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CHAPTER 5
RHEOLOGY
Dr. Masdouq Al-Taj
Introduction
• Rheology is the study of the flow of material.
• In Other words: The relation between stress and
strain with time.
• Do rocks flow?
• Geologically speaking and due to the
availability of time, rocks are able to flow. Not
in the same physical meaning, but with the final
result, which can be achieved by rocks.
Strain Rate
Strain rate (ė) is the time interval it takes to accumulate a
certain amount of strain; and it is defined as the elongation
per time.
• ė = e/t = l/l0t
t in second
Example: 30% finite longitudinal strain (|e|= 0.3) is achieved
in an experiment that lasts one hour (3600 s). The
correspond strain rate is ė = 0.3/3600 = 8.3 x 10–5/s
Now let’s see what happens to the strain rate when we change
the time interval, but maintain the same amount of finite
strain of 30%.
t=1 day (86.4 x 103 s)
ė =3.5 x 10–6/s
t=1 year (3.15 x 107 s)
ė =9.5 x 10–9/s
t=1 m.y. (3.15 x 1013 s)
ė =9.5 x 10–15/s
Q: What is the strain rate of a 200 km long
thrust sheet moving 50 km in 1 m.y.?
Answer: Now we can calculate the strain rate
ė = e/t =  l/l0t = 50/200 x 3.15 x 1013
ė = 0.8 x 10-14/s
In some cases, shear strain rate preferred to
be used rather than strain rate.The relation
between shear strain rate (γׂ) and strain
rate is: γׂ= 2 ė
• A widely used estimate is based on the
Quaternary displacement along the San Andreas
Fault in California, which gives a strain rate on
the order of 10–14/s (moderate geological strain).
• Which agrees well with present-day observation
of plate velocities.
• For geological processes, the typical range of
strain rate is between 10-12/s to 10-15/s. There are
some exceptions.
General behavior: the creep curve
•
Three creep regimes are obtained:
I. Primary or transient creep
(decreasing strain rate)
II. Secondary or
steady-state creep
(constant strain rate)
III. Tertiary or
accelerated creep
(increase strain rate)
Rheologic Relationships
•
The behavior of materials can be divided into
three types:
1. Elastic behavior: recoverable and
instaneous response to stress (time independent)
2. Viscous behavior (Plastic behavior in rock)
3. Combination of elastic and viscous behaviors
(visco-elastic and elastico-viscous)
The stress-strain relation is commonly expressed
in graphs known as stress-strain diagrams.
1. Curve A is a brittle substance.
2. Curve B is an ideal plastic substance.
3. Curve C more normal type of plastic behavior
4. Curve D another
common type of
plastic deformation.
•
Stress-strain curve showing three types of rheological
behavior:
1. Elastic region: Stress is proportional to strain (Hook’s law)
2. Plastic region: Behave according to ductile flow.
3. Brittle Failure region : fracture.
* Elastic limit is the maximum
stress under which the material
exhibits elastic strain. Beyond
this value, the material
undergoes permanent
deformation by ductile flow
or by brittle fracture.
Elastic Behavior
- Modulus of Elasticity
• We can express elastic behavior by the
equation:
 σ = E • e,
 where E is the modulus of elasticity or Young’s
Modulus
1. Young’s Modulus Description
• Another description of Young’s modulus is the
slope of the line in a σ - e diagram
• The slope is the tangent of angle 
• Figure 5.3a in text
Dimensions of Young’s Modulus
• Since strain I dimensionless, Young’s
modulus has the same units as stress,
Pascals
• A typical value of Young’s modulus for
rocks is -1011 Pa
• The sign is negative because we apply a
positive stress (compression) to produce a
negative elongation (shortening)
2. Rigidity
• Another expression for elastic behavior is
given by
 σs = G • γ
 where G is the rigidity and γ is shear strain
3. Bulk Modulus and
Compressibility
• We can also write equations for dilation:
 σ = K • ((V - V0)/V0)
 K is the bulk modulus, or incompressibility
 Its reciprocal is the compressibility, β
4. Poisson’s Ratio
• This ratio may be expressed as:
ν = e ┴ /e //
 where e┴ is the elongation perpendicular
to the compressive stress
 e// is the elongation parallel to
compressive stress
Plastic Behavior
• After stress is
removed, there is a
rapid drop in strain,
followed by a period
of relaxation, during
which the strain
decreases a bit more
before
Viscous Behavior
• The symbol to the left is a “dashpot”, a type of
leaky piston-cylinder apparatus
• Resistance encountered by the piston depends on
the viscosity of the fluid
Definition of Viscosity
• Viscosity is defined by the term η in the
equation relating stress and strain rate:
 σ=ηė
• Viscosity dimensions may be deduced from
the equation
• Stress is pressure per unit area, or Pa
• Strain rate has units of reciprocal time, or s-1
• So, viscosity must have units of Pa. s
Low Viscosity
Position after short
period of flow
Initial position
Moderate
Viscosity
• Videos of pahoehoe
flows, Hawaii
Relative Viscosities
Substance
Air (at 18 oC)
Water (at 20 oC)
Canola Oil at room temp.
