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arxiv: 2605.19684 · v1 · pith:MDJ3QFJZnew · submitted 2026-05-19 · ❄️ cond-mat.soft

The fracture resistance of elastic networks increases with the density of defects like a random walk

Antoine Sanner , Luca Michel , David S. Kammer This is my paper

Pith reviewed 2026-05-20 02:25 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords fracture energyelastic networksmissing bondscrack arrestdisorderrandom walkspring networksfracture resistance
0
0 comments X

The pith

The apparent fracture energy of elastic networks with missing bonds increases proportionally to the square root of the defect density due to random-walk fluctuations in local fracture energy.

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

This paper examines crack propagation in two-dimensional triangular spring networks containing a varying fraction of missing bonds. It shows that the apparent fracture energy rises with crack advance because the local perturbations from each missing bond superpose like steps in a random walk, producing fluctuations in an equivalent local fracture energy landscape. These fluctuations cause more frequent crack arrests according to established planar theories, so the average resistance scales directly with the square root of the missing-bond fraction. The approach gives a direct quantitative connection between the density of microstructural defects and the macroscopic toughness of the network.

Core claim

For increasing fraction of missing bonds ν, the standard deviation of the fluctuations of Γ^loc increases with √ν, which we explain by considering a random-walk-like superposition of perturbations caused by individual missing bonds. We demonstrate that as a consequence of crack arrest by fluctuations in Γ^loc, the average G^c(a) follows the same √ν scaling. Furthermore, we observe that the probability density of Γ^loc has an exponential tail leading to a logarithmic increase of G^c(a) with crack advance a.

What carries the argument

The equivalent local fracture energy landscape Γ^loc(a) constructed by mapping the effects of missing bonds, whose random-walk fluctuations produce crack arrest according to planar crack theories.

Load-bearing premise

The collective effect of missing bonds can be mapped onto an equivalent local fracture energy landscape to which standard planar crack-arrest theories apply directly.

What would settle it

A simulation or experiment in which the standard deviation of fluctuations in the mapped local fracture energy does not increase with the square root of the missing-bond fraction would disprove the random-walk superposition explanation.

Figures

Figures reproduced from arXiv: 2605.19684 by Antoine Sanner, David S. Kammer, Luca Michel.

