Fracture mechanics · Phase-field model

Phase-Field Fracture Model of a Growing Crack

Can a smooth damage field decide on its own where a crack goes? Yes, within its model. A phase-field crack is a narrow damage band that grows wherever doing so lowers stored energy plus surface cost, so it can bend toward a hole or turn under shear without a path rule. In this notched square, the sharp-crack onset load falls between the load peak and the load at which the band has grown two widths.

Pull or shear a notched square and watch a damage band grow, turn toward a hole or heal without memory. Compare its onset with the sharp-crack prediction.

Model assumptions

A two-dimensional quasi-static AT2 phase-field model in plane strain, Poisson ratio 0.3, on a 40 by 40 element mesh. The tensile-energy split keeps compression from breaking the material; with it, the damage step is not exactly variational. The page runs three alternating solves per load step, so a running crack spreads over several steps; the recorded benchmarks iterate each step to convergence. Band width, mesh and toughness are dimensionless model choices.

Grip displacement over side

0.0367

Grip force, model units

0.0247

Crack tip advance over side

0

New surface energy, share of work

0\%

The gold curve is the simulated grip force. The dashed line is the sharp-crack onset computed separately for the same notch.

Save the run and read the model

The JSON lists the settings, model version, sharp-crack onset and every load step with force, energies and crack tip advance.

The strength formula and the damaged-notch comparison follow Tanné and coworkers, 2018. Read the source

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

Start with the essentials

Focus question: Can a smooth damage field decide on its own where a crack goes?
One-sentence intuition: The band grows where stored energy plus surface cost can drop. The sharp-crack onset prediction lies between the load peak and two band widths of growth.

Core mathematical model

AT2 regularized energy

E_{\ell}(u,d)=\int_{\Omega}g(d)\,\psi(\varepsilon(u))\,dx+\frac{G_c}{2}\int_{\Omega}\left(\frac{d^{2}}{\ell}+\ell\,|\nabla d|^{2}\right)dx

The first integral is stored elastic energy, weakened by the damage. The second is the surface cost of the band. For a straight band it equals the toughness times the crack length when the mesh is fine.

Degradation and memory

g(d)=(1-d)^{2}+k,\qquad \mathcal H(x,t)=\max_{s\le t}\psi^{+}(x,s)

A small residual stiffness keeps the equations solvable in broken regions. The history field drives damage with the largest tensile energy seen so far, so damage cannot heal when the load drops.

Homogeneous strength of the AT2 model

\sigma_c=\frac{3}{16}\sqrt{\frac{3G_cE'}{\ell}}

Without a notch, a uniform bar reaches this peak stress. A narrower band means a stronger material, so the length scale is also a strength parameter, as Tanné and coworkers emphasize.

Common difficulties

The band width is not a crack opening

Typical misconceptionThe damage band shows how wide the crack has opened.

Better mental modelThe band width is set by the length scale and the mesh, not by the load. The opening is the jump in displacement across the band.

What this model leaves out

Typical misconceptionThe onset, path and load curve here are mesh-independent predictions.

Better mental modelA two-dimensional quasi-static AT2 phase-field model in plane strain, Poisson ratio 0.3, on a 40 by 40 element mesh. The tensile-energy split keeps compression from breaking the material; with it, the damage step is not exactly variational. The page runs three alternating solves per load step, so a running crack spreads over several steps; the recorded benchmarks iterate each step to convergence. Band width, mesh and toughness are dimensionless model choices.

Run the experiment

01
Pull a notched square
Choose Tension grows a straight crack and watch the force against grip displacement in the data row.
What to observe: Force rises, peaks, then falls as the band runs across. The dashed line marks the sharp-crack onset.
02
Change the loading
Choose Shear turns the crack, then Hole pulls the crack.
What to observe: No path rule is given. Under shear the band turns down toward the corner; near a hole it bends toward the hole.
03
Remove the memory
Choose Without memory the crack heals. The load rises and then falls back.
What to observe: Without the history field the band shrinks back as the load drops, which no real crack does.

Phase-Field Fracture Model of a Growing Crack

Let the damage field find the crack path

Core mathematical model

AT2 regularized energy

Degradation and memory

Homogeneous strength of the AT2 model

Common difficulties

The band width is not a crack opening

What this model leaves out

Run the experiment

Pull a notched square

Change the loading

Remove the memory

Related experiments and articles

Fracture mechanics from Griffith’s energy balance to branching cracks