Kinetics: Stress

Prof. Seong Jin ParkMechanical Engineering, POSTECH

CONTINUUM & FINITE ELEMENT METHOD

• Coordinate System• Covariant• Contravariant

• Body Force• Surface Force• Direction of Surface

covariant contravariantiv iv

Stress Analysis

0F

Må = 0

E

F

I I

H H

F

E

x

y

z

A A

CC

BB

DD

RAy

RCy

RAzRCz

RBy

RDy

RBzRDz

G1

G20

T0

w I ,TI

Ti

FN

G1

RAyRAz

B

RByRBz

C

D

RDz

RDy

FN

G2

RCyRCz

T0

Load Analysis: Free Body Diagram

L1

L2

L3L4

L1

L2

• Engineering Stress: Piola-Kirchhoff• True Stress: Cauchy Stress

• Normal Stress and Shear Stress

zzzyzx

yzyyyx

xzxyxx

i

j

ijA

F

Stress

jiij

ljklikij QQ *

ijkjikQQ

y

z

x

yxyz

xy

xzzx

zy

x

y

z

t xy

t xy

t xy

t xy

s x s x

s y

s y

x

y

x

y

t xy

t xy

s x

s y

dxdy

n

Three Dimensional Stress Tensor

( angular momentum conservation)∵

Traction (Stress) vector (Cauchy’s Law)

Action and Reaction Law

Normal Stress

Shear Stress

jiji nt

from Total Stress Tensor for given direction vector ni

Cauchy Stress Tensor (True Stress Tensor)

jiijn nn

inis nt

jiji nt

Stress Tensor

e1

xi = xi X j( )

nx

ds

t

TX

dS

N

0

¶W0

t = 0

t

e2

e3

x3 , x3

x1, x1

x2 , x2

time

time

(deformed)

(unreformed)

Cauchy Stress Tensor(True Stress)

Kirchhoff Stress Tensor

1st Piola-Kirchhoff Stress Tensor(Engineering Stress, not symmetric)

2nd Piola-Kirchhoff Stress Tensor

s ijn jds = PijN jdS

dSNJFdsn jjii

1

jii Xxx

ijFJ detji

j

iij x

X

xF ,

Nanson’s formula

Stress Tensor

t

J -1t J -1PFt FSFtJ -1

PFt FSFt

F–ttJs F–t P FS

SF-1PF-1 F–ttJF–1 F–t s

s

Js t

Principal Stresses & Principal Directions

jiij nn

0322

13 III

Traction Free

Pure Shear

12

2/1

3/1

3/2

3

1

2

2/1

Decomposition into Hydrostatic and Distortional Parts

Spherical Part (Pressure)of Stress Tensor

Volume Deformation

Dilatation

Total Stress Tensor Deviatoric Part ofStress Tensor

Shear Deformation

1

s 2 s avs 2 -s av

s av

s av

s1 -s av

s 3 -s avs 3

ij ijp ij = +

Octahedral Stresses

Normal Stress

Shear Stress (von Mises)

)(,ˆ 33 xn

)(,ˆ 22 xn

)(,ˆ 11 xn

)(n

octts oct

t octn

i

n

ioct nt)(

2n

n

i

n

ioct tt )()(