Sheet-metal parts are lightweight and versatile. Low-carbon steel is most common (cheap, good strength/formability).

Shearing

A blank is cut from a sheet by shear stress; cracks start at top and bottom edges; the sheared edge has shiny burnished + rough fracture zones. Parameters: punch speed, lubrication, clearance $c$.

• ↑Clearance → rougher edge, larger deformation zone, taller burr.
• ↑Punch speed → smoother surface.
• Burnished/rough ratio ↑ with ductility, ↓ with thickness and clearance.

Punch force:

$F = 0.7\,T\,L\,(\text{UTS})$

$T$ = sheet thickness, $L$ = total sheared length, UTS = ultimate tensile strength.

Operations: punching (slug discarded), blanking (slug is the part); die cutting = perforating, parting, notching, lancing; fine blanking (smooth square edges); slitting (circular blades). Nesting minimizes scrap. Clearances ~2–8% of thickness; smaller clearance → better edge. Compound / progressive / transfer dies combine operations.

Sheet Characteristics

• Elongation: uniform elongation ↑ with $n$ (good formability).
• Yield-point elongation (low-C steel) → Lueder’s bands; avoid by temper-rolling 0.5–1.5%.
• Grain size affects properties and surface appearance.
• Anisotropy (from processing) causes wavy/eared edges.

Formability Tests

• Cupping test: push a ball/punch into clamped sheet until cracking; greater punch depth = more formable (but it’s axisymmetric, unlike real forming).
• Forming-limit diagram (FLD): mark a grid of circles (2.5–5 mm); after stretching they become ellipses — major and minor engineering strains plot safe vs failure zones. Thicker sheet → higher (more formable) curve. Major strain can’t be negative.

Bending

Outer fibers in tension, inner in compression. As $R/T$ ↓, outer-fiber strain ↑ until cracking.

Springback (elastic recovery after bending):

$\frac{R_i}{R_f} = 4\left(\frac{R_i Y}{E T}\right)^3 - 3\left(\frac{R_i Y}{E T}\right) + 1$

$Y$ = yield stress. Higher $Y$ or larger $R$ → more springback; thicker sheet → less.

Bending force:

$P = \frac{k\,Y\,L\,T^2}{W}$

$L$ = bend length, $T$ = thickness, $W$ = die opening; $k \approx 0.3$ (wiping), $0.7$ (U-die), $1$–$3$ (V-die).

Deep Drawing

A blank is held by a blankholder while a punch forms a cup. Variables: sheet properties, $D_0/D_p$, clearance, punch & die radii, blankholder force, friction, lubrication.

• Failure = wall thinning under tension; wavy edges = earing (planar anisotropy).
• Blankholder force too high → wall tears; too low → wrinkling. Drawbeads control flow.
• Avoid tearing: large die radii, good lubrication, drawbeads, proper blank size, 45° corner cut-offs. Ironing makes wall thickness uniform.

Other Processes

• Rubber forming: one die is a flexible polyurethane membrane.
• Spinning: form axisymmetric parts over a rotating mandrel.
• Stretch forming: clamp edges, stretch over a form block (aircraft wings, door panels).
• Tube bending: pack with sand, or bulge with a rubber plug in a split die.

Worked Examples

Strains. A 7 mm circle → ellipse 13 × 4.5 mm: $varepsilon_{text{maj}} = ln(13/7) = 0.619$, $varepsilon_{text{min}} = ln(4.5/7) = -0.44$. Volume is conserved:

$\varepsilon_{\text{maj}} + \varepsilon_{\text{min}} + \varepsilon_{\text{thick}} = 0 \;\Rightarrow\; \varepsilon_{\text{thick}} = -0.177$

so a 1 mm sheet thins to $e^{-0.177} = 0.84$ mm.

Punch force. $L = 0.4$ m, $T = 0.001$ m, UTS = 190 MPa (5052-O Al):

$F = 0.7(0.001)(0.4)(190\times 10^6) = 53.2\ \text{kN}$