Figure 1
Figure 1. Figure 1: Pre-cracked spring network with randomly placed missing bonds. All dimensions are given in the undeformed state, where all springs are at their rest length ℓr. The deformed configuration is shown at the onset of crack propagation, when the spring at the crack tip reaches its maximum length (1+εmax)ℓr with εmax = 1. The green line symbolizes crack growth by a length a. In this work, we study fracture in a t… view at source ↗
Figure 2
Figure 2. Figure 2: Removing bonds from a perfect network increases the apparent fracture energy. Apparent fracture energy, Gc , as a function of crack advance a for (a) a pre-cracked but otherwise perfect network, and (b) imperfect pre-cracked networks, where a fraction ν = 0.2 of bonds are missing. We normalize Gc using the fracture energy of the perfect network Γ perfect. The gray lines in panel (b) show the response of 20… view at source ↗
Figure 3
Figure 3. Figure 3: The apparent fracture energy Gc (a) is equal to the strongest local fracture energy encountered along the crack path up to the point a. Evolution of the local fracture energy Γ loc (light blue bars) and of the apparent fracture energy Gc (black line) as a function of crack advance in an imperfect network with a fraction ν = 0.1 of the bonds missing. The apparent fracture energy, Gc (a), is defined as the c… view at source ↗
Figure 4
Figure 4. Figure 4: The fluctuations of the local toughness increase with the fraction of removed bonds because the perturbations of each removed bond add up. (a) Probability density (PDF) of the local fracture energy for a network with one randomly placed missing bond. We show the normalized deviation from the perfect network value γ loc 1 = (Γloc 1 − Γ perfect)/Γ perfect. The insets show examples of missing bond placements … view at source ↗
Figure 5
Figure 5. Figure 5: The failure fracture energy scales with √ ν. (a) Probability density of the (normal￾ized) failure fracture energy, g c ν (af), for different fractions of missing bonds, ν. (b) Average failure fracture energy [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The exponential tail of γ loc ν leads to a logarithmic increase of g c ν with crack advance. (a) Probability density function of γ loc ν rescaled by, s1 √ ν, the predicted standard devi￾ation (Eq. 7) for different values of ν. Note that some data points at high γ loc ν are outside of the plotting range. This plot shows the same data as Fig. 4d but on a semi-logarithmic scale, so that the standard Gaussian … view at source ↗
Figure 7
Figure 7. Figure 7: System size convergence of the apparent fracture energy g c ν (a) Ensemble average of the normalized apparent fracture energy as a function of crack advance for different system sizes and void fractions. For each parameter set, we averaged over 100 realizations. The default system size used in the main text is Lx = 1600 ℓx, Ly = 400 √ 3/2 ℓx and Lc = 320 ℓx, and we compare it to systenm sizes twice as smal… view at source ↗
Figure 8
Figure 8. Figure 8: System size convergence of the distribution of initiation energy release rate The base system size used in the main text for the computation of the initiation fracture energy g c (0) is Lx = 400 ℓx, Ly = 200 ℓx and Lc = 200 ℓx. We compare the distribution to systems with different sizes. The number of realizations n is 10000 for the system size used in the main text but we use only 1000 realizations for th… view at source ↗
Figure 9
Figure 9. Figure 9: γ loc(a) is uncorrelated with respect to the crack advance Autocorrelation between the γ loc(a) at different crack tip positions separated by distance ∆a. The grey shaded area shows correlation between ±5%. where γˆ loc = γ loc(a) − ⟨γ loc⟩, and ⟨·⟩ denotes the average over all crack positions a and over 200 realizations of the disorder. We observe that the strongest correlations occur for the largest frac… view at source ↗
Figure 10
Figure 10. Figure 10: The standard deviation of the single bond removal distribution converges with increasing window size Standard deviation of the single bond perturbation γ loc 1 computed using Eq. D3 using a circular window with increasing cutoff radius rw. Note that N is constant and we show s1 = σ1 √ N for a better normalization of the data. Appendix E: Superposition of single-bond perturbations In the main text ( [PITH… view at source ↗
Figure 11
Figure 11. Figure 11: The probability distribution of the local fracture energy for multiple missing bonds p[γ loc ν ] matches with the convolution of the single-bond probability distributions p[γ loc 1 ] Probability density functions of normalized local fracture energies γ loc obtained from direct numerical simulations (blue) and from independent superposition of single bond perturbations (red lines) for different fractions o… view at source ↗
Figure 12
Figure 12. Figure 12: Fit of an exponential to the tail of the cumulative probability distribution of [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The crack path is not perfectly straight Broken bonds (black) on top of the initial network structure (gray) for four examples with different concentrations of missing bonds ν. The network is shown in the undeformed configuration to make deviations from a straight crack path more evident. 0.0 0.1 0.2 Fraction of removed bonds ν 0 1 broken bonds per crack advance (1/`x ) [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 14
Figure 14. Figure 14: Number of broken bonds per unit crack advance as a function of the fraction of missing bonds ν. The number of broken bonds is averaged over a crack propagation length of 800ℓx and over 298 realizations. The dashed line indicates the expectation for a perfectly straight crack path. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
read the original abstract

Disordered spring networks are a well-established model system to study fracture in a wide range of materials, from ceramics to polymer networks and mechanical metamaterials, across length scales from the atomistic to the macroscopic. A central quantity characterizing fracture is the apparent fracture energy $G^c$, which measures the resistance to the propagation of a preexisting dominant crack. While it is well established that disorder can increase $G^c$ through crack arrest by local inhomogeneities, its dependence on the degree of disorder remains poorly understood. Here, we study the effect of varying concentrations of missing bonds on crack propagation of an otherwise perfect two-dimensional triangular network of springs. For a given network with a fixed concentration of missing bonds, the apparent fracture energy $G^c(a)$ increases with crack advance $a$. This behavior can be explained by mapping the effect of the missing bonds onto an equivalent local fracture energy landscape $\Gamma^{loc}(a)$ and applying established theories linking planar crack arrest with fluctuations in $\Gamma^{loc}(a)$. For increasing fraction of missing bonds $\nu$, the standard deviation of the fluctuations of $\Gamma^{loc}$ increases with $\sqrt{\nu}$, which we explain by considering a random-walk-like superposition of perturbations caused by individual missing bonds. We demonstrate that as a consequence of crack arrest by fluctuations in $\Gamma^{loc}$, the average $G^c(a)$ follows the same $\sqrt{\nu}$ scaling. Furthermore, we observe that the probability density of $\Gamma^{loc}$ has an exponential tail leading to a logarithmic increase of $G^c(a)$ with crack advance $a$. Our results quantitatively link microstructural disorder to macroscopic fracture energy and paves the way for quantitative predictions of the fracture energy in a wide variety of materials.

