Limitation of The Concept of Stress in Structural Geology
Stress (σ)
- Stress (σ) = F/A dimana A=luas permukaan
Unit stress yang umum adalah pascal (KPa, MPa, GPa), bar atau dalam •
2 skala luas seperti psi (pound per square inch) dan kg/cm Stress untuk batuan didalam bumi: (lithostatic stress)
σ = ρgh
•
Stress pada suatu titik dapat dibagi menjadi normal ( ) dan shear ( ) σ σ•
n s
stress komponen Stress dapat bersifat compressive (+) dan tensile (-) • Shear stress dalam system kopel akan positive bila searah jarum jam dan • negative bila berlawanan arah jarum jam
- Stress 2D disuatu titik digambarkan sebagai stress ellipse Stress 3D disuatu titik digambarkan sebagai stress ellipsoid •
Principles stress : σ > σ > σ •
1
2
3 Koordinat sumbu utama stress (x ,x ,x ) adalah sejajar dengan stress •
1
2
3 utama
Stress-1 BS/03/02 Stress-2 BS/03/02
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STRESS vs. STRAIN STRESS vs. STRAIN Stress Stress Strain Strain
Relationship Between Stress and Strain Limitation of The Concept of Stress Limitation of The Concept of Stress in Structural Geology in Structural Geology
- Evaluate Using Experiment of Rock
Deformation
- Rheology of The Rocks • Using Triaxial Deformation Apparatus • Measuring Shortening • Measuring Strain Rate • Strength and Ductility
Stress-5 BS/03/02 Stress-6 BS/03/02
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TECTONICS AND STRUCTURAL GEOLOGY
- Study of rock Deformation as Response to Forces and Stresses • Involving Motion of Rigid Body
Deformation = Translation + Rotation + Dilation + Distortion FACTOR CONTROLING DEFORMATION
- SCALE FACTOR
- RHEOLOGY>TIME FACTOR
- DESCRIPTIVE ANALYSIS>KINEMATIC ANALYSIS
- DYNAMIC ANALYSIS
- LINKED FAULT AND FOLD SYSTEMS >PROGRESSIVE DEFORMATION
- SCALE INDEPENDENCE IN BRITTLE DEFORMATION
- STRUCTURAL INHERITANCE
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Stress-9 BS/03/02 TECTONICS AND STRUCTURAL GEOLOGY NEW CONCEPTS IN TECTONIC AND STRUCTURAL GEOLOGY
1. Geometric
2. Kinematic
3. Dynamic
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Stress-10 BS/03/02
Twiss and Moores, 1992
SCALE FACTOR STRUCTURAL GEOLOGY DATA FOLLOW FRACTAL RELATIONSHIP Plates Aerial Photograph
Km-Scale Fold m-Scale Fold
Geologic Cross-Section and Seismic Section
5 Km
10Km
Deformation of rock in various scale
EVOLUTION OF STRUCTURE
- Force history
- Movement history
Single Particle Particles
(Modified from Means, 1976)
Stress-13 BS/03/02 Stress-14 BS/03/02
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DESCRIPTIVE ANALYSIS
RHEOLOGY
- DUCTILE
- BRITTLE
THREE TYPES OF STRUCTURES
- CONTACTS
- PRIMARY STRUCTURES
- SECONDARY STRUCTURES
Force Equilibrium
BASIC CONCEPTS
- Force is any action which alters, or tends to alter
(A) Balance
- Newton II law of motion : F = M a
2
2
5
- Unit force : kgm/s = newton (N) or dyne = gram cm/s ; N = 10 dynes
(B) Torque (C) Static Equilibrium (a). Force: vector quantity with magnitude and direction (D) Dynamic Equilibrium (b). Resolving by the parallelogram of forces Two Types of Force
- Body Forces (i.e. gravitational force)
- Contact Forces (i.e. loading)
(Davis and Reynolds, 1996)
Modified Price and Cosgrove (1990)
Stress-17 BS/03/02 Stress-18 BS/03/02
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STRESS Stress defined as force per unit area: σ = F/A
Z A = area, Stress units = Psi, Newton (N),
R
V
5 Pascal (Pa) or bar (10 Pa)
W
V W
(Davis and Reynolds, 1996) (Twiss and Moores, 1992)
) STRESS n
STRESS on PLANE σ
( s s e
- Stress at a point in 2D tr
S σ) l
- Types of stress
a rm o N Normal stress (σ )
N Stress (
She ar S tres s (σ ) s (+) Compressive (-) Tensile Shear stress (σ )
S
(+) (-)
- Coordinate System
Stress-21 BS/03/02 Stress-22 BS/03/02
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Σ z σ 1 σ 3 The State of Σ x Two-Dimensional Stress Ellipsoid Stress at Point
A. Stress elipse 3 σ 1 σ
X 3 X 1
a) Triaxial stress Principal Stress:
σ > σ
1
3
b) Principal planes of
B. Principal stress components Principal coordinate the ellipsoid (top) (top) axes and planes σ Σ zz z
, Σ = Surface Stress Σ (top) (rt) Z x z
Σ (lft) σ x zx (lft) xz dz σ σ xx dx (rt) (rt) X σ (lft) (bot) σ x x xz σ zx Σ x (bot) (bot) Σ σ z zz Arbitrary coordinate axes and planes
