Metal Forming and Machining: Principles, Mechanics, and Processes

Fundamentals of Metal Forming and Plasticity

Plastic deformation allows the shaping of metallic workpieces through permanent displacement without volume change ($\Delta V = 0$, $\nu = 0.5$). Under uniaxial tension, engineering stress $s = \frac{P}{A_0}$ and engineering strain $e = \frac{l - l_0}{l_0}$ relate to true stress $\sigma = s(1+e)$ and true strain $\varepsilon = \ln(1+e)$. In the plastic regime, the material flow curve follows Hollomon's power law $\sigma = K\varepsilon^n$, where $K$ is the strength coefficient and $n$ is the strain-hardening exponent. Diffuse necking instability initiates when the work-hardening rate matches true stress, $\frac{d\sigma}{d\varepsilon} = \sigma$, dictating that the uniform true strain at necking equals the strain-hardening exponent ($\varepsilon_u = n$). Classical plasticity theory relies on five core assumptions: strain-rate independent response, plastic incompressibility, absence of the Bauschinger effect, yield stress independence from hydrostatic pressure, and material isotropy.

Any three-dimensional stress state can be decomposed into a hydrostatic stress $\sigma_m = \frac{\sigma_{11} + \sigma_{22} + \sigma_{33}}{3}$ and a deviatoric stress tensor $s_{ij} = \sigma_{ij} - \sigma_m \delta_{ij}$. Yielding is driven strictly by deviatoric stress components. The stress invariants of the deviatoric stress tensor include $J_1 = 0$ and the second deviatoric invariant $J_2 = \frac{1}{6}[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]$. According to the Tresca (Maximum Shear Stress) criterion, yielding occurs when the maximum shear stress equals the shear yield stress in uniaxial tension: $\tau_{\max} = \frac{\sigma_1 - \sigma_3}{2} = \frac{S_{yt}}{2}$. The von Mises (Distortion Energy) criterion states that yielding begins when the second invariant reaches a critical value, expressed as $\frac{1}{\sqrt{2}}[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]^{1/2} = S_{yt}$, predicting a pure shear yield strength of $S_{sy} = \frac{S_{yt}}{\sqrt{3}} \approx 0.577\,S_{yt}$. Plastic flow increments are governed by the Levy-Mises flow equations, which state that strain increments are proportional to deviatoric stress components: $\frac{d\varepsilon_1}{s_1'} = \frac{d\varepsilon_2}{s_2'} = \frac{d\varepsilon_3}{s_3'} = d\lambda$.

Bulk Forming Processes

Bulk deformation processes alter cross-sectional geometries through large plastic strains while maintaining a low surface-area-to-volume ratio. Working temperature strongly dictates deformation mechanics: cold working occurs below the recrystallization temperature ($T_{\text{rec}} \approx 0.3\,T_m$ for pure metals, $0.5\,T_m$ for alloys), producing high dimensional accuracy, superior surface finish, and strain hardening at the expense of higher tool loads. Warm forming operates between $550^\circ\text{C}$ and $950^\circ\text{C}$ to balance ductility and forming force. Hot working occurs above $T_{\text{rec}}$, eliminating porosity and refining grain structures through dynamic recrystallization, though surface oxidation causes scale formation. Isothermal forming maintains dies at the exact workpiece temperature to eliminate surface chilling and ensure uniform metal flow in heat-sensitive superalloys and titanium.

Forging shapes discrete parts through compressive forces using open dies, impression dies, or closed flashless dies. In open-die upsetting, friction along the die-workpiece interface and thermal chilling produce outward barreling. The forging force for cylindrical upsetting is calculated as $F = Y_f \pi r^2 \left(1 + \frac{0.4 \mu D}{h}\right)$, where $Y_f$ is flow stress, $\mu$ is the coefficient of friction, $D$ is diameter, and $h$ is height. Impression-die forging uses a restricted flash gutter; the rapidly cooling flash creates high hydraulic pressure, forcing the metal to fill intricate die cavities. Precision or flashless forging eliminates flash entirely through strict billet volume control. Specialized forging techniques include orbital forging, roll forging for tapered profiles, and skew rolling for ball bearings. Common forging defects include cold shuts from sharp corners or excessive chilling, surface cracks from hot shortness or low temperature, flash cracking, and incomplete penetration.

Rolling reduces slab, bloom, or billet thickness using counter-rotating cylindrical rolls. Maximum draft in a single pass is governed by roll radius $R$ and friction: $(\Delta h)_{\max} = \mu^2 R$. Within the contact arc, the strip accelerates from entrance velocity $v_0$ to exit velocity $v_f$; at the neutral plane, strip velocity matches roll peripheral velocity, causing frictional shear forces to reverse direction. Total rolling force is $F = \bar{Y}_f w L$, where $w$ is width and contact length is $L = \sqrt{R \Delta h}$. Rolling mill arrangements include two-high (reversing or continuous), three-high, four-high (with large backup rolls providing rigidity), and Sendzimir cluster mills for ultra-thin foil. Rolling defects include wavy edges from roll deflection, center zipper cracks, edge cracks from low ductility, and alligatoring due to non-uniform tensile stresses.

