In the dynamic lattice model, a crack driven at 1.86 times the continuum fracture energy runs perfectly straight when every site is identical. Give the breaking strain a frozen random variation of 5 percent and the path wanders by about half a process-zone size; at 15 percent it wanders by about 1.6. Branching stays rare at this load. These are statistics of one two-dimensional model, from six seeds per disorder strength.
Sandbox Physics ·
Put the disorder into the breaking strain
Each lattice site gets its own critical strain: the uniform value times one plus a disorder strength times a random number. The random field is Gaussian with unit variance, smoothed over a chosen number of sites, and frozen: it is fixed before the crack starts and does not change in time. Because the fracture energy grows with the critical strain, weak and tough patches follow.
Each random field is generated from a seed and from the global position of the site, so a strip that slides forward with the crack creates new columns that agree with any earlier view of the same sample. That makes runs repeatable: the same seed gives the same path.
ϵc(x)=ϵc[1+ηζ(x)],⟨ζ⟩=0,⟨ζ2⟩=1
Worked example
Measure the path, not a single picture
For every column the crack has crossed, find the centre of the broken sites. The lateral spread is the standard deviation of that centre along the path, in process-zone sizes. The branch fraction is the share of crossed columns that hold more than one separate broken run.
The ensemble uses a strip of half-width 15 at spacing 0.5, the relaxation ratio 2, a load of 1.86 times the continuum fracture energy, disorder correlated over two sites and seeds 11 to 16. Without disorder all six runs are identical and straight. At disorder strength 0.05 the mean spread is 0.52, with single runs from 0.28 to 0.87. At 0.15 the mean spread is 1.62, and branching appears in 0.15 percent of columns. On average the cracks advanced 72 to 74 process-zone sizes at every disorder strength.
The spread between seeds is as large as the mean at small disorder, so a single run says little. Six seeds are enough for a mean; describing the shape of the distribution would need many more.
sy=N1j=1∑N(yˉj−⟨yˉ⟩)2,⟨yˉ⟩=N1j=1∑Nyˉj
What the numbers show and what they leave open
The ensemble shows that, in this model, frozen disorder moves the path sideways by an amount that grows with disorder strength, while the crack keeps nearly the same speed and length. It does not show how the spread grows with distance travelled, whether it saturates, or how it depends on the strip width. Each of those needs longer strips and more seeds.
Real fracture surfaces are three-dimensional. Bouchbinder, Fineberg and Marder describe how micro-branching marks fracture surfaces in fast cracks, with dynamics that leave linear elastic fracture mechanics behind. A two-dimensional antiplane model has no crack front that can bend, so its path statistics cannot be compared directly with measured surface roughness. A comparison with measured roughness exponents would need a three-dimensional model and a review of that literature, both left for later work.
An experiment to think through
Predict, change, explain
In the dynamic crack branching Lab, choose the rough-path preset and predict whether three successive runs will follow the same line.
Let the Lab restart with new seeds and compare the overlaid paths, then set the disorder to zero.
Use the spread readout to say how much of the path difference comes from the disorder strength and how much from the particular seed.
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. Two runs with the same seed and settings give different paths. What does that tell you?
Read an explained answer
Something other than the frozen disorder differs, such as the time step or the initial seed crack. With the same seed, settings and code, the run is deterministic and the paths must be identical.
2. At disorder strength 0.05 the six spreads range from 0.28 to 0.87. Is a mean of 0.52 a precise number?
Read an explained answer
Not very. The six values have a sample standard deviation of about 0.22, so the standard error of the mean is about 0.22 over the square root of six, roughly 0.09. More seeds would narrow it.
3. Why can a two-dimensional antiplane model not predict the roughness exponent of a real fracture surface?
Read an explained answer
It has no crack front. Surface roughness in a solid comes from a front that can bend and leave different heights along its length; a two-dimensional model only records one line.
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.
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.
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