Lab.15.2011.pptx - 12/5/11 15. OPENING ­MODE...

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Unformatted text preview: 12/5/11 15. OPENING ­MODE FRACTURES (JOINTS AND DIKES) I Main Topics A Why are opening mode fractures important? B Geometry C KinemaOcs D Mechanics E Key take ­away points 12/5/11 GG303 1 15. OPENING ­MODE FRACTURES Joint Sets Map of Joints. From Segall and Pollard, 1983. 12/5/11 hTp://en.wikipedia.org/wiki/File:Joints_Caithness.JPG GG303 2 1 12/5/11 15. OPENING ­MODE FRACTURES Radial joints in a stock (from Rogers and Gerla and, 1988) 12/5/11 GG303 3 15. OPENING ­MODE FRACTURES Radial Joints 12/5/11 GG303 4 2 12/5/11 15. OPENING ­MODE FRACTURES •  Columnar Joints 12/5/11 GG303 5 15. OPENING ­MODE FRACTURES Mudcracks 12/5/11 GG303 6 3 12/5/11 15. OPENING ­MODE FRACTURES Shee$ng Joints 12/5/11 Columnar Joints GG303 7 15. OPENING ­MODE FRACTURES Plumose Structure 12/5/11 GG303 8 4 12/5/11 15. OPENING ­MODE FRACTURES Stylolites:"anO ­cracks" in carbonate rocks From Rispoli (1981) via Pollard and Segall (1987) 12/5/11 GG303 9 15. OPENING ­MODE FRACTURES III Why are opening ­mode fractures important? A Most common geologic structures on earth ( joints) 12/5/11 GG303 10 5 12/5/11 15. OPENING ­MODE FRACTURES III Why are fractures important? (cont.) B Exceedingly important in heat & fluid flow (all sorts of fluids) hTp://volcanoes.usgs.gov/Imgs/Jpg/Photoglossary/fissure4_large.JPG 12/5/11 GG303 11 15. OPENING ­MODE FRACTURES III Why are fractures important? (cont.) C Important in determining geologic & engineering rock strength Joints at rockfall site, Utah 12/5/11 GG303 12 6 12/5/11 15. OPENING ­MODE FRACTURES III Why are fractures important? (cont.) D Impact on landscape and surficial processes Half Dome. Photo from Greg Stock 12/5/11 Joints, Utah GG303 13 15. OPENING ­MODE FRACTURES A Geometry 1 Bounded in extent 2 RelaOvely planar 12/5/11 GG303 14 7 12/5/11 15. OPENING ­MODE FRACTURES A Geometry (cont.) 3 Shape of an "isolated” joint a Massive rocks: joints are probably ellipOcal or circular b Layered rocks: joints are probably rectangular 12/5/11 GG303 15 15. OPENING ­MODE FRACTURES III KinemaOcs A RelaOve (not absolute) displacement of originally neighboring points is perpendicular to the fracture 12/5/11 GG303 16 8 12/5/11 15. OPENING ­MODE FRACTURES III KinemaOcs B RelaOve displacement << in ­plane dimensions 12/5/11 GG303 17 15. OPENING ­MODE FRACTURES III KinemaOcs B Surface textures (Plumose structure) EssenOally idenOcal to surface textures of opening mode fractures produced in engineering tests 12/5/11 GG303 18 9 12/5/11 15. OPENING ­MODE FRACTURES IV Mechanics SuperposiOon 12/5/11 GG303 19 15. OPENING ­MODE FRACTURES IV Mechanics Stresses (plain strain) arising from opening 2D elasOc model (from Pollard and Segall, 1987) “Driving Pressure” (over ­pressure) ∞ σ xx = σ xx + Δσ I ⎤ r⎡ R ⎛ a2 ⎞ cos (θ − Θ ) − − ⎜ 2 ⎟ sin θ sin 3Θ ⎥ ⎢ R⎣ r ⎝R ⎠ ⎦ ∞ σ yy = σ yy + Δσ I ⎤ r⎡ R ⎛ a2 ⎞ ⎢ cos (θ − Θ ) − + ⎜ 2 ⎟ sin θ sin 3Θ ⎥ R⎣ r ⎝R ⎠ ⎦ ⎡⎛ a 2 r ⎞ ⎤ ∞ σ xy = σ xy + Δσ I ⎢⎜ 3 ⎟ sin θ cos 3Θ ⎥ ⎣⎝ R ⎠ ⎦ 12/5/11 GG303 20 10 12/5/11 15. OPENING ­MODE FRACTURES IV Mechanics: EvaluaOon of principal stresses τs τmax (τxn, τxs) σ 1 = σ + radius = σ + τ max σ 2 = σ − radius = σ − τ max τxs σ = τ τ xn + τ yn σ xx + σ yy 1 x’n σ = = 2 2 σ  ­2θxx’ τn 2 τ max = σ xx − σ yy + 2σ xy τxn ­τyn 2σ xy −2θ xx ′ = tan −1 σ xx − σ yy σ2 = τy’n ( (τyn, τys) 12/5/11 )( GG303 ) 2 21 15. OPENING ­MODE FRACTURES IV Mechanics: Most tensile stress Unit pressurized mode ­I crack; no remote stress field: σ1 12/5/11 Trac$on ­free mode ­I crack under unit uniaxial tension: σ1 GG303 22 11 12/5/11 15. OPENING ­MODE FRACTURES IV Mechanics: Trajectories normal to most tensile stress Pressurized mode ­I crack; no remote stress field 12/5/11 Trac$on ­free mode ­I crack under uniaxial tension GG303 23 15. OPENING ­MODE FRACTURES IV Mechanics Near ­&p stresses arising from opening of a mode ­I crack, 2D elasOc model (from Pollard and Segall, 1987) Near the crack Op (r1<<a) r → a, r2 → 2 a, R → ( r1 2 a ) 1/ 2 ( 2r ) r a a1/ 2 R → → 1/2 1/ 2 → 1 / 2 , R a1 ( r1 2a ) ( 2r1 ) r 1/ 2 →0 θ→0 θ2 → 0 Θ → θ1 2 ⎛ a2 ⎞ ⎛ a2 ⎞ a ⎜ R2 ⎟ → ⎜ r 2a ⎟ → 2r ⎝⎠ ⎝1 ⎠ 1 12/5/11 GG303 24 12 12/5/11 15. OPENING ­MODE FRACTURES IV Mechanics θ→0 θ2 → 0 Θ → θ1 2 Near ­&p stresses near the Op of a mode ­I crack, 2D elasOc model (from Pollard and Segall, 1987) ∞ σ xx = σ xx + Δσ I ⎤ r⎡ R ⎛ a2 ⎞ ⎢ cos (θ − Θ ) − − ⎜ 2 ⎟ sin θ sin 3Θ ⎥ R⎣ r ⎝R ⎠ ⎦ ⎛a⎞ ∞ σ xx = σ xx + Δσ I ⎜ ⎟ ⎝ 2 r1 ⎠ 1/ 2 ⎡ ⎛ θ1 ⎞ ⎛ a ⎞ ⎛ 3θ1 ⎞ ⎤ ⎢ cos ⎜ ⎟ + ⎜ ⎟ sin θ sin ⎜ ⎝ 2 ⎟⎥ ⎠⎦ ⎣ ⎝ 2 ⎠ ⎝ 2 r1 ⎠ a sin θ ≈ y → r1 sin θ1 ⎛a⎞ σ xx = σ + Δσ I ⎜ ⎟ ⎝ 2r ⎠ 1/ 2 ∞ xx 1 12/5/11 ⎛ θ1 ⎞ ⎛ 1 ⎞ ⎛ 3θ1 ⎞ ⎤ ⎢ cos ⎜ 2 ⎟ + ⎜ 2 ⎟ sin θ1 sin ⎜ 2 ⎟ ⎥ ⎝ ⎠⎝⎠ ⎝ ⎠⎦ ⎣ r a a1/ 2 → , 1/ 2 → R ( r1 2a ) ( 2r1 )1/ 2 ( 2r ) R → 1/2 → 0 r a1 2 ⎛ a2 ⎞ ⎛a ⎞ a ⎜ R2 ⎟ → ⎝ r 2a ⎠ → 2r ⎜ ⎟ ⎝⎠ 1 1 1/ 2 GG303 25 15. OPENING ­MODE FRACTURES IV Mechanics θ→0 θ2 → 0 Θ → θ1 2 Near ­&p stresses near the Op of a mode ­I crack, 2D elasOc model (from Pollard and Segall, 1987) ∞ σ yy = σ yy + Δσ I ⎤ r⎡ R ⎛ a2 ⎞ ⎢ cos (θ − Θ ) − − ⎜ 2 ⎟ sin θ sin 3Θ ⎥ R⎣ r ⎝R ⎠ ⎦ ⎛a⎞ σ yy = σ + Δσ I ⎜ ⎟ ⎝ 2r ⎠ 1/ 2 ∞ yy 1 ⎡ ⎛ θ1 ⎞ ⎛ a ⎞ ⎛ 3θ1 ⎞ ⎤ ⎢ cos ⎜ ⎟ − ⎜ ⎟ sin θ sin ⎜ ⎝ 2 ⎠ ⎝ 2 r1 ⎠ ⎝ 2 ⎟⎥ ⎠⎦ ⎣ a sin θ ≈ y → r1 sin θ1 ⎛a⎞ ∞ σ yy = σ yy + Δσ I ⎜ ⎟ ⎝ 2 r1 ⎠ 12/5/11 1/ 2 ⎡ ⎛ θ1 ⎞ ⎛ 1 ⎞ ⎛ 3θ1 ⎞ ⎤ ⎢ cos ⎜ 2 ⎟ − ⎜ 2 ⎟ sin θ1 sin ⎜ 2 ⎟ ⎥ ⎝ ⎠⎦ ⎣ ⎝ ⎠⎝⎠ GG303 r a a1/ 2 → , 1/ 2 → R ( r1 2a ) ( 2r1 )1/ 2 ( 2r ) R → 1/2 → 0 r a1 2 ⎛ a2 ⎞ ⎛a ⎞ a →⎜ ⎜ R2 ⎟ ⎟ → 2r ⎝⎠ ⎝ r1 2 a ⎠ 1 1/ 2 26 13 12/5/11 15. OPENING ­MODE FRACTURES IV Mechanics θ→0 θ2 → 0 Θ → θ1 2 Near ­&p stresses near the Op of a mode ­I crack, 2D elasOc model (from Pollard and Segall, 1987) ∞ σ xy = σ xy + Δσ I ⎤ r ⎡⎛ a 2 ⎞ ⎢⎜ 2 ⎟ sin θ cos 3Θ ⎥ R ⎣⎝ R ⎠ ⎦ ⎛a⎞ ∞ σ xy = σ xy + Δσ I ⎜ ⎟ ⎝ 2 r1 ⎠ 1/ 2 ⎡⎛ a ⎞ ⎛ 3θ1 ⎞ ⎤ ⎢⎜ ⎟ sin θ cos ⎜ ⎝ 2 ⎟⎥ ⎠⎦ ⎣⎝ 2 r1 ⎠ a sin θ ≈ y → r1 sin θ1 ⎛a⎞ σ xy = σ + Δσ I ⎜ ⎟ ⎝ 2r ⎠ ∞ xy 1 1/ 2 ⎡⎛ 1 ⎞ ⎛ 3θ1 ⎞ ⎤ ⎢⎜ 2 ⎟ sin θ1 cos ⎜ 2 ⎟ ⎥ ⎝⎠ ⎝ ⎠⎦ ⎣ 12/5/11 r a a1/ 2 → , 1/ 2 → R ( r1 2a ) ( 2r1 )1/ 2 ( 2r ) R → 1/2 → 0 r a1 2 ⎛ a2 ⎞ ⎛a ⎞ a ⎜ R2 ⎟ → ⎝ r 2a ⎠ → 2r ⎜ ⎟ ⎝⎠ 1 1 1/ 2 GG303 27 15. OPENING ­MODE FRACTURES IV Mechanics: Most tensile stress Unit pressurized mode ­I crack; no remote stress field: σ1 12/5/11 Trac$on ­free mode ­I crack under unit uniaxial tension: σ1 GG303 28 14 12/5/11 15. OPENING ­MODE FRACTURES IV Mechanics: Trajectories normal to most tensile stress Pressurized mode ­I crack; no remote stress field 12/5/11 Trac$on ­free mode ­I crack under uniaxial tension GG303 29 26. OPENING ­MODE FRACTURES V Key take ­away points •  RelaOve displacements perpendicular to crack Dikes at Shiprock, New Mexico •  Finite extent •  Thin relaOve to length and height •  Strong stress concentraOon at Op (σ~r ­1/2, σ~a) •  Tend to grow perpendicular to the most tensile total stress along stress trajectories •  Can be used to deduce stress trajectory paTern 12/5/11 GG303 30 15 ...
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This note was uploaded on 12/05/2011 for the course GEOLOGY 300 taught by Professor Stephenmartel during the Fall '11 term at University of Hawaii, Manoa.

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