Scaling laws of shrinkage induced fragmentation phenomena

This paper presents a model for surface cracking based on beams breaking between polygons. The authors determine a phase transition between a damaged and a fragmented surface and determine the critical exponents using finite size scaling. The obtained results are very interesting and the presentation is clear. There are just a few points that should be taken into account before publication.
Since the finite size scaling is performed by changing the system size just by a factor four and the sizes considered are rather small, the data collapse of Figs.7 and 8 is not very powerful. Therefore, I am convinced that the error bars given in Fig.11 for the exponents beta, gamma and nu obtained in Figs.7 and 8 are far too optimistic. The authors should present a more profound error estimation.
The authors claim that the obtained exponents are in the universality class of the 2d Ising model and state “the adhesion strength …. does not affect the universality class of the transition”. But in Fig.11 one can clearly see a strong dependence of gamma and nu on the adhesion strength (if one takes the error bars seriously).
The numerical evidence for the exponents being equal to the ones of the 2d Ising model is very weak, because of the large error bars discussed before, which allows for the numerical exponents to be in fact consistent with several other universality classes. A percolation transition would be expected if the bond breaking would be completely random. The elastic field induces correlations, which when reaching the critical point, become short range. So, percolation exponents seem a reasonable guess. The authors however put forward an argument for the Ising exponents, which is: “The reason of this universality is that cracks always advance through the failure of nearest neighbor cohesive contacts.” This argument is very superficial and applies equally well to percolation, 3 state Potts model, etc. The authors should remove this empty argument and eventually replace it, if they can, by a more convincing argument which should explicitly include the Ising interaction between binary variables.
The authors should define what is a “regularized Voronoi tessellation”.
The authors should add the missing references to the original work in which surface cracking was introduced for the first time as: P. Meakin, Thin Solid Films 151, 165 (1987) or Colina et al Phys. Rev. B 48, 3666 (1993).
Typos: loading -> leading, strength -> strengths (2x)

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1) Novel theoretical framework: Clear identification of damage vs fragmentation phases with well-characterized critical transition

2) Comprehensive methodology: Systematic parameter study across multiple system sizes with proper statistical sampling

3) Universal scaling laws: Finite size scaling analysis with critical exponents consistent with 2D Ising model

4) Physical insights: Demonstrates how adhesion strength controls stress field heterogeneity and crack patterns

1) Critical experimental gap: Fails to address Kooij et al.'s highly relevant sugar glass experiments showing exponential fragment distributions under slow thermal fracture vs power-law under impact - directly relevant to their damage/fragmentation transition

2) Limited validation: No direct comparison with experiments beyond citing previous work

3) Universality claims: Insufficient justification for 2D Ising assignment over alternative mechanisms like percolation

4) Model limitations: 2D constraint and simplified adhesion model may not capture realistic 3D behavior

This paper investigates shrinkage-induced fragmentation using discrete element modeling, identifying a critical transition between damage and fragmentation phases. The authors claim this transition belongs to the 2D Ising universality class and demonstrate different fragment size distributions in each phase.

The theoretical approach is sophisticated and the numerical work comprehensive, but the paper suffers from a fundamental disconnect with experimental reality. The most significant oversight is the complete failure to engage with the sugar glass experiments by Kooij et al., which demonstrate exactly the phenomenon being studied theoretically. These experiments show the same material producing different fragment size distributions under different loading conditions - exponential tails for slow fracture versus power-law for impact fracture. This provides direct experimental support for the authors' theoretical predictions about different fragmentation regimes.

The discrete element model itself is well-constructed and captures important physics, particularly the role of adhesion strength in controlling stress field heterogeneity. The identification of two distinct phases separated by a critical transition is a valuable contribution. However, the claim that this transition belongs to the 2D Ising universality class requires stronger theoretical justification. While the critical exponents are "reasonably close" to Ising values, the authors don't adequately explain why this particular fragmentation process should exhibit Ising-like behavior rather than, for example, percolation-type criticality.

The finite size scaling analysis is technically sound and demonstrates continuous phase transition behavior. The transition from power-law fragment distributions in the subcritical regime to log-normal distributions in the supercritical regime is well-documented and physically reasonable, arising from the binary splitting mechanism that dominates post-critical dynamics.

Recommendation

Major Revision Required

This work addresses an important problem with a novel theoretical perspective, but requires significant improvements.

Essential Revisions:

1) Address experimental validation: Thoroughly discuss Kooij et al.'s sugar glass experiments and their connection to the theoretical predictions

2) Strengthen universality arguments: Provide rigorous justification for the 2D Ising classification

3) Improve statistical analysis: Better quantify uncertainties in critical exponents

Additional Improvements:

4) Expand discussion of 3D limitations

5) Include parameter sensitivity analysis

6) Enhance figure quality and statistical presentation

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