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

6. s = he.  (dyne/cm2)/(1/s)  poise
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 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

10. 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

11. 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)

12. 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

13. 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)

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

15. 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

16. e. = de/dt = (dl/lo)/dt [T-1]
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

17. Shear Strain Rate g . = 2 e. [T-1] Shear strain rate:
Typical geological strain rates are on the order of s-1 to s-1
Strain rate of meteorite impact is on the order of 102 s-1 to 10-4 s-1

18. 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

19. 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. = s-1 = 8.3 x 10-5 s -1

20. 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/( )(365 x 24 x 3600 s)= 9.5 x 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

21. 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

22. 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

23. Strength Rupture Strength (breaking strength) Fundamental 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

24. 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

25. 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

26. 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)

27. 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