Question

Hall–Petch equation relates yield strength (\(\sigma_y\)) to:

• A. Strain rate
• B. Grain size
• C. Stacking fault energy
• D. Poisson's ratio

Hint:

Smaller grain sizes result in higher yield strength, which is a key concept in materials science and solid mechanics. This is why materials with fine-grained structures are often stronger than those with larger grains.

Answer 1

The Correct Option is B

The Hall–Petch equation is a well-known relationship that connects the yield strength (\(\sigma_y\)) of materials to their grain size. According to this equation, the yield strength increases as the grain size decreases. This relationship highlights the strengthening effect caused by grain boundary obstacles to dislocation motion. The Hall–Petch equation is given by: \[ \sigma_y = \sigma_0 + k_y d^{-1/2} \] Where:
- \(\sigma_y\) is the yield strength,
- \(\sigma_0\) is the friction stress (a constant),
- \(k_y\) is the Hall-Petch constant, and
- \(d\) is the grain size.
- Strain rate: This is not directly related to the Hall–Petch equation. The equation focuses on the relationship between yield strength and grain size, not strain rate.
- Grain size: This is the correct answer. The Hall–Petch equation indicates that smaller grain sizes lead to higher yield strengths, due to the grain boundary strengthening mechanism.
- Stacking fault energy: This is not the focus of the Hall–Petch equation. Stacking fault energy relates to dislocation behavior in crystals, not directly to yield strength and grain size.
- Poisson's ratio: This is also not related to the Hall–Petch equation. Poisson's ratio is a measure of the material's ability to deform laterally under axial stress, not its yield strength dependence on grain size.
Thus, the Hall–Petch equation describes how yield strength depends on grain size.