Fracture mechanics, Mohr circles, and the Coulomb criterion (Stress and failure)
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Transcript of Fracture mechanics, Mohr circles, and the Coulomb criterion (Stress and failure)
Fracture mechanics, Mohr circles, and the Coulomb criterion
(Stress and failure)
Introduction to Fracture Mechanics
In this lecture, we will be focusing on faults:How they form
Definitions
Stress states
Fault strain
How we can use faults to tell us things about the geologic history
Representative structures on planetary bodies
Definitions
Fracture: a pair of distinct surfaces that are separated in a material; ‘‘a structure defined by two surfaces or a zone across which displacement occurs
A joint is a fracture that exhibits only opening displacement (Mode I)
A fault is a planar zone along which shear displacement occurs (Mode II)
Fracture modesMode I: opening displacement (joints)
Mode II: in-plane shear (faults)
Mode III: out-of-plane shear (scissors, tearing)
Stress
Stress is force/area, units of pressure (most often bars or pascals)
σ = F/A
Normal stress (σ) acts perpendicular to a surface
Shear stress (τ) acts parallel to a surface
StressThe “on-in” convention says that the stress component σij acts normal to the ‘i’ direction and parallel to the ‘j’ direction:
Normal stresses i = j
Shear stresses i ≠ j
StressPrinciple stresses have magnitudes and directions
σ1: maximum compressive stress
σ2: intermediate compressive stress
σ3: minimum compressive stress
Principle stresses act on planes that do not ‘feel’ shear stress, i.e., they are normal stresses
StressCalculate principle stresses from an arbitrary remote stress state with normal stresses σxand σy, parallel to the x and y axis, respectively
And the orientation of the 2 mutually perpendicular principle planes
Stress exampleGiven values of σx = 1.2 MPa (compression positive), σy= –0.85 MPa, and τxy = 0.45 MPa, find (a) the principal stresses, and (b) predict the orientations of the principal planes on which the principal stresses act.
SOLUTION
Substitute the values of normal and shear stress into equation 1 to obtainσmax,min=1.294MPa and –0.944MPa
Now substitute the same initial values of stress into equation 2 to obtain θp=11.6° and 101.6°
Add 90º to get second plane orientation
Resolving stresses onto planes
To resolve the stresses on a plane is to calculate the magnitude and direction of the normal and shear stresses onto a plane of particular orientation (here, a fault plane) due to the principle stresses
Example
σ1 = 5 MPa, σ2 = 1 MPa, α = 30¡ (to the plane), and θ = 60º (to the plane’s normal).
What is the magnitude of the normal and shear stresses?
σn = 2 MPa (compressive) and τ = 1.7 MPa
Which way is the fault going to slip?Left-lateral
Mohr CirclesMean stress: describes the confining stress experienced by rock at some depth
(σ1 + σ3)/2
Differential stress: describes the greatest amount of stress change that a rock can withstand without breaking
(σ1 – σ3)
Mohr circles is a geometric representation of these equations used to determine when rock will fracture or when faults will slip
Mohr circles
Mohr circles
X and Y axes are normal and shear stress, respectively.
This method only works for compressional normal stress (i.e., compression vs. tension; faults)
Plot σ1 and σ3 on the X axis as points. The difference between these values is the differential stress.
We’ll revisit this when we talk about the Coulomb Criterion.
Mohr circles and effective stress
Effective stress is the normal stress reduced by the pore fluid pressure
σn*=σn – pf
Pore pressure counteracts the effects of normal stress, reducing the magnitude of the principle stresses, but not of the differential stress.
Pore fluid must not support shear stress (i.e., is a fluid like water or air)
Coulomb criterionDefines the amount of shear stress needed to overcome the frictional resistance of a fault, leading to slip
Where C0 is the rock cohesion, μ is the coefficient of friction (typically between 0.6 and 0.85), ρ is the material density, g is the gravitational acceleration, and z is the depth
Cohesion describes the minimum amount of shear stress needed to start slip when the normal stress is tiny but compressive.
