GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Earth Materials
Lecture 13
Earth Materials
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equationsThese 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
LengthExtensionE
AreaForce
×=
σn = E εn
where E is material constant, the Young’s Modulus
Units are force/area – N/m2 or Pa
Hooke’s law
klijklij C εσ =
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Shear modulus and bulk modulus
Shear or rigidity modulus:
sSS G εµεσ ==Bulk modulus (1/compressibility):
vKP ε=−Can write the bulk modulus in terms of the Laméparameters λ, µ:
K = λ + 2µ/3
and write Hooke’s law as:
σ = (λ +2µ) ε
Young’s or stiffness modulus:
nn Eεσ =
Mt Shasta andesite
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Young’s Modulus or stiffness modulusYoung’s Modulus or stiffness modulus:
nn Eεσ =
Interatomic distance
Interatomic force
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Shear Modulus or rigidity modulusShear modulus or stiffness modulus:
ss Gεσ =
Interatomic distance
Interatomic force
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Hooke’s Law
In the isotropic case this tensor reduces to just two independent elastic constants, λ and µ.
So the general form of Hooke’s Law reduces to:
ijkkijij µεελδσ 2+=
1212
1133221111
22)(
µεσµεεεελσ
=+++=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 so Cijkl 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’s Law
In terms of principal stresses and principal strains:
ijkkijij µεελδσ 2+=
3333221133
2233221122
1133221111
2)(2)(2)(
µεεεελσµεεεελσµεεεελσ
+++=+++=+++=
Hooke’s Law:
Consider normal stresses and normal strains:
3213
3212
3211
)2()2(
)2(
εµλελελσελεµλελσελελεµλσ
+++=+++=+++=
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Hooke’s Law
where E is the Young’s Modulus and υ is 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 λ and µ, or E and υ.
Can write in inverse form:
3213
3212
3211
1
1
1
σσυσυε
συσσυε
συσυσε
EEE
EEE
EEE
+−−=
−+−=
−−=
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equation: uniaxial elastic deformationAll components of stress zero except σ11:
3333221133
2233221122
1133221111
2)(02)(0
2)(
µεεεελσµεεεελσ
µεεεελσ
+++==+++==
+++=
11113322
111111
)(2
)23(
νεεµλ
λεε
εεµλ
µλµσ
−=+
−==
=++
= E
where E is Young’s Modulus and ν is Poisson’s ratio.
The solution to this set of simultaneous equations is:
σ11
ε11
dσ11/dε11 = E
σ11
σ22 = 0
σ11
σ33 = 0
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equations: isotropic compressionNo shear or strain; all normal stresses
equal to –p; all normal strains equal to εv /3.
VV KP εεµλ =⎟⎠⎞
⎜⎝⎛ +=−
32
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 ++=∆
=VV
v -p
εv
-dp/dεv = K
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Typical ERubber 7 MPaNormally consolidated clays 0.2 ~ 4 GPaBoulder clay (oversolidated) 10 ~20 GPaConcrete 20 GPaSandstone 20 GPaGranite 50 GPaBasalt 60 GPaSteel 205 GPaDiamond 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
FractureCalculate the stress which will just separate two adjacent layers of atoms x layers apart x
σ
σ
ε
strain energy / m2 = ½ stress x strain x vol
Ue = ½ σn εn x
σ
εHooke’s law: εn = σn / E
Ue = σn2 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 strain energy between two layers of atoms is potentially convertible to surface energy:
sn UE
x 22
2
≈σ
or xEU
xEU ss
n ≈≈ 2σ
For steel: Us = 1 J/m; E = 200 GPa;
x = 2 x 10-10 m⇒ σmax = 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
rss G
εεεεσ
+= 0
εr is the reference strain
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equation for soilsSoils are fractal materialsThere 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 µ by their definitions in terms of shear stress σs and shear strain εs :
d
s
s
s
s cdd
⎟⎟⎠
⎞⎜⎜⎝
⎛=
εσ
εσ
constitutive equation for soils
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equation for soilsd
s
s
s
s cdd
⎟⎟⎠
⎞⎜⎜⎝
⎛=
εσ
εσ
From fractals:
Integrating and setting d = 2:rs
rss G
εεεεσ
+= 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/dε of 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 transitionThis aspect of soil behaviour is completely different from crystalline rock
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Constitutive equation: viscous flowIncompressible viscous fluidsFor viscous fluids the deviatoric stress is proportional to strain-rate:
where η is the shear viscosityijij
•
= '' 2 εησ
Viscosity is 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/m3) 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 lava flow 100 Pa s, an a'a flow has a viscosity of 1000 Pa s. We can mentally imagine a sphere dropping through them and how long it might take.
ε
σ1/2η
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Experimental techniques to study friction
Shear box
Rotary shearTriaxial test
Direct shear
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Experimental resultsAt low normal stresses (σN < 200 MPa)
Linear friction law observed: σS = µ σNA significant amount of variation between rock types: µcan vary between 0.2 and 2.0 but most commonly between 0.5 – 0.9Average 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 obey Amonton’s Law (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 = µ N - µ is the coefficient of friction
2nd: Friction force (F) is independent of the areas in contact
So in terms of stresses: σS = µ σN = σN tanφ
May be simply represented on a Mohr diagram:
σS
σN
φµ= tan φ
φ is the “angle of friction”slope µ
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Field observationsWe are concerned with friction related to earthquakes, i.e., friction on faultsFaults 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’s Friction Laws
All data may be fitted by two straight lines:σN < 200 MPa σS = 0.85 σN
σN > 200 MPa σS = 50 + 0.6 σN
These are largely independent of rock typeIndependent of roughness of contacting surfacesIndependent of rock strength or hardnessIndependent of sliding velocityIndependent 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
ConfiningPressure PC
Differential Stress (σ1 - σ3) Total
AxialStress σ1
PCHydrostaticPC applied inall directionsprior to thedifferentialloading.
PC PC = σ2 = σ3
Modes of brittle fracture in a triaxial system
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Bottom steelFv520 piston
Pressure Vessel
Fibrous alumina insulation
Bottom plug
Bottom waveguide
Top wave-guide
Pore fluid inlet
Rock Specimen
Load Cell
Alumina coil support
Alumina Disc
Top steelFv520 piston
Top pyrophillite enclosing disc
Bottomenclosing pyrophillite
block
Insulating filler
To AE transducer
Fluid outlet fitting Thermocouple feedthrough
Pressure fittings
Actuator applying axial load
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Experimental resultsSchematic 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
slope µ i
Tensile fracture
(σ2 = -σT)
Shear fracture
CσT – tensile strength