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Home , Bulk modulus, Earth materials, Shear modulus

Materials

Lecture 13

Earth Materials

GNH7/GG09/GEOL4002 SEISMOLOGY AND EARTHQUAKE HAZARD Hooke’s law of

Force Extension = E × Area Length Hooke’s law

σn = E εn

where E is material constant, the Young’s Modulus Units are force/area – N/m2 or Pa Robert Hooke (1635-1703) was a virtuoso scientist contributing to geology, σ = C ε palaeontology, biology as well as mechanics ij ijkl kl ß Constitutive equations These are relationships between forces and in a continuum, which define the material behaviour.

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD and Young’s or modulus:

σ n = Eε n Shear or rigidity modulus:

σ S = Gε S = µε s Bulk modulus (1/): Mt Shasta andesite − P = Kεv 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

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Young’s Modulus or stiffness modulus Young’s Modulus or stiffness modulus: σ n = Eε n

Interatomic force

Interatomic distance

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Shear Modulus or rigidity modulus Shear modulus or stiffness modulus: σ s = Gε s

Interatomic force

Interatomic distance

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Hooke’s Law

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. 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:

σ ij = λδ ij ε kk + 2 µε ij

This can be deduced from substituting into the Taylor expansion for and differentiating.

For example: σ = λ(ε + ε + ε ) + 2µε Normal stress σ 11 11 22 33 11

12 = 2µε12

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Hooke’s Law

Hooke’s Law: σ = λδ ε + 2 µε σ ij ij kk ij Consider normal stressesλ εand normal strains: σ σ = λ(ε +εε + ε ) + 2µε 11 λ ε11 22 ε 33 11 = ( + ε + ) + 2µε 22 11 22 ε 33 22

33 = ( 11 + 22 + 33 ) + 2µε 33 σ In terms of principal stressesλε and principal strains: σ λ σ 1 = (λλε+ 2µ)ε1 µ+ λε 2 + λε 3 λ ε = + ( ε + 2 ) +λε 2 1 λ 2 3 µ 3 = 1 + 2 + ( + 2 )ε 3

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Hooke’s Law Can write in inverse form: ε ε υ1 υ σ υ = σ − σ − 1 E σ 1 E 2 E 3 σ υ ε 1 = −υ + −σ 2 Eσ 1 υE 2 E 3 σ 1 = − − + σ 3 E 1 E 2 E 3 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.

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD : uniaxial elastic deformation σ All components of stress zero except σ11: σ11 σ λ ε σ = λ(ε + ε ε+ ε ) + 2µε 11 11λ ε 22 33ε 11 = 0 = ( + ε + ) + 2µε dσ /dε = E 22 11 22 ε 33 22 11 11

33 = 0 = ( 11 + 22 + 33 ) + 2µε 33

ε11 The solutionλ to this set of simultaneous equations is: µ ε σ = 0 ε 33 µ(3λ + 2µ) ε σ22 = 0 σ = ε = E 11 + λ 11 11 λ µ ε = = − = −νε σ11 σ11 22 33 2( + ) 11 11

where E is Young’s Modulus and ν is Poisson’s ratio.

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Constitutive equations: isotropic compression

No shear or strain; all normal stresses σ33 = -p equal to –p; all normal strains equal to εv /3. σ = -p λ 22 µ ε σ = -p ⎛ 2 ⎞ 11 σ11 = -p − P = ⎜ + ⎟ V = KεV ε ⎝ 3 ⎠ σ22 = -p ε ∆V ε σ33 = -p = = + +ε v V 11 22 33 -p

P = - 1/3 (σ11 + σ22 + σ33 ) = - 1/3 σii

-dp/dεv = K where K is the bulk modulus; hence K = λ + 2/3µ

εv

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Young’s Modulus (initial tangent) of Materials Typical E Rubber 7 MPa Normally consolidated clays 0.2 ~ 4 GPa Boulder (oversolidated) 10 ~20 GPa Concrete 20 GPa Sandstone 20 GPa Granite 50 GPa Basalt 60 GPa 205 GPa 1,200 GPa

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD “Strength” of Materials

Uniaxial tensile Compressive strength strength - unconfined 300 kPa 1 MPa

Sandstone 1 MPa 10 MPa

Concrete 4 MPa 40 MPa Basalt 4 MPa 40 MPa

Granite 5 MPa 50 MPa

Rubber 30 MPa 2,000 MPa Spruce along/across grain 100 / 3 MPa 100 / 3 MPa

Steel piano wire 3,000 MPa 3,000 MPa

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Fracture σ ε Calculate the stress which will just separate two adjacent layers of atoms x layers apart σ x strain energy / m2 = ½ stress x strain x vol

