Stress: Force per unit area across an arbitrary plane.

Slides:



Advertisements
Similar presentations
Surface Waves and Free Oscillations
Advertisements

The Asymptotic Ray Theory
Earthquake Seismology: The stress tensor Equation of motion
Types, characteristics, properties
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.
Distribution of Microcracks in Rocks Uniform As in igneous rocks where microcrack density is not related to local structures but rather to a pervasive.
Physical processes within Earth’s interior Topics 1.Seismology and Earth structure 2.Plate kinematics and geodesy 3.Gravity 4.Heat flow 5.Geomagnetism.
Seismic Wave Demonstrations and Animations L. Braile, Purdue University  Copyright L. Braile. Permission.
Seismic Wave Demonstrations and Animations L. Braile, Purdue University  Copyright L. Braile. Permission.
Chapter 3 Rock Mechanics Stress
Chapter 16 Wave Motion.
A disturbance that propagates Examples Waves on the surface of water
Stress: Force per unit area across an arbitrary plane.
The stresses that cause deformation
Wave Type (and names) Particle MotionOther Characteristics P, Compressional, Primary, Longitudinal Dilatational Alternating compressions (“pushes”) and.
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.
1 MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 2: FLUID STATICS Instructor: Professor C. T. HSU.
Stress, Strain, and Viscosity San Andreas Fault Palmdale.
4.4.1 Wave pulse: a wave pulse is a short wave with no repeated oscillations Progressive wave: a wave that moves through a medium transferring energy as.
Geology 5640/6640 Introduction to Seismology 18 Feb 2015 © A.R. Lowry 2015 Last time: Spherical Coordinates; Ray Theory Spherical coordinates express vector.
MODERN GLOBAL SEISMOLOGY BODY WAVES AND RAY THEORY-2.
Force on Floating bodies:
GG 450 March 19, 2008 Stress and Strain Elastic Constants.
Basic dynamics  The equations of motion and continuity Scaling Hydrostatic relation Boussinesq approximation  Geostrophic balance in ocean’s interior.
Kinematic Representation Theorem KINEMATIC TRACTIONS Time domain representation Frequency domain representation Green Function.
Unit 12, Presentation 2. Simple Pendulum  The simple pendulum is another example of simple harmonic motion  The force is the component of the weight.
CE 1501 CE 150 Fluid Mechanics G.A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University,
Energy momentum tensor of macroscopic bodies Section 35.
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.
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.
Copyright © 2009 Pearson Education, Inc. Lecture 1 – Waves & Sound b) Wave Motion & Properties.
Hooke’s Law F s = - k x F s is the spring force k is the spring constant It is a measure of the stiffness of the spring A large k indicates a stiff spring.
Quiz answers 1: Compositional descriptions b)Mantle c)Crust 2: The mantle is made up primarily of: c)Olivine and Silicate minerals 3: The thickness of.
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.
GG450 April 1, 2008 Huygen’s Principle and Snell’s Law.
Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Earthquake Engineering GE / CEE - 479/679 Topic 11. Wave Propagation 1 John G. Anderson Professor.
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.
Chapter 16 Waves-I Types of Waves 1.Mechanical waves. These waves have two central features: They are governed by Newton’s laws, and they can exist.
Chapter 16 Waves-I Types of Waves 1.Mechanical waves. These waves have two central features: They are governed by Newton’s laws, and they can exist.
Seismology Part V: Surface Waves: Rayleigh John William Strutt (Lord Rayleigh)
Basic dynamics ●The equations of motion and continuity Scaling Hydrostatic relation Boussinesq approximation ●Geostrophic balance in ocean’s interior.
Basic dynamics The equation of motion Scale Analysis
The stresses that cause deformation
1 Linear Wave Equation The maximum values of the transverse speed and transverse acceleration are v y, max =  A a y, max =  2 A The transverse speed.
Outline Force, vectors Units Normal, shear components Pressure
Seismology Part II: Body Waves and Ray Theory. Some definitions: Body Waves: Waves that propagrate through the "body" of a medium (in 3 dimensions) WRONG!
Structure of Earth as imaged by seismic waves
Chapter 14 Vibrations and Waves. Hooke’s Law F s = - k x F s is the spring force k is the spring constant It is a measure of the stiffness of the spring.
Seismic waves Mathilde B. Sørensen and Jens Havskov.
Introduction to Seismology
Lecture 11 WAVE.
Continuum Mechanics (MTH487)
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
Introduction to Seismology
Let s be the 3x3 stress matrix and let n be a 3x1 unit vector
P-SV Waves and Solution to Elastic Wave Equation for a ½ Space and 2-Layer Medium and Reflection Coefficients.
Reflection & Standing waves
Chapter 3 Force and Stress
MEE …. WAVE PROPAGATION IN SOLIDS
Mechanical Principles
Identification of seismic phases
Fluid statics Hydrostatics or Fluid Statics is the study of fluids at rest. It's practical applications are numerous. Some of which are Fluid Manometers,
Making Waves: Seismic Waves Activities and Demonstrations
Phys2120 Basic Principles of Contemporary Physics Waves, Optics, and Modern Physics Alexander Dzyubenko © 2014.
A disturbance that propagates Examples Waves on the surface of water
Reflection and refraction Dispersion
1st Week Seminar Sunryul Kim Antennas & RF Devices Lab.
Presentation transcript:

