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Rheology II

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Rheology II. Ideal Viscous Behavior. Viscosity theory deals with the behavior of a ... For viscous material, stress, s, is a linear function of strain rate e. ... – PowerPoint PPT presentation

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Title: Rheology II

1
Rheology II
2
Ideal Viscous Behavior
• Viscosity theory deals with the behavior of a
liquid
• For viscous material, stress, s, is a linear
function of strain rate e.?e/?t, i.e.,
• s he. where h is the viscosity
• Implications
• 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)

3
Viscous Deformation
• Viscous deformation is a function of time
• This means that strain accumulates over time
• Viscous behavior is essentially dissipative
• Hence deformation is irreversible, i.e. strain is
• Non-recoverable
• Permanent
• Flow of water is an example of viscous behavior.
• Some parts of Earth behave viscously given the
large amount of geologic time available

4
Ideal Viscous Behavior
• Integrate the equation s he. with respect to
time, t
• ?sdt ?he. dt ? st he or s he/t or e
st/h
• 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
• Analog Dashpot a leaky piston that moves
inside a fluid-filled cylinder. The resistance
encountered by the moving perforated piston
reflects the viscosity

5
Viscosity, h
• 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-2T or ML-1 T-1

6
Viscosity, h
• Units of h Pa s (kg m -1 s -1 )
• s he. ? (N/m2)/(1/s) ? Pa s
• s he. ? (dyne/cm2)/(1/s) ? poise
• 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
• poise 0.1 Pa s
• h of water is 10-3 Pa s
• Water is about 20 orders of magnitude less
viscous than most rocks
• h of mantle is on the order 1020-1022 Pa s

7
Nonlinear Behavior
• 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 lt granite lt
basalt-gabbro lt olivine.
• So strength generally increases as you go from
crust into mantle, from granitic dominated
lithologies to ultramafic rocks.

8
Plastic Deformation
• 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 (yield
stress elastic limit) where strain is no
longer linear with stress
• Plastic strain is shear strain at constant
volume, and can only be caused by shear stress
• Is dissipative and irreversible. If applied
stress is removed, only the elastic strain is
reversed
• Time does not appear in the constitutive equation

9
Elastic vs. Plastic
• The terms elastic and plastic describe the nature
of the material
• Brittle and ductile describe how rocks behave.
• Rocks are both elastic and plastic materials,
depending on the rate of strain and the
environmental conditions (stress, pressure,
temperature), and we say that rocks are
viscoelastic materials.

10
Plastic Deformation
• For perfectly plastic solids, deformation does
not occur unless the stress is equal to the
threshold strength (at yield stress)
• Deformation occurs indefinitely under constant
stress (i.e., threshold strength cannot be
exceeded)
• For plastic solids with work hardening, stress
must be increased above the yield stress to
obtain larger strains
• Neither the strain (e) nor the strain rate (e. )
of a plastic solid is related to stress (s)

11
Brittle vs. Ductile
• Brittle rocks fail by fracture at less than 3-5
strain
• Ductile rocks are able to sustain, under a given
set of conditions, 5-10 strain before
deformation by fracturing

12
Recall Strain or Distortion
• 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 1e l½ no dimension
• Quadratic elongation l s2 (1e)2
• Natural strain (logarithmic strain)
• e S dl/lo ln l/lo ln s ln (1e) 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)

13
Factors Affecting Deformation
• Confining pressure, Pc
• Effective confining pressure, Pe
• Pore pressure, Pf is taken into account
• Temperature, T
• Strain rate, e.

14
Effect of T
• 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

15
Strain Rate, e.
• 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
• e. de/dt (dl/lo)/dt T-1 e.g., s-1

16
Shear Strain Rate
• Shear strain rate
• g. 2 e. T-1
• Typical geological strain rates are on the order
of 10-12 s-1 to 10-15 s-1
• Strain rate of meteorite impact is on the order
of 102 s-1 to 10-4 s-1

17
Effect of strain rate e.
• Decreasing strain rate
• decreases rock strength
• increases ductility
• Effect of slow e. is analogous to increasing T
• Think about pressing vs. hammering a silly putty
• Rocks are weaker at lower strain rates
• Slow deformation allows diffusional
crystal-plastic processes to more closely keep up
with applied stress

18
Strain Rate (e.) Example
• 30 extension (i.e., de 0.3) in one hour (i.e.,
dt 3600 s) translates into
• e. de/dt 0.3/3600 s
• e. 0.000083 s-1 8.3 x 10-5 s -1

19
Strain Rate (e.) More Examples
• 30 extension (i.e., de 0.3) in 1 my (i.e., dt
1000,000 yr ) translates into
• e. de/dt
• 0.3/1000,000 yr
• 0.3/(1000000)(365 x 24 x 3600 s) 9.5 x 10-15
s-1
• If the rate of growth of your finger nail is
nail is
• e (l-lo) / lo (1-0)/0 1 (no units)
• e. de/dt 1/yr 1/(365 x 24 x 3600 s)
• 3.1 x 10-8 s-1

20
Effect of Pc
• Increasing confining pressure
• Greater amount of strain accumulates before
failure
• i.e., increases ductility
• increases the viscous component and enhances flow
• resists opening of fractures
• i.e., decreases elastic strain

21
Effect of Fluid Pressure Pf
• Increasing pore fluid pressure
• reduces rock strength
• reduces ductility
• The combined reduced ductility and strength
promotes flow under high pore fluid pressure
• Under wet conditions, rocks deform more readily
by flow
• Increasing pore fluid pressure is analogous to
decreasing confining pressure

22
Strength
• Rupture Strength (breaking strength)
• Stress necessary to cause rupture at room
temperature and pressure in short time
experiments
• Fundamental Strength
• 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

23
Factors Affecting Strength
• Increasing temperature decreases strength
• 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
• This effect is much more pronounced at low T (lt
100o) where frictional processes dominate, and
diminishes at higher T (gt 350o) where ductile
deformation processes, that are temperature
dominated, are less influenced by pressure

24
Factors Affecting Strength
• Increasing time decreases strength
• Solutions (water) decrease strength, particularly
in silicates by weakening bonds (hydrolytic
weakening)
• High fluid pressure weakens rocks because it
reduces effective stress

25
Flow of Solids
• Flow of solids is not the same as flow of
liquids, and is not necessarily constant at a
given temperature and pressure
• A fluid will flow with being stressed by a
surface stress - it does response to gravity (a
body stress).
• A solid will flow only when the threshold stress
exceeds some level (yield stress on the Mohr
diagram)

26
Rheid
• 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.
• 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)
• A Rheid fold, therefore, is a flow fold - a fold,
the layers of which, have deformed as if they
were fluid