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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
    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

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
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