# Auxetic metamaterials from disordered networks

^{a}Institute for Molecular Engineering, University of Chicago, Chicago, IL 60637;^{b}Department of Physics, University of Chicago, Chicago, IL 60637;^{c}Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL 60439;^{d}Department of Physics, University of Pennsylvania, Philadelphia, PA 19104;^{e}Materials Science Division, Argonne National Laboratory, Lemont, IL 60439

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Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved December 27, 2017 (received for review October 4, 2017)

## Significance

Recent work indicates that selective pruning of disordered networks of nodes connected by bonds can generate materials with nontrivial mechanical properties, including auxetic networks having a negative Poisson’s ratio ν. Until now, auxetic networks created based on this strategy have not been successfully realized in experiment. Here a model that includes angle-bending forces and the experimental boundary conditions is introduced for pruning-based design of auxetic materials. By pruning the appropriate bonds, ν can be tuned to values approaching the lower mechanical limit of −1, and the corresponding laboratory networks exhibit good agreement with model predictions. Optimization algorithms are then used to show that highly auxetic materials can be engineered from inhomogeneous bonds and nodes that exhibit distinct mechanical characteristics.

## Abstract

Recent theoretical work suggests that systematic pruning of disordered networks consisting of nodes connected by springs can lead to materials that exhibit a host of unusual mechanical properties. In particular, global properties such as Poisson’s ratio or local responses related to deformation can be precisely altered. Tunable mechanical responses would be useful in areas ranging from impact mitigation to robotics and, more generally, for creation of metamaterials with engineered properties. However, experimental attempts to create auxetic materials based on pruning-based theoretical ideas have not been successful. Here we introduce a more realistic model of the networks, which incorporates angle-bending forces and the appropriate experimental boundary conditions. A sequential pruning strategy of select bonds in this model is then devised and implemented that enables engineering of specific mechanical behaviors upon deformation, both in the linear and in the nonlinear regimes. In particular, it is shown that Poisson’s ratio can be tuned to arbitrary values. The model and concepts discussed here are validated by preparing physical realizations of the networks designed in this manner, which are produced by laser cutting 2D sheets and are found to behave as predicted. Furthermore, by relying on optimization algorithms, we exploit the networks’ susceptibility to tuning to design networks that possess a distribution of stiffer and more compliant bonds and whose auxetic behavior is even greater than that of homogeneous networks. Taken together, the findings reported here serve to establish that pruned networks represent a promising platform for the creation of unique mechanical metamaterials.

When one stretches a material along one axis, intuition suggests that the material will contract in the orthogonal lateral directions. For most natural and synthetic materials, this intuition is confirmed by experiment. This behavior is quantified by Poisson’s ratio, ν, which for a deformed material is defined as the negative ratio of the material’s lateral strain to its axial strain. In linear elastic theory for an isotropic sample, Poisson’s ratio is a monotonic function of the ratio of the material’s shear modulus, G, to its bulk modulus, B. In two dimensions

Auxetic materials have been formed through a variety of preparation protocols. Under special processing conditions, polymer foams and fibers, for example, can exhibit negative Poisson’s ratios (4, 11⇓⇓–14). Auxetic foams, in particular, can be formed through a process of heating and sintering fine particles of ultrahigh molecular weight polyethylene (11, 15), leading to structures of nodes connected by thin fibrils which collapse isotropically when compressed. Such structures are termed “reentrant” and are a common motif in auxetic materials (11, 16). When compressed uniaxially, these nodes and fibrils undergo complex rearrangements that give rise to their auxetic behavior. As materials approach the lower limit of Poisson’s ratio, their hardness, or resistance to a small indentation, is predicted to increase rapidly (17). This prediction is confirmed in the case of ultrahigh molecular weight polyethylene, where the hardness of the auxetic material far exceeds that of a nonauxetic but otherwise equivalent foam (4).

The node and fibril structures common in auxetic polymer foams can be thought of as networks consisting of nodes connected by bonds. A central, common feature of past efforts to design auxetic materials in both theory and experiment, however, has been a reliance on regular, ordered lattices. Such lattices include the double-arrowhead structure (8, 18), star honeycomb structures (19), reentrant honeycombs (20, 21), and others (22). Building on recent theoretical arguments (23⇓–25), in this work we focus on disordered, random networks.

