Question

The Hall–Petch relation correlates yield stress with

• A. Temperature
• B. Strain rate
• C. Grain size
• D. Dislocation density

Hint:

To remember the Hall–Petch behavior, connect it with grain boundary strengthening: Smaller grains mean more boundaries, more boundaries lead to more dislocation pile-ups, and more pile-ups require a higher yield stress to cause deformation.

Answer 1

The Correct Option is C

Concept: The mechanical strength of polycrystalline materials depends significantly on the dimensions of their individual microstructural grains. The Hall–Petch equation provides a mathematical formulation describing how the yield stress of a metal scales inversely with the square root of its average grain diameter. The mathematical model is expressed as: \[ \sigma_y = \sigma_0 + \frac{k_y}{\sqrt{d}} \] Where:
• \(\sigma_y\) represents the material's yield strength or yield stress.
• \(\sigma_0\) is a materials-specific constant known as the friction stress or resistance of the lattice to dislocation motion.
• \(k_y\) is the strengthening coefficient, a parameter unique to each material that quantifies the grain boundary's effectiveness in impeding dislocation movement.
• \(d\) represents the average grain diameter or grain size of the material.

Step 1: Analyzing the relationship with each microstructural property.
Let us analyze how grain boundaries alter mechanical performance:
• Grain Boundaries as Obstacles: Microscopic grains possess different crystallographic orientations. When a dislocation moves along a slip plane within a single grain and encounters a grain boundary, it must alter its path of propagation due to the mismatch in crystal lattices.
• Dislocation Pile-up: As stress is applied, multiple dislocations accumulate at the boundary, forming a pile-up. This accumulation creates a localized stress concentration that can eventually activate dislocation sources in the neighboring grain.
• Effect of Grain Size: In materials with a smaller grain size (\(d\)), there is a larger surface area of grain boundaries per unit volume. The distance a dislocation travels before hitting an obstacle is small, reducing the number of dislocations that can pile up in a single queue. Since the stress concentration at the front of a shorter pile-up is lower, a larger externally applied macroscopic stress (\(\sigma_y\)) is necessary to force plastic deformation across the boundaries. Therefore, refining the grain size directly increases the yield stress of the metal.

Step 2: Verification of given choices.

• Temperature: Yield stress generally decreases with an increase in temperature due to thermally activated dislocation movement, but this is not defined by the Hall–Petch relation.
• Strain rate: Higher strain rates increase yield strength via dynamic effects, typically modeled by power laws or thermal activation models rather than Hall–Petch.
• Grain size: As established by the relation \(\sigma_y \propto d^{-1/2}\), grain size is the primary independent physical variable correlated here.
• Dislocation density: The relation between yield stress and dislocation density is described by the Taylor equation (\(\sigma_y \propto \sqrt{\rho}\)), not the Hall–Petch relation. Thus, the correct choice is Option (C).