Pull a bar and record the truth it tells: a straight elastic line whose slope is Young's modulus E, a yield knee where the metal forgets its shape, a climb through plastic flow to the ultimate strength, and then fracture. A material's whole life story in one curve.
Engineering stress σ = F/A₀ against engineering strain ε = ΔL/L₀. Four regimes stitched into one function σ(ε):
elastic: σ = E·ε (ε ≤ ε_y) yield: (ε_y, σ_y), σ_y = E·ε_y plastic: hardening → UTS at ε_u necking: decline to fracture at ε_fConstants: E = 200000 MPa, σ_y = 240 MPa, UTS = 400 MPa. E is exactly the slope of the elastic line — recovered from two points, not stored twice.
The straight part of this curve is the-hookes-law region: σ = Eε, slope E. Where this sphere ends — the yield point σ_y — the neighbouring sphere the-hall-petch begins: σ_y = σ₀ + k/√d, so a finer grain raises the knee and moves the whole plastic story upward. One curve hands off to the next law at the exact strain where linearity dies.
Live re-check: recover E from two low-strain points and confirm the elastic slope is positive and equal to E. If window 6 flips the slope sign, this badge catches it within one frame.
WITNESS: —A strain sweep ε : 0 → ε_f (0 to 0.25) fed one point at a time into σ(ε). Plus two probe strains for slope recovery and one unload command per point.
Proven at boot: E recovered as the elastic slope; resilience = ½σ_y²/E; elastic loading reversible; plastic unload leaves permanent set; UTS is the curve's maximum.
wall "Engineering stress is the true stress." No. σ=F/A₀ uses the original area A₀; as the bar necks the real area shrinks, so true stress keeps rising even while the engineering curve falls after UTS. The decline past the peak is an area artifact, not softening. This engine reports the engineering curve and says so.
amber The hardening shape (parabola to UTS, linear necking) is a plausible model, not this exact alloy — the laws (E=slope, resilience, reversibility, UTS=max) are exact; the specific ε_u, ε_f are chosen.
Yield strength is where the curve stops being straight. → Departure from linearity is gradual and hard to read; engineers define yield by the 0.2% offset line parallel to the elastic slope.
UTS is the stress at which the bar breaks. → UTS is the maximum of the curve; fracture happens later, at lower engineering stress, after necking.
A stiffer material (high E) is a stronger material. → E is stiffness (slope); strength is σ_y / UTS (heights). Rubber is weak but compliant; glass is stiff but brittle.
Planted void (disclosed): flip the sign of the elastic slope so E comes out negative — stress would fall as you pull. Physically impossible; the witness in 7 must catch it.