In the linear regime, the bulk modulus, B, or the shear modulus, G, of a network is proportional to the sum of the potential energies that are stored in each bond when the network is compressed or sheared. The decrease in B or G when the

Furthermore, there is little correlation between the values of

Here we address that question by introducing a mechanical model of disordered networks that incorporates the effects of angle bending in a unique way. The model is minimally complex, and it is parameterized by comparison with experimental data for simple, random disordered networks. By adopting a pruning strategy that identifies and removes select bonds from these networks, it is shown that it is possible to reach Poisson’s ratios as low as

## Models

### Simulation Model.

To have well-defined starting configurations, we base our networks on jammed packings of frictionless spheres at zero temperature (27). We note, however, that the results are not necessarily confined to this choice of starting conditions. Spherical particles are initially placed at random positions within the simulation area. Particles i and j experience harmonic repulsions

Two particles are considered to be in contact when *Methods*.

The total energy of a network under stress is the sum of two terms: a compressive component given by Eq. **2** and a bending component, given by Eq. **3**. Note that the compressive strength is scaled by

Fig. 2 shows representative realizations of 2D disordered networks consisting of nodes connected by bonds, before and after pruning.

In two dimensions, there are two independent shear moduli—one associated with simple shear and one with pure shear. The modulus associated with simple shear influences the value of ν that is measured when the material is deformed by pulling or pushing from opposite corners. The modulus associated with pure shear relates to the value of ν measured when the material is uniaxially compressed or expanded in x or y, as shown in *SI Appendix*, Fig. S5. In this study we focus primarily on algorithms that influence only the modulus associated with pure shear since this can be more easily measured in our experiments. However, we also show that isotropic auxetic networks can be created using similar algorithms which consider bond contributions to both pure and simple shear. Such materials are auxetic with respect to any uniaxial deformation. G and B are measured as described in *Methods*.

## Results

### Bond Response Distributions.

In an amorphous network the distributions of

A second crucial condition for successful pruning is that

Fig. 3 *A* and *B* shows the probability distributions *Methods*. As

To facilitate effective pruning, bond response distributions must be not only broad, but also uncorrelated. Fig. 3*C* shows the Pearson correlation coefficient for

### Pruning.

For the iterative pruning strategy adopted here, at each iteration the lowest

Fig. 4 shows Poisson’s ratios, ν, for networks having different values of

Poisson’s ratio is determined by introducing a small strain of magnitude *SI Appendix*, Fig. S2,

Several interesting features are apparent in the pruning progression shown in Fig. 4. First, even before there is any pruning, Poisson’s ratio of the networks decreases from 0.51 at *C*. Networks with

To explore further how ν changes with pruning, we examine G and B of networks as they are pruned, as shown in Fig. 5. In two dimensions, linear elastic theory states

### Structural Features.

Fully pruned networks (*B*. Here, a reentrant node is defined as one having an angle between adjacent bonds that is greater than 180°. As can be seen in Fig. 2, reentrant nodes manifest as concave angles in polygons within the network. Such polygons tend to collapse inward at reentrant nodes when compressed. A sufficient number of such polygons could lead to globally auxetic behavior. As can be seen in Fig. 4*B*, more auxetic networks exhibit a higher percentage of reentrant nodes. This structural motif therefore provides a basis for design of amorphous or otherwise disordered networks that are auxetic and isotropic. In this calculation we did not classify nodes with only two bonds as reentrant, although we arrive at qualitatively the same conclusions if they are included.

### Experimental Validation.

Experimental pruned networks are made of laser-cut sheets of rubber (26) as described in *Methods*. The strength of bond bending, *Inset*. We focus on the bond shape shown in Fig. 6. The deformation of such networks can be described quantitatively by our model with the value

We now examine the response of a particular network formed with *A* shows a network compressed with *B* directly compares experimental and simulated configurations at *B* is isotropic and will be auxetic with respect to any strain. By iteratively pruning the minimum

### Angle-Bending Stiffness.

We have focused only on values of *SI Appendix*, Fig. S1.

