- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Criticality Enhances the Reinforcement of Disordered Networks by Rigid Inclusions

Jordan L. Shivers1,2,3,4, Jingchen Feng2, and Fred C. MacKintosh1,2,5,6

PDF

Phys. Rev. X 15, 031061 – Published 2 September, 2025

DOI: https://doi.org/10.1103/b9bh-rrj1

Abstract

The mechanical properties of biological materials are spatially heterogeneous. Typical tissues are made up of a spanning fibrous extracellular matrix in which various inclusions, such as living cells, are embedded. While the influence of embedded inclusions on the stiffness of common elastic materials such as rubber has been studied for decades and can be understood in terms of the volume fraction and shape of inclusions, the same is not true for disordered filamentous and fibrous networks. Recent work has shown that, in isolation, such networks exhibit unusual viscoelastic behavior indicative of an underlying mechanical phase transition controlled by network connectivity and strain. How this behavior is modified when inclusions are present is unclear. Here, we present a theoretical and computational study of the influence of rigid inclusions on the mechanics of disordered elastic networks near the connectivity-controlled central-force rigidity transition. Combining scaling theory and coarse-grained simulations, we predict and confirm an anomalously strong dependence of the composite stiffness on inclusion volume fraction, beyond that seen in ordinary composites. This stiffening exceeds the well-established volume-fraction-dependent stiffening expected in conventional composites, e.g., as an elastic analog of the classic volume-fraction-dependent increase in the viscosity of liquids first identified by Einstein. We show that this enhancement is a consequence of the interplay between interparticle spacing and an emergent correlation length, leading to an effective finite-size scaling imposed by the presence of inclusions. We outline the expected scaling of the linear shear modulus and strain fluctuations with the inclusion volume fraction and network connectivity, confirm these predictions in simulations, and discuss potential experimental tests and implications for our predictions in real systems.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Many biological and synthetic materials use a mix of soft networks and stiff inclusions to tune their mechanical properties. Classical composite theory says stiffness should increase steadily with the amount of rigid material added. But soft fibrous networks—like collagen scaffolds in living tissue—often operate near a mechanical critical point, where small changes can lead to dramatic effects. In this study, we show that adding even a very small number of stiff inclusions near this point causes a much larger-than-expected increase in stiffness.

We use a combination of scaling theory and computer simulations to explore this behavior. The rigid inclusions disrupt the network’s structure and limit how far deformations can spread. Normally, near the critical point, deformation correlations grow very large. But these inclusions act like barriers, reducing the correlation length and making the material stiffer. Our analysis shows that the spacing between inclusions and their volume fraction are key factors and that the resulting stiffening goes well beyond the predictions of traditional models.

This work reveals how small numbers of embedded structures—like cells in tissue—can greatly influence a material’s mechanics. Our findings suggest new ways to design materials with highly controllable stiffness for applications like tissue engineering, soft robotics, and responsive materials that can adapt to their environment.

Article Text

References (136)