Motor Oil at room temp.
Corn syrup at room temp.
Pahoehoe lava
A'a lava
Andesite lava
Rhyolite lava
Viscosity (Pa s)
1.9 x 10-5 (0.000019)
1 x 10-3 (0.001)
0.1
1
8
100 to 1,000
1000 to 10,000
106 to 107
1011 to 1012
Strain Rate of the Mantle
• We can make an estimate of the strain rate
for the mantle using a viscosity of 1021 Pa.s
and a differential stress of 50 MPa
 50 x 106 Pa ÷ 1021 Pa • s = 5 x 10-14 sec-1
Combinations
Viscoelastic Behavior
• The equation for visco-elastic behavior is:
 σ = η ė +E • e
• An example is a sponge filled with water
Viscoelastic Diagram
• When a weight is placed on the sponge, the sponge
loses water (viscous behavior) until the water is gone.
• Then the sponge supports the load elastically
Example of Viscoelastic
Behavior
• Ropy pahoehoe surface Upper crust behaves in a
viscoelastic fashion before
solidifying into a brittle
crust
• The soft crust can deform
into folds and ropy
structures as the result of
flow beneath the
viscoelastic layer
Elasticoviscous
• The spring responds instantly to stress, whereas the
dashpot moves and keeps moving as long as stress is
present
• When stress is removed, the dashpot stops moving
and stays put.
Elasticoviscous Equation
• Elasticoviscous behavior is given
mathematically as:
 ė = σ/E + σ/η
it
Initial sigma should have a dot symbol above
Relaxation Time of Mantle
• Thus, the elasticoviscous model seems to fit
the mantle.
• What is the relaxation time of the mantle?
• If the viscosity is 1021 Pa • s and G = 1011
Pa, we get a relaxation time of 1010 seconds,
or about 300 years
General Linear Behavior
• General linear behavior is a combination of
elasticoviscous and viscoelastic behavior in series.
Comparison of Rock Strengths
• General linear behavior can be rewritten as
a function of stress:
 σ = (ė • A)1/nexp(E*/RT)
• Substituting known values of A, n, and E* at
constant T allows us to compare the relative
strength of different rock types.
Creep Strength versus Depth
Natural Rocks
•
It is very important to examine the relationships
between stress, strain and strain rate by using
natural rocks to have better understanding of the
flow of rock (Rheology).
• What are the main reasons of doing
experiments on natural rocks?
1. We observe the actual behavior of natural rocks.
2. We can vary several parameters in our
experiments such as pressure, temperature,
time and fluid pressure, and to examine their
roles in rock deformation.
The effect of the following parameters in the
triaxial tests are:
Effective Pressure
• pc - pf is equal to the
effective pressure
• Pistons at both ends of the
cylinder allow the
experimenter to impose
pressure along the vertical
axis, called the axial stress
1. Confining (Lithostatic) Pressure
(Pc)
Acts equally in all direction (1 = 2 = 3)
By changing the confining pressure during
the experiments, we observe a very important
characteristic.
1. With increasing confining pressure greater
amounts of strain accumulate before
failure occurs.
2. Increasing confining pressure, increases the
viscous component and the rock’s ability to
flow.
3. Higher confining pressures increasingly
resist the opening of fractures.
