Question 3 of 7: Cold Work and Annealing — Hardness Response of Brass

Question 3: Cold Work and Annealing — Hardness Response of Brass (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Original thickness $t_0=0.250''$; nine rolling passes reduce thickness in $0.025''$ steps down to $0.050''$. Percent cold work $\%\text{CW}=100(t_0-t_f)/t_0$ (area reduction in rolling is proportional to thickness reduction for a fixed width). The sample rolled to $0.100''$ ($\%\text{CW}=100(0.250-0.100)/0.250=60\%$) is subsequently annealed at seven temperatures, $100^\circ$C to $700^\circ$C, $10$ min each.

Find. Sketch and explain (a) hardness vs. $\%$ cold work; (b) hardness vs. annealing temperature for the $60\%$-cold-worked stock.

Approach

Part (a) is governed by strain (work) hardening: dislocation density rises with plastic strain, and dislocations increasingly obstruct each other's motion, so the stress needed to continue deforming — and hence hardness — rises steeply at first and then levels off as the dislocation network saturates. Part (b) is governed by the three sequential annealing stages (recovery, recrystallization, grain growth), each with a distinct hardness signature.

1. (a) Hardness vs. cold work — shape of the curve. Hardness rises steeply over roughly the first $20$–$30\%$ CW, where dislocation density is still low and each additional dislocation created by rolling substantially increases the tangle/pile-up density, then the curve flattens toward a plateau at higher $\%$CW as the microstructure approaches a saturation dislocation density (further deformation mostly rearranges existing tangles into cells rather than adding proportionally as much new obstruction). This is the classic strain-hardening (work-hardening) curve, $\sigma\propto\varepsilon^n$-type behaviour reflected in hardness.
2. (b) Hardness vs. annealing temperature — three stages. Over $100^\circ$–$250^\circ$C (recovery), hardness stays nearly flat or drops only slightly — dislocations rearrange into lower-energy configurations (polygonization, climb/annihilation of opposite-sign dislocations) but the overall dislocation density, and hence hardness, changes little. Between roughly $250^\circ$ and $420^\circ$C (recrystallization), hardness falls sharply as strain-free new grains nucleate and consume the deformed microstructure by grain-boundary migration, eliminating most of the dislocation density essentially in one step. Above $\sim\!420^\circ$C (grain growth), hardness continues to decrease, but only gradually, as the now strain-free grains coarsen (larger average grain size, fewer grain boundaries) and the Hall–Petch strengthening contribution from grain boundaries correspondingly weakens.