Limitation of The Concept of Stress in Structural Geology

(1)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

Institut Teknologi Bandung

Stress-1 BS/03/02

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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

=

ρ

gh

(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)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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

(4)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

Stress-13 BS/03/02

(Modified from Means, 1976)

Deformation of rock in various scale

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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

(6)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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

)

(+)

(-)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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) xx

σ(bot)

zx

(lft)

xz

σ

Σ(rt) x

X3

σ3

Σ(top) z

A. Stress elipse

Σz σ

1

σ3 _Σ

x

The State of

Two-Dimensional

Stress at Point

(Twiss and Moores, 1992)

Principal Stress:

σ

1

> σ

3

(7)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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 σyy yz 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)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

Stress-26 BS/03/02

Geologic Sign

Convention of

Stress Tensor

σ

n

r

n

(p)

σ

n (p)

σ

s

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

σ1

σ

n

σ3

(p)

(σ

n , (p)

σ )

s

2θ

A. Physical Diagram

B. Mohr Diagram

(8)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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

σ1

σ

n

2σ

xz

(θ + 90º)

α

σ1

σ3

σz z

σzx

z

σ3

θ

x

3

x

1

x

σ

xz

2 (θ + 90º) 2α

A. Physical Diagram

B. Mohr Diagram

Mohr Diagram 2-D

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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

σ

s

=

σ

1

–

σ

3

Sin 2

θ

2

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

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 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

(10)

Program Studi Teknik Geologi Fakultas Ilmu dan Teknologi Kebumian

Stress-37 BS/03/02

•

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

1

>

σ

2

>

σ

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