When metals solidify from the molten state, atoms arrange into orderly crystals. The smallest group of atoms showing the characteristic lattice is a unit cell. Three common structures:

• Body-centered cubic (BCC)
• Face-centered cubic (FCC)
• Hexagonal close-packed (HCP)

Atoms are held by metallic bonding — attracting but repelling if too close; the stable structure minimizes total energy. Interatomic spacing ≈ $0.1$ nm.

Allotropism (polymorphism) — a metal can take different crystal structures at different temperatures; this underlies heat treatment. For steel:

• Quenching (rapid): Austenite (FCC) → Martensite (very hard, strong)
• Slow cooling: Austenite → Pearlite (softer, ductile)

Deformation of Single Crystals

• Elastic — recovers on unloading.
• Plastic (permanent) — does not recover.

Two mechanisms:

1. Slip — one atomic plane slides over an adjacent one under shear stress, along planes of maximum atomic density and closely packed directions. Direction-dependent behavior makes a single crystal anisotropic.
2. Twinning.

Slip Systems

A slip system = slip plane + slip direction. Metals with ≥ 5 slip systems are generally ductile; fewer are not.

Structure	Slip systems	Behavior
BCC	48	High required shear stress → good strength, moderate ductility
FCC	12	Low required shear → moderate strength, good ductility
HCP	3	Low slip probability → brittle at room T (more systems activate when hot)

Imperfections

Actual strength is ~1–2 orders of magnitude below theoretical, due to dislocations / defects.

• They lower mechanical & electrical properties (yield, fracture strength, conductivity).
• Physical & chemical properties (melting point, specific heat, thermal expansion, $E$, $G$) are not sensitive to defects.

Point defects: vacancy (missing atom), interstitial atom (extra), interstitial impurity atom (foreign).

A slip plane containing a dislocation needs less shear stress to slip than a perfect lattice — the main reason actual < theoretical strength.

Work (Strain) Hardening

Dislocations entangle and are blocked by grain boundaries, impurities, and inclusions, raising the shear stress for further slip — work / strain hardening.

Examples: rolling car-body sheet (Ch13), forging bolt heads (Ch14), drawing wire (Ch15).

Grains & Grain Boundaries

Crystals nucleate independently with random orientations and grow into grains.

• High nucleation rate → many grains → small grain size.
• Rapid cooling → smaller grains; slow cooling → larger grains.

Grain Size (ASTM)

ASTM number $n$ vs grains $N$ per in² at 100×:

$N = 2^{n-1}$

where $N$ is grains per $0.01text{ in} times 0.01text{ in}$. In grains/mm² (that area $= 0.0645text{ mm}^2$):

$N' = \frac{1}{0.0645}\cdot 2^{n-1} = 16\cdot 2^{n-1} = 2^{n+3}$

• $n = 5$–8 → fine grains; $n \ge 7$ acceptable for sheet (car bodies, appliances, utensils).
• Large grains → low strength/hardness/ductility and a rough orange-peel surface after stretching.

Recovery, Recrystallization & Grain Growth

Cold work → ↑strength, ↓ductility, anisotropy. Heating (annealing) reverses it.

• Pb, Sn, Cd, Zn recrystallize near room temperature → they don’t work-harden when cold-worked.
• More prior cold work → lower recrystallization temperature.
• Greater deformation → finer recrystallized grains (refines a coarse structure).
• Grain growth: further heating enlarges grains → orange peel on stretched sheet.

Cold-, Warm-, Hot-Working

With $T$ and $T_m$ in kelvin ($T(text{K}) = 273 + {}^{circ}text{C}$):

• $T/T_m < 0.3$ → cold-working
• $0.3 < T/T_m < 0.5$ → warm-working
• $T/T_m > 0.6$ → hot-working