Mechanical Behavior · Tensile testing

Reading a tensile curve: the stress-strain curve, yield, UTS and necking

Press Play and watch the specimen stretch, neck and break while its curve draws. Then pick mild steel and look for the small drop at yield.

What does each part of the stress-strain curve mean?

Engineering stress s = load ÷ starting area. Engineering strain e = change in length ÷ starting gauge length. Both use the starting size of the specimen.

Left: the specimen (grips grey, gauge section coloured) and a zoom of the start of the curve. Right: the engineering stress-strain curve as it is measured.
Material
Run the test

Try it: press Play with mild steel. The stress rises, falls a little, then runs flat while a band crosses the specimen. Then compare 6061-T6 with 304 stainless: 6061-T6 yields higher, but 304 hardens far more and ends much stronger.

Why do true stress and engineering stress differ?

True stress σ = load ÷ the area now. True strain ε = ln(1 + e). Before necking the whole gauge shrinks evenly, so σ = s(1 + e). After necking only the neck shrinks, and those formulas stop working.

Coral: engineering curve (s against e). Blue: true curve (σ against ε); dashed blue is the true stress inside the neck. Grey dots: s(1+e) against ln(1+e), the formula used past its limit. Purple: the slope dσ/dε.
Try it: move n from 0.1 to 0.5. The necking point (where the purple slope line meets the blue curve) slides right, and the true strain there always equals n. Then drag the cursor past necking and watch the grey dots fall below the dashed blue line.

Why is the modulus from my tensile test too low?

The load frame, grips and load cell stretch too. If strain is worked out from the crosshead movement, their stretch is counted as if it were the specimen's.

Left: the load frame and the specimen are two springs in series; the bars show how much each stretches at the yield load. Right: the first 2% of the curve, measured three ways.
Material
Strain measured by
Try it: with mild steel at 100 kN/mm the crosshead gives a modulus of about 34 GPa, not 207. Push the stiffness to 2000 kN/mm: still only about 165 GPa. Then switch to the extensometer.

Why does elongation at fracture depend on the gauge length?

Up to the UTS the whole gauge stretches evenly. After it, only the neck stretches, and the neck is about as long as the specimen is thick, whatever the gauge length (Barba's law).

Left: the broken specimen to scale; blue is the even stretch, coral the extra stretch at the neck. Right: elongation at fracture against L0/√A0, with the ASTM and ISO proportional gauges.
Material
Try it: with annealed copper, shorten the gauge from 50 mm to 20 mm: elongation at fracture rises from 45% to about 59%, while the uniform elongation stays at 36%. Then move the strain rate from 10-5 to 10-1 /s: the UTS moves by a few percent, the elongation hardly at all.

Key points

Yield is read with a rule

Most metals bend away from the straight elastic line gradually. The 0.2% offset line, drawn parallel to it, gives one repeatable number.

The peak is where necking starts

At the UTS, hardening can no longer make up for the shrinking area (Considère). For σ = Kεn the true strain there equals n.

True-stress formulas stop at the neck

σ = s(1+e) and ε = ln(1+e) assume the gauge shrinks evenly. After the UTS they underestimate the stress in the neck.

Measure strain on the specimen

Crosshead strain includes the machine's stretch, so E comes out far too low. Elongation at fracture means little without its gauge length.

More detail: the model behind the page and its limits

Flow law. Each preset uses the Swift form of the Hollomon law, σ = K(ε0 + εp)n, where εp is the true plastic strain. K, ε0 and n are fitted so the drawn curve gives the handbook yield strength and UTS and an assumed uniform elongation. The fitted n values are not the handbook n values, which come from fits over other strain ranges. Widget 2 uses the plain Hollomon law, ε0 = 0, with no elastic part.

Necking. The load F = σA is largest where dF = 0, that is dσ/dε = σ (Considère, 1885). For σ = Kεn this gives εu = n; for the Swift form, εu = n − ε0. With strain-rate sensitivity m, Hart's condition dσ/dε = (1 − m)σ delays necking to n/(1 − m) − ε0 (widget 4 only).

After the UTS. The load is carried by the neck: s = σ(εneck) exp(−εneck). The neck is taken to be √A0 long, so it adds (εneck − εu)√A0 to the length. This is Barba's law, ef = eu + β√A0/L0, with β = εfracture − εu. The fracture strain in the neck is chosen so the standard ASTM round specimen (d = 12.5 mm, L0 = 50 mm) gives the handbook elongation. The neck's three-axis stress state (Bridgman correction) is ignored, so the dashed "true stress in the neck" is the average stress, not the flow stress. The reduction of area that follows (last column) is lower than typical measured values for copper and 304.

Mild steel. Upper yield point 12% above the lower one and a Lüders strain of 1.8% are chosen values; real ones depend on the test machine stiffness, the grain size and the specimen alignment. The small wiggles on the plateau are drawn, not modelled.

Load frame. Crosshead strain = e + F/(k L0), with F = sA0. The whole specimen is taken as the gauge section; real specimens also stretch in the shoulders and slip a little in the grips, which lowers the measured modulus further. The corrected curve subtracts F/k exactly; in practice k is measured and the correction is only as good as that measurement.

Strain rate. All stresses scale by (rate ÷ 10-3 s-1)m. The post-uniform part is not changed by m here, although real post-uniform elongation also grows with m.

