Question

The Hall-Petch equation relates grain size to:

• A. Electrical conductivity
• B. Thermal expansion
• C. Yield strength
• D. Magnetic susceptibility

Hint:

Smaller grains = more barriers to dislocation motion = stronger material. This is what the Hall-Petch equation is all about!

Answer 1

The Correct Option is C

The Hall-Petch equation is a fundamental principle in materials science that explains how the strength of a metal is influenced by its grain size.
Hall-Petch Relationship:

• The equation is given by: \[ \sigma_y = \sigma_0 + k \cdot d^{-\frac{1}{2}} \]
• Where:
  • \( \sigma_y \): yield strength of the material,
  • \( \sigma_0 \): material constant (friction stress),
  • \( k \): strengthening coefficient,
  • \( d \): average grain diameter.
• This shows that as grain size \( d \) decreases, the yield strength \( \sigma_y \) increases. This is because smaller grains create more grain boundaries, which block the movement of dislocations (the main carriers of plastic deformation).

Why the Other Options Are Incorrect:

• (A) Electrical conductivity: Primarily affected by free electron flow and impurities, not directly by grain size in this context.
• (B) Thermal expansion: Depends on bonding and lattice vibrations, unrelated to grain size.
• (D) Magnetic susceptibility: Related to the magnetic domains and electron spin, not directly connected to grain size through Hall-Petch.

Thus, the Hall-Petch equation is specifically used to explain the increase in yield strength with decreasing grain size.