ANS:B - d^-1/2

The yield strength of a polycrystalline metal with an average grain size, d, is inversely proportional to the square root of the grain size, represented as d^−1/2. Explanation:

• In polycrystalline materials, the presence of grain boundaries affects mechanical properties such as yield strength.
• As the grain size decreases, the number of grain boundaries increases. These grain boundaries act as barriers to dislocation movement, which strengthens the material.
• The Hall-Petch equation describes the relationship between the yield strength (σy​) and the average grain size (d) in polycrystalline materials:

• σ0​ is the initial yield strength of the material (i.e., when the grain size approaches infinity).
• k is the Hall-Petch constant.
• d is the average grain size.
• From the Hall-Petch equation, it's clear that the yield strength is inversely proportional to the square root of the grain size, represented by the term d−1/2.