Question

Two randomly oriented polycrystalline copper samples with average grain sizes of 10 $\mu$m (Sample A) and 100 $\mu$m (Sample B) were tested at room temperature. 
Given: $E_A$ = Young's modulus of Sample A $E_B$ = Young's modulus of Sample B $Y_{SA}$ = Yield strength of Sample A $Y_{SB}$ = Yield strength of Sample B 
Which one of the following statements is CORRECT?

• A. $E_A>E_B$ and $Y_{SA}>Y_{SB}$
• B. $E_A = E_B$ and $Y_{SA}<Y_{SB}$
• C. $E_A>E_B$ and $Y_{SA} = Y_{SB}$
• D. $E_A = E_B$ and $Y_{SA}>Y_{SB}$

Hint:

According to the Hall-Petch relationship, materials with smaller grain sizes typically exhibit higher yield strength due to increased resistance to dislocation motion.

Answer 1

The Correct Option is D

In polycrystalline materials like copper, the grain size plays a significant role in determining both the Young's modulus and yield strength. The key points to consider are:
Step 1: Effect of grain size on Young's modulus - Young's modulus is generally not significantly affected by the grain size for a material like copper, especially in the typical range of grain sizes for polycrystalline materials. Hence, we expect $E_A = E_B$.
Step 2: Effect of grain size on yield strength - Yield strength typically increases with decreasing grain size, due to the Hall-Petch relationship. Smaller grains impede the movement of dislocations, making the material stronger. Therefore, for copper with a smaller grain size, Sample A (10 $\mu$m) is expected to have a higher yield strength than Sample B (100 $\mu$m).
Step 3: Conclusion Thus, the correct answer is Option (D), which states that $E_A = E_B$ and $Y_{SA}>Y_{SB}$. This is consistent with the general behavior of materials with varying grain sizes.