Lec.28b.pptx - 12/2/11 28. Folds (II) I Main Topic:...

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Unformatted text preview: 12/2/11 28. Folds (II) I Main Topic: Mechanics of folds above intrusions A Background B G.K. Gilbert’s idealizaGon C SuperposiGon D Displacements around an opening ­mode crack (sill) E Dimensional analysis of governing eq. for bending F Idealized form of folds over a laccolith G Development of laccoliths and saucer ­shaped sills 12/2/11 GG303 1 28. Folds (II) Laccolith, Montana hTp://upload.wikimedia.org/wikipedia/en/a/a6/Laccolith_Montana.jpg 12/2/11 GG303 2 1 12/2/11 28. Folds (II) II Mechanics of folds above intrusions Half ­stereogram of Mount Ellsworth From Gilbert, 1877, Report on the geology of the Henry Mountains hTp://www.nps.gov/history/history/online_books/geology/publicaGons/bul/707/images/fig53.jpg 12/2/11 GG303 3 28. Folds (II) II Mechanics of folds above intrusions G.K. Gilbert 12/2/11 David Pollard GG303 4 2 12/2/11 28. Folds (II) II Mechanics of folds above intrusions (beam theory) “Ideal Cross ­sec7on of a Mountain of Erup7on” “Ideal Cross ­sec7on of a Laccolite, showing the typical form and the arching of the overlying strata” Figures from Gilbert, 1887 12/2/11 GG303 5 28. Folds (II) II Mechanics of folds above intrusions SuperposiGon 12/2/11 GG303 6 3 12/2/11 28. Folds (II) II Mechanics of folds above intrusions SuperposiGon •  Opening ­mode crack modeled by opening ­mode displacement disconGnuiGes (dds) of different apertures •  Openings [X(i)] of dds set so that sum of tracGon changes matches boundary condiGon [B(j)] on crack walls = Δσyyc •  [A(ij)][X(i)] = [B(j)], where A(ij) is effect of unit opening at element i on tracGons at element j 12/2/11 Displacement DisconGnuiGes GG303 7 28. Folds (II) II Mechanics of folds above intrusions SuperposiGon •  Total stress field around crack equals sum of stress contribuGons of all dds: σt = Σσi •  Total displacement field around crack equals sum of displacement contribuGons of all dds: ut = Σui 12/2/11 GG303 Displacement DisconGnuiGes 8 4 12/2/11 28. Folds (II) II Mechanics of folds above intrusions Displacements arising from opening of a mode ­I crack, 2D elasGc model (from Pollard and Segall, 1987) ux = Δσ I ⎧ ⎡r ⎤⎫ ⎨(1 − 2ν ) ( R cos Θ − r cosθ ) − r sin θ ⎢ sin (θ − Θ ) ⎥ ⎬ 2G ⎩ ⎣R ⎦⎭ “Driving Pressure” (over ­pressure) uy = Δσ I 2G ⎧ ⎡r ⎤⎫ ⎨2 (1 − ν ) ( R sin Θ − r sin θ ) − r sin θ ⎢ cos (θ − Θ ) − 1⎥ ⎬ ⎣R ⎦⎭ ⎩ Shear modulus of host rock 12/2/11 GG303 9 28. Folds (II) II Mechanics of folds above intrusions Displacements arising from opening of a mode ­I crack, 2D elasGc model (from Pollard and Segall, 1987) Now specialize to the crack walls. Rsinθ = y 0, hence ux = Δσ I ⎧ ⎡r ⎤ ⎫ Δσ I {(1 − 2ν )( R cos Θ − r cosθ )} ⎨(1 − 2ν ) ( R cos Θ − r cosθ ) − r sin θ ⎢ sin (θ − Θ ) ⎥ ⎬ → 2G ⎩ 2G ⎣R ⎦⎭ uy = Δσ I 2G 12/2/11 Δσ I ⎧ ⎡r ⎤⎫ {2 (1 − ν )( R sin Θ )} ⎨2 (1 − ν ) ( R sin Θ − r sin θ ) − r sin θ ⎢ cos (θ − Θ ) − 1⎥ ⎬ → 2G ⎣R ⎦⎭ ⎩ GG303 10 5 12/2/11 28. Folds (II) II Mechanics of folds above intrusions Displacements arising from opening of a mode ­I crack, 2D elasGc model (from Pollard and Segall, 1987) r1 is distance from right end r2 is distance from len end So the remaining key terms are: R, cosΘ, sinΘ, and rcosθ. Along the crack, these terms are simple: 12/2/11 R = [(a - x )(a + x )]1/ 2 = a 2 − x 2 θ ±π Θ= 1 = , so cos Θ = 0, sin Θ = ±1 2 2 r cosθ = x GG303 11 28. Folds (II) II Mechanics of folds above intrusions Displacements arising from opening of a mode ­I crack, 2D elasGc model (from Pollard and Segall, 1987) Along the crack, R=[(a ­x)(a+x)]1/2,rcosθ = x uc ( x ≤ a) = x uc ( x ≤ a) = y 12/2/11 Δσ I Δσ {(1 − 2ν )( R cos Θ − r cosθ )} → u cx = 2GI {(1 − 2ν )( − x )} 2G { ( Δσ I Δσ {2 (1 − ν )( R sin Θ )} → u cy = 2GI 2 (1 − ν ) ± a 2 − x 2 2G GG303 )} 12 6 12/2/11 28. Folds (II) II Mechanics of folds above intrusions Displacements arising from opening of a mode ­I crack, 2D elasGc model (from Pollard and Segall, 1987) y x Now consider the displacements normal to the crack: c uy = { ± Δσ I 2 (1 − ν ) 2G ( c uy u c y (max) a2 − x 2 = ± ( )} → u a2 − x 2 a 12/2/11 c y (max) ( x = 0) = ) = ± ⎛⎜ ⎛ x⎞ 1− ⎜ ⎟ ⎝ a⎠ ⎜ ⎝ + Δσ I {2 (1 − ν ) a} 2G 2 ⎞ ⎟ ⎟ ⎠ GG303 13 28. Folds (II) II Mechanics of folds above intrusions Displacements arising from opening of a mode ­I crack, 2D elasGc model (from Pollard and Segall, 1987) y x Now consider the displacements parallel to the crack: uc ( x ≤ a) = x Δσ I + Δσ {(1 − 2ν )( − x )}, and u cy(max) ( x = 0 ) = 2G I {2 (1 − ν ) a} 2G c ux (1 − 2ν ) ( − x ) = c u y (max) 2 (1 − ν ) a For ν = 0.25, 12/2/11 c x c y (max) u u = (1 2 )( − x ) = (1 2 )( − x ) = −1 x 2(3 4)a ( 3 2) a 3 a GG303 14 7 12/2/11 28. Folds (II) II Mechanics of folds above intrusions Displacements arising from opening of a mode ­I crack, 2D elasGc model (from Pollard and Segall, 1987) Normalized Crack wall displacements (v = 0.25) Along the crack (|x/a| ≤ 1) c uy u c y (max) = ± ( a2 − x 2 For ν = 0.25, 12/2/11 a c x c y (max) u u ) = ± ⎛⎜ = ⎛ x⎞ 1− ⎜ ⎟ ⎝ a⎠ ⎜ ⎝ 2 ⎞ ⎟ ⎟ ⎠ (1 2 )( − x ) = (1 2 )( − x ) = −1 x 2(3 4)a ( 3 2) a 3 a GG303 15 28. Folds (II) II Mechanics of folds above intrusions Sketch from field notes of Gilbert hTp://pangea.stanford.edu/~annegger/images/colorado%20plateau/laccolith_sketch.jpg 12/2/11 GG303 16 8 12/2/11 28. Folds (II) II Mechanics of folds above intrusions Dimensional analysis of terms in governing equa2on for bending of an elasGc layer (from Pollard and Fletcher, 2005) d 4 v 12 p = dx 4 BH 3 y H v x v = verGcal deflecGon of mid ­plane {Length} x = horizontal distance {Length} L = length of flexed part of layer {Length} p = overpressure {Force/area} B = sGffness {Force/area} H = thickness of layer {Length} Dimensions check 12/2/11 Mid ­plane of layer L Overpressure (p) GG303 17 28. Folds (II) II Mechanics of folds above intrusions Dimensional analysis of terms in governing