### Physical Insights from Model Improvement.

Three features distinguish the model used in this work from that of previous attempts: (*i*) the use of finite rather than periodic simulations, (*ii*) the use of fixed boundary conditions as used in experiment, and (*iii*) most importantly the addition of an angle-bending potential. To demonstrate the importance of the angle-bending term, we study a network which has been pruned to *SI Appendix*, Fig. S8. By picking a value of

Our results suggest that weaker angle-bending forces allow for more dramatic deformations of the concave polygons present in these networks. Assigning a larger

### Stress–Strain Behavior.

For a variety of impact-mitigation applications, it is of interest to develop materials that display a relatively constant stress–strain behavior. Such materials can absorb more energy while maintaining lower applied forces and thus reduce the possibility of damage. As shown in *SI Appendix*, Fig. S3, pruned networks display nearly constant stress past *SI Appendix*, Fig. S2. We find that the linear response framework applies well until roughly

### Bond Strength Optimization.

Up to this point, we have relied on homogeneous materials, with identical bonds, for all calculations and experiments. In what follows, we modify the strength of individual bonds as a means for decreasing ν in networks composed of bonds with different stiffnesses. This process can be mimicked in experiment by modifying the thickness or material of a given bond. We implement a simple optimization algorithm that iteratively strengthens by 10% the bond leading to the greatest decrease in ν. Both the compressive modulus and the bending modulus of a particular bond are increased when a bond is strengthened. We examine a particular network with *Inset*.

To validate the predictions of our simulations, we also prepared an experimental realization of this optimized network. For simplicity, bonds strengthened by a factor of 5 or greater were made thicker, and others were left unchanged. The corresponding experimental values of ν are shown in Fig. 9, showing a decrease in ν of 0.059 at

## Conclusion

In summary, we have established that it is possible to create designer auxetic materials from amorphous networks. The pruning strategy that we have proposed does not depend on the initial configuration but rather relies on measuring aspects of local response to a globally applied deformation. As such, it may apply more generally to networks based on a variety of initial preparation protocols and not just those based on jamming. The models and concepts introduced in this work have been validated through a concerted program of design, computation, and laboratory experimentation. Amorphous networks are shown to offer a number of control parameters that can be tuned to achieve particular mechanical responses. It is found, for example, that a network’s propensity to be made auxetic depends on both the network’s original coordination number and the relative resistance to angle bending. More pliable networks yield the lowest Poisson’s ratios due to their wide bond response distribution and their low response correlation. Stiffer networks are less amenable to pruning and show only limited changes of their Poisson’s ratio through pruning. By relying on bond-strength optimization schemes, however, it is possible to alter Poisson’s ratio of networks with stiff bonds considerably, thereby providing a strategy to alter not only how auxetic a material is, but also its intrinsic stiffness. While the results presented here have been limited to 2D networks, the concepts and strategies proposed should be equally applicable to three dimensions, where we can use 3D printing to realize our computer models. We therefore anticipate that these networks could be potentially useful for applications involving additive manufacturing. Using appropriately designed nanoparticles, it is also conceivable that one could form auxetic materials through a self-assembly process.

## Methods

### Simulation Methods.

Simulated networks are generated as described in *Models*. A harmonic wall coefficient of 2.0 is used to compress particles. To measure ν,

### Experimental Methods.

Experimental networks are constructed of laser-cut silicone rubber sheets with a Shore value of *A*. Poisson’s ratio is determined by applying a uniaxial compression in the y direction and measuring the resulting lateral strain.

## Acknowledgments

We acknowledge Carl Goodrich and Daniel Hexner for their helpful discussions. The design and fabrication of mechanical metamaterials based on random networks was supported by the University of Chicago Materials Research Science and Engineering Center, which is funded by the National Science Foundation under Award DMR-1420709. The development of auxetic systems for impact mitigation applications and the corresponding materials optimization strategies presented here are supported by the Center for Hierarchical Materials Design (CHiMaD), which is supported by the National Institute of Standards and Technology, US Department of Commerce, under financial assistance award 70NANB14H012.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: depablo{at}uchicago.edu.

Author contributions: D.R.R., N.P., J.M.W., H.M.J., A.J.L., S.R.N., and J.J.d.P. designed research; D.R.R. and N.P. performed research; D.R.R. and N.P. contributed new reagents/analytic tools; D.R.R., N.P., S.R.N., and J.J.d.P. analyzed data; J.M.W. assisted with parallel software implementation; and D.R.R., N.P., S.R.N., and J.J.d.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1717442115/-/DCSupplemental.

Published under the PNAS license.

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