FOLDS (I) A Flexure (deformation-induced curvature) in rock (esp. layered) B All kinds of rocks can be folded, even granites

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GG303 Lectue 26 8/19/05 1 FOLDS (I) I II Main Topics A What is a fold? B Fold geomety C Fold teminology and classification What is a fold? A Flexue (defomation-induced cuvatue) in ock (esp.

1 GG303 Lectue 26 8/19/05 1 FOLDS (I) I II Main Topics A What is a fold? B Fold geomety C Fold teminology and classification What is a fold? A Flexue (defomation-induced cuvatue) in ock (esp. layeed) B All kinds of ocks can be folded, even ganites III Fold geomety A Tangents Conside a cuve (t), whee t is any paamete, and is a vecto function that gives points on the cuve 1 Tangent vecto: '= d dt 2 Unit tangent = T = ' ' Stephen Matel 26-1 Univesity of Hawaii

2 GG303 Lectue 26 8/19/ Example 1: paabola y = x 2 ( x) = x '= d i + x 2 dx = d x i + x 2 dx i + 2x T = ' ' = 1 ( ) x ( ) 2 = = i + 2x 1 i + 2x 1+ 4x 2 at x =1, 4 Example 2: unit cicle ( x = cosθ, y = sinθ) ( θ) = cosθ i + sinθ '= d T = ' ' dθ = d cosθ ( i + sinθ ) dθ sinθ i + cosθ = ( sinθ) 2 + cosθ = sinθ i + cosθ ( ) 2 = sinθ i + 2 T = 5 i + cosθ Stephen Matel 26-2 Univesity of Hawaii

3 GG303 Lectue 26 8/19/05 3 B Cuvatue = deviation fom a staight line Stephen Matel 26-3 Univesity of Hawaii

4 GG303 Lectue 26 8/19/ Cuvatue along a cuve is the fist deivative (i.e., ate of change) of the unit tangent (i.e., slope) with espect to distance (s) along the cuve; 2 Cuvatue vecto (K) = K (t) = d T d dt dt 3 Cuvatue (K) = K(t) = K = d T d dt dt 4 K(s) = T'(s) = "(s) (simple vesion of (3), d = ds) ( ) 5 Example 2: cicle x = ρcosθ, y = ρsinθ a ( θ) = ρcosθ i + ρsinθ b '= d dθ = d ρcosθ ( i + ρsinθ ) dθ c T = ' ρsinθ i + ρcosθ = ' ρsinθ d K = dt dθ ' = ( ) 2 + ( ρcosθ) 2 = d i sinθ + cosθ dθ ρ = ρsinθ i + ρcosθ ρsinθi + ρcosθ = sinθ i + cosθ ρ = i cosθ sin θ = ρ ρ 2 e K = K = 1 ρ ( cosθ ) 2 + ( sin θ ) 2 = 1 ρ So K points opposite, and the cicle cuvatue = 1/ρ. 6 Cuvatue = 1/(adius of cuvatue) = 1/ρ Stephen Matel 26-4 Univesity of Hawaii

5 GG303 Lectue 26 8/19/ The cuvatue at a point on a cuved suface depends on the diection of the path along the suface. The deivative of the cuvatue can be taken to yield the maximum and minimum cuvatues. These tun out to be at ight angles and ae called pincipal cuvatues. Gaussian cuvatue = (Cmax)(Cmin). Fo a waped but unstetched sheet, CG = const. 8 Cuvatue has an associated sign: "U" > 0; "Λ" < 0 9 Inflection point: Cuvatue = zeo. Cuvatue changes fom concave up to concave down. IV Fold teminology and classification A B C D D Hinge point: point of local maximum cuvatue. Hinge line: connects hinge points along a given laye. Axial suface: locus of hinge points in all the folded layes. Limb: suface of low cuvatue. Cylindical fold: a suface swept out by moving a staight line paallel to itself 1 Fold axis: line that can geneate a cylindical fold 2 Paallel fold: top and bottom of layes ae paallel and laye thickness is peseved (assumes bottom and top of laye wee oiginally paallel). a b Cuved paallel fold: cuvatue is faily unifom. Angula paallel fold: cuvatue is concentated nea the hinges and the limbs ae elatively plana. 3 Non-paallel fold: top and bottom of layes ae not paallel; laye thickness is not peseved (assumes bottom and top of laye wee oiginally paallel). Hinges typically thin and limbs thicken. E Non-cylindical fold example: dome Stephen Matel 26-5 Univesity of Hawaii

