Strain Hardening and Annealing

Chapter Learning Objectives

• Quantify key deformation parameters
  • Strain-hardening exponent $n$ → gauges strength increase produced by plastic strain
  • Strain-rate sensitivity $m$ → gauges strength increase produced by higher deformation speed
• Explain why both parameters control the forming force required for metals
• Predict evolution of strength, ductility, texture & residual stress when a cold-worked polycrystal is annealed or further deformed
• Design sequential cold-work + annealing schedules that deliver target thicknesses & mechanical properties

Fundamentals: Cold Working & the Stress–Strain Curve

• Cold working / strain hardening
  • Plastic deformation imposed below recrystallisation temperature
  • Requires an applied stress > \sigma_y of the undeformed metal
  • Leaves a permanent (residual) plastic strain → raises flow stress
• Flow stress $\sigma_f$
  • Current stress necessary to continue plastic deformation in a previously strained metal
  • Empirical power law: $\sigma_f = K \epsilon^n$ where
  • $K$ = strength coefficient (material constant)
  • $\epsilon$ = true plastic strain
  • $n$ = strain-hardening exponent (0–0.6 for metals)
• Springback
  • Elastic strain that recovers after removal of load; vital for die design in sheet-metal stamping & polymer extrusion
• Bauschinger effect
  • Plastic tension lowers subsequent compressive yield stress (and vice versa)
  • Originates from back stresses produced by dislocation structures

Key Definitions & Formulae

• Percent cold work (area based):
  $h = \Big( \frac{A<em>0 - A</em>f}{A<em>0} \Big)\times100\,\%$ where $A</em>0$ = original cross-sectional area, $A_f$ = final area
• Strain-rate dependent flow law (Johnson–Cook form simplified):
  $\sigma = C\, \dot\epsilon^{\,m}$ with m>0 → suppresses necking under dynamic loading

Manufacturing Processes Employing Cold Working (also used hot)

• Rolling (flat or shape)
• Forging (open-die & closed-die)
• Extrusion (direct & indirect)
• Wire / rod drawing
• Stamping, deep drawing & press forming
• Reference visuals & supplementary videos: aluminium extrusion, steel forgings, stamping dies

Quantitative Descriptors of Deformation Behaviour

• High $n$ (e.g., some Cu alloys) → greater strengthening per increment strain
• High positive $m$ (superplastic alloys, polymers at elevated T) →
  • Higher forming loads at high speed
  • Delays onset of diffuse necking → large uniform elongations

Strain-Hardening Mechanisms

• Metals: multiplication & tangling of dislocations
  • Frank–Read source continually emits new dislocation loops
  • Dislocation density $\rho$ rises by ~2–3 orders of magnitude during heavy cold work
• Ceramics & covalent solids: limited strain hardening (brittle, scarce dislocations)
• Thermoplastic polymers: strength rise comes from chain alignment not dislocation activity
  • Neck initiates, propagates; gauge length converts from random coils → oriented chains

Properties vs % Cold Work

• As $h$ ↑:
  • $\sigma<em>y$, $\sigma</em>{UTS}$ ↑ roughly linearly at first then plateau
  • Ductility (% elongation) ↓ toward 0
  • Electrical conductivity & corrosion resistance ↓
• Practical ceiling set by required formability in subsequent operations

Representative Example Calculations

Example 1 (copper plate thinning)

1 cm → 0.50 cm → 0.16 cm

• Percent cold work step 1: $h_1 = (1-0.5)/1 = 50\%$
• Step 2: $h_2 = (0.5-0.16)/0.5 = 68\%$
• Total (sequential area changes): $A<em>0=1, A</em>1=0.5, A<em>2=0.16 \Rightarrow h</em>{total} = (1-0.16)/1 = 84\%$
• Using Cu property chart: $\sigma_{UTS}\,(84\%) \approx 67{,}000\,\text{psi}$ (value supplied in tables)

Example 3 (wire drawing 0.30 in → 0.25 in Cu, $\sigma_y=20\,000\,\text{psi}$)

• Required draw force (no friction):
  $F = \sigma<em>y A</em>0 = 20\,000\,\text{psi} \times \frac{\pi}{4}(0.30\,\text{in})^2 = 4.24\,\text{kips}$
• Stress in final wire during drawing:
  $\sigma<em>f = \frac{F}{A</em>f} = \frac{4.24}{\pi(0.25)^2/4} = 27\,000\,\text{psi}$
• Because strain hardening during drawing raises $\sigma_y$ of final wire above $27\,000\,\text{psi}$, breakage is avoided (Bauschinger & example chart corroborate)

Microstructure, Texture Strengthening & Residual Stress

• Grain shape evolution:
  • 10 % CW → slight elongation
  • 30 % CW → cigar-shaped grains
  • 90 % CW → long fibre-like grains spanning part thickness
• Crystallographic (sheet / rolling) texture
  • Preferred orientation of slip planes/dirs leads to anisotropic $E$, $\sigma_y$, magnetic & electrical properties
  • Thin films develop growth textures independent of applied stress
• Residual stresses
  • Elastic portion of applied stress locked-in after unloading
  • Removed by stress-relief anneal (sub-recrystallisation temperature)
• Glass analogues
  • Annealed glass: stress-free
  • Tempered glass: surface in compression via quench → safe fracture behaviour

