1 Structural Geology Deformation and Strain – Mohr Circle for Strain, Special Strain States, and Representation of Strain – Lecture 8 – Spring 2016

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

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

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

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

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

7 Circle Parameters The radius of the circle is thus ½ (λ 3 ́ - λ 1 ́ ) = ½( ) = ½ (3.75) = 1.9 Centered at ½ (λ 1 ́ + λ 3 ́ ) = ½( ) = ½ (4.25) = 2.1 on the λ’ axis

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

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°

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

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

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

13 Prolate Shapes

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

15 Oblate Shapes

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

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)

18 Cases Without Dilation General strain, axially symmetric strain, and plane strain do not involve a volume change, implying that XYZ = 1 (Δ = 0)

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

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

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

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,

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

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 )

25 Strain Ellipsoid Description The shape of the strain ellipsoid is described by a parameter, k, defined as:  k = (a - 1)/(b - 1)

26 Flinn Diagram Modification A modification of the Flinn diagram is the Ramsey diagram, named after structural geologist John Ramsey ( ) Ramsey used the natural log of (X/Y) and (Y/Z)

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

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

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

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

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

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

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)

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

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

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

37 Intensity of Strain The intensity of strain, represented by i, is given by:  i = (((X/Y) - 1) 2 + ((Y/Z) - 1) 2 ) ½

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

39 Magnitude- Orientation Example Flinn diagram identifying position of each ellipsoid 1-8 = prolate, 9-12 – plane strain, and oblate  Figure 4.17 in text