When a growing crack stays under control

Once a crack has started, it stays under control if the energy release rate falls as the crack lengthens, so that each step needs a little more loading. If the energy release rate rises instead, every step makes the next one easier and the crack runs away. Whether it rises depends on the specimen and on whether the load or the grip position is held.

Sandbox Physics · 2026-10-29

Compare two slopes

At the moment of initiation, the energy release rate equals the fracture energy. Let the crack advance a little with the loading device held as it is. If the energy release rate drops below the fracture energy, the crack stops and waits for more load. If it stays above, the crack keeps going.

For a material whose fracture energy does not depend on how far the crack has grown, stability therefore depends only on the sign of the slope of the energy release rate along the crack. Materials whose resistance rises with growth are more forgiving, but the comparison is the same.

\left.\frac{\partial G}{\partial a}\right|_{P}<\frac{dG_c}{da}\qquad\qquad\left.\frac{\partial G}{\partial a}\right|_{\Delta}<\frac{dG_c}{da}

Worked example

The same beam tears stably under grips and runs away under a weight

Chapter two found that a double cantilever beam under a fixed load has an energy release rate proportional to the square of the crack length. Its slope is positive, so a hanging weight that starts the crack will pull it through the whole beam.

Hold the total opening fixed instead. The load equals the opening divided by the compliance, and the compliance grows as the cube of the crack length. Substituting gives an energy release rate that falls as the fourth power of the crack length. Each advance now needs a wider opening: the crack length at which growth stops grows only as the square root of the opening.

P=\frac{\Delta}{C}\ \Rightarrow\ G=\frac{3}{16}\frac{E\,\Delta^2h^3}{a^4},\qquad a_{\rm arrest}=\left(\frac{3E\,\Delta^2h^3}{16\,G_c}\right)^{1/4}\propto\Delta^{1/2}

A gripped strip tears at one steady opening

Grip a long strip along its top and bottom edges and pull the grips apart. Far ahead of a long crack the material is stretched uniformly; far behind it the two halves carry nothing. When the crack advances by one unit, a slice of uniformly stretched material turns into an unloaded slice. The energy release rate is the stored energy of that slice, independent of crack length.

This is the geometry Bouchbinder, Fineberg and Marder use to define the Griffith point: an ideally brittle strip carries load until the grip opening reaches one critical value, then separates. For a strip of total height twice the half-height, in plane strain, the stretched material ahead has the stiffness of uniaxial strain. The Lab’s finite-element solution stays within 0.1 percent of this plateau for cracks between two and six half-heights long. Karma and Lobkovsky load their fast-crack simulations the same way, with the strip sheared out of plane; the plateau is then the shear modulus times the grip displacement squared, divided by the half-width.

G=\frac{(\lambda+2\mu)\,\Delta^2}{h}

Where the extra energy goes when a crack runs away

In a quasi-static calculation, a crack that runs away releases more energy than the new surfaces need. The Lab reports that difference as excess energy. In a real body it becomes kinetic energy of the moving material and elastic waves, which chapter seven follows.

The Lab shows the three cases side by side. A centre crack in a plate under steady stress starts within 2 percent of the predicted load and then runs; when it stops at the end of the modelled plate, about 31 percent of the work done by the load is excess, about 6 percent paid for surface and the rest is still stored. A long crack in the gripped strip turns 95 percent of the work into surface and leaves under 1 percent as excess. An edge notch between rigid grips grows gradually with the opening and leaves under 1 percent too.

An experiment to think through

Predict, change, explain

In the Griffith crack growth Lab, predict which preset leaves the largest excess energy: the plate, the gripped strip or the edge notch.
Run each preset to the end of its load ramp and read the bars that split the work into stored energy, surface energy and excess.
Connect each result to the slope of the energy release rate curve shown for that specimen.

Grow a crack and track where the work goes

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 double cantilever beam is held at a fixed opening, and the crack has stopped at a length of 40 millimetres. The opening is then increased by 21 percent. Where does the crack stop?

Read an explained answer

The arrest length grows as the square root of the opening. The square root of 1.21 is 1.1, so the crack stops at 44 millimetres.

2. The gripped strip is made twice as tall while the grip displacement stays the same. What happens to the steady energy release rate?

Read an explained answer

It halves, because the strain ahead of the crack halves and the energy density falls by four, while the stretched slice is twice as tall.

3. Why does a centre crack in a plate under steady stress run away even after the stress stops rising?

Read an explained answer

Its energy release rate grows with crack length at fixed stress. Once it equals the fracture energy, every advance raises it further above that value.

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.