# Work hardening **Work hardening**, also known as strain hardening, is the strengthening of a metal by plastic deformation: every increment of permanent strain leaves the crystal harder to deform than it was before. [[Rolling_(metalworking)|Rolling]], drawing, bending, hammering and machining all do it, whether or not the shop wants it, and [[Cold_working|cold working]] is the same family of processes seen from the shop floor. The mechanism is the multiplication and entanglement of [[Dislocation|dislocations]], the line defects that carry [[Slip_(materials_science)|slip]] through a crystal; as their density grows they obstruct one another, and the stress needed to keep them moving rises. The effect is undone by [[Annealing_(materials_science)|annealing]], a heat treatment in which [[Recrystallization_(metallurgy)|recrystallization]] replaces the tangled grains with fresh, nearly dislocation-free ones. In the microsim below the reader cold-rolls a bar by a thickness reduction *r* between 0 and 80 %. The flow stress climbs along the Hollomon curve `σ = K·εⁿ`, with the rolling strain `ε = −ln(1 − r)`; the dislocation density implied by Taylor's relation `σ = α·G·b·√ρ` is read out beside it; the ductility left in the bar falls toward zero; and an *anneal* button recrystallizes the bar and returns the curve to its annealed start. On the [[Materials_science|Materials science]] flagship this article serves the *Work hardening and annealing* section of Part IV, Fundamentals › Processing, as a sibling of the [[Stress–strain_curve|stress–strain curve]] sim from which it borrows its Hollomon fit. ## Undesirable work hardening A forming operation deforms metal on purpose, but the hardening that comes with the deformation is often a nuisance. A sheet drawn into a cup is stronger afterwards but has less [[Ductility|ductility]] left for the next draw, so deep parts are drawn in stages with anneals between them. Drawn wire hardens with every die pass and is annealed between passes for the same reason. The everyday demonstration is a paper clip bent back and forth until it snaps: each bend hardens the metal there, the ductility is used up, and a crack opens. A cutting edge shears the layer ahead of it and drags the layer below it, so the surface it leaves can be harder than the bulk, and the next pass cuts into hardened metal. Austenitic [[Stainless_steel|stainless steels]] and nickel alloys are the notorious cases: the rapid hardening that makes them tough to dent makes them tough to cut, and the shop's cutting-speed table shows the price. *Manufacturing Processes 4-5* lists 60 surface feet per minute for 302 and 304 stainless and only 5–10 for Inconel, against 40–140 for low-carbon [[Steel|steel]] and 100–500 for [[Copper|copper]].[^virasak-u2] Where a hardened surface must still be finished to size, the shop turns from a cutting edge to an abrasive wheel, the subject of the same book's surface-grinder chapter.[^virasak-ch5] The cutting-speed arithmetic is the subject of [[Speeds_and_feeds|Speeds and feeds]]. ## Intentional work hardening Cold-rolled sheet and strip, hard-drawn wire and tube, cold-headed bolts and cold-coiled springs get much of their strength from the same mechanism applied deliberately. The producer controls the amount of cold work and sells the result as a *temper*. For [[Aluminium|aluminium]] the American National Standard temper system spells this out: an H1x temper is strain-hardened only, H2x is strain-hardened and then partially annealed, H3x is strain-hardened and then stabilized by a low-temperature treatment, and the second digit records the degree of hardening, with 2, 4, 6 and 8 standing for quarter-, half-, three-quarter- and full-hard.