What is the effect of decreasing grain size on yield strength according to Hall-Petch?

• A. Decrease yield strength.
• B. No effect.
• C. Increase yield strength.
• D. Always increases then decreases.

Answer: (C)

The Hall-Petch relationship shows that yield strength increases as grain size decreases because grain boundaries impede dislocation motion. When grains are smaller, there are more boundaries per unit volume to block dislocations, so more stress is needed to move them and reach yielding.

Mathematically, the idea is captured by sigma_y = sigma_0 + k d^(-1/2), where d is the average grain diameter. As d gets smaller, d^(-1/2) gets larger, raising the yield stress. So, decreasing grain size strengthens the material.

This describes the common behavior in metals within typical grain-size ranges. (Note: at extremely small sizes, other effects can appear, but the standard Hall-Petch trend is an increase in yield strength with smaller grains.)