2. 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
3. Term for Stress & Strain
*) Important distinction between two quantities
6. FORCES & VECTORSFORCES & VECTORS
• ForceForce is any action which alters, or tends to alteris any action which alters, or tends to alter
• Newton II law of motion :Newton II law of motion : F = M aF = M a
• Unit force : kgm/sUnit force : kgm/s22
= newton (N) or dyne = gram cm/s= newton (N) or dyne = gram cm/s22
; N = 10; N = 1055
dynesdynes
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)
7. 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)
8. STRESSSTRESS
• Stress at a point in 2DStress at a point in 2D
• Types of stressTypes of stress
Stress(Stress(σσ))
NormalStress(
NormalStress(σσ
nn))
Shear Stress (
Shear Stress (σσ
ss ))
Normal stress (Normal stress (σσNN))
(+) Compressive(+) Compressive (-) Tensile(-) Tensile
Shear stress (Shear stress (σσSS))
(+)(+) (-)(-)
9. STRESS ON A PLANE AND AT A POINT
Stress Tensor Notation
σ11 σ12 σ13
σ = σ21 σ22 σ23
σ31 σ32 σ33
10. Stress EllipsoidStress Ellipsoid
FUNDAMENTAL STRESS EQUATIONSFUNDAMENTAL STRESS EQUATIONS
Principal Stress:Principal Stress:
σσ11 >> σσ22 >> σσ33
• All stress axes are mutuallyAll stress axes are mutually
perpendicularperpendicular
• Shear stress are zero in theShear stress are zero in the
direction of principal stressdirection of principal stress
Stress Tensor NotationStress Tensor Notation
σσ1111 σσ1212 σσ1313
σσ == σσ2121 σσ2222 σσ2323
σσ3131 σσ3232 σσ3333
σσ1212 == σσ2121,, σσ1313 == σσ3131,, σσ2323 == σσ3232
11. Stress EllipsoidStress Ellipsoid
a) Triaxial stressa) Triaxial stress
b) Principal planes ofb) Principal planes of
the ellipsoidthe ellipsoid
(Modified from Means, 1976)(Modified from Means, 1976)
13. B. Principal stress components
σ1
z
x
σ3
x1
x3
y
y
x2
x
x
y
z
σ2
x
σzy
σxy σyy
σyz
σyx
σxx
σzx
σzz
σxz
z
y
Arbitrary
coordinate planes
A. Stress elipsoid
C. General stress components
z
Principal
coordinate planes
The State ofThe State of
3-Dimensional3-Dimensional
Stress at PointStress at Point
(Twiss and Moores, 1992)(Twiss and Moores, 1992)
Principal Stress:Principal Stress:
σσ11 >> σσ22 >> σσ33
14. n
-
Planes of maximum
shear stress
Clockwise
shear stress
x3
x1
σs σs
Counterclockwise
shear stress
θ' = +45º
σ1
x3σ3
σ1
n
+
σs
x1
θ = +45º
σ1
σ3
2θ = +90º σn
σs max
Clockwise
2θ = −90' º
σs max
Counter clockwise
σ3
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)
15. σσcc == σσoo + tan+ tan θθ ((σσnn))
The Coulomb Law of FailureThe Coulomb Law of Failure
σσcc = critical shear stress= critical shear stress
σσoo = cohesive strength= cohesive strength
tantan θθ = coefficient= coefficient
of internal frictionof internal friction
σσnn = normal stress= normal stress
(Modified from Davis and Reynolds, 1996)(Modified from Davis and Reynolds, 1996)
Compressive FracturesCompressive Fractures
16.
17.
18.
19. • Body force works from distance and depends on the amount of materialsBody 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 theSurface 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 componentStress 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 andStructural geology assumed that force at point are isotropic and
homogenoushomogenous
• Stress vector around a point in 3-D as stress ellipsoid which have threeStress 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 stressPrincipal stress σσ11>>σσ22>>σσ33
• The inequant shape of the ellipsoid has to do with forces in rock and hasThe 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
28. 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
θ
θ
r
θ
θ
r = Sn
T
R
e tans = 1/2 ψt
ψ
γ ψ= tan
ψ
Shear Strain ( )γ
40. 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
41. (Modified from Park, 1989)
Deformation and Material
A. Elastic strain
B. Viscous strain
C. Viscoelastic strain
D. Elastoviscous
E. Plastic strain
Hooke’s Law: e = σ/E, E = Modulus Young or elasticity
Newtonian : σ = ηε, η = viscosity, ε = strain-rate
43. 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
