Download - Lec.20.pptx Rheology & Linear Elasticity
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20. Rheology & Linear Elas7city
I Main Topics
A Rheology: Macroscopic deforma7on behavior B Linear elas7city for homogeneous isotropic materials
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Viscous (fluid) Behavior
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Duc7le (plas7c) Behavior
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Elas7c Behavior
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BriIle Behavior (fracture)
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II Rheology: Macroscopic deforma7on behavior A Elas7city
1 Deforma7on is reversible when load is removed
2 Stress (σ) is related to strain (ε)
3 Deforma7on is not 'me dependent if load is constant
4 Examples: Seismic (acous'c) waves, rubber ball
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20. Rheology & Linear Elas7city
II Rheology: Macroscopic deforma7on behavior A Elas7city
1 Deforma7on is reversible when load is removed
2 Stress (σ) is related to strain (ε)
3 Deforma7on is not 'me dependent if load is constant
4 Examples: Seismic (acous'c) waves, rubber ball
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II Rheology: Macroscopic deforma7on behavior B Viscosity
1 Deforma7on is irreversible when load is removed
2 Stress (σ) is related to strain rate (ε)
3 Deforma7on is 7me dependent if load is constant
4 Examples: Lava flows, corn syrup
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II Rheology: Macroscopic deforma7on behavior B Viscosity
1 Deforma7on is irreversible when load is removed
2 Stress (σ) is related to strain rate (ε)
3 Deforma7on is 7me dependent if load is constant
4 Examples: Lava flows, corn syrup
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II Rheology: Macroscopic deforma7on behavior C Plas7city
1 No deforma7on un7l yield strength is locally exceeded; then irreversible deforma7on occurs under a constant load
2 Deforma7on can increase with 7me under a constant load
3 Examples: plas7cs, soils
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II Rheology: Macroscopic deforma7on behavior C BriIle Deforma7on
1 Discon7nuous deforma7on
2 Failure surfaces separate
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II Rheology: Macroscopic deforma7on behavior D Elasto-‐plas7c rheology
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II Rheology: Macroscopic deforma7on behavior E Visco-‐plas7c rheology
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II Rheology: Macroscopic deforma7on behavior F Power-‐law creep
1 ė = (σ1 − σ3)n e(−Q/RT)
2 Example: rock salt
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II Rheology: Macroscopic deforma7on behavior G Linear vs. nonlinear behavior
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II Rheology: Macroscopic deforma7on behavior H Rheology=f (σij ,fluid pressure, strain rate, chemistry, temperature)
I Rheologic equa7on of real rocks = ?
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II Rheology (cont.) J Experimental results
Compression test data on Tennessee marble II from Wawersik, 1968
Axial stress
Pc = confining pressure Ela
s7c
Plas7c
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III Linear elas7city A Force and displacement
of a spring (from Hooke, 1676): F= kx 1 F = force 2 k = spring constant
Dimensions:F/L 3 x = displacement
Dimensions: length L)
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x
F
F
x
k
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III Linear elas7city (cont.)
B Hooke’s Law for uniaxial stress: σ = Eε
1 σ = uniaxial stress 2 E = Young’s modulus
Dimensions: stress 3 ε = strain Dimensionless
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σ
ε
E
σ
ε=ΔL/L0 L0 L0+ΔL
20. Rheology & Linear Elas7city Typical rock moduli and strengths
Young’s Poisson’s Modulus Ra0o
(GPa)
Rock type Emin Emax ν Ν
Quartzite 70 105 0.11 0.25
Gneiss 16 103 0.10 0.40
Basalt 16 101 0.13 0.38
Granite 10 74 0.10 0.39
Limestone 1 92 0.08 0.39
Sandstone 10 46 0.10 0.40
Shale 10 44 0.10 0.19
Coal 1.5 3.7 0.33 0.37
Uniaxial Strengths (MPa)
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Rock type Tensile (low)
Tensile (high)
Comp. (low)
Comp. (high)
Quartzite 17 28 200 304
Gneiss 3 21 73 340
Basalt 2 28 42 355
Granite 3 39 30 324
Limestone 2 40 48 210
Sandstone 3 7 40 179
Shale 2 5 36 172
Coal 1.9 3.2 14 30
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III Linear elas7city (cont.) B Hooke’s Law for uniaxial
stress (cont.): ε1 = σ1/E 1 σ2 = σ3 = 0 2 ε2 = ε3 = -‐νε1
a ν = Poisson’s ra7o b ν is dimensionless c Strain in one
direc7on tends to induce strain in another direc7on
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ε
σ1
ε=ΔL/L0 L0 L0+ΔL
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III Linear elas7city (cont.)
C Linear elas7city in 3D for homogeneous isotropic materials By superposi7on: 1 εxx = σxx/E – (σyy+σzz)(ν/E) 2 εyy = σyy/E – (σzz+σxx)(ν/E) 3 εzz = σzz/E – (σxx+σyy)(ν/E)
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ε
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III Linear elas7city (cont.) C Linear elas7city in 3D for
homogeneous isotropic materials (cont.) 4 Direc7ons of principal
stresses and principal strains coincide
5 Extension in one direc7on can occur without tension
6 Compression in one direc7on can occur without shortening
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ε
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III Linear elas7city
E Special cases 1 Isotropic (hydrosta7c) stress
a σ1 = σ2 = σ3 b No shear stress
2 Uniaxial strain a εxx= ε1≠ 0 b εyy = εzz = 0
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x
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III Linear elas7city E Special cases
3 Plane stress (2D) σz = 0 “Thin plate” case
4 Plane strain (2D) εz = 0 a Displacement in z-‐direc7on is constant (e.g., zero)
b Plate is confined between rigid walls
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III Linear elas7city
E Special cases 5 Pure shear stress (2D)
σxx= -‐σyy; σzz=0
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III Linear elas7city
F Strain energy (W0) for uniaxial stress
1
2 W = (1/2)(σxxdydz) (εxxdx)
3 W = (1/2) (σxxεxx) (dxdydz)
4 W0 = W/(dxdydz)
5 W0 = (1/2)(σxxεxx) 10/29/12 GG303 27
εxxdx
W = Fdu0
u
∫ = σ xxdydz( ) dudx
⎛⎝⎜
⎞⎠⎟dx
0
du
∫
W0 = strain energy density
σxxdydz
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III Linear elas7city G Strain energy (W0) in 3D
W0 = (1/2)(σ1ε1+σ2ε2+σ3ε3)
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εxxdx
σxxdydz
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III Linear elas7city D Rela7onships among
different elas7c moduli 1 G = μ = shear modulus G = E/(2[1+ν])
εxy = σxy/2G 2 λ = Lame' constant
λ = Ev/([1 + ν][1 -‐ 2ν]) 3 K = bulk modulus K = E/(3[1 -‐ 2ν])
4 β = compressibility β = 1/K
∆ = εxx+εyy+ εzz= -‐p/K p = pressure
5 P-‐wave speed: Vp
6 S-‐wave speed: Vs
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Vp = K + 43µ⎛
⎝⎜⎞⎠⎟ ρ
Vs = µ ρ