Question

A steel bar exhibits a Hall-Petch relationship with a yield strength of 300 MPa when the average grain size is 10 micrometers. What trend in yield strength would you expect if the grain size is reduced to 5 micrometers?

• A. Decrease
• B. Increase
• C. Stay the same
• D. Initially increase, then decrease

Hint:

Hall-Petch Relationship. Yield Strength \(\sigma_y = \sigma_0 + k_y d^{-1/2\). Strength increases as grain size (d) decreases (grain boundary strengthening). Finer grains lead to higher strength.

Answer 1

The Correct Option is B

The Hall-Petch relationship describes how the yield strength (\(\sigma_y\)) of a polycrystalline material changes with its average grain size (\(d\)).
The relationship is typically expressed as: $$ \sigma_y = \sigma_0 + k_y d^{-1/2} $$ where \(\sigma_0\) is the friction stress (intrinsic strength) and \(k_y\) is the Hall-Petch coefficient (a material constant).
This equation shows that the yield strength (\(\sigma_y\)) is inversely proportional to the square root of the grain size (\(d^{-1/2}\)).
Therefore, as the grain size (\(d\)) decreases, the term \(d^{-1/2}\) increases, and consequently, the yield strength (\(\sigma_y\)) increases.
Reducing the grain size from 10 micrometers to 5 micrometers will lead to an increase in yield strength.
This phenomenon is known as grain boundary strengthening, as grain boundaries act as barriers to dislocation movement.