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Lecture 13: Earth Materials

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Lecture 13: Earth Materials Earth Materials Lecture 13 Earth Materials GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Hooke’s law of elasticity 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 deformation in a continuum, which define the material behaviour. GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Shear modulus and bulk modulus Young’s or stiffness modulus: σ n = Eε n Shear or rigidity modulus: σ S = Gε S = µε s Bulk modulus (1/compressibility): 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 stress and differentiating. For example: Normal stress σ 11 = λ(ε11 + ε 22 + ε 33 ) + 2µε11 σ 12 = 2µε12 Shear stress GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Hooke’s Law Hooke’s Law: σ ij = λδ ij ε kk + 2 µε ij Consider normal stresses and normal strains: σ 11 = λ(ε11 + ε 22 + ε 33 ) + 2µε11 σ 22 = λ(ε11 + ε 22 + ε 33 ) + 2µε 22 σ 33 = λ(ε11 + ε 22 + ε 33 ) + 2µε 33 In terms of principal stresses and principal strains: σ 1 = (λ + 2µ)ε1 + λε 2 + λε 3 σ 2 = λε1 + (λ + 2µ)ε 2 + λε 3 σ 3 = λε1 + λε 2 + (λ + 2µ)ε 3 GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Hooke’s Law Can write in inverse form: 1 υ υ ε = σ − σ − σ 1 E 1 E 2 E 3 υ 1 υ ε = − σ + σ − σ 2 E 1 E 2 E 3 υ υ 1 ε = − σ − σ + 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 Constitutive equation: uniaxial elastic deformation All components of stress zero except σ11: σ11 σ 11 = λ(ε11 + ε 22 + ε 33 ) + 2µε 11 σ 22 = 0 = λ(ε11 + ε 22 + ε 33 ) + 2µε 22 dσ11/dε11 = E σ 33 = 0 = λ(ε11 + ε 22 + ε 33 ) + 2µε 33 ε11 The solution to this set of simultaneous equations is: σ33 = 0 µ(3λ + 2µ) σ22 = 0 σ = ε = Eε 11 λ + µ 11 11 λ ε = ε = − ε = −νε σ11 σ11 22 33 2(λ + µ) 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 ε /3. v σ22 = -p σ = -p ⎛ 2 ⎞ 11 σ11 = -p − P = ⎜λ + µ ⎟εV = KεV ⎝ 3 ⎠ σ22 = -p ∆V σ33 = -p ε = = ε +ε +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 clay (oversolidated) 10 ~20 GPa Concrete 20 GPa Sandstone 20 GPa Granite 50 GPa Basalt 60 GPa Steel 205 GPa Diamond 1,200 GPa GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD “Strength” of Materials Uniaxial tensile Compressive strength strength - unconfined Soil 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 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 • Deformation is strongly non-linear • The curvature of the stress-strain is largest near the origin • Properties are highly dependent on Uniaxial deformation water 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 and shear strain εs : power law: d dσ s ⎛σ s ⎞ G = c µd = c⎜ ⎟ dε 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 d dσ s ⎛σ s ⎞ From fractals: = c⎜ ⎟ dε 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 earthquakes 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 fluids ε For viscous fluids the deviatoric stress is proportional to strain-rate: • ' ' σ ij = 2η ε ij 1/2η where η is the shear viscosity σ 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. 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.
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