Transcript Rheology

Rheology I
Rheology
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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 flow given time and other conditions!
Linear Rheologies
The ratios of stress over strain or stress over
strain rate is constant, e.g.:
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Elastic behavior: s = Ee
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Viscous behavior: s = ηe.
Rheology Explains Behavior
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Drop onto a concrete floor four objects:
 a gum eraser
 a cube of halite
 a ball of soft clay
 one cm3 of honey
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When they fall, they behave the same by following the
Newton’s Second Law (F = mg)
Their difference 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
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Rheology depends on:
 Extrinsic (external) conditions such as:
 P, T, t, chemistry of the environment
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Intrinsic (internal) material properties such as:
 rock composition, mass, density
Material Parameters
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Are actually not purely “material constants”
Are related to the rheological properties of a body, e.g.:
 rigidity
 compressibility
 viscosity, fluidity
 elasticity
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These depend on external parameters
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Are scalars in isotropic material and tensors of higher
order in anisotropic material
Constitutive Equations
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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, σ
Constitutive Equations, Example
F = ma
s=Ee
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The constitutive equations involve both
mechanical and material parameters:
.
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f (e, e , s, s , ……, M ) = 0
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M is material property depending on P, T, etc.
Law of Elasticity - Hooke’s Law
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A linear equation, with no intercept, relating
stress (s) to strain (e)
For longitudinal strain:
s=Ee
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(de/dt = 0)
The proportionality constant ‘E’ between stress and
longitudinal strain is the Young’s 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)
Characteristics of Elasticity
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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 returning
the system to its original state (internal strain
energy)
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It does not dissipate into heat; i.e., strain is recoverable
Typically, elastic strains are less than a few percents
of the total strain
Law of Elasticity
.
Shear Modulus
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For shear stress and strains
ss = Gg
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The proportionality constant G between stress
and shear strain is the shear modulus
(rigidity)
Bulk Modulus
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For volume change under pressure:
P = Kev
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K = P/ev is the bulk modulus; ev is dilation
K is the proportionality constant between
pressure and volumetric strain
The inverse of the bulk modulus is the
compressibility:
k = 1/K
Units of the proportionality constants
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The proportionality constants ‘E’, ‘G’, and ‘K’
are the slope of the line in the s-e diagram
(slope = s/e)
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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
Poisson Ratio, n (nu)
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Under uniaxial load, an elastic rock will shorten under
compression while expanding in orthogonal direction
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Poisson ratio: The ratio of the elongation perpendicular to
the compressive stress (called: transverse, et, or lateral
strain, elat) and the elongation parallel to the compressive
stress (longitudinal strain, el)
n = elat/elong = et/el [no dimension]
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It shows how much a core of rock bulges as it is
shortened
http://silver.neep.wisc.edu/~lakes/PoissonIntro.html
| ½ el
_ ½ et
n = et /el
http://en.wikipedia.org/wiki/Poisson's_ratio
Poisson Ratio …
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Because rocks expand laterally in response to an
axially applied stress, they exert lateral stress
(Poisson effect) on the adjacent material
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If no lateral expansion is allowed, such as in a
confined sedimentary basin or behind a retaining
wall, the tendency to expand laterally produces
lateral stress
Poisson Ratio n = et/el
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By setting the lateral (i.e., transverse, et) strains to zero, and
loading a column of earth, describing its tendency to expand by
Poisson's ratio and translating these lateral strains into stresses
by Young's modulus we can show that (assume s1 is vertical):
s2 = s3 = slateral = svertical n/(1-n)
or
sh = sv n/(1-n)
(h =horizontal, v =vertical)
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For a material that expands as much as it is compressed (fully
incompressible), for example a fluid (n = 0.5), this leads to:
sh = sv
(hydrostatic response)
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The second equation is used by engineers in calculating
stresses behind retaining walls to estimate lateral stresses in
mine shafts or in sedimentary basins. This is an elastic model,
other options can be used to estimate stress at plastic failure
Material become narrower when they are stretched!
http://silver.neep.wisc.edu/~lakes/PoissonIntro.html
Poisson ratio, n =et/el ranges
between 0.0 and 0.5
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n = 0.0 for fully compressible material, i.e.,
those that change volume under stress without
extending laterally (i.e., et=0):
if et=0.0  n =et/el=0.0
Note: Sponge has a low n!
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n = 0.5 for fully incompressible material (e.g.,
fluid) which maintain constant volume irrespective
of stress (material extends laterally): i.e.,
n =et/el=0.5  et=0.5el
Note: lead cylinder a high n!
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Values of the Poisson ratio in natural rocks
range between 0.25 and 0.35
(n  0.25 for most rocks)
For a n  0.25 , the magnitude of lateral stress
(sh = s2 = s3) for most rocks (i.e., the Poisson
effect) is  1/3 of the greatest principal stress
(sl is vertical), i.e., s3 = 1/3 sl because:
sh = sv n/(1-n)
s3 = 0.25/(1-0.25)
or
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s3 = n/(1-n) sl
s3 = 1/3 sl