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Rheology II

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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Rheology II

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  1. Rheology II

  2. 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)

  3. 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

  4. 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

  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-2][T] or [ML-1 T-1]

  6. 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

  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 < granite < basalt-gabbro < 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 (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

  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, temp.) • Rocks are viscoelastic materials at certain conditions

  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. 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)

  13. Factors Affecting Deformation • Confining pressure, Pc • Effective confining pressure, Pe • Pore pressure, Pfis 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-1to 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 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

  20. 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

  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 (< 100o) where frictional processes dominate, and diminishes at higher T (> 350o) where ductile deformation processes, that are temperature dominated, are less influenced by pressure

  24. 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

  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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