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MAT 33 – Fall 2007
Homework #8 – Solutions
Due:
11/05/07
1.)
The yield stress for an iron that has an average grain diameter of 5 x 10
2
mm is
135MPa. At a grain diameter of 8 x 10
3
mm, the yield stress increases to
260MPa. At what grain diameter will the yield stress be 200MPa?
The best way to solve this problem is to first establish two simultaneous expressions of the Hall
Petch equation, solve for
σ
o
and
k
y
, and finally determine the value of
d
when
σ
y
= 200
MPa.
The data pertaining to this problem may be tabulated as follows:
σ
y
d (mm)
d
1/2
(mm)
1/2
135 MPa
5 x 10
2
4.47
260 MPa
8 x 10
3
11.18
The two equations thus become
135 MPa =
σ
o
+ (4.47)k
y
260 MPa =
σ
o
+ (11.18)k
y
Which yield the values,
σ
o
= 51.7 MPa and
k
y
= 18.63 MPa(mm)
1/2
.
At a yield strength of
200 MPa
200 MPa =
51.7MPa + [18.63 MPa(mm)
1/2
] d
–1/2
or
d
1/2
= 7.96(mm)
1/2
, which gives
d = 1.57 x 10
2
mm
.
2).
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 Fall '07
 keith

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