Stress–strain_curve

# Stress–strain curve A **stress–strain curve** is the record of a tension test: a bar of known cross-section is pulled at a controlled rate, the force and the extension are logged, and each is divided by an original dimension to give stress and strain. The curve that results is the single most informative measurement in [[Solid_mechanics|solid mechanics]], because almost every number an engineer needs about a material — its stiffness, the load at which it stops springing back, the load at which it breaks, how far it stretches first, and how much energy it absorbs on the way — is a feature of one graph.[^univphys-ch12][^astm-e8] In the microsim below the reader pulls the strain. A three-dimensional bar elongates and then necks while the curve draws itself: first the elastic line `σ = E·ε`, then the departure marked by the 0.2 % offset yield point, then the plastic rise fitted by Hollomon's law `σ = K·ε^n` in true quantities, until Considère's condition `ε_u = n` puts the maximum of the engineering curve at the [[Ultimate_tensile_strength|ultimate tensile strength]], after which deformation localises into the neck and the bar fractures. Presets are mild steel, aluminium 6061, Ti-6Al-4V, PMMA and rubber; the readouts are `E`, the yield strength `σ_y`, the UTS, the elongation at fracture and the toughness, which is the area under the curve. On the [[Materials_science]] flagship this page serves Part III, Fundamentals › Properties, as the shared anchor of that part: it is where the structural mechanisms of Part II — [[Dislocation|dislocations]], [[Grain_boundary|grain boundaries]], second phases — are finally read off as numbers a designer can use. ## Definition Two definitions of each axis are in circulation, and the difference between them is the difference between the two halves of the curve. *Engineering stress* is the force divided by the original cross-sectional area, `σ_e = F/A_0`, and *engineering strain* is the extension divided by the original gauge length, `ε_e = ΔL/L_0`. Both use dimensions the specimen had before the test, which makes them directly computable from a load cell and an extensometer, and makes the curve a description of the specimen rather than of the material.[^univphys-ch12] *True stress* uses the instantaneous area, `σ_t = F/A`, and *true strain* is the integral of the increments of length, `ε_t = ln(L/L_0)`. While deformation stays uniform and the volume is conserved — a good approximation once flow is plastic — the two pairs convert exactly: `σ_t = σ_e·(1 + ε_e)` and `ε_t = ln(1 + ε_e)`. At the sim's mild-steel preset, 22 % engineering strain is 0.20 in true strain, and an engineering stress of 400 MPa there is a true stress of 489 MPa. The distinction matters because the engineering curve turns over and falls while the true curve keeps rising. Nothing weakens at the maximum of the engineering curve; the specimen simply stops getting longer everywhere and starts getting thinner in one place, so the force falls although the material at the neck is still hardening. The sim draws the engineering curve, because that is what a testing machine produces, and reports true values at the cursor. Stress has the units of pressure, and the numbers are large: a pascal is one newton per square metre, so engineering work is done in megapascals and gigapascals. Strain is dimensionless and small in the elastic range — a few parts in a thousand for a metal — which is why testing standards specify an extensometer on the gauge length rather than the crosshead displacement, which includes the machine's own stretch.[^astm-e8] ## Stages Every ductile curve in the sim passes through the same three stages, and each is governed by a different mechanism: reversible stretching of bonds, then the motion and multiplication of [[Dislocation|dislocations]], then a geometric instability that has nothing to do with the material weakening. The stages are separated by two events, yield and the tensile maximum, and neither is found by the specimen: yield is fixed by an agreed construction on the graph, and the maximum is where two competing rates happen to cross. The order is the same for every metal, and the differences between the presets are differences in where the boundaries fall. A hardened [[Titanium_alloys|titanium alloy]] spends most of its curve in the first stage; [[Annealing_(materials_science)|annealed]] copper spends almost none of it there. Reading a curve well is mostly a matter of knowing which stage a feature belongs to, because a designer who confuses the second boundary with failure will size a part against a number the material never actually reaches in service. ### Linear elastic region At small strains a solid behaves as Robert Hooke described in 1678, *ut tensio sic vis*: the extension is proportional to the force.