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

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

Step-by-step 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/√(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/√(0.01)` -----2

And ii) represented as

`275 = a +b / √(0.006)` ------3

Further simplifying since
`1/√(0.01)` = 10

And
`1/ √(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/√(d)` -----6

Therefore for a yield point of 310, eqn 6 becomes

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