Fast cracks and branching in a dynamic phase-field model

A fast crack carries kinetic energy and elastic waves, and above a limiting speed a single straight crack is no longer a steady solution. In Karma and Lobkovsky’s phase-field model, raising the load turns a straight crack into a snake-like path and then into chaotic branching. A lattice version of their model, rerun at their nominal settings, reproduces the straight crack and a wandering path but not the tip splitting; in a strip half as wide it does split into two branches.

Sandbox Physics · 2026-10-29

What experiments show

Bouchbinder, Fineberg and Marder summarize the measurements. A single crack accelerates toward a limiting speed set by the Rayleigh surface-wave speed. Its motion follows the equation of motion of linear elastic fracture mechanics up to about 0.4 of that speed in PMMA and in glass, whose Rayleigh speeds are about 930 and 3300 metres per second.

Near that speed the single crack becomes unstable to micro-branching: short side branches sprout and die, the speed oscillates and the fracture surface roughens. The review states that no first-principles theory of this onset exists, and that numerical studies with cohesive zones found micro-branching that depended on mesh size.

A phase-field model with inertia

Karma and Lobkovsky write the energy of an antiplane displacement and a phase field that is one in intact material and zero in broken material. The elastic term is softened by an interpolating function and shifted by a critical strain, so broken material is favoured once the strain exceeds it. The displacement obeys the wave equation with the softened stress; the phase field relaxes toward lower energy at a finite rate.

Scaling lengths by the process-zone size and time by that size over the shear-wave speed leaves two numbers: one sets the scaled surface energy, the other compares the relaxation time with the time for a wave to cross the process zone. A strip of half-width W, held by grips displaced in opposite directions, delivers the same energy release rate at every crack length, as in the gripped strip of chapter three.

E=\int\!\Big[\frac{\rho}{2}\dot u^2+\frac{\kappa}{2}|\nabla\phi|^2+h\,f(\phi)+\frac{\mu}{2}g(\phi)\big(|\nabla u|^2-\epsilon_c^2\big)\Big]dx\,dy,\quad f=16\phi^2(1-\phi)^2,\quad g=4\phi^3-3\phi^4,\quad G=\frac{\mu\Delta^2}{W}

Worked example

Check the energy flow before reading the paths

Equation (5) of their arXiv version says that, for a crack running at steady speed, the energy release rate in excess of the fracture energy, times the speed, equals the rate at which the phase field dissipates energy. The lattice model reproduces this balance to 3.1 percent. Over a whole run, the work done equals stored, kinetic and dissipated energy to within 2.6, 1.5 and 0.8 percent as the time step is halved twice.

One difference matters for every later comparison. On a lattice with spacing one half of the process-zone size, the energy needed to break a straight line of sites is 1.262 times the continuum value; at spacing 0.4 it is 1.202. A seed crack loaded at 0.9 times the lattice value heals, while at 1.1 and 1.3 times it runs at 0.17 and 0.25 of the shear-wave speed. Loads quoted against the continuum fracture energy are therefore lower when measured against what the lattice actually charges.

v\,(G-G_c)=\chi^{-1}\int\dot\phi^{\,2}\,dx\,dy,\qquad \frac{G_c^{\rm lattice}}{G_c}=1.262\ \ (\Delta x=0.5\,\xi)

Rerun Figure 3 at its nominal settings

Figure 3 of Karma and Lobkovsky uses a strip of half-width 30 process-zone sizes, the relaxation ratio 2 and loads of 1.56, 1.86 and 2.90 times the continuum fracture energy. Their panels show transient branching with a straight crack, a weak periodic “snake”, and chaotic branching. They report the snake’s period as close to four half-widths over the wave speed.

The lattice rerun used the same half-width, ratio and loads, a spacing of 0.4, a straight seed crack of 30 and a small random threshold variation to break symmetry. At 1.56 the crack stayed straight, with a lateral spread of 0.11. At 1.86 a single crack wandered with a spread of 1.26. At 2.90 it wandered more, with a spread of 2.57, and did not split. The rise of path disorder with load agrees qualitatively; the branching does not.

Three differences are known. The lattice charges 1.20 times the continuum fracture energy at this spacing, so the loads are effectively 1.30, 1.55 and 2.41. Karma and Lobkovsky started from relaxed stationary cracks whose length the paper does not state. And a square lattice has preferred directions that a continuum does not. In the Lab’s narrower strip, half-width 15 at spacing 0.5, the load 2.90 does split the tip into two symmetric branches, and the 1.56 crack runs straight. Which of these differences controls branching in the lattice is open.

Predict, change, explain

In the dynamic crack branching Lab, predict how the path changes when the load goes from the steady-crack preset to the symmetric-branches preset.

Drag the load from about 1.5 toward 3 and watch the speed readout and the path behind the tip.

Relate the change to the energy that the tip has to dissipate at a given speed, and say which features of the path you would compare with Figure 3 of Karma and Lobkovsky.

The fracture Labs run simplified two-dimensional models in scaled units. Each Lab states its model, units and limits in its notes and can export the record behind its readouts.

Try explaining it yourself

1. A crack in the Lab runs at 0.2 of the shear-wave speed with an excess drive of 0.1 in scaled units. What dissipation rate does the energy balance predict?

Read an explained answer

Speed times excess drive, 0.2 times 0.1, which is 0.02 in the same units. The phase-field dissipation integral should give the same value within the model’s tolerance.

2. Why is a load of 1.1 times the continuum fracture energy too small for a crack to run on the lattice with spacing 0.5?

Read an explained answer

The lattice charges 1.262 times the continuum value. A load of 1.1 is only about 0.87 of what the lattice needs, so the seed crack heals.

3. Karma and Lobkovsky relate the snake period to four half-widths over the wave speed. What period would you expect for half-width 30 in scaled units?

Read an explained answer

Four times 30 divided by one, which is 120 scaled time units.

Sources to read alongside

Bouchbinder, Fineberg & Marder, Dynamics of Simple Cracks, arXiv:0911.0173; Annual Review of Condensed Matter Physics 1, 371–395 (2010)

Section 1, equations (1) to (3), defines the Griffith point of a rigidly gripped strip and the surface energy per unit area. Section 2 states the energy balance, the inverse-square-root stress near the tip, the relation between energy release rate and stress intensity, the principle of local symmetry and the assumptions of linear elastic fracture mechanics. It also reports the micro-branching onset near 0.4 of the Rayleigh speed in PMMA and glass. Section 3 discusses small scales at the tip, including cohesive zones and phase-field models. The course quotes these statements; it does not use the review’s data.

Karma & Lobkovsky, Unsteady crack motion and branching in a phase-field model of brittle fracture, Physical Review Letters 92, 245510 (2004); arXiv:cond-mat/0401056

Equations (1)–(7) of the arXiv version define the antiplane phase-field model, its scaled units and the energy flow into the tip; equation (5) is the steady-state energy balance. Figure 3 shows a straight crack with transient branching, a snake-like path and chaotic branching at three loads. The course reruns those nominal settings on a lattice discretization and compares only the path shapes.