Stress: Force per unit area across an arbitrary plane

Stress Defined as a Vector ^ N = unit vector normal to plane t(n) = (tx,ty,tz) = traction vector ^ The part of t that is perpendicular to the plane is normal stress The part of t that is parallel to the plane is shear stress

Stress Defined as a Tensor z txz y x z tzx ^ ^ t(y) t(z) x t(x) ^ No net rotation txx txy txz t = t T = tyx tyy tyz tzx tzy tzz

Relation Between the Traction Vector and the Stress Tensor z txz y x z tzx ^ ^ t(y) t(z) x t(x) ^ No net rotation ^ tx(n) txx txy txz nx t(n) = t n = ty(n) = tyx tyy tyz ny tz(n) tzx tzy tzz nz ^ ^ ^ ^ ^ ^ ^

Relation Between the Traction Vector and the Stress Tensor That is, the stress tensor is the linear operator that produces the traction vector from the normal unit vector…. ^ tx(n) txx txy txz nx t(n) = t n = ty(n) = tyx tyy tyz ny tz(n) tzx tzy tzz nz ^ ^ ^ ^ ^ ^ ^

Principal Stresses Most surfaces has both normal and tangential (shear) traction components. However, some surfaces are oriented so that the shear traction = 0. These surfaces are characterized by their normal vector, called principal stress axes The normal stress on these surfaces are called principal stresses Principal stresses are important for source mechanisms

Stresses in a Fluid -P 0 0 t = 0 -P 0 0 0 -P If t1=t2=t3, the stress field is hydrostatic, and no shear stress exists -P 0 0 t = 0 -P 0 0 0 -P P is the pressure

Pressure inside the Earth Stress has units of force per area: 1 pascal (Pa) = 1 N/m^2 1 bar = 10^5 Pa 1 kbar = 10^8 Pa = 100 MPa 1 Mbar = 10^11 Pa = 100 GPa Hydrostatic pressures in the Earth are on the order of GPa Shear stresses in the crust are on the order of 10-100 MPa

Pressure inside the Earth At depths > a few km, lithostatic stress is assumed, meaning that the normal stresses are equal to minus the pressure (since pressure causes compression) of the overlying material and the deviatoric stresses are 0. The weight of the overlying material can be estimated as rgz, where r is the density, g is the acceleration of gravity, and z is the height of the overlying material. For example, the pressure at a depth of 3 km beneath of rock with average density of 3,000 kg/m^3 is P = 3,000 x 9.8 x 3,000 ~ 8.82 10^7 Pa ~ 100 MPa ~ 0.9 kbar

Mean (M) and Deviatoric (D) Stress txx txy txz t = tyx tyy tyz tzx tzy tzz M = txx + tyy + tzz = tii/3 txx-M txy txz D = tyx tyy-M tyz tzx tzy tzz-M

Strain: Measure of relative changes in position (as opposed to absolute changes measured by the displacement) U(ro)=r-ro E.g., 1% extensional strain of a 100m long string Creates displacements of 0-1 m along string

J can be divided up into strain (e) and rotation (Ω) is the strain tensor (eij=eji) ux ux uy ux uz x y x z x ½( + ) ½( + ) e = uy ux uy uy uz x y y z y ½( + ) ½( + ) uz ux uz uy uz x z y z z ½( + ) ½( + )

J can be divided up into strain (e) and rotation (Ω) ux uy ux uz y x z x 0 ½( - ) ½( - ) Ω= uy ux uy uz x y z y -½( - ) 0 ½( - ) uz ux uz uy x z y z -½( - ) -½( - ) 0 is the rotation tensor (Ωij=-Ωji)

Volume change (dilatation) = 1/3 ( + + ) = tr(e) = div(u) > 0 means volume increase < 0 means volume decrease ux uy uz x y z ux x ux x >0 <0