•
•
Clapping Underwater
• Clapping one’s
hand in the air is
easy, but doing it
under water is
much harder
• The water resists
movement
• By increasing the experimental temperatures the
effects of confining pressure become clear.
4. Larger strain can be achieved before failure
with increasing depths in the Earth (lithostatic).
• The amount of strain before failure (ductility)
differs significantly among the various rock types
2. Temperature
• A change in temperature conditions also
produces a marked change in response.
1. Most rocks are ruptured at low T.
2. At these conditions most of the strain prior to
rupture is recoverable (elastic).
3. When T increases, the elastic portion of the
strain decreases while the plasticity
increases.
4. Rocks have lower strength and become more
ductile with depth in the Earth, where we find
higher T.
Differential Stress vs.
Temperature
• Effect of increasing temperature
on the ability of various rocks
to withstand a given differential
stress
• In all cases, the rocks grow
weaker as temperature
increases, but the effect is much
higher on pure calcite than
other materials.
3. Strain Rate
1. Decreasing the strain rate results in decreased
rock strength and increased ductility (viscosity)
2. T. changes produce similar effects as strain rate
variations in rocks experiments (h. T  s. ė)
Young Experimentalist
• Human factors also play a
role
• Old experimenters may die
or become incapacitated
• Younger experimenters,
who face lower odds of
either event happening,
have to obtain results
quickly to justify continued
funding
4. Pore-fluid Pressure (Pf)
• Some rock types (sandstone, shale) contain a
significant fluid component that affects their
behavior under stress.
• Experiments show that
increasing the pore-fluid
pressure produces a drop
in the strength and
reduces the ductility
of the sample.
Quartz, Wet and Dry
• Effect of water on the
mineral quartz
• Strength decreases as
temperature increases,
while the quartz is dry,
but when it is wet
……….
• Rocks are weaker when the pore-fluid pressure
is high.
• Increasing the pore-fluid pressure will have the
same effect as decreasing the confining pressure
of the experiment.
Effect pressure = Confining pressure – fluid pressure
(Pe = Pc – Pf)
Significance of Experiments to
Natural Conditions
• Increasing the confining pressure (Pc) and fluid
pressure (Pf) have an opposite effects.
• Increasing temperature (T) and lowering strain
rate (ė) have the same effects.
• Confining pressure and temperature, which both
increase with depth in the Earth, result in rocks
that increasingly resist failure, while at the same
time they allow larger strain accumulation, and
increase the ability for rocks to flow.
• High fluid content is more complex and may
promote fracturing if Pf is high.
Strain Softening and Hardening
• D no elastic component, with
strain softening
• E is elastic-plastic behavior,
with permanent strain at
constant stress above the yield
stress
• F is elastic strain followed by
permanent strain that requires
increasingly higher stresses to
accumulate (strain hardening)
Terminology
• We also need to introduce some other terms
– Strength is the stress a rock can withstand
without failure
– Competency compares the resistance of
various rock to flow
• Qualitative competency guides have been
developed from field observations and from
experimental data
Competency Guides,
Sedimentary Rocks
• In increasing order of competency:






Salt (Low)
Shale
Limestone
Graywacke
Sandstone
Dolomite (High)
Competency Guides,
Crystalline Rocks
• In increasing order of competency:






Schist (Low)
Marble
Quartzite
Gneiss
Granite
Basalt (High)
Rheological Properties of Minerals
• Experimental data can
also be used to categorize
rheological properties on
the basis of properties of
the major minerals which
comprise the rocks.
• From the previous observation we would
predict that:
1. Brittle behavior (fracturing) is largely
restricted to the upper crust. (faulting and
earthquakes < 15km depth).
2. Ductile behavior (flow) dominates at greater
depth.
• One of the geological structure’s classification
based on the cohesiveness during deformation:
Brittle
 Fracture system
Ductile
 Fold system
Brittle-ductile  Shear zones
Brittle-Ductile Classification Scheme
•
•
•
•
(a) brittle (Faults)
(b) brittle-ductile (faults and folds)
(c ) brittle-ductile (small crack faulting and folding)
(d) ductile (folding)