Coulomb criterion
Equation of a line!C0 is the y-intercept
μ is the slope of the line
If shear stress is greater than frictional resistance, the fault will slip. If not, no slip.
Coulomb criterion and Mohr circles
Combining the Coulomb criterion and Mohr circles allows you to predict if failure will occur with given stress magnitudes and if so, what the orientation of the plane will be when failure occurs.
Coulomb criterion practice
Given the values of remote principal normal stresses σ1 = 4.8 MPa (compression positive), σ2 = 1.15 MPa, and an angle to the normal to the plane from the σ1 direction of 58°, with values for cohesion of 0.001 MPa and friction coefficient of 0.58, determine whether the fracture implied will slip (using the Coulomb criterion), and if so, what its sense of shear will be.
SOLUTION
We find that σn = 2.175 MPa (compressive) and τ = 1.640 MPa (left-lateral).
Plugging these numbers into the Coulomb criterion, we find that 1.640 MPa > 0.001 + (0.58) (2.175) Mpa
1.640 MPa > 1.363 MPa. So the left-hand side (the driving forces) is greater than the right-hand side (the resisting forces) and the surface will fail by frictional sliding, and in a left-lateral sense because the calculated shear stress was positive.
Griffith CriteriaTensile failure (i.e., Mode I) occurs when the local stress at the most optimally oriented flaw attains a value characteristic of the material
Uniaxial tensile failure strength:E: Young’s modulus
ν: Poisson’s ratio
γ: energy required to create new crack walls
a: flaw half length
Gc: critical strain energy release rate
Griffith CriteriaThe Griffith criterion and Mohr circles showing (a) Uniaxial tensile failure, σ1 = 3T0 (at σ2 = T0); (b) uniaxial compressive failure, σ1 = 8T0 (at σ2 = 0); and (c) transition stress from crack growth to frictional sliding, σ1 ≅ 4.5T0.
Pure Mode I failure is predicted where the Mohr circle touches the Griffith criterion at exactly one point
Andersonian Fault Mechanics
E. M. Anderson’s first work on the subject of faulting was in 1905, but he is probably best known for his 1951 book (at right).
First proposed that faults are brittle fractures that occur according to the Coulomb criterion.
Divided faults into 3 classes that form depending on the ratio and orientation of principle stresses:
Normal
Thrust or reverse
Strike-slip
Andersonian Fault Mechanics
For all faults:σ1 > σ2 > σ3
Fault forms at an angle θ from σ1
Thrust fault:σ1 = σH
σ2 parallel to fault strike
σ3 = σv
Strike-slip faultσ1 = σH
σ2= σv
σ3 = σh
Normal faultσ1 = σv
σ2 parallel to fault strike
σ3 = σh
Andersonian Fault Mechanics
σv = weight of overburden or ρgz
Cube of rock subjected to vertical stress from overburden will extend in x and y directions (horizontal): Poisson ratio
ν = horizontal expansion/vertical shortening
ν< 0.5
The coefficient of friction and fault dip are related in order to minimize horizontal stress resolved on the fault plane.
To determine the fault dip from vertical for each dip-slip fault type, we use this equation:
Tan2θ = 1/μ
θ = 0.5(tan-1(1/μ))
This plot shows the minimum differential stress required to initiate sliding on a normal, strike-slip, and thrust fault.
Which line is which? Why?
Thrust faults require 16x more stress to rupture than normal faults and 2.6 x more stress than strike-slip faults.
How does this relate to earthquakes?
THRUST
STRIKE-SLIP
NORMAL
StrainStress σ causes strain ε
Strain is non-dimensional
Strain is any change in shape, volume, or orientation of a rock volume
Contractional normal strain perpendicular to σ1 and extensional normal strain perpendicular to σ3
Normal faulting results in extensional strain (horizontal)
Thrust faulting results in contractional strain (horizontal)
Extensional strain (vertical)
StrainStrain can be calculated by doing a 1D traverse across a set of faults
Calculate strain based on fault geometry
StrainExtension
Need depth of graben (d)
Fault dip angle (θ)
h = d/tanθ
Add these up for every fault in the traverse
Strain
This method of calculating strain can miss some faults.
In addition, the traverse may miss locations of maximum fault displacement (usually near the center of the fault, but not always).
Thus, better, more accurate methods exist to calculate strain from fault populations.
Seismology to the rescue!
Strain, the right way
Where D is the average displacement, L is the fault length, and H is the down-dip fault height (faulting depth/sinθ), V is the volume of the deformed region (depth of faulting*area of faults), and δ is the fault dip angle
This method includes all mapped faults and is not dependent on the location of 1D traverses.
Tour of Tectonics
http://mesic.astronomie.cz/Prohlidka/mesicni-brazda.php
NORMAL FAULTING
Moon
Planetary Tectonics:Normal Faults
Primary extensional morphologies on the planets:• graben• single normal faults
Graben are the down-dropped blocks between two antithetic NF.
Single normal faults are rare.
Where are NF found?• Moon• Mars• Venus• Earth• Mercury• Asteroids• Icy satellites
Planetary Tectonics:Normal Faults
How do these NF form?• rifting• dike intrusion• basin loading• regional-scale uplift• tidal stresses• impacts
Planetary Tectonics:Normal Faults
• Rare individual NF
• Basin loading or from Imbrium?
• Grew from small segments from S to N• opposite direction if formed by Imbrium
impact
• Likely not from basin loading, since it is on the edge of Nubium, too young, and straight
Moon
Planetary Tectonics:Normal Faults
• Valles Marineris: the largest canyon system in the solar system
• rift valley• 10 km wide• 5 km deep• 3000 km long
• Crustal extension along large-scale normal faults related to Tharsis volcanism and heat production
Mars
Planetary Tectonics:Normal Faults
• Pantheon Fossae • set of ~radial graben near
center of Caloris basin, Mercury
• Hypotheses:• Formed as a result of the
impact• Dike intrusion• Basin interior uplift
• PF formed in response to a dome with a R = 300 km, T = 150 km, and a maximum uplift of 10 km [Klimczak et al. (2011)]
Mercury
40 km
Planetary Tectonics:Normal Faults
http://apod.nasa.gov/apod/image/0911/PSP_007769_9010_IRB_Stickney.jpg
• Grooves• long• parallel• narrow
• tidal or thermal stresses?• impact-related?
Phobos (Mars)
Planetary Tectonics:Normal Faults
http://nssdc.gsfc.nasa.gov/imgcat/hires/mgn_c130n279_1.gif
• Venusian chasma (rift)• very long• high relief• relatively young features?
• Regional-scale crustal extension along antithetic normal faults
• Associated with volcanoes, likely due to uplift from underlying plume
Venus
~100 km
Planetary Tectonics:Normal Faults
Basin and Range province formed by crustal extension along antithetic normal faults forming graben (basin) and horsts (range) from subduction of the East Pacific Rise ~15 Ma
http://rst.gsfc.nasa.gov/Sect6/nv.jpg http://rst.gsfc.nasa.gov/Sect6/38Basin_Range_aerialsm.jpg
Earth
Planetary Tectonics:Normal Faults
THRUST FAULTING
Mercury
~40 km
Primary contractional morphologies on the planets:• wrinkle ridges• lobate scarps
Wrinkle ridges are blind thrust faults, often in areas of interbedded lava and regolith and/or pyroclastic material.
Lobate scarps are surface-cutting thrust faults that occur in mechanically homogenous terranes.
http://www.nasa.gov/images/content/475507main_lobate_scarp_thrust_fault_graphic.jpg
Planetary Tectonics:Thrust Faults
Where are TF found?• Moon• Mars• Venus• Earth• Mercury• Asteroids
Planetary Tectonics:Thrust Faults
How are these TF formed?• basin loading• lava cooling and contraction• impacts• (global contraction)
Planetary Tectonics:Thrust Faults
http://www.lpi.usra.edu/publications/slidesets/redplanet2/slide_9.html
• Hesperia Planum• type locality for Hesperian
epoch• Lava plains likely
interbedded with pyroclastic deposits (from Tyrrhena Patera)
• Multiple generations of WR indicate several temporally and spatially distinct episodes of compression
Mars
Planetary Tectonics:Thrust Faults
~25 km
• Wrinkle ridges in Mare Crisium
• Show a buried crater• Can give an idea of depth
of mare fill in Crisium Basin
Moon
~15 km
Planetary Tectonics:Thrust Faults
http://spacefellowship.com/news/art19041/crisium-s-region-of-interest.html
http://www.sciencemag.org/content/286/5437/87/F1.large.jpg
• Venusian wrinkle ridges appear to preferentially form in topographic lows
• Compressional stress likely from cooling and contraction in a topographic low
• Some WR terranes show more than one orientation of WR• more complicated and varied
stress field
Venus
Planetary Tectonics:Thrust Faults
~40 km
Watters et al., GRL 2011
• Lobate scarp (Hinks Dorsum) on the asteroid Eros
• Modeling gives fault parameters• Depth = 250 m• 90 m of offset
• Near-surface shear strength ~1 – 6 MPa
• Formed by impact-induced compression
Eros
Planetary Tectonics:Thrust Faults
~6 km
http://www.visit-himalaya.com/gifs/nepal-everest2.jpg
• Himalayan fold-and-thrust belt
• Large-scale continental collision
• Began ~50 Ma• Peak of Everest is
LIMESTONE• ~2400 km of India already ‘lost’• 1500 km of India subducted
over the next 10 Myr
Earth
Planetary Tectonics:Thrust Faults
~300 km
STRIKE-SLIP FAULTING
Earth
~ 110 km
http://photojournal.jpl.nasa.gov/catalog/PIA06661
Strike-slip fault morphology• linear• long
Strike-slip faults are rare on planets other than Earth.
Planetary Tectonics:Strike-slip Faults
Where are SSF found?• Earth• Mars• Venus• Icy satellites
Planetary Tectonics:Strike-slip Faults
How are these SSF formed?• plate motion (Earth only)• lateral movement of the lithosphere
• tidal stresses• accommodation structures w/WR• impacts?
Planetary Tectonics:Strike-slip Faults
• San Andreas: best-studied fault zone on Earth
• Farallon plate and spreading center subducted under N. American plate
• development right-lateral transform (SSF) that propagated along the continental margin
• Formed • ~28 Ma• 470 km of offset inferred
Earth
Planetary Tectonics:Strike-slip Faults
Andrews-Hanna et al., JGR 2008
• Martian strike-slip faults• Noachian in age and
continued for ~ 1 Gyr• used to estimate crustal
deviatoric stress magnitude and orientation during early Mars’ history
• Strike-slip faulting caused by loading and a background compressional stress
Mars
Planetary Tectonics:Strike-slip Faults
• Ovda Regio, in Aphrodite Terra
• Formed as a transform fault from collisional zone in the north (‘escape tectonics,’ a la Tibet)
• ~75 km of left-lateral strike-slip motion
http://www.lpi.usra.edu/publications/slidesets/venus/images/sven_s13.gif
Venus
Planetary Tectonics:Strike-slip Faults
• Androgeos Linea• 350 m high
• Strike-slip movement from tidal deformation and internal convection
http://www.lpi.usra.edu/galileoAnniv/img/hiRes/eplains-mos-KH.jpg
Europa
Planetary Tectonics:Strike-slip Faults
• Tiger stripes • 130 km long• 35 km apart• 2 km wide• 500 m deep
• Strike-slip faults with displacements of 0.5 m/event (Smith-Konter and Pappalardo, 2008)
• Associated with high heat flow and water-vapor plumes
• Tidal stresses from Saturn at periapsis
Enceladus
http://photojournal.jpl.nasa.gov/catalog/PIA06247
Planetary Tectonics:Strike-slip Faults