Rheology II

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Rheology II
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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)

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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

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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

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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

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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

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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.

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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

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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.

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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)

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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

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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)

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Factors Affecting Deformation

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

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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

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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

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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

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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

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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

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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
• 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
  about 1 cm/year, the strain rate of your finger
  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

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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

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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

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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

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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

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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

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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)

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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