  1. H. M. Smallwood, Limiting law of the reinforcement of rubber, J. Appl. Phys. 15, 758 (1944).
  2. E. Guth, Theory of Filler reinforcement, J. Appl. Phys. 16, 20 (1945).
  3. A. Einstein, Eine neue Bestimmung der Moleküldimensionen, Ann. Phys. (Berlin) 324, 289 (1906).
  4. A. Einstein, Berichtigung zu meiner Arbeit: “Eine neue Bestimmung der Moleküldimensionen,” Ann. Phys. (Berlin) 339, 591 (1911).
  5. S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Interdisciplinary Applied Mathematics Vol. 16 (Springer, New York, 2002).
  6. Y. Zhao, H. Hu, Y. Huang, H. Liu, C. Yan, C. Xu, R. Zhang, Y. Wang, and Q. Xu, Elasticity-controlled jamming criticality in soft composite solids, Nat. Commun. 15, 1691 (2024).
  7. J. Song, E. Deiss-Yehiely, S. Yesilata, and G. H. McKinley, Strain-stiffening universality in composite hydrogels and soft tissues, Nat. Phys. 21, 1125 (2025).
  8. M. C. Gandikota, K. Pogoda, A. Van Oosten, T. A. Engstrom, A. E. Patteson, P. A. Janmey, and J. M. Schwarz, Loops versus lines and the compression stiffening of cells, Soft Matter 16, 4389 (2020).
  9. J. L. Shivers, J. Feng, A. S. G. Van Oosten, H. Levine, P. A. Janmey, and F. C. MacKintosh, Compression stiffening of fibrous networks with stiff inclusions, Proc. Natl. Acad. Sci. U.S.A. 117, 21037 (2020).
  10. K. Pogoda, L. Chin, P. C. Georges, F. J. Byfield, R. Bucki, R. Kim, M. Weaver, R. G. Wells, C. Marcinkiewicz, and P. A. Janmey, Compression stiffening of brain and its effect on mechanosensing by glioma cells, New J. Phys. 16, 075002 (2014).
  11. M. Perepelyuk, L. Chin, X. Cao, A. Van Oosten, V. B. Shenoy, P. A. Janmey, and R. G. Wells, Normal and fibrotic rat livers demonstrate shear strain softening and compression stiffening: A model for soft tissue mechanics, PLoS One 11, e0146588 (2016).
  12. A. S. G. Van Oosten, X. Chen, L. Chin, K. Cruz, A. E. Patteson, K. Pogoda, V. B. Shenoy, and P. A. Janmey, Emergence of tissue-like mechanics from fibrous networks confined by close-packed cells, Nature (London) 573, 96 (2019).
  13. A. Parker, M. C. Marchetti, M. L. Manning, and J. M. Schwarz, How does the extracellular matrix affect the rigidity of an embedded spheroid?, Phys. Rev. E 111, 044410 (2025).
  14. D. Vader, A. Kabla, D. Weitz, and L. Mahadevan, Strain-induced alignment in collagen gels, PLoS One 4, e5902 (2009).
  15. K. A. Jansen, R. G. Bacabac, I. K. Piechocka, and G. H. Koenderink, Cells Actively stiffen fibrin networks by generating contractile stress, Biophys. J. 105, 2240 (2013).
  16. H. Wang, A. S. Abhilash, C. S. Chen, R. G. Wells, and V. B. Shenoy, Long-range force transmission in fibrous matrices enabled by tension-driven alignment of fibers, Biophys. J. 107, 2592 (2014).