[^ansi-h35] Copper and [[Brass|brass]] are sold in a parallel ladder of tempers. The method has two attractions. It costs nothing but the deformation the part was going to receive anyway, and it works on metals that cannot be strengthened by [[Heat_treating|heat treatment]]: pure metals, single-phase alloys such as commercially pure aluminium and 70/30 brass, and austenitic stainless steel, none of which form [[Martensite|martensite]] or an age-hardening precipitate. Its weakness is that heat lets the dislocation tangle escape. A cold-worked part that is [[Welding|welded]] loses its temper in the [[Heat-affected_zone|heat-affected zone]] beside the seam, which is why hard-tempered aluminium and copper structures are riveted or bonded more often than welded. Shot peening and burnishing use the effect at small scale, hardening a thin skin that also carries a compressive residual stress against [[Fatigue_(material)|fatigue]]. ## Theory Plastic deformation of a crystal is not the sliding of whole planes of atoms over one another. It is the passage of dislocations, and work hardening is what happens when those dislocations get in each other's way. ### Elastic and plastic deformation Under a small load a metal deforms elastically: stress is proportional to strain through [[Young's_modulus|Young's modulus]] *E* by [[Hooke's_law|Hooke's law]], `σ = E·ε`, and the strain disappears when the load is removed. Beyond the elastic limit the deformation becomes plastic and permanent; the stress–strain curve bends over, continues to rise, reaches a maximum and ends at fracture.[^openstax-up1-12-4] *University Physics* tabulates copper at a Young's modulus of about 110 GPa and a shear modulus of about 44 GPa, aluminium at about 70 GPa and 25 GPa, and steel at about 200 GPa and 75 GPa,[^openstax-up1-12-3] so a stress of 100 MPa stretches copper elastically by less than 0.1 %, while plastic strains of tens of percent are the daily business of a rolling mill. Elastic strain stretches the bonds between atoms; plastic strain changes their neighbours, as one plane of atoms slips over the next by a whole lattice spacing and leaves the crystal as perfect as before, only sheared. Estimates from the 1920s of the stress needed to slide a whole plane rigidly came out at a sizeable fraction of the shear modulus, orders of magnitude above the yield stress of a soft metal, and that discrepancy is what the dislocation was invented to explain.[^hull-bacon][^taylor1934] ### Dislocations and lattice strain fields A dislocation is the boundary line of a region that has slipped, inside a region that has not. In an edge dislocation an extra half-plane of atoms ends in the crystal; in a screw dislocation the lattice spirals around the line. The [[Burgers_vector|Burgers vector]] *b* measures the slip the line carries, one lattice translation, about a quarter of a nanometre in copper.[^hull-bacon] Because the atoms near the line are squeezed on one side and stretched on the other, every dislocation carries a long-range elastic strain field with a stored energy per unit length of the order of `G·b²/2`.[^hull-bacon] Parallel dislocations feel each other through these fields, like signs repelling and opposite signs attracting, and a dislocation gliding on its slip plane must also cut through the "forest" of dislocations that thread the plane from other slip systems. These interactions are the raw material of work hardening; the dislocation itself was proposed in 1934 in three independent papers, one of them by G. I. Taylor.[^taylor1934][^hull-bacon] ### Increase of dislocations and work hardening An annealed metal contains dislocations at a density *ρ*, measured as line length per unit volume, of roughly 10¹⁰ to 10¹² m⁻²; heavy cold work raises it to 10¹⁵ to 10¹⁶ m⁻².[^hull-bacon] The new lines are generated during slip itself, most famously by the [[Frank–Read_source|Frank–Read source]], in which a pinned segment bows out under stress and closes on itself to emit a fresh loop, again and again.[^hull-bacon] The mean spacing between dislocations scales as `1/√ρ`, so a thousand-fold rise in density brings the lines thirty times closer, and a gliding dislocation meets an obstacle thirty times sooner. In face-centred cubic metals the lines gather into tangles and then into cell walls, a [[Microstructure|microstructure]] visible in the electron microscope. Only a small fraction of the work done on the metal is stored in these strain fields; the rest leaves as heat. ### Quantification of work hardening Taylor's 1934 theory turned the picture into a number. If the obstacles to glide are other dislocations spaced about `1/√ρ` apart, the shear stress needed to push a dislocation past them scales as `τ = α·G·b·√ρ`, with *G* the shear modulus, *b* the Burgers vector and *α* a constant of order 0.2 to 0.5 that absorbs the geometry of the obstacle array; assuming that the density grows in proportion to strain, Taylor obtained the parabolic hardening law `τ ∝ √γ` that single crystals of soft metals roughly follow.