This preview shows page 1. Sign up to view the full content.
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 ...
View
Full
Document
 Fall '11
 StephenMartel

Click to edit the document details