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1 Structural Geology Deformation and Strain – Mohr Circle for Strain, Special Strain States, and Representation of Strain – Lecture 8 – Spring 2016
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2 Math for Mohr Circle λ = λ 1 cos 2 φ + λ 3 sin 2 φ λ = ½ (λ 1 + λ 3 ) + ½ (λ 1 - λ 3 )cos2φ γ = ((λ 1 /λ 3 ) - λ 3 /λ 1 - 2) ½ cosφ sinφ γ = ½ (λ 1 - λ 3 ) sinφ
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3 Transform to Deformed State So we need to further manipulate the equations to get expressions in terms of the deformed state Let λ́ = 1/λ and γ́ = γ/λ
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4 Mohr Circle Equations Then: λ́ = ½ (λ 1 ́ + λ 3 ́ ) - ½ (λ 3 ́ - λ 1 ́ ) cos2φ γ́ = ½ (λ 3 ́ - λ 1 ́ ) sin2φ These equations describe a circle, with radius ½ (λ 3 ́ - λ 1 ́ ) located at ½ (λ 1 ́ + λ 3 ́ ) on a Cartesian system with the horizontal axis labeled λ' and the vertical axis labeled γ́
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5 Unit Square Deformation Figure 4_13 shows an example A unit square is deformed so that is shortened in one direction by 50% and lengthened in the other by 100% Hence, e 1 = 1 and e 3 = -0.5 λ 1 = 4 and λ 3 = 0.25
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6 Constant Area The area is constant since λ 1 ½ x λ 3 ½ = 1 λ 1 ́ = 0.25 and λ 3 ́ = 4 Figure 4.13a in text
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7 Circle Parameters The radius of the circle is thus ½ (λ 3 ́ - λ 1 ́ ) = ½(4 - 0.25) = ½ (3.75) = 1.9 Centered at ½ (λ 1 ́ + λ 3 ́ ) = ½(0.25 + 4) = ½ (4.25) = 2.1 on the λ’ axis
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8 Mohr Circle for Strain Figure shows the plot and a line OP’ with an angle of 25° to the maximum strain axis From the graph we can determine: λ́ = 0.9 and γ’ = 1.4, so that λ = 1.1 and γ = 1.5 Figure 4.13b in text
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9 Significance of Line OP OP can represent the long axis of any significant geologic feature, such as a fossil We can gain further information: φ = tan -1 ((λ 1 /λ 3 ) ½ ) tanφ́) = 62°
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10 Illustration of Angular Relationships α = 62° - 25° = 37° which is the angle the long axis rotated from the undeformed to the deformed state The angular shear, ψ, is 56° (since ψ = tan -1 γ) Figure 4.13c in text
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11 General Strain X > Y > Z Also known as triaxial strain NOT the same as general shear Unshaded figure is the original cube, shaded figure is the deformed structure Figure 4.14a in text
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12 Axially Symmetric Elongation X > Y = Z Produces prolate strain ellipsoid with extension in the X direction and shortening in Y and Z Hotdog or football shaped ellipsoid Figure 4.14b in text
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13 Prolate Shapes
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14 Axially Symmetric Shortening X = Y > Z Produces an oblate ellipsoid with equal amounts of extension in the directions perpendicular to the shortening direction The strain ellipsoid resembles a hamburger Figure 4.14c in text
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15 Oblate Shapes
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16 Plane Strain X > 1 > Z One axis remains the same as before deformation, and commonly this is Y The description is often that of a two-dimensional ellipse in the XZ plane, with extension along X and contraction along Z Figure 4.14d in text
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17 Simple Elongation X > Y = Z = 1 (Prolate elongation) or X = Y = 1 > Z (oblate shortening) Figure 4.14e in text Prolate elongation produces a volume increase (Δ > 0) Oblate shortening produces a volume decrease (Δ < 0)
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18 Cases Without Dilation General strain, axially symmetric strain, and plane strain do not involve a volume change, implying that XYZ = 1 (Δ = 0)
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Comparison Strain analysis often seeks to compare strain From one place in an outcrop to another Between regions Strain is often heterogenesis on a scale of a single structure, and is always heterogeneous on larger scale (mountains, orogens) A large spatial distribution of data points may allow conclusions to be drawn on the state of strain in a region 19
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20 Helvetic Alps, Switzerland Map Region has undergone thrusting to the NW, so the greatest extension is parallel to that direction Figure 4.15a in text
