 Solving for τ yields the principal stress magnitudes (Most tensile σ 1, Intermediate σ 2, least tensile σ 3 ) 4 Solving for [X] yields the principal stress direc1ons Principal stresses are normal stresses and mutually perpendicular ([σ] is symmetric) 11/6/16 GG
Topics. GG612 Structural Geology Sec3on Steve Martel POST 805 Lecture 4 Mechanics: Stress and Elas3city Theory
GG612 Structural Geology Sec3on Steve Martel POST 805 smartel@hawaii.edu Lecture 4 Mechanics: Stress and Elas3city Theory 11/6/15 GG611 1 Topics 1. Stress vectors (trac3ons) 2. Stress at a point 3. Cauchy
More informationGG612 Lecture 3. Outline
GG61 Lecture 3 Strain and Stress Should complete infinitesimal strain by adding rota>on. Outline Matrix Opera+ons Strain 1 General concepts Homogeneous strain 3 Matrix representa>ons 4 Squares of line
More information19. Principal Stresses
19. Principal Stresses I Main Topics A Cauchy s formula B Principal stresses (eigenvectors and eigenvalues) C Example 10/24/18 GG303 1 19. Principal Stresses hkp://hvo.wr.usgs.gov/kilauea/update/images.html
More information23. Disloca0ons. 23. Disloca0ons. I Main Topics
I Main Topics A Disloca0ons and other defects in solids B Significance of disloca0ons C Planar disloca0ons D Displacement and stress fields for a screw disloca0on (mode III) 11/10/16 GG303 1 hhp://volcanoes.usgs.gov/imgs/jpg/photoglossary/fissure4_large.jpg
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Surface and body forces Tensors, Mohr circles. Theoretical strength of materials Defects Stress concentrations Griffith failure
More informationGG612 Lecture 3. Outline
GG612 Lecture 3 Strain 11/3/15 GG611 1 Outline Mathema8cal Opera8ons Strain General concepts Homogeneous strain E (strain matri) ε (infinitesimal strain) Principal values and principal direc8ons 11/3/15
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body
More informationLecture Topics. Structural Geology Sec3on Steve Martel POST Introduc3on. 2 Rock structures. 3 Stress and strain 4 Isostacy
GG611 Structural Geology Sec3on Steve Martel POST 805 smartel@hawaii.edu Lecture 1 Philosophy Orienta3on of Lines and Planes in Space 1 1 Introduc3on A. Philosophy Lecture Topics B. Orienta3on of lines
More information14. HOMOGENEOUS FINITE STRAIN
I Main Topics A vectors B vectors C Infinitesimal differences in posi>on D Infinitesimal differences in displacement E Chain rule for a func>on of mul>ple variables F Gradient tensors and matri representa>on
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 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 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 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 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 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 informationPEAT SEISMOLOGY Lecture 12: Earthquake source mechanisms and radiation patterns II
PEAT8002 - SEISMOLOGY Lecture 12: Earthquake source mechanisms and radiation patterns II Nick Rawlinson Research School of Earth Sciences Australian National University Waveform modelling P-wave first-motions
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 informationTensor Visualization. CSC 7443: Scientific Information Visualization
Tensor Visualization Tensor data A tensor is a multivariate quantity Scalar is a tensor of rank zero s = s(x,y,z) Vector is a tensor of rank one v = (v x,v y,v z ) For a symmetric tensor of rank 2, its
More informationMechanics of materials Lecture 4 Strain and deformation
Mechanics of materials Lecture 4 Strain and deformation Reijo Kouhia Tampere University of Technology Department of Mechanical Engineering and Industrial Design Wednesday 3 rd February, 206 of a continuum
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 2 Governing Equations
Chapter Governing Equations Abstract In this chapter fundamental governing equations for propagation of a harmonic disturbance on the surface of an elastic half-space is presented. The elastic media is
More informationStress transformation and Mohr s circle for stresses
Stress transformation and Mohr s circle for stresses 1.1 General State of stress Consider a certain body, subjected to external force. The force F is acting on the surface over an area da of the surface.
