Load 0.00 kNExtension 0.00 mmEng. stress 0 MPaEng. strain 0.0 %State elasticFPS 0

Drag to orbit, wheel or pinch to zoom. The bar is drawn 20 times larger than life; the dark rings are 10 mm gauge marks painted on the bar. Colour of the bar: grey = barely strained, amber = heavily strained (local true strain).

Tensile Test: Hollomon Hardening, Considère Necking and Cup-and-Cone Fracture

Pull a round bar in a universal testing machine and watch the load, the stress-strain curve and the bar itself. Everything on screen comes from one constitutive law: elastic up to yield, then power-law work hardening until the neck forms and the bar breaks.

Test and specimen

Live readouts

Engineering stress against strain

Solid: engineering stress F/A0. Dashed: true stress F/A. Grey dashed: the 0.2 % offset line (slope E). The curve is drawn as the test runs; "Unload" retraces the elastic line and leaves a permanent set.

Load against crosshead extension

Experiments

Each button runs full tests numerically with the current bar size and tabulates the result against the closed-form prediction.

Independent numerical checks

Controls

Drag or one finger: orbit; wheel or pinch: zoom. "Follow the neck" keeps the camera on the thinnest section.
The material buttons restart the test. Bar diameter and gauge length restart it too: the stress-strain curve does not change with size, but the load does.
Test pace sets how fast the animation runs. The crosshead speeds up as the bar stretches so that the short elastic stage stays visible; a real machine pulls at a constant speed of a few mm per minute.
"Unload" (before necking) reverses the crosshead: the load falls along a line parallel to the elastic one and a permanent set stays in the bar. Press it again to reload.

The science

Engineering and true quantities. The machine records load F and extension ΔL. Engineering stress and strain use the original size: σ_e = F/A₀, e = ΔL/L₀. True stress and strain use the current size: σ = F/A, ε = ln(1+e). While the bar stretches evenly its volume stays constant (plastic flow), A·L = A₀L₀, so σ = σ_e(1+e).

Material law. Elastic: σ = Eε up to the yield stress σ_y. Beyond it the strain is ε = σ/E + (σ/K)^1/n − ε₀ with ε₀ = (σ_y/K)^1/n, a Hollomon power law σ = K ε_pⁿ shifted so the curve is continuous at yield. n is the work-hardening exponent. The 0.2 % offset yield is the stress at plastic strain 0.002, σ = K(0.002+ε₀)ⁿ.

Necking (Considère). Load is F = σA. Hardening raises σ, shrinking area lowers A. The load peaks, and a neck starts, when dσ/dε = σ. With the law above this gives ε_p + ε₀ = n(1 − σ/E), which is almost exactly n − ε₀ plastic strain. After that the neck alone keeps straining, the rest of the bar unloads elastically, and the engineering stress falls until fracture at the true strain ε_f.

What the model simplifies. The neck is a smooth bell of fixed width (0.35 L₀) at the same place every time; the triaxial stress inside a real neck (Bridgman correction) is ignored, as are strain-rate and temperature effects, machine compliance and the yield-point drop with Lüders bands of real mild steel. Fracture occurs at a fixed true strain ε_f per material. The brittle iron breaks elastically at a fixed stress. All material constants are typical textbook-order values chosen for illustration, not data for a specific grade.

Cup and cone. Voids form in the necked centre, join into a crack across the middle, and the outer ring then shears at about 45 degrees, leaving a cone on one half and a cup on the other. The break surface here is drawn that way for ductile metals and flat for the brittle one.

Toughness. The area under the engineering curve is the work per unit volume needed to break the bar, u = ∫σ_e de in MJ/m³ - why a ductile steel absorbs far more energy than a stronger but brittle cast iron.