Lec.20.pptx - 10/30/11 20. Rheology & Linear...

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Unformatted text preview: 10/30/11 20. Rheology & Linear Elas7city I Main Topics A Rheology: Macroscopic deforma7on behavior B Linear elas7city for homogeneous isotropic materials 10/30/11 GG303 1 20. Rheology & Linear Elas7city Viscous (fluid) Behavior hHp://manoa.hawaii.edu/graduate/content/slide ­lava 10/30/11 GG303 2 1 10/30/11 20. Rheology & Linear Elas7city Duc7le (plas7c) Behavior hHp://www.hilo.hawaii.edu/~csav/gallery/scien7sts/LavaHammerL.jpg hHp://hvo.wr.usgs.gov/kilauea/update/images.html 10/30/11 GG303 3 hHp://upload.wikimedia.org/wikipedia/commons/8/89/Ropy_pahoehoe.jpg 20. Rheology & Linear Elas7city Elas7c Behavior hHps://thegeosphere.pbworks.com/w/page/24663884/Sumatra hHp://www.earth.ox.ac.uk/__data/assets/image/0006/3021/seismic_hammer.jpg 10/30/11 GG303 4 2 10/30/11 20. Rheology & Linear Elas7city BriHle Behavior (fracture) 10/30/11 GG303 5 hHp://upload.wikimedia.org/wikipedia/commons/8/89/Ropy_pahoehoe.jpg 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 10/30/11 GG303 hHp://www.fordogtrainers.com 6 3 10/30/11 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 10/30/11 GG303 7 20. Rheology & Linear Elas7city 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 10/30/11 hHp://wholefoodrecipes.net GG303 8 4 10/30/11 20. Rheology & Linear Elas7city 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 10/30/11 GG303 9 20. Rheology & Linear Elas7city 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 10/30/11 hHp://www.therapypuHy.com/images/stretch6.jpg GG303 10 5 10/30/11 20. Rheology & Linear Elas7city II Rheology: Macroscopic deforma7on behavior C BriHle Deforma7on 1  Discon7nuous deforma7on 2  Failure surfaces separate hHp://www.thefeeherytheory.com 10/30/11 GG303 11 20. Rheology & Linear Elas7city II Rheology: Macroscopic deforma7on behavior D Elasto ­plas7c rheology 10/30/11 GG303 12 6 10/30/11 20. Rheology & Linear Elas7city II Rheology: Macroscopic deforma7on behavior E Visco ­plas7c rheology 10/30/11 GG303 13 20. Rheology & Linear Elas7city II Rheology: Macroscopic deforma7on behavior F Power ­law creep 1 ė = (σ1 − σ3)n e(−Q /RT) 2 Example: rock salt 10/30/11 GG303 14 7 10/30/11 20. Rheology & Linear Elas7city II Rheology: Macroscopic deforma7on behavior G Linear vs. nonlinear behavior 10/30/11 GG303 15 20. Rheology & Linear Elas7city II Rheology: Macroscopic deforma7on behavior H Rheology=f (σij ,fluid pressure, strain rate, chemistry, temperature) I Rheologic equa7on of real rocks = ? 10/30/11 GG303 16 8 10/30/11 20. Rheology & Linear Elas7city 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) 10/30/11 x F k F x GG303 17 20. Rheology & Linear Elas7city σ III Linear elas7city (cont.) B Hooke’s Law for L0+ΔL uniaxial stress: σ = Eε 1 σ = uniaxial stress 2 E = Young’s modulus Dimensions: stress σ 3 ε = strain Dimensionless 10/30/11 GG303 L0 ε=ΔL/L0 E ε 18 9 10/30/11 20. Rheology & Linear Elas7city III Linear elas7city (cont.) B Hooke’s Law for uniaxial L0+ΔL 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 10/30/11 σ1 L0 ε=ΔL/L0 ε GG303 19 20. Rheology & Linear Elas7city 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) ε 10/30/11 GG303 20 10 10/30/11 20. Rheology & Linear Elas7city 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 10/30/11 ε GG303 21 20. Rheology & Linear Elas7city 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 10/30/11 GG303 x 22 11 10/30/11 20. Rheology & Linear Elas7city 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 c “ Thick plate” case 10/30/11 GG303 23 20. Rheology & Linear Elas7city III Linear elas7city E Special cases 5  Pure shear stress (2D) σxx=  ­σyy; σzz=0 10/30/11 GG303 24 12 10/30/11 20. Rheology & Linear Elas7city III Linear elas7city F Strain energy (W0) for uniaxial stress σxxdydz 1 W = ∫ F du = ∫ (σ xx dydz ) ⎛ ⎜ 0 0 u 2 3 4 5 10/30/11 du ⎞ dx ⎝ dx ⎟ ⎠ du W = (1/2)(σxxdydz) (εxxdx) εxxdx W = (1/2) (σxxεxx) (dxdydz) W0 = strain energy density W0 = W/(dxdydz) W0 = (1/2)(σxxεxx) GG303 25 20. Rheology & Linear Elas7city III Linear elas7city G Strain energy (W0) in 3D W0 = (1/2)(σ1ε1+σ2ε2+σ3ε3) σxxdydz εxxdx 10/30/11 GG303 26 13 10/30/11 20. Rheology & Linear Elas7city 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]) 10/30/11 GG303 4 β = compressibility β = 1/K ∆ = εxx+εyy+ εzz=  ­p/K p = pressure 5  P ­wave speed: Vp 4⎞ ⎛ Vp = ⎜ K + µ ⎟ ρ ⎝ 3⎠ 6 S ­wave speed: Vs Vs = µ ρ 27 14 ...
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