Metal Forming: Rolling, Forging and Extrusion

Forming shapes metal by plastic deformation without removing material. GATE asks about rolling limits, forming forces and sheet-metal calculations.

Hot and cold working

• Hot working is done above the recrystallisation temperature (about 0.4–0.5 of the melting point in kelvin): low forces, no strain hardening, but poorer finish and accuracy.
• Cold working is done below it: strain hardening raises strength, good finish and accuracy, but higher forces.
• Flow stress of a strain-hardening metal: σ_f = K εⁿ (n = strain-hardening exponent). Average flow stress over a strain ε: K εⁿ/(n + 1).
• True strain ε = ln(L/L₀) = ln(A₀/A).

Rolling

• Draft Δh = h₀ − h_f.
• Maximum possible draft (bite condition): Δh_max = μ² R (R = roll radius).
• Roll force ≈ σ_f,avg × w × L, with contact length L = √(R Δh).
• Neutral point: where the strip moves at the roll speed. Before it the strip is slower, after it faster.

Forging

• Open-die: simple shapes; force for upsetting a cylinder F = σ_f × A × (1 + 2μr/(3h)) (friction raises it).
• Closed (impression)-die: flash forms around the parting line.

Extrusion and drawing

• Extrusion ratio R = A₀/A_f. Extrusion pressure ≈ σ_f ln R (ideal, no friction); real presses add friction and redundant work.
• Direct extrusion: billet moves with the ram, friction with the container. Indirect: die moves, lower friction.
• Wire drawing: the wire is pulled, so the drawing stress must stay below the flow stress of the drawn wire. Ideal limit: ln(A₀/A_f) = 1, a maximum reduction of 1 − 1/e ≈ 63% per pass.

Sheet metal

• Blanking / punching force: F = L t τ_s (perimeter × thickness × shear strength).
• Blanking: the die is the blank size; clearance is put on the punch. Punching (piercing): the punch is the hole size; clearance on the die.
• Deep drawing: limiting draw ratio D/d ≈ 2. Blank diameter for a cup of diameter d and height h (thin): D = √(d² + 4dh).
• Bending: spring-back after unloading; minimum bend radius depends on ductility.

Practice