Extrusion forces a billet through a shaped die opening under high compressive pressure. The extrusion ratio is $r_x = \frac{A_0}{A_f}$. Under Johnson's empirical relationship, extrusion strain is $\varepsilon_x = a + b \ln(r_x)$, yielding a direct extrusion ram pressure of $p = \bar{Y}_f \left(\varepsilon_x + \frac{2L}{D_0}\right)$. Direct extrusion involves high container friction and leaves butt-end scrap; indirect extrusion mounts the die on a hollow ram, eliminating container friction and decreasing ram force by $25\% - 30\%$. Hydrostatic extrusion surrounds the billet with pressurized fluid to eliminate container friction and suppress brittle fracture. Extrusion defects comprise center burst (chevron cracking from axial tensile stresses), piping (surface skin drawn into the billet center), and surface cracking from excessive extrusion speeds.

Wire, bar, and tube drawing pull material through a converging die to reduce cross-sectional area. The draw die features four sequential zones: entry, a conical approach angle ($6^\circ - 20^\circ$), a bearing surface (land) that sizes the product, and back relief ($\approx 30^\circ$). According to Schey's equation, drawing stress is $\sigma_d = \bar{Y}_f \left(1 + \frac{\mu}{\tan(\alpha)}\right) \phi \ln\left(\frac{A_0}{A_f}\right)$, where $\phi = 0.88 + 0.12 \frac{D}{L_c}$ accounts for redundant work. Continuous wire drawing utilizes multi-die capstan benches with intermediate annealing steps. Tube drawing controls inner diameter and wall thickness using fixed mandrels, floating plugs, or mandrel-free tube sinking.

Sheet Metal Forming and Shearing Mechanics

Sheet metalworking encompasses cutting and shaping operations performed on sheets ranging from $0.4\,\text{mm}$ to $6\,\text{mm}$ in thickness. Key material properties include total elongation, planar anisotropy $r$, strain-hardening exponent $n$, and yield point elongation. Yield point elongation creates visible surface defects called Lüders bands or stretcher strains, which are eliminated prior to forming via temper or skin-pass rolling. Shearing severs metal via localized shear stresses between punch and die edges.

In blanking, the sheared slug is the desired part, while in punching (piercing), the slug is scrap. Clearance $c$ between punch and die typically ranges between $2\% - 10\%$ of sheet thickness $t$. For blanking, die diameter equals blank diameter $D$, and punch diameter is $D - 2c$; for punching, punch diameter equals hole diameter $H$, and die diameter is $H + 2c$. Maximum cutting force is given by $F = L t S_s \approx 0.7 t L (\text{UTS})$, where $L$ is the perimeter of the cut and $S_s$ is shear strength. Sheared part edges display four distinct regions: rollover, burnish (smooth shear zone), rough fracture zone, and burr. Applying a shear angle to the punch or die reduces instantaneous cutting force by extending deformation over time. Fine blanking uses a $V$-shaped impingement ring, triple-action counterforce, and minimal clearance ($\approx 0.5\% t$) to achieve fully sheared, fracture-free edges.

Deep drawing transforms flat circular sheet blanks into cylindrical or contoured hollow cups. As the punch descends, the sheet experiences bending over the die radius, straightening into the wall, and severe circumferential compressive stress in the flange. A blank holding force $F_h \approx \frac{1}{3} F$ prevents flange wrinkling. Maximum drawing force is given by $F = \pi D_p t (\text{TS}) \left(\frac{D_b}{D_p} - 0.7\right)$. The limiting draw ratio (LDR) is defined as $\frac{D_b}{D_p} \approx 2.0$. Redrawing accomplishes deeper draws across multiple steps ($40\% - 45\%$ reduction in the first draw, $30\%$ in the second, $16\%$ in the third). Ironing intentionally squeezes the drawn cup wall through restricted clearances to enforce uniform wall thickness. Draw defects include flange wrinkling, wall tearing at the punch radius, and earing resulting from planar anisotropy.

Sheet stretchability and failure limits are quantified through Forming Limit Diagrams (FLD). Using circular grid analysis, major strain $\varepsilon_1$ and minor strain $\varepsilon_2$ are tracked across deformation paths ranging from uniaxial tension to plane strain ($\varepsilon_2 = 0$) and equi-biaxial tension. The lowest limit strain occurs under plane strain conditions. High Energy Rate Forming (HERF) processes—including explosive forming, electrohydraulic forming, and electromagnetic pulse forming—deliver rapid energy surges over microseconds, utilizing high deformation velocities to form large or high-strength components with minimal tooling.