[^taylor1934] For a polycrystal in tension, the differently oriented grains raise the flow stress above the single-crystal shear stress by the Taylor factor, about 3 for face-centred cubic metals.[^taylor1938] The microsim folds that factor into its constant and displays the tensile form `σ = α·G·b·√ρ`, so its *α* is of order 1, an ILLUSTRATIVE bookkeeping choice rather than a measured constant. Engineers also track the hardening rate `θ = dσ/dε`, highest just after yield and decaying as recovery removes dislocations almost as fast as slip creates them. ### Example The sim's annealed-copper preset runs the arithmetic. A cold-rolled strip whose thickness falls from *t*₀ to *t* has a true thickness strain `ε = ln(t₀/t) = −ln(1 − r)`, so reductions of 20, 50 and 80 % give ε = 0.22, 0.69 and 1.61. With the Hollomon constants for annealed copper, K = 530 MPa and n = 0.44,[^callister-nk] the flow stress after each reduction is 274, 451 and 653 MPa (derived). The Taylor relation then reads the dislocation density off that stress: with G = 44 GPa,[^openstax-up1-12-3] b = 0.256 nm[^hull-bacon] and the sim's α = 1, `G·b` is 11.3 N/m and `ρ = (σ/(G·b))²`. | reduction *r* | true strain ε | Hollomon σ (MPa) | implied ρ (m⁻²) | uniform strain left, n − ε | |---|---|---|---|---| | 20 % | 0.22 | 274 | 5.9 × 10¹⁴ | 0.22 | | 50 % | 0.69 | 451 | 1.6 × 10¹⁵ | none | | 80 % | 1.61 | 653 | 3.4 × 10¹⁵ | none | All entries but the constants are derived on the page from the cited values. The last column is the ductility readout. In a tensile test a bar necks when the hardening rate can no longer keep up with the shrinking cross-section, `dσ/dε = σ`, and for the Hollomon law that happens at a true strain equal to *n*; a bar that already carries a prior strain ε has only `n − ε` of uniform stretch left, and none once ε exceeds *n*.[^hollomon1945] The 50 %-rolled strip therefore necks as soon as it yields, which is how a full-hard temper behaves. Two cautions belong on the readout. The Hollomon fit is measured between yield and necking, so its use at rolling strains of 0.7 and 1.6 is an ILLUSTRATIVE extrapolation, and real cold-rolled copper saturates below 653 MPa. And the power law's zero stress at zero strain hides the yield stress that [[Grain_boundary|grain boundaries]], solutes and lattice friction give a real annealed bar, so the sim's annealed density is the textbook range, not a number from the fit. ## Empirical relations The curve the sim draws is Hollomon's 1945 power law, `σ = K·εⁿ`, in true stress and true strain.[^hollomon1945] On log–log axes it is a straight line of slope *n*, the strain-hardening exponent, and *K*, the strength coefficient, is the stress at a true strain of 1. Because *n* is also the uniform strain at which necking begins, a high *n* marks a metal that spreads its deformation evenly and forms well. Textbook tabulations of the fit give annealed low-carbon steel n = 0.21 and K = 600 MPa, annealed 304 stainless steel n = 0.44 and K = 1,400 MPa, annealed copper n = 0.44 and K = 530 MPa, annealed naval brass n = 0.21 and K = 585 MPa, 2024-T3 aluminium n = 0.17 and K = 780 MPa, and a quenched-and-tempered 4340 steel n = 0.12 and K = 2,650 MPa.[^callister-nk] Other relations bracket Hollomon's. Ludwik's 1909 form `σ = σ₀ + K·εⁿ` adds a yield stress so that the curve does not start at zero.[^ludwik1909] Saturating forms such as Voce's, in which the stress approaches a limit exponentially, describe the late stages of cold rolling better than a power law that rises without bound.