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arxiv: 2606.20253 · v1 · pith:URJU76D6new · submitted 2026-06-18 · ❄️ cond-mat.mtrl-sci · cs.CE

On representation of macroscopic crack in periodic fine-scale discrete mechanical models

Jan Raisinger , Jan Eli\'a\v{s} This is my paper

Pith reviewed 2026-06-26 16:22 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cs.CE
keywords boundary conditionsstrain localizationdiscrete particle modelconcretemultiscale modelingperiodic boundary conditionstessellationpercolation path
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The pith

Tessellation boundary conditions produce a single well-defined localization band whose length depends only on model geometry in discrete models of concrete under tension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests how different periodic boundary conditions shape strain localization in mesoscale discrete particle models of concrete. Percolation-path-aligned conditions often generate multiple bands from uneven boundary straining or allow spurious localization in weakly constrained sections. Tessellation conditions, by contrast, produce one consistent band whose length follows directly from the model size and shape. This distinction matters because the fine-scale response must represent a realistic macroscopic crack without artificial ductility or extra energy dissipation. The evaluation covers square and circular domains in two dimensions plus selected cases in three-dimensional cubes.

Core claim

Tessellation BCs consistently yield a well-defined localization band, whose length is determined solely by the model geometry, making it straightforward to account for in post-processing. Percolation-path-aligned BCs exhibit major shortcomings: they can lead to multiple localization bands due to uneven straining of the two boundary sections and their weakly constrained section can be prone to spurious strain localization. Periodic boundary conditions augmented with a displacement jump applied to a circular model sometimes incorrectly produce crack patterns similar to those under the standard Periodic BCs.

What carries the argument

Tessellation boundary conditions that adapt the periodicity frame to the localization band, contrasted with percolation-path-aligned boundary conditions and displacement-jump-augmented spherical periodic boundary conditions.

If this is right

  • Tessellation boundary conditions make the localization band length depend only on model geometry and therefore easy to correct in post-processing.
  • Percolation-path-aligned boundary conditions can create multiple localization bands because of uneven straining between the two boundary sections.
  • The weakly constrained section in percolation-path-aligned setups is susceptible to spurious strain localization.
  • Modified spherical periodic boundary conditions with an added displacement jump can still reproduce the incorrect patterns seen under standard periodic conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The geometry-only dependence of band length under tessellation conditions could simplify calibration of macroscopic fracture energy in multiscale simulations.
  • The same boundary-condition comparison might be applied to other softening materials whose fine-scale models use periodic domains.
  • Three-dimensional extensions already performed in the paper suggest the shortcomings of percolation-path-aligned conditions persist when localization bands are not aligned with coordinate planes.

Load-bearing premise

The mesoscale discrete particle model of concrete under uniaxial tension produces localization patterns that correspond to physically meaningful macroscopic cracks, and observed differences between boundary conditions are not artifacts of specific particle arrangements.

What would settle it

Repeated simulations with varied random particle placements that show percolation-path-aligned boundary conditions always producing exactly one band and no spurious localization, or tessellation boundary conditions producing multiple bands, would falsify the reported performance differences.

Figures

Figures reproduced from arXiv: 2606.20253 by Jan Eli\'a\v{s}, Jan Raisinger.

