# Earth Materials

Mechanics

Jul 18, 2012 (6 years ago)

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GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Earth Materials
Lecture 13
Earth Materials
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equations
These are relationships between forces and deformation in a continuum, which
define the material behaviour.
Hooke’s law of elasticity
Robert Hooke (1635-1703) was a virtuoso
scientist contributing to geology,
palaeontology, biology as well as mechanics
Length
Extension
E
Area
Force
×=
σn
= E εn
where E is material constant, the
Young’s Modulus
Units are force/area –N/m
2
or Pa
Hooke’s law
klijklij
C
ε
σ
=
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Shear modulus and bulk modulus
Shear or rigidity modulus:
sSS
G
ε
µ
ε
σ
=
=
Bulk modulus (1/compressibility):
v
KP
ε
=

Can write the bulk modulus in terms of the Lamé
parameters λ, µ:
K = λ+ 2µ/3
and write Hooke’s law as:
σ= (λ+2µ) ε
Young’s or stiffness modulus:
nn
E
ε
σ
=
Mt Shasta andesite
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Young’s Modulus or stiffness modulus
Young’s Modulus or stiffness modulus:
nn
E
ε
σ
=
Interatomic distance
Interatomic force
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Shear Modulus or rigidity modulus
Shear modulus or stiffness modulus:
ss
G
ε
σ
=
Interatomic distance
Interatomic force
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Hooke’sLaw
In the isotropic case this tensor reduces to just two independent elastic
constants, λand µ.
So the general form of Hooke’sLaw reduces to:
ijkkijij
µ
ε
ε
λδ
σ
2
+
=
ㄲㄲ
ㄱ㌳㈲ㄱㄱ
2
2)(
µεσ
µ
ε
ε
ε
ε
λ
σ
=
+
+
+
=
For example:
Normal stress
Shear stress
This can be deduced from substituting into the Taylor expansion
for stress and differentiating.
σij
and εkl
are second-rank tensors soC
ijkl
is a fourth-rank tensor.
For a general, anisotropic material there are 21 independent elastic moduli.
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Hooke’sLaw
In terms of principal stresses and principal strains:
ijkkijij
µ
ε
ε
λδ
σ
2+
=
3333221133
2233221122
1133221111
2)(
2)(
2)(
µεεεελσ
µεεεελσ
µ
ε
ε
ε
ε
λ
σ
+++=
+++=
+
+
+
=
Hooke’sLaw:
Consider normal stresses and normal strains:
3213
3212
3211
)2(
)2(
)2(
εµλελελσ
ελεµλελσ
ε
λ
ε
λ
ε
µ
λ
σ
+++=
+++=
+
+
+
=
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Hooke’sLaw
where E is the Young’s Modulus and υis the Poisson’s ratio.
Poisson’s ratio varies between 0.2 and 0.3 for rocks.
A principal stress component σi
produces a strain σI
/E in the
same direction and strains (-υ.σi
/ E) in orthogonal directions.
Elastic behaviour of an isotropic material can be characterized
either by specifying either λand µ, or E and υ.
Can write in inverse form:
3213
3212
3211
1
1
1
σσ
υ
σ
υ
ε
σ
υ
σσ
υ
ε
σ
υ
σ
υ
σε
E
E
E
EEE
EEE
+−−=
−+−=
−−=
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equation: uniaxial elastic deformation
All components of stress zero except σ11:
3333221133
2233221122
1133221111
2)(0
2)(0
2)(
µεεεελσ
µεεεελσ
µε
ε
ε
ε
λ
σ
+++==
+++==
+
+
+
=
11113322
111111
)(2
)23(
νεε
µλ
λ
εε
εε
µλ
µ
λ
µ
σ
−=
+
−==
=
+
+
=E
where E is Young’s Modulus and νis Poisson’s ratio.
The solution to this set of simultaneous equations is:
σ
11
ε11
dσ11/dε11
= E
σ11
σ22
= 0
σ
11
σ33
= 0
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equations: isotropic compression
No shear or strain; all normal stresses
equal to –p; all normal strains
equal to εv /3.
VV
KP
εεµλ
=