  17. Y. L. Han, A. F. Pegoraro, H. Li, K. Li, Y. Yuan, G. Xu, Z. Gu, J. Sun, Y. Hao, S. K. Gupta, Y. Li, W. Tang, H. Kang, L. Teng, J. J. Fredberg, and M. Guo, Cell swelling, softening and invasion in a three-dimensional breast cancer model, Nat. Phys. 16, 101 (2020).
  18. I. Muntz, M. Fenu, G. J. V. M. Van Osch, and G. H. Koenderink, The role of cell–matrix interactions in connective tissue mechanics, Phys. Biol. 19, 021001 (2022).
  19. T. Zhang, S. Ameen, S. Ghosh, K. Kim, M. Pandey, B. Cheung, M. Thanh, A. E. Patteson, M. Wu, and J. Schwarz, Enhanced extracellular matrix remodeling due to embedded spheroid fluidization, New J. Phys. 27, 073301 (2025).
  20. J. V. Shah and P. A. Janmey, Strain hardening of fibrin gels and plasma clots, Rheol. Acta 36, 262 (1997).
  21. W. A. Lam, O. Chaudhuri, A. Crow, K. D. Webster, T.-D. Li, A. Kita, J. Huang, and D. A. Fletcher, Mechanics and contraction dynamics of single platelets and implications for clot stiffening, Nat. Mater. 10, 61 (2011).
  22. S. J. Pathare, W. Eng, S.-J. J. Lee, and A. K. Ramasubramanian, Fibrin prestress due to platelet aggregation and contraction increases clot stiffness, Biophys. Rep. 1, 100022 (2021).
  23. Y. Qiu, D. R. Myers, and W. A. Lam, The biophysics and mechanics of blood from a materials perspective, Nat. Rev. Mater. 4, 294 (2019).
  24. A. Zakharov, M. Awan, T. Cheng, A. Gopinath, S.-J. J. Lee, A. K. Ramasubramanian, and K. Dasbiswas, Clots reveal anomalous elastic behavior of fiber networks, Sci. Adv. 10, eadh1265 (2024).
  25. P. Friedl and D. Gilmour, Collective cell migration in morphogenesis, regeneration and cancer, Nat. Rev. Mol. Cell Biol. 10, 445 (2009).
  26. C. Bonnans, J. Chou, and Z. Werb, Remodelling the extracellular matrix in development and disease, Nat. Rev. Mol. Cell Biol. 15, 786 (2014).
  27. A. S. Abhilash, B. M. Baker, B. Trappmann, C. S. Chen, and V. B. Shenoy, Remodeling of fibrous extracellular matrices by contractile cells: Predictions from discrete fiber network simulations, Biophys. J. 107, 1829 (2014).
  28. P. Provenzano, Cell force–mediated collagen remodeling in cancer metastasis, in Cell and Matrix Mechanics, edited by R. Kaunas and A. Zemel (CRC Press, Taylor & Francis Group Group, Boca Raton, 2015).
  29. A. Naylor, Y. Zheng, Y. Jiao, and B. Sun, Micromechanical remodeling of the extracellular matrix by invading tumors: Anisotropy and heterogeneity, Soft Matter 19, 9 (2022).
  30. I. Levental, P. C. Georges, and P. A. Janmey, Soft biological materials and their impact on cell function, Soft Matter 3, 299 (2007).
  31. A. J. Engler, S. Sen, H. L. Sweeney, and D. E. Discher, Matrix elasticity directs stem cell lineage specification, Cell 126, 677 (2006).
  32. O. Chaudhuri, L. Gu, D. Klumpers, M. Darnell, S. A. Bencherif, J. C. Weaver, N. Huebsch, H.-p. Lee, E. Lippens, G. N. Duda, and D. J. Mooney, Hydrogels with tunable stress relaxation regulate stem cell fate and activity, Nat. Mater. 15, 326 (2016).
