Answer:
grain diameter is [tex]d=4.34*10^-^3\\d =0.00434 mm[/tex]
Explanation:
i) Lower yield point =230MPa when grain diameter is [tex]1*10^-^2[/tex]mm
ii) Lower yield point =275MPa when grain diameter is [tex]6*10^-^3[/tex] mm
The relation between yield stress and grain size is
[tex]Yp = a + b/\sqrt{d}[/tex]-----1
This is the Hall–Petch equation
where Yp is the yield stress,
a is a materials constant
b is the strengthening coefficient (material specific)
and d is the average grain diameter
therefore (i) can be represented as
[tex]230 = a+b/\sqrt{0.01}[/tex] -----2
And ii) represented as
[tex]275 = a +b / \sqrt{0.006}[/tex] ------3
Further simplifying since [tex]1/\sqrt{0.01}[/tex] = 10
And [tex]1/ \sqrt{0.006}[/tex] = 12. 91
We have that eqn 2 and 3 becomes respectively
[tex]230 = a + 10b -----4\\275 = a + 12.91b - - - - - -5\\[/tex]
Solving the simultaneous eqns 4 and 5
Eqn 5 – eqn 4 gives
[tex]230 = a + 10b -----4\\-\\275 = a + 12.91b - - - - - -5\\[/tex]
[tex]45 =2.91b[/tex]
Therefore b = 15.46
Substituting the value of b in eqn 4 we have that
[tex]230 =a + (10*15.46) = a +154.6[/tex]
Therefore a becomes 230 -154.6 = 75.4
a=75.4
Eqn 1 now becones
[tex]Yp = 75.4 +15.46/\sqrt{d}[/tex] -----6
Therefore for a yield point of 310, eqn 6 becomes
[tex]310=75.4 + 15.46/\sqrt{d}\\234.6 = 15.46/\sqrt{d}\\ Cross multiplying,\\\sqrt{d} = 15.46/234.6\\d =(15.46/234.6)^2\\d=4.34*10^-^3\\d =0.00434 mm in 3 significant figures[/tex]
