Question 2 of 8: Cold-Worked Brass — Hardness vs. Cold Work and Annealing
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exam 04-BS-11, Properties of Materials — May 2014. 3 hours,
closed-book examination (approved Casio or Sharp calculator only). Any five questions constitute
a complete paper; only the first five questions as they appear in the answer book are marked. All
eight questions are solved below for completeness.
Given. Annealed C26000 (70Cu-30Zn) brass strip, original thickness
$t_0=0.250$ in; nine samples rolled to gaps $0.250, 0.225,\dots,0.050$ in (0.025 in steps); the
0.100 in sample is re-annealed at seven temperatures, $100$–$700^\circ$C in $100^\circ$C
steps, each for 10 minutes, then water-quenched.
Find. Sketch and explain (a) hardness vs. % cold work, (b) hardness vs.
annealing temperature.
Fig. Q2 — (a) hardness rises steeply then saturates as % cold work
increases (strain hardening); (b) hardness stays roughly flat through recovery, drops sharply through
the recrystallization range, then declines slowly during grain growth.
Approach
Part (a) is a strain-hardening problem: hardness is tracked purely as a function of the amount
of plastic deformation (percent cold work), computed from the thickness reduction at each roll
gap. Part (b) is the classic three-stage annealing response (recovery, recrystallization, grain
growth) of the single heavily-cold-worked (60% CW) sample as a function of temperature at a fixed
10-minute hold.
(a) Percent cold work at each gap. Rolling reduces thickness at essentially
constant width, so $\%CW=\dfrac{t_0-t_f}{t_0}\times100$ with $t_0=0.250$ in. The nine gaps
(0.250 down to 0.050 in, 0.025 in steps) give $\%CW=0,10,20,30,40,50,60,70,80\%$ — a uniform
ladder of increasing deformation.
Nature of the hardness-vs-%CW curve. Hardness increases
monotonically with %CW because each pass multiplies the dislocation density; but the curve
rises steeply at low %CW and flattens (saturates) at high %CW, since the material
work-hardens fastest while dislocations are still relatively free to move, and progressively less
new hardening accrues per additional percent of cold work as the dislocation network becomes dense
and mutually tangled (dislocation forest hardening approaching a plateau). This is the same
strain-hardening curve shape seen in any cold-worked FCC metal (brass, copper, aluminum).
(b) The 0.100 in sample's cold work. Before annealing, that sample already
carries $\%CW=(0.250-0.100)/0.250\times100=60\%$ cold work — a heavily worked, high-hardness,
high-stored-energy starting condition, which is what drives the subsequent annealing response.
Nature of the hardness-vs-annealing-temperature curve. Three regimes appear
as the (fixed 10-minute) anneal temperature rises: (i) Recovery (roughly
$100$–$200^\circ$C) — dislocations rearrange into lower-energy configurations
(polygonization) but their overall density barely drops, so hardness stays nearly flat,
close to the as-cold-worked value. (ii) Recrystallization (roughly
$300$–$450^\circ$C for a heavily-worked 70/30 brass) — new, strain-free grains
nucleate and consume the deformed structure, so hardness drops sharply over a
narrow temperature band as the sample transitions from cold-worked to fully recrystallized.
(iii) Grain growth (above roughly $450$–$500^\circ$C) — the new
strain-free grains coarsen, which slightly further reduces hardness (more grain-boundary
area, more free surface for dislocation glide) but far more gradually than the recrystallization
drop.
