Question

The lower yield point for an iron that has an average grain diameter of 1 × 10–2 mm is 230 MPa. At a grain diameter of 6 × 10–3 mm, the yield point increases to 275 MPa. At what grain diameter will the lower yield point be 310 MPa. Express the answer in mm to three significant figures

Answer 1

Answer:

grain diameter is $d=4.34*10^-^3\\d =0.00434 mm$

Explanation:

i) Lower yield point =230MPa when grain diameter is $1*10^-^2$mm

ii) Lower yield point =275MPa when grain diameter is $6*10^-^3$ mm

The relation between yield stress and grain size is  

$Yp = a + b/\sqrt{d}$-----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  

$230 = a+b/\sqrt{0.01}$ -----2

And ii) represented as  

$275 = a +b / \sqrt{0.006}$ ------3

Further simplifying since $1/\sqrt{0.01}$ = 10

And $1/ \sqrt{0.006}$ = 12. 91

We have that eqn 2 and 3 becomes respectively

$230 = a + 10b -----4\\275 = a + 12.91b - - - - - -5$

Solving the simultaneous eqns 4 and 5

Eqn 5 – eqn 4 gives

$230 = a + 10b -----4\\-\\275 = a + 12.91b - - - - - -5$

$45 =2.91b$

Therefore b = 15.46

Substituting the value of b in eqn 4 we have that

$230 =a + (10*15.46) = a +154.6$

Therefore a becomes 230 -154.6 = 75.4

a=75.4

Eqn 1 now becones

$Yp = 75.4 +15.46/\sqrt{d}$ -----6

Therefore for a yield point of 310, eqn 6 becomes

$310=75.4 + 15.46/\sqrt{d}\\234.6 = 15.46/\sqrt{d}\\ Cross multiplying,\\\sqrt{d} = 15.46/234.6\\d =(15.46/234.6)^2\\d=4.34*10^-^3\\d =0.00434 mm in 3 significant figures$