2/27/2014
1
STRAIN ANALYSIS
UNDEFORMED DEFORMED
KINEMATIC ANALYSISKinematic analysis is the reconstruction of movements
cf
A. Rigid Body Translation
ba
f
a
B. Rigid Body Rotation
cf
cde
ba
cf
d
e b
E. Nonrigid Deformation by Distortion
C. Original Object
e
b
d a ba
cb
e d
f c
d
f
eD. Nonrigid Deformation by Dilation
(Davis and Reynolds, 1996)
2/27/2014
2
TYPES OF STRAIN
H
I
H
B. Inhomogeneous strain
A. Homogeneous strain
HH
l = 5 cmo
L' = 3 cm
L
l 8
Fundamental Strain Equations
Strain
R e = n
Deformed StateUndeformed State
R = 1θ θ
r = Sn
L
l = 8 cmf
L' = 4.8 cm
Extension (e) = (lf – lo)/lo
Stretch (S) = lf/lo = 1 + e
Lengthening e>0 and shortening e<0
Strain
B. Shear strain
Deformed StateUndeformed State
A. Extension and stretch
θr
θ
T
Re tans = 1/2 ψt
ψ
γ ψ = tan ψ
Shear Strain ( ) γ
Quadratic elongation (λ) = S2
λ’ = 1/λ = 1/S2
2/27/2014
3
S1
Strain Ellipsoid
S1 = Maximum Finite StretchS3 = Minimum Finite Stretch
S2
S2S3
S3
S3
S1
S1(Davis and Reynolds, 1996)
Mohr Strain Diagram
Ad
θd = +15º
S1
1 Unit
A
S1
B
γ/λ
.49
1.01.0
2 = +30ºθd
(λ γ λ', / )
C
Distorted Clay Cake
γ/λ
1.01.0
2 θd
λ −λ
2' '3 1
λ'3.0
.56
0 C
d
λ' 2.43 = λ'1 = .42 2.01.0
A
γ λ = λ −λ . 2θ
2/ ' ' SIN 3 1
d
λ'33.0
Minus
C
λ λ' + ' 2
1 3
2.01.0
λ −λ . 2θ2
' ' COS 3 1 d
0 A'λ'1
Equalsλ'
(Davis and Reynolds, 1996)
2/27/2014
4
HOMOGENOUS DEFORMATION
O
N
Simple Shear(Noncoaxial Strain)
A B
ML
Pure Shear(Coaxial Strain)
Progressive Deformation
M
S1
25% FlatteringS3
S1
S3 S1+ 22º
+ 31º S3S1 S3
30% Flattering
S3S3
S1
S1+ 45º
40% Flattering
(Davis and Reynolds, 1996)
2/27/2014
5
A. Regional scale
100 m
perpendicu larto layer ^^
S1
S2S3
S1^
^ ^ ^S2 < 1 S2 = 1 S2 >1
STRAIN HISTORY Scale Factor
100 mμ
B. Outcrop scale
10 mm
C. Hand sample scale
D.
A.
perpendicu larto layer
perpendicu larto layer
C.B.
^
^
S1^
S1^
S2^
S2^
S2S3^
S3^
S3
D. Microscope scale
E. F.
Structural development in competent layerbased on orientation of S1, S2 and S3
Strain Measurement
• Geological Map • Geologic Cross-sectionGeologic Cross section• Seismic Section• Outcrop• Thin Section
Knowing the initial objects• Shape• Size
• Orientation
2/27/2014
6
Field of Expansion
Field ofNo Strain
Strating Sizeand Shape
S3
Strain Field Diagram
Field of Compensation
Fieldof
LinearShortening
1.0 Field of Linier Strecthing
1.0
Field of Contraction
S1
Z
X
Y
A
Z
K = 1
k = χ
Special Types of Homogenous Strain
X
Z
Y
Y
X
B
^1
S2
^a =
SS
1
2^
K = 0
ConstrictionalStrain
FlatteringStrain
Plane S
train
Sim
ple
Ext
ensi
on
Simple Flattering1
Xb =
SS
2
3^
A. Axial symmetric extension (X>Y=Z) or Prolate uniaxialB. Axial symmetric shortening (X=Y>Z) or Oblate uniaxial
C. Plane strain (X>Y=1>Z) or Triaxial ellipsoid
Flinn Diagram
2/27/2014
7
Strain Measurement from Outcrop
Δ
Δ
Δ = gap
2/27/2014
8
STRESS vs. STRAIN
Relationship Between Stress and Strain
• Evaluate Using Experiment of Rock Deformation
• Rheology of The Rocks• Using Triaxial Deformation Apparatus• Measuring Shorteningg g• Measuring Strain Rate • Strength and Ductility
2/27/2014
9
C
MP
a)
Repture
400
300Yield
Strength
UltimateStrength
Yield StrengthAfter StrainHardening D
A
EB
Stress – Strain Diagram
2 3 4 61S ( %)
Diff
eren
tial S
tress
(in
M ReptureStrength
5
100
200
Strength
Strain (in %)
A. Onset plastic deformationB. Removal axial loadC. Permanently strained D. Plastic deformationE. Rupture
140130
2000
25ºC
Effects of Temperature and Differential Stress
0 2 4 6 8 10 12 14 16
Diff
eren
tial S
tress
, MPa
300
200
100
70
20
Crown Point Limestone
40
60
801500
1000
800ºC
700ºC
500ºC
300ºC
500Diff
eren
tial S
tress
(in
MP
a)
25 C
Strain, percent5 10 15 0 Strain (in %)
2/27/2014
10
Deformation and Material
A. Elastic strainB. Viscous strainC Viscoelastic strain
(Modified from Park, 1989)
C. Viscoelastic strainD. ElastoviscousE. Plastic strain
Hooke’s Law: e = σ/E, E = Modulus Young or elasticityNewtonian : σ = ηε, η = viscosity, ε = strain-rate
Effect increasing stress to strain-rate
(Modified from Park, 1989)
2/27/2014
11
Stress Strain
Limitation of The Concept of Stress in Structural Geology
• No quantitative relationship between t d t t istress and permanent strain
• Paleostress determination contain errors• No implication equation relating stress to strain rate that causes the deformation
2/27/2014
12