Deformation and Strain
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1 Deformation and Strain Processes in Structural Geology & Tectonics Ben van der Pluijm WW Norton+Authors, unless noted otherwise 2/13/ :13
2 We Discuss Deformation and Strain Deformation Components Homogeneous vs. Heterogeneous Strain Strain Ellipse (and Ellipsoid) Coaxial and Non-coaxial Strain Superposed Strain Strain Quantities Representations of Strain (Finite) Strain Analysis Spherical Objects Angular Changes Length Changes Understanding Strain Values Deformation&Strain PSG&T 3
3 The Components of Deformation 1. Strain (distortion) a) Extension (or stretch) length changes b) Internal rotation (vorticity) finite strain axes rotate relative to instantaneous strain axes c) Volume change 2. Rigid-body rotation (or spin) instantaneous strain axes and finite strain axes rotate together 3. Rigid-body translation Deformation&Strain PSG&T 4
4 Homogeneous vs. Heterogeneous Strain Homogeneous strain: Straight lines remain straight Parallel lines remain parallel Circles become ellipses (or spheres become ellipsoids) vs. (deck of cards) Heterogeneous strain Deformation&Strain PSG&T 5
5 Homogeneous Strain: Principal Strain Axes Homogeneous strain: transformation of square to rectangle, circle to ellipse ( strain ellipse ). Two material lines perpendicular before and after strain are principal axes of strain ellipse (red lines): X Y Z. Dashed lines are material lines that do not remain perpendicular after strain; they rotate toward long axis of strain ellipse (X). Deformation&Strain PSG&T 6
6 Strain path Incremental strain (steps) Finite strain (difference between unstrained and final shape) Infinitely small strain increments (mathematical): instantaneous strain Deformation&Strain PSG&T 7
7 Strain Accumulation: non-coaxial and coaxial strain a. (progressive) simple shear or non-coaxial strain b. (progressive) pure shear, or coaxial strain Compare instantaneous strain (X i, Y i ) and finite strain (X 1,2,3, Y 1,2,3 ): vorticity Deformation&Strain PSG&T 8
8 Progressive Strain Pure Shear DePaor, 2002 Deformation&Strain PSG&T 9
9 Progressive Strain Simple Shear DePaor, 2002 Deformation&Strain PSG&T 10
10 Vorticity (internal rotation) and Particle Paths pure shear general shear simple shear RBR Rotation of material lines wrt to finite strain is kinematic vorticity. Kinematic vorticity number: W k = cos a a. W k = 0 (pure shear, coaxial strain end-member) b. 0 < W k < 1 (general shear, non-coaxial strain) c. W k = 1 (simple shear, non-coaxial strain end-member) d. W k = (rigid-body rotation) Deformation&Strain PSG&T 11
11 Transtension and Transpression Transtension Transpression Adding simple shear (non-coaxial strain) and pure shear (coaxial strain) Deformation&Strain PSG&T 12
12 Superimposed Strain I. lines continue to be extended II. lines continue to be shortened III. shortening during event 1 is followed by extension during event 2. I. to III. IV. extension is followed by shortening during event 2 Deformation&Strain PSG&T 13
13 Strain quantification Deformation&Strain PSG&T 15
14 StrainSim (R. Allmendinger) Deformation&Strain PSG&T 16
15 Strain Quantities Y X e = elongation = (l-l o )/l o = dl/l o (e 1 e 2 e 3 ; e 3 usually negative) s = stretch = l/l o (X Y Z) = [(l-l o )/l o + l o /l o ] = e+1 g = shear strain = tan y y = angular shear = quadratic elongation = s 2 = (e+1) misnomer Deformation&Strain PSG&T 17
16 Relationship between s, and angles Assume constant area: X.Y = 1, so Y = 1/X Deformation&Strain PSG&T 18
17 Homework: Strain A diagrammatic brachiopod before (a) and after strain (b). Determine elongation (e) of hinge line and angular shear (y) and shear strain (g) of the deformed shell. A sequence of tilted sandstone beds is unconformably overlain by a unit containing ellipsoidal inclusions (clasts in a conglomerate). The strain ratio of inclusions in sectional view is X/Y = 4, and dip of underlying beds is 50 o. What was angle of dip for beds in sectional view if inclusions were originally spherical? Name: Date: Deformation&Strain PSG&T 19
18 Extra: Increments and Natural Strain, ε For vanishingly small strain increment (or infinitessimal strain), elongation is: The natural strain, ε (epsilon), is the summation of all increments: Integrating gives: Deformation&Strain PSG&T 20
19 Extra: Mohr Construction for Strain Deformation&Strain PSG&T 21
20 3D Strain States (a) General strain (X > Y > Z) (b) Axially symmetric extension (X > Y = Z) (c) axially symmetric shortening (X = Y > Z) (d) plane strain (X > 1 > Z) (e) simple shortening (1 > Z) Deformation&Strain PSG&T 23
21 Representation of Strain Section, Map Strain in Helvetic nappes (large recumbent folds), Switzerland. Deformation&Strain PSG&T 24
22 Representation of Strain 3D in 2D-Space Δ + 1 = X Y Z 3D strain geometry in 2D plot: axial ratios and volume change Deformation&Strain PSG&T 25
23 Strain Methods: Variably-shaped Objects Deformation&Strain PSG&T 26
24 Strain Methods: Initially Spherical Objects Deformed ooids after: (b) 25% (X/Z = 1.8) shortening (c) 50% (X/Z = 4.0) shortening Deformation&Strain PSG&T 27
25 Strain Methods: Non-spherical Objects, c-t-c Center-to-center method (or Fry method): spacing varies as function of finite strain Deformation&Strain PSG&T 28
26 Strain Methods: Non-spherical Objects, R f /f R f /f method Deformation&Strain PSG&T 29
27 Strain Methods: Angular Changes Breddin method: Strain ratio determined at maximum angular shear, angle ϕ = 45 Substituting tan ϕ = 1 Y/X = tan ϕ' max Deformation&Strain PSG&T 30
28 Strain Methods: Length Changes X/Y=1.23 Deformation&Strain PSG&T 31
29 Finite Strain Compilation k=1, Δ=0 k=1, Δ=-0.5 Typical values: 1 < X/Z < 20 1 < X < < Z < 1 k=1 is plane strain (X>1>Z) Δ is volume change Where does lost rock volume go? Deformation&Strain PSG&T 32
30 Understanding Strain Values Strain and Mechanical Contrast: Passive markers have no mechanical contrast: bulk rock strain (clay and clay inclusion; oolite) Active markers have mechanical contrast: marker strain (clay and marble or fluid; conglomerate) Deformation&Strain PSG&T 33
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