Inverse Hall-Petch Effect: Nanograin Strength Peak

Microstructure

Grain size d 15.0 nm

GB layer thickness t 1.0 nm

Loading rate 1.0×

Mechanical Response

Dislocation pile-up (Hall-Petch)

Yield strength σ_y

— MPa

GB volume fraction

—

Critical grain size d꜀

— nm

Grains modeled

—

How it works

Below ~10 nm, nanocrystalline metals stop obeying the classical Hall-Petch law and start getting weaker as grains shrink further. This simulator uses the two-phase "core-shell" composite model: every grain is split into a dislocation-carrying interior and a thin, disordered grain-boundary (GB) shell of fixed thickness t.

GB volume fraction:  φ = 1 − ((d − t) / d)³
Interior strength:   σᵢ(d) = σ₀ + k_HP / √d      (dislocation pile-up)
GB-shell strength:   σ_gb  ≈ const               (boundary sliding / Coble-type diffusion)
Rule of mixtures:     σ_y(d) = (1 − φ)·σᵢ(d) + φ·σ_gb

As d shrinks, φ grows (more of the material is boundary, not lattice) while σᵢ(d) keeps rising via 1/√d — the two effects trade off and σ_y(d) peaks at a critical grain size d꜀, typically a few nanometres. Below d꜀ the soft, sliding GB phase dominates and strength falls: the inverse Hall-Petch effect.

Grain size d — rebuilds the lattice of grains and recomputes σ_y(d) live.
GB layer thickness t — a thicker disordered shell pushes d꜀ larger and deepens the softening.
Loading rate — speeds up the dislocation glide (blue regime) or GB shear oscillation (red regime) shown in the 3D view; it is a visualization speed, not a new physical input.
Shading toggle — colors each grain by its own GB volume fraction contribution instead of a flat material color.

The small chart traces σ_y against grain size across the whole range and marks the current point, so the peak is always visible.

Grain interior (dislocations) Grain boundary (sliding)

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