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21 Depth Profile (Section) Depth profile showing the XZ ellipses plotted We see that extension increases with depth The marks plotted are called sectional strain ellipses Figure 4.15b in text
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22 Shape and Intensity – Flinn Diagram The Flinn diagram, named for British geologist Derek Flinn, is a plot of axial ratios In strain analysis, we typically use strain ratios, so this type of plot is very useful The horizontal axis is the ratio Y/Z = b (intermediate stretch/minimum stretch) and the vertical axis is X/Y = a (maximum stretch divided by intermediate stretch) Derek Flinn, 1922-2012
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23 Flinn Diagram On the β = 45° line, we have plane strain Above this line is the field of constriction, and below it is the field of flattening Figure 4.16a in text
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24 Flinn Parameters The parameters a and b may be written: a = X/Y = (1 + e 1 )/(1 + e 2 ) b = Y/Z = (1 + e 2 )/(1 + e 3 )
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25 Strain Ellipsoid Description The shape of the strain ellipsoid is described by a parameter, k, defined as: k = (a - 1)/(b - 1)
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26 Flinn Diagram Modification A modification of the Flinn diagram is the Ramsey diagram, named after structural geologist John Ramsey (1931 - ) Ramsey used the natural log of (X/Y) and (Y/Z)
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27 Mathematics of Modification Mathematically, ln a = ln (X/Y) = ln (1 + e 1 )/(1 + e 2 ) ln b = ln (Y/Z) = (1 + e 2 )/(1 + e 3 ) From the definition of a logarithm, ln (X/Y) = ln X - ln Y
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28 Use of Natural Strain Natural strain is defined as ε = ln (1 + e) Thus, we can simplify the equations to: ln a = ε 1 - ε 2 ln b = ε 2 - ε 3
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29 Definition of K k is redefined as K: K = ln a/ ln b = (ε 1 - ε 2 )/(ε 2 - ε 3 ) Some geologists use plots to the base ten instead of e, but the plot is always log-log
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30 Ellipsoid Descriptions Using K Above the line K = 1 we have the field of apparent constriction Below the line we find the field of apparent flattening Figure 4.16b in text
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31 Plotting Dilation Another advantage of the Ramsey diagram is the ability to plot lines showing the effects of dilation The previous discussion assumed dilation was zero (XYZ = 1 (Δ = 0)) Δ = (V - V 0 )/ V 0 and V 0 = 1
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32 Zero Dilation If Δ = 0, then (Δ + 1) = XYZ = (1 + e 1 )(1 + e 2 )(1 + e 3 ) which can be expressed in terms of natural strains ln (Δ + 1) = ε 1 + ε 2 + ε 3
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33 Mathematical Rearrangement We can rearrange this into the axes of the Ramsey diagram as follows: (ε 1 - ε 2 ) = (ε 2 - ε 3 ) - 3 ε 2 + ln (Δ + 1)
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34 ε 2 = 0 If ε 2 = 0 (plane strain) then: (ε 1 - ε 2 ) = (ε 2 - ε 3 ) + ln (Δ + 1) This is the equation of a straight line with unit slope If Δ > 0, the line intersects the (ε 1 - ε 2 ) axis, and if Δ < 0, it intersects the (ε 2 - ε 3 ) axis
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Why “Apparent”? The diagram makes it clear that, if K = 1, the volume change must be known to determine the actual strain state of a body A strain ellipsoid below the solid line may present true flattening but, depending on Δ, could represent plane strain or even constriction 35
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Ellipsoid Shape and Degree of Strain The further a point in a Flinn/Ramsey diagram is located from the origin, the more the strain ellipsoid deviates from a sphere The same degree of deviation from a sphere (same degree of strain) occurs for different shapes of the ellipsoid (different k or K) The same shape of the ellipsoid may occur for different degrees of strain 36
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37 Intensity of Strain The intensity of strain, represented by i, is given by: i = (((X/Y) - 1) 2 + ((Y/Z) - 1) 2 ) ½
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38 Intensity and Natural Strain We can rewrite this in terms of natural strains, I = (ε 1 - ε 2 ) 2 + (ε 2 - ε 3 ) 2 Listing the corresponding shape (k or K) and intensity (i or I) allows numerical comparisons of strain in the same structure, or over large regions
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39 Magnitude- Orientation Example Flinn diagram identifying position of each ellipsoid 1-8 = prolate, 9-12 – plane strain, and 13-20 - oblate Figure 4.17 in text
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