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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: - PowerPoint PPT Presentation

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Rheology II

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Rheology II

Ideal (Newtonian) Viscous Behavior• Viscosity theory deals with the behavior of a liquid• For viscous material, stress, is a linear function of

strain rate, e.=e/t, i.e.,

=e. where is the viscosity

• Implications:

• The - 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

=e.= 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

Ideal Viscous Behavior

Integrate the equation =e

. with respect to time, t:

dt =e. dt t = e or = e/t or e = t/

• For a constant stress, strain will increase linearly with time, e = t/ (with slope: /)

• Thus, stress is a function of strain and time!

= e/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,

An ideally viscous body is called a Newtonian fluid

Newtonian fluid has no shear strength, and its viscosity is independent of stress

From = e/t we derive viscosity (

= t/e

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

Units of Viscosity, Units of : Pa s (kg m -1 s -1 )

=e. (N/m2)/(1/s) Pa s

=e. (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 of 1 poise

poise = 0.1 Pa s of water is 10-3 Pa s

• Water is about 20 orders of magnitude less viscous than most rocks

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 < granite < basalt-gabbro < 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

(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

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

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 ()

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

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 = ½ [no dimension]• Quadratic elongation: = s2 = (1+e)2

• Natural strain (logarithmic strain): =l/lo = ln l’/lo= ln s = ln (1+e) and since s = ½ then = ln s = ln ½ = ½ ln • Volumetric strain:

ev = (v’-vo) / vo = v/vo [no dimension]• Shear strain (Angular strain)= 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. = e/t = (l/lo)/t [T-1]

e.g., s-1

Shear Strain Rate Shear strain rate:

.= 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., e = 0.3) in one hour (i.e., t =3600 s) translates into:

e. = e/t = 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., e = 0.3) in 1 my

(i.e., t = 1000,000 yr ) translates into:

e. = e/t

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. = e/t = 1/yr = 1/(365 x 24 x 3600 s)

e. = 3.1 x 10-8 s-1

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

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

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

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

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