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Auxetic metamaterials from disordered networks
Daniel R Reid, Nidhi Pashine, Justin M Wozniak, Heinrich M Jaeger, Andrea J Liu, Sidney R Nagel, Juan J de Pablo
TL;DR
Pruning-based auxetic designs lacked successful experimental realization, motivating a more realistic disordered-network model with angle bending and experimental boundary conditions. The paper develops sequential pruning and optimization strategies, validates designed networks computationally and experimentally, and reports tunable Poisson’s ratios and enhanced auxetic behavior.
Problem
Experimental attempts to create auxetic materials from pruning-based disordered-network theories had not been successful, despite the potential for tunable mechanical responses.
Method
The paper combines disordered networks with angle-bending forces, experimental boundary conditions, sequential bond pruning, optimization algorithms, simulations, and laser-cut physical realizations.
Results
The approach produces designer auxetic materials whose Poisson’s ratio can be tuned, with physical realizations behaving as predicted.
Takeaways & Limitations
Amorphous pruned networks provide control parameters for engineering mechanical responses, including auxetic behavior and networks with tailored bond stiffness distributions.
Takeaways & Limitations
The study focuses primarily on algorithms influencing pure shear because that response is easier to measure experimentally; isotropic auxetic networks are treated separately.
Abstract
from arXiv · showhide
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 the 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 new and 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 non-linear regimes. In particular, it is shown that the 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 two-dimensional 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 posses 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 novel mechanical metamaterials.
Models
The model represents amorphous networks as mechanically stable configurations of particles connected by compressive bonds and angle-bending constraints. It measures their mechanical response under boundary conditions designed to represent experiments.
- Network construction: Networks are generated from disordered jammed packings of frictionless spheres and relaxed into zero-temperature, mechanically stable configurations.Particles are assigned one of four radii, producing an amorphous packing.
- Bond model: Bonds are placed between contacting particle centers after removing the soft-sphere potential, and their compression energy is scaled by the bond’s equilibrium length.This scaling represents a physical strut of constant thickness.
- Angle-bending model: Angle-bending constraints are introduced with a unit vector at each node and a quadratic energy cost for deviations from equilibrium angles.The bending coefficient kang sets the angular energy scale and is calibrated against experimental networks.
- Mechanical response: The model distinguishes pure and simple shear, which govern Poisson’s-ratio responses under uniaxial deformation and opposite-corner loading, respectively.The study primarily targets pure shear because it is easier to measure experimentally, while also constructing isotropic auxetic networks.
Results
Selective pruning exploits broad, weakly correlated bond responses to tune the Poisson’s ratio of disordered networks. The resulting networks exhibit auxetic behavior in simulations and experiments, with structural and mechanical properties controlled by coordination and bond-bending stiffness.
- Bond Response Distributions: Broad distributions of ∆Bi and ∆Gi, together with weak correlation, enable pruning bonds that minimally affect G while strongly reducing B.This selective removal increases G/B and decreases ν toward a chosen target.
- Bond Response Distributions: As Z0 increases, bond-response distributions narrow and correlations rise after Z0 = 5.2, reducing the networks’ amenability to pruning.Networks pruned from Z0 = 5.2 provide the most favorable balance for tuning.
- Pruning: During pruning, B initially decreases while G remains nearly constant; after substantial pruning, G also decreases, and ν reaches its minimum near Z = 3.0.The modulus progression explains the changing slope of ν during pruning.
- Pruning: Networks initialized at Z0 = 5.2 reach an average ν = −0.62 after pruning, while one network reaches ν = −0.79.The pruning protocol removes the lowest ∆Gi bond at each iteration; higher initial coordination does not necessarily improve the final response because distributions narrow and correlations increase.
- Pruning: The networks’ initial ν varies from 0.51 at Z0 = 4.0 to 0.21 at Z0 = 5.9, a range of ∆ν = 0.3.The initial pruning slope also decreases from dν/dZ = 0.47 to 0.14 across this Z0 range.
Conclusion
The study establishes that amorphous networks can be engineered into designer auxetic materials through modeling, computation, optimization, and laboratory validation. Network coordination, angle-bending resistance, and bond-strength optimization provide control over Poisson’s ratio and stiffness.
- Amorphous networks can be designed into auxetic materials and validated through coordinated modeling, computation, and laboratory experiments.
- More pliable networks reach lower Poisson’s ratios because they have broader bond-response distributions and weaker response correlations.
- Stiffer networks are less responsive to pruning, but bond-strength optimization can substantially alter their Poisson’s ratio and intrinsic stiffness.
- The study’s applications include mechanical metamaterial design and impact-mitigation systems.
Methods
The methods combine simulations of disordered networks with experimentally fabricated laser-cut silicone sheets. Poisson’s ratio is measured under controlled compression while angle-bending resistance is varied through bond geometry.
- Simulations measure Poisson’s ratio by applying a small y-direction strain and recording transverse strain at the network edges.The applied strain is ϵy = 1 × 10−4.
- Bulk and shear moduli are measured using small uniform, pure-shear, and simple-shear deformations.The protocols use strains of magnitude 1 × 10−4.
- Experimental networks are laser-cut from 1.5 mm silicone rubber sheets with Shore A70 hardness.
- Angle-bending resistance is tuned by narrowing or widening bond sections near the nodes.
- Experimental Poisson’s ratio is obtained from uniaxial y-direction compression and the resulting lateral strain.
Supplementary Information
The supplementary analyses show how bending stiffness, pruning, deformation mode, boundary conditions, and optimization shape network mechanics. They identify nonlinear behavior, directional auxeticity, isotropic designs, and limits of linear or collision-free modeling.
- Bond response distributions with bending stiffness: Lower angle-bending stiffness produces broader, more pruneable bond-response distributions and larger changes in Poisson’s ratio.For unpruned networks, the reported ∆Gi–∆Bi correlations are 0.134, 0.125, and 0.843 for kang = 10−4, 10−2, and 100.
- Non-linear behavior: Linear-regime predictions become less accurate with increasing strain, reaching a median relative error of 30% at ϵy = 5%.The relative error rises from roughly 0% at ϵy = 0.5%, while individual nodes can exceed 800% error.
- Non-linear behavior: Pruned networks show sublinear stress–strain behavior and nearly constant stress beyond 3% strain, unlike unpruned networks’ linear response.
- Directional auxeticity: Low-∆Gp pruning produces auxeticity for edge pulling, whereas low-∆Gs pruning produces auxeticity for diagonal pulling.
- Isotropic networks: Isotropic pruning yields ν = −0.25 in the linear regime but produces less negative Poisson’s ratios than anisotropic pruning.The isotropic design is formed by iteratively pruning the lowest ∆Gp + ∆Gs bond.
- Directional auxeticity: Pure- and simple-shear moduli govern auxetic responses under edge-normal and corner loading, respectively, so optimizing one does not guarantee the other.
- Bending-stiffness mechanisms: Increasing angle-bending stiffness can switch the mechanism of auxeticity from concave-polygon collapse to shape retention, with ν = 0 at the crossover.
- Boundary conditions: Fixed experimental-style boundary conditions do not significantly change ν relative to free boundary conditions.