STRAIN
RATE & RHEOLOGY
Strain
Rate=Strain per unit time
Time
required to accumulate a given strain
Expressed as the elongation per time
=change in length/ (original length)(time)
Geologic
strain rates typically ~ 10�12 - 10�15.
Note
that 3.15 X 1013 seconds are in 1 my
Rheology-
study of flow of matter
Ideal
Materials: Elastic, Viscous, Plastic
1.Elastic
Behavior- linear plot of stress vs strain.
When stress is applied, strain is instantaneous; i.e., not time dependent.
Furthermore, instantaneous recovery ensues upon removal of stress. Some rocks at
shallow depths and for short periods of time, approach ideal elastic behavior
during small magnitudes of deformation. Seismic waves are an example of elastic
behavior. Recoverable or Reversible strain
=
(E)(
)
=
Stress 
= strain
=
Strain= elongation of a line=
=
change in length/ original length of line
E= Young�s Modulus of Elasticity= constant of proportionality that describes the slope of the line
Young's
modulus (constant;stress vs strain slope line)
is a measure of resistance to elastic distortion.
E~
10�11 Pa for crustal rocks. As E increases, the slope becomes
steeper, i.e. the rock is �stiffer�.
Elastic
behavior is also known as Hookean
behavior in which stress is proportional to strain.
Shear
Strain
s= G
G=rigidity
or resistance to change in shape (constant of proportionality)
=
shear strain
Dilation
=
K (V-Vo)/Vo
K= Bulk Modulus= ratio of the pressure change to the resultant dilation
K=
p/v
= change in pressure divided by change in volume
Bulk Modulus (K)= incompressibility
incompressibility= measure of resistance to change in shape
Poissons
Ratio= v=-1
/
3
1=
elongation normal to compressive stress
3=
elongation parallel to compressive stress
Check out the following website:
http://silver.neep.wisc.edu/~lakes/Poisson.html
Akin
to Bulk Modulus (K), Poisson's ratio
(v, nu) is another means to describe the
relationship between volume change and stress. v
may range from 0- 0.5 (compressible (0) to (0.5) incompressible). Incompressible
materials maintain constant volume regardless of the stress applied. Most rocks
have a Poisson's ratio ~0.25-0.35. Poisson's ratio describes the ability of a
material to shorten parallel to
1
without corresponding elongation in the
3
direction.
2.Viscous
Strain- Stress is proportional to strain
rate (Newtonian behavior; not really
observed in rocks); characteristic of some fluids. Low viscosity fluids deform
more rapidly than high viscosity fluids, under a given set of conditions. No
critical stress value is required to achieve strain. The constant of
proportionality is the viscosity of the fluid. Permanent strain
=
=
Stress= (shear strain rate)(Viscosity)
3.Plastic
Behavior- continuous deformation after
some critical stress (
c)
value is achieved and maintained. Many rocks exhibit plastic behavior. Permanent
Strain
Three
Megascale Types of Deformation-
Visible
effects of strain in rocks are usually of plastic or rupture variety as elastic
strain produces little long term features.
a.Elastic
Deformation
b.Plastic
Deformation
c.Rupture
Deformation
Elastic
Deformation-Occurs when a body is deformed
in response to a stress, but returns to its original shape when stress is
removed. Stress is totally reversible or recoverable.
Viscoelastic
(Anelastic) Strain- strain totally
recoverable but not instantaneous recovery; time dependent, describe in terms of
strain rate. Most rocks have elastic and anelastic properties at small stress
magnitudes.
Plastic
Deformation- Irreversible strain without
visible fractures. Stress is applied to a rock body and deformation occurs. When
stresses are removed, a portion of the strain remains. That portion of the rock
that is deformed has experienced plastic strain. Permanent plastic deformation precludes
visible fractures. Material deforms but does
not break and produce visible fractures. Microscopic fracturing may occur,
however. Plastic strain is not recoverable or reversible.
Rupture
Deformation- visible fractures form.
Irreversible, not recoverabe strain. Material loses cohesion.
Terms describing Behavior of
Materials during Deformation:
Ductile-
Rocks experience large amounts of plastic deformation before rupturing.
Plastic-flow
without mesoscopic brittle behavior
Brittle-
Rocks that exhibit elastic behavior followed by rupture.
Rupture-
loss of cohesion; occurs prior to significant amounts of plastic deformation.
Elastic
Limit- ductile rocks deform elastically to
a point (stress value of which is the yield strength),
beyond this point, plastic deformation ensues with increasing stress.
Rupture
point- (rupture strength)
brittle rocks experience elastic deformation until a rupture point is attained,
whereat the rock deforms by brittle rupture.
Failure-
point when a brittle rock loses all resistance to stress and crumbles.
Failure
is difficult to discern in plastic deformation.
Ultimate
Strength- maximum stress that a rock can
support before failure.
Competency-
relative term that compares the resistance of
rocks to flow.
Generalized
Stress-Strain Curve for Rocks
Brittle Rocks-
exhibit elastic behavior before rupture
Ductile Rocks- exhibit elastic-plastic behavior before rupture
Rock Tests
= Stress - Strain Behavior
Triaxial
Test- used to simulate behavior of rocks
at depth. A cylindrical sample is enclosed in a jacket through which a radial
confining pressure can be applied using gases or liquids. An axial load is then
applied. Repeated tests are conducted to determine failure at different axial
(normal) and confining (shear) values. Thus a failure envelope
may be constructed on a Mohr's diagram by constructing tangential lines to Mohr's
circle representing the failure values for each test result. Note that failure
does not occur at the maximum shear level.
Effects
of Confining Pressure (Pc): The unconfined
compression test is applicable to rock engineering where rock masses are exposed
at the Earth�s surface. However, in the design of tunnels, mines, or other
underground excavations (waste repositories), confining pressures are important.
At depth, the minimum principal stress is 3
is no longer zero as in unconfined compression tests.
Pc=
g h
=
density
Pc= lithostatic pressure
g= gravity
h= depth
Confining pressure- due to the weight of the surrounding rock. Unlike hydrostatic pressure
which is equal in all directions, lithostatic pressure is not always equal in
all directions. The principal compressive stress 1
may be oriented vertically or laterally.
As
confining pressure increases:
1.Rock
strain proceeds from brittle to ductile behavior. Ductile deformation dominates
at Pc > 700 kg/cm2.
2.Higher
confining pressures resists opening of fractures
2.Rock
strength increases (greater amounts of strain accumulate before failure occurs).
________________________________________
Rock
strength decreases with:
1.Increasing
temperatures (reduces rock strength and increases ductility)
2.decreasing
strain rate (increasing time); rocks more ductile at lower strain rates.
3.Foliations
4.Increased
porosity
5.Presence
of water
_________________________________________
Effective Pressure= Pc-Pf
Pc= confining pressure
Pf= fluid
pressure
Work
Hardening- Stress necessary to continue
deformation increases as strain increases
Work
Softening- Stress necessary to continue
deformation decreases as strain increases
Thus,
Rock Strength determined by:
Degree of interlocking of mineral grains
Presence of discontinuities
Degree of weathering
Mineral properties
Climate
Grain size and variability
Rock density
Cementation and compaction
and
temperature
confining pressure
fluid pressure
porosity
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