  33. O. Chaudhuri, J. Cooper-White, P. A. Janmey, D. J. Mooney, and V. B. Shenoy, Effects of extracellular matrix viscoelasticity on cellular behaviour, Nature (London) 584, 535 (2020).
  34. M. L. Gardel, I. C. Schneider, Y. Aratyn-Schaus, and C. M. Waterman, Mechanical integration of actin and adhesion dynamics in cell migration, Annu. Rev. Cell Dev. Biol. 26, 315 (2010).
  35. P. Lu, V. M. Weaver, and Z. Werb, The extracellular matrix: A dynamic niche in cancer progression, J. Cell Biol. 196, 395 (2012).
  36. T. M. Koch, S. Münster, N. Bonakdar, J. P. Butler, and B. Fabry, 3D traction forces in cancer cell invasion, PLoS One 7, e33476 (2012).
  37. C. D. Paul, P. Mistriotis, and K. Konstantopoulos, Cancer cell motility: Lessons from migration in confined spaces, Nat. Rev. Cancer 17, 131 (2017).
  38. F. Kai, A. P. Drain, and V. M. Weaver, The extracellular matrix modulates the metastatic journey, Dev. Cell 49, 332 (2019).
  39. S. Feng and P. N. Sen, Percolation on elastic networks: New exponent and threshold, Phys. Rev. Lett. 52, 216 (1984).
  40. C. P. Broedersz and F. C. MacKintosh, Modeling semiflexible polymer networks, Rev. Mod. Phys. 86, 995 (2014).
  41. J. C. Maxwell, On the calculation of the equilibrium and stiffness of frames, London, Edinburgh, Dublin Philos. Mag. J. Sci. 27, 294 (1864).
  42. M. Wyart, H. Liang, A. Kabla, and L. Mahadevan, Elasticity of floppy and stiff random networks, Phys. Rev. Lett. 101, 215501 (2008).
  43. A. Zaccone and E. Scossa-Romano, Approximate analytical description of the nonaffine response of amorphous solids, Phys. Rev. B 83, 184205 (2011).
  44. C. P. Broedersz, X. Mao, T. C. Lubensky, and F. C. MacKintosh, Criticality and isostaticity in fibre networks, Nat. Phys. 7, 983 (2011).
  45. K. G. Wilson, Problems in physics with many scales of length, Sci. Am. 241, No. 2, 158 (1979).
  46. K. Binder, Finite size effects on phase transitions, Ferroelectrics 73, 43 (1987).
  47. J. Cardy, Finite-Size Scaling (Elsevier Science, Amsterdam, 2014).
  48. C. P. Goodrich, A. J. Liu, and J. P. Sethna, Scaling ansatz for the jamming transition, Proc. Natl. Acad. Sci. U.S.A. 113, 9745 (2016).
  49. C. P. Goodrich, A. J. Liu, and S. R. Nagel, Finite-size scaling at the jamming transition, Phys. Rev. Lett. 109, 095704 (2012).
  50. J. L. Shivers, S. Arzash, A. Sharma, and F. C. MacKintosh, Scaling theory for mechanical critical behavior in fiber networks, Phys. Rev. Lett. 122, 188003 (2019).
  51. K. E. Kasza, A. C. Rowat, J. Liu, T. E. Angelini, C. P. Brangwynne, G. H. Koenderink, and D. A. Weitz, The cell as a material, Curr. Opin. Cell Biol. 19, 101 (2007).
  52. F. Burla, Y. Mulla, B. E. Vos, A. Aufderhorst-Roberts, and G. H. Koenderink, From mechanical resilience to active material properties in biopolymer networks, Nat. Rev. Phys. 1, 249 (2019).
