Strain

StrainLecture #11

STRAIN ANALYSIS

UNDEFORMED DEFORMED

Strain is defined as the change in size and shape of a body resulting from the action of an applied stress field

KINEMATIC ANALYSISKinematic analysis is the reconstruction of movements

cf

a

cde

ba

cf

A. Rigid Body Translation

ba

f

d

e b

a

B. Rigid Body Rotation

E. Nonrigid Deformation by Distortion

C. Original Object

c

e

b

e d

f c

d

ad

f

e

b

D. Nonrigid Deformation by Dilation

(Davis and Reynolds, 1996)

Eastic strain if the body of rock returns to its previous shape after the stress has been removed. A good example is the slow rebound of the North American crust after having been downwarped by the great weight of the Pleistocene glaciers.

Brittle strain occurs when the stress is great enough to break (fracture) the rock.

Plastic strain results in a permanent change in the shape of the rock. A ductile rock is one that “flows plastically” in response to stress. Whether the strain is plastic or brittle depends on both the magnitude of the stress and how quickly the stress is applied. A great stress that is slowly applied often folds rocks into tight, convoluted patterns without breaking them.

Type of Deformation

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 ( )

Quadratic elongation (l) = S2

l’ = 1/l = 1/S2

S2

S2

S3

S3

S3

S1

S1

S1

Strain Ellipsoid

S1 = Maximum Finite StretchS3 = Minimum Finite Stretch

(Davis and Reynolds, 1996)

Mohr Strain Diagram Ad

d = +15º

C

'3

Distorted Clay Cake

S1

1 Unit

A

S1

3.0

Minus

1.01.0

C

2 d

' '

' + ' 2

1 3

2.01.0

B

'3.0

.56

.49

0

1.01.0

C

2 = +30ºd

', / )

' 2.43 = '1 = .42 2.01.0

' 'COS

d

0

A

A''1

Equals

/

' 'SIN

d

'

(Davis and Reynolds, 1996)

HOMOGENOUS DEFORMATION

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)

D. Microscope scale

100 m

A. Regional scale

100 m

B. Outcrop scale

10 mm

C. Hand sample scale

D.

A.

perpendicu larto layer

perpendicu larto layer

perpendicu larto layer

E.

C.

F.

B.

^^

S1

S2S3

S1^

^

^ ^ ^

^

S1^

S1^

S2^

S2^

S2S3^

S3^

S3

S2 < 1 S2 = 1 S2 >1

STRAIN HISTORY

Structural development in competent layerbased on orientation of S1, S2 and S3

Scale Factor

Strain Measurement

• Geological Map • Geologic Cross-section• Seismic Section• Outcrop• Thin Section

Knowing the initial objects• Shape• Size • Orientation

Field of Compensation

Field of Expansion

1.0

Field ofNo Strain

Strating Sizeand Shape

Fieldof

LinearShortening

Field of Contraction

S1

1.0

S3

Field of Linier Strecthing

Strain Field Diagram

X

Z

Y

Z

XY

A

Z

YX

B

^1

b = S

S

2

3^

^a =

S

S

1

2^

K = 1

K = 0

ConstrictionalStrain

FlatteringStrain

Plane

Stra

in

Sim

ple

Ext

ensi

on

Simple Flattering

1

k =

Special Types of Homogenous Strain

A. Axial symmetric extension (X>Y=Z) or Prolate uniaxial

B. Axial symmetric shortening (X=Y>Z) or Oblate uniaxial

C. Plane strain (X>Y=1>Z) or Triaxial ellipsoid

Flinn Diagram

Strain Measurement from Outcrop

D= gap

D

D

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

2 3 4 61

C

Strain (in %)

Diff

eren

tial S

tres

s (in

MP

a)

ReptureStrength

400

5

100

200

300Yield

Strength

UltimateStrength

Yield StrengthAfter StrainHardening

D

A

EB

Stress – Strain Diagram

A. Onset plastic deformationB. Removal axial loadC. Permanently strained D. Plastic deformationE. Rupture

0 2 4 6 8 10 12 14 16

Diff

eren

tial S

tres

s, M

Pa

Strain, percent

300

200

100

70

20

Crown Point Limestone

40

140130

60

80

5 10 15

2000

1500

1000

0 Strain (in %)

800ºC

700ºC

500ºC

300ºC

500Diff

eren

tial S

tres

s (in

MP

a)

25ºC

Effects of Temperature and Differential Stress

(Modified from Park, 1989)

Deformation and Material

A. Elastic strainB. Viscous strainC. Viscoelastic strainD. ElastoviscousE. Plastic strain

Hooke’s Law: e = s/E, E = Modulus Young or elasticityNewtonian : s = h ,e =h viscosity, e = strain-rate

(Modified from Park, 1989)

Effect increasing stress to strain-rate

Stress Strain

Limitation of The Concept of Stress in Structural Geology

• No quantitative relationship between stress and permanent strain• Paleostress determination contain errors• No implication equation relating stress to strain rate that causes the deformation

Questions…