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Rheology II

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)

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

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

- 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

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

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

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

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 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

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

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
- 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

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

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

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

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

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

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

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)

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