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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hIp://manoa.hawaii.edu/graduate/content/slide-‐lava

Viscous (fluid) Behavior

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Duc7le (plas7c) Behavior


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hIps://thegeosphere.pbworks.com/w/page/24663884/Sumatra

Elas7c Behavior

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BriIle Behavior (fracture)


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 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


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


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


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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E Special cases 5  Pure shear stress (2D)

σxx= -‐σyy; σzz=0

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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


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

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Lec.20.pptx Rheology & Linear Elasticity