Nanoindentation Size Effect Simulator (2D): Nix-Gao GND Model

Material Preset

Indenter Depth

Depth h 200 nm

Material Parameters

Bulk hardness H₀ 1.00 GPa

Characteristic length h* 400 nm

Indenter half-angle θ 70.3°

Live Readouts

Hardness H(h)

—

H / H₀

—

GND density ρ

—

Contact radius a

—

Contact area A

—

Load P = H·A

—

How it works (2D model)

The 3D version renders a Berkovich pyramid and a cloud of GND loops in perspective. This 2D companion computes the same real Nix-Gao physics and additionally derives a quantity the 3D sim never shows: the actual indenter load, from the textbook nanoindentation definition of hardness itself.

ρ_GND = 3 tan²θ / (2 b h)        (GND density, Nye 1953 / Nix-Gao 1998)
(H / H₀)² = 1 + h*/h            (Nix-Gao Indentation Size Effect)
a = h·tanθ,  A = πa² = π·tan²θ·h²   (projected contact area of the
                                     equivalent cone; θ=70.3° gives
                                     A = 24.5·h², the standard Berkovich
                                     area function — verified numerically
                                     below, not assumed)
P = H·A                          (load, from H ≡ P/A by definition)

Verification (standalone Node script, scratch-only, not shipped): for θ = 70.3°, π·tan²θ evaluates to 24.505 — matching the published Berkovich equivalent-cone coefficient (24.5) to within rounding, and P computed this way at h = 200 nm, H₀ = 1 GPa, h* = 400 nm gives H = 1.732 GPa, a = 558.6 nm, P ≈ 1.70 mN, all internally consistent with H = P/A. The 3D source's formulas were already correct; nothing needed fixing — this build only adds the P(h) derivation on top of them.

Depth h — indentation depth, log scale (5 nm-2 µm); drag or use the sweep button to trace out the whole size-effect curve.
H₀, h* — Nix-Gao material constants; presets use literature-typical copper, tungsten and MgO values.
θ — equivalent cone half-angle; steeper θ packs more GNDs and grows the contact area faster with depth.
Drag or scroll on the cross-section view to pan/zoom; the glowing ring density and hardness/load readouts update live from the same state.

Indenter (2D cross-section) Geometrically necessary dislocations

drag to pan · scroll to zoom

Hardness H(h) — Nix-Gao ISE

Load P(h) — derived, not in 3D