Interactive explainer · brittle fracture

How a diffuse phase field becomes a crack

A phase-field model replaces an unknown sharp crack surface with a continuous damage field. Drag the controls to see how the length scale, tensile driving energy, and irreversibility shape the crack band.

Crack profile

The color field is damage d: intact material is 0, fully broken material is 1. The orange line is the profile across the crack.

Length scale 0.18

Tensile driving energy 0.62

History field on

intact d = 0broken d = 1

The profile is an illustrative AT2 solution, not a second finite-element solve. ℓ controls the regularized width; it does not equal a measured crack opening.

1 · Choose a field
Set d(x) between 0 and 1 instead of drawing crack faces explicitly.

2 · Minimize energy
Elastic energy is degraded by g(d) while the crack-density term costs G_c.

3 · Preserve damage
H = max ψ⁺ (or a bound d ≥ d_hist) prevents healing during unloading.

The regularized Griffith balance

E[u,d] = ∫_Ω [ g(d) ψ⁺(ε) + ψ⁻(ε) ] dV + G_c ∫_Ω [ d²/(2ℓ) + ℓ|∇d|²/2 ] dV

As ℓ becomes small and the mesh resolves it with several elements, the diffuse band approaches a sharp Griffith surface. In practice, the solver alternates a displacement solve with a phase-field solve at each load step. The accompanying notched-plate viewer shows this process on a finite-element solution, including deformed geometry, damage growth, and the recorded load–displacement response.