Ue = ½ σn εn x ε Hooke’s law: εn = σn / E 2 Ue = σn x / 2E σ

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: σ 2 x U E U E n ≈ 2U σ ≈ 2 s ≈ s 2E s or n x x For steel: U = 1 J/m; E = 200 GPa; s ⇒σmax = 30 GPa ≈ E / 10 x = 2 x 10-10 m 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 (line defects) in shear

The reason why rocks don’t reach their theoretical shear strength is because they contain dislocations models are also used in modelling earthquake faulting

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Engineering behaviour of • Soils are granular materials – their behaviour is quite different to crystalline • Deformation is strongly non-linear • The curvature of the stress-strain is largest near the origin • Properties are highly dependent on Uniaxial deformation content • The constitutive relation for shear deformation, found from hundreds of experiments is: ε s ε r σ s = G0 ε s + ε r ε is the reference strain r Shear deformation

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 replacing G and µ by their surfaces. So the shear modulus definitions in terms of shear stress and rigidity must be related by a σs andσ shear strain εs : d power law: ⎛σ ⎞ d ε s s G = c µd = c⎜ ⎟ d s ⎝ ε s ⎠ where d is the fractal dimension of the grain size distribution constitutive equation for soils

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Constitutive equation for soils σ d ⎛σ ⎞ d ε s s From fractals: = c⎜ ⎟ d s ⎝ ε s ⎠

ε s ε r Integrating and setting d = 2: σ s = G0 ε s + ε r

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 Liquefaction of soils: phase transition This aspect of soil behaviour is completely different from crystalline rock

Soil liquefaction: Kobe port area Motion on soft ground to strong earthquake is fundamentally different to small Stress-strain curve of a soil as because sediments go through a compared with that of a crystalline phase transition and liquefy rock – note different definition of rigidity

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Constitutive equation: viscous flow Incompressible viscous ε For viscous fluids the deviatoric stress is proportional to strain-rate: σ • ' η ' ij = 2 ε ij 1/2η where η is the shear σ

Viscosity is an internal property of a that offers resistance to flow. Viscosity is measured in units of Pa s ( seconds), which is a unit of times a unit of time. This is a force applied to the fluid, acting for some length of time. A marble ( 2800 kg/m3) and a steel ball bearing (7800 kg/m3) will both measure the viscosity of a 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.

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Experimental techniques to study friction

Shear box Direct shear

Triaxial test Rotary shear

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Experimental results

ß At low normal stresses (σN < 200 MPa)

a Linear friction law observed: σS = µσN a 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

a Average for all data given by: σS = 0.85 σN

ß At higher normal stresses (σN > 200 MPa) a Very little variation between wide range of rock types (with some notable exceptions – esp. clay minerals which can have unusually low µ a But friction does not obey Amonton’s Law (i.e. straight line through origin) but Coulomb’s Law a Best fit to all data given by:

a 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 = µσN = σN tanφ May be simply represented on a Mohr diagram:

σ S µ= tan φ µ pe φ slo φ is the “angle of friction”

σN

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Field observations

ß We are concerned with friction related to earthquakes, i.e., friction on faults ß Faults are interfaces that have already fractured in previously intact material and have subsequently been displaced in shear (i.e., have slipped) ß Hence they are not “mated” surfaces (unlike joints)

Joint

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Summary: Byerlee’s Friction Laws

ß All data may be fitted by two straight lines:

a σN < 200 MPa σS = 0.85 σN

a σN > 200 MPa σS = 50 + 0.6 σN ß These are largely independent of rock type ß Independent of roughness of contacting surfaces ß Independent of rock strength or ß Independent of sliding velocity ß Independent of (up to 400oC)

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Experimental results of triaxial deformation tests

Differential Stress (σ1 - σ3) Total Axial Modes of brittle fracture in a triaxial system Stress

Confining σ1 Pressure P C σ1 σ1 σ1 σ1 Hydrostatic PC PC applied in all directions prior to the differential loading.

PC PC = σ2 = σ3

σ3 σ3

σ σ σ 1 1 1 σ1

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Actuator applying To AE transducer axial load

Thermocouple Fluid outlet fitting feedthrough

Top wave-guide

Pressure Vessel

Load Cell

Insulating filler Top steel Fv520 piston

Top pyrophillite enclosing disc Alumina coil support

Alumina Disc

Rock Specimen

Pore fluid inlet

Fibrous alumina insulation

Bottom steel Fv520 piston

Bottom enclosing pyrophillite block Bottom wave guide

Pressure fittings Bottom plug

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 Strength of Westerly granite as confining a function of confining pressure for a pressure. Also shown is wide range of frictional strength. igneous rocks

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

Tensile fracture Shear fracture

σS µ i φi pe slo

σN C (σ2 = -σT) µi = tan φi σT – tensile strength φi is the “angle of internal friction” GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD