GY403 Structural Geology. The general equations of the Mohr Circle for strain

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1 GY403 Structural Geology The general equations of the Mohr Circle for strain

2 Strain Ellipsoid A three-dimensional ellipsoid that describes the magnitude of dilational and distortional strain P Assume a perfect sphere before deformation P Three mutually perpendicular axes X, Y, and Z P X is maximum stretch (S X ) and Z is minimum stretch (S Z ) P There are unique directions corresponding to values of S X and S Z, but an infinite number of directions corresponding to S Y Z X Y

3 Strain The results of deformation via distortion and dilation P Heterogeneous strain: strain ellipsoid varies from point-to-point in deformed body P Homogenous strain: strain ellipsoid is equivalent from point-to-point in deformed body P Although hetereogenous strain is the rule in real rocks, often portions of a deformed body behave as homogenous with respect to strain

4 Homogeneous Strain Ground Rules Characteristics of homogenous strain P Straight lines that exist in the non-rigid body remain straight after deformation P Lines that are parallel in the non-rigid body remain parallel after deformation P In a special case of homogenous strain termed Plane Strain, volume and area are conserved

5 General Strain Equations Extension (e), Stretch (S), and Quadratic Elongation (λ) These equations measure linear strain : l O = original length l F = final length e = l F -l O l O l O = 5cm ouch! S = λ = l F l O l F l O 2 l F = 12cm S = l F l O = 12cm 5cm = 2.4 e = (S-1) = = 1.4 λ = S 2 = (2.4) 2 = 5.76

6 Rotational Strain Equations quantifying angular shear (ψ) and shear strain (γ) θ = angle between reference line (L) and maximum stretch (X) measured from X to A (clockwise=+; anticlockwise=-) M θ=-35 L X M ψ Z L θ d =-25 X deformation Strain ellipse M ψ L (perpendicular to L relative to M) = -40 γ L = tan(ψ L ) = tan(-40) = α L = θ d - θ = (-25) - (-35) = +10 angle of internal rotation

7 Mohr Circle for Strain General equations as a function of λ X, λ Z, and θ d 1 λ = λ λ X = quadratic elongation parallel to X axis of finite strain ellipse λ Z = quadratic elongation parallel to Z axis of finite strain ellipse λ = λ Z +λ X -λ Z -λ X cos(2θ d ) 2 2 γ λ Z -λ X sin(2θ d ) λ = 2 tan θ d = tan θ S Z SX α = θ d - θ (internal rotation)

8 Mohr Circle for Strain Geometric relations between the finite strain ellipse and the Mohr Circle for strain 1.0 A Z A Strain ellipse S X = (l F /l O )=(1.414/1.0)=1.414 S Z = (l F /l O )=(0.816/1.0)=0.816 X θ d =+30 2θ d =60 λ X λ 1.0 λ Z 2.0 λ λ X = (S x ) 2 =(1.414) 2 =2.0 λ X = 1/λ= 1/2.0 = 0.5 λ Z = (S z ) 2 = (0.816) 2 = λ Z = 1/λ = 1/0.666 = 1.50

9 Mohr Circle for Strain Reference lines in the undeformed and deformed state a b cde f g h i j k l m n srq po a b cd e f g S X =1.936 S Z =0.707 h i j s r q p o n m l k

10 Mohr Circle Strain Relationships Values of quadratic elongation (λ), shear strain (γ), original θ angle, angular shear (ψ), and angle of internal rotation (α) as a function of θ d Line a b c d e f g h i j k l m n o p q r s θ d λ γ θ ψ α S X =1.936 S Z =0.707 a b cde f g s r q p o n m h l i k j

11 Strain Ellipse General Equation Values for quadratic elongation (λ) and shear strain (γ) as a function of θ d l k m n o p q a b j r s c d e f g i h d

12 Strain Ellipse General Equation Values for angular shear (ψ) and internal rotation (α) as a function of θ d a b c d e f g h i j k l m n o p q r s θ d Internal Rotation(α) Angular Shear(ψ)

13 Example strain problem Given a finite strain ellipse of S X =1.936 and S Z =0.707, find for direction θ d =-20E values of S, λ, γ, ψ, and α λ X =(1.936) 2 = 3.750; λ Z = (0.707) 2 = 0.500; λ X =0.267; λ Z =2.0 λ = Cos(-40) = (0.866)(0.766) = λ = 1/λ = 1/0.470 = ˆ S = (2.128) 0.5 = γ = Sin(-40) λ = (0.866)(-0.643)(2.128) = ψ = tan -1 (γ) = tan -1 (-1.185) = -49.8E Tan(θ d ) = tan(θ) S Z S X α = θ d -θ = (-20) - (-44.9) = +24.9E ˆ Tan(θ) = tan(θ d ) S X S Z ˆ θ = -44.9E

14 Application of Plane Strain Deformed oolids from the study of Cloos (1947) Assuming plane strain: no dilational component to strain, therefore, constant volume applies: V S = 4/3πr 3 where r is the radius of the sphere V e = 4/3πabc where (a,b,c) are the ½ axial legths of the ellipsoid V s =V e 4/3πr 3 = 4/3πabc Because of plane strain r = b ˆ r 2 = ac r = (ac) 0.5 Example: a=4.2mm; c=2.5mm; r=(4.2*2.5) 0.5 = 3.3 ˆ S x = 4.2/3.3 = 1.27

15 Application to Deformed Strain Markers PMarkers may be original spheres or ellipsoids PPebbles, sand grains, reduction spots, ooids, fossils, etc. PAssume homogenous strain domain

16 Measuring Length/Width Ratios (R f ) P Measure major and minor axis of each strain ellipse P Rf = (Major/minor) (yields a unitless ratio) P φ = Angle from reference direction (usually foliation or cleavage), positive angles are clockwise, negative counterclockwise

17 Spreadsheet setup for Rf/ φ analysis Ellipse Length Width Rf φ P Note: φ is measured relative to a chosen reference direction such as foliation

18 Hyperbolic Net P Used to plot strain markers that were originally ellipsoidal P Statistically the Rf ratios will tend to fall along one of the hyperbolic curves N φ=+30 Rf=1.6