# Lab.14 - 14. Folds I Main Topics A Local...

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Unformatted text preview: 11/23/11 14. Folds I Main Topics A Local geometry of a plane curve (cylindrical fold) B Local geometry of a curved surface (3D fold) C Numerical evaluaFon of curvature (geometry) D KinemaFcs of folding E Fold terminology and classiﬁcaFon (geometry) 11/23/11 GG303 1 14. Folds hMp://upload.wikimedia.org/wikipedia/commons/a/ae/Caledonian_orogeny_fold_in_King_Oscar_Fjord.jpg 11/23/11 GG303 2 1 11/23/11 14. Folds AnFcline, New Jersey Syncline, Rainbow Basin, California hMp://en.wikipedia.org/wiki/File:NJ_Route_23_anFcline.jpg 11/23/11 hMp://en.wikipedia.org/wiki/File:Rainbow_Basin.JPG GG303 3 14. Folds Folds, New South Wales, Australia Folds in granite, Sierra Nevada, California hMp://en.wikipedia.org/wiki/File:Folded_Rock.jpg 11/23/11 GG303 4 2 11/23/11 14. Folds Energy Resources and an AnFcline 11/23/11 hMp://www.wou.edu/las/physci/Energy/graphics/OilAnFcline.jpg GG303 5 14. Folds II Local geometry of a plane curve (cylindrical fold) in a tangenFal reference frame A Express the plane curve as a power series: 1 y = ⎡… + C−2 x −2 + C−1 x −1 ⎤ + ⎡C0 x 0 ⎤ + ⎡C1 x1 + C2 x 2 + C3 x 3 + …⎤ ⎣ ⎦⎣ ⎦⎣ ⎦ At x= 0, y = 0, so all the coeﬃcients for terms with non ­posiFve exponents must be zero 2 y = C1 x1 + C2 x 2 + C3 x 3 + … 11/23/11 GG303 6 3 11/23/11 14. Folds II Local geometry of a plane curve (cylindrical fold) in a tangenFal reference frame 2 y = C1 x1 + C2 x 2 + C3 x 3 + … Now examine y’ 3 y′ = C1 x 0 + 2C2 x1 + 3C3 x 2 + … = 0 At x = 0, y’ = 0, so C1 = 0, so 4 y = C2 x 2 + C3 x 3 + … = 0 As x 0, higher ­order terms vanish 2 m 5 lxi→ 0 y = C2 x 6 lim k = y( s )′′ = y( x )′′ = 2C2 So all plane curves are locally second ­order (parabolic). x→0 11/23/11 GG303 7 14. Folds III Local geometry of a curved surface in a At (x= 0, y = 0), z = 0, ∂z/∂x = 0, ∂z/∂y = 0 tangenFal reference frame A Plane curves are formed by intersecFng a curved surface with a plane containing the surface normal B These plane curves z = z(x,y) are locally all of second ­order, so any conFnuous surface is locally 2nd order. The general form of such a surface in a tangenFal frame is z = Ax 2 + Bxy + Cy 2 This is the equaFon of a paraboloid: all surfaces are locally either elllipFc or hyperbolic paraboloids Parabolic plane curves C Example: curve (normal sec+on) in the arbitrary plane y = mx 2 y = lim z = Ax 2 + Bx ( mx ) + C ( mx ) = A + Bm + Cm 2 x 2 x → 0, y→ 0 11/23/11 GG303 ( Sum of constants ) 8 4 11/23/11 14. Folds III Local geometry of a curved surface … (cont.) D Dilemma 1 EvaluaFng curvatures of a surface zL=zL (xL,yL), where“zL” is normal to the surface, is easy 2  The “global” reference frame, zG= zG (xG,yG), in which data are collected are usually misaligned with the tangenFal local reference frame 3  Alignment is generally diﬃcult 11/23/11 GG303 9 14. Folds III Local geometry of a curved surface … (cont.) E “resoluFon” 1 At certain places the local and global reference frames are easily aligned though: at the summits or boMoms of folds 2 We will evaluate the curvatures there, leaving the more general problem to “later” 11/23/11 GG303 10 5 11/23/11 14. Folds III Local geometry of a curved surface … (cont.) F Example (analyFcal) zG = 4xGyG 1 First plot and evaluate zG near (0,0) >> [X,Y] = meshgrid([ ­2:0.1:2]); >> Z=4*X.*Y; >> surf(X,Y,Z); >> xlabel('x'); ylabel('y'); >> zlabel('z'); Ftle('z= 4xy') This is a saddle 11/23/11 GG303 11 14. Folds F Example (analyFcal) (cont.) zG = 4xGyG 2 Now evaluate the ﬁrst derivaFves a ∂zG/∂xG = 4yG b ∂zG/∂yG = 4xG c Both derivaFves equal zero at (0,0) 3 The local tangenFal and global reference frames are aligned at (0,0) 11/23/11 GG303 >> hold on >> plot3(X(21,:),Y(21,:),Z(21,:),'r') >> plot3(X(:,21),Y(:,21),Z(:,21),'y') 12 6 11/23/11 14. Folds F Example (analyFcal) (cont.) zG = 4xGyG 4 Now evaluate the second derivaFves a ∂2zG/∂xG2 = 0 b ∂2zG/∂xG∂yG = 4 c ∂2zG/∂yG∂xG = 4 >> c=contour(X,Y,Z); clabel(c); d ∂2zG/∂yG2 = 0 >> xlabel('x'); ylabel('y'); >> Ftle('z=4xy') >> hold on >> plot([0 0],[ ­2 2],[ ­2 2],[ ­2,2],[ ­2 2],[0 0],[ ­2 2],[2  ­2]) >> axis equal 11/23/11 GG303 13 14. Folds F Example (analyFcal) (cont.) zG = 4xGyG 5 Now form the Hessian matrix ⎡0 4⎤ H=⎢ ⎥ ⎣4 0⎦ 6 Find its eigenvectors and eigenvalues >> H=[0 4;4 0] H = 0  ­4  ­4 0 Saddle >> [v,k]=eig(H) geometry much v = more clear in  ­0.7071 0.7071 principal 0.7071 0.7071 reference frame k =  ­4 0 0 4 11/23/11 GG303 14 7 11/23/11 14. Folds IV EvaluaFon of curvature from discrete data (geometry) A Three (non ­colinear) points deﬁne a plane – and a circle. B Locate three discrete non ­ colinear points along a curve (e.g., L, M, N) C Draw the perpendicular bisectors to line segments LM and MN D Intersect perpendicular bisectors at the center of curvature C. E The radius of curvature (ρ) equals the distance from C to L, M, or N. F The curvature is reciprocal of the radius of curvature (k = 1/ρ) G Local geometry of a curve also is circular! 11/23/11 GG303 15 14. Folds IV KinemaFcs of folding (strain) A Curvature of a plane curve k = dϕ/ds, where ϕ = orientaFon of tangent t to curve s = distance along curve B Curvature of a circular arc φ = θ + 90°, so dφ = dθ s = ρθ , so ds = ρdθ k = dφ /ds = dθ /ds = 1/ρ Large curvature = small radius Small curvature = large radius C Curvature can be assigned a sign + = concave up  ­ = concave down 11/23/11 GG303 16 8 11/23/11 14. Folds V KinemaFcs of folding (cont.) D Layer ­parallel normal strain (εθθ) for cylindrical folds 1 Mid ­plane of layer (y = 0) maintains length L0 2  Layer maintains thickness t during folding εθθ = ΔL L1 − L0 ( ρ + y )θ − ρθ y = = = = yk L0 L0 ρθ ρ t ⎞ +tk ⎛ εθθ ⎜ y = ⎟ = (elongation ) ⎝ 2⎠ 2 −t ⎞ −tk ⎛ εθθ ⎜ y = ⎟ = (contraction ) ⎝ 2⎠ 2 11/23/11 Note: If convex curvature is considered negaFve, then all the equaFons here should minus signs on the right side GG303 17 14. Folds V KinemaFcs of folding (cont.) E Layer ­parallel normal strain for three ­dimensional folds 1 Gauss’ Theorem: If the product of the principal curvatures (i.e., the Gaussian curvature K = k1k2) a deformed surface remains unstrained* 2  For geologic folds, the Gaussian curvature invariably changes during folding, so layer ­parallel strains will occur on the surfaces as well as interiors of folded layers 11/23/11 * * * GG303 18 9 11/23/11 14. Folds VI Fold terminology and classiﬁcaFon A Hinge point: point of local maximum curvature. B Hinge line: connects hinge points along a given layer. C Axial surface: locus of hinge points in all the folded layers. D Limb: surface of low curvature. 11/23/11 GG303 19 14. Folds VI Fold terminology and classiﬁcaFon (cont.) D Cylindrical fold: a surface swept out by moving a straight line parallel to itself 1 Fold axis: line that can generate a cylindrical fold 2  Parallel fold: top and boMom of layers are parallel and layer thickness is preserved* 3  Non ­parallel fold: top and boMom of layers are not parallel; layer thickness is not preserved* * AssumpFon: boMom and top of layer were originally parallel 11/23/11 GG303 20 10 11/23/11 14. Folds VI Fold terminology and classiﬁcaFon (cont.) E Fleuty’s ClassiﬁcaFon 1 Based on orientaFon of axial surface and fold axis 2 First modiﬁer (e.g., "upright") describes orientaFon of axial surface 3  Second modiﬁer (e.g., "horizontal") describes orientaFon of fold axis 11/23/11 GG303 21 14. Folds Fold terminology and classiﬁcaFon (cont.) F Inter ­limb angle Interlimb angle ClassiﬁcaFon 180°  ­ 120° Gentle 120°  ­ 70° Open 70°  ­ 30° Close 30°  ­ 0° Tight "0°” Isoclinal NegaFve Mushroom 11/23/11 GG303 22 11 ...
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