Lec.27.pptx - 11/23/11 27. Folds (I) I Main Topics...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 11/23/11 27. Folds (I) I Main Topics A What is a fold? B Curvature of a plane curve C Curvature of a surface 11/23/11 GG303 1 27. Folds (I) hEp://upload.wikimedia.org/wikipedia/commons/a/ae/Caledonian_orogeny_fold_in_King_Oscar_Fjord.jpg 11/23/11 GG303 2 1 11/23/11 27. Folds (I) AnScline, New Jersey Syncline, Rainbow Basin, California hEp://en.wikipedia.org/wiki/File:NJ_Route_23_anScline.jpg 11/23/11 hEp://en.wikipedia.org/wiki/File:Rainbow_Basin.JPG GG303 3 27. Folds (I) Folds, New South Wales, Australia Folds in granite, Sierra Nevada, California hEp://en.wikipedia.org/wiki/File:Folded_Rock.jpg 11/23/11 GG303 4 2 11/23/11 27. Folds (I) Energy Resources and an AnScline 11/23/11 hEp://www.wou.edu/las/physci/Energy/graphics/OilAnScline.jpg GG303 5 27. Folds (I) Three ­dimensional Fold, Salt Dome, Zagros Mountains hEp://upload.wikimedia.org/wikipedia/commons/2/2c/ZagrosMtns_SaltDome_ISS012 ­E ­18774.jpg 11/23/11 GG303 6 3 11/23/11 27. Folds (I) Complex Folds hEp://upload.wikimedia.org/wikipedia/en/2/2d/SaltTectonics1.jpg 11/23/11 GG303 7 27. Folds (I) II What is a fold? A DefiniSon: a surface (in a rock body) that has undergone a change in its curvature (at least locally) B All kinds of rocks can be folded, even granites C Consider a folded piece of paper… 11/23/11 Folded dike in granite near fault Sierra Nevada, California GG303 8 4 11/23/11 27. Folds (I) III Curvature of a plane curve D Tangents Consider a curve r(t), where r is a vector funcSon that gives points on the curve, and t is any parameter dr 1 Tangent vector: r′ = dt r′ 2  Unit tangent vector: T = r′ 3  Tangent gives the slope 11/23/11 GG303 9 27. Folds (I) D Tangents (cont.) 5 Example 1: parabola y = x2 → r ( x ) = x i + x2 j 2 dr d x i + x j r'( x ) = = = i + 2xj dx dx ( ) 1i + 2 xj 1i + 2 xj = 1 + 4 x2 12 + ( 2 x )2 i +2j At x = 1, T = 5 r' T ( x) = = r' 11/23/11 GG303 PosiSon vectors in black Unit tangent vectors in red 10 5 11/23/11 27. Folds (I) D Tangents (cont.) 4 Example 2: unit circle r (θ ) = cosθ i + sinθ j dr d cosθ i + sin θ j r' = = = − sin θ i + cosθ j dθ dθ ( r' T= = r' 11/23/11 ) − sin θ i + cosθ j ( − sin θ )2 + ( cosθ )2 PosiSon vectors in black Unit tangent vectors in red = − sin θ i + cosθ j GG303 Note that T•r = 0 here 11 27. Folds (I) III Curvature of a plane curve A Tangents (cont.) 6 If origin is on curve and reference axis is tangent to curve, then local slope = 0 11/23/11 GG303 12 6 11/23/11 27. Folds (I) III Curvature of a plane curve B Curvature = deviaSon from a straight line 1  Curvature is the first derivaSve (i.e., rate of change) of the unit tangent (i.e., slope) with respect to distance (s) along the curve 2  Curvature vector is normal to tangent vector lim Δφ = tan ( Δφ ) = ΔT T1 = ΔT 1 = ΔT s→ 0 11/23/11 GG303 13 27. Folds (I) III Curvature of a plane curve B Curvature (cont.) dT 3 K ( s ) = T′ ( s ) = ds 4 T ( s ) = dr ds dr dr ds s In a local tangenSal reference frame, dr is in the direcSon of ds, |dr| = ds , and |dr/ds|=1 dT 5 K ( s ) = T′ ( s ) = = ds 11/23/11 ⎛ dr ⎞ d⎜ ⎟ ⎝ ds ⎠ d 2r = 2 = r ′′( s ) ds ds GG303 14 7 11/23/11 27. Folds (I) B Curvature of a plane curve (cont.) 6  Curvature vector (K) a K(t) = dT dr dT b K(t) = dt dr dt 7 Curvature magnitude dT dr K (t) = K = dt dt 11/23/11 GG303 15 27. Folds (I) B Curvature of a plane curve (cont.) r (θ ) = ρcosθ i + ρsinθ j 8 Example: circle of radius ρ dr = dθ dT dr K(θ ) = dθ dθ d ( ρcosθ i + ρ sin θ j ) r' T= = r' dθ ρ ( − ρ sin θ ) + ( ρ cosθ ) 2 = − sin θ i + cosθ j −r K(θ ) = ( − cosθ i − sin θ j ) ρ = 2 −r 1 K= 2 = ρ 11/23/11 θ = − ρ sin θ i + ρcosθ j − ρ sin θ i + ρ cosθ j 2 r ρ GG303 ρ 16 8 11/23/11 27. Folds (I) B Curvature of a plane curve (cont.) 9  One can assign a sign to the curvature a PosiSve = concave (curve opens up) b NegaSve = convex (curve opens down) 11/23/11 GG303 InflecSon pt Convex Concave 17 27. Folds (I) IV Curvature of a surface A Consider a local x,y,z “tangenSal” reference frame, where the x and y axes are tangent to the surface and z is perpendicular to a folded surface that was originally planar 11/23/11 GG303 18 9 11/23/11 27. Folds (I) IV Curvature of a surface B The first parSal derivaSves ∂z/∂x and ∂z/∂y are the slopes of the curves formed by intersecSng the surface with xz ­plane (black curve) and the yz ­plane (white white), respecSvely. At the local origin, these derivaSves equal zero. 11/23/11 GG303 19 27. Folds (I) IV Curvature of a surface C The second parSal derivaSves ∂2z/∂x2, ∂2z/∂x∂y, ∂2z/∂y∂x, and ∂2z/∂y2 can be arranged in a symmetric matrix (a Hessian matrix). ⎡ ⎢ ⎢ H=⎢ ⎢ ⎢ ⎣ 11/23/11 ∂2 z ∂x 2 ∂2 z ∂y∂x ∂2 z ⎤ ⎥ ∂x∂y ⎥ ⎥ ∂2 z ⎥ ∂y 2 ⎥ ⎦ ⎡ ⎢ ⎡ dx ⎤ ⎢ If [ X ] = ⎢ ⎥ , [ H ][ X ] = ⎢ ⎢ dy ⎦ ⎥ ⎣ ⎢ ⎢ ⎣ GG303 ∂2 z ∂x 2 ∂2 z ∂y∂x ⎡ ∂z ∂2 z ⎤ ⎥ ⎢ ∂x∂y ⎥ ⎡ dx ⎤ ⎢ ∂x ⎢ ⎥=⎢ ⎥ ∂z ∂ 2 z ⎥ ⎣ dy ⎦ ⎢ ⎥ ⎢ 2 ∂y ∂y ⎥ ⎢ ⎣ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 20 10 11/23/11 27. Folds (I) IV Curvature of a surface D The principal values of the symmetric Hessian matrix are the greatest and least normal curvatures E The principal direcSons X of the symmetric Hessian matrix are the direcSons of the principal curvatures; these direcSons are perpendicular ⎡ ⎢ ⎢ H=⎢ ⎢ ⎢ ⎣ 11/23/11 ∂2 z ∂x 2 ∂z ∂y∂x 2 ∂2 z ⎤ ⎥ ∂x∂y ⎥ ⎡ k1 ⎥→ ⎢ ∂2 z ⎥ ⎢ 0 ⎣ ∂y 2 ⎥ ⎦ 0⎤ ⎥ k2 ⎥ ⎦ Analogous to principal stresses [ H ][ X ] = k [ X ] X gives direcSons in which slope increases or decreases most rapidly GG303 21 27. Folds (I) Curvature ­based Three ­dimensional Fold ClassificaSon Scheme 11/23/11 GG303 22 11 ...
View Full Document

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.

Ask a homework question - tutors are online