Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
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 )()(