  53. F. C. MacKintosh, J. Käs, and P. A. Janmey, Elasticity of semiflexible biopolymer networks, Phys. Rev. Lett. 75, 4425 (1995).
  54. K. Kroy and E. Frey, Force-extension relation and plateau modulus for wormlike chains, Phys. Rev. Lett. 77, 306 (1996).
  55. R. Satcher and C. Dewey, Theoretical estimates of mechanical properties of the endothelial cell cytoskeleton, Biophys. J. 71, 109 (1996).
  56. J. W. Weisel, The mechanical properties of fibrin for basic scientists and clinicians, Biophys. Chem. 112, 267 (2004).
  57. M. L. Gardel, J. H. Shin, F. C. MacKintosh, L. Mahadevan, P. Matsudaira, and D. A. Weitz, Elastic behavior of cross-linked and bundled actin networks, Science 304, 1301 (2004).
  58. C. Storm, J. J. Pastore, F. C. MacKintosh, T. C. Lubensky, and P. A. Janmey, Nonlinear elasticity in biological gels, Nature (London) 435, 191 (2005).
  59. P. A. Janmey, J. P. Winer, and J. W. Weisel, Fibrin gels and their clinical and bioengineering applications, J. R. Soc. Interface 6, 1 (2009).
  60. R. H. Pritchard, Y. Y. Shery Huang, and E. M. Terentjev, Mechanics of biological networks: From the cell cytoskeleton to connective tissue, Soft Matter 10, 1864 (2014).
  61. K. A. Jansen, A. J. Licup, A. Sharma, R. Rens, F. C. MacKintosh, and G. H. Koenderink, The role of network architecture in collagen mechanics, Biophys. J. 114, 2665 (2018).
  62. D. A. Fletcher and R. D. Mullins, Cell mechanics and the cytoskeleton, Nature (London) 463, 485 (2010).
  63. L. Blanchoin, R. Boujemaa-Paterski, C. Sykes, and J. Plastino, Actin dynamics, architecture, and mechanics in cell motility, Physiol. Rev. 94, 235 (2014).
  64. M. Bonucci, T. Shu, and L. J. Holt, How it feels in a cell, Trends Cell Biol. 33, 924 (2023).
  65. J. W. Weisel, Enigmas of blood clot elasticity, Science 320, 456 (2008).
  66. H.-C. Flemming and J. Wingender, The biofilm matrix, Nat. Rev. Microbiol. 8, 623 (2010).
  67. C. D. Nadell, K. Drescher, N. S. Wingreen, and B. L. Bassler, Extracellular matrix structure governs invasion resistance in bacterial biofilms, ISME J. 9, 1700 (2015).
  68. H.-C. Flemming, E. D. van Hullebusch, T. R. Neu, P. H. Nielsen, T. Seviour, P. Stoodley, J. Wingender, and S. Wuertz, The biofilm matrix: Multitasking in a shared space, Nat. Rev. Microbiol. 21, 70 (2023).
  69. T. R. Cox, The matrix in cancer, Nat. Rev. Cancer 21, 217 (2021).
  70. C. F. Guimarães, L. Gasperini, A. P. Marques, and R. L. Reis, The stiffness of living tissues and its implications for tissue engineering, Nat. Rev. Mater. 5, 351 (2020).
  71. D. A. Head, A. J. Levine, and F. C. MacKintosh, Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks, Phys. Rev. E 68, 061907 (2003).
  72. D. A. Head, A. J. Levine, and F. C. MacKintosh, Mechanical response of semiflexible networks to localized perturbations, Phys. Rev. E 72, 061914 (2005).
