For various engineering and industrial applications it is desirable to realize mechanical systems with broadly adjustable elasticity to respond flexibly to the external environment. Here we discover a topology-correlated transition between affine and non-affine regimes in elasticity in both two- and three-dimensional packing-derived networks. Based on this transition, we numerically design and experimentally realize multifunctional systems with adjustable elasticity. Within one system, we achieve solid-like affine response, liquid-like non-affine response and a continuous tunability in between. Moreover, the system also exhibits a broadly tunable Poisson’s ratio from positive to negative values, which is of practical interest for energy absorption and for fracture-resistant materials. Our study reveals a fundamental connection between elasticity and network topology, and demonstrates its practical potential for designing mechanical systems and metamaterials.
This is a preview of subscription content, access via your institution
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 per month
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$259.00 per year
only $21.58 per issue
Rent or buy this article
Get just this article for as long as you need it
Prices may be subject to local taxes which are calculated during checkout
All the raw data of the figures presented in this manuscript can be obtained by accessing the Open Science Framework (https://doi.org/10.17605/OSF.IO/7EQ5Z) or directly visiting https://osf.io/7eq5z/ (ref. 47).
All custom computer code and algorithms used to generate the results reported in the paper are available upon request.
Kadic, M., Bückmann, T., Schittny, R. & Wegener, M. Metamaterials beyond electromagnetism. Rep. Prog. Phys. 76, 126501 (2013).
Berger, J. B., Wadley, H. N. G. & McMeeking, R. M. Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness. Nature 543, 533–537 (2017).
Rocks, J. W. et al. Designing allostery-inspired response in mechanical networks. Proc. Natl Acad. Sci. USA 114, 2520–2525 (2017).
Reid, D. R. et al. Auxetic metamaterials from disordered networks. Proc. Natl Acad. Sci. USA 115, 1384–1390 (2018).
Nicolaou, Z. G. & Motter, A. E. Mechanical metamaterials with negative compressibility transitions. Nat. Mater. 11, 608–613 (2012).
Coulais, C., Sounas, D. & Alù, A. Static non-reciprocity in mechanical metamaterials. Nature 542, 461–464 (2017).
Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2014).
Zheng, X. et al. Ultralight, ultrastiff mechanical metamaterials. Science 344, 1373–1377 (2014).
Coulais, C., Sabbadini, A., Vink, F. & van Hecke, M. Multi-step self-guided pathways for shape-changing metamaterials. Nature 561, 512–515 (2018).
Goodrich, C. P., Liu, A. J. & Nagel, S. R. The principle of independent bond-level response: tuning by pruning to exploit disorder for global behavior. Phys. Rev. Lett. 114, 225501 (2015).
Florijn, B., Coulais, C. & van Hecke, M. Programmable mechanical metamaterials. Phys. Rev. Lett. 113, 175503 (2014).
Majmudar, T. S., Sperl, M., Luding, S. & Behringer, R. P. Jamming transition in granular systems. Phys. Rev. Lett. 98, 058001 (2007).
Dauchot, O., Marty, G. & Biroli, G. Dynamical heterogeneity close to the jamming transition in a sheared granular material. Phys. Rev. Lett. 95, 265701 (2005).
Keys, A. S., Abate, A. R., Glotzer, S. C., & Durian, D. J. Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material. Nat. Phys. 4, 260–264 (2007).
Alexander, S. Amorphous solids: their structure, lattice dynamics and elasticity. Phys. Rep. 296, 65–236 (1998).
Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C. & Levine, D. Geometry of frictionless and frictional sphere packings. Phys. Rev. E 65, 031304 (2002).
Oswald, L., Grosser, S., Smith, D. M. & Kas, J. A. Jamming transitions in cancer. J. Phys. D 50, 483001 (2017).
Mongera, A. et al. A fluid-to-solid jamming transition underlies vertebrate body axis elongation. Nature 561, 401–405 (2018).
Liu, A. J. & Nagel, S. R. Jamming is not just cool any more. Nature 396, 21–22 (1998).
Olsson, P. & Teitel, S. Critical scaling of shear viscosity at the jamming transition. Phys. Rev. Lett. 99, 178001 (2007).
Head, D. A. Critical scaling and aging in cooling systems near the jamming transition. Phys. Rev. Lett. 102, 138001 (2009).
Somfai, E., Roux, J. N., Snoeijer, J. H., Van Hecke, M. & Van Saarloos, W. Elastic wave propagation in confined granular systems. Phys. Rev. E 72, 021301 (2005).
Makse, H. A., Gland, N., Johnson, D. L. & Schwartz, L. M. Why effective medium theory fails in granular materials. Phys. Rev. Lett. 83, 5070 (1999).
Makse, H. A., Johnson, D. L. & Schwartz, L. M. Packing of compressible granular materials. Phys. Rev. Lett. 84, 4160 (2000).
Zhang, Z. X. et al. Thermal vestige of the zero-temperature jamming transition. Nature 459, 230–233 (2009).
Weitz, D. Packing in the spheres. Science 303, 968–969 (2004).
Tkachenko, A. V. & Witten, T. A. Stress propagation through frictionless granular material. Phys. Rev. E 60, 687 (1999).
Ellenbroek, W. G., van Hecke, M. & van Saarloos, W. Jammed frictionless disks: connecting local and global response. Phys. Rev. E 80, 061307 (2009).
Wyart, M., Nagel, S. R. & Witten, T. A. Geometric origin of excess low-frequency vibrational modes in weakly connected amorphous solids. Europhys. Lett. 72, 486 (2005).
Wyart, M., Silbert, L. E., Nagel, S. R. & Witten, T. A. Effects of compression on the vibrational modes of marginally jammed solids. Phys. Rev. E 72, 051306 (2005).
Ellenbroek, W. G., Somfai, E., van Hecke, M. & Van Saarloos, W. Critical scaling in linear response of frictionless granular packings near jamming. Phys. Rev. Lett. 97, 258001 (2006).
Donev, A., Torquato, S. & Stillinger, F. H. Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings. Phys. Rev. E 71, 011105 (2005).
Silbert, L. E., Liu, A. J. & Nagel, S. R. Vibrations and diverging length scales near the unjamming transition. Phys. Rev. Lett. 95, 098301 (2005).
Zhao, C., Tian, K. & Xu, N. New jamming scenario: from marginal jamming to deep jamming. Phys. Rev. Lett. 106, 125503 (2011).
Mao, X., Xu, N. & Lubensky, T. C. Soft modes and elasticity of nearly isostatic lattices: randomness and dissipation. Phys. Rev. Lett. 104, 085504 (2010).
Xu, N., Vitelli, V., Liu, A. J. & Nagel, S. R. Anharmonic and quasi-localized vibrations in jammed solids—modes for mechanical failure. Europhys. Lett. 90, 56001 (2010).
Liu, A. J. & Nagel, S. R. The jamming transition and the marginally jammed solid. Annu. Rev. Condens. Matter Phys. 1, 347–369 (2010).
van Hecke, M. Jamming of soft particles: geometry, mechanics, scaling and isostaticity. J. Phys. Condens. Matter 22, 033101 (2010).
Calladine, C. R. Buckminster Fuller tensegrity structures and Clerk Maxwell rules for the construction of stiff frames. Int. J. Solids Struct. 14, 161–172 (1978).
Wyart, M., Liang, H., Kabla, A. & Mahadevan, L. Elasticity of floppy and stiff random networks. Phys. Rev. Lett. 101, 215501 (2008).
Lubensky, T. C., Kane, C. L., Mao, X., Souslov, A. & Sun, K. Phonons and elasticity in critically coordinated lattices. Rep. Prog. Phys. 78, 073901 (2015).
Trudeau, R. J. Introduction to Graph Theory 64–116 (Dover, 1993).
Nakamura, N. Geometry, Topology and Physics 67–91 (Institute of Physics Publishing, 2008).
Giménez, O. & Noy, M. Asymptotic enumeration and limit laws of planar graphs. J. Am. Math. Soc. 22, 309–329 (2009).
Ellenbroek, W. G., Zeravcic, Z., Van Saarloos, W. & Van Hecke, M. Non-affine response: jammed packings vs spring networks. Europhys. Lett. 87, 34004 (2009).
Li, S. et al. Liquid-induced topological transformations of cellular microstructures. Nature 592, 386–391 (2021).
Shen, X. et al. Achieving Adjustable Elasticity with Non-affine to Affine Transition https://doi.org/10.17605/OSF.IO/7EQ5Z (OSF, 2021).
The experiments were performed at The Chinese University of Hong Kong, and we acknowledge the computational support from the Beijing Computational Science Research Center. L.X. acknowledges financial support from NSFC-12074325, Guangdong Basic and Applied Basic Research Fund 2019A1515011171, GRF-14306518, CRF-C6016-20G, CRF-C1018-17G, CUHK United College Lee Hysan Foundation Research Grant and Endowment Fund Research Grant, CUHK direct grant 4053354. X.X. acknowledges financial support from NSFC 11974038 and U1930402. X.S. acknowledges financial support from Guangdong Basic and Applied Basic Research Foundation 2019A1515110211, and project funding by China Postdoctoral Science Foundation 2020M672824.
The authors declare no competing interests.
Peer review informationNature Materials thanks Larry Howell, Yang Jiao, Zachary Nicolaou and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Figs. 1–11 and discussion.
Supplementary Video 1
Producing negative Poisson’s ratio in our system.
Supplementary Video 2
Adding intersecting bonds does not change the Poisson’s ratio.
Supplementary Video 3
Illustration of bond removal and addition operations in our 2D spring network.
Supplementary Video 4
The comparison of internal strain fields between two 3D networks at z = 7.696 and z = 9.312.
Supplementary Video 5
The comparison of internal strain fields between two 3D networks at z = 9.312 and z = 10.432.
Supplementary Video 6
Illustration of 3D-printed detachable bonds.
Rights and permissions
About this article
Cite this article
Shen, X., Fang, C., Jin, Z. et al. Achieving adjustable elasticity with non-affine to affine transition. Nat. Mater. 20, 1635–1642 (2021). https://doi.org/10.1038/s41563-021-01046-8