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Deformation:Deformation:STRESS & STRESS & STRAINSTRAIN
DeformationDeformation
Dilation: Dilation: a change in volumea change in volume
Translation: Translation: a change in placea change in place
Rotation: Rotation: a change in orientationa change in orientation
Distortion: Distortion: a change in forma change in form
Term for Stress & Strain
*) Important distinction between two quantities
SCALARSSCALARS
temperaturetemperature
speedspeed
volumevolume
timetime
lengthlength
VECTORSVECTORS
force and stressforce and stress(on a surface)(on a surface)
temperaturetemperaturegradientgradient accelerationacceleration
Earth’sEarth’sgravitygravity
fieldfield Earth’sEarth’smagneticmagnetic
fieldfield
velocityvelocity
MantleMantleconvectionconvection
flowflow
oceanoceancurrentscurrents
Scalars Scalars vs.vs. Vectors Vectors
VECTOR & COORDINATE SYSTEMVECTOR & COORDINATE SYSTEM
FORCES & VECTORSFORCES & VECTORS
• ForceForce is any action which alters, or tends to alter is any action which alters, or tends to alter• Newton II law of motion :Newton II law of motion : F = M a F = M a • Unit force : kgm/sUnit force : kgm/s2 2 = newton (N) or dyne = gram cm/s= newton (N) or dyne = gram cm/s22; N = 10; N = 1055 dynes dynes
BASIC CONCEPTSBASIC CONCEPTS
(a). Force: vector quantity with magnitude and direction(a). Force: vector quantity with magnitude and direction
(b). Resolving by the parallelogram of forces(b). Resolving by the parallelogram of forces
Modified Price and Cosgrove (1990)Modified Price and Cosgrove (1990)
Two Types of ForceTwo Types of Force
• Body Forces (i.e. gravitational force)Body Forces (i.e. gravitational force)
• Contact Forces (i.e. loading)Contact Forces (i.e. loading)
STRESSSTRESS
Stress defined as force per unit area:Stress defined as force per unit area:
σ = F/Aσ = F/A
A = area, Stress units = Psi, Newton (N), A = area, Stress units = Psi, Newton (N),
Pascal (Pa) or bar (10Pascal (Pa) or bar (1055 Pa) Pa)
(Davis and Reynolds, 1996)(Davis and Reynolds, 1996)
(Twiss and Moores, 1992)(Twiss and Moores, 1992)
STRESS STRESS
• Stress at a point in 2D Stress at a point in 2D • Types of stressTypes of stress
Str
ess
(S
tres
s (
))
Nor
mal
Str
ess
(
Nor
mal
Str
ess
(nn))
Shear Stress (
Shear Stress (ss ))
Normal stress (Normal stress (NN))
(+) Compressive(+) Compressive (-) Tensile(-) Tensile
Shear stress (Shear stress (SS))
(+)(+) (-)(-)
STRESS ON A PLANE AND AT A POINT
Stress Tensor Notation
11 12 13
= 21 22 23
31 32 33
Stress EllipsoidStress Ellipsoid
FUNDAMENTAL STRESS EQUATIONSFUNDAMENTAL STRESS EQUATIONS
Principal Stress:Principal Stress:11
• All stress axes are mutually All stress axes are mutually perpendicularperpendicular• Shear stress are zero in the Shear stress are zero in the direction of principal stress direction of principal stress
Stress Tensor NotationStress Tensor Notation
1111 1212 1313
= = 2121 22 22 2323
3131 3232 3333
1212 = = 2121, , 1313 = = 3131, , 2323 = = 3232
Stress EllipsoidStress Ellipsoid
a) Triaxial stressa) Triaxial stress
b) Principal planes ofb) Principal planes of the ellipsoid the ellipsoid
(Modified from Means, 1976)(Modified from Means, 1976)
σ2
σ1
σ3
σ1
σ1
σ1
σ1
σ1
σ2
σ2
σ2 σ2
σ3σ3
σ3
σ3σ2
ELIPSOID TEGASAN
σ1 > σ2 = σ3
σ1 = σ2 > σ3
σ1 > σ2 > σ3
B. Principal stress components
z
x
x1
x3
y
yx2
x
x
y
z
x
zy
xy yyyz
yx
xx
zx
zz
xz
z
y
Arbitrarycoordinate planes
A. Stress elipsoid
C. General stress components
z
Principalcoordinate planes
The State of The State of 3-Dimensional 3-Dimensional Stress at PointStress at Point
(Twiss and Moores, 1992)(Twiss and Moores, 1992)
Principal Stress:Principal Stress:11
n-
Planes of maximumshear stress
Clockwiseshear stress
x
x
s s
Counterclockwiseshear stress
' = +45º
x
n+
s
x
= +45º
º n
s max
Clockwise
' º
s max
Counter clockwise
B. Mohr DiagramB. Mohr DiagramA. Physical DiagramA. Physical Diagram
Planes of maximum shear stressPlanes of maximum shear stress
