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Rheology II


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

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.?e/?t, i.e.,
  • s he. where h is the viscosity
  • Implications
  • The s - e. plot is linear, with viscosity as the
  • 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 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

Ideal Viscous Behavior
  • Integrate the equation s he. with respect to
    time, t
  • ?sdt ?he. dt ? st he or s he/t or e
  • 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
  • An ideally viscous body is called a Newtonian
  • 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

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
  • 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.

Plastic Deformation
  • Plasticity theory deals with the behavior of a
  • Plastic strain is continuous - the material does
    not rupture, and the strain is irreversible
  • 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
  • Time does not appear in the constitutive equation

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.

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

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)

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

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
  • 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
  • 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
  • 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

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
  • 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
  • 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
  • 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

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

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
  • 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
  • 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
  • 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

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

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

  • 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
  • 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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