Materials
Crack Branching: Why a Fast Crack Splits Into a Tree
Crack Branching is the abrupt splitting of a single, fast-running crack into two or more cracks — and then into a branching tree — once its speed climbs past a critical fraction of the material's Rayleigh wave speed. It is why a smashed windscreen webs into a network of forks rather than one clean line, and why a fast crack in glass or acrylic never reaches the sound speed it appears to be chasing. Instead of accelerating further, the surplus elastic energy is spent tearing new surfaces sideways.
- Branching onset (glass)≈1500 m/s (~0.5 c_R)
- Rayleigh speed, soda-lime glass≈3100 m/s
- Microbranch instabilityv_c ≈ 0.36 c_R
- Yoffe stress bifurcationθ ≈ ±60° above ~0.6 c_R
- PMMA critical velocity≈340 m/s
- First explainedYoffe 1951; instability, Fineberg 1991
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A Crack That Runs Out of Room to Accelerate
Picture a brittle plate — glass or PMMA (acrylic) — pulled in tension with a small flaw at one edge. Once the flaw runs, its speed is set by an energy balance: the dynamic energy release rate G(v), the elastic energy flowing to the tip per unit of new crack area, must equal the fracture energy Γ needed to make surface. Freund's dynamic-fracture result makes the speed dependence explicit:
G(v) ≈ G_static · (1 − v/c_R)
Here c_R is the Rayleigh (surface) wave speed — about 3100 m/s in soda-lime glass, ~930 m/s in PMMA. The factor (1 − v/c_R) is the trap: as the crack nears c_R, the energy actually reaching the tip falls toward zero. The tip goes 'stiff' — you keep pumping in strain energy, but it can no longer be spent on going faster. c_R is a hard ceiling because the crack-tip stress field is carried by elastic waves, and no signal can outrun the surface wave. So the surplus energy has to go somewhere else.
Yoffe's Clue: the Stress Peak Steps Off the Crack Line
Where that surplus goes was first hinted at by Elizabeth Yoffe in 1951. She solved for the field around a crack of fixed length gliding at constant speed and tracked the circumferential (hoop) stress σ_θθ around the tip — the stress that would open a new crack at angle θ. At low speed, σ_θθ peaks straight ahead (θ = 0): the crack goes straight. Above roughly v ≈ 0.6 c_R, that single forward peak splits into two off-axis maxima near θ ≈ ±60°.
The physics: the moving field is contracted and spread sideways, so the maximum tension is no longer dead ahead. The tip now 'prefers' to open at an angle — the seed of branching. Yoffe's model over-predicts both the onset speed (real branching starts nearer 0.4–0.5 c_R) and the angle (real branches are far shallower), because it fixes crack length and ignores the extra surface energy of side cracks. But its message survives: a fast tip has two off-axis directions competing with straight-ahead.
The Real Trigger: a Dynamic Instability
The sharper picture came from Fineberg, Gross, Marder and Swinney (1991–92), who measured PMMA crack speed at ~MHz using a grid of fine conductive lines the crack severs one by one. Below a critical velocity v_c ≈ 0.36 c_R (~340 m/s in PMMA) the crack runs smoothly on a mirror-flat plane. Cross v_c and it goes unstable: the speed begins to oscillate, the surface roughens, and tiny microbranches — side cracks a fraction of a millimetre long that start, then die — sprout behind the tip.
This is the escape valve. Two (or many) slower cracks create more total surface per unit of advance than one fast crack, so the excess G(v) is burned as new area rather than speed. Measured fracture energy rises steeply above v_c — often threefold or more — precisely because of all the hidden microbranch area. When one microbranch grows enough to survive, the front forks macroscopically; each fork can fork again → the tree.
Reading the Tree on the Fracture Surface
Every stage leaves a fingerprint, and forensic engineers read it backwards to the fracture origin. Radiating out from the source you find a smooth mirror, then a matte mist, then coarse hackle ridges, then full branching (see the table above). The mirror–mist boundary marks v_c, and the mirror radius r_m encodes the driving stress through the mirror constant:
σ_f · √r_m ≈ A_m
with A_m ≈ 2 MPa·√m for soda-lime glass. Because branches always open away from the source, tracing the forks and river lines back through the mist to the mirror centre pinpoints the initiation site — and even the stress at failure. This is the basis of ASTM C1322 fractography for glass and ceramics.
