Model a crack as a narrow band of damage

Replace the sharp crack by a damage field that equals one on the crack and falls to zero over a short length. The total energy is the elastic energy, softened where the material is damaged, plus a fracture energy spread over the band. Minimizing that energy decides initiation, growth and path. As the length shrinks, the band costs the Griffith fracture energy per unit crack length.

Sandbox Physics · 2026-10-29

Spread the crack over a band

Give every point a damage value between zero, intact, and one, broken. Multiply the elastic energy density by a degradation function that falls from one to almost zero as damage goes from zero to one. A tiny residual stiffness keeps the equations solvable in fully broken material.

Charge the fracture energy through two terms. One penalizes damage itself and one penalizes its gradient, and the length scale sets their balance. This is the AT2 model in the form studied by Tanné and coauthors. Bouchbinder, Fineberg and Marder describe phase-field models as one way to give the crack tip a built-in length.

\mathcal E[u,d]=\int_\Omega\Big[\big((1-d)^2+k\big)\,\psi(\varepsilon)+\frac{G_c}{2}\Big(\frac{d^2}{\ell}+\ell\,|\nabla d|^2\Big)\Big]\,dA

Worked example

Check that a straight band costs the fracture energy

Take a straight crack along a line and look across it. Minimizing the fracture terms alone gives, as Bourdin, Francfort and Marigo show, a damage profile that decays exponentially with distance from the crack, with decay length equal to the length scale. Insert it into the two fracture terms and integrate across the band. Each term contributes one half of the fracture energy, so the band costs exactly the fracture energy per unit crack length.

This calculation is the reason the regularized model reduces to Griffith’s balance as the length scale goes to zero. On a mesh, the band is resolved only approximately. In the Lab’s model a straight band held fully broken costs 1.118, 1.031 and 1.008 times the fracture energy when the element size is 1, 1/2 and 1/4 of the length scale. The initiation predictions in chapter six use the measured value for the mesh they run on.

d(x)=e^{-|x|/\ell}\ \Rightarrow\ \frac{G_c}{2}\int_{-\infty}^{\infty}\Big(\frac{d^2}{\ell}+\ell\,d'^2\Big)dx=\frac{G_c}{2}(1+1)=G_c

A finite band gives a finite strength

Pull a uniform bar without a notch. In the AT2 model the stress first rises, then damage spreads through the whole bar and the stress falls. The peak stress is finite and depends on the length scale. Tanné and coauthors give its value in their Table 1, together with the characteristic length that combines fracture energy, modulus and strength.

So the length scale is more than a numerical device. Shrinking it raises the strength as one over its square root. If the length scale is matched to a measured strength, it becomes a material length. For unit modulus, a fracture energy of one thousandth and a length scale of 0.05, the formula gives a strength of 0.0459. The browser model’s uniform bar reaches the same peak to a relative difference of 1.5 parts in a million.

\sigma_c=\frac{3}{16}\sqrt{\frac{3G_cE'}{\ell}},\qquad \ell_{\rm ch}=\frac{G_cE'}{\sigma_c^2}=\frac{256}{27}\,\ell

Three rules the model cannot do without

Damage must not heal. The Lab drives damage with the largest tensile energy reached so far at each point, so unloading leaves the crack in place. This memory is the history field of Miehe and coauthors; Tanné and coauthors instead forbid damage from decreasing between load steps. With this memory switched off, the band shrinks back as the load drops, which no real crack does. In the benchmark run, the remembered crack keeps its full surface after unloading, while the forgetful one returns to the initial notch.

Compression must not break the material. The Lab splits the strain energy by the sign of the principal strains and lets only the tensile part drive damage, while the stress is softened as a whole. That hybrid choice, proposed by Ambati and coauthors, has no exact energy balance: the global work balance closes to about 1 percent with the split and to 0.07 percent without it on the finest load stepping.

The mesh must resolve the band. Element sizes much larger than the length scale make the band too expensive and the crack path follows the mesh. The Lab’s mesh has 40 elements per side, so the element size stays below the smallest length scale its slider allows.

Paths with no path rule

Nothing in the energy says which way to go. The path comes out of the minimization. Pull the notched square in tension and the band runs straight across. Shear it and the band turns downward toward the corner. Put a hole near the notch and the band bends into the hole, where the stress is higher and part of the crack already exists.

These are qualitative checks. Testing the turning angle under shear against the principle of local symmetry would need a finer mesh and a smaller length scale than a browser model can afford.

An experiment to think through

Predict, change, explain

In the phase-field crack Lab, predict which way the band turns when the square is sheared instead of pulled.
Switch between the tension, shear and hole presets, then reduce the length scale with the slider.
Explain each path from where the tensile energy is largest. Then note where the force peak sits relative to the dashed sharp-crack line as the length scale changes.

Watch a damage band choose its path

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. The length scale is halved, all else equal. How does the uniform-bar strength change?

Read an explained answer

It rises by the square root of 2, about 1.41 times, because it scales as one over the square root of the length scale.

2. Why should the element size not exceed the length scale?

Read an explained answer

The band would be represented by too few elements. Its discrete energy then exceeds the fracture energy noticeably, about 12 percent at an element size equal to the length scale in this model, and the path can lock onto mesh lines.

3. A specimen is loaded until a crack has grown, then unloaded. What does the damage field look like after unloading, with and without irreversibility?

Read an explained answer

With irreversibility the crack stays at its full length. Without it, the damage relaxes back toward the initial notch, because the band is then held only by the current, lower elastic energy.

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.
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.
Bourdin, Francfort & Marigo, The variational approach to fracture, Journal of Elasticity 91, 5–148 (2008)
Section 8.1, equation (8.2), writes the Ambrosio–Tortorelli energy used for numerical fracture; with their regularization parameter equal to half the course’s length scale it is the AT2 energy of chapter five. Remark 8.2 gives the exponential optimal profile across a crack. The same section estimates how a linear-element mesh inflates the effective toughness when the crack occupies a band of broken elements, and Section 8.1.2 enforces irreversibility at the discrete level.
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.