Mohr Diagram 2-DMohr Diagram 2-D
(Twiss and Moores, 1992)(Twiss and Moores, 1992)
c c = = oo + tan + tan ( (nn))
The Coulomb Law of FailureThe Coulomb Law of Failure
cc = critical shear stress = critical shear stress
oo = cohesive strength = cohesive strength
tan tan = coefficient = coefficient of internal frictionof internal frictionnn = normal stress = normal stress
(Modified from Davis and Reynolds, 1996)(Modified from Davis and Reynolds, 1996)
Compressive FracturesCompressive Fractures
• Body force works from distance and depends on the amount of materials Body force works from distance and depends on the amount of materials
affected (i.e. gravitational force).affected (i.e. gravitational force).• Surface force are classes as compressive or tensile according to the Surface force are classes as compressive or tensile according to the
distortion they produce.distortion they produce.• Stress is defined as force per unit area.Stress is defined as force per unit area.• Stress at the point can be divided as normal and shear component Stress at the point can be divided as normal and shear component
depending they direction relative to the plane.depending they direction relative to the plane.• Structural geology assumed that force at point are isotropic and Structural geology assumed that force at point are isotropic and
homogenoushomogenous• Stress vector around a point in 3-D as stress ellipsoid which have three Stress vector around a point in 3-D as stress ellipsoid which have three
orthogonal principal directions of stress and three principal planes. orthogonal principal directions of stress and three principal planes.
• Principal stress Principal stress 11>>22>>33
• The inequant shape of the ellipsoid has to do with forces in rock and has The inequant shape of the ellipsoid has to do with forces in rock and has
nothing directly to do with distortions. nothing directly to do with distortions. • Mohr diagram is a graphical representative of state of stress of rockMohr diagram is a graphical representative of state of stress of rock
STRESSSTRESS
STRAIN
UNDEFORMED DEFORMED
Strain is defined as the change (in size and shape) of a body resulting from the action of an applied stress field
TYPES OF STRAIN
B. Inhomogeneous strain
A. Homogeneous strain
H
I
H
L
l = 5 cmo
L' = 3 cm
L
l = 8 cmf
L' = 4.8 cm
Fundamental Strain Equations
Extension (e) = (lf – lo)/lo
Stretch (S) = lf/lo = 1 + e
Lengthening e>0 and shortening e<0 Strain
B. Shear strain
Deformed State
Strain
R e = n
Deformed State
Undeformed State
A. Extension and stretch
Undeformed State
R = 1
r
r = Sn
T
Re tans t
= tan
Shear Strain ( )
SHEAR STRAIN
S2
S2
S3
S3
S3
S1
S1
S1
Strain Ellipsoid
S1 = Maximum Finite StretchS3 = Minimum Finite Stretch
(Davis and Reynolds, 1996)
τ1
τ3τ2
ELIPSOID TERAKAN τ1
τ3τ2
τ1
τ3
τ2
τ1
τ3
τ2
τ1
τ3
τ2
τ1
τ3
τ2
τ1 > τ2 = τ3
τ1 = τ2 > τ3
τ1 > τ2 > τ3
ON
Simple Shear(Noncoaxial Strain)
A B
M
S1
ML
Pure Shear(Coaxial Strain)
S3S3
S1
25% FlatteringS3
S1
S3 S1+ 22º
+ 31º S3S1
S1
S3
30% Flattering
+ 45º
40% Flattering
Progressive Deformation
(Davis and Reynolds, 1996)
Strain Measurement
• Geological Map • Geologic Cross-section• Seismic Section• Outcrop• Thin Section
Knowing the initial objects• Shape• Size
• Orientation
Strain Measurement from Outcrop
= gap
STRESS vs. STRAIN
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
(Modified from Park, 1989)
Deformation and Material
A. Elastic strainB. Viscous strainC. Viscoelastic strainD. ElastoviscousE. Plastic strain
Hooke’s Law: e = /E, E = Modulus Young or elasticityNewtonian : = viscosity, = strain-rate
Stress EllipsoidStrain Ellipsoid
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
STRESS – STRAIN RELATIONS
BRITTLE & DUCTILE DEFORMATIONS
DEFORMATION MECHANISMS
THANK YOU
GEOLOGY CARTESIAN COORDINATE SYSTEM GEOLOGY CARTESIAN COORDINATE SYSTEM
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