Fracture mechanics · Dynamic phase field

Dynamic Crack Branching in a Stretched Strip

Why does a fast crack split or wander instead of running straight? A fast crack gets more energy than one straight path can absorb. Above a threshold load the extra energy feeds side branches, so the crack splits; frozen variations in strength push it sideways, so even a single crack wanders.

Drive an antiplane crack through a stretched strip. Low loads give a steady straight crack, high loads split it, and frozen disorder roughens its path.

Model assumptions

Karma and Lobkovsky mode-III phase-field model in scaled units on a square lattice of spacing 0.5, strip half-width 15, damping 2 and no extra double-well barrier. At this spacing the lattice raises the fracture energy about 1.26 times above the continuum value, so loads are shown against both. Branching thresholds depend on strip width, spacing and seed; this is a qualitative reproduction, not the published figure.

Tip speed over shear wave speed

0.25

Load over lattice toughness

1.24

Broken columns with two branches

0\%

Lateral spread of the path

0

Both plots come from the simulated lattice. Lengths are in units of the process-zone size and time in units of that size over the shear wave speed.

Save the run and read the model

The JSON lists the settings, seed, model version, speed trace, path statistics and every broken site of the run.

The model and the strip geometry follow Karma and Lobkovsky, 2004. Their published branching map used a wider strip and finer lattice than this page. Read the source

Read the fracture course: from Griffith energy to phase-field cracks and branching

Start with the essentials

Focus question: Why does a fast crack split or wander instead of running straight?
One-sentence intuition: Energy balance fixes how much is dissipated per unit length; the tip chooses its shape. Above threshold, two branches dissipate the energy that one straight crack cannot.

Core mathematical model

Karma–Lobkovsky energy

E=\int\left[\frac{\rho}{2}\dot u^{2}+\frac{\kappa}{2}|\nabla\phi|^{2}+h\,f(\phi)+\frac{\mu}{2}\,g(\phi)\left(|\nabla u|^{2}-\epsilon_c^{2}\right)\right]dx\,dy

Antiplane displacement $u$ carries kinetic and elastic energy. The phase field $\phi$ is one in intact material and zero where it is broken. The double well $f$ and the softening function $g$ shape the broken state.

Driving force of the strip

G=\frac{\mu\Delta^{2}}{W},\qquad G_c=2\gamma

Grips at $y=\pm W$ move by $\pm\Delta$ . Far ahead of the tip each unit length stores this energy; far behind, none. The Lab reports loads against the continuum and the lattice toughness.

Speed just above threshold

v\,(G-G_c)=\chi^{-1}\int\left(\frac{\partial\phi}{\partial t}\right)^{2}d\mathbf x

Energy released per unit time by the moving tip equals the rate of dissipation in the process zone. The Lab checks this balance on its steady cracks to within a few percent.

Common difficulties

Branching is not just noise

Typical misconceptionA crack branches only because the material is imperfect.

Better mental modelIn this model a perfectly uniform strip still branches above a threshold load. Disorder changes the details, not the existence of the instability.

What this model leaves out

Typical misconceptionThe thresholds shown here match a real material.

Better mental modelKarma and Lobkovsky mode-III phase-field model in scaled units on a square lattice of spacing 0.5, strip half-width 15, damping 2 and no extra double-well barrier. At this spacing the lattice raises the fracture energy about 1.26 times above the continuum value, so loads are shown against both. Branching thresholds depend on strip width, spacing and seed; this is a qualitative reproduction, not the published figure.

Run the experiment

01
A steady crack
Choose Steady straight crack and watch the speed trace in the data row.
What to observe: The crack accelerates, then runs straight at an almost constant speed.
02
Feed it more energy
Choose Two symmetric branches, or drag a grip outward to raise the load.
What to observe: At high load the tip splits into two branches that run side by side.
03
Add frozen disorder
Choose Disorder roughens the path and compare the path record with earlier runs.
What to observe: Random variations in the critical strain push the crack sideways; the lateral spread grows with disorder strength.

Dynamic Crack Branching in a Stretched Strip

A fast crack has energy to spare

Core mathematical model

Karma–Lobkovsky energy

Driving force of the strip

Speed just above threshold

Common difficulties

Branching is not just noise

What this model leaves out

Run the experiment

A steady crack

Feed it more energy

Add frozen disorder

Related experiments and articles

Fracture mechanics from Griffith’s energy balance to branching cracks