AE 321
–
Homework 8 solution
Chapter 7:
Failure and Fatigue
1a.
We start with the following trial solution.
0
0
0
0
0
0
0
P
P
Note that the entries in the upper left of the stress tensor, i.e.
21
12
22
11
,
,
,
arise from the inplane hydrostatic case, and that all other entries are zero because
the cylinder is tractionfree in the
x
3
direction.
It can be easily shown that equilibrium is satisfied since
P
is constant.
Using the stressstrain relation,
E
P
E
P
P
E
P
P
2
0
0
0
0
0
0
Compatibility is also satisfied because all entries in the strain tensor are constant.
Boundary conditions:
end faces
T
n
1
0
0
satisfied
0
0
0
33
23
13
Boundary conditions:
lateral surface
T
n
0
sin
cos
sin
cos
sin
sin
cos
cos
22
12
2
12
11
1
P
T
P
T
Since the boundary conditions are satisfied, the trial solution is the
solution.
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1b.
The stress state from part (a) is already the principal stress.
Therefore,
P
3
2
1
0
Using the Von Mises criterion,
y
y
oct
P
P
P
P
3
2
3
2
0
3
1
3
1
2
2
2
1
3
2
3
2
2
2
1
max
1c.
When the pressure
P
were also applied on the end faces, then the stress state
becomes
P
P
P
0
0
0
0
0
0
Nothing that
P
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 Spring '08
 Dutton,J
 Materials Science, Force, Yield surface, Boundary conditions, MZ

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