Indentation Size Effect: Nix-Gao Strain Gradient Plasticity

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)

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H / H₀

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GND density ρ

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Contact radius a

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H(h) Curve

How it works

Pushing a sharp pyramidal indenter into a crystal forces the material below it to bend around the tip. That bending cannot be accommodated by statistically stored dislocations alone — it requires extra, geometrically necessary dislocations (GNDs) whose density scales with the indenter's geometry and inversely with depth:

ρ_GND = 3 tan²θ / (2 b h)

where θ is the indenter's effective half-angle, b is the Burgers vector (~0.25 nm here), and h is indentation depth. Shallower indents pack far more GNDs per unit volume — the plastic zone shrinks faster than the dislocations needed to accommodate the imposed strain gradient.

Nix & Gao (1998) showed this geometric argument leads directly to a depth-dependent hardness — the Indentation Size Effect:

(H / H₀)² = 1 + h*/h

H₀ is the depth-independent "macroscopic" hardness set by statistically stored dislocations alone, and h* is a characteristic length (tens to hundreds of nm, material-dependent) at which the GND contribution equals the SSD contribution. As h → h*, hardness rises well above H₀ — the same crystal reads as intrinsically harder under a shallow indent than a deep one, purely from geometry.

Depth h — indentation depth on a log scale (5 nm–2 µm); the indenter and the visible dislocation cloud both scale with it.
H₀, h* — the two Nix-Gao material constants; the presets use literature-typical values for copper, tungsten and MgO.
θ — effective cone half-angle (70.3° reproduces a Berkovich pyramid's projected area); steeper angles pack more GNDs at a given depth.
Chip color and count of the glowing loops around the indenter tip visualize ρ_GND directly — denser and brighter as h shrinks.

Indenter (pyramidal tip) Geometrically necessary dislocations

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