Answer:
The grain diameter should be 0.01477mm
Explanation:
At an average gram diameter of 5 x 10⁻² mm the lower yeidl point of an iron is 135 MPa
At a grain diameter of 8 x 10⁻³mm the yield point increases to 260 MPa
We are asked to determine the grain diameter for an iron which will give yield strength of 205 MPa.
The best way to solve this problem is to
(1) first establish two simultaneous expressions of σ(y) = σ(i) + k(y) x (d)^-1/2
(2) solve for σ(i) and k(y) , and
(3) Finally determine. value of d when σ(y) at 205 MPa
σ(y)=135 MPa, d = 5 x 10⁻² mm, d^{-1/2} =4.472,
σ(y) = σ(i) + k(y)d-1/2 can be written as 135 = σ(i) + 4.472k(y), equation 1
σ(y) = 260 MPa, d = 8 * 10-3 mm, d^{-1/2} = 11.8,
σ(y) = σ(i) + k(y)d-1/2 can be written as 260 = σ(i) + 11.18k(y), equation 2
From the above equation 1 and 2 we get,
6.708 x k(y) = 125
k(y) = 18.634 MPa. mm^1/2
We have,
σ(i) = 135 - (4.472 k(y))
σ(i) = 135 - (4.472 x 18.634)
σ(i) = 5.67 MPa
From the general equation
d = (k(y)/σ(y) – σ(i))²
At σ(y) = 205 MPa
d = (18.634/205 - 51.67)²
d = 0. 01477 mm
Therefore, the grain diameter is d = 0. 01477 mm.