A Worked Example: How Fast, How Soon
Take soda-lime glass: E ≈ 70 GPa, ν ≈ 0.22, ρ ≈ 2500 kg/m³. The shear-wave speed is c_s = √(μ/ρ) with μ = E ÷ [2(1+ν)] ≈ 28.7 GPa, giving c_s ≈ 3390 m/s, so c_R ≈ 0.92·c_s ≈ 3120 m/s. Microbranching should switch on near 0.36·c_R ≈ 1120 m/s, and in practice branching in glass is seen around 1500 m/s — only about 0.48 c_R, confirming the crack never gets near the Rayleigh ceiling.
Timescale: a crack crossing a 100 mm pane at ~1500 m/s takes ≈ 67 µs, so the whole mirror → mist → hackle → branch cascade unfolds in tens of microseconds. That is why it took 10⁶-frame/s spark cameras and resistive-grid probes to catch it at all.
Where Engineers Design For (and Against) It
The most familiar case is thermally toughened (tempered) glass. Rapid surface cooling locks in ~80–150 MPa of surface compression balanced by interior tension — a large reservoir of stored elastic energy. When a flaw finally reaches the tension zone, that energy drives runaway branching, dicing the pane into thousands of small, blunt cubes instead of long shards. High stored energy is deliberately traded for extreme branching density — a safety feature mandated for car side windows and shower doors.
The reverse problem is controlled fracture: cutting glass, silicon wafers or display panels means keeping a single crack below v_c by scoring a shallow guide flaw and pulling gently, so the crack stays in the mirror regime and never branches. And in pressure vessels and pipelines, 'leak-before-break' design and crack arrestors exist to stop a fast, branching brittle crack from running the length of a structure.
| Zone | Crack speed (of c_R) | Surface texture | What is happening |
|---|---|---|---|
| Mirror | < ~0.3 c_R | Optically smooth | One planar crack; energy makes a single surface |
| Mist | ~0.36 c_R (v_c) | Faint matte stipple | Microbranching instability begins; speed oscillates |
| Hackle | ~0.4–0.5 c_R | Coarse ridges, river lines | Dense microbranches; fracture energy climbs steeply |
| Macro-branch | ≳ 0.5 c_R | Crack forks visibly | Surplus energy splits the front into 2+ cracks → tree |
| Ceiling | never > ~0.6–0.65 c_R | — | g(v)→0 near c_R; the crack cannot reach the Rayleigh speed |
Frequently asked questions
Why can't a crack just keep accelerating up to the Rayleigh wave speed?
Because the energy actually delivered to the tip scales like g(v) ≈ (1 − v/c_R), which falls to zero as v approaches c_R. Near the ceiling the tip becomes effectively rigid to further acceleration, so pumping in more energy no longer buys speed — it feeds new surface (roughening, microbranches, branches) instead. Mode-I (opening) cracks therefore stall and split at ~0.5–0.65 c_R and never reach c_R.
What angle do the branches make?
Macroscopic branches in glass and PMMA open at only about ±10–15° from the parent path (~30° included angle), and microbranches start even shallower, a few degrees. Yoffe's continuum prediction of ±60° marks where the stress field first favours off-axis opening, but it overstates the real geometry because it ignores the extra energy cost of the side surfaces.
Does branching need a flaw or impurity to start?
No. Branching is an intrinsic dynamic instability of a fast crack in an otherwise perfect, homogeneous material — the microbranching instability appears above v_c even in pristine PMMA. Flaws and inhomogeneities only bias where a branch happens to nucleate; they are not the cause of the tree.
Why does tempered glass shatter into tiny cubes but ordinary window glass into long shards?
Tempered glass stores a large amount of residual elastic energy (surface compression of order 100 MPa over interior tension). When it fails, that energy far exceeds what one crack can absorb, so it drives intense, repeated branching that dices the pane. Annealed window glass holds little stored energy, so its crack branches sparsely and leaves long, dangerous shards.
Can a crack ever exceed the Rayleigh speed?
In tensile (mode-I) fracture, no — c_R is the limit. But shear-dominated (mode-II) cracks, such as some earthquake ruptures and weak-interface debonds, can go 'intersonic' or supershear, running between the shear speed c_s and the longitudinal speed c_l. That is a shear phenomenon, distinct from opening-mode branching.
How is crack branching used to find where a fracture started?
Branches and river lines always diverge away from the origin, so an investigator follows the forks backward, converging through the hackle and mist zones to the smooth mirror at the initiation site. The mirror radius even gives the stress at failure via σ_f·√r_m ≈ A_m. This branch-tracing is standard fractography (ASTM C1322) for glass and ceramic failure analysis.