  73. C. Heussinger and E. Frey, Floppy modes and nonaffine deformations in random fiber networks, Phys. Rev. Lett. 97, 105501 (2006).
  74. A. S. Shahsavari and R. C. Picu, Size effect on mechanical behavior of random fiber networks, Int. J. Solids Struct. 50, 3332 (2013).
  75. H. Yang, E. Berthier, C. Li, P. Ronceray, Y. L. Han, C. P. Broedersz, S. Cai, and M. Guo, Local response and emerging nonlinear elastic length scale in biopolymer matrices, Proc. Natl. Acad. Sci. U.S.A. 120, e2304666120 (2023).
  76. L. C. Davis, K. C. Hass, J. Chen, and M. F. Thorpe, Elastic moduli of composites with random, rigid inclusions, Appl. Mech. Rev. 47, S5 (1994).
  77. J. F. Brady, The Einstein viscosity correction in n dimensions, Int. J. Multiphase Flow 10, 113 (1984).
  78. J. L. Shivers, S. Arzash, and F. C. MacKintosh, Nonlinear Poisson effect governed by a mechanical critical transition, Phys. Rev. Lett. 124, 038002 (2020).
  79. H. S. Seung and D. R. Nelson, Defects in flexible membranes with crystalline order, Phys. Rev. A 38, 1005 (1988).
  80. S. Arzash, J. L. Shivers, and F. C. MacKintosh, Finite size effects in critical fiber networks, Soft Matter 16, 6784 (2020).
  81. R. Rens and E. Lerner, Rigidity and auxeticity transitions in networks with strong bond-bending interactions, Eur. Phys. J. E 42, 114 (2019).
  82. A. W. Lees and S. F. Edwards, The computer study of transport processes under extreme conditions, J. Phys. C 5, 1921 (1972).
  83. J. Nocedal and S. J. Wright, Numerical Optimization (Springer Science & Business Media, New York, 2006).
  84. E. A. Gosselin, H. B. Eppler, J. S. Bromberg, and C. M. Jewell, Designing natural and synthetic immune tissues, Nat. Mater. 17, 484 (2018).
  85. T. A. Vilgis, Soft matter food physics—the physics of food and cooking, Rep. Prog. Phys. 78, 124602 (2015).
  86. B. I. Zielbauer, N. Schönmehl, N. Chatti, and T. A. Vilgis, Networks: From Rubbers to Food, in Designing of Elastomer Nanocomposites: From Theory to Applications, Advances in Polymer Science, edited by K. W. Stöckelhuber, A. Das, and M. Klüppel (Springer International Publishing, Cham, 2017), pp. 187–233.
  87. L. A. MacQueen, C. G. Alver, C. O. Chantre, S. Ahn, L. Cera, G. M. Gonzalez, B. B. O’Connor, D. J. Drennan, M. M. Peters, S. E. Motta, J. F. Zimmerman, and K. K. Parker, Muscle tissue engineering in fibrous gelatin: Implications for meat analogs, npj Sci. Food 3, 20 (2019).
  88. A. J. T. M. Mathijssen, M. Lisicki, V. N. Prakash, and E. J. L. Mossige, Culinary fluid mechanics and other currents in food science, Rev. Mod. Phys. 95, 025004 (2023).
  89. M. Das, D. A. Quint, and J. M. Schwarz, Redundancy and cooperativity in the mechanics of compositely crosslinked filamentous networks, PLoS One 7, e35939 (2012).

