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Midterm GLY 420 G review
STRAIN
Change in shape
Elongation: e = (l – lo/lo)
(final length – initial length)/initial length
Principal elongations: e1 > e2 > e3
Dilation – volume change:  =(V – Vo)/Vo
Pure shear- irrotational
Simple shear– rotational: = tan ()

e2 = 0 in simple shear (plane strain)
Homogeneous strain - straight lines remain straight
Inhomogeneous-do not remain straight
Strain ellipsoid- major strain axes long, intermediate, short: X= 1+e1, Y= 1 + e2,
Z = 1+ e3.
Strain rate: strain/time (e/t)
geologic strain rate 10-14 s-1
3.3 x 1013 seconds in 1 million yrs.
STRAIN measurement
Need strain marker: initial shape known, behaves same as rock
e.g. pebbles, reduction spots, fossils, vesicles, ooids
R/ method
R – ellipicity
- angle between long axis of ellipse and x axis
Allows initially non-spherical strain markers to be used
FLINN diagram
Measure X, Y and Z lengths of object and plot k value
Ordinate: X/Y; abscissa: Y/Z
K = (a-1)/b-1); a = X/Y; b = Y/Z
Flattening field (k = small values) and elongation field (k = large values). Cannot plot
simple shear deformation.
Ramsay diagram: log or ln axes. Can plot volume change ().
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STRESS
Stress = force/area
Dimensions = ML/L2T2
Units : Pascals; kilobars; 100 Mpa = 1 kilobar
Pascal = Newton/m2
Vector components: normal stress (n) and shear stress (
s)
3 Principal stress directions:  , 2, 3
no shear stress across plane normal to principal stress
Stress at depth = density x gravity x depth (gh)
1 =2 = 3
Single point on Mohr diagram
Lithostatic stress: due to weight overlying rock
stress at 1 km depth: 2700 kg/m3 x 9.8 m/s2 x 1000 m
=
2700. 9.8. 1000 = 26.4 x 106 Pa = 26.4 MPa = 0.264 kbars
Mean stress : (1+ 3)/2
Differential or deviatoric stress: (1- 3)/2 related to rock strength.
Effective stress: (n – Pfluid). Pluid pressure weakens rock. Promotes tensile fractures.
Anderson theory of faulting
Assumes one principal stress is vertical near earth surface.
Normal faults: sigma max. vertical; sigma min. horizontal
Thrust faults: sigma max horizontal, sigma min. vertical
Strike slip: sigma intermed. vertical
Mohr circle
Center of circle = mean stress
Radius of circle = differential stress
Positive shear stress = left lateral
Negative shear stress = right lateral
Positive x direction = compression
Negative x axis = tension
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Plane makes angle with normal to sigma one (1)
What is the Mohr circle good for?
1) Predicts rock failure with Coulomb line:
( = c + n)
2) Predicts effect of fluid pressure (circle moves left)
3) Predicts angle of fault to normal to max. stress (2)
4) Distinguishes shear fractures versus tensional fractures.
5) Predicts fault re-activation or new fault will form.
RHEOLOGY
Elastic- stress proportional to strain-recoverable
Hooke’s law: 
(E: Young’s modulus; units Pa)
Elastic shear strain:  = G
ocks 1011
Poisson's ratio:  = ehor/etransv. (rocks 0.25-0.3)
Related to seismic wave velocities.
Viscous behavior
= e’
stress = viscosity x strain rate
Linear viscous = Newtonian
Earth’s mantle: solid during earthquakes, but liquid during plate motions (e.g. Sillyputty).
Viscosity: units of poise or Pascal seconds (Pa.s)
Mantle viscosity: 1020 Pa.s
Water: 10-3
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Elastico-viscous: combination rheology
= Ee (elastic)+ e’(viscous)
Piston-spring in seriesApply stress, instantaneous strain, followed by slower (time dependant) strain.
Remove stress, instantaneous partial recovery, followed by slower full recovery.
Time to full recovery (Maxwell relaxation time)
t = /G (viscosity/rigidity)
1021/1011 = 1010s = 1000 years
e.g. uplift of continents after last ice age- isostatic uplift.
The lithospheric plates behave as elastico-viscous bodies. During thrust emplacement;
sediment loading; or ice ages.
Strain rate (e/t)
(units: s-1)
1 yr. = 3 x 107 s.
1 my = 3 x 1013 s
Typical geological rate: 10-14 s-1
Earthquakes, volcanic eruption, meteoric impact: 10-2 - 10+2
10-1 s-1 : 10% strain in 1 second
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Effects of temperature, pressure, strain rate, Pfluid, and water on rock strength.
Increasing temperature: decreases strength, promotes ductility.
Increasing confining pressure: increases strength; promotes ductility. Inhibits opening of
fractures.
Fast strain rate: increases strength; promotes brittle behavior.
Slow strain rates: promotes ductility.
Increasing fluid pressure (decreasing effective stress): decreases strength; promotes
brittle behavior. Reduces frictional resistance- (Beer can experiment)
Water: chemical effect weakens Si-O bonds – weakens rock (especially quartz).
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BRITTLE DEFORMATION
Griffith theory: assumes all materials contain microcracks.
Stress concentration at crack tip causes crack to lengthen, leading to rock failure.
Hydraulic fractures (tensile cracks): produced by high fluid pressure
Form parallel to 1
Pfluid must exceed 3 + T (tensile strength)
Types of cracks: Mode I (tensile), Modes II, II- shear type
Coulomb failure criterion:  = c +n (c is rock cohesion).
c = 0 for faults.
Failure occurs when Mohr circle touches line. Fault occurs at 2 to normal to 1. Faults
occur 30o to 1.
Straight line; slope = tan  =  = coefficient of friction.
At higher confining pressure, straight line becomes curved = Mohr envelope.
FRICTION
Amonton’s laws:
1) Frictional force depends on normal stress
2) Frictional force depends on real area of contact, not size of surfaces in contact.
Real area of contact = total area of asperities
Byleree’s law: coefficient of friction () constant for most rock types (0.6 to 0.85) except
clays/micas:
 = 50 MPa + 0.6 n (> 200 MPa)
 = 0.85n (< 200 MPa)
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FAULTING
Experiments: faults form at 30 to 1
Listric faults: due to curvature of stress trajectories with depth.
Thrust fault paradox: If friction coefficient is normal (
-0.85), then thrust blocks
should deform internally, rather than slide along basal detachment.
Hubbert-Rubey hypothesis: high fluid pressure reduces effective stress, hence basal shear
stress: as Pfluid increases,  goes to zero:
 =  (n – Pfluid)
Alternatives: weak basal layer- shale, salt. Not plausible for basement thrusts (e.g. Blue
Ridge).
Sandbox experiment
Normal faults in left compartment;
min horiz.
max vertical
3 decreases; 1 constant.
Thrust in right compartment: 1 horiz., min vertical
1 increases; 3 constant
Faults make 30o to normal to sigma 1.
Thrusts dip 30o
Normal faults dip 60o
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Beer can experiment
Principal of effective stress- angle of slope goes to zero as ratio of fluid pressure to
normal stress increase to unity.
Thrust faults
Sigma 1 horizontal; dip shallow angle (30o)
Thrust paradox: need very steep dip for frictional sliding downhill by gravity
Cannot push block > 5.5 km before it fails internally
Solution: high fluid pressure (Hubert and Rubey, 1959).
Second solution: thrust moves like caterpillar- one step at a time (Price, 1988)
Coloumb critical wedge hypothesis: thrust belts are wedge shaped
1) weak wedge- deforms internally- Mohr circle in contact with failure envelope
2) strong wedge- deforms only along basal detachment- Mohr circle far from
envelope
3) Critical wedge- Mohr circle barely touches envelope- due to fluid pressure, circle
always in contact with envelope
Joints
Three types:
Mode I: tensile
Mode II: shear fractures
Mode III: tear fractures
Form parallel to sigma 1 and normal to sigma 3
Origin: tectonic (e.g. folding); topographic (valley walls); hydraulic (high fluid pressure);
erosional uplift.
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DEFORMATION MECHANISMS
1. Cataclasis: low temperature, high stress (brecciation)
2. Pressure solution: low temperature, low stress (common in limestone)
3. Dislocation creep (crystal plasticity): high temperature
Involves motion of dislocations (edge and screw)
Requires diffusion of atoms (D = Do exp(-E/RT)
Leads to recrystallization at high stress.
Leads to recovery and annealing at high temperature.
Mylonites display these deformation mechanisms.
4. Grain boundary sliding (superplasicity)
v. high temperature, small grain size
Brittle upper crust: breccias
Ductile lower crust: mylonites- fine grained
Intermediate depths: complex mix of mechanisms
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