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Example No. 01: Design of a Mild Steel
The yield strength of mild steel with an average grain size of
0.05 rom is 20,000 psi. The yield stress of the same steel with a grain size of 0.007 mm is 40,000 psi. What will be the average grain size of the same steel with a yield stress of 30,000 psi?
Assume the Hall-Petch equation is valid and that changes in the observed yield stress are due to changes in grain size.
Solution:
σ y
=
σ
0
+ Kd -1/2
Where σ y
is the yield strength, d is the average dia of the grains, and σ
0 and K are the constants for the metal.
Thus, for a grain size of 0.05 mm the yield stress is
20 x 6.895 MPa = 137.9 MPa.
(Note: 1,000 psi = 6.895 MPa). Using the Hall-Petch equation
137.9 =
σ
0
+ K / √0.05
For the grain size of 0.007 mm, the yield stress is 40 x 6.895 MPa =
275.8 MPa. Therefore, again using the Hall-Petch equation:
275.8 =
σ
0
+ K / √0.007
Solving these two equations K = 18.43 MPa_mm
σ y
= 55.5 + Kd
I
/
2
, and σ
0
=
55.5 MPa. Now we have the Hall-Petch equation as
-1/2
If we want a yield stress of 30,000 psi or 30 x 6.895 = 206.9 MPa, the grain size will be 0.0148 mm or 14.8 µ-m.
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The Effect of grain size on the yield strength of steel at room temperature
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