  90. It is important to note, however, that the same is not true for strain-controlled criticality.

  91. M. Sheinman, C. P. Broedersz, and F. C. MacKintosh, Nonlinear effective-medium theory of disordered spring networks, Phys. Rev. E 85, 021801 (2012).
  92. S. Arzash, J. L. Shivers, A. J. Licup, A. Sharma, and F. C. MacKintosh, Stress-stabilized subisostatic fiber networks in a ropelike limit, Phys. Rev. E 99, 042412 (2019).
  93. B. Cui, G. Ruocco, and A. Zaccone, Theory of elastic constants of athermal amorphous solids with internal stresses, Granular Matter 21, 69 (2019).
  94. H. Hatami-Marbini and M. Rohanifar, Nonlinear mechanical properties of prestressed branched fibrous networks, Biophys. J. 120, 527 (2021).
  95. M. Vahabi, A. Sharma, A. J. Licup, A. S. G. Van Oosten, P. A. Galie, P. A. Janmey, and F. C. MacKintosh, Elasticity of fibrous networks under uniaxial prestress, Soft Matter 12, 5050 (2016).
  96. G. H. Koenderink, Z. Dogic, F. Nakamura, P. M. Bendix, F. C. MacKintosh, J. H. Hartwig, T. P. Stossel, and D. A. Weitz, An active biopolymer network controlled by molecular motors, Proc. Natl. Acad. Sci. U.S.A. 106, 15192 (2009).
  97. C. P. Broedersz and F. C. MacKintosh, Molecular motors stiffen non-affine semiflexible polymer networks, Soft Matter 7, 3186 (2011).
  98. M. Sheinman, C. P. Broedersz, and F. C. MacKintosh, Actively stressed marginal networks, Phys. Rev. Lett. 109, 238101 (2012).
  99. M. Sarkar, B. M. Burkel, S. M. Ponik, and J. Notbohm, Unexpected softening of a fibrous matrix by contracting inclusions, Acta Biomater. 177, 253 (2024).
  100. S. Goren, Y. Koren, X. Xu, and A. Lesman, Elastic anisotropy governs the range of cell-induced displacements, Biophys. J. 118, 1152 (2020).
  101. H. Hatami-Marbini and M. Rohanifar, Mechanical response of composite fiber networks subjected to local contractile deformation, Int. J. Solids Struct. 228, 111045 (2021).
  102. S. Heyden, R. W. Style, and E. R. Dufresne, Strain stiffening elastomers with swelling inclusions, Soft Matter 19, 4385 (2023).
  103. G. Chaudhary, N. A. Bharadwaj, P. V. Braun, and R. H. Ewoldt, Exploiting nonlinear elasticity for anomalous magnetoresponsive stiffening, ACS Macro Lett. 9, 1632 (2020).
  104. G. Chaudhary, A. Ghosh, N. A. Bharadwaj, J. G. Kang, P. V. Braun, K. S. Schweizer, and R. H. Ewoldt, Thermoresponsive stiffening with microgel particles in a semiflexible fibrin network, Macromolecules 52, 3029 (2019).
  105. E. Straube and D. Matzen, Determination of crosslink-densities by stress-strain measurements and modelling of the stress-strain behavior, Kautsch. Gummi Kuntst. 45, 264 (1992).
  106. G. Huber and T. A. Vilgis, Universal properties of filled rubbers: Mechanisms for reinforcement on different length scales, Kautschuk + Gummi Kunststoffe 52, 102 (1999).
  107. G. Huber, On the mechanism of hydrodynamic reinforcement in elastic composites, Macromolecules 35, 9204 (2002).
  108. M. Klüppel, The role of disorder in filler reinforcement of elastomers on various length scales, in Filler-Reinforced Elastomers Scanning Force Microscopy, Advances in Polymer Science, edited by B. Capella, M. Geuss, M. Klüppel, M. Munz, E. Schulz, and H. Sturm (Springer, Berlin, Heidelberg, 2003), pp. 1–86.
  109. M. Mermet-Guyennet, J. Gianfelice De Castro, H. Varol, M. Habibi, B. Hosseinkhani, N. Martzel, R. Sprik, M. Denn, A. Zaccone, S. Parekh, and D. Bonn, Size-dependent reinforcement of composite rubbers, Polymer 73, 170 (2015).
  110. H. S. Varol, F. Meng, B. Hosseinkhani, C. Malm, D. Bonn, M. Bonn, A. Zaccone, and S. H. Parekh, Nanoparticle amount, and not size, determines chain alignment and nonlinear hardening in polymer nanocomposites, Proc. Natl. Acad. Sci. U.S.A. 114, E3170 (2017).
  111. I. Dellatolas, M. Bantawa, B. Damerau, M. Guo, T. Divoux, E. Del Gado, and I. Bischofberger, Local mechanism governs global reinforcement of nanofiller-hydrogel composites, ACS Nano 17, 20939 (2023).
  112. J. Steck, J. Kim, Y. Kutsovsky, and Z. Suo, Multiscale stress deconcentration amplifies fatigue resistance of rubber, Nature (London) 624, 303 (2023).
  113. L. Mullins, The thixotropic behavior of carbon black in rubber., J. Phys. Colloid Chem. 54, 239 (1950).
  114. L. Mullins and N. R. Tobin, Stress softening in rubber vulcanizates. Part I. Use of a strain amplification factor to describe the elastic behavior of filler-reinforced vulcanized rubber, J. Appl. Polym. Sci. 9, 2993 (1965).
  115. M. R. Islam and R. C. Picu, Random fiber networks with inclusions: The mechanism of reinforcement, Phys. Rev. E 99, 063001 (2019).
  116. M. R. Islam and R. C. Picu, Random fiber networks with inclusions: The effect of the inclusion stiffness, Mech. Soft Mater. 1, 14 (2019).
  117. R. W. Style, R. Boltyanskiy, B. Allen, K. E. Jensen, H. P. Foote, J. S. Wettlaufer, and E. R. Dufresne, Stiffening solids with liquid inclusions, Nat. Phys. 11, 82 (2015).
  118. X. Wei, J. Zhou, Y. Wang, and F. Meng, Modeling elastically mediated liquid-liquid phase separation, Phys. Rev. Lett. 125, 268001 (2020).
  119. K. A. Rosowski, T. Sai, E. Vidal-Henriquez, D. Zwicker, R. W. Style, and E. R. Dufresne, Elastic ripening and inhibition of liquid–liquid phase separation, Nat. Phys. 16, 422 (2020).
  120. D. S. W. Lee, A. R. Strom, and C. P. Brangwynne, The mechanobiology of nuclear phase separation, APL Bioeng. 6, 021503 (2022).
  121. P. Ronceray, S. Mao, A. Košmrlj, and M. P. Haataja, Liquid demixing in elastic networks: Cavitation, permeation, or size selection?, Europhys. Lett. 137, 67001 (2022).
  122. J. X. Liu, M. P. Haataja, A. Košmrlj, S. S. Datta, C. B. Arnold, and R. D. Priestley, Liquid–liquid phase separation within fibrillar networks, Nat. Commun. 14, 6085 (2023).
  123. M. Kothari and T. Cohen, The crucial role of elasticity in regulating liquid–liquid phase separation in cells, Biomech. Model. Mechanobiol. 22, 645 (2023).
  124. C. Fernández-Rico, S. Schreiber, H. Oudich, C. Lorenz, A. Sicher, T. Sai, V. Bauernfeind, S. Heyden, P. Carrara, L. D. Lorenzis, R. W. Style, and E. R. Dufresne, Elastic microphase separation produces robust bicontinuous materials, Nat. Mater. 23, 124 (2024).
  125. Y. Qiang, C. Luo, and D. Zwicker, Nonlocal elasticity yields equilibrium patterns in phase separating systems, Phys. Rev. X 14, 021009 (2024).
  126. K. Gersh, C. Nagaswami, and J. Weisel, Fibrin network structure and clot mechanical properties are altered by incorporation of erythrocytes, Thrombosis Haemostasis 102, 1169 (2009).
  127. Y. Jiang, Y. Cui, Y. Li, Z. Liu, C. Ness, and R. Seto, Filled colloidal gel rheology: Strengthening, stiffening, and tunability, J. Rheol. 69, 35 (2025).
  128. A. Zaccone, Elastic deformations in covalent amorphous solids, Mod. Phys. Lett. B 27, 1330002 (2013).
  129. A. Sharma, A. J. Licup, K. A. Jansen, R. Rens, M. Sheinman, G. H. Koenderink, and F. C. MacKintosh, Strain-controlled criticality governs the nonlinear mechanics of fibre networks, Nat. Phys. 12, 584 (2016).
  130. S. Chen, T. Markovich, and F. C. MacKintosh, Field theory for mechanical criticality in disordered fiber networks, Phys. Rev. Lett. 133, 028201 (2024).
  131. J. L. Shivers, A. Sharma, and F. C. MacKintosh, Strain-controlled critical slowing down in the rheology of disordered networks, Phys. Rev. Lett. 131, 178201 (2023).
  132. Y. Song and Q. Zheng, A guide for hydrodynamic reinforcement effect in nanoparticle-filled polymers, Crit. Rev. Solid State Mater. Sci. 41, 318 (2016).
  133. T. Shen, C. S. O’Hern, and M. D. Shattuck, Contact percolation transition in athermal particulate systems, Phys. Rev. E 85, 011308 (2012).
  134. M. G. Yucht, M. Sheinman, and C. P. Broedersz, Dynamical behavior of disordered spring networks, Soft Matter 9, 7000 (2013).
  135. D. T. Lee and B. J. Schachter, Two algorithms for constructing a Delaunay triangulation, Int. J. Comp. Inf. Sci. 9, 219 (1980).
  136. R. Rens, C. Villarroel, G. Düring, and E. Lerner, Micromechanical theory of strain stiffening of biopolymer networks, Phys. Rev. E 98, 062411 (2018).

Outline

Information

Phys. Rev. X 15, 031061– Published 2 September, 2025
  • Received 14 August 2024
  • Accepted 11 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation