Fracture mechanics · Energy release rate

Griffith Criterion for Crack Growth

When does a crack start to grow, and does it then stop or run away? A crack grows when the elastic energy freed by a small advance at least pays for the new surface. Under a fixed stress the payment grows with crack length, so a plate crack runs away; between fixed grips the payment stays level or falls, so the crack tears steadily or stops.

Load a cracked plate, a gripped strip or an edge notch. The crack grows once released elastic energy pays for new surface. Export the energy record.

Crack length over width

0.3

Load over predicted onset

0.91

Energy release rate over toughness

0.828

Excess release, share of work

0\%

Blue is the simulated energy release rate at this load; the red dashed line is the toughness. Bars split the work done by the load.

Save the run and read the model

The JSON lists the specimen, mesh, units, model version, predicted onset and every recorded load step with the four ledger terms.

The plate case is checked against the secant formula for a center crack quoted by Sargado and coworkers, and the gripped strip against its exact steady energy release rate. Read the source

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

Start with the essentials

Focus question: When does a crack start to grow, and does it then stop or run away?
One-sentence intuition: Growth starts when the energy release rate reaches the toughness. It stays stable only if a longer crack releases less energy at the same load.

Core mathematical model

Griffith criterion

G=-\frac{\partial\Pi}{\partial a}\ge G_c

The potential energy of the loaded body falls by $G$ for each unit of crack advance per unit thickness. The crack can grow when that release reaches the toughness $G_c$ , the energy needed per unit of new crack area.

The same rate from the compliance

G=\frac{P^{2}}{2}\frac{dC}{da}

For one crack tip and unit thickness, the energy release rate follows from how fast the compliance grows with crack length. It holds both at fixed load and at fixed grips. The Lab checks it against a crack-closure estimate that uses only nodal values at the tip.

Two reference cases

G_{\text{plate}}=\frac{\pi\sigma^{2}a}{E}F\!\left(\frac{a}{w}\right)^{2},\qquad G_{\text{strip}}=\frac{(\lambda+2\mu)\Delta^{2}}{h}

A center crack in a plate under remote stress releases more energy the longer it gets. A long crack between rigid grips, each moved by the displacement shown, releases a steady amount that no longer depends on its length.

Where the work goes

W=U+\Gamma+D,\qquad D\ge 0

Work done by the loads goes into stored elastic energy, new crack surface and an excess that a quasi-static model cannot track. That excess appears when a crack jumps ahead at a load above its onset.

Common difficulties

Stress at the tip is not the criterion

Typical misconceptionThe crack grows when the stress at its tip exceeds the strength.

Better mental modelA sharp elastic crack has unbounded stress at its tip at any load. Griffith compares energies instead, which gives a finite onset load.

What this model leaves out

Typical misconceptionThe predicted onset load is what a real specimen will show.

Better mental modelA two-dimensional linear elastic finite-element model with a straight crack on its symmetry plane. Crack-plane nodes are released one at a time once the energy release rate reaches the toughness. Units: Young modulus 1, toughness 1, specimen width 1. The model has no plasticity, process zone, inertia or crack turning.

Run the experiment

01
Watch a plate crack run away
Choose Plate crack runs away and let the load rise. Watch the release curve in the data row as the crack starts to grow.
What to observe: Once the white point reaches the toughness line, the curve keeps rising with length. The crack crosses the plate at the same load and the excess bar grows.
02
Compare a short crack
Choose Short crack needs more stress, or drag the crack tip toward the centre.
What to observe: A shorter crack needs a larger stress before it grows, because it releases less energy per step.
03
Hold the grips
Choose Gripped strip tears steadily, then Edge notch grows under grips. Drag the grips to raise or lower the displacement.
What to observe: Between fixed grips the crack grows over a narrow range of grip displacement and stops when the grips stop. The excess bar stays near zero, while the plate under fixed stress leaves about a third of the work as excess.

Griffith Criterion for Crack Growth

A crack grows when the energy bill is paid

Core mathematical model

Griffith criterion

The same rate from the compliance

Two reference cases

Where the work goes

Common difficulties

Stress at the tip is not the criterion

What this model leaves out

Run the experiment

Watch a plate crack run away

Compare a short crack

Hold the grips

Related experiments and articles

Fracture mechanics from Griffith’s energy balance to branching cracks