Question

Hall-Petch equation predicts the relation between grain size and

• A. Yield Strength
• B. Ductility
• C. %Elongation
• D. Tensile Strength

Hint:

Grain refinement is the only strengthening mechanism in metals that simultaneously increases both strength and toughness.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks to identify the mechanical property that is related to grain size via the Hall-Petch equation.

Step 2: Key Formula or Approach:
The Hall-Petch relationship is mathematically expressed as:
\[ \sigma_y = \sigma_0 + k_y d^{-1/2} \]
where:
$\sigma_y$ is the yield strength.
$\sigma_0$ is the friction stress (resistance of the lattice to dislocation movement).
$k_y$ is a material-specific strengthening coefficient.
$d$ is the average grain diameter.

Step 3: Detailed Explanation:

• Grain boundaries act as barriers to the motion of dislocations because the crystallographic orientation changes abruptly across the boundary.

• A smaller grain size ($d$) means there is a higher density of grain boundaries within the material.

• As dislocations pile up at these boundaries, higher applied stress is required for the dislocations to slip and pass through into the neighboring grain.

• Since plastic deformation begins when dislocations start to move globally, a smaller grain size increases the stress required to initiate yield.

• Therefore, as grain size decreases ($d \rightarrow 0$), the yield strength ($\sigma_y$) of the metal increases according to the $d^{-1/2}$ relation.

Step 4: Final Answer:
The Hall-Petch equation relates grain size to the Yield Strength of a material.