[^hooke1678] Written for a material rather than a spring, that is `σ = E·ε`, and the constant `E`, named for Thomas Young, is [[Young's_modulus|Young's modulus]], the slope of the first part of the curve.[^young1807][^univphys-ch12] It is a stiffness, not a strength: it says how much stress a given strain costs, and says nothing about when the material will fail. `E` varies over four orders of magnitude across the sim's presets, and the shape of the whole curve follows. The mechanics sub-manual's landing-strut example uses 4340 [[Steel|steel]] with `E` = 30 × 10⁶ psi, which is 207 GPa, and a yield strength of 230 ksi, or 1.59 GPa.[^aero-strut] For ordinary mild steel with `E` = 200 GPa and `σ_y` = 250 MPa, the elastic range ends at a strain of 250/200,000 = 0.00125, or 0.125 %. Since the same bar will stretch 25 % before it breaks, the elastic part occupies about one two-hundredth of the curve's width, which is why the sim has to magnify it. [[Aluminium_alloy|Aluminium]] 6061-T6, at 69 GPa and 275 MPa, yields at 0.40 %; [[Titanium_alloys|Ti-6Al-4V]], at 114 GPa and 880 MPa, at 0.77 %; PMMA, at about 3 GPa, reaches 70 MPa at 2.3 %; and rubber leaves the linear region almost immediately.[^presets-cn] Elastic deformation is stretching of interatomic bonds, so `E` is fixed by bonding and by crystal structure and is barely changed by anything a metallurgist does. Cold work, [[Heat_treating|heat treatment]] and [[Precipitation_hardening|precipitation hardening]] can multiply the yield strength of a steel by five and will not move its modulus at all. The related [[Elastic_energy|elastic energy]] stored per unit volume at yield, the resilience `σ_y²/(2·E)`, is 0.16 MJ/m³ for the mild-steel preset. ### Strain hardening region Past yield the curve keeps climbing, because plastic flow makes the material harder to deform further. The mechanism is on the [[Dislocation]] page: dislocations multiply, obstruct one another and pile up at [[Grain_boundary|grain boundaries]], so the stress needed to keep them moving rises with the density of those already made — [[Work_hardening|work hardening]]. Where yield begins is a matter of definition. Mild steel obliges with a genuine discontinuity, an upper and lower yield point followed by a plateau along which a band of deformation sweeps the gauge length; most metals do not, and their departure from linearity is gradual. The convention is the 0.2 % offset: draw a line of slope `E` from `ε` = 0.002 and take its intersection with the curve as [[Yield_(engineering)|yield]].[^astm-e8] It is an arbitrary choice that everyone makes the same way, which is all a specification needs. The rise is fitted by Hollomon's power law in true quantities, `σ_t = K·ε_t^n`, where `K` is the strength coefficient and `n` the strain-hardening exponent.[^hollomon1945] `n` runs from near zero for a heavily cold-worked or precipitation-hardened alloy to about 0.5 for [[Annealing_(materials_science)|annealed]] brass, and it is the more important of the two, because it fixes where the curve turns over. The fit is ILLUSTRATIVE: it is a two-parameter description of the plastic range chosen because it is convenient, not a law derived from the mechanism, and real curves depart from it at both ends. ### Necking region The tensile maximum is a competition. As the specimen extends, the load-bearing area shrinks; as it deforms, the material hardens. While hardening wins the deformation stays uniform, and when area loss wins it concentrates. Setting the derivative of the load to zero gives Considère's condition, `dσ_t/dε_t = σ_t`, which for Hollomon's law reduces to the clean result that necking begins when the true strain equals the hardening exponent, `ε_u = n`.[^considere1885] That single equality is why the sim can predict the whole shape from two numbers. With `n` = 0.2 the mild-steel preset necks at a true strain of 0.20, an engineering strain of `e^0.2 − 1` = 22 %, and the corresponding engineering stress `K·n^n·e^(−n)` is the UTS: taking the preset's 400 MPa UTS back through that expression gives `K` ≈ 675 MPa. Aluminium 6061-T6, with `n` near 0.05, necks at 5.1 % engineering strain — its curve is nearly flat after yield, and its UTS of about 310 MPa sits only a little above its yield strength. Annealed brass at `n` = 0.5 would stay uniform to 65 %. After the neck forms the test is no longer a measurement of uniaxial behaviour. The reduced section pulls the material around it inward, so the stress state at the neck becomes triaxial and the axial stress required is raised above the true flow stress; Bridgman's correction recovers the uniaxial curve from the neck geometry.