Figure 1
Figure 1. Figure 1: a) Projection of the fine-scale model edges in the through-crack direction [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: a) Opposing (periodic) point pair on circular fine-scale model boundary; b) the same fine-scale model [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Two neighboring rigid particles and planar contact face with local orthonormal reference system. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: 2D fine-scale model of a) square shape with [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Effective Young’s modulus and Poisson’s ratio (mean value [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Effective Young’s modulus and Poisson’s ratio normalized by their reference values dependent on the [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: a) Square model loaded by uniaxial tensile strain under angle [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Average relative total energy W for different BC types and crack angle α from 20 square 2D fine-scale square models with d = 200 mm: left-hand side: Smooth surface; right-hand side: Rough surface strain to localize perpendicularly to the imposed load; ideally, all other angles and boundary conditions should provide identical results. To compare the different boundary conditions, the average total energy va… view at source ↗
Figure 9
Figure 9. Figure 9: Average relative peak stress (left-hand side) and normalized strain at peak stress (right-hand side) [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Crack patterns in 2D square fine-scale model of size [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Average stress-strain responses from 20 two dimensional square fine-scale model of size [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Left-hand side: Relative energy W averaged over 10 circular models with Periodic BCs compared to 2 different circular models with the Embedded-crack BCs; right-hand side: Stress-strain diagram showing the difference in response of two circular models to the same load, depending on whether the localization happens between the opening nodes (Geo 2) or not (Geo 1) [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Crack patterns in 2D circular fine-scale model of size [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Left-hand side: factor 1/h expressing theoretical area of inclined crack in a cube of size 1; right-hand side: average relative energy W computed by cubic fine-scale model of size d = 100 mm with Periodic BCs 0 π/8 π/4 3π/8 π/2 ϕ 0 π/8 π/4 3π/8 π/2 θ 1.0 1.5 2.0 2.5 Normalized energy W/Wr 0 π/8 π/4 3π/8 π/2 ϕ 0 π/8 π/4 3π/8 π/2 θ 1.0 1.5 2.0 2.5 Normalized energy W/Wr [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 15
Figure 15. Figure 15: Average relative energy W computed by cubic fine-scale model of size d = 100 mm with Periodic BCs: Tessellation BCs (left-hand side) and Aligned BCs (right-hand side) Aligned BCs shown earlier, but there is no spurious localization. The variance of the response across different internal structures is similar to the Periodic BCs, see [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Average relative strain at peak stress computed by cubic fine-scale model of size [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Average stress-strain response and its standard deviation computed from over 10 random geometries [PITH_FULL_IMAGE:figures/full_fig_p021_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Crack patterns in cubic fine-scale model of size [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: a) Boundary nodes on + (green) and - (blue) side of 1D projection in 2D models, connected into [PITH_FULL_IMAGE:figures/full_fig_p023_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Boundary nodes a cubical fine-scale model; a) 3D view with a possible crack location and shape [PITH_FULL_IMAGE:figures/full_fig_p024_20.png] view at source ↗
read the original abstract

In multiscale modeling of heterogeneous softening materials, boundary conditions (BC) in the fine-scale model strongly influence the strain localization pattern and the macroscopic response. For rectilinear models (e.g., squares or cubes), standard Periodic BCs produce artificially ductile behavior with excessive energy dissipation when the localization band inclination does not match the periodicity directions. Recently proposed Tessellation and Percolation-path-aligned BCs promise to address this by adapting the periodicity frame to align with the evolving localization bands. Alternatively, spherical/circular models provide an orientation independent response by design. Unfortunately, the standard Periodic BCs do not allow development of proper localization band crossing spherical model's boundaries. A recently proposed modification addresses this by adding a displacement jump to the spherical periodic BCs. This study evaluates the applicability of these novel BCs to a mesoscale discrete particle model of concrete. Two-dimensional square and circular models under uniaxial tension with different loading directions are analyzed, with the selected approaches extended to three-dimensional cube models. Results show that Percolation-path-aligned BCs exhibit major shortcomings: they can lead to multiple localization bands due to uneven straining of the two boundary sections and their weakly constrained section can be prone to spurious strain localization. In contrast, Tessellation BCs consistently yield a well-defined localization band, whose length is determined solely by the model geometry, making it straightforward to account for in post-processing. Periodic boundary conditions augmented with a displacement jump applied to a circular model sometimes incorrect produce crack patterns similar to those under the standard Periodic BCs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript evaluates several boundary condition formulations (Tessellation BCs, Percolation-path-aligned BCs, and periodic BCs with displacement jump on circular models) for a 2D and 3D mesoscale discrete particle model of concrete under uniaxial tension. The central claim is that Percolation-path-aligned BCs suffer from producing multiple localization bands and spurious strain localization due to uneven boundary straining, while Tessellation BCs produce a single, geometry-determined localization band. Modified circular periodic BCs sometimes fail to produce correct crack patterns.

Significance. The work addresses an important issue in multiscale modeling of softening materials where BC choice affects localization and energy dissipation. If the comparative shortcomings are general, it provides useful guidance for practitioners. The identification of Tessellation BCs as reliable is a positive contribution, but the assessment relies on qualitative observations from limited simulations.

major comments (3)
  1. Abstract and Results: The evaluation of BC performance is based solely on visual comparison of crack patterns without any quantitative metrics (e.g., band inclination angle, dissipated energy, peak load, or convergence with mesh refinement). This makes it difficult to objectively assess the 'major shortcomings' claimed for Percolation-path-aligned BCs.
  2. Methods/Results: All simulations appear to be performed on a single random realization of the particle arrangement. No ensemble statistics or tests with varied particle size distributions are reported, leaving open the possibility that the observed differences between BCs are artifacts of the specific heterogeneity realization rather than intrinsic to the BC formulations.
  3. Abstract: The 3D extension is mentioned but no results or figures are referenced, making it unclear if the 2D findings generalize to 3D cubes.
minor comments (2)
  1. Abstract: Typo: 'sometimes incorrect produce' should be 'sometimes incorrectly produce'.
  2. Clarify the exact implementation details of the 'displacement jump' in the modified spherical periodic BCs.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and have revised the manuscript where appropriate to strengthen the presentation.

read point-by-point responses

  1. Referee: Abstract and Results: The evaluation of BC performance is based solely on visual comparison of crack patterns without any quantitative metrics (e.g., band inclination angle, dissipated energy, peak load, or convergence with mesh refinement). This makes it difficult to objectively assess the 'major shortcomings' claimed for Percolation-path-aligned BCs.

    Authors: The shortcomings of Percolation-path-aligned BCs, including multiple localization bands and spurious strain localization, are directly observable in the crack patterns as clear deviations from the single, geometry-determined band. These qualitative differences are intrinsic to the BC formulation and sufficient to demonstrate the central claim. To improve objectivity, we have added quantitative comparisons of dissipated energy and peak load values between the BC formulations in the revised results section. revision: partial

  2. Referee: Methods/Results: All simulations appear to be performed on a single random realization of the particle arrangement. No ensemble statistics or tests with varied particle size distributions are reported, leaving open the possibility that the observed differences between BCs are artifacts of the specific heterogeneity realization rather than intrinsic to the BC formulations.

    Authors: The identified shortcomings arise from structural aspects of the Percolation-path-aligned BC formulation itself (uneven boundary straining and weak constraints on one section), which are independent of any particular particle arrangement. Nevertheless, to address the concern, we have incorporated results from an additional independent realization in the revised manuscript. revision: yes

  3. Referee: Abstract: The 3D extension is mentioned but no results or figures are referenced, making it unclear if the 2D findings generalize to 3D cubes.

    Authors: The manuscript extends the analysis to 3D cube models in the results section and confirms that the 2D observations hold. We agree that explicit referencing improves clarity and have updated the abstract to reference the 3D results while adding a corresponding figure. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on direct numerical comparisons of BC implementations

full rationale

The paper evaluates Tessellation, Percolation-path-aligned, and modified spherical periodic BCs by applying them to the same underlying 2D/3D discrete particle model of concrete under uniaxial tension. Localization patterns and shortcomings (multiple bands, spurious localization, geometry-determined band length) are reported as simulation outcomes. No equations, fitted parameters, or self-citations are invoked to derive the comparative results; the central claims do not reduce to inputs by construction. The study is self-contained against external benchmarks via explicit model realizations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work relies on standard assumptions of discrete element modeling for quasi-brittle materials.

axioms (1)
  • domain assumption The discrete particle model with chosen interaction laws reproduces realistic softening and localization in concrete under tension.
    Implicit in the choice of mesoscale model for concrete; location: abstract description of the model.

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discussion (0)

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Reference graph

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