Stress (σ) Stress. Strain

(1)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

Institut Teknologi Bandung

Stress-1 BS/03/02

Stress-2 BS/03/02

Stress (

σ)

• Stress (σ) = F/A

dimana A=luas permukaan

• Unit stress yang umum adalah pascal (KPa, MPa, GPa), bar atau dalam

skala luas seperti psi (pound per square inch) dan kg/cm

2

• Stress untuk batuan didalam bumi:

σ = ρgh

(lithostatic stress)

• Stress pada suatu titik dapat dibagi menjadi normal (σ

_n

) dan shear (σ

_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

₁

>

σ

₂

>

σ

₃

• Koordinat sumbu utama stress (x

₁

,x

₂

,x

₃

) adalah sejajar dengan stress

utama

Stress

(2)

Stress-5 BS/03/02

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-6 BS/03/02

Limitation of The Concept of Stress

in Structural Geology

TECTONICS AND STRUCTURAL GEOLOGY

• Study of rock

Deformation

as Response to Forces and Stresses

• Involving Motion of Rigid Body

FACTOR CONTROLING DEFORMATION

• SCALE FACTOR

• RHEOLOGY

• TIME FACTOR

Deformation = Translation + Rotation + Dilation + Distortion

• DESCRIPTIVE ANALYSIS

• KINEMATIC ANALYSIS

(3)

Stress-9 BS/03/02

TECTONICS AND STRUCTURAL GEOLOGY

NEW CONCEPTS IN TECTONIC AND STRUCTURAL GEOLOGY

• LINKED FAULT AND FOLD SYSTEMS

1. Geometric

2. Kinematic

3. Dynamic

• PROGRESSIVE DEFORMATION

• SCALE INDEPENDENCE IN BRITTLE DEFORMATION

• STRUCTURAL INHERITANCE

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

(4)

Stress-13 BS/03/02

(Modified from Means, 1976)

Deformation of rock in various scale

Stress-14 BS/03/02

EVOLUTION OF STRUCTURE

Single Particle

Particles

• Force history

• Movement history

DESCRIPTIVE ANALYSIS

• CONTACTS

• PRIMARY STRUCTURES

• SECONDARY STRUCTURES

THREE TYPES OF STRUCTURES

RHEOLOGY

(5)

Stress-17 BS/03/02

FORCES AND VECTORS

• Force is any action which alters, or tends to alter

• Newton II law of motion :

F = M a

• Unit force : kgm/s

2

_{= newton (N) or dyne = gram cm/s}

2

_{; N = 10}

5

_dynes

BASIC CONCEPTS

(a). Force: vector quantity with magnitude and direction

(b). Resolving by the parallelogram of forces

Modified Price and Cosgrove (1990)

Two Types of Force

• Body Forces (i.e. gravitational force)

• Contact Forces (i.e. loading)

Stress-18 BS/03/02

Force Equilibrium

(A) Balance

(B) Torque

(C) Static Equilibrium

(D) Dynamic Equilibrium

(Davis and Reynolds, 1996)

STRESS

Stress defined as force per unit area:

σ = F/A

A = area, Stress units = Psi, Newton (N),

Pascal (Pa) or bar (10

5

_Pa)

(Davis and Reynolds, 1996) (Twiss and Moores, 1992)

Z W V W R V

(6)

Stress-21 BS/03/02

STRESS

• Stress at a point in 2D

• Types of stress

Stress (

σ

)

N

o

rm

a

l

S

tr

e

s

(

σ

)

n

She

_{ar S}

tres

_{s (σ}

s

)

Normal stress (

σ

_N

)

(+) Compressive

(-) Tensile

Shear stress (

σ

_S

)

(+)

(-)

Stress-22 BS/03/02

STRESS on PLANE

• Coordinate System

Stress Ellipsoid

a) Triaxial stress

b) Principal planes of

the ellipsoid

(Modified from Means, 1976)

Arbitrary coordinate axes and planes C. General stress components

B. Principal stress components

X

Principal coordinate axes and planes

Z X1 σ1 Σ (lft) xx (lft) x σ (top) zz σ dx σ(bot) zz dz σ(top) zx σ(rt) xz Σ(bot) z σ(rt) x x σ(bot) zx (lft) xz σ Σ(rt) x X3 σ3 Σ(top) z A. Stress elipse Σz _σ1 σ3 _Σx

The State of

Two-Dimensional

Stress at Point

(Twiss and Moores, 1992)

Principal Stress:

σ

₁

> σ

₃

(7)

Stress-25 BS/03/02

B. Principal stress components

σ1 z x σ3 x1 x3 y y x2 x x y z σ2 x σzy σxy _σ yy σyz σyx σxx σzx σzz σxz z y Arbitrary coordinate planes A. Stress elipsoid