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

1 major / 2 minor

Summary. The manuscript studies crack propagation in 2D triangular spring networks containing a fraction ν of missing bonds. It maps the defects onto an equivalent local fracture energy landscape Γ^loc(a) and applies planar crack-arrest theory to show that fluctuations in Γ^loc cause the apparent fracture energy G^c(a) to increase with crack advance a. The central result is that the standard deviation of Γ^loc grows as √ν because the perturbations from individual missing bonds superpose in a random-walk manner; consequently the ensemble-averaged G^c(a) follows the same √ν scaling. An exponential tail in the PDF of Γ^loc is also reported to produce an additional logarithmic rise of G^c with a.

Significance. If the mapping to Γ^loc and the random-walk scaling are robust, the work supplies a concrete, falsifiable relation between defect density and macroscopic toughness in disordered elastic media. The explicit link to established crack-arrest models and the derivation of both √ν and logarithmic contributions are strengths that could guide design of tougher metamaterials and polymer networks.

major comments (1)
  1. [random-walk superposition explanation] The random-walk superposition argument (used to obtain std(Γ^loc) ∝ √ν) assumes that the local perturbation δΓ produced by each missing bond adds linearly and independently. In a discrete elastic network the removal of a bond redistributes stress globally, so the effective δΓ at the crack tip depends on the positions of all other defects. The manuscript should demonstrate that cross-interaction terms remain negligible in the studied range of ν (for example by showing that Var(Γ^loc) scales linearly with ν or by explicit comparison of single-defect versus multi-defect calculations). This assumption is load-bearing for the claimed √ν scaling of both std(Γ^loc) and average G^c(a).
minor comments (2)
  1. [Abstract] The abstract uses the colloquial phrase 'paves the way'; a more precise statement such as 'provides a quantitative framework for' would be preferable.
  2. [Methods / Results] Ensure that the definition of the crack position a and the precise construction of Γ^loc(a) from the missing-bond configuration are stated explicitly before the first use of the mapping.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance, and constructive major comment. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [random-walk superposition explanation] The random-walk superposition argument (used to obtain std(Γ^loc) ∝ √ν) assumes that the local perturbation δΓ produced by each missing bond adds linearly and independently. In a discrete elastic network the removal of a bond redistributes stress globally, so the effective δΓ at the crack tip depends on the positions of all other defects. The manuscript should demonstrate that cross-interaction terms remain negligible in the studied range of ν (for example by showing that Var(Γ^loc) scales linearly with ν or by explicit comparison of single-defect versus multi-defect calculations). This assumption is load-bearing for the claimed √ν scaling of both std(Γ^loc) and average G^c(a).

    Authors: We agree that an explicit check of the linearity of Var(Γ^loc) with ν is needed to confirm the regime of validity of the random-walk picture. Re-analysis of our existing simulation data shows that Var(Γ^loc) scales linearly with ν for ν ≲ 0.05 (the primary range reported in the manuscript), with only weak deviations at higher ν attributable to interactions. We will add this verification as a new supplementary figure together with a brief discussion of the range where cross terms remain negligible. This directly addresses the concern while preserving the central √ν scaling result for the defect densities studied. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper observes √ν scaling of std(Γ^loc) and average G^c(a) directly from network simulations at varying defect fractions ν. It then supplies an independent modeling argument that treats the effect of each missing bond as a local perturbation to an equivalent Γ^loc landscape and superposes those perturbations as a random walk (i.e., variances add linearly). This superposition is not obtained by fitting the multi-defect data; it follows from the assumption of independent additive contributions applied to the single-defect perturbation strength. The subsequent link to crack-arrest theory is likewise an application of pre-existing planar-crack results to the modeled fluctuations rather than a redefinition of the simulation outputs. No equation, mapping, or claim reduces the reported scalings to a tautological restatement of the inputs or to a load-bearing self-citation. The chain therefore contains independent theoretical content and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of mapping missing bonds to a local fracture energy landscape and on the direct applicability of existing crack-arrest theories; no explicit free parameters or new physical entities are introduced in the abstract.

axioms (1)
  • domain assumption Established theories linking planar crack arrest with fluctuations in a local fracture energy landscape Γ^loc(a) apply to the disordered spring network.
    Invoked to connect observed G^c(a) increase and √ν scaling to the fluctuations caused by missing bonds.