+=−
3
2
where K is the bulk modulus;
hence K = λ+ 2/3µ
σ11
= -p
σ22
= -p
σ33
= -p
σ22
= -p
σ11
= -p
σ33
= -p
P = -1/3 (σ11
+ σ22
+ σ33
) = -1/3 σii
332211
εεεε
++=

=
V
V
v
-p
εv
-dp/dεv
= K
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Typical E
Rubber
7 MPa
Normally consolidated clays
0.2 ~ 4 GPa
Boulder clay (oversolidated)
10 ~20 GPa
Concrete
20 GPa
Sandstone
20 GPa
Granite
50 GPa
Basalt
60 GPa
Steel
205 GPa
Diamond
1,200 GPa
Young’s Modulus (initial tangent) of Materials
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
50 MPa5 MPaGranite
40 MPa4 MPaBasalt
40 MPa4 MPaConcrete
10 MPa1 MPaSandstone
1 MPa300 kPaSoil
2,000 MPa30 MPaRubber
3,000 MPa3,000 MPaSteel piano wire
100 / 3 MPa100 / 3 MPaSpruce along/across grain
Compressive strength
-unconfined
Uniaxial tensile
strength
“Strength” of Materials
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Fracture
Calculate the stress which will just separate two
adjacent layers of atoms x layers apart
x
σ
σ
ε
strain energy / m
2
= ½ stress x strain x vol
Ue
= ½ σn
εn
x
σ
ε
Hooke’s law: εn
= σn
/ E
Ue
= σn
2
x / 2E
If Us
is the surface energy of the solid per square metre, then the total
surface energy of the solid per square metre would be 2Us
per square metre
Suppose that at the theoretical strength the whole of the strainenergy
between two layers of atoms is potentially convertible to surface energy:
s
n
U
E
x
2
2
2

σ
or
x
EU
x
EU
ss
n
≈≈2
σ
For steel: Us = 1 J/m; E = 200 GPa;
x = 2 x 10-10
m
⇒σ
max
= 30 GPa≈E / 10
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Griffith energy balance
Microcrack in lava
The reason why rocks don’t reach their theoretical strength is because they
contain cracks
Crack models are also used in modelling earthquake faulting
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Dislocations (line defects) in shear
The reason why rocks don’t reach their theoretical shear strength is because
they contain dislocations
Dislocation models are also used in modelling earthquake faulting
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Engineering behaviour of soils
•Soils are granular materials –their
behaviour is quite different to crystalline
rock
Uniaxial deformation
Shear deformation
•Properties are highly dependent on
water content
•The curvature of the stress-strain is largest
near the origin
•Deformation is strongly non-linear
•The constitutive relation for shear
deformation, found from hundreds
of experiments is:
rs
rs
s
G
εε
ε
ε
σ
+
=
0
εr
is the reference strain
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equation for soils
Soils are fractal materials
There is a lognormal distribution
of grain sizes (c.f. crack lengths
in rocks)
Suppose we subject a soil to a
simple shear strain. The shear
forces applied to each grain must
be lognormally distributed since
they are proportional to the grain
surfaces. So the shear modulus
and rigidity must be related by a
power law:
G = c µd
where d is the fractal dimension
of the grain size distribution
replacing G and µby their
definitions in terms of shear stress
σs
and shear strain εs
:
d
s
s
s
s
c
d
d

=
ε
σ
ε
σ
constitutive equation for soils
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equation for soils
d
s
s
s
s
c
d
d

=
ε
σ
ε
σ
From fractals:
Integrating and setting d = 2:
rs
rs
s
G
εε
ε
ε
σ
+
=
0
This is the same as the empirical constitutive equation!
This is a hyperbolic stress-strain relation (i.e., like a deformation stress-strain curve)
It may be interpreted as saying that the shear modulus G = dσ/dεof a soil decays
inversely as (1 + τ) where τ= εs
/ εr
is the normalised strain
Note that the stress-strain behaviour of soils cannot be linearized at small
strain
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Stress-strain curve of a soil as
compared with that of a crystalline
rock –note different definition of
rigidity
Soil liquefaction: Kobe port area
Motion on soft ground to strong
earthquake is fundamentally
different to small earthquakes
because sediments go through a
phase transition and liquefy
Liquefaction of soils: phase transition
This aspect of soil behaviour is
completely different from
crystalline rock
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equation: viscous flow
Incompressible viscous fluids
For viscous fluids the deviatoric stress
is proportional to strain-rate:
where ηis the shear viscosity
ij
ij

