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Achieving adjustable elasticity with non-affine to affine transition


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.

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Fig. 1: Non-affine to affine transition in packing-derived networks.
Fig. 2: Understanding the variation of moduli at the single-particle level.
Fig. 3: The numerical design of a network with both affine and non-affine tunability.
Fig. 4: Experimental realization of affine and non-affine tunability.
Fig. 5: Extending the 2D results into three dimensions.

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Data availability

All the raw data of the figures presented in this manuscript can be obtained by accessing the Open Science Framework ( or directly visiting (ref. 47).

Code availability

All custom computer code and algorithms used to generate the results reported in the paper are available upon request.


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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.

Author information

Authors and Affiliations



X.S. and C.F. contributed equally to this research. L.X. conceived the research. X.S., C.F., J.H.Y.L., X.X. and L.X. designed the research. X.S. performed most of the theoretical and numerical analysis. C.F. performed most of the experiments. Z.J., S.T., H.S., H.T. and N.X. helped in the experiments or the simulations. X.S., C.F. and L.X. prepared the manuscript. L.X. and X.X. supervised the research.

Corresponding authors

Correspondence to Jack Hau Yung Lo, Xinliang Xu or Lei Xu.

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The authors declare no competing interests.

Additional information

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.

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Supplementary information

Supplementary Information

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.

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Shen, X., Fang, C., Jin, Z. et al. Achieving adjustable elasticity with non-affine to affine transition. Nat. Mater. 20, 1635–1642 (2021).

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