#### Transcript Rheology II

```Rheology II
Ideal (Newtonian) Viscous Behavior
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Viscosity theory deals with the behavior of a liquid
For viscous material,
stress, s, is a linear function
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of strain rate, e =e/t, i.e.,
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s = he
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where h is the viscosity
Implications:
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The s - e plot is linear, with viscosity as the slope
The higher the applied stress, the faster the
material will deform
A higher rate of flow (e.g., of water) is associated
with an increase in the magnitude of shear stress
(e.g., on a steep slope)
Viscous Deformation
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Viscous deformation is a function of time
s = he = he/t
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Meaning that strain accumulates over time (next slide)
Viscous behavior is essentially dissipative
Hence deformation is irreversible, i.e. strain is
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Non-recoverable and permanent
Flow of water is an example of viscous behavior
Some parts of Earth behave viscously given the large
amount of geologic time available
Ideal Viscous Behavior
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Integrate the equation s = he with respect to time, t:
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sdt = he dt  st = he or s = he/t or e = st/h
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For a constant stress, strain will increase linearly
with time, e = st/h (with slope: s/h)
Thus, stress is a function of strain and time!
s = he/t
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Analog: Dashpot; a leaky piston that moves inside
a fluid-filled cylinder. The resistance encountered
by the moving perforated piston reflects the viscosity
Viscosity, h
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An ideally viscous body is called a Newtonian
fluid
Newtonian fluid has no shear strength, and its
viscosity is independent of stress
From s = he/t we derive viscosity (h)
h = st/e
Dimension of h: [ML-1 T-2][T] or [ML-1 T-1]
Units of Viscosity, h
Units of h : Pa s (kg m -1 s -1 )
s = he  (N/m2)/(1/s)  Pa s
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s = he  (dyne/cm2)/(1/s)  poise
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If a shear stress of 1 dyne/cm2 acts on a liquid, and gives
rise to a strain rate of 1/s, then the liquid has a h of 1
poise (pronounced in French as in Poiseuille)
poise= P = 0.1 Pa s
 h of water is 10-3 Pa s
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Water is about 20 orders of magnitude less viscous than
most rocks
h of mantle is on the order 1020-1022 Pa s
Nonlinear Behavior
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Viscosity usually decreases with temperature
(effective viscosity)
Effective viscosity: not a material property but a
description of behavior at specified stress, strain rate,
and temperature
Most rocks follow nonlinear behavior and people
spend lots of time trying to determine flow laws for
these various rock types
Generally we know that in terms of creep threshold,
strength of salt < granite < basalt-gabbro < olivine
So strength generally increases as you go from crust
into mantle, from granitic-dominated lithologies to
ultramafic rocks
Plastic Deformation
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Plasticity theory deals with the behavior of a solid
Plastic strain is continuous - the material does not
rupture, and the strain is irreversible (permanent)
Occurs above a certain critical stress
(sy, yield stress = elastic limit) where strain is no
longer linear with stress
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Plastic strain is shear strain at constant volume,
and can only be caused by shear stress
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Is dissipative and irreversible. If applied stress is
removed, only the elastic strain is reversed
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Time does not appear in the constitutive equation
Elastic vs. Plastic
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The terms elastic and plastic describe the nature of
the material
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Brittle and ductile describe how rocks behave
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Rocks are both elastic and plastic materials,
depending on the rate of strain and the
environmental conditions (stress, pressure, temp.)