Questions people ask

Why is the 0.2% offset used for yield strength?

Most metals do not have a sharp point where plastic flow starts. The curve bends away from the elastic line slowly, so "the first departure" depends on how closely you look. A line parallel to the elastic line, shifted by 0.2% strain, cuts the curve at one repeatable stress. The 0.2% is a convention set in the test standards (ASTM E8, ISO 6892-1).

What is the difference between engineering stress and true stress?

Engineering stress divides the load by the starting area. True stress divides it by the area at that moment. As a metal stretches its area shrinks, so true stress is higher: σ = s(1+e) until necking. For the copper preset at the UTS, s = 220 MPa and σ is about 300 MPa.

When does necking start in a tensile test?

At the maximum load, the UTS. There the metal's hardening rate dσ/dε has fallen to the true stress itself (the Considère criterion). For a metal that follows σ = Kεn, that happens at a true strain equal to n.

Why does the stress drop after the UTS if the metal is still hardening?

The stress in the neck keeps rising. The drop is in engineering stress, which divides the load by the starting area. The neck's area is shrinking faster than its strength rises, so the load falls.

Why is the Young's modulus from my tensile test too low?

Usually because strain was worked out from crosshead movement. The frame, grips and load cell stretch in series with the specimen. For the mild steel preset in a frame of 100 kN/mm, the crosshead slope is about 34 GPa instead of 207. Use a clip-on or video extensometer, or subtract a measured frame compliance. For an accurate E, use ASTM E111 or a resonance method.

What causes the yield point drop in mild steel?

Carbon and nitrogen atoms gather at dislocations and pin them (Cottrell atmospheres). It takes a higher stress to pull the first dislocations free (upper yield point). Once free, they multiply and move at a lower stress (lower yield point). Yielding spreads along the specimen as a Lüders band, at nearly constant load.

Why does percent elongation depend on gauge length?

The extra stretch at the neck is about the same length in millimetres whatever the gauge length, so it adds more percent to a short gauge than a long one. That is why standards use proportional gauges (L0 = 4d in ASTM E8, 5.65√A0 in ISO 6892-1) and why an elongation must be quoted with its gauge length.

Is the area under the stress-strain curve the toughness?

It is the energy absorbed per unit volume up to fracture, often called tensile toughness, in MJ/m3. It is not the fracture toughness KIc, which measures resistance to a crack that is already there. See fracture toughness.

On this site: How a hardness number gets made (modulus and frame compliance in indentation) · Beyond hardness: yield and flow from indentation · Strengthening mechanisms · Fracture toughness · Fatigue and S-N curves · Dislocations and Burgers vectors · Creep and deformation maps · KAM and GNDs in EBSD · Weak-beam dark field of dislocations

References

Show the 11 references
  1. W. D. Callister Jr. and D. G. Rethwisch, Materials Science and Engineering: An Introduction, 10th ed., Wiley (2018): chapter 6 (mechanical properties of metals) and Appendix B, Tables B.2 (modulus) and B.4 (yield strength, tensile strength, % elongation in 50 mm). Source of E, yield, UTS and elongation for the four presets.
  2. G. E. Dieter, Mechanical Metallurgy, SI metric ed., McGraw-Hill (1988): chapter 8 (the tension test: true stress, Considère construction, Barba's law) and the section on the yield-point phenomenon in chapter 6.
  3. W. F. Hosford, Mechanical Behavior of Materials, 2nd ed., Cambridge University Press (2010): chapters 3 (tensile testing), 4 (strain hardening and necking) and 6 (strain-rate dependence).
  4. ASTM E8/E8M-22, Standard Test Methods for Tension Testing of Metallic Materials, ASTM International (2022). doi:10.1520/E0008_E0008M-22
  5. ISO 6892-1:2019, Metallic materials: Tensile testing, Part 1: Method of test at room temperature, ISO (2019).
  6. ASTM E111-17, Standard Test Method for Young's Modulus, Tangent Modulus, and Chord Modulus, ASTM International (2017). doi:10.1520/E0111-17
  7. A. Considère, Mémoire sur l'emploi du fer et de l'acier dans les constructions, Annales des Ponts et Chaussées 9, 574 (1885).
  8. J. H. Hollomon, Tensile deformation, Transactions of the AIME 162, 268 (1945).
  9. H. W. Swift, Plastic instability under plane stress, Journal of the Mechanics and Physics of Solids 1, 1 (1952). doi:10.1016/0022-5096(52)90002-1
  10. E. W. Hart, Theory of the tensile test, Acta Metallurgica 15, 351 (1967). doi:10.1016/0001-6160(67)90211-8
  11. A. H. Cottrell and B. A. Bilby, Dislocation theory of yielding and strain ageing of iron, Proceedings of the Physical Society A 62, 49 (1949). doi:10.1088/0370-1298/62/1/308
Cite this page: Tripathy, Manisha. “Reading a Tensile Curve.” untethered atom, 2026, https://untetheredatom.com/mechanical-behavior/reading-a-tensile-curve.
BibTeX 
@misc{tripathy2026tensilecurve,
  author = {Tripathy, Manisha},
  title  = {Reading a Tensile Curve},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/mechanical-behavior/reading-a-tensile-curve}},
  note   = {Interactive web tool}
}
Last updated 24 September 2026.