[^voce1948] All of these are fits to tensile data chosen for convenience, not laws of nature. The sim's use of the rolling thickness strain as the Hollomon strain is a further simplification, since the effective strain of plane-strain rolling differs from it by a constant factor that the sim ignores. The trade behind them is not a fit: the [[Ultimate_tensile_strength|tensile strength]] and [[Yield_(engineering)|yield]] of a cold-worked metal rise together while its elongation falls. ## Work hardening in specific materials ### Steel Plain low-carbon steel hardens moderately, with n near 0.21,[^callister-nk] and is sold as cold-rolled strip in tempers up to full-hard for panels, brackets and stampings. It softens again cheaply, because pure iron recrystallizes at about 450 °C, far below its melting point of 1,538 °C,[^callister-recryst] so a process anneal restores formability between draws. Austenitic stainless steel is the opposite case: its n of 0.44 and K of 1,400 MPa[^callister-nk] make 304 sheet excellent for deep-drawn sinks and hard on the tools that draw and cut it; the cutting-speed table's 60 surface feet per minute for 302 and 304 against 40–140 for low-carbon steel is the machinist's version of the same fact.[^virasak-u2] For hardenable [[Carbon_steel|carbon steels]], work hardening competes with a different route to strength, the quench to martensite and the temper that follows, described in the heat-treating chapter of *Manufacturing Processes 4-5* alongside annealing and normalizing.[^virasak-ch6] A hard austenitic part, by contrast, can only be cold worked. ### Copper Copper is the sim's default because it hardens strongly, anneals easily and is familiar. Annealed, it flows with n = 0.44 and K = 530 MPa;[^callister-nk] hard-drawn, it holds its shape as tube and busbar. High-purity copper recrystallizes at about 120 °C against a melting point of 1,085 °C,[^callister-recryst] which is why a plumber can soften a hard-drawn tube with a hand torch before bending it. The rule of thumb places a metal's recrystallization temperature between one-third and one-half of its absolute melting temperature, and the cluster's recrystallization variant uses 0.4 T_m, about 270 °C for copper, within the range that commercial purity and prior cold work move it.[^callister-recryst] Alloying raises the threshold sharply: 60/40 brass recrystallizes at about 475 °C,[^callister-recryst] so a brass spring keeps its temper where a copper one would not. At the lathe, copper's softness and rapid hardening combine into a cutting-speed band of 100–500 surface feet per minute, the widest in the table.[^virasak-u2] ### Gold and other precious metals Jewellers have always hardened [[Gold|gold]], [[Silver|silver]] and [[Platinum|platinum]] by working them. Pure gold is too soft to hold a setting, and while alloying to a lower carat helps, much of the strength of a finished ring or chain comes from the drawing, rolling, twisting and planishing that shaped it. The craft also knows the limit: soldering a joint anneals the metal around it, so a piece that must stay springy is worked last, after the last torch. Some precious-metal alloys can be age-hardened as well, a separate mechanism treated under [[Precipitation_hardening|precipitation hardening]], but for the pure metals and the simple alloys cold work is the only strengthening available. ### Aluminum Commercially pure aluminium and its non-heat-treatable alloys are the textbook case of the strain-hardened temper, and the H designations above were written for them.[^ansi-h35] High-purity aluminium recrystallizes at about 80 °C against a melting point of 660 °C,[^callister-recryst] so close to room temperature that a freshly cold-worked strip can soften on the shelf; the H2x partial anneal and the H3x stabilizing treatment settle the temper before it settles itself. The heat-treatable [[Aluminium_alloy|aluminium alloys]] combine both routes: a T3 temper is solution-treated, cold-worked and naturally aged, and a T8 temper is solution-treated, cold-worked and artificially aged,[^ansi-h35] so that the dislocation tangle and the precipitates obstruct slip together. The 2024-T3 fit, n = 0.17 and K = 780 MPa,[^callister-nk] shows the result: a lower hardening exponent than the pure metal, because the precipitates have already done part of the hardening's work. *See also:* [[Recrystallization_(metallurgy)]] · [[Rolling_(metalworking)]] · [[Cold_working]] · [[Forging]] · [[Stress–strain_curve]] · [[Dislocation]] · [[Heat_treating]] · [[Precipitation_hardening]] ## References [^taylor1934]: Taylor, G. I. (1934). "The mechanism of plastic deformation of crystals. Part I. Theoretical." *Proceedings of the Royal Society A* 145 (855): 362–387. https://doi.org/10.1098/rspa.1934.0106 [^taylor1938]: Taylor, G. I. (1938). "Plastic strain in metals." *Journal of the Institute of Metals* 