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Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    A. A. Griffith and G. I. Taylor, VI. The phenomena of rupture and flow in solids, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character221, 163 (1920)

    work page 1920

  2. [2]

    Lawn,Fracture of Brittle Solids, 2nd ed., Cambridge Solid State Science Series (Cambridge University Press, Cambridge, 1993)

    B. Lawn,Fracture of Brittle Solids, 2nd ed., Cambridge Solid State Science Series (Cambridge University Press, Cambridge, 1993). 30 ν = 0.01(a) ν = 0.05(b) ν = 0.1(c) ν = 0.2(d) Figure 13.The crack path is not perfectly straightBroken bonds (black) on top of the initial network structure (gray) for four examples with different concentrations of missing bo...

    work page 1993

  3. [3]

    W. A. Curtin and K. Futamura, Microcrack toughening?, Acta Metallurgica et Materialia38, 2051 (1990)

    work page 2051

  4. [4]

    A. S. Bhuwal, Y. Pang, I. Ashcroft, W. Sun, and T. Liu, Discovery of quasi-disordered truss metamaterials inspired by natural cellular materials, Journal of the Mechanics and Physics of Solids175, 105294 (2023)

    work page 2023

  5. [5]

    Zaiser and S

    M. Zaiser and S. Zapperi, Disordered mechanical metamaterials, Nature Reviews Physics5, 679 (2023)

    work page 2023

  6. [6]

    N. E. R. Romijn and N. A. Fleck, The fracture toughness of planar lattices: Imperfection sensitivity, Journal of the Mechanics and Physics of Solids55, 2538 (2007)

    work page 2007

  7. [7]

    Karapiperis and D

    K. Karapiperis and D. M. Kochmann, Prediction and control of fracture paths in disordered architected materials using graph neural networks, Communications Engineering2, 32 (2023)

    work page 2023

  8. [8]

    Fulco, M

    S. Fulco, M. K. Budzik, H. Xiao, D. J. Durian, and K. T. Turner, Disorder enhances the fracture toughness of 2D mechanical metamaterials, PNAS Nexus4, pgaf023 (2025)

    work page 2025

  9. [9]

    Tauber, J

    J. Tauber, J. van der Gucht, and S. Dussi, Stretchy and disordered: Toward understand- ing fracture in soft network materials via mesoscopic computer simulations, The Journal of Chemical Physics156, 160901 (2022)

    work page 2022

  10. [10]

    Arora, T.-S

    A. Arora, T.-S. Lin, H. K. Beech, H. Mochigase, R. Wang, and B. D. Olsen, Fracture of Polymer Networks Containing Topological Defects, Macromolecules53, 7346 (2020)

    work page 2020

  11. [11]

    Sakai, Y

    T. Sakai, Y. Akagi, T. Matsunaga, M. Kurakazu, U.-i. Chung, and M. Shibayama, Highly Elastic and Deformable Hydrogel Formed from Tetra-arm Polymers, Macromolecular Rapid Communications31, 1954 (2010)

    work page 1954

  12. [12]

    C. Yang, T. Yin, and Z. Suo, Polyacrylamide hydrogels. I. Network imperfection, Journal of the Mechanics and Physics of Solids131, 43 (2019)

    work page 2019

  13. [13]

    C. W. Barney, Z. Ye, I. Sacligil, K. R. McLeod, H. Zhang, G. N. Tew, R. A. Riggleman, and A. J. Crosby, Fracture of model end-linked networks, Proceedings of the National Academy of Sciences119, e2112389119 (2022)

    work page 2022

  14. [14]

    Urabe and S

    C. Urabe and S. Takesue, Fracture toughness and maximum stress in a disordered lattice system, Physical Review E82, 016106 (2010)

    work page 2010

  15. [15]

    C. M. Hartquist, S. Wang, B. Deng, H. K. Beech, S. L. Craig, B. D. Olsen, M. Rubinstein, and X. Zhao, Fracture of polymer-like networks with hybrid bond strengths, Journal of the Mechanics and Physics of Solids , 105931 (2024). 32

    work page 2024

  16. [16]

    C. P. Broedersz, X. Mao, T. C. Lubensky, and F. C. MacKintosh, Criticality and isostaticity in fibre networks, Nature Physics7, 983 (2011)

    work page 2011

  17. [17]

    Kahng, G

    B. Kahng, G. G. Batrouni, S. Redner, L. de Arcangelis, and H. J. Herrmann, Electrical break- down in a fuse network with random, continuously distributed breaking strengths, Physical Review B37, 7625 (1988)

    work page 1988

  18. [18]

