chap_03nnnnn

8/20/2019 chap_03nnnnn

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Stress and Strain Tensors – Stress

MCEN 5023/ASEN 5012

Chapter 3

Fall, 2006


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2

Stress Tensor

Traction

ΒuS

1P

u

2P

nP

2 X

1 X

3 X

v

Area: ΔS

Δ F

S S ΔΔ=

→Δ

FTv

0limTraction:

Traction is a vector, whose direction and magnitude

depend on how the surface S is obtained, or the

direction of S.


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Stress Tensor

Traction and Stress Tensor

323222121

2

eeeT

e

σ σ σ ++=

2 X

1 X

3 X

2e

T

22σ 21σ

23σ

Stress at a material point

2 X

1 X

3 X

Six faces passing a material point.

2 X

1 X

3 X

To better visualize, we move

these six faces to form a cube.


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Stress Tensor


2 X 1 X

3 X

22σ 21σ

23σ

12σ

11σ

13σ

32σ

31σ

33σ

323222121

2

eeeTe

σ σ σ ++=

313212111

1

eeeTe

σ σ σ ++=

333232131

3

eeeTe

σ σ σ ++=

jij

i

eTe

σ =

ijσ represents the stress state of a material point.


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Stress Tensor

Stress Tensor

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

⊗=

333231

232221

131211

σ σ σ

σ σ σ

σ σ σ

σ jiij eeσ

2 X

1 X

3 X

22σ

21σ

23σ

12σ

11σ

13σ

32σ 31σ

33σ

11σ 22σ 33σ Normal Stresses:

Shear Stresses:12σ 21σ 31σ 23σ 32σ 13σ

jiij σ σ =


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Stress Tensor

Stress Tensor: Sign Rules

Positive surface:

Negative surface:

The normal of a surface:

2 X

1 X

3 X

22σ

21σ

23σ

12σ

11σ

nn

22σ

21σ

23σ

32

σ 31σ

33σ

32σ

33σ

31σ

13σ

12σ 11σ

13σ


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S S Δ

Δ=

→Δ

FTv

0limTraction:

Stress Tensor


ΒuS

1P

2P

nP

2 X

1 X

3 X

v

Area: ΔS

Δ F

True Stress: if both force and area are measuredin deformed configuration.

Nominal Stress: if area is measured in

undeformed configuration.

Stress Tensor

L0

A0

L

A F F

True Stress: Nominal Stress: A

F

0 A

F

Undeformed Deformed

Small deformation:0 A A ≈


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Stress Tensor

Traction on an arbitrary surface

2 X

1 X

v

T

2n

1nv

θ

12σ

11σ

22

σ

21σ 2S Δ

1S Δ

vS Δ


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Stress Tensor


2 X

1 X

v

T

2n

1n v

θ

12σ

11σ

22σ

21σ 2S Δ

1S Δ

vS Δ


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Stress Tensor


In general 3D case:

iij j vT σ =

v

vσTv

= Cauchy’s Formula

Cauchy’s formula ensures that it is necessary and sufficient

to use a stress tensor, , to describe traction on any surface.


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Stress Tensor

Transformation of Coordinates

2 X

1 X

1e

T′

2n

1

n

12σ

11σ

22σ

21σ

1e′

2 X

1 X

2n

1n

12σ

11σ

22σ

21σ

1e′

2e′

1e

T′


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Stress Tensor


2 X

1 X

2n

1n

12σ

11σ

22σ

21σ

1e′

2e′

1e

T′


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Stress Tensor


2 X

1 X

2n

1n

12σ

11

σ

1e′

2e′

Previous slide

2 X ′

1 X ′

1

e′2e′

2 X

1 X

1e

2

e

22σ

21σ

22σ

21σ

12σ

11σ 12σ

11σ

12σ ′ 11σ ′

12σ ′11

σ ′

22σ ′

21σ ′

22σ ′

21σ ′

1e

T′

ijkjik σ β β 1

e1

T =′


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Stress Tensor


kl jlik ij σ β β σ =′

( )( )k iik k iik eeeeee ,cos ′=′•=•′= θ β

2 X ′

1 X ′

1e′

2e′

2 X

1 X 1e

2e


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Stress Tensor


kl jlik ij σ β β σ =′


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Stress Tensor

Transformation of Coordinates – 2D

2 X ′

1 X ′

1e′

2e′

2 X

1 X 1e

2e

θ


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Stress Tensor

Transformation of Coordinates – 2D


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Stress Tensor

Transformation of Coordinates – 2D Mohr’s Circle

Normal stress

Shear Stress

2

2211 σ σ +

2θ

2

2211 σ σ −=CD

B

D

C 2θ

2θ 12σ A

E

12σ = DE 11σ

22σ

2

2211 σ σ −

12σ

( )θ σ σ

2cos2

2211 −

( ) ( )θ σ θ

σ σ σ σ σ 2sin2cos

2212

2211221111 +

−+

+=′

( )θ σ θ

σ σ σ 2cos2sin

212

221112 +

−−=′

2

12

2

2211

2

σ σ σ

+⎟

⎠

⎞⎜

⎝

⎛ −=CE


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Stress Tensor


11σ

22σ

12σ

Normal stress

Shear Stress

22211

σ σ +

12σ

2θ

11σ ′

22σ ′

12σ ′

12σ ′

2 X ′1

X ′

2 X

1 X

θ


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Stress Tensor


Steps to construct and use Mohr’s circle

1. On the normal stress-shear stress plot, find

the following three points:

(σ11, -σ12)(σ22, σ12)( (σ11 + σ22)/2, 0)

2. Draw a circle based on these three points.

( (σ11 + σ22)/2, 0) as center, and (σ11, -σ12)and (σ22, σ12) as two points on the circle.

3. Rotate the line connecting points (σ11, -σ12)

and (σ22, σ

12) by 2θ.4. The newly obtained two points on the circleare

(σ’ 11, -σ’ 12) and (σ’ 22, σ12’ )

11σ 22σ

12σ

Normal

stress

Shear Stress

2

2211 σ σ +

12σ

2θ

11σ ′22σ ′

12σ ′

12σ ′

For any two points on the Mohr’s

circle, if the line connecting them

passes the center, they are stress

components under the same

coordinate system.


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Normal stress

Shear Stress

Stress Tensor

Observations from 2D Mohr’s Circle

2

2211 σ σ +

C

E

2

12

2

2211

2σ

σ σ +⎟

⎠

⎞⎜⎝

⎛ −=CE

1σ 2σ

maxτ

2

12

2

221122111

22σ

σ σ σ σ σ +⎟

⎠

⎞⎜⎝

⎛ −+⎟

⎠

⎞⎜⎝

⎛ +=

2

12

2

221122112

22σ

σ σ σ σ σ +⎟

⎠

⎞⎜⎝

⎛ −−⎟

⎠

⎞⎜⎝

⎛ +=

212

2

2211max

2σ σ σ τ +⎟

⎠ ⎞⎜

⎝ ⎛ −=


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Stress Tensor


Normal

stress

Shear Stress

1. The maximum normal stress and minimum

normal stress occur under the same

coordinate system.2. Under the coordinate system where the

maximum/minimum normal stresses are

reached, there is no shear stress.

Normal stress

Shear stress

1. The maximum shear stress is reached by

rotating the coordinate system by 45 degree

from the principle directions.

2. At maximum shear stress, the two normal

stresses are σ11=σ22=(σ1 +σ2)/2 3. Pure Shear, if σ11=σ22=0, or σ1 = -σ2.


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F

Stress Tensor


F

FF

Pure Shear


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Stress Tensor


F

F

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