6 GG303 Lectue 26 8/19/05 6 Stephen Matel 26-6 Univesity of Hawaii

7 GG303 Lectue 26 8/19/05 7 % Matlab scipt GG303_27_1.m % Matlab scipt to poduce plots fo tangents to a paabola and a cicle % fo Figue 27.2 of Lectue 27 of GG303 % Paabola poblem (Example 1, cuve 1, y = x^2) x = -2:0.1:2; % This sets the ange in x; O1x = zeos(size(x)); % Defines oigin fo plotting position vectos; O1y = zeos(size(x)); % Defines oigin fo plotting position vectos; 1i = x; % The "i" component of position 1 as a function of x; 1 = x.^2; % The "" component of position 1 as a function of x; T1i = 1./sqt(1+4*x.^2); % The "i" component of unit tangent T1 as a function of x; T1 = 2*x./sqt(1+4*x.^2);% The "" component of unit tangent T1 as a function of x; % Cicle poblem (Example 2, cuve 2, unit cicle centeed at oigin) thetad = 0:1:360; % This sets the ange of theta in adians; theta = thetad*pi/180; % This sets the ange of theta in adians; O2x = zeos(size(thetad)); % Defines oigin fo plotting position vectos; O2y = zeos(size(thetad)); % Defines oigin fo plotting position vectos; 2i = cos(theta); % The "i" component of position 2 as a function of theta; 2 = sin(theta); % The "" component of position 2 as a function of theta; T2i = -sin(theta); % The "i" component of unit tangent T2 as a function of theta; T2 = cos(theta); % The "" component of unit tangent T2 as a function of theta; % Now plot the figues on one page figue(1) clf subplot(2,1,1) % Pepae the fist plot in a 2-ow, 1column set of plots; plot (1i,1) % Plot the cuve hold on; % Get eady fo anothe plot index1a = find(x == 1); % Find whee x = 1; index1b = find(x == -2); % Find whee x = -2; index1 = [index1a, index1b]; % Daw black aows fom the oigin to cuve with heads at the indexed points; % The "stock" vesion of quive poduces aow heads of diffeent size: % quive ( O1x(index1),O1y(index1),1i(index1),1(index1),0,'k' ); % So I fist plot black lines connecting the oigin to the indexed points... line ([O1x(index1a),1i(index1a)],[O1y(index1a),1(index1a)],'Colo',[0 0 0] ); line ([O1x(index1b),1i(index1b)],[O1y(index1b),1(index1b)],'Colo',[0 0 0] ); % and then use quive to daw aows of UNIT length with heads whee I want heads scalefacto = sqt( ( 1i(index1) - O1x(index1) ).^2 + ( 1(index1) - O1y(index1) ).^2 ); newdx = ( 1i(index1) - O1x(index1) )./scalefacto; newdy = ( 1(index1) - O1y(index1) )./scalefacto; quive ( 1i(index1)-newdx,1(index1)-newdy,newdx,newdy,0,'k' ); Stephen Matel 26-7 Univesity of Hawaii

8 GG303 Lectue 26 8/19/05 8 % Daw ed aows fo unit tangents to cuve 1 with tails at the indexed points; quive ( 1i(index1),1(index1),T1i(index1),T1(index1),0,'' ); axis('equal') % Set the x-scale and y-scale equal xlabel('x','fontsize',14) ylabel('y','fontsize',14) title('positions and Unit Tangents Along a Paabolic Cuve Fig. 27.2a','FontSize',18) subplot(2,1,2) % Pepae the second plot in a 2-ow, 1column set of plots; plot (2i,2) % Plot the cuve hold on; % Get eady fo anothe plot index2a = find(thetad == 30); % Find whee thetad = 30 ; index2b = find(thetad == 120); % Find whee thetad = 120 ; index2 = [index2a, index2b]; % Daw black aows fom the oigin to cuve with heads at the indexed points; quive ( O2x(index2),O2y(index2),2i(index2),2(index2),0,'k'); % Daw ed aows fo unit tangents to cuve with tails at the indexed points; quive ( 2i(index2),2(index2),T2i(index2),T2(index2),0,'' ); axis('equal') % Set the x-scale and y-scale equal xlabel('x','fontsize',14) ylabel('y','fontsize',14) title('positions and Unit Tangents Along a Cicula Cuve Fig. 27.2b','FontSize',18) pint -dill Fig_27.1.ill % Save figue as an Adobe Illustato file Stephen Matel 26-8 Univesity of Hawaii

Chapter 10 Sample Exam

Chapter 10 Sample Exam Chapte Sample Exam Poblems maked with an asteisk (*) ae paticulaly challenging and should be given caeful consideation.. Conside the paametic cuve x (t) =e t, y (t) =e t, t (a) Compute the length of the