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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 o t
) 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 10 5 /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: = E
e
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.
Y
oung’s modulus, How much stress is required to achieve a given amount of length-parallel elastic shortening of a rock. Stress and strain are directly and linearly related = the slope of the line.
Poisson’s Ratio (
n
)
Describes the relationship between lateral strain and longitudinal strain.
n
=
e lat / e long
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 = e lat /e long 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): =
D hydrostatic stress
/
D dilation
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
Rheologic Behavior
2) Viscous behavior
: of time (e.g., strain rate) h e (h is a constant) Strain accumulation is a function 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. [E, elasticity)
Rheologic Behavior
4) Elastico-viscous behavior
:
e
/
E
/ h Elastic deformation with initial stress Viscous behavior Maxwell relaxation time – stress relaxation decays exponentially
Nature rocks and deformation
Deformation experiments
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 jacketed with weak material copper or plastic.
Specimens are drilled out cores that are ‘machined’ to have perfectly parallel and smooth ends. Specimens are carefully measured to determine their initial length (l o ) and diameter (to get initial cross-sectional area, A o ).
Nature rocks and deformation
Deformation experiments
Pressure chamber – confining pressure (P c ) Pore-fluid pressure (P f ) Difference between Pc and Pf (P c – P f ) is effective pressure, P e Adjust pressure
Nature rocks and deformation
Deformation experiments What is confining pressure
P c
g
h
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 c
g
h
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 c
g
h
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
P e = P c - P f
Nature rocks and deformation
Deformation experiments
Pore-fluid pressure Effective pressure is less than confining pressure.
Effective pressure equals confining pressure – pore-fluid pressure P e = P c - P f
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