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Rheology
Relations between stress and strain.
Not easy to define
Rheology: describes the ability of stressed materials to deform.
strain rate
creep regimes
elastic behavior
viscous types of behavior
Strain rate
The time it takes material to accumulate a certain amount of
strain.
.
e  e /t  l /(l t)
o
Elongation (e) per time.
Dimensionless, [t]-1 – unit second-1
For example 30% finite strain e = 0.3 in one hour (3600 sec), 8.3 x 105/sec
One day, 3.5 x 10-6/sec
One year, 9.5 x 10-9/sec
One m.y., 3.15 x 10-15/sec
1.5 cm long fingernail
grows 1 cm/yr
0.67/yr or 2 x 10-8/s
Typical geologic rates are 10-12/s – 10-15/s
Creep curve
Behavior of rocks to
compression is not simple.
Three creep regimes:
1) Primary or transient creep:
strain rate decreases with time
following rapid initial
accumulation
2) Secondary or steady state
creep: strain accumulation is
linear with time
3) Tertiary or accelerated
creep: strain rate increases
with time.
Creep curve
Behavior of rocks to
compression is not simple.
Removing stress in steady
state creep.
1) Drop in strain
2) Permanent strain remains
Rheologic Behavior
Two types of behavior
1) Elastic behavior
2) Viscous behavior
Rheologic Behavior
1) Elastic behavior:
Stress and strain are linear
  Ee
The equation is known as Hookes Law
E = Young’s modulus (slope of stress/strain diagram)
 Seismic waves travel thru elastic medium
 Rubber band analogy
Rheologic Behavior
Hooke’s Law
 = Ee Stress is linearly related to strain by the
constant E, known as Young’s modulus
Rheologic Behavior
Hookes Law
1) This straight line relation between stress
and strain is called Hookes law (e µ ).
Add proportionality constant to get
Hookes law:  = Ee
Strain (e) is linearly proportional to stress ()
where
E = Young’s modulus
E = /e = stress/strain
The value of E, or Young’s modulus describes
the slope of a straight line, stress-strain
curve.
Stress and strain are directly and linearly
related = the slope of the line.
Young’s modulus, How
much stress is required to
achieve a given amount of
length-parallel elastic
shortening of a rock.
Poisson’s Ratio (n)
Describes the relationship between lateral strain and longitudinal
strain.
n = elat / elong
n, another elastic modulus.
Vertical loading will produce horizontal stresses because of the
Poisson effect.
The degree to which a specimen will widen upon shortening is
a function of it’s Poisson’s ratio.
2 = 3 = (n / (1 - n)) 1
For common rocks, Poisson’s ratio tends to be around n = 0.25
Poisson’s ratio, Greek
letter nu (n).
This describes the amount
that a rock bulges as it
shortens.
The ratio describes the ratio
of lateral strain to longitudinal
strain:
n = elat/elong
Poisson’s ratio is unit-less,
since it is a ratio of extension.
What does this ratio mean?
Typical values for n are:
Fine-grained limestone: 0.25
Apilite: 0.2
Oolitic limestone: 0.18
Granite: 0.11
Calcareous shale: 0.02
Biotite schist: 0.01
Poisson’s ratio
If we shorten a granite and
measure how much it bulges,
we see that we can shorten a
granite, but it may not be
compensated by an increase
in rock diameter.
So stress did not produce the
expected lateral bulging.
Somehow volume decreases
and stress was stored until the
rock exploded!
Thus low values of Poisson’s
ratio are significant.
Bulk and Shear Moduli
Bulk modulus (K): = Dhydrostatic stress / Ddilation
Shear modulus (G): = s / g
The two other parameters that describe the elastic relationship between
stress and strain are:
1) Bulk modulus (K): resistance that elastic solids to changes in volume.
Divide the change of hydrostatic pressure by the amount of dilation
produced by pressure changes.
K = bulk modulus = hydrostatic stress /dilation
2) Shear modulus (G): resistance that elastic solids to shearing:
Divide shear stress (s) by shear strain (g)
G = shear modulus = s/g
Rheologic Behavior
1) Elastic behavior:
  Ee
Stress and strain are linear
Reversible. Once stress is removed, the material returns to its
original shape – strain is recoverable
Instantaneous
response to strain

Rheologic Behavior
2) Viscous behavior:   h  e
of time (e.g., strain rate)
Strain accumulation is a function
(h is a constant)
Non-recoverable
strain and permanent.

Leaky hydraulic cylinder: the resistance to flow
Examples:
Upper mantle, lower mantle, magmas, ice, salt domes
Rheologic Behavior
3) Viscoelastic behavior:
  E  e +h  e
Reversible deformation
Strain accumulation and recovery is delayed.
 that is loaded on top
Water soaked sponge
[E, elasticity)
Rheologic Behavior
4) Elastico-viscous behavior: e   / E   / h
Elastic deformation with initial stress
Viscous behavior
 recovery is delayed.
Strain accumulation and
Maxwell relaxation time – stress relaxation decays exponentially
Nature rocks and deformation
Deformation experiments
 Specimens are jacketed
with weak material copper or plastic.