=
''
2
εησ
Viscosityis an internal property of a fluid that offers resistance to flow.
Viscosity is measured in units of Pa s (Pascal seconds), which is a unit of
pressure times a unit of time. This is a force applied to the fluid, acting for
some length of time. A marble (density 2800 kg/m
3) and a steel ball bearing
(7800 kg/m3) will both measure the viscosity of a liquid with different
velocities. Water has a viscosity of 0.001 Pa s, a Pahoehoe lavaflow 100 Pa
s, an a'aflow has a viscosity of 1000 Pa s. We can mentally imagine a sphere
dropping through them and how long it might take.
ε
σ
1/2η
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Experimental techniques to study friction
Shear box
Rotary shear
Triaxial test
Direct shear
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Experimental results
At low normal stresses (σN
< 200 MPa)
Linear friction law observed: σS
= µσ
N
A significant amount of variation between rock types: µ
can vary between 0.2 and 2.0 but most commonly
between 0.5 –0.9
Average for all data given by: σS
= 0.85 σN
At higher normal stresses (σN
> 200 MPa)
Very little variation between wide range of rock types (with
some notable exceptions –esp. clay minerals which can have
unusually low µ
But friction does not obeyAmonton’sLaw (i.e. straight line
through origin) but Coulomb’s Law
Best fit to all data given by:
σS
= 50 + 0.6 σN
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Simple failure criteria
(a)Friction –Amonton’s Law
1st: Friction is proportional normal load (N)
Hence: F = µN-µis the coefficient of friction
2nd: Friction force (F) is independent of the areas in contact
So in terms of stresses: σS
= µσ
N
= σN
tanφ
May be simply represented on a Mohr diagram:
σS
σN
φ
µ= tan φ
φis the “angle of friction”
s
l
o
p
e

µ
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Field observations
We are concerned with friction related to earthquakes, i.e.,
friction on faults
Faults are interfaces that have already fractured in previously
intact material and have subsequently been displaced in shear
(i.e., have slipped)
Hence they are not “mated” surfaces (unlike joints)
Joint
Fault
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Summary:Byerlee’sFriction Laws
All data may be fitted by two straight lines:
σN
< 200 MPaσS
= 0.85 σN
σN
> 200 MPaσS
= 50 + 0.6 σN
These are largely independent of rock type
Independent of roughness of contacting surfaces
Independent of rock strength or hardness
Independent of sliding velocity
Independent of temperature (up to 400oC)
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Experimental results of triaxial deformation tests
σ3
σ3
σ1
σ1
σ1
σ1
σ1
σ1
σ1
σ1
Confining
Pressure PC
Differential Stress
(σ1 - σ3)
Total
Axial
Stress
σ
1
PC
Hydrostatic
PC applied in
all directions
prior to the
differential
PC
PC = σ2 = σ3
Modes of brittle fracture in a triaxial system
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Bottom steel
Fv520 piston
Pressure Vessel

Fibrous alumina
insulation
Bottom plug
Bottom wave
guide
Top wave-guide
Pore fluid inlet
Rock Specimen
A
lumina coil
support
A
lumina Disc
Top steel
Fv520 piston
Top pyrophillite
enclosing disc
Bottom
enclosing pyrophillite
block
Insulating filler
To AE transducer
Fluid outlet fitting
Thermocouple
feedthrough
Pressure fittings
Actuator
applying
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Experimental results
Schematic
stress-strain
curves for rock
deformation
over a range of
confining
pressure
Dependence of
differential
stress at shear
failure in
compression on
confining
pressure for a
wide range of
igneous rocks
Strength of Westerly granite as
a function of confining
pressure. Also shown is
frictional strength.
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Simple failure criteria
(b) Faulting –Coulomb’s Law
σS
= C + µi
σN
= σN
tanφi
C is a constant –the cohesion µi
is the coefficient of “internal”friction
µi
= tan φi
φi
is the “angle of internal friction”
σS
σN
φi
s
l
o
p
e

µ
i
Tensile fracture
(σ2
= -σT)
Shear fracture
C
σT
–tensile strength