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Samples were manufactured using fused deposition modeling, a facile extrusion-based 3D printing technique. In this widely available technique, thermoplastic polymers are extruded in a layer-by-layer manner on a heated plate. The standard MakerBot Replicator was used as the desktop 3D printer and 1.75 mm diameter polylactic acid (PLA) filament was extruded through a tapered nozzle. These viscoelastic filament polymers exhibit shear-thinning behavior that enables facile extrusion from the nozzle and subsequent cooling into the desired structure on the print bed. A heated build plate was used in conjunction with a 50 micron layer of Kapton® tape to improve the adhesion of the printed filament to the build plate. Figure S1A demonstrates the 3D printing set up used in this study.

The samples were designed using SketchUp, a freely available 3D modeler, and written into a stereolithography file format (STL), which specifies tool path, print speed, retraction amount, and other factors that influence print quality. To create the 3D structure, models were made by printing unit strips of multiple cells and then solvent welding them together to form an interdigitated 3D material.

For load-displacement measurements, a Tinius Olsen benchtop Hounsfield 5 kN tension-compression machine with a 250 N load cell was used. The specimens were compressed using flat aluminum plates under quasi-static conditions at a rate of 1 mm/min. To enable lateral movement of the samples due to auxetic property, the samples were lubricated with oil. It was found that this lubrication reduced the friction at the sample edges, and by comparing the experimental force-displacement results with the analytical equations, the reduced coefficient of static friction was found to be 0.2556.

In our tests, the samples were not attached to the compression plates. As a result, in some of the measurements, there is a brief nonlinearity before the elastic response in the force-displacement curve. This nonlinearity is not present in each measurement and indicates the brief change in contact between the sample and the compression plate. Each test was repeated multiple times and the results were in good agreement, demonstrating good repeatability.

To characterize the Poisson's ratio, the inwards movement of markers placed at the ends of the auxetic samples were measured as the samples were compressed a predetermined amount.

The commercial finite element (FE) software ABAQUS (SIMULA, version 6.14) was used for the simulation of the elastic and plastic responses in the auxetic models. 3D FE models were constructed by importing AutoCAD drawing exchange format models of the auxetic into ABAQUS and using element type CPE6MH and each instance was seeded to an “Approximate global size” 1.

The response of the material was captured using a linear elastic model with PLA filament values reported in literature; a Young's modulus of 3500 MPa and Poisson's ratio of 0.36 (Tymrak et al., 2014). These values were used both in this finite element simulation and in the analytical calculations.

We applied a boundary condition of a downward 20 mm movement to one edge of the auxetic. This set simulated the movement of the compression plates in our force-displacement testing. The bottom edge of the structure was constrained with an ENCASTRE constraint to prevent vertical or horizontal movement, simulating the stationary bottom edge during the force-displacement testing.

In each simulation, the spatial displacement and reaction force in the relative x or y direction was monitored at the centralized reference point. This resulted in the simulation force-displacement curves shown above. Poisson's ratios were extracted in a similar manner by monitoring the displacement of the edges of the structure.

The finite element simulation accurately models the elastic region of the auxetic deformations. Figures 2A,B demonstrates the comparison between the experimental and simulation results. As depicted for both the honeycomb and the s-hinged auxetic deformed in the straight-edge direction, the simulation is in agreement with the experimental data until the yield point of PLA.

The novel stress-delocalization design was inspired by a smooth hinge geometry in nature, which minimizes stress concentration by distributing it through the length of the hinge. We applied the concept to replace the straight edges of the conventional reentrant honeycomb with optimally deforming redesigned arcs between the solid ribs (Figure 1A). These arcs were calculated with an appropriate radius and arc length that enable the greatest range of deformation, while staying within the elastic limit of PLA printer filament, and exhibit a much larger range of deformation compared to the conventional honeycomb (Figures 1B,C).

To illustrate the benefits of this new auxetic design, we also fabricated a conventional auxetic honeycomb structure with the same unit cell size and reentrant angle for valid comparison. We note that this 3D printing procedure is particularly well suited to demonstrate advantages of our s-hinge geometric design because common defects, such as rounded corners or poorly connected nodes that occur when using one continuous filament to print sharp edges, are prevented by printing the curved structure.

We combined experimental testing and finite element (FE) simulations to determine both the macroscopic and local stress and strain behavior in the s-hinge compared to the conventional honeycomb.

The experimental and FE models are in good agreement (Figures 2A,B), showing that the model faithfully captures the physics of the deformation. In the conventional reentrant honeycomb, stress is concentrated at the corners of the honeycomb, limiting macroscopic elastic deformation to only 3%. At this strain the local von Mises stress in the joint corners reaches the critical stress of 54 MPa for PLA (Figure 2C) (Tymrak et al., 2014). In contrast, the s-hinge structure delocalizes the stress concentration, reaching a maximum local von Mises stress of 8 MPa at that same macroscopic compression (Figure 2D). This delocalization of stress enables a larger elastic macroscopic strain for the structure, increasing from 3% in the conventional honeycomb to 30% for the s-hinged auxetic. The s-hinge structure also enables a lower equivalent plastic strain (Figure S3).

