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. Let \(E_A, E_B\) be the Young's moduli and \(YS_A, YS_B\) the yield strengths of A and B. Which statement is \emph{correct?}

• A. \(E_A > E_B\) and \(YS_A > YS_B\)
• B. \(E_A = E_B\) and \(YS_A < YS_B\)
• C. \(E_A > E_B\) and \(YS_A = YS_B\)
• D. \(E_A = E_B\) and \(YS_A > YS_B\)

Hint:

Hall–Petch strengthens as grains get finer; elastic modulus is bonding-controlled and essentially grain-size independent for the same, randomly oriented metal.

Answer 1

The Correct Option is D

Step 1: Effect of grain size on yield strength (Hall-Petch).
For polycrystals, \[ \sigma_y = \sigma_0 + k\,d^{-1/2}, \] where \(d\) is the average grain size. Smaller \(d \Rightarrow\) larger \(d^{-1/2}\Rightarrow\) higher \(\sigma_y\). Here \(d_A=10\,\mu\text{m} < d_B=100\,\mu\text{m} \Rightarrow YS_A > YS_B\). Step 2: Effect of grain size on Young's modulus.
Elastic modulus \(E\) depends mainly on interatomic bonding and crystal structure. For the same material (Cu) with random texture, changing grain size does \emph{not} change \(E\) appreciably. \(\Rightarrow E_A \approx E_B\) (take \(=\) within experimental scatter). Step 3: Conclude.
Combining Steps 1-2 gives \[ \boxed{E_A = E_B \quad \text{and} \quad YS_A > YS_B,} \] which matches option (D).