62: 307–324. (No DOI; page to pin against the journal record.) [^hollomon1945]: Hollomon, J. H. (1945). "Tensile deformation." *Transactions of the American Institute of Mining and Metallurgical Engineers* 162: 268–290. (No DOI; not a Portal Book; page to pin against the journal record.) [^ludwik1909]: Ludwik, P. (1909). *Elemente der technologischen Mechanik*. Berlin: Springer. (Page to pin.) [^voce1948]: Voce, E. (1948). "The relationship between stress and strain for homogeneous deformation." *Journal of the Institute of Metals* 74. (Page to pin.) [^hull-bacon]: Hull, D.; Bacon, D. J. (2011). *Introduction to Dislocations*, 5th ed. Oxford: Butterworth-Heinemann. Chapter 1 (Defects in crystals), Chapter 4 (Elastic properties of dislocations), Chapter 8 (Origin and multiplication of dislocations) and Chapter 10 (Strength of crystalline solids). Not a Portal Book; page to pin. [^callister-nk]: Callister, W. D.; Rethwisch, D. G. (2010). *Materials Science and Engineering: An Introduction*, 8th ed. Hoboken: Wiley. Chapter 6, Mechanical Properties of Metals, table of *n* and *K* values for the true-stress–true-strain power law. Not a Portal Book; page to pin. [^callister-recryst]: Callister, W. D.; Rethwisch, D. G. (2010). *Materials Science and Engineering: An Introduction*, 8th ed. Hoboken: Wiley. Chapter 7, Dislocations and Strengthening Mechanisms, table of recrystallization and melting temperatures and the accompanying rule of thumb. Not a Portal Book; page to pin. [^openstax-up1-12-3]: Ling, S. J.; Sanny, J.; Moebs, W. (2016). *University Physics Volume 1*. OpenStax (Portal Book 077). Chapter 12, Static Equilibrium and Elasticity, §12.3 Stress, Strain, and Elastic Modulus, Table 12.1, chapter pp. 565–610 (table page to pin). https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus [^openstax-up1-12-4]: Ling, S. J.; Sanny, J.; Moebs, W. (2016). *University Physics Volume 1*. OpenStax (Portal Book 077). Chapter 12, §12.4 Elasticity and Plasticity, chapter pp. 565–610 (page to pin). https://openstax.org/books/university-physics-volume-1/pages/12-4-elasticity-and-plasticity [^virasak-u2]: Virasak, LamNgeun (2019). *Manufacturing Processes 4-5*. Open Oregon Educational Resources (Portal Book 094). Chapter 1, Unit 2: Speeds, Feeds, and Tapping, Table 1 (cutting speeds in surface feet per minute), pp. 29–31. https://open.umn.edu/opentextbooks/textbooks/manufacturing-processes-4-5 [^virasak-ch5]: Virasak, LamNgeun (2019). *Manufacturing Processes 4-5* (Portal Book 094). Chapter 5: Surface Grinder, pp. 133–144. https://open.umn.edu/opentextbooks/textbooks/manufacturing-processes-4-5 [^virasak-ch6]: Virasak, LamNgeun (2019). *Manufacturing Processes 4-5* (Portal Book 094). Chapter 6: Heat Treating, pp. 147–150. https://open.umn.edu/opentextbooks/textbooks/manufacturing-processes-4-5 [^ansi-h35]: American National Standards Institute / The Aluminum Association. *ANSI H35.1/H35.1(M), American National Standard Alloy and Temper Designation Systems for Aluminum*. Arlington, Virginia: The Aluminum Association. (Edition year and clause to pin.) ### Bibliography - Virasak, LamNgeun (2019). *Manufacturing Processes 4-5*. Open Oregon Educational Resources. Portal Book 094. https://open.umn.edu/opentextbooks/textbooks/manufacturing-processes-4-5 - Ling, S. J.; Sanny, J.; Moebs, W. (2016). *University Physics Volume 1*. OpenStax. Portal Book 077. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 - Hull, D.; Bacon, D. J. (2011). *Introduction to Dislocations*, 5th ed. Butterworth-Heinemann. Not a Portal Book. - Callister, W. D.; Rethwisch, D. G. (2010). *Materials Science and Engineering: An Introduction*, 8th ed. Wiley. Not a Portal Book. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Work_hardening.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Work hardening* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Work_hardening.html" data-title="Work hardening"></div> *Built from `MICROSIM_GUIDE/specs/sims/Work_hardening.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Work_hardening) : [Wikitube](https://en.wikitube.io/wiki/Work_hardening) · pinned revision [1351372972](https://en.wikipedia.org/w/index.php?oldid=1351372972) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M20 · sim pending (matter/Work_hardening).*