equaGon for bending of an elasGc layer (from Pollard and Fletcher, 2005) Find constant length scales and non ­dimensionalize x* = y H Mid ­plane of layer v x v , v* = L vmax x L 12/2/11 GG303 18 9 12/2/11 28. Folds (II) II Mechanics of folds above intrusions Dimensional analysis … (cont.) Now non ­dimensionalize the differenGal operator 1 dx * 1 x→ = L dx L d d dx * d1 = = dx dx * dx dx * L x* = d2 ⎛ d ⎞ ⎛ d ⎞ ⎛ d 1 ⎞ ⎛ d 1 ⎞ ⎛ 1 ⎞ ⎛ d2 ⎞ =⎜ ⎟⎜ ⎟ =⎜ ⎟⎜ ⎟ =⎜ ⎟ dx 2 ⎝ dx ⎠ ⎝ dx ⎠ ⎝ dx * L ⎠ ⎝ dx * L ⎠ ⎝ L ⎠ ⎜ dx *2 ⎟ ⎝ ⎠ 2 2 3 d3 ⎛ d ⎞ ⎛ d2 ⎞ ⎛ d 1 ⎞ ⎛ ⎛ 1 ⎞ ⎛ d2 ⎞ ⎞ ⎛ 1 ⎞ ⎛ d3 ⎞ =⎜ ⎟⎜ 2⎟ =⎜ =⎜ ⎟ ⎜ ⎟ ⎜⎜ ⎟ ⎜ 3 2 ⎟⎟ ⎝ dx ⎠ ⎝ dx ⎠ ⎝ dx * L ⎠ ⎝ ⎝ L ⎠ ⎝ dx * ⎠ ⎠ ⎝ L ⎠ ⎝ dx *3 ⎟ dx ⎠ 3 4 d4 ⎛ d ⎞ ⎛ d3 ⎞ ⎛ d 1 ⎞ ⎛ ⎛ 1 ⎞ ⎛ d3 ⎞ ⎞ ⎛ 1 ⎞ ⎛ d4 ⎞ = ⎜ ⎟⎜ 3⎟ = ⎜ =⎜ ⎟ ⎜ , etc. ⎟⎜ ⎟ dx 4 ⎝ dx ⎠ ⎝ dx ⎠ ⎝ dx * L ⎠ ⎜ ⎝ L ⎠ ⎜ dx *3 ⎟ ⎟ ⎝ L ⎠ ⎝ dx *4 ⎟ ⎝ ⎠⎠ ⎠ ⎝ 12/2/11 GG303 19 28. Folds (II) II Mechanics of folds above intrusions Dimensional analysis of terms in governing equaGon for bending of an elasGc layer (from Pollard and Fletcher, 2005) SubsGtute into governing eq. y H v d 4 ( v ) 12 p = dx 4 BH 3 v = v * vmax Mid ­plane of layer x L d4 d4 1 = dx 4 dx *4 L4 4 d 4 (v) 1 d 4 1 ( v ) d ( v *) 12 p =4 ( v * vmax ) = 4 max 4 = 3 dx 4 L dx *4 L dx * BH 12/2/11 GG303 20 10 12/2/11 28. Folds (II) II Mechanics of folds above intrusions Dimensional analysis of terms in governing equaGon for bending of an elasGc layer (from Pollard and Fletcher, 2005) y H v d 4 ( v ) 12 p = dx 4 BH 3 d ( v *) 12 p L = dx *4 B vmax H 3 4 12/2/11 x L d 4 ( v ) 1 ( vmax ) d 4 ( v *) 12 p =4 = dx 4 L dx *4 BH 3 4 Mid ­plane of layer GG303 Right side contains only constants v* ~ L4 v* ~ 1/H3 Long thin layers will deflect much more than short thick layers 21 28. Folds (II) II Mechanics of folds above intrusions Asymmetric Pressurized Crack Parallel to a Surface Symmetric Pressurized Crack in an Infinite Body Thin beam “Thick beam” Bending of layer over laccolith should cause shearing at laccolith perimeter. This suggests laccoliths should propagate up towards the surface as they grow. 12/2/11 GG303 22 11 12/2/11 28. Folds (II) II Mechanics of folds above intrusions TheoreGcal form of soluGon y 4 H d v 12 p = = C4 dx 4 BH 3 Mid ­plane of layer v x 3 dv = C4 x + C3 dx 3 d 2 v C4 2 = x + C 3 x + C2 dx 2 2 L dv C4 3 C3 2 = x+ x + C2 x + C1 dx 6 2 v= C 4 4 C 3 3 C2 2 x+ x+ x + C1 x + C0 24 6 2 v(x=0) = C0 The funcGon v is even: v( ­x) = v(x) By symmetry, the odd coefficients (C3 and C1) must equal zero That leaves C2; C2 is set so that v = 0 at x=L/2 12/2/11 GG303 23 28. Folds (II) II Mechanics of folds above intrusions The Golden Valley Sill, South Africa – a saucer ­shaped sill From Polteau et al., 2008 12/2/11 GG303 24 12 12/2/11 28. Folds (II) II Mechanics of folds above intrusions 1 3 Bending of overlying layers Laccolith Dike 2 4 Sill 12/2/11 Laccolith and saucer ­shaped sill GG303 25 13 ...
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