20. Rheology & Linear Elasticity

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1 I Main Topics A Rheology: Macroscopic deformation behavior B Linear elasticity for homogeneous isotropic materials 10/29/18 GG303 1

2 Viscous (fluid) Behavior 10/29/18 GG303 2

3 Ductile (plastic) Behavior /29/18 GG303 3

4 Elastic Behavior data/assets/image/0006/3021/seismic_hammer.jpg 10/29/18 GG303 4

5 Brittle Behavior (fracture) 10/29/18 GG303 5

6 II Rheology: Macroscopic deformation behavior A Elasticity 1 Deformation is reversible when load is removed 2 Stress (σ) is related to strain (ε) 3 Deformation is not time dependent if load is constant 4 Examples: Seismic (acoustic) waves, rubber ball 10/29/18 GG303 6

7 II Rheology: Macroscopic deformation behavior A Elasticity 1 Deformation is reversible when load is removed 2 Stress (σ) is related to strain (ε) 3 Deformation is not time dependent if load is constant 4 Examples: Seismic (acoustic) waves, rubber ball 10/29/18 GG303 7

8 II Rheology: Macroscopic deformation behavior B Viscosity 1 Deformation is irreversible when load is removed 2 Stress (σ) is related to strain rate (!ε ) 3 Deformation is time dependent if load is constant 4 Examples: Lava flows, corn syrup 10/29/18 GG303 8

9 II Rheology: Macroscopic deformation behavior B Viscosity 1 Deformation is irreversible when load is removed 2 Stress (σ) is related to strain rate (!ε ) 3 Deformation is time dependent if load is constant 4 Examples: Lava flows, corn syrup 10/29/18 GG303 9

10 II Rheology: Macroscopic deformation behavior C Plasticity 1 No deformation until yield strength is locally exceeded; then irreversible deformation occurs under a constant load 2 Deformation can increase with time under a constant load 3 Examples: plastics, soils 10/29/18 GG303 10

11 II Rheology: Macroscopic deformation behavior C Brittle Deformation 1 Discontinuous deformation 2 Failure surfaces separate 10/29/18 GG303 11

12 II Rheology: Macroscopic deformation behavior D Elasto-plastic rheology 10/29/18 GG303 12

13 II Rheology: Macroscopic deformation behavior E Visco-plastic rheology 10/29/18 GG303 13

14 II Rheology: Macroscopic deformation behavior F Power-law creep!ε 1 = (σ 1 σ 3 ) n e ( Q/RT) 2 Example: rock salt 10/29/18 GG303 14

15 II Rheology: Macroscopic deformation behavior G Linear vs. nonlinear behavior 10/29/18 GG303 15

16 II Rheology: Macroscopic deformation behavior H Rheology=f (σ ij,fluid pressure, strain rate, chemistry, temperature) I Rheologic equation of real rocks =? 10/29/18 GG303 16

17 II Rheology (cont.) J Experimental results Axial stress Plastic P c = confining pressure Elastic Increasing confining pressure Compression test data on Tennessee marble II from Wawersik and Fairhurst, /29/18 GG303 17

18 III Linear elasticity 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) F x k x F 10/29/18 GG303 18

19 III Linear elasticity (cont.) B Hooke s Law for uniaxial stress: σ = Eε 1 σ = uniaxial stress 2 E = Young s modulus Dimensions: stress 3 ε = strain (elongation) Dimensionless L 0 +ΔL 10/29/18 GG σ L 0 ε σ E ε=δl/l 0

20 Typical rock moduli and strengths Young s Poisson s Modulus Ratio (GPa) Rock type E min E max ν min ν max Quartzite Gneiss Basalt Granite Limestone Sandstone Shale Coal Uniaxial Strengths (MPa) Rock type Tensile (low) Tensile (high) Comp. (low) Comp. (high) Quartzite Gneiss Basalt Granite Limestone Sandstone Shale Coal /29/18 GG303 20

21 III Linear elasticity (cont.) B Hooke s Law for uniaxial stress (cont.): ε 1 = σ 1 /E 1 σ 2 = σ 3 = 0 2 ε 2 = ε 3 = -νε 1 a ν = Poisson s ratio b ν is dimensionless c Strain in one direction tends to induce strain in another direction L 0 +ΔL L 0 σ 1 ε=δl/l 0 10/29/18 GG303 21

22 III Linear elasticity (cont.) C Linear elasticity in 3D for homogeneous isotropic materials By superposition: 1 ε xx = σ xx /E (σ yy +σ zz )(ν/e) 2 ε yy = σ yy /E (σ zz +σ xx )(ν/e) 3 ε zz = σ zz /E (σ xx +σ yy )(ν/e) L 0 +ΔL L 0 σ 1 ε=δl/l 0 10/29/18 GG303 22

23 III Linear elasticity (cont.) C Linear elasticity in 3D for homogeneous isotropic L 0 +ΔL materials (cont.) 4 Directions of principal stresses and principal strains coincide 5 Extension in one direction can occur without tension 6 Compression in one direction can occur without shortening L 0 σ 1 ε=δl/l 0 10/29/18 GG303 23

24 III Linear elasticity E Special cases 1 Isotropic (hydrostatic) stress a σ 1 = σ 2 = σ 3 b No shear stress 2 Uniaxial strain a ε xx = ε 1 0 b ε yy = ε zz = 0 x 10/29/18 GG303 24

25 III Linear elasticity E Special cases 3 Plane stress (2D) σ z = 0 Thin plate case 4 Plane strain (2D) ε z = 0 a Displacement in z- direction is constant (e.g., zero) b Plate is confined between rigid walls c Thick plate case 10/29/18 GG303 25

26 III Linear elasticity E Special cases 5 Pure shear stress (2D) σ xx = -σ yy ; σ zz =0 10/29/18 GG303 26

27 III Linear elasticity F Strain energy (W 0 ) for uniaxial stress σ xx dydz 1 2 W = (1/2)(σ xx dydz) (ε xx dx) 3 W = (1/2) (σ xx ε xx ) (dxdydz) 4 W 0 = W/(dxdydz) ε xx dx W 0 = strain energy density 5 W 0 = (1/2)(σ xx ε xx ) 10/29/18 GG303 27

28 IIILinear elasticity G Strain energy (W 0 ) in 3D W 0 = (1/2)(σ 1 ε 1 +σ 2 ε 2 +σ 3 ε 3 ) σ xx dydz ε xx dx 10/29/18 GG303 28

29 III Linear elasticity D Relationships among different elastic moduli 1 G = μ = shear modulus G = E/(2[1+ν]) ε xy = σ xy /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 = ε xx +ε yy + ε zz = -p/k p = pressure 5 P-wave speed: V p V p = K µ 6 S-wave speed: V s ρ 10/29/18 GG303 29