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Rocks are viscoelastic materials at certain conditions
Plastic Deformation
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For perfectly plastic solids, deformation does not
occur unless the stress is equal to the threshold
strength (at yield stress)
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Deformation occurs indefinitely under constant
stress (i.e., threshold strength cannot be exceeded)
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For plastic solids with work hardening, stress must
be increased above the yield stress to obtain larger
strains
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Neither the strain (e) nor the strain rate (e ) of a
plastic solid is related to stress (s)
Brittle vs. Ductile
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Brittle rocks fail by fracture at less than
3-5% strain
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Ductile rocks are able to sustain, under a
given set of conditions, 5-10% strain
before deformation by fracturing
Recall: Strain or Distortion
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A component of deformation dealing with shape and
volume change
– Distance between some particles changes
– Angle between particle lines may change
Extension: e=(l’-lo) / lo = l/ lo [no dimension]
Stretch: s = l’/lo =1+e = l½ [no dimension]
Quadratic elongation: l = s2 = (1+e)2
Natural strain (logarithmic strain):
e =S dl/lo = ln l’/lo= ln s = ln (1+e) and since s = l½ then
e = ln s = ln l½ = ½ ln l
Volumetric strain:
ev = (v’-vo) / vo = v/vo [no dimension]
Shear strain (Angular strain) g = tan 
 is the angular shear (small change in angle)
Factors Affecting Deformation
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Confining pressure, Pc
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Effective confining pressure, Pe
 Pore pressure, Pf is taken into account
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Temperature, T
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Strain rate, e
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Effect of T
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Increasing T increases ductility by activating
crystal-plastic processes
Increasing T lowers the yield stress (maximum
stress before plastic flow), reducing the elastic
range
Increasing T lowers the ultimate rock strength
Ductility: The % of strain that a rock can take
without fracturing in a macroscopic scale
Strain Rate, e
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Strain rate:
• The time interval it takes to accumulate a
certain amount of strain
– Change of strain with time (change in length
per length per time). Slow strain rate means
that strain changes slowly with time
– How fast change in length occurs per unit time
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e = de/dt = (dl/lo)/dt
e.g., s-1
[T-1]
Shear Strain Rate
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Shear strain rate:
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g =2e
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[T-1]
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Typical geological strain rates are on the
order of 10-12 s-1 to 10-15 s-1
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Strain rate of meteorite impact is on the
order of 102 s-1 to 10-4 s-1
Effect of strain rate e
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Decreasing strain rate:
 decreases rock strength
 increases ductility
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Effect of slow e is analogous to increasing T
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Think about pressing vs. hammering a silly putty
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Rocks are weaker at lower strain rates
Slow deformation allows diffusional crystal-plastic
processes to more closely keep up with applied stress
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Strain Rate (e ) – Example
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30% extension (i.e., de = 0.3) in one
hour (i.e., dt =3600 s) translates into:
e = de/dt = 0.3/3600 s
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e = 0.000083 s-1 = 8.3 x 10-5 s -1
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Strain Rate (e ) – More Examples
30% extension (i.e., de = 0.3) in 1 my
(i.e., dt = 1000,000 yr ) translates into:
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e = de/dt
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e = 0.3/1000,000 yr
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e = 0.3/(1000000)(365 x 24 x 3600 s)= 9.5 x 10-15 s-1
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If the rate of growth of your
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1 cm/yr, the strain rate, e , of your finger nail is:
e = (l-lo) / lo = (1-0)/0 = 1 (no units)
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e = de/dt = 1/yr = 1/(365 x 24 x 3600 s)
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e = 3.1 x 10-8 s-1
Effect of Pc
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Increasing confining pressure:
– inreases amount of strain before failure
• i.e., increases ductility
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increases the viscous component and
enhances flow
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resists opening of fractures
• i.e., decreases elastic strain
Effect of Fluid Pressure Pf
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Increasing pore fluid pressure
– reduces rock strength
– reduces ductility
• The combined reduced ductility and strength
promotes flow under high pore fluid pressure
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Under ‘wet’ conditions, rocks deform more
Increasing pore fluid pressure is analogous to
decreasing confining pressure
Strength
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Rupture Strength (breaking strength)
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Stress necessary to cause rupture at room
temperature and pressure in short time
experiments
Fundamental Strength
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Stress at which a material is able to withstand,
regardless of time, under given conditions of T,
P, and presence of fluids, without fracturing or
deforming continuously
Factors Affecting Strength
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Increasing temperature decreases strength
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Increasing confining pressure causes significant
 increase in the amount of flow before rupture
 increase in rupture strength
 (i.e., rock strength increases with confining
pressure
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This effect is much more pronounced at low T (< 100o)
where frictional processes dominate, and diminishes at
higher T (> 350o) where ductile deformation processes,
that are temperature dominated, are less influenced by
pressure
Factors Affecting Strength
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Increasing time decreases strength
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Solutions (e.g., water) decrease strength,
particularly in silicates by weakening bonds
(hydrolytic weakening) (OH- substituting for O-)
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High fluid pressure weakens rocks because it
reduces effective stress
Flow of Solids
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Flow of solids is not the same as flow of liquids, and is not
necessarily constant at a given temperature and pressure
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A fluid will flow with being stressed by a surface stress –
it does response to gravity (a body stress)
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A solid will flow only when the threshold stress exceeds
some level (yield stress on the Mohr diagram)
Rheid
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A name given to a substance (below its melting point) that
deforms by viscous flow (during the time of applied stress)
at 3 orders of magnitude (1000 times) that of elastic
deformation at similar conditions.
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Rheidity is defined as when the viscous term in a
deformation is 1000 times greater than the elastic term (so
that the elastic term is negligible)
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A Rheid fold, therefore, is a flow fold - a fold, the layers of
which, have deformed as if they were fluid
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