[^bridgman1952] The elongation at fracture reported from the test is therefore partly a property of the specimen, which is why standards fix the ratio of gauge length to diameter.[^astm-e8] ## Classification The useful first division is not by material class but by how much plastic strain the curve contains before it ends, because that decides whether a part gives warning before it fails. The labels are properties of a test, not of a substance: the same [[Steel|steel]] is ductile at room temperature and brittle at −60 °C, ductile when pulled slowly and brittle under impact, ductile as a smooth bar and brittle when notched. What the classification really separates is two ways of spending the work done on a specimen — spread through the volume as plastic flow, or concentrated at one crack tip as new surface — and the sim shows the difference as an area rather than as a height. Two presets can share an [[Ultimate_tensile_strength|ultimate tensile strength]] and differ by a factor of fifty in the energy their curves enclose. ### Ductile materials A ductile material yields well before it fractures and absorbs a great deal of energy between the two. Toughness is the area under the whole engineering curve, and for the mild-steel preset the trapezoid estimate `(σ_y + UTS)/2 × ε_f` gives (250 + 400)/2 × 0.25 ≈ 81 MJ/m³ — about five hundred times its resilience of 0.16 MJ/m³, so essentially all the energy a steel part can absorb is absorbed plastically. This is the whole argument for using ductile metals in structures: an overloaded member deforms visibly, redistributes its load to its neighbours, and takes a large amount of work to separate. [[Ductility]] is reported two ways from the same test, as percentage elongation `(L_f − L_0)/L_0` and as percentage reduction of area `(A_0 − A_f)/A_0`, the second being the more sensitive because it is measured at the neck. Both fall as strength rises, and the trade is the central fact of alloy design: the mechanisms that raise `σ_y` — [[Work_hardening|cold work]], [[Solid_solution_strengthening|solute]] additions, precipitates, [[Martensite|martensite]] — all leave less room for plastic strain, so the tough, weak annealed condition and the strong, brittle hardened one are two readings of the same curve after different [[Heat_treating|heat treatments]]. Ductility also depends on [[Temperature|temperature]] and on loading rate, and body-centred cubic metals such as mild steel lose it abruptly on cooling. ### Brittle materials A brittle material fractures at or barely past the elastic line, so its curve is short and almost straight and its toughness is small: the PMMA preset, reaching about 70 MPa at 5 % strain, stores roughly 1.8 MJ/m³, some forty-five times less than the steel. [[Glass]], most [[Ceramic|ceramics]], [[Cast_iron|cast iron]] and cold [[Carbon_steel|carbon steel]] behave this way, and so do many [[Polymer|polymers]] below their glass transition. The distinguishing feature of brittle behaviour is not its low strength but its scatter. Fracture starts at the worst pre-existing flaw, and Griffith showed in 1921 that the stress to propagate a crack of length `a` varies as `1/√a`, so a population of specimens with different flaws gives a population of strengths: increasing the largest flaw from 1 µm to 100 µm lowers the fracture stress tenfold.[^griffith1921][^aero-ch13] Brittle strengths are therefore quoted as a distribution with a [[Weibull_modulus|Weibull modulus]] rather than as a single number, and they depend on the volume and the surface state of the test piece. A ductile metal blunts its own cracks by yielding at the tip, which is why its curve is reproducible and why [[Fracture_toughness|fracture toughness]] rather than tensile strength is the design quantity for a cracked part.[^aero-ch13] Rubber is the reminder that neither class describes everything. Its curve is far from linear from the start, rises steeply only after several hundred per cent extension as the molecular network pulls taut, and is recoverable throughout: it is not elastic in Hooke's sense and not plastic either.[^presets-cn] [[Viscoelasticity|Viscoelastic]] and [[Creep_(deformation)|creeping]] materials add a further dimension the sim's single quasi-static pull cannot show, since for them the curve depends on how fast it was drawn. Where a full tension test is impractical, [[Hardness|hardness]] is used as a proxy for strength, an indentation measurement that correlates with `σ_y` and is fast enough for production control.[^manuf-hardness] ## See also - [[Young's_modulus]] — the variant sim, `E` alone with a moduli table - [[Hooke's_law]] - [[Yield_(engineering)]] - [[Ultimate_tensile_strength]] - [[Ductility]] - [[Poisson's_ratio]] — the variant `G = E/(2·(1 + ν))`, `K = E/(3·(1 − 2·ν))` - [[Strength_of_materials]] - [[Solid_mechanics]] - [[Hardness]] - [[Creep_(deformation)]] ## References [^univphys-ch12]: Sanny, J.; Ling, S. et al. *University Physics Volume 1* (2016), OpenStax, Ch. 12 Static Equilibrium and Elasticity, pp. 565–610 (stress, strain, Young's, shear and bulk moduli; Table 12.1 of tabulated moduli) (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 [^aero-ch13]: Johnson, Eric. *Aerospace Structures* (2022), Ch. 13 Fracture, pp. 389–416 (yielding against fracture; flaw-controlled strength and the stress-intensity approach) (page to pin). https://open.umn.edu/opentextbooks/textbooks/aerospace-structures [^aero-strut]: Johnson, *Aerospace Structures* (2022), landing-strut sizing example: 4340 steel, `E` = 30 × 10⁶ psi (207 GPa) [p. 419], yield 230 ksi (1.59 GPa) and allowable 160 ksi (1.10 GPa) [p. 420], density 0.284 lb/in³ [p. 421]. Page cites as carried by `MICROSIM_GUIDE/manuals/02_mechanics.md` §10.4. [^manuf-hardness]: Virasak, LamNgeun. *Manufacturing Processes 4-5* (2019), Ch. 6 Unit 2 Hardness Testing, pp. 151–154 (indentation hardness as a shop-floor proxy for strength) (page to pin). https://open.umn.edu/opentextbooks/textbooks/manufacturing-processes-4-5 [^hooke1678]: Hooke, Robert (1678). *Lectures de Potentia Restitutiva, or of Spring, Explaining the Power of Springing Bodies*. London: John Martyn ("ut tensio sic vis"). [^young1807]: Young, Thomas (1807). *A Course of Lectures on Natural Philosophy and the Mechanical Arts*. London: Joseph Johnson (lecture and page to pin). [^hollomon1945]: Hollomon, J. H. (1945). "Tensile Deformation." *Transactions of the American Institute of Mining and Metallurgical Engineers* 162: 268–290 (pages to pin; no DOI asserted). [^considere1885]: Considère, Armand (1885). "Mémoire sur l'emploi du fer et de l'acier dans les constructions." *Annales des Ponts et Chaussées* (series 6) 9: 574–775 (the instability condition; pages to pin). [^bridgman1952]: Bridgman, P. W. (1952). *Studies in Large Plastic Flow and Fracture, with Special Emphasis on the Effects of Hydrostatic Pressure*. New York: McGraw-Hill (the triaxiality correction at a neck; chapter and page to pin). [^griffith1921]: Griffith, A. A. (1921). "The Phenomena of Rupture and Flow in Solids." *Philosophical Transactions of the Royal Society of London A* 221: 163–198 (DOI to pin). [^astm-e8]: ASTM International. *ASTM E8/E8M, Standard Test Methods for Tension Testing of Metallic Materials* (gauge length and diameter ratios, extensometry, the 0.2 % offset construction) (edition and clause to pin). [^presets-cn]: *Citation needed.* The preset property sets — mild steel (`E` = 200 GPa, `σ_y` = 250 MPa, UTS 400 MPa, `n` = 0.2, elongation 25 %), aluminium 6061-T6 (69 GPa, 275 MPa, 310 MPa, `n` ≈ 0.05), Ti-6Al-4V (114 GPa, 880 MPa), PMMA (≈ 3 GPa, ≈ 70 MPa at 5 %) and rubber — are the values supplied by the M11 sim row and are typical handbook figures, not measurements of a particular heat or lot. Everything computed from them on this page (yield strains, `K` ≈ 675 MPa, uniform elongations, resilience and toughness estimates) inherits that status. A tabulated source for each preset, with its temper and specimen geometry, would settle them; 077 Ch. 12 Table 12.1 is the Portal Book's modulus table and should be read first.[^univphys-ch12]  **Microsim — three.js (Wikitube framework):** *Stress–strain curve* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Stress–strain_curve.html" data-title="Stress–strain curve"></div> *Built from `MICROSIM_GUIDE/specs/sims/Stress–strain_curve.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).*   **Microsim — three.js (Wikitube framework):** *Stress–strain curve* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Stress–strain_curve.html" data-title="Stress–strain curve"></div> *Built from `MICROSIM_GUIDE/specs/sims/Stress–strain_curve.json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Stress–strain_curve) : [Wikitube](https://en.wikitube.io/wiki/Stress–strain_curve) · pinned revision [1352317719](https://en.wikipedia.org/w/index.php?oldid=1352317719) · 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 M11 · sim pending (matter/Stress–strain_curve).*