More informationUnit IV State of stress in Three Dimensions
Unit IV State of stress in Three Dimensions State of stress in Three Dimensions References Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength
More informationLecture 8. Stress Strain in Multi-dimension
Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]
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 informationGG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS
GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS I Main Topics A Why deal with tensors? B Order of scalars, vectors, and tensors C Linear transformation of scalars and vectors (and tensors) II Why
More informationResolving Stress Components and Earthquake Triggering
Resolving Stress Components and Earthquake Triggering Earthquake Triggering Do certain events make an earthquake more likely to occur? Earthquakes Slow Slip Wastewater Fluids Dams The focus of this presentation
More information! EN! EU! NE! EE.! ij! NN! NU! UE! UN! UU
A-1 Appendix A. Equations for Translating Between Stress Matrices, Fault Parameters, and P-T Axes Coordinate Systems and Rotations We use the same right-handed coordinate system as Andy Michael s program,
More informationSmart Materials, Adaptive Structures, and Intelligent Mechanical Systems
Smart Materials, Adaptive Structures, and Intelligent Mechanical Systems Bishakh Bhattacharya & Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 19 Analysis of an Orthotropic Ply References
More informationRaymond A. Serway Chris Vuille. Chapter Seven. Rota9onal Mo9on and The Law of Gravity
Raymond A. Serway Chris Vuille Chapter Seven Rota9onal Mo9on and The Law of Gravity Rota9onal Mo9on An important part of everyday life Mo9on of the Earth Rota9ng wheels Angular mo9on Expressed in terms
More informationARC 341 Structural Analysis II. Lecture 10: MM1.3 MM1.13
ARC241 Structural Analysis I Lecture 10: MM1.3 MM1.13 MM1.4) Analysis and Design MM1.5) Axial Loading; Normal Stress MM1.6) Shearing Stress MM1.7) Bearing Stress in Connections MM1.9) Method of Problem
More informationCombined Stresses and Mohr s Circle. General Case of Combined Stresses. General Case of Combined Stresses con t. Two-dimensional stress condition
Combined Stresses and Mohr s Circle Material in this lecture was taken from chapter 4 of General Case of Combined Stresses Two-dimensional stress condition General Case of Combined Stresses con t The normal
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 informationRHEOLOGY & LINEAR ELASTICITY
GG303 Lecture 20 10/25/09 1 RHEOLOGY & LINEAR ELASTICITY I Main Topics A Rheology: Macroscopic deformation behavior B Importance of fluids and fractures in deformation C Linear elasticity for homogeneous
More informationTectonics. Lecture 12 Earthquake Faulting GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Tectonics Lecture 12 Earthquake Faulting Plane strain 3 Strain occurs only in a plane. In the third direction strain is zero. 1 ε 2 = 0 3 2 Assumption of plane strain for faulting e.g., reverse fault:
More informationUNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich
UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST
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 informationMechanical Properties of Materials
Mechanical Properties of Materials Strains Material Model Stresses Learning objectives Understand the qualitative and quantitative description of mechanical properties of materials. Learn the logic of
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 informationStrain Transformation equations
Strain Transformation equations R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur-613 401 Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 8 Table of Contents 1. Stress transformation
More informationRHEOLOGY & LINEAR ELASTICITY. B Importance of fluids and fractures in deformation C Linear elasticity for homogeneous isotropic materials
GG303 Lecture 2 0 9/4/01 1 RHEOLOGY & LINEAR ELASTICITY I II Main Topics A Rheology: Macroscopic deformation behavior B Importance of fluids and fractures in deformation C Linear elasticity for homogeneous
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting Lectures & 3, 9/31 Aug 017 www.geosc.psu.edu/courses/geosc508 Discussion of Handin, JGR, 1969 and Chapter 1 Scholz, 00. Stress analysis and Mohr Circles Coulomb Failure
More informationBasic Concepts of Strain and Tilt. Evelyn Roeloffs, USGS June 2008
Basic Concepts of Strain and Tilt Evelyn Roeloffs, USGS June 2008 1 Coordinates Right-handed coordinate system, with positions along the three axes specified by x,y,z. x,y will usually be horizontal, and
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 informationMechanics of Solids (APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY) And As per Revised Syllabus of Leading Universities in India AIR WALK PUBLICATIONS
Advanced Mechanics of Solids (APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY) And As per Revised Syllabus of Leading Universities in India Dr. S. Ramachandran Prof. R. Devaraj Professors School of Mechanical
