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Rheology II. Ideal (Newtonian) 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 = h e . where h is the viscosity Implications:

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ideal newtonian viscous behavior
Ideal (Newtonian) 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)
viscous deformation
Viscous Deformation
  • Viscous deformation is a function of time

s = he.= he/t

  • Meaning that strain accumulates over time (next slide)
  • Viscous behavior is essentially dissipative
  • Hence deformation is irreversible, i.e. strain is
    • 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
Ideal Viscous Behavior
  • Integrate the equation s = he. with respect to time, t:

sdt =he. dt st = he ors = 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
viscosity h
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-2][T] or [ML-1 T-1]

units of viscosity h
Units of 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
nonlinear behavior
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 < granite < basalt-gabbro < olivine
  • So strength generally increases as you go from crust into mantle, from granitic-dominated lithologies to ultramafic rocks
plastic deformation
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 (y, 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
elastic vs plastic
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, temp.)
    • Rocks are viscoelastic materials at certain conditions
plastic deformation1
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)
brittle vs ductile
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
strain or distortion
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 =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
Factors Affecting Deformation
  • Confining pressure, Pc
  • Effective confining pressure, Pe
    • Pore pressure, Pfis taken into account
  • Temperature, T
  • Strain rate, e.
effect of t
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
strain rate e
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

shear strain rate
Shear Strain Rate
  • Shear strain rate:

g .= 2 e. [T-1]

  • Typical geological strain rates are on the order of 10-12 s-1to 10-15 s-1
  • Strain rate of meteorite impact is on the order of 102 s-1 to 10-4 s-1
effect of strain rate e
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
strain rate e example
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

strain rate e more examples
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

e.= 0.3/1000,000 yr

e.= 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/yr, the strain rate, e., 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)

e.= 3.1 x 10-8 s-1

effect of p c
Effect of Pc
  • Increasing confining pressure:
    • inreases amount of strain before failure
      • i.e., increases ductility
    • increases the viscous component and enhances flow
    • resists opening of fractures
      • i.e., decreases elastic strain
effect of fluid pressure p f
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
  • 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
factors affecting strength
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 (< 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 strength1
Factors Affecting Strength
  • Increasing time decreases strength
  • Solutions (e.g., water) decrease strength, particularly in silicates by weakening bonds (hydrolytic weakening) (OH- substituting for O-)
  • High fluid pressure weakens rocks because it reduces effective stress
flow of solids
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)
  • 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