C. General stress components (Modified from Means, 1976) (Twiss and Moores, 1992)
The State of
3-Dimensional Principal Stress: Stress at Point
σ > σ > σ
1
2
3 x 3 A. Stress elipsoid x 1 z σ σ 1 3 Principal coordinate planes z Stress Tensor Notation
σ 2 σ σ σ
11
12
13
y = σ 2 σ σ σ x
21
22
23
y x σ σ σ
31
32
33 B. Principal stress components
x z Arbitrary coordinate planes
σ z zz = σ , σ = σ , σ = σ
σ
12
21
13
31
23
32
σ zx σ zy Geologic Sign
σ yx y
σ xy σ yy Convention of
σ σ xz yz σ xx Stress Tensor y x x
C. General stress components
(Twiss and Moores, 1992)
(Twiss and Moores, 1992)
Stress-25 BS/03/02 Stress-26 BS/03/02
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Mohr Diagram 2-D Mohr Diagram 2-D
A. Physical Diagram
A. Mohr Diagram
A. Physical Diagram
B. Mohr Diagram x
σ s σ n x
1 s θ n
σ
1 α θ
(p) (p) (p) (p')
(σ n , σ ) s (p') (p) (p)
(σ σ ) σ s p' n , s
(σ σ ) − α n , s
2α n'
1 3 r (p) sin σ σ −σ 2θ n
2 (p)
2θ −2α
(p) p
σ 3 x 3 σ σ σ n 3 σ n n σ
1 σ s
2θ σ n
σ σ
3
1 x
3 σ n
Plane P σ +σ
1
3 σ −σ
1
3
2
2 σ −σ
1
3 cos 2θ
(Twiss and Moores, 1992)
2
(Twiss and Moores, 1992)
Mohr Diagram 2-D Mohr Diagram 2-D Planes of maximum shear stress
A. Physical Diagram
B. Mohr Diagram x
1 σ s x
A. Physical Diagram
B. Mohr Diagram θ
σ s (θ + 90º)
σ xx x
1 x
1 z
(σ σ ) xx' xz
Planes of maximum σ s max θ = +45º θ' = +45º
σ σ xz z z shear stress Counter clockwise
σ
- + -
α
1 n n
2θ σ 1 σ
1 2σ
σ 2 (θ + 90º) zx xz σ σ 2α σ
3 σ x
3 1 n
3 2θ = +90º
σ 3 x 3 σ 3 x 3 σ 3 σ
1 σ s
σ n σ s ' º 2θ = −90
(σ σ ) zz' zx
(σ −σ ) xx z z σ s max
Counterclockwise C lockwise (σ + σ ) shear stress xx zz shear stress Clockwise
2 (Twiss and Moores, 1992)
(Twiss and Moores, 1992)
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Mohr Diagram 3-D Mohr Diagram 3-D Geometry of a three-dimensional
Maximum Shear Stress Stress on a Mohr diagram (Twiss and Moores, 1992) (Twiss and Moores, 1992)
FUNDAMENTAL STRESS EQUATIONS
Principal Stress: σ > σ > σ
1
2
3
- All stress axes are mutually perpendicular
- Shear stress are zero in the direction of
principal stress σ + σ - σ – σ
1
3
1
3 σ = cos 2 θ
N
2
2 (Davis and Reynolds, 1996)
σ
- – σ
1
3
- Mohr diagram is a graphical representative of state of stress
Sin 2 σ θ
= s Mean stress is hydrostatic component which tends to produce dilation
- Deviatoric stress is non hydrostatic which tends to produce distortion • Differential stress , if greater is potential for distortion
2
- Stress-33 BS/03/02
Stress Ellipsoid
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Image of Stress Image of Stress
σ σ s s σ σ σ s s s
σ n Deviatoric Applied
σ 1 σ σ 3 σ 2 σ 1 σ n Δ 3 σ Δ 3 σ σ 3 n σ = σ 1 2 σ = σ = σ 1 2 3 σ σ = σ n n n 2 3 σ 1 σ σ 3 σ σ n p a p a Δ 1 σ σ − σ 1 n p
= σ − σ 3
- a Δ 1 σ n
p
- a
G. Pure shear stress
H. Deviatoric stress
A. Hydros tatic stress
B. Uniaxial compression
C. Uniaxial tension (two-dimensional) σ σ
σ σ σ s s s s s Effective Applied
σ 3 σ 1 σ 3 3 E E σ 2 σ σ 3 σ 1 σ σ 3 σ 3 σ n σ 3 σ 1 σ 3 σ 1 σ σ 1 σ E σ 2 σ σ 2 σ 1 n n n n σ 1 σ 2 σ 3 σ 3 a a a D D D σ σ σ p f E f σ 1 σ − 1 p b a b D = σ − σ = σ − σ 1 3 2 p E σ 2 f b b c
D. Axial or confined
E. Axial extension or
F. Triaxial stress E σ 3 f σ − 3 p
I. Differential stress J. Effective stress compression extensional stress (Three examples)
STRESS
- Body force works from distance and depends on the amount of materials affected (i.e. gravitational force).
- Surface force are classes as compressive or tensile according to the distortion they produce.
- Stress is defined as force per unit area.
- Stress at the point can be divided as normal and shear component depending they direction relative to the plane.
- Structural geology assumed that force at point are isotropic and
homogenous
- Stress vector around a point in 3-D as stress ellipsoid which have three orthogonal principal directions of stress and three principal planes.
- Principal stress >σ >σ σ
1
2
3
- The inequant shape of the ellipsoid has to do with forces in rock and has nothing directly to do with distortions.
- Mohr diagram is a graphical representative of state of stress of rock
Stress-37 BS/03/02 Stress-38 BS/03/02
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