More informationStress/Strain. Outline. Lecture 1. Stress. Strain. Plane Stress and Plane Strain. Materials. ME EN 372 Andrew Ning
Stress/Strain Lecture 1 ME EN 372 Andrew Ning aning@byu.edu Outline Stress Strain Plane Stress and Plane Strain Materials otes and News [I had leftover time and so was also able to go through Section 3.1
More informationM E 320 Professor John M. Cimbala Lecture 10
M E 320 Professor John M. Cimbala Lecture 10 Today, we will: Finish our example problem rates of motion and deformation of fluid particles Discuss the Reynolds Transport Theorem (RTT) Show how the RTT
More informationApplications of Eigenvalues & Eigenvectors
Applications of Eigenvalues & Eigenvectors Louie L. Yaw Walla Walla University Engineering Department For Linear Algebra Class November 17, 214 Outline 1 The eigenvalue/eigenvector problem 2 Principal
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 informationExercise: concepts from chapter 6
Reading: Fundamentals of Structural Geology, Chapter 6 1) The definition of the traction vector (6.7) relies upon the approximation of rock as a continuum, so the ratio of resultant force to surface area
More informationExercise: concepts from chapter 8
Reading: Fundamentals of Structural Geology, Ch 8 1) The following exercises explore elementary concepts associated with a linear elastic material that is isotropic and homogeneous with respect to elastic
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 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 information6. SCALARS, VECTORS, AND TENSORS (FOR ORTHOGONAL COORDINATE SYSTEMS)
(FOR ORTHOGONAL COORDINATE SYSTEMS) I Main Topics A What are scalars, vectors, and tensors? B Order of scalars, vectors, and tensors C Linear transformaoon of scalars and vectors (and tensors) D Matrix
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 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 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 informationEigen decomposition for 3D stress tensor
Eigen decomposition for 3D stress tensor by Vince Cronin Begun September 13, 2010; last revised September 23, 2015 Unfinished Dra ; Subject to Revision Introduction If we have a cubic free body that is
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 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 informationVYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA
VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA FAKULTA METALURGIE A MATERIÁLOVÉHO INŽENÝRSTVÍ APPLIED MECHANICS Study Support Leo Václavek Ostrava 2015 Title:Applied Mechanics Code: Author: doc. Ing.
More informationGG303 Lecture 15 10/6/09 1 FINITE STRAIN AND INFINITESIMAL STRAIN
GG303 Lecture 5 0609 FINITE STRAIN AND INFINITESIMAL STRAIN I Main Topics on infinitesimal strain A The finite strain tensor [E] B Deformation paths for finite strain C Infinitesimal strain and the infinitesimal
More informationGG303 Lab 12 11/7/18 1
GG303 Lab 12 11/7/18 1 DEFORMATION AROUND A HOLE This lab has two main objectives. The first is to develop insight into the displacement, stress, and strain fields around a hole in a sheet under an approximately
More informationSummary so far. Geological structures Earthquakes and their mechanisms Continuous versus block-like behavior Link with dynamics?
Summary so far Geodetic measurements velocities velocity gradient tensor (spatial derivatives of velocity) Velocity gradient tensor = strain rate (sym.) + rotation rate (antisym.) Strain rate tensor can
More informationDEFORMATION PATTERN IN ELASTIC CRUST
DEFORMATION PATTERN IN ELASTIC CRUST Stress and force in 2D Strain : normal and shear Elastic medium equations Vertical fault in elastic medium => arctangent General elastic dislocation (Okada s formulas)
More informationLecture 3: The Concept of Stress, Generalized Stresses and Equilibrium
Lecture 3: The Concept of Stress, Generalized Stresses and Equilibrium 3.1 Stress Tensor We start with the presentation of simple concepts in one and two dimensions before introducing a general concept
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 informationBrittle Deformation. Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm
Lecture 6 Brittle Deformation Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm WW Norton, unless noted otherwise Brittle deformation EarthStructure (2 nd
More informationEquilibrium of Deformable Body
Equilibrium of Deformable Body Review Static Equilibrium If a body is in static equilibrium under the action applied external forces, the Newton s Second Law provides us six scalar equations of equilibrium
More informationReview of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis
uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods
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 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 informationExercise: concepts from chapter 5
Reading: Fundamentals of Structural Geology, Ch 5 1) Study the oöids depicted in Figure 1a and 1b. Figure 1a Figure 1b Figure 1. Nearly undeformed (1a) and significantly deformed (1b) oöids with spherulitic
