Predict when a notch starts to crack, then test the prediction

Yes, to within an interval that narrows as the damage band gets thinner. For a notched square pulled by rigid grips, a separate sharp-crack solver predicts initiation at a grip opening of 0.0408 of the side length. Phase-field runs with three band widths start to soften before that opening and have grown a crack of two band widths after it; the upper end of the interval moves from 1.23 to 1.165 times the prediction as the band narrows.

Sandbox Physics · 2026-10-29

Fix the test problem before computing anything

The specimen is a unit square in plane strain with Young’s modulus one and Poisson’s ratio 0.3. A straight notch of length one half runs in from the left edge along the mid-height line. The top and bottom edges are held by rigid grips that move apart; the opening is the total relative displacement of the grips. Lengths are in units of the side, energies per unit thickness in units of the modulus times the side squared.

The fracture energy is one thousandth in these units. The phase-field runs use the AT2 model of chapter five with the tensile-energy split and the history field. Damage on the notch is held at one. Rows are graded so that the elements near the mid-plane are half the length scale.

Predict with a sharp crack solved separately

The prediction uses a different program path from the phase-field solver: the sharp-crack finite-element model of chapter two, with 64 elements across the width. Mirror symmetry about the mid-plane turns the square into an edge-notched half held between rigid grips, with half the opening. Its energy release rate per unit squared grip displacement, at the notch length, follows from the slope of compliance.

Use the fracture energy that the phase-field mesh actually charges for a straight band, 1.031 times the nominal value at element size one half the length scale. Without that correction the prediction would be 1.5 percent lower. Refining the sharp-crack mesh to 128 elements changes the prediction by 0.04 percent.

G(\Delta)=G_1(a_0)\left(\frac{\Delta}{2}\right)^2,\qquad \Delta_c=2\sqrt{\frac{G_c^{\rm eff}}{G_1(a_0)}}=0.0408\,L

Measure initiation as an interval

A phase-field run has no single instant at which the crack starts. The reaction force on the grips rises, peaks while damage concentrates ahead of the notch, then falls as a band forms and runs. Record two openings: the peak of the reaction force, and the opening at which the fully damaged region, damage at least 0.9, has advanced two length scales beyond the notch.

The runs used length scales of 0.05, 0.035 and 0.025 on meshes of 40, 57 and 80 elements per side, with load steps of 0.005 times the prediction near initiation. The force peaked at 0.820, 0.785 and 0.775 times the predicted opening. The crack had advanced two length scales at 1.230, 1.195 and 1.165 times. Every interval contains the prediction, and its upper end moves toward it as the band narrows. The lower end does not: the force peak moves slightly further below the prediction.

The characteristic length of chapter five explains why the interval is wide. It equals 0.95, 0.66 and 0.47 times the notch length for the three runs. With a strength length comparable to the notch, damage softens the region ahead of the notch well before a sharp crack would grow, and the sharp-crack picture becomes accurate only as that ratio shrinks.

\frac{\Delta_{\rm peak}}{\Delta_c}\le1\le\frac{\Delta_{2\ell}}{\Delta_c},\qquad \left.\frac{\Delta_{2\ell}}{\Delta_c}\right|_{\ell_{\rm ch}/a_0=0.95,\,0.66,\,0.47}=1.230,\ 1.195,\ 1.165

Worked example

List every source of error and its size

Sharp-crack energy release rate: changing from 64 to 128 elements moves the prediction by 0.04 percent. The same solver matches the handbook value for a centre crack to within 2 percent at that resolution and 1 percent at 128 elements.

Discrete fracture energy: measured on the same mesh and included, a 3.1 percent energy correction that shifts the prediction by 1.5 percent. Load stepping: the openings are resolved to 0.005 of the prediction. Staggered solution: each load step iterates to a damage change below one ten-thousandth, with at most 80 iterations.

Model form: the finite length scale is the dominant term, tens of percent at these settings, which is why the result is reported as an interval. A run at a smaller length scale needs a finer mesh; halving the length scale again would quadruple the element count.

Records, versions and rerunning

The benchmark run saves every load step: opening over prediction, reaction force and crack tip position, for each length scale. The quasi-static model is labelled fr28-quasi-static-2026-10-29. The calculation is deterministic and uses no random seed. The default run covers the two larger length scales in about three minutes; the full run adds the smallest and takes about eight.

Status of the claims: the sharp-crack energy release rate is a numerical result checked against an analytic strip limit and a published handbook formula. The phase-field interval is a numerical result of this model. Whether the lower end converges as the length scale shrinks further is open.

An experiment to think through

Predict, change, explain

In the phase-field crack Lab, start the tension preset and predict where the force peak will sit relative to the dashed sharp-crack line.
Change the band width slider from 0.08 to 0.03 and replay the load.
Use the ratio of the characteristic length to the notch to explain why the peak and the dashed line agree better for some settings than others.

Watch a damage band choose its path 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. Why does the prediction use the discrete fracture energy of the phase-field mesh?

Read an explained answer

Because the phase-field run actually charges the discrete value for a straight crack on that mesh. Using it compares like with like; leaving it out would put a mesh error into the comparison and lower the prediction by 1.5 percent.

2. The peak force arrives at 0.78 of the predicted opening. Does this mean the sharp-crack prediction is wrong by 22 percent?

Read an explained answer

No. The peak marks softening ahead of the notch, not crack growth. With a characteristic length comparable to the notch, that softening comes early. The prediction is tested by whether it lies inside the interval and whether the interval narrows as the length scale shrinks.

3. Grip opening scales with the square root of the fracture energy at fixed geometry. If the fracture energy is doubled, what is the predicted opening?

Read an explained answer

It rises by the square root of 2, from 0.0408 to about 0.0577 side lengths.

\Delta_c'=\sqrt2\times0.0408\approx0.0577

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.
Jeong, Signetti, Han & Ryu, Phase field modeling of crack propagation under combined shear and tensile loading with hybrid formulation, arXiv:1807.03151v2 (2018)
Section 2, equations (7) to (14), restates the isotropic and the spectrally split phase-field formulations of Miehe and coauthors, with the history field that stops healing, and the hybrid formulation of Ambati and coauthors, which softens the stress as a whole but lets only the tensile energy drive damage. The course takes these equations from this restatement; the original papers by Miehe and coauthors (2010) and Ambati and coauthors (2015) were not consulted. The Lab omits the extra hybrid condition that forbids damage where the compressive energy dominates.