Understanding Folds: A Study of Geometry, Curvature, and Classification, Study notes of Geology

Download Understanding Folds: A Study of Geometry, Curvature, and Classification and more Study notes Geology in PDF only on Docsity! 8/19/05 1 FOLDS (I) I Main Topics A What is a fold? B Fold geometry C Fold terminology and classification II What is a fold? A Flexure (deformation-induced curvature) in rock (esp. layered) B All kinds of rocks can be folded, even granites III Fold geometry A Tangents Consider a curve r(t), where t is any parameter, and r is a vector function that gives points on the curve 1 Tangent vector: € r'= dr dt 2 Unit tangent = T = € r' r' Docsity.com 8/19/05 2 3 Example 1: parabola € y = x2    € r r x( ) = x r i + x2 r j € r'= d r r dx = d x r i + x 2 r j ( ) dx = r i + 2x r j € r T = r' r' = 1 r i + 2x r j 12 + 2x( )2 = 1 r i + 2x r j 1+ 4x2 at x =1, r T = r i + 2 r j 5 4 Example 2: unit circle € x = cosθ, y = sinθ( ) € r r θ( ) = cosθ r i + sinθ r j € r '= d r r dθ = d cosθ r i + sinθ r j ( ) dθ = − sinθ r i + cosθ r j € r T = r ' r ' = −sinθ r i + cosθ r j −sinθ( )2 + cosθ( )2 = − sinθ r i + cosθ r j Docsity.com 8/19/05 5 7 The curvature at a point on a curved surface depends on the direction of the path along the surface. The derivative of the curvature can be taken to yield the maximum and minimum curvatures. These turn out to be at right angles and are called principal curvatures. Gaussian curvature = (Cmax)(Cmin). For a warped but unstretched sheet, CG = const. 8 Curvature has an associated sign: "U" > 0; "Λ" < 0 9 Inflection point: Curvature = zero. Curvature changes from concave up to concave down. IV Fold terminology and classification 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. 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 bottom of layers are parallel and layer thickness is preserved (assumes bottom and top of layer were originally parallel). a Curved parallel fold: curvature is fairly uniform. b Angular parallel fold: curvature is concentrated near the hinges and the limbs are relatively planar. 3 Non-parallel fold: top and bottom of layers are not parallel; layer thickness is not preserved (assumes bottom and top of layer were originally parallel). Hinges typically thin and limbs thicken. E Non-cylindrical fold example: dome Docsity.com Limbs and Hinges along Folds Fig. 26.4 Positive curvature = concave up Negative curvature = concave down & inflection point; V7 curvature = 0, & Radius of curvature is smail(est) at the hinge, largjest) on the limbs Curved Parallel Fold eo Wavelength >| Nt No hinge points in this example! Angular Parallel Fold Ke Wavelength | Hinge Limb Amplitude Docsity.com 8/19/05 7 % Matlab script GG303_27_1.m % Matlab script to produce plots for tangents to a parabola and a circle % for Figure 27.2 of Lecture 27 of GG303 % Parabola problem (Example 1, curve r1, y = x^2) x = -2:0.1:2; % This sets the range in x; O1x = zeros(size(x)); % Defines origin for plotting position vectors; O1y = zeros(size(x)); % Defines origin for plotting position vectors; r1i = x; % The "i" component of position r1 as a function of x; r1j = x.^2; % The "j" component of position r1 as a function of x; T1i = 1./sqrt(1+4*x.^2); % The "i" component of unit tangent T1 as a function of x; T1j = 2*x./sqrt(1+4*x.^2);% The "j" component of unit tangent T1 as a function of x; % Circle problem (Example 2, curve r2, unit circle centered at origin) thetad = 0:1:360; % This sets the range of theta in radians; thetar = thetad*pi/180; % This sets the range of theta in radians; O2x = zeros(size(thetad)); % Defines origin for plotting position vectors; O2y = zeros(size(thetad)); % Defines origin for plotting position vectors; r2i = cos(thetar); % The "i" component of position r2 as a function of theta; r2j = sin(thetar); % The "j" component of position r2 as a function of theta; T2i = -sin(thetar); % The "i" component of unit tangent T2 as a function of theta; T2j = cos(thetar); % The "j" component of unit tangent T2 as a function of theta; % Now plot the figures on one page figure(1) clf subplot(2,1,1) % Prepare the first plot in a 2-row, 1column set of plots; plot (r1i,r1j) % Plot the curve hold on; % Get ready for another plot index1a = find(x == 1); % Find where x = 1; index1b = find(x == -2); % Find where x = -2; index1 = [index1a, index1b]; % Draw black arrows from the origin to curve r with heads at the indexed points; % The "stock" version of quiver produces arrow heads of different size: % quiver ( O1x(index1),O1y(index1),r1i(index1),r1j(index1),0,'k' ); % So I first plot black lines connecting the origin to the indexed points ... line ([O1x(index1a),r1i(index1a)],[O1y(index1a),r1j(index1a)],'Color',[0 0 0] ); line ([O1x(index1b),r1i(index1b)],[O1y(index1b),r1j(index1b)],'Color',[0 0 0] ); % and then use quiver to draw arrows of UNIT length with heads where I want heads scalefactor = sqrt( ( r1i(index1) - O1x(index1) ).^2 + ( r1j(index1) - O1y(index1) ).^2 ); newdx = ( r1i(index1) - O1x(index1) )./scalefactor; newdy = ( r1j(index1) - O1y(index1) )./scalefactor; quiver ( r1i(index1)-newdx,r1j(index1)-newdy,newdx,newdy,0,'k' ); Docsity.com