This document provides information about strain analysis and the relationship between stress and strain. Some key points: - Strain is defined as the change in size and shape of a body resulting from an applied stress. Kinematic analysis is used to reconstruct deformation. - There are different types of strain including elastic, brittle, and plastic, which depend on the magnitude and rate of applied stress. Homogeneous and inhomogeneous strain can occur. - Strain is measured using various techniques at different scales from regional to microscopic. Equations relate changes in length, shear, and elongation to strain. - The relationship between stress and strain in rocks is evaluated experimentally. Stress-strain diagrams show properties like strength and duct

1. Lecture #11

2. STRAIN ANALYSIS
UNDEFORMED DEFORMED
Strain is defined as the change in size and shape of a
body resulting from the action of an applied stress
field

3. KINEMATIC ANALYSIS
Kinematic analysis is the reconstruction of movements
f c
a
b a
c
e d
A. Rigid Body
Translation
a b
f c
a b
f
d e
B. Rigid Body
Rotation
E. Nonrigid Deformation
by Distortion
C. Original Object
c
e
b
f c
e d
d a
d
f
e
b
D. Nonrigid Deformation
by Dilation
(Davis and Reynolds, 1996)

4. Type of Deformation
Eastic strain if the body of rock returns to its previous shape after the
stress has been removed. A good example is the slow rebound of the
North American crust after having been downwarped by the great
weight of the Pleistocene glaciers.
Brittle strain occurs when the stress is great enough to break
(fracture) the rock.
Plastic strain results in a permanent change in the shape of the rock.
A ductile rock is one that “flows plastically” in response to stress.
Whether the strain is plastic or brittle depends on both the magnitude
of the stress and how quickly the stress is applied. A great stress that is
slowly applied often folds rocks into tight, convoluted patterns
without breaking them.

5. TYPES OF STRAIN
A. Homogeneous strain
B. Inhomogeneous strain
H
I
H

6. Fundamental Strain Equations
L
l = 5 cm o
L' = 3 cm
L
l = 8 cm f
L' = 4.8 cm
Extension (e) = (lf – lo)/lo
Lengthening e>0 and shortening e<0
Stretch (S) = lf/lo = 1 + e
Strain
Undeformed State
R = 1

A. Extension and stretch
Strain
Deformed State

Deformed State
B. Shear strain
R = en
Undeformed State

r

r = Sn
T
R

e tan st  
 = tan 

Shear Strain ()
Quadratic elongation (l) = S2
l’ = 1/l = 1/S2

7. S2
S2
S3
S3
S3
S1
S1
S1
Strain Ellipsoid
S1 = Maximum Finite Stretch
S3 = Minimum Finite Stretch
(Davis and Reynolds, 1996)

8. Mohr Strain Diagram
Ad
d = +15º
C
l'3
Distorted Clay Cake
S1
1 Unit
A
S1
3.0
1.0
l
 l
' '   
2 = +30º d
1.0 2.0
Minus
1.0
2 d
C

l l

l' + l'
1 3
2
B
l'
3.0
.56
l
.49
0
C
l', /l)
l' 2.4 3 l' = 1 = .42 1.0 2.0
' '
l  l   
 COS 

d
0
A
l' A' 1
Equals
l l  

/
' '
SIN   
d
l'
(Davis and Reynolds, 1996)

9. HOMOGENOUS DEFORMATION

10. Progressive Deformation
A B
O
N
Simple Shear
(Noncoaxial Strain)
M
S1
L M
Pure Shear
(Coaxial Strain)
30% Flattering
S3 S3
S1
25% Flattering
S3
S1
S3 S1 + 22º
+ 31º
S3
S1
S1
S3
+ 45º
40% Flattering
(Davis and Reynolds, 1996)

11. D. Microscope scale
100 m
A. Regional scale
100 m
B. Outcrop scale
10 mm
C. Hand sample scale
STRAIN HISTORY
D.
A.
^
perpendicular
to layer
perpendicular
to layer
perpendicular
to layer
E.
C.
F.
B.
^
S 1
^ ^
S2
S3
S1
^
^ ^ ^
^
^
S1
^
S1
^
S2
^
S2
S2
^
S3
^
S3
S3
S2< 1 S2 = 1 S2 >1
Structural development in competent layer
based on orientation of S1, S2 and S3
Scale Factor

12. Strain Measurement
• Geological Map
• Geologic Cross-section
• Seismic Section
• Outcrop
• Thin Section
Knowing the initial objects
• Shape
• Size
• Orientation

13. Strain Field Diagram
Field of
Field of Compensation
Expansion
Field of
No Strain
1.0
Strating Size
and Shape
Field
of
Linear
Shortening
Field
of Contract ion
S1
S3
1.0
Field of Linier Strecthing

14. X
Z
Y
Z
X
Y
A
Z
Y
X
B
Special Types of
Homogenous Strain
^
k = 
1
b =
S
S
2
3
^
^
a =
S
S
1
2
^
K = 1
K = 0
Constrictional
Strain
Flattering
Strain
Plane Strain
Simple Extension
Simple Flattering
1
A. Axial symmetric extension (X>Y=Z) or Prolate uniaxial
B. Axial symmetric shortening (X=Y>Z) or Oblate uniaxial
C. Plane strain (X>Y=1>Z) or Triaxial ellipsoid
Flinn Diagram

15. Strain Measurement from Outcrop

16. D
D
D = gap

17. STRESS vs. STRAIN

18. Relationship Between Stress and Strain
• Evaluate Using Experiment of Rock
Deformation
• Rheology of The Rocks
• Using Triaxial Deformation Apparatus
• Measuring Shortening
• Measuring Strain Rate
• Strength and Ductility

19. Stress – Strain Diagram
C
1 2 3 4 6
Strain (in %)
Differential Stress (in MPa)
Repture
Strength
400
5
300
200
100
Yield
Strength
Ultimate
Strength
Yield Strength
After Strain
Hardening
D
A
B E
A. Onset plastic deformation
B. Removal axial load
C. Permanently strained
D. Plastic deformation
E. Rupture

20. Effects of Temperature and
Differential Stress
40
80
0 2 4 6 8 10 12 14 16
Differential Stress, MPa
Strain, percent
300
200
100
70
20
Crown Point Limestone
140
130
60
300ºC
500ºC
700ºC
5 10 15
2000
1500
1000
0
Strain (in %)
800ºC
500
Differential Stress (in MPa)
25ºC

21. Deformation and Material
A. Elastic strain
B. Viscous strain
C. Viscoelastic strain
D. Elastoviscous
E. Plastic strain
(Modified from Park, 1989)
Hooke’s Law: e = s/E, E = Modulus Young or elasticity
Newtonian : s = he, hviscosity, e = strain-rate

22. Effect increasing stress to strain-rate
(Modified from Park, 1989)

23. Stress Strain

24. Limitation of The Concept of Stress in Structural Geology
• No quantitative relationship between
stress and permanent strain
• Paleostress determination contain
errors
• No implication equation relating
stress to strain rate that causes the
deformation