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