3 rheology i

7/27/2019 3 Rheology I

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Rheology I


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Rheology Part of mechanics that deals with the flow of rocks,

or matter in general

Deals with the relationship of the following:

(in terms of constitutive equations): stress, s strain, e

strain rate e.(hence time, t)

material properties

other external conditions

Rocks f low given time and other conditions!


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Linear Rheologies

The ratios of stress over strain or stress over

strain rate is constant, e.g.:

Elastic behavior: s = Ee Viscous behavior: s = e.


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Rheology Explains Behavior

Drop onto a concrete floor four objects:

a gum eraser a ball of soft clay

a cube of halite

one cm3 of honey

When they fall, they behave the same by following the

Newtons Second Law (F = mg)

Their dif ference is when they reach the ground:

The eraser rebounds and bounces (elastic)

The clay flattens and sticks to the floor (ductile)

The halite fractures and fragments scatter (brittle)

The honey slowly spreads on the floor (viscous)


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Material Parameters

Rheology depends on:

Extrinsic (external) conditions such as:

P, T, t, chemistry of the environment

Intrinsic (internal) material properties such as:

rock composition, mass, density


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Material Parameters

Are actually not purely mater ial constants

Are related to the rheological properties of a body, e.g.:

rigidity

compressibility

viscosity, fluidity elasticity

These depend on external parameters

Arescalars in isotropic material and tensors of higherorder in anisotropic material


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Constitutive Equations

Mechanical state of a body is specified by:

Kinematic quantities such as:

strain, e

displacement, d velocity, v

acceleration, a

Dynamic quantities such as:

force, F

stress,


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Constitutive Equations, Example

F = mas = E e The constitutive equations involve both

mechanical and material parameters:

f (e, e., s, s. , , M ) = 0

M is material property depending on P, T, etc.


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Law of Elasticity -Hookes Law

A linear equation, with no intercept, relating

stress (s) to strain (e) For longitudinal strain:

s = E e (de/dt = 0) The proportionality constant E between stress and

longitudinal strain is the Youngs modulus

Typical values of E for crustal rocks are on the

order of 10-11 Pa

Elasticity is typical of rocks at room T and pressures

observed below a threshold stress (yield stress)


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Characteristics of Elasticity

Instantaneous deformation upon application of a

load Instantaneous and total recovery upon removal of

load (rubber band, spring)

It is the only thermodynamically reversible

rheological behavior

Stress and strains involved are small

Energy introduced remains available for returningthe system to its original state (internal strainenergy)

It does not dissipate into heat; i.e., strain is recoverable

Typically, elastic strains are less than a few percents

of the total strain


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Law of Elasticity

.


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Shear Modulus

For shear stress and strains

ss = Gg The proportionality constant G between stress

and shear strain is the shear modulus

(rigidity)


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Bulk Modulus

For volume change under pressure:

P = Kev

K = P/ev is the bulk modulus; ev is dilation

K is the proportionality constant betweenpressure and volumetric strain

The inverse of the bulk modulus is thecompressibility:

k = 1/K


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Units of the proportionality constants

The proportionality constants E, G, and Kare the slope of the line in the s-e diagram(slope = s/e)

Since E, ,G, and K are the ratio of stress

over strain (s/e), their units are stress (e.g., Pa,

Mpa, bar) because e is dimensionless


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Poisson Ration (nu) Under uniaxial load, an elastic rock will shorten under

compression while expanding in orthogonal direction

Poisson ratio: The ratio of the elongation perpendicular to

the compressive stress (called: transverse, et, orlateral

strain, elat) and the elongation parallel to the compressive

stress (longitudinal strain,el)

n = elat

/elong

= et/e

l[no dimension]

It shows how much a core of rock bulges as it is

shortened


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Poisson Ratio

Because rocks expand laterally in response to an

axially applied stress, they exert lateral stress

(Poisson effect) on the adjacent material

If no lateral expansion is allowed, such as in aconfined sedimentary basin or behind a retaining

wall, the tendency to expand laterally produces

lateral stress


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Poisson Ratio n = et/el By setting the lateral (i.e., transverse, et) strains to zero, and

loading a column of earth, describing its tendency to expand byPoisson's ratio and translating these lateral strains into stresses

by Young's modulus we can show that (assume s1is vertical):s2= s3=slateral = sverticaln/(1-n) ors

h

= sv

n/(1-n) (h =horizontal v =vertical) For a material that expands as much as it is compressed, for

example a fluid (n = 0.5), this leads to:sh=sv (hydrostatic response) This relationship is used by engineers in calculating stresses

behind retaining walls to estimate lateral stresses in mine shaftsor in sedimentary basins. This is an elastic model, otheroptions can be used to estimate stress at plastic failure


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Poisson ratio,n=et/elranges between 0.0 and 0.5

n= 0.0for fully compressible rocks, i.e., thosethat change volume under stress without extendinglaterally (i.e., et=0):

if et =0.0 thenn= et/el=0.0

n= 0.5 for fully incompressible rocks whichmaintain constant volume irrespective of stress(material extends laterally):

if et = 0.5el then n= et/el=0.5

Note: asponge has a low Poisson ratioa lead cylinder has a high Poisson ratio


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Values of the Poisson ratio in natural rocks

range between 0.25 and 0.35(n0.25 for most rocks)

The magnitude of lateral stress (sh

= s2

= s3)

for most rocks (i.e., the Poisson effect) is1/3 of the greatest principal stress (sl is

vertical), i.e., s3 = 1/3slRecall: sh = svn/(1-n) or s3=n/(1-n) sls3= 0.25/(1-0.25)s3 = 1/3sl

3 rheology i

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