More informationLast Time: Finish Ch 5, Start Ch6 Today: Finish Ch 6
Last Time: Finish Ch 5, Start Ch6 Today: Finish Ch 6 Monday Finish Chapter 5 o Circular mo6on o Dynamics of circular mo6on Chapter 6 o Work done by forces o Kine6c energy o Work and KE prelecture Today
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 informationChapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship
Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction
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 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 informationSEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by
SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e
More informationCHAPTER 4 Stress Transformation
CHAPTER 4 Stress Transformation ANALYSIS OF STRESS For this topic, the stresses to be considered are not on the perpendicular and parallel planes only but also on other inclined planes. A P a a b b P z
More informationProperties of the stress tensor
Appendix C Properties of the stress tensor Some of the basic properties of the stress tensor and traction vector are reviewed in the following. C.1 The traction vector Let us assume that the state of stress
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 Thermodynamics Derivation Hooke s Law: Anisotropic Elasticity
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 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 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 informationLecture Notes #10. The "paradox" of finite strain and preferred orientation pure vs. simple shear
12.520 Lecture Notes #10 Finite Strain The "paradox" of finite strain and preferred orientation pure vs. simple shear Suppose: 1) Anisotropy develops as mineral grains grow such that they are preferentially
More informationELASTICITY (MDM 10203)
ELASTICITY () Lecture Module 3: Fundamental Stress and Strain University Tun Hussein Onn Malaysia Normal Stress inconstant stress distribution σ= dp da P = da A dimensional Area of σ and A σ A 3 dimensional
More informationMECE 3321 MECHANICS OF SOLIDS CHAPTER 1
MECE 3321 MECHANICS O SOLIDS CHAPTER 1 Samantha Ramirez, MSE WHAT IS MECHANICS O MATERIALS? Rigid Bodies Statics Dynamics Mechanics Deformable Bodies Solids/Mech. Of Materials luids 1 WHAT IS MECHANICS
More informationFINAL EXAMINATION. (CE130-2 Mechanics of Materials)
UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130- Mechanics of Materials) Problem 1: (15 points) pinned -bar structure is shown in Figure 1. There is an external force, W = 5000N,
More informationTHE CAUSES: MECHANICAL ASPECTS OF DEFORMATION
17 THE CAUSES: MECHANICAL ASPECTS OF DEFORMATION Mechanics deals with the effects of forces on bodies. A solid body subjected to external forces tends to change its position or its displacement or its
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 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 informationYour Advisor Said, Get the GPS Data and Plot it. Jeff Freymueller Geophysical University of Alaska Fairbanks
Your Advisor Said, Get the GPS Data and Plot it Jeff Freymueller Geophysical Ins@tute University of Alaska Fairbanks Actual GPS Data GPS Data are phase and pseudorange data that measure distances to satellites.
More informationTentamen/Examination TMHL61
Avd Hållfasthetslära, IKP, Linköpings Universitet Tentamen/Examination TMHL61 Tentamen i Skademekanik och livslängdsanalys TMHL61 lördagen den 14/10 2000, kl 8-12 Solid Mechanics, IKP, Linköping University
More informationBy drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ.
Mohr s Circle By drawing Mohr s circle, the stress transformation in -D can be done graphically. σ = σ x + σ y τ = σ x σ y + σ x σ y cos θ + τ xy sin θ, 1 sin θ + τ xy cos θ. Note that the angle of rotation,
More informationUnravelling earthquake dynamics with SeisSol: Megathrust ruptures, off- fault plas?city and rough faults Elizabeth H.
Unravelling earthquake dynamics with SeisSol: Megathrust ruptures, off- fault plas?city and rough faults Elizabeth H. Madden (Betsy) Stephanie Wollherr, Thomas Ulrich, Alice- Agnes Gabriel 1 Earthquake
More informationA DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any
Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»
More informationLast Time: Chapter 6 Today: Chapter 7
Last Time: Chapter 6 Today: Chapter 7 Last Time Work done by non- constant forces Work and springs Power Examples Today Poten&al Energy of gravity and springs Forces and poten&al energy func&ons Energy
More informationDEFORMATION PATTERN IN ELASTIC CRUST
DEFORMATION PATTERN IN ELASTIC CRUST Stress and force in 2D Strain : normal and shear Elastic medium equations Vertical fault in elastic medium => arctangent General elastic dislocation (Okada s formulas)
More informationIntroduction to Strain and Borehole Strainmeter Data
Introduction to Strain and Borehole Strainmeter Data Evelyn Roeloffs U. S. Geological Survey Earthquake Science Center March 28, 2016 Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016
More information