Visual glossary for phase-field fracture

This page is a visual companion to the phase-field primer. It focuses on the terms that usually slow down first-time users: energy splits, AT1 versus AT2, and the regularisation length scale l0.

Why this page exists

The continuum equations in Physics, units, and formulation and the phase-field primer are the right place for the mathematics. This page provides a compact visual guide:

  • AT1 vs AT2: how the local crack-density term changes nucleation.

  • Energy splits: why compressive energy should not be degraded.

  • Length scale l0: how a diffuse crack band replaces a sharp crack.

Energy splits in bending

The schematic below compares two common choices for the degraded elastic energy. The left panel shows the problem with degrading the full energy: damage can appear in a compressed region. The right panel shows a spectral split, where only the tensile part of the strain energy is degraded and the crack follows the tensile notch tip.

Schematic comparison of isotropic and spectral energy splits under bending

Conceptual comparison of phase-field damage under bending for two elastic energy formulations. Isotropic degradation lets the total strain energy drive damage and can therefore accumulate damage in a compressive zone. A spectral split restricts the driving force to the tensile strain contribution. This diagram is an idealized explanation and is not a direct PhAST simulation result.

AT1 and AT2 at a glance

The local dissipation density is the other common source of confusion. The table below summarizes the public choices in PhAST.

Model

Local term w(d)

c_w

Visual interpretation

AT1

d

8/3

Damage stays at zero until a threshold is reached.

AT2

d^2

1/2

Damage can begin smoothly once the crack driving field is nonzero.
AT1 and AT2 local crack-density terms

The plot compares only the local dissipation term. The full crack-surface density also includes the gradient penalty and the normalization constant c_w.

What the regularisation length scale does

The length scale l0 controls how wide the diffuse crack band becomes. Smaller values give a narrower band and a sharper transition, but they require finer meshes. In practice, the mesh spacing near the crack should be fine enough to resolve the diffuse band.

As a rule of thumb, the public fracture examples are written so that users can see the relationship between l0, mesh density, and the visible damage field directly in the result plots.

Where to go next