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Structural Geology

Deformation and Strain – Mohr Circle for Strain, Special Strain

States, and Representation of Strain – Lecture 8 – Spring 2005

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Math for Mohr Circle

• λ = λ1cos2φ + λ3sin2φ

• λ = ½ (λ1 + λ3) + ½ (λ1 - λ3)cos2φ

• γ = ((λ1/λ3) - λ3/λ1 - 2)½cosφ sinφ

• γ = ½ (λ1 - λ3) sinφ

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Transform to Deformed State

• So we need to further manipulate the equations to get expressions in terms of the deformed state

• Let λ2 = 1/λ and γ2 = γ/λ

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Mohr Circle Equations

• Then: λ2 = ½ (λ1 2 + λ3 2 ) - ½ (λ3 2 - λ1 2 ) cos2φ2

γ2 = ½ (λ3 2 - λ1 2 ) sin2φ2

• These equations describe a circle, with radius ½ (λ32 - λ12 ) located at ½ (λ12 + λ32 ) on a Cartesian system with the horizontal axis labeled λ' and the vertical axis labeled γ2

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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, e1 = 1 and e3 = -0.5

λ1 = 4 and λ3 = 0.25

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Constant Area

• The area is constant since λ1

½ + λ3½ = 1

λ1 2 = 0.25 and λ3 2 = 4 Figure 4.13a in text

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Circle Parameters

• The radius of the circle is thus ½ (λ32 - λ12 ) = ½(4 - 0.25) = ½ (3.75) = 1.9

• Centered at ½ (λ12 + λ32 ) = ½(0.25 + 4) = ½ (4.25) = 2.1

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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: λ2 = 0.9 and γ2 = 1.4, so that λ = 1.1 and γ = 1.5

Figure 4.13b in text

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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 ((λ1/λ3)½) • tanφ2) = 62°

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Illustration of Angular Relationships

α - 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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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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Axially Symmetric Extension

• X > Y = Z • Produces prolate strain

ellipsoid with extension in the X direction and shortening in Y and Z

• Hotdog to football shaped

Figure 4.14b in text

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Prolate Shapes

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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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Oblate Shapes

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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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Simple Elongation

• X > Y = Z = 1 (Prolate elongation) or

• X = Y = 1 > Z (oblate shortening)

Figure 4.14e in text

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Cases Without Dilation

• General strain, axially symmetric strain, and plane strain do not involve a volume change, implying that X•Y•Z = 1 (Δ = 0)

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Helvetic Alps, Switzerland

• 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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Depth Profile

• 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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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)

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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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Flinn Parameters

• The parameters a and b may be written: a = X/Y = (1 + e1)/(1 + e2)

b = Y/Z = (1 + e2)/(1 + e3)

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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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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)

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Mathematics of Modification

• Mathematically, ln a = ln (X/Y) = ln (1 + e1)/(1 + e2)

ln b = ln (Y/Z) = (1 + e2)/(1 + e3)

• From the definition of a logarithm, ln (X/Y) = ln X - ln Y

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Use of Natural Strain

• We also so that natural strain is defined as ε = ln (1 + e)

• Thus, we can simplify the equations to: ln a = ε1 - ε2

ln b = ε2 - ε3

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Definition of K

• k is redefined as K: K = ln a/ ln b = (ε1 - ε2)/(ε2 - ε3)

• Some geologists use plots to the base ten instead of e, but the plot is always log-log

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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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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 (X•Y•Z = 1 (Δ = 0))

Δ = (V - V0)/ V0 and V0 = 1

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Zero Dilation

• If Δ = 0, then (Δ + 1) = X•Y•Z = (1 + e1)•(1 + e2)•(1 + e3) which can be expressed in terms of natural strains ln (Δ + 1) = ε1 + ε2 + ε3

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Mathematical Rearrangement

• We can rearrange this into the axes of the Ramsey diagram as follows: (ε1 - ε2) = (ε2 - ε3) - 3 ε2 + ln (Δ + 1)

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ε2 = 0

• If ε2 = 0 (plane strain) then:

(ε1 - ε2) = (ε2 - ε3) + ln (Δ + 1)

• This is the equation of a straight line with unit slope

• If Δ > 0, the line intersects the line intersects the (ε1 - ε2) axis, and if Δ < 0, it intersects the (ε2 - ε3) axis

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Intensity of Strain

• The same shape of ellipsoid may occur for different degrees 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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Intensity and Natural Strain

• We can rewrite this in terms of natural strains, I = (ε1 - ε2)2 + (ε2 - ε3)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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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