∂2ui/∂t2 = ∂jij + fi = Equation of motion = Homogeneous eom when fi=0

Seismic Wave Equation (one version) For (discrete) homogeneous media and ray theoretical methods, we have  ∂2ui/∂t2 = ()·u-xx u

^ Plane Waves: Wave propagates in a single direction u(x,t) = f(tx/c) travelling along x axis = A()exp[-i (t-s•x)] = A()exp[-i(t-k•x)] where k= s = (/c)s is the wave number ^

s x sin = vt, t/x = sin/v = u sin = p u = slowness, p = ray parameter (apparent/horizontal slowness) rays are perpendicular to wavefronts x  s wavefront at t+t wavefront at t

p = u1 sin 1 = u2 sin 2 u = slowness, p = ray parameter (apparent/horizontal slowness) 1 v1 2 v2

p = u1sin 1= u2sin 2= u3sin 3 Fermat’s principle: travel time between 2 points is stationary (almost always minimum) 1 v1 2 v2 3 v3

Continuous Velocity Gradients p = u0sin 0= u sin  = constant along a single ray path X v z 0   =90o, u=utp T dT/dX = p = ray parameter X

X(p) generally increases as p decreases -> dX/dp < 0 v z  p decreasing  =90o, u=utp T Prograde traveltime curve X

X(p) generally increases as p decreases but not always v X z Prograde Retrograde T caustics Prograde X

Reduced Velocity Prograde Retrograde T caustics Prograde X T-X/Vr X

X(p) generally increases as p decreases -> dX/dp < 0 Shadow zone v X z lvz T  X p

Traveltime tomography j Traveltime tomography T = ∫ 1/v(s)ds = ∫u(s)ds Tj = ∑ Gij ui Tj = ∑ Gij ui i=1 d=Gm GTd=GTGm mg=(GTG)-1GTd j-th ray i=1

Earthquake location uncertainty 2 = ∑ [ti-tip]2/i2 i expected standard deviation 2 (mbest)= ∑ [ti-tip(mbest)]2/ndf mbest is best-fitting station 2(m) = ∑ [ti-tip]2/2 - contour! n i=1   n i=1

Fast location: S-P times: D ~ 8 x S-P(s)

Other sources of error: Lateral velocity variations slow fast Station distribution

Emean = 1/2 A2 2 (higher frequencies carry more E!) A2/A1= (1c1/2c2)1/2 ds2 ds1

1cos1-2cos2 S’S’’= 1cos1+2cos2 2 1cos1 S’S’ = since ucoscos For vertical incidence ( 1 - 2 A1’’= 1 + 2 2 1 A2’ = 1-2 2 1 S’S’’vert= S’S’vert= 1+2 1+2

S waves vertical incidence P waves vertical incidence: 1-2 2 1 P’P’’vert= - P’P’vert= 1+2 1+2 1-2 2 1 S’S’’vert= S’S’vert= 1+2 1+2

E1flux = 1/2 c1 A12 2 cos1 E2flux = 1/2 c2 A22 2 cos2 Anorm = [E2flux/E1flux]1/2 = A2/A1 [c2cos2/ c1cos1]1/2 = Araw [c2cos2/ c1cos1]1/2 1cos1-2cos2 S’S’’norm= = S’S’’raw 1cos+2cos 2 1cos (c2cos2)1/2 S’S’norm = x 1cos1+2cos2 (c1cos1)1/2

2 1cos S’S’ = 1cos+2cos What happens beyond c ? There is no transmitted wave, and cos = (1-p2c2)1/2 becomes imaginary. No energy is transmitted to the underlying layer, we have total internal reflection. The vertical slowness =(u2-p2)1/2 becomes imaginary as well. Waves with Imaginary vertical slowness are called inhomogeneous or evanescent waves.

Phase changes: Vertical incidence, free surface: S waves - no change in polarity P waves - polarity change of  Vertical incidence, impedance increases: S waves - opposite polarity P waves - no change in polarity Fig 6.4 Phase advance of /2 - Hilbert Transform

Attenuation: scattering and intrinsic attenuation Scattering: amplitudes reduced by scattering off small-scale objects, integrated energy remains constant Intrinsic: 1/Q() = -E/2E E is the peak strain energy, -e is energy loss per cycle Q is the Quality factor A(x)=A0exp(-x/2cQ) X is distance along propagation distance C is velocity

Ray methods: t* = ∫dt/Q ( r ), A()=A0()exp(-t*/2) i.e., higher frequencies are attenuated more! pulse broadening