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

loading.

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

Load Cell

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

axial load

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

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