Structural Geology
Deformation and Strain – Mohr
Circle for Strain, Special Strain
States, and Representation of Strain –
Lecture 8 – Spring 2016
1
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
2
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
3
Mohr Circle Equations
• Then:
 λ́ = ½ (λ1́ + λ3́ ) - ½ (λ3́ - λ1́ ) cos2φ́
 γ́ = ½ (λ3́ - λ1́ ) sin2φ́
• These equations describe a circle, with radius
½ (λ3́ - λ1́ ) located at ½ (λ1́ + λ3́ ) on a
Cartesian system with the horizontal axis
labeled λ' and the vertical axis labeled γ́
4
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
5
Constant Area
• The area is constant since
 λ1½ x λ3½ = 1
 λ1́ = 0.25 and λ3́ = 4
 Figure 4.13a in text
6
Circle Parameters
• The radius of the circle is thus ½ (λ3́ - λ1́ ) =
½(4 - 0.25) = ½ (3.75) = 1.9
• Centered at ½ (λ1́ + λ3́ ) = ½(0.25 + 4) = ½
(4.25) = 2.1 on the λ’ axis
7
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
8
Significance of Line OP
• OṔ 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) • tanφ́) = 62°
9
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
10
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
11
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
12
Prolate Shapes
13
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
14
Oblate Shapes
15
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
16
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)
17
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)
18
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
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
20
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
21
Shape and Intensity
– Flinn Diagram
Derek Flinn, 1922-2012
• 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)
22
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
23
Flinn Parameters
• The parameters a and b may be written:
 a = X/Y = (1 + e1)/(1 + e2)
 b = Y/Z = (1 + e2)/(1 + e3)
24
Strain Ellipsoid Description
• The shape of the strain ellipsoid is described
by a parameter, k, defined as:
 k = (a - 1)/(b - 1)
25
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)
26
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
27
Use of Natural Strain
• Natural strain is defined as ε = ln (1 + e)
• Thus, we can simplify the equations to:
 ln a = ε1 - ε2
 ln b = ε2 - ε3
28
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
29
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
30
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
31
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
32
Mathematical Rearrangement
• We can rearrange this into the axes of the
Ramsey diagram as follows:
 (ε1 - ε2) = (ε2 - ε3) - 3 ε2 + ln (Δ + 1)
33
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 (ε1 - ε2) axis,
and if Δ < 0, it intersects the (ε2 - ε3) axis
34
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
Intensity of Strain
• The intensity of strain, represented by i, is
given by:
 i = (((X/Y) - 1)2 + ((Y/Z) - 1)2)½
37
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
38
MagnitudeOrientation
Example
Figure 4.17 in text
• Flinn diagram identifying
position of each ellipsoid
• 1-8 = prolate, 9-12 – plane39
strain, and 13-20 - oblate