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

grain diameter is

Explanation:

i) Lower yield point =230MPa when grain diameter is mm

ii) Lower yield point =275MPa when grain diameter is mm

The relation between yield stress and grain size is  

-----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  

-----2

And ii) represented as  

------3

Further simplifying since = 10

And = 12. 91

We have that eqn 2 and 3 becomes respectively

Solving the simultaneous eqns 4 and 5

Eqn 5 – eqn 4 gives

Therefore b = 15.46

Substituting the value of b in eqn 4 we have that

Therefore a becomes 230 -154.6 = 75.4

a=75.4

Eqn 1 now becones

-----6

Therefore for a yield point of 310, eqn 6 becomes

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