The yield strength of mild steel is 150 MPa for an average grain diameter of 0.038 mm; yield strength is 250 MPa for average grain diameter 0.009 mm.

What is the yield strength for the same steel having an average grain diameter 0.004mm?

Hint: Assume Hall-Petch equation is valid.

Hall-Petch equation provides direct relations between the strength of the material and the grain size:

σ=σ0+k/√d, where d - grain size, σ - strength for the given gran size, σ0 and k are the equation constants.

As in this problem, we don't know the constants of the equation, but we know two properties of the material, we are able to find them from the system of equations:

σ1=σ0+k/√d1

σ2=σ0+k/√d2, where 1 and 2 represent 150MPa and 250MPa strength of the steel.

Note, that for the given problem, there is no need to convert units to SI, as constants can have any units, which are convenient for us.

From the system of equations calculations, we can find constant: σ0=55.196 MPa, k=18.48 MPa*mm^ (0.5)

Now we are able to calculate strength for the grain diameter of 0.004 mm:

σ=55.196+18.48 / (√0.004) = 347.39 MPa

The strength of the steel with the grais size of 0.004 mm is 347.39 MPa.