Stress-Strain Curve & Hall-Petch Yield

Test Specimen

Material

Grain size d 20 µm

Applied load rate 1.0×

Stress σ

0 MPa

Strain ε

0.0%

Yield σ_y (Hall–Petch)

–

State

Elastic

How it works

Below the yield stress the specimen deforms elastically — atomic bonds stretch like springs and fully recover if the load is removed, following Hooke's law. Past yield, dislocations glide through the crystal lattice and the deformation becomes permanent (plastic). Smaller grains pack more grain-boundary area into the same volume, and each boundary blocks dislocation motion, so the yield stress rises as grains shrink — the Hall–Petch relation.

σ = E·ε              (elastic region)
σ_y = σ₀ + k·d⁻¹⁢⁄⁢²   (Hall–Petch)
K = Yσ√(πa)         (fracture, K≥K_IC breaks)

Material — sets Young's modulus E, base yield σ₀ and whether the curve necks (ductile) or snaps without warning (brittle ceramic).
Grain size d — smaller d raises σ_y through the Hall–Petch term k·d⁻¹⁢⁄⁢², exactly as grain refinement strengthens real alloys.
Applied load rate — how fast the bar is pulled; only changes playback speed, not the physics.
Pull / Reset — starts or restarts the loading history so you can trace the whole curve from zero to fracture.

Real-world relevance: engineers pick grain size (via rolling, annealing or micro-alloying) and material together specifically to raise σ_y without sacrificing the ductility a structure needs to absorb overload safely.

Stress-strain curve Yield point Fracture