Traditional and Abrasive Machining Operations

Machining is a subtractive manufacturing process that generates precise surfaces by shearing away unwanted material as chips. Conventional machine tools rely on coordinated primary cutting speed and feed motions. The engine lathe rotates the cylindrical workpiece while single-point tools perform turning, facing, boring, reaming, threading, knurling, and parting. Shaping machines use a reciprocating horizontal ram driven by a quick-return mechanism (cutting stroke angle $\approx 220^\circ$, return stroke $\approx 140^\circ$), where a clapper box lifts the tool to prevent surface scuffing during the idle return stroke. Planers move large workpieces beneath stationary tools. Drilling operations utilize twist drills characterized by two flutes, a chisel edge, and standard point angles of $118^\circ$ with $30^\circ$ helix angles ($20^\circ$ slow helix for brass, $40^\circ$ quick helix for aluminum).

Milling uses multi-tooth rotating cutters to generate flat or contoured surfaces. In peripheral (slab) milling, the cutting edges lie on the cylinder periphery; in face milling, cutting occurs at both the face and periphery; in end milling, cutters machine slots, pockets, and profiles. Up-milling (conventional) starts chip formation at zero thickness and increases to a maximum, pulling the workpiece upward and causing tool chatter. Down-milling (climb) initiates contact at maximum chip thickness, forcing the workpiece firmly downward into the fixture, yielding superior tool life and surface finish provided table backlash is eliminated. Circular divisions are machined using dividing heads via direct indexing, simple indexing ($n_{cr} = \frac{40}{N_{\text{div}}}$), or compound indexing.

Abrasive grinding utilizes bonded wheels comprising hard abrasive grains held within a vitrified (glass/ceramic), resinoid (phenolic resin), rubber, or metallic matrix. Conventional abrasives include aluminum oxide ($\text{Al}_2\text{O}_3$) and silicon carbide ($\text{SiC}$); superabrasives include Cubic Boron Nitride (CBN, Knoop hardness $4000 - 5000$) and diamond (Knoop hardness $7000 - 8000$). Grains have random geometries with large negative rake angles ($\approx -60^\circ$), removing material through three distinct stages: rubbing (elastic contact), plowing (plastic displacement), and chip cutting. High friction produces sparks and extreme local flash temperatures (up to $1600^\circ\text{C}$), which can induce surface burning, tempering, and heat-checking cracks. Grinding wheel wear occurs via attritious wear (flank wear flats), grain fracture (friability/self-sharpening), and bond fracture. When wheels suffer from glazing (dull flattened grains) or loading (pores clogged with chips), wheel sharpness is restored through dressing using single-point diamond dressers inclined at a $10^\circ - 15^\circ$ drag angle, star-disc dressers, or abrasive sticks.

Mechanics of Metal Cutting

In orthogonal metal cutting, the cutting edge of the tool is oriented perpendicular to the direction of cutting velocity, constraining deformation to two dimensions. The primary shear zone extends from the tool tip to the free surface at a shear plane angle $\phi$. Uncut chip thickness $t$ and cut chip thickness $t_c$ define the cutting ratio $r = \frac{t}{t_c}$. The shear plane angle is calculated as $\tan(\phi) = \frac{r \cos(\alpha)}{1 - r \sin(\alpha)}$, where $\alpha$ is the tool rake angle. Shear strain during chip formation is given by $\gamma = \tan(\phi - \alpha) + \cot(\phi)$.

Cutting forces are resolved analytically using Merchant's composite force circle. Dynamometers experimentally measure the horizontal cutting force $F_P$ (or $F_t$) and vertical thrust/feed force $F_Q$ (or $F_f$). The resultant force $R$ balances forces along the tool face and shear plane:

$F_C = F_P \sin(\alpha) + F_Q \cos(\alpha)$

$N_C = F_P \cos(\alpha) - F_Q \sin(\alpha)$

$F_S = F_P \cos(\phi) - F_Q \sin(\phi)$

$N_S = F_Q \cos(\phi) + F_P \sin(\phi)$

The coefficient of friction along the tool rake face is $\mu = \tan(\beta) = \frac{F_C}{N_C} = \frac{F_Q + F_P \tan(\alpha)}{F_P - F_Q \tan(\alpha)}$, where $\beta$ is the friction angle. Under Ernst and Merchant's minimum energy hypothesis, the shear angle maximizes shear stress according to $\phi = \frac{\pi}{4} - \frac{1}{2}(\beta - \alpha)$. Total cutting power is determined by cutting force and speed as $P = \frac{F_P V}{60 \times 1000}\,\text{kW}$, with material removal rate calculated as $\text{MRR} = V t f$.