Advantages & Limitations of Cold Working

• Simultaneous shaping & strengthening
• Excellent surface finish, tight tolerances
• Economical for large volumes of small parts
  − Limited formability of brittle alloys
  − Reduces ductility, conductivity, corrosion resistance
  − Cold-worked state unacceptable for high-T service (recrystallisation)
  − Some processes (wire drawing) feasible only cold

Wire Drawing Mechanics (Fig 8-12)

• Pull force $F<em>d$ acts on both entry (diameter $d</em>o$) and exit (diameter $d_f$)
• Stress in exit section: $\sigma<em>f = \dfrac{4F</em>d}{\pi d_f^{\,2}}$
• Without strain hardening, $\sigma<em>f > \sigma</em>y$ → fracture; CW raises $\sigma_y$ concurrently

Annealing: Purpose & Three Stages

• Heat treatment to erase some or all effects of cold work

1. Recovery
   • T just below recryst.
   • Dislocations rearrange into low-angle walls → polygonised subgrains
   • $\rho$ unchanged, mechanical strength unchanged, residual stress & electrical resistivity drop
2. Recrystallisation
   • New nuclei form along subgrain boundaries & grow into strain-hardened matrix
   • Sharp fall in $\rho$, $\sigma_y$; big rise in ductility (%EL)
   • Recrystallisation temperature $T_R$ variables:
     • ↑%CW → ↓$T_R$
     • Smaller initial grains → ↓$T_R$
     • Pure metals < alloys
     • Longer hold time → lower necessary $T_R$
     • Approximate rule: $T<em>R \approx 0.4T</em>m$
3. Grain growth
   • Continued heating/coarsening once strain-free grains impinge
   • Average grain size $d$ increases; some properties (creep, toughness) worsen

Controlling Annealing Outcomes

• Grain size after recrystallisation ↓ by:
  • Lower anneal temperature
  • Shorter or rapid-heating cycles
  • Higher prior %CW
  • Second-phase particles (Zener pinning) inhibiting growth
• Boundary between cold vs hot working: temperature slightly below $T_R$

Interaction with Manufacturing

• Cycle cold work ↔ anneal to extend total attainable reduction (strip, foil)
• Welding heat-affected zone (HAZ) in cold-worked metal may locally recrystallise → softened band; mitigate by low-heat-input, fast-cool processes
• High-T service rapidly recrystallises cold-worked alloys → catastrophic strength loss

Hot Working (T > $T_R$)

• Plastic deformation concurrent with dynamic recovery & recrystallisation → no net strengthening
• Enables massive strain without cracking; vital for
  • Primary breakdown of cast ingots
  • Forming of HCP alloys (Mg, Ti) that are brittle at room T
• Defect healing: welds internal voids, homogenises segregation, collapses pores
• Anisotropy remains: surface cools faster → finer grains than core; rolls imprint texture
• Surface finish & accuracy poorer than cold working
  • Oxide scale forms → acid pickling
  • Thermal contraction & elastic springback demand oversizing

Comparative Process Design Example (Al 3105 Plate, Example 4)

Objective: produce 0.3 in plate from 3 in stock, $\sigma_{UTS}\ge25\,000\,\text{psi}$, %EL ≥ 5%, max per-pass CW = 80 %
Cold-work route

1. Step 1: CW 80 % → 3 in → 0.6 in, intermediate anneal (full recryst.)
2. Step 2: CW 67 % → 0.6 in → 0.20 in, anneal (partial, to retain strength)
3. Step 3: CW 33 % → 0.20 → 0.30 in thickness increase not possible; so reverse: final reduction 0.20 → 0.30 unrealistic. Correct sequence: multiple passes with anneal after ~70 % cumulative reduction to balance strength/ductility.
   Hot-work alternative

• Initial hot rolling 3 → 0.5 in in one or two passes
• Finish cold roll 0.5 → 0.3 in (40 % CW) to achieve target strength, followed by low-temperature recovery anneal to reach %EL requirement
• Fewer passes, lower force, less intermediate annealing

Glass & Shot-Peening Analogies

• Annealed glass: reheated to remove cooling stresses; analogous to metallic stress-relief
• Tempered / laminated glass: rapid quench → surface compression; mirrors shot peening of steels which bombards surface with shot to create compressive residual stress → boosts fatigue life

Consolidated Takeaways

• Strength–ductility trade-off is tunable via % cold work and subsequent heat treatment
• FCC metals exhibit the widest useful cold-work range; HCP often require hot working
• Residual & texture-induced anisotropy must be considered in design (springback, directional properties)
• Key rules of thumb:
  • $\sigma_f = K\epsilon^n$; higher $n$ = more strengthening
  • $T<em>R \approx 0.4T</em>m$; higher %CW or purity lowers $T_R$
  • Percent cold work $h$ governs property changes more directly than true strain
• Optimal manufacturing chains strategically alternate deformation and annealing to achieve final geometry, surface, and mechanical targets