This stress and strain minimization through the s-hinge design allows a wider range of materials to be used for the auxetic structure. For a 10% macroscopic strain of the structure, one achieves a maximum of 0.70% logarithmic strain locally, which is within the elastic limit of a wider range of materials. This property of the s-hinged structure would enable more materials, such as glass fibers, ceramics, and other brittle materials with small elastic regions that were previously difficult to apply in auxetic geometries, to be used as auxetics, while still achieving a large macroscopic reversible strain.

To investigate how this increased elastic range of the s-hinged auxetic would affect repeated cyclability, we carried out fatigue testing to determine strain recovery after cycling both structures to 7% macroscopic strain, which falls within the elastic region of the s-hinged structure and the plastic region of the conventional honeycomb (Figure 2E). Plastic deformation, even in the absence of a large deviation from elastic behavior, leads to damage and its effects can be seen upon repeated compressive cycling. After 35 cycles, the s-hinge is able to recover nearly all of its strain (96%), while the conventional honeycomb only recovers to 85%, demonstrating the benefit of increased elastic macroscopic deformation gained from the s-hinged structure.

The stress versus strain curve in Figure 2F summarizes the mechanical behavior of the two geometries in compression, for which our structure is optimized. The initial linear region in both curves demonstrates the structure's elastic regime. The conventional honeycomb is much stiffer and quickly reaches its elastic limit, where the sharp peak denotes plastic collapse of the beams and the subsequent spikes show softening and hardening of the structure that occur during the breaking and collapse of different sets of beams. In contrast to this behavior, the much larger apparently elastic region of the s-hinge structure extends up to 33% macroscopic strain, compared to 3% in the conventional honeycomb. We note that for extending these principles to extension, one should smoothen the connection where the s-hinges meet with the straight-edge linking segments. Though not in the scope of this work, introducing another smooth joint would only increase the design complexity somewhat, yet could prevent stress concentration in tension as well. Additional improvements, as beautifully shown by Masoumi Khalil Abad et al. (2012), could be made by providing smoothening through G² continuity though here the choice was made to keep the structural elements simple for the purpose of both design and analytical calculations.

When loaded along the direction of the rigid ribs, the force-displacement curve closely matches that of the honeycomb characterized by Gibson et al. in their seminal work on this type of structure (Figure 3A, Movie S1) (Gibson et al., 1982). In the first region up to 5% macroscopic strain, the deformation is elastic and reversible. In this region, the edges transverse to the direction of compression contract inwards, demonstrating auxetic behavior. Upon further compression, the structure exhibits buckling, during which the planes of the honeycombs slip in opposite directions. This slipping of the rigid straight edges results in a metastable state with plane dislocations caused by shear stress and enables a large deformation at a nearly constant load. At the end of this plateau region, continued loading reaches contact between elements leading to densification of the structure, increasing the macroscopic “solid-like” stress, but also coupled with plastic irreversible yielding deformations as well. In this region, load required for further deformation increases rapidly as the opposing cell walls impinge upon one another.

When loaded along the direction of the hinges, the force-displacement curve displays a behavior where the deformation pattern goes directly from the elastic region to densification (Figure 3B, Movie S2). The geometry of structure enables the force to be evenly transferred along the s-hinge and thus deform nearly-linearly without plane dislocations.

In contrast to the conventional honeycomb where densification occurs unevenly throughout the structure (Figure S4), densification occurs uniformly in both loading directions of the s-hinge. We note that this stress-strain profile for the 3D printed s-hinged auxetic closely matches that of less well-defined open-cell polyurethane foams currently researched as isotropic auxetics, and could improve our understanding of their behavior, including the change to a quasi-exponential stress-strain curve (Cadamagnani et al., 2009). This deformation behavior can be tuned by changing the arc angle of the s-hinge, in a controllable manner similar to that shown for kagome lattices (Wu et al., 2015). By changing the arc angle, the macroscopic strain and position at which the first row begins to buckle changes (Figure S5).

To demonstrate how the deformation of these structures can be tuned, we designed a double s-hinge structure (Figure 3E, inset). The double s-hinge rigidifies the structure to inhibit lateral movement of the unit cells. Figures 3C,D show the three deformation regions of the 160-degree single s-hinge structure. The introduction of extra hinges limits the amount of linear elastic deformation one could achieve before contact with neighboring elements. The double hinge structure (Figures 3E,F) has significantly higher stress for the same macroscopic strain and deforms monotonically from the elastic to the densifying deformation region. This has the added benefit of avoiding the buckling instability modes present in the single hinge, leading to higher predictability of stress at high strain deformations in this structure.

