Why a crack needs energy to grow

A crack grows when the elastic energy released by a small extension at least pays for the two new surfaces. Griffith’s energy balance explains why long cracks fail at low stress: the released energy grows with crack length, while the cost of each new piece of surface stays the same.

Sandbox Physics · 2026-10-29

A sharp tip piles up stress

Drill a circular hole in a wide plate and pull the plate. At the two points of the hole edge that face sideways to the load, the stress is three times the remote stress. A slot that is long across the load and narrow along it concentrates stress more strongly, and an ideal crack with a perfectly sharp tip makes the elastic stress grow without limit as the tip is approached.

Taken literally, that would mean a cracked body breaks under any load at all. Real tips are blunted at some small scale by atomic spacing, plastic flow or damage. A rule based on peak stress then depends on a tip radius that is rarely known. Griffith avoided the tip details by counting energy instead.

Compare the energy released with the energy spent

Let the crack grow by a small area while the load is held steady. The body becomes more compliant, so its potential energy drops. The material spends energy to separate the two new faces and to deform the small region at the tip. Growth becomes possible when the energy released per unit of new crack area equals the energy spent per unit area.

The released energy per unit area is called the energy release rate. The energy spent per unit area is the fracture energy. In an ideally brittle solid the fracture energy is twice the surface energy, because one cut makes two faces. In real materials it also includes plastic work and other losses near the tip, so it is obtained from fracture tests.

G=-\frac{\partial\Pi}{\partial A}\ge G_c,\qquad G_c^{\rm ideal}=2\gamma

Worked example

Derive the failure stress of a cracked plate

Take a large plate in plane stress with a straight through crack of total length twice the half-length, perpendicular to a remote tensile stress. The elastic solution gives the energy released by opening this crack, per unit plate thickness. It grows as the square of the stress and the square of the half-length, and falls with Young’s modulus.

The crack has two tips, so a small extension of the half-length adds twice that amount of crack length. Divide the change of released energy by the change of crack length to get the energy release rate per tip. Set it equal to the fracture energy and solve for the stress. The failure stress falls as one over the square root of the crack length: a crack four times longer fails at half the stress.

Put in round illustrative numbers. With a Young’s modulus of 70 gigapascals, a fracture energy of 8 joules per square metre and a half-length of 1 micrometre, the failure stress is about 420 megapascals. A half-length of 100 micrometres brings it down to about 42 megapascals. These are round numbers chosen to show the scaling.

Griffith’s 1921 paper printed this breaking stress with Poisson’s ratio under the square root. A note at the end of the same paper says the strain-energy calculation behind it needs correction. The formula above is the corrected form, as later papers state it.

\Delta U=\frac{\pi\sigma^2a^2}{E},\qquad G=\frac{1}{2}\frac{d\,\Delta U}{da}=\frac{\pi\sigma^2a}{E},\qquad \sigma_f=\sqrt{\frac{E\,G_c}{\pi a}}

What the energy balance leaves open

The balance says whether a crack can grow. It does not say which way the crack turns, or how fast it runs once it starts. A crack in a uniform plate under pure tension stays straight by symmetry. Under mixed loading a direction rule is needed. A common one, the principle of local symmetry, turns the tip until the shear at the tip vanishes.

The balance also assumes that the region where the material actually breaks is tiny compared with the crack and the specimen. Bouchbinder, Fineberg and Marder list this small-scale assumption among the conditions under which linear elastic fracture mechanics works. When the broken region becomes comparable to the crack, the material’s strength enters as a second scale. Chapter five builds that scale into a phase-field model.

An experiment to think through

Predict, change, explain

Open the Griffith crack growth Lab with the plate preset. Before the load ramps up, predict whether the crack stops after it starts to grow under a steady remote stress.
Drag the crack tip to make the starting crack longer, then let the load ramp again.
Compare the stress at which growth begins with the inverse-square-root rule, and use the bar chart to say where the released energy went.

Grow a crack and track where the work goes See stress concentrate around a hole Measure modulus and yield stress

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 plate fails at 200 megapascals with a crack of half-length 0.1 millimetre. At what stress does an identical plate fail if the half-length is 1.6 millimetres?

Read an explained answer

The failure stress scales as one over the square root of crack length. The crack is 16 times longer, so the stress is 200 divided by 4, which is 50 megapascals.

2. Why does doubling the plate thickness leave the Griffith failure stress unchanged?

Read an explained answer

The released energy and the surface cost both scale with thickness, so their balance per unit thickness is the same. Thickness matters only through the change from plane stress to plane strain and through the size of the broken region at the tip.

3. In the worked example, the fracture energy is doubled. By what factor does the failure stress change?

Read an explained answer

The failure stress scales as the square root of the fracture energy, so it rises by the square root of 2, about 1.41 times.

\frac{\sigma_f'}{\sigma_f}=\sqrt{\frac{2G_c}{G_c}}=\sqrt2\approx1.41

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.
Sargado, Welch & Lüthje, Approximate formulas for the energy release rate of a crack perpendicular to a material interface, arXiv:2210.06315v1 (2022)
Equations (1) to (3) restate the energy released by a centre crack in a large plate and its energy release rate, with the effective modulus for plane stress and plane strain. Equation (29) quotes the Tada, Paris and Irwin secant correction for a centre-cracked strip. The course uses it as the finite-width reference for the centre-crack benchmark. The 1973 handbook itself was not consulted.
Tanné, Li, Bourdin, Marigo & Maurini, Crack nucleation in variational phase-field models of brittle fracture, Journal of the Mechanics and Physics of Solids 110, 80–99 (2018); preprint hal-01568702
Section 2 gives the AT2 energy (equation 2 with the AT2 choice), the characteristic length and the relation between critical stress intensity factor and fracture energy (equation 5), and the uniform-bar strength (Table 1). Section 3.1, equations (7) to (11), gives the singular field at a V-notch; when the notch closes into a crack, its amplitude is the mode-I stress intensity factor. Section 4 treats the regularization length as a material length through size effects. Benchmark B4 checks the browser model’s uniform-bar strength against Table 1.
Griffith, The phenomena of rupture and flow in solids, Philosophical Transactions of the Royal Society A 221, 163–198 (1921)
Section 2 states the energy criterion: a crack spreads if the potential energy, including the surface energy of the new faces, can keep falling. Section 3 computes the energy of a straight crack in a plate. The printed plane-stress breaking stress has Poisson’s ratio under the square root, and a note at the end of the paper says this strain-energy calculation needs correction. The course uses the corrected form, which Sargado and coauthors restate in their equations (1) to (3).