The simulations show two distinct crack-front regimes. At lower toughness contrast, the penny-shaped front remains stable. Above a critical threshold, the model shows pinned regions alongside advancing fingers. The paper also reports finite front stiffness at low contrast and vanishing stiffness above the threshold. This is a model-based finding about the behavior specified in the simulations, not a general result for all brittle materials.
The model behind the result
The study examines finite-amplitude stability, or the response of the front to changes beyond a tiny perturbation, in heterogeneous brittle solids. The calculations vary the model's toughness contrast, meaning the difference in assigned resistance to crack growth, and its obstacle number. The comparison is between weak-pinning, lower-contrast conditions and higher-contrast conditions in which the simulations show fingering.
The investigation comprises hundreds of simulations across the phase diagram, although the exact simulation count is not reported. That scale allows the authors to map the change between regimes within the numerical model, but it does not turn the study into an experimental survey of physical cracks.
To update the crack shape, the numerical method incrementally adds linear approximations of the stress-intensity-factor perturbation and of the kernel perturbation as the front deforms. The stress-intensity factor is the model's measure of the load acting at the crack front. At each time step, the applied stress is adjusted to maintain quasi-static propagation, with K = Kc at at least one front point and K ≤ Kc elsewhere.
A boundary in the phase diagram
The boundary between stable and fingering behavior is described by a fitted phase-diagram relation: ∆c(1/k) ≃ (1 − 1/k)/(1 + 3.65/k). Here, ∆c denotes the critical contrast and k is the obstacle number in the model. The fit approaches a contrast of one as the obstacle number increases. No uncertainty is reported for this fitted threshold.
The fingering regime has a second notable feature. The simulations report daughter cracks, or secondary cracks, reaching a size similar to the original mother crack. The paper also reports finite front stiffness at low contrast and vanishing stiffness above the threshold.
The shape of the transition
The stationary shape of the fingered front follows its own scaling law. Near the threshold, the normalized stationary petal aspect ratio, a measure of the final petal's shape, follows a power law with exponent γ = 0.51 ± 0.02. The paper describes this as square-root scaling near the threshold.
Time behavior shows the same critical approach. As the fingering threshold is approached, relaxation toward a stationary crack-front configuration slows, and the reported relaxation time follows τ ∝ (∆c − ∆)^−1/2. In ordinary terms, the closer the simulated contrast is to the threshold, the longer the model takes to recover its stationary configuration.
A landscape without a stationary state
The energy calculation offers a way to read the transition. Under weak pinning, the computed energy landscape has a minimum, corresponding to a stationary state. In the fingering regime, the energy decreases monotonically instead. The simulations therefore show different equilibrium behavior on the two sides of the threshold.
From these results, the authors interpret fingering as a global loss of Griffith-compatible equilibria, rather than a local stable-unstable branch annihilation. Put more simply, the proposed mechanism is the disappearance of stationary configurations that satisfy the model's fracture condition, with no detectable unstable counterpart. This conclusion is drawn from the simulated energy landscape and front behavior, so it is an interpretation of the model's simulated behavior.
A supplemental test examined whether the outcome depended on the starting geometry. Distinct initial crack geometries evolved to the same stationary Griffith-compatible configuration, which the authors identify as the unique attractor in that test. At the fingering threshold, the tested normalized stationary front profiles also collapsed onto a single master curve independent of obstacle number.
What the result does and does not establish
The document is an arXiv version-1 preprint dated 26 August 2026. Its evidence is limited to the quasi-static penny-shaped crack-front model specified in the paper, and the exact number of simulations is not given. The paper does not provide experimental validation, and its interpretation of a missing unstable branch is not an exhaustive inventory of every possible equilibrium.
It therefore does not establish that the mechanism occurs in other materials, under dynamic or rate-dependent propagation, or when cracks can nucleate. The reported result is a set of simulated behaviors and scaling laws under the model's stated assumptions.
Paper data and sources
Original title: Beyond linear stability: Heterogeneity-induced fingering of crack fronts
Authors: Manish Vasoya, Laurent Ponson, Veronique Lazarus
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text