 Specimens are drilled
out cores that are
‘machined’ to have
perfectly parallel and
smooth ends.
 Experiments are carried out in steel pressure
vessels.
 Confining pressure (2 = 3) is often supplied by
fluid that surrounds the specimen.
 Temperature can be varied.
 Pore-fluid pressure can also be varied.
 Specimens are carefully
measured to determine
their initial length (lo) and
diameter (to get initial
cross-sectional area, Ao).
Nature rocks and deformation
Deformation experiments
 Pressure chamber –
confining pressure (Pc)
 Pore-fluid pressure (Pf)
 Difference between Pc
and Pf (Pc – Pf ) is
effective pressure, Pe
 Adjust pressure
Nature rocks and deformation
Deformation experiments
What is confining pressure
P  gh
c
Lithostatic pressure
High confining pressure & rock
strength
Compression stress-strain curves at
various confining pressure at 25°C
Nature rocks and deformation
Deformation experiments
What is confining pressure
P  gh
c
Lithostatic pressure
High confining pressure & rock
strength
Compression stress-strain curves at
various confining pressure at 400°C
Nature rocks and deformation
Deformation experiments
What is confining pressure
P  gh
c
Lithostatic pressure
High confining pressure and
rock strength
Changing confining pressure on
various rock types
Nature rocks and deformation
Deformation experiments
Role of temperature and
rock strength
Compression stress-strain curves at various confining pressure
at 400°C
Nature rocks and deformation
Deformation experiments
Role of temperature and
rock strength
Yield strength decreases
with increasing
temperatures
Yield strength: the
maximum stress that a
rock can support until is
fails (flows)
Temperature & rock strength
Nature rocks and deformation
Deformation experiments
Summary:
Experiments
demonstrate that rocks
have higher strength with
increasing depth.
At higher pressures,
rocks have lower
strength in the Earth’s
crust, where we find
higher temperatures.
Temperature & rock strength
Nature rocks and deformation
Deformation experiments
Role of strain rate and
rock strength
Decreasing strain rates
causes decreased rock
strength
Silly putty analogy
At 400° C, differential stress is 20 mpa at 10-14/s
At 400° C, at 10-6/s, differential stress is 160 mpa
Nature rocks and deformation
Deformation experiments
Pore-fluid pressure
Acts in all directions
Increase of pore-fluid
pressure = drop in rock
strength
Rocks are weaker with
high pore-fluid pressure
Effective pressure equals
confining pressure – pore-fluid pressure
Pe = Pc - Pf
Nature rocks and deformation
Deformation experiments
Pore-fluid pressure
Effective pressure is less
than confining pressure.
Effective pressure equals
confining pressure – pore-fluid pressure
Pe = Pc - Pf
Elastic deformation
What is the state of stress on a
Mohr diagram?
The state of stress plots as a
single point on the Mohr diagram,
because the axial stress equals
the confining pressure.
Differential stress: d = 1 - 3
The state of stress appears on the Mohr
diagram as successively large circle, of
diameter 3 - 1, sharing on the
confining pressure 3, as a common
point.
Eventually the sample starts to
deform plastically.
Its elastic behavior is surpassed,
and non-recoverable
deformation begins to
accumulate in the rock.
Plastic deformation produces
deformation in a rock without
failure by rupture.
The onset of plastic deformation
begins when the stress-strain
curve departs from the straight
line elastic mode.
Below its yield
strength the rock
behaves as an elastic
solid.
The point of departure from elastic
behavior is called the elastic limit.
Its value is known as yield strength.
Faulting finally takes place at
about 120 MPa and the stress
drops to zero.
Some of the elastic energy is
expended making the fracture,
some in sound, some in the
frictional heating due to sliding.
When we remove the sample,
we notice that the fracture lies
about 24° to the axis of the
cylinder.
1) Brittle rocks first shorten
a elastically during these
tests.
2) Then they fail abruptly
by discrete fractures.
3) Sometimes plastic
deformation occurs before
failure, called strain
softening.
Just prior to failure, what if we raised the confining pressure and repeated
the experiment on the same sample? How would the limestone respond?
Work hardening & softening
When the load is reapplied at
to Point C, the elastic limit is
greater than during the first
test.
The yield strength is also
greater, because the original
fabric of the rock was changed
slightly by the plastic
deformation.
This rock has undergone
strain hardening.
The yield strength increases
due to modification of original
rock.
Applying more load, the
limestone displays an
increase of plastic behavior
before fracturing, unlike the
previous experiment.
This accelerated plastic
deformation is called strain
softening, because less
stress is required for each
new increment of strain.
Eventually the rock fractures,
but the rupture strength is
greater in this experiment.
Rupture strength is
the stress level of
failure by fracturing.
Rocks become
stronger at higher
levels of confining
pressure.
Deformation in the lithosphere
Rheologic stratification in the
lithosphere
Brittle-ductile transition
Strength: stress that a material can
support before failure
Competency: Resistance of rocks to
flow.
Interplay of lithospheric strength, rock
composition, and depth (temperature)
Deformation in the lithosphere
Faulting and folding with brittle to ductile behavior