C. General stress components

z

Principal coordinate planes

The State of

3-Dimensional

Stress at Point

Principal Stress:

σ

₁

> σ

₂

> σ

₃

Stress Tensor Notation

σ

11

σ

12

σ

13

σ

=

σ

₂₁

σ

₂₂

σ

₂₃

σ

₃₁

σ

₃₂

σ

₃₃

σ

₁₂

=

σ

₂₁

,

σ

₁₃

=

σ

₃₁

,

σ

₂₃

=

σ

₃₂

(Twiss and Moores, 1992)

Stress-26 BS/03/02

Geologic Sign

Convention of

Stress Tensor

σ

n

r

n

(p)

σ

n (p)

σ

s

2

2 σ −σ

1 3

2 σ +σ

1 3

σ

n

(σ

n , (p)

σ

1 (p)

σ )

s

σ −σ

1 3

_cos

_2θ

2

σ −σ 2θ

1 3

_sin

σs

2θ

x

3 (p)

σ

s (p)

σ

n

σ

3

θ

σ

1

Plane P

x

σ

3

Mohr Diagram 2-D

A. Physical Diagram

A. Mohr Diagram

− α

x

3

n'

p

(p')

θ

p'

n

x

1

α

−2α

(σ

n , (p')

σ

s

σ

n 2α

σ )

s

σ

1

σ

n

σ

3 (p)

(σ

n , (p)

_{σ )}

s 2θ

A. Physical Diagram

B. Mohr Diagram

Mohr Diagram 2-D

(8)

Stress-29 BS/03/02

(σ σ )

xx' xz 2θ

σ

xx

(σ σ )

zz' zx

2 (σ + σ )

xx zz (σ −σ )xx z z

σ

s

σ

1

σ

n

2σ

xz (θ + 90º) α

σ

1

σ

3 σz z σz x

z

σ

3 θ

x

3

x

1

x

σ

xz 2 (θ + 90º) 2α

A. Physical Diagram

B. Mohr Diagram

Mohr Diagram 2-D

Stress-30 BS/03/02 n -Planes of maximum shear stress C lockwise shear stress

x

3

x

1

σ

s

σ

s Counterclockwise shear stress θ' = +45º

σ

1

x

3

σ

3

σ

1 n+

σ

s

x

1 θ = +45º

σ

1

σ

3 2θ = +90º

σ

n

σ

smax Clockwise 2θ = −90' º

σ

smax Counter clockwise

σ

3

B. Mohr Diagram

A. Physical Diagram

Planes of maximum shear stress

Mohr Diagram 2-D

Mohr Diagram 3-D

Geometry of a three-dimensional

Stress on a Mohr diagram

Mohr Diagram 3-D

Maximum Shear Stress

(9)

Stress-33 BS/03/02

Stress Ellipsoid

FUNDAMENTAL STRESS EQUATIONS

Principal Stress:

σ

₁

> σ

₂

> σ

₃

• All stress axes are mutually perpendicular

• Shear stress are zero in the direction of

principal stress

σ

₁

+

σ

₃

-

σ

₁

–

σ

₃

σ

_N

=

cos 2

θ

2

2 σ

_s

=

σ

1 –

σ

3 Sin 2

θ

2

Stress-34 BS/03/02

• Mohr diagram is a graphical representative of state of stress

• 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

(Davis and Reynolds, 1996)

0 0 0 0 0 0 a b c 0 0 0 0 0 a a b 0 0 0 0 0 a b b 0 0 0 0 0 a p 0 0 0 p 0 0 0 p 0 0 0 0 -a 0 0 0 0 F. Triaxial stress D. Axial or confined compression E. Axial extension or extensional stress σn p σs σn σs σ = σ = σ1 2 3 σ2 0 0 σ = σ2 3 σs σn σ1 σ = σ1 2 σs σn σ3 σ1 σ3 0 σs σn σ3 σ3 σ3 0 0 σs σn σ2 σ1 σ3 C. Uniaxial tension A. Hydros tatic stress B. Uniaxial compression

Image of Stress

0 0 0 0 -a 0 0 0 a σs Δ 3σ Δ 1σ σn σn σ1 σ3 σn σs σ3 0 Δ 3σ σ − σ3 n = 0 0 Δ 1σ σ − σ1 n 0 σs σ1 σn σ3 σ2 σ3 σ1 σ3 σ1 σn Dσ = σ − σ1 3 Dσ Dσ Dσ σs σ1 σn σ1 σ2 σ3 Eσ2 σ −2pf Eσ3 Eσ1 pf Eσ2 0 Eσ3 Eσ1 0 = 0 0 0 0 0 0 0 0 0 0 σ −3pf σ −1pf Applied

G. Pure shear stress H. Deviatoric stress (two-dimensional) I. Differential stress (Three examples) J. Effective stress Effective Applied Deviatoric