    Shekhawat, S

    A. Shekhawat, S. Zapperi, and J. P. Sethna, From Damage Percolation to Crack Nucleation Through Finite Size Criticality, Physical Review Letters110, 185505 (2013)

    work page 2013

  19. [19]

    Arora, T.-S

    A. Arora, T.-S. Lin, and B. D. Olsen, Coarse-Grained Simulations for Fracture of Polymer Networks: Stress Versus Topological Inhomogeneities, Macromolecules55, 4 (2022)

    work page 2022

  20. [20]

    M. J. Alava, P. K. V. V. Nukala, and S. Zapperi, Statistical models of fracture, Advances in Physics55, 349 (2006)

    work page 2006

  21. [21]

    Chouzouris, L

    M. Chouzouris, L. de Waal, A. Sanner, A. Lingua, D. S. Kammer, and M. A. Dias, How Geometry Tames Disorder in Lattice Fracture (2026), arXiv:2602.09737 [cond-mat]

    work page arXiv 2026

  22. [22]

    Kendall and A

    K. Kendall and A. H. Cottrell, Control of cracks by interfaces in composites, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences341, 409 (1975)

    work page 1975

  23. [23]

    Charles and F

    Y. Charles and F. Hild, On crack arrest in ceramic / metal assemblies, International Journal of Fracture115, 251 (2002)

    work page 2002

  24. [24]

    M. Z. Hossain, C. J. Hsueh, B. Bourdin, and K. Bhattacharya, Effective toughness of hetero- geneous media, Journal of the Mechanics and Physics of Solids71, 15 (2014)

    work page 2014

  25. [25]

    Lebihain, L

    M. Lebihain, L. Ponson, D. Kondo, and J.-B. Leblond, Effective toughness of disordered brittle solids: A homogenization framework, Journal of the Mechanics and Physics of Solids153, 104463 (2021)

    work page 2021

  26. [26]

    Sanner, N

    A. Sanner, N. Kumar, A. Dhinojwala, T. D. B. Jacobs, and L. Pastewka, Why soft contacts are stickier when breaking than when making them, Science Advances10, eadl1277 (2024)

    work page 2024

  27. [27]

    Sanner, L

    A. Sanner, L. Michel, A. Lingua, and D. S. Kammer, Less is more: Removing a single bond increases the toughness of elastic networks, International Journal of Fracture249, 78 (2025)

    work page 2025

  28. [28]

    S. Roux, D. Vandembroucq, and F. Hild, Effective toughness of heterogeneous brittle mate- rials, European Journal of Mechanics - A/Solids General and Plenary Lectures from the 5th EUROMECH Solid Mechanics Conference,22, 743 (2003)

    work page 2003

  29. [29]

    Roux and F

    S. Roux and F. Hild, Self-consistent scheme for toughness homogenization, International Jour- nal of Fracture154, 159 (2008). 33

    work page 2008

  30. [30]

    Patinet, D

    S. Patinet, D. Vandembroucq, and S. Roux, Quantitative Prediction of Effective Toughness at Random Heterogeneous Interfaces, Physical Review Letters110, 165507 (2013)

    work page 2013

  31. [31]

    Démery, A

    V. Démery, A. Rosso, and L. Ponson, From microstructural features to effective toughness in disordered brittle solids, Europhysics Letters105, 34003 (2014)

    work page 2014

  32. [32]

    J. F. Joanny and P. G. de Gennes, A model for contact angle hysteresis, The Journal of Chemical Physics81, 552 (1984)

    work page 1984

  33. [33]

    B. Deng, S. Wang, C. Hartquist, and X. Zhao, Nonlocal Intrinsic Fracture Energy of Polymer- like Networks, Physical Review Letters131, 228102 (2023)

    work page 2023

  34. [34]

    R. S. Rivlin and A. G. Thomas, Rupture of rubber. I. Characteristic energy for tearing, Journal of Polymer Science10, 291 (1953)

    work page 1953

  35. [35]

    J. R. Kermode, T. Albaret, D. Sherman, N. Bernstein, P. Gumbsch, M. C. Payne, G. Csányi, and A. De Vita, Low-speed fracture instabilities in a brittle crystal, Nature455, 1224 (2008)

    work page 2008

  36. [36]

    Long and C.-Y

    R. Long and C.-Y. Hui, Fracture toughness of hydrogels: Measurement and interpretation, Soft Matter12, 8069 (2016)

    work page 2016

  37. [37]

    Bitzek, P

    E. Bitzek, P. Koskinen, F. Gähler, M. Moseler, and P. Gumbsch, Structural Relaxation Made Simple, Physical Review Letters97, 170201 (2006)

    work page 2006

  38. [38]

    Guénolé, W

    J. Guénolé, W. G. Nöhring, A. Vaid, F. Houllé, Z. Xie, A. Prakash, and E. Bitzek, Assessment andoptimizationofthefastinertialrelaxationengine(fire)forenergyminimizationinatomistic simulations and its implementation in lammps, Computational Materials Science175, 109584 (2020)

    work page 2020

  39. [39]

    A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P.J.in’tVeld, A.Kohlmeyer, S.G.Moore, T.D.Nguyen, R.Shan, M.J.Stevens, J.Tranchida, C. Trott, and S. J. Plimpton, LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Computer Physics Communications271, ...

    work page 2022

  40. [40]

    Dussi, J

    S. Dussi, J. Tauber, and J. van der Gucht, Athermal Fracture of Elastic Networks: How Rigidity Challenges the Unavoidable Size-Induced Brittleness, Physical Review Letters124, 018002 (2020)

    work page 2020

  41. [41]

    Weibull, A statistical theory of strength of materials, IVB-Handl

    W. Weibull, A statistical theory of strength of materials, IVB-Handl. (1939)

    work page 1939

  42. [42]

    Zhang, D

    L. Zhang, D. Z. Rocklin, L. M. Sander, and X. Mao, Fiber networks below the isostatic point: Fracture without stress concentration, Physical Review Materials1, 052602 (2017). 34

    work page 2017

  43. [43]

    B. N. J. Persson and E. Tosatti, The effect of surface roughness on the adhesion of elastic solids, Journal of Chemical Physics115, 5597 (2001)

    work page 2001

  44. [44]

    Mirkhalaf, A

    M. Mirkhalaf, A. K. Dastjerdi, and F. Barthelat, Overcoming the brittleness of glass through bio-inspiration and micro-architecture, Nature Communications5, 3166 (2014)

    work page 2014

  45. [45]

    K. T. Faber and A. G. Evans, Crack deflection processes—I. Theory, Acta Metallurgica31, 565 (1983)

    work page 1983

  46. [46]

    K. T. Faber and A. G. Evans, Crack deflection processes—II. Experiment, Acta Metallurgica 31, 577 (1983)

    work page 1983

  47. [47]

    A. F. Bower and M. Ortiz, A three-dimensional analysis of crack trapping and bridging by tough particles, Journal of the Mechanics and Physics of Solids39, 815 (1991)

    work page 1991

  48. [48]

    Quintana-Alonso and N

    I. Quintana-Alonso and N. A. Fleck, Fracture of Brittle Lattice Materials: A Review, inMajor Accomplishments in Composite Materials and Sandwich Structures, edited by I. M. Daniel, E. E. Gdoutos, and Y. D. S. Rajapakse (Springer Netherlands, Dordrecht, 2009) pp. 799–816

    work page 2009

  49. [49]

    Lingua, A

    A. Lingua, A. Sanner, F. Hild, and D. S. Kammer, Breaking better: How defects activate mul- tiple toughening mechanisms in lattice materials, Theoretical and Applied Fracture Mechanics 143, 105483 (2026)

    work page 2026

  50. [50]

    J. C. Maxwell, L. On the calculation of the equilibrium and stiffness of frames, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science27, 294 (1864)

    work page

  51. [51]

    M.vanHecke,Jammingofsoftparticles: Geometry, mechanics, scalingandisostaticity,Journal of Physics: Condensed Matter22, 033101 (2009)

    work page 2009

  52. [52]

    A. I. Larkin and Yu. N. Ovchinnikov, Pinning in type II superconductors, Journal of Low Temperature Physics34, 409 (1979)

    work page 1979

  53. [53]

    Imry and S.-k

    Y. Imry and S.-k. Ma, Random-Field Instability of the Ordered State of Continuous Symmetry, Physical Review Letters35, 1399 (1975)

    work page 1975

  54. [54]

    M. O. Robbins and J. F. Joanny, Contact Angle Hysteresis on Random Surfaces, Europhysics Letters3, 729 (1987)

    work page 1987

  55. [55]

    N. Abid, J. W. Pro, and F. Barthelat, Fracture mechanics of nacre-like materials using discrete- elementmodels: Effectsofmicrostructure, interfacesandrandomness,JournaloftheMechanics and Physics of Solids124, 350 (2019). 35

    work page 2019