Stress-Strain Lab (3D)

What this page shows

A uniaxial tensile test of a round bar under displacement control: the crosshead moves at constant speed, the load cell reads the force, and the page plots engineering stress s = F/A0 against engineering strain e = ΔL/L0. The material is described by five numbers: Young's modulus E, yield strength σy, hardening exponent n and the true strain at fracture, plus the specimen size.

Material law

True stress and true strain are σ = F/A and ε = ln(L/L0). The law used is elastic up to the yield strain εy = σy/E, then power-law hardening, σ = σy (ε/εy)n (a Hollomon law with an elastic branch). Plastic flow keeps volume constant, so A = A0 e-ε and the engineering quantities follow from s = σ e-ε and e = eε - 1.

Necking (Considere)

The load F = σA is largest where dσ/dε = σ, which for the power law means ε = n. At that point the engineering stress peaks (the ultimate tensile strength K nn e-n) and deformation localises in a neck: the page models a neck of 20 percent of the gauge length that takes all further plastic strain while the rest of the bar unloads elastically. Fracture occurs when the neck's true strain reaches the fracture strain. A brittle material (fracture strain below the yield strain) breaks while still on the elastic line.

Unloading and energy

If the load is reversed, the bar unloads along a line of slope E A/L0 and keeps a permanent set; reloading follows the same line until it meets the original curve. The work done by the machine is W = ∫ F dL; the elastic part returned on unloading is F²/(2k) and the remainder is plastic work, turned into heat. The area under the engineering curve to fracture divided by the volume is the toughness in MJ/m³.

Checks run in the page

1. Slope of the first points of the engineering curve against E. 2. Strain and stress at the load maximum against en - 1 and K nn e-n. 3. Numerical integral of F dL up to yield against ½ E ε² A0 L0. 4. True stress s(1+e) against the power law at a point on the curve.

Honest simplifications

Material numbers are typical textbook values, not data for a named alloy. Real metals show a yield point drop, Luders bands, strain-rate and temperature effects, the Bauschinger effect on reversal and a triaxial stress state in the neck, none of which are modelled; the neck is a fixed-length region rather than a solved geometry. The 3D bar's diameter change is real (exp(-ε/2)); the length scale is drawn compressed.