Force and Stress Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm © WW Norton, unless noted otherwise.

Slides:



Advertisements
Similar presentations
Earthquake Seismology: The stress tensor Equation of motion
Advertisements

III. Strain and Stress Strain Stress Rheology Reading Suppe, Chapter 3 Twiss&Moores, chapter 15.
Micromechanics Macromechanics Fibers Lamina Laminate Structure Matrix.
Mohr Circle for stress In 2D space (e.g., on the s1s2 , s1s3, or s2s3 plane), the normal stress (sn) and the shear stress (ss), could be given by equations.
Mechanics of Materials – MAE 243 (Section 002) Spring 2008
Read Chapter 1 Basic Elasticity - Equilibrium Equations
Chapter Outline Shigley’s Mechanical Engineering Design.
CONTINUUM MECHANICS (STATE OF STRESS)
III. Strain and Stress Basics of continuum mechanics, Strain Basics of continuum mechanics, Stress Reading Suppe, Chapter 3 Twiss&Moores, chapter 15 Additional.
Chapter 3 Rock Mechanics Stress
B. NOWAK, Stress – Principles…., CASA Seminar, 8 th March 2006 Stress and Stress Principles CASA Seminar.
Geology 3120 Powerpoint notes available online at:
Stress: Force per unit area across an arbitrary plane.
The stresses that cause deformation
Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros.
An Introduction to Stress and Strain
1 MAE 5130: VISCOUS FLOWS Momentum Equation: The Navier-Stokes Equations, Part 1 September 7, 2010 Mechanical and Aerospace Engineering Department Florida.
Announcements Next week lab: 1-3 PM Mon. and Tues. with Andrew McCarthy. Please start on lab before class and come prepared with specific questions Cottonwood.
Joints and Shear Fractures
Mechanics of Rigid Bodies
Deformation and Strain
Expectations after today’s lecture
Stress, Strain, and Viscosity San Andreas Fault Palmdale.
Chapter 2: Load, Stress and Strain
Stress II Cauchy formula Consider a small cubic element of rock extracted from the earth, and imagine a plane boundary with an outward normal, n, and an.
Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius.
IV. Basics of continuum mechanics, Stress Reading Suppe, Chapter 3 Twiss&Moores, chapter 15 Additional References : Jean Salençon, Handbook of continuum.
Give the expression for the velocity of an object rolling down an incline without slipping in terms of h (height), M(mass), g, I (Moment of inertia) and.
Theories of Stress and Strain
Mechanics 105 Kinematics – answers the question “how?” Statics and dynamics answer the question “why?” Force Newton’s 1 st law (object at rest/motion stays.
Force and Stress I. Fundamental Quantities & Units of Rocks Mass: Dimension: [M]Unit: g or kg Length: Dimension: [L] Unit: cm or m Time: Dimension: [T]
Stress II. Stress as a Vector - Traction Force has variable magnitudes in different directions (i.e., it’s a vector) Area has constant magnitude with.
Transformations of Stress and Strain
APPLICATIONS/ MOHR’S CIRCLE
III. Strain and Stress Strain Stress Rheology Reading Suppe, Chapter 3
1 MAE 5130: VISCOUS FLOWS Momentum Equation: The Navier-Stokes Equations, Part 2 September 9, 2010 Mechanical and Aerospace Engineering Department Florida.
FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS.
This week: PS #1 assigned (due next Tuesday) Lab #1 due tomorrow at 11:00 Lectures: ch. 3- Force and Stress Lab #2: Field trip to Mt Baldy- Will collect.
This is the trace of the strain tensor. In general the trace of the strain tensor gives area change in 2-D and volume change in 3-D The principal axes.
Introduction to Seismology
Definitions ! Forces and Motion. Acceleration ! Acceleration is the rate of change of velocity, with respect to time. Velocity ! Velocity is the rate.
Elasticity I Ali K. Abdel-Fattah. Elasticity In physics, elasticity is a physical property of materials which return to their original shape after they.
Chapter 3 Force and Stress. In geology, the force and stress have very specific meaning. Force (F): the mass times acceleration (ma) (Newton’s second.
NB: Uniaxial strain is a type a non-rotational transformation Uniaxial strain.
Stress and Strain ( , 3.14) MAE 316 – Strength of Mechanical Components NC State University Department of Mechanical & Aerospace Engineering Stress.
The stresses that cause deformation
Today: Back to stress stress in the earth Strain Measuring Strain Strain Examples Today: Back to stress stress in the earth Strain Measuring Strain Strain.
Outline Force, vectors Units Normal, shear components Pressure
Transformations of Stress and Strain
Strain I. Recall: Pressure and Volume Change The 3D stresses are equal in magnitude in all directions (as radii of a sphere) The magnitude is equal to.
Lecture TWO Lecture TWO Definition, Limits and Agents of Metamorphism.
1 Structural Geology Force and Stress - Mohr Diagrams, Mean and Deviatoric Stress, and the Stress Tensor Lecture 6 – Spring 2016.
Force and Stress – Normal and Shear Stress Lecture 5 – Spring 2016
Mohr-Coulomb failure Goal: To understand relationship between stress, brittle failure, and frictional faulting and to use this relationship to predict.
Transformation methods - Examples
STRESS SUMMARY Stress= amount of force per unit area (units Pa) Force= mass * acceleration (units N)
Continuum Mechanics (MTH487)
A graphical method of constructing the shear and normal stress tractions on any plane given two principal stresses. This only works in 2-D. Equations.
Mohr Circle In 2D space (e.g., on the s1s2 , s1s3, or s2s3 plane), the normal stress (sn) and the shear stress (ss), could be given by equations (1) and.
Stress = Force/Area Force is measured in units of mass*acceleration 1 N (Newton) = 1 kg * m * s-2 another common unit for force is the pound.
Introduction to Seismology
Lecture 16 Newton Mechanics Inertial properties,Generalized Coordinates Ruzena Bajcsy EE
Transformations of Stress and Strain
Continuum Mechanics (MTH487)
Dynamics The Study of Force.
Chapter 4 Laws of Motion.
Chapter 3 Force and Stress
The Mohr Stress Diagram
Concepts of stress and strain
Presentation transcript:

Force and Stress Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm © WW Norton, unless noted otherwise

© EarthStructure (2 nd ed)2 5/28/2016 Mechanics (a) Newtonian mechanics: displacements between bodies 1 st Law: No force on object means constant velocity (inertia law) 2 nd Law: F = m.a 3 rd Law: F = -F (b) Continuum mechanics: displacement between and WITHIN bodies

© EarthStructure (2 nd ed)3 5/28/2016 Units and conversions Stress = Force/Area = (m. a)/Area = kg.m.s -2.m -2 = N.m -2 = Pa (Pascal) 1E5 = 0.1MPa = 1bar ; 1kbar = 100 MPa

© EarthStructure (2 nd ed)4 5/28/2016 Stress on a plane: Tractions Stress on a two- dimensional plane is defined by a stress acting perpendicular to plane (normal stress) and a stress acting along plane (shear stress).

© EarthStructure (2 nd ed)5 5/28/2016 Relationship between Force and Stress F = . area (c) normalized values of Fn and σn on plane with angle θ; (d) normalized values of Fs and σs on a plane with angle θ.

© EarthStructure (2 nd ed)6 5/28/2016 3D stress 3 rd Law: six independent components:

© EarthStructure (2 nd ed)7 5/28/2016 Infinitessimal stress and stress ellipsoid (a) two-dimensions (stress ellipse); (b) three dimensions (stress ellipsoid) Principal stresses:  ≥  ≥ 

© EarthStructure (2 nd ed)8 5/28/2016 Interactive stress module

© EarthStructure (2 nd ed)9 5/28/2016 Normal and shear stress relationships

© EarthStructure (2 nd ed)10 5/28/2016 Deriving normal and shear stress F = . area

© EarthStructure (2 nd ed)11 5/28/2016 Mohr diagram for stress Rearrange: Circle with radius r, centered on x-axis at distance a from origin Radius is 1/2(  1-  3) =  s, or half the differential stress

© EarthStructure (2 nd ed)12 5/28/2016 Planes in stress space For each value of the shear stress and the normal stress there are two corresponding planes, as shown in the Mohr diagram (a). The corresponding planes in σ1 – σ3 space are shown in (b).

© EarthStructure (2 nd ed)13 5/28/2016 Homework To estimate the normal and shear stresses on the six planes shown in (a) apply the Mohr construction in the graph (b). The principal stresses and angles θ are given. You should check your estimates from the construction by using the derived Equations for σn and σs:

© EarthStructure (2 nd ed)14 5/28/2016 Stress states

© EarthStructure (2 nd ed)15 5/28/2016 Isotropic and non-isotropic stress (a) volume change and (b) shape change, reflecting mean stress and deviatoric stresses, respectively:

© EarthStructure (2 nd ed)16 5/28/2016 Stress tensor The transformation of point P defined by coordinates P(x, y, z) to point P′(x′, y′, z′). We describe the transformation of the three coordinates of P as a function of P′ by The tensor that describes the transformation from P to P′ is the matrix: In matrix notation, the nine components of a stress tensor are: with σ 11 oriented parallel to the 1-axis and acting on a plane perpendicular to the 1-axis, σ 12 oriented parallel to the 1-axis and acting on a plane perpendicular to the 2-axis, and so on. Mean stress and deviatoric stress:

© EarthStructure (2 nd ed)17 5/28/2016 Stress measurement Fossen (2010)

© EarthStructure (2 nd ed)18 5/28/2016 World Stress Map (2008) Heidbach, O., Tingay, M., Barth, A., Reinecker, J., Kurfeß, D., and Müller, B., The World Stress Map database release 2008 doi: /GFZ.WSM.Rel2008, 2008:

World Stress Map © EarthStructure (2 nd ed)19 5/28/2016 Fossen (2010)

© EarthStructure (2 nd ed)20 5/28/2016 Global stress fields

© EarthStructure (2 nd ed)21 5/28/2016 Stress at depth P(litho) = ρ ⋅ g ⋅ h If ρ (density is 2700 kg/m3, g (gravity) is 9.8 m/s2, and h (depth) is 3000 m: P(litho) = 2700 ⋅ 9.8 ⋅ 3000 = 79.4 ⋅ 106 Pa ≈ 80 MPa (or 800 bars) For every kilometer in Earth’s crust, lithostatic pressure increases by approximately 30 MPa. Differential stress increases to a few hundred MPa until Brittle-Plastic transition

© EarthStructure (2 nd ed)22 5/28/2016 Lithospheric stress a) Cold lithosphere (cratons; Precambrian rocks) b) Hot lithosphere (orogenic belts, ocean floor; Cenozoic rocks)  1 -  3 is differential stress)

© EarthStructure (2 nd ed)23 5/28/2016 StressMohr

Extra © EarthStructure (2 nd ed)24 5/28/2016

© EarthStructure (2 nd ed)25 5/28/2016 Stress trajectories