We developed a parametric numeric model and incorporated the samples' dimensions and materials properties to predict elastic deformation and Poisson's ratios of the s-hinged structures (Figure 4A). The methods used in this semi-analytical model are applicable to a wider range of complex initial geometries and force applications.

To begin our model, we assumed bending deformation as the primary method of deformation during compression of the structure. In conventional beam bending analysis, the beam is assumed to be straight initially. However, for an initially curved beam, there is a change in radius from the initial shape

where R_o is the initial radius curvature and R is the subsequent radius of curvature.

Our analytical calculation is additionally complicated by the axial application of force. In conventional analysis, the beam experiences a transverse force. In our buckling calculations however, the axial force application results in a bending moment that depends upon the local deflected shape of the beam, yielding a complex relationship. By combining these two conditions together—the initial curved structure and axially applied force—the simple analytical solution yields accurate predictions for the experimental stress-strain relations.

Using Microsoft Excel, we developed an iterative system of analytical formulas, which use the initial curved structure of the beam to calculate where it moves upon applying an incremental force. The deflections of the beam can be calculated by using an intrinsic relationship between the length along the curved beam, s, and the tangential angle, α,

By using an incremental form of this relationship, we carried out a finite integration of the deflection of the beam and iterated to find the appropriate fixed end moments. The ends of the beam are constrained not to rotate, as in our physical model the hinges are constrained by the rigid ribs. Therefore, the fixed end moments are found by applying an incremental force less than 10% of the maximum compressive force

where M₀ is the bending moment at the start of the curve, Fy is the force applied in the hinge direction, Fx is the force applied in the rib direction, X and Y are coordinates of the curve, and the primary index i represents the time step and the secondary index n represents the position along the curve. For the fixed end moments, n = 1, the end position of the hinge.

Using the fixed end moments, moments along the remainder of the beam are calculated, used to find the change in radius of curvature, and used to find the next position. This process is repeated several times to yield the deformation of the hinge over several applied forces (Figure 4A), resulting in a force-displacement curve. The remainder of the model is presented in the supplementary information.

This semi-analytical model closely matches the experimental and numerical simulation data in the elastic deformation region (95% accuracy for a 10% unit cell strain) and provides a powerful way to generalize the behavior of an s-hinged unit cell (Figure 4B). Further, the analytically predicted Poisson's ratios match with the experimental Poisson's ratios for the double s-hinge structures (Figure 4D). The maximum achievable negative Poisson's ratio from this structure is −1.5 when compressed along the hinge direction. With this model, we easily obtain the effect of changing beam thicknesses, structure size, or hinge arc angle on the stress-strain curves of the structure.

Using the deformation ratios found in the analytical model, we discovered a critical arc-angle of 139 degrees in the s-hinge, at which point the structure can achieve bistable transformations between auxetic and non-auxetic states. The structure designed with this critical angle exhibits close to zero Poisson's ratio with axial load, but upon application of a transverse load, it can switch between states with either negative or positive Poisson's ratio (Movie S3). Above a 139 degree arc angle, the structure has a negative Poisson's ratio and below, it has a positive one (Figure 4C). This critical behavior is remarkably similar to that exhibited by the Venus Fly Trap where a minimal transverse energy input results in a large deformation snap-through. Using the analytical model we can tune the energy barriers for switching between different structures, with small orthogonal stimuli.

The s-hinged design was easily assembled into 3D structures. For demonstration purposes, a 3D 160-degree s-hinged auxetic was designed and printed as a rectangular prism of 4 × 4 × 3 unit cells and a unique x-axis direction (Figure 5a) and Poisson's ratios were measured as a function of strain when compressed along each of the three different axes (Figure S1C). The Poisson's ratios are negative upon compression in each direction and can be tuned by changing the arcs on the unit cells making up the structure. When compressed along the x-axis near-symmetrical contraction was achieved along two 4 × 3 faces as expected, and the Poisson's ratio changes from approximately −0.5 to −1 depending on strain (Figure 5b). Upon compression along the different y- and z-axes, the Poisson's ratio changes from approximately −0.5 to −0.2 (Figure 5b). Because the rigid ribs of the unit cells are connected both axially and transversely to their neighboring unit cells, the 3D structure is very robust and exhibits near uniformity without plane dislocations upon compression, eliminating rotations associated with buckling in the 2D auxetics, while maintaining easy cyclability.

To demonstrate the control of deformation our s-hinge structure enables, we varied arc-angles in a 3D printed Batman logo outline, which has curves similar to the critical angles we discovered in our unit cell (Figure 5c). While the original Batman logo (top), has a positive Poisson's ratio when compressed at the wings, a slight change of arc angle in the bottom re-designed logo makes it auxetic (see Movie S4). Such materials would lead to novel functionalities, such as designed shape-change upon mechanical compression. This tunability also enables the development of gradient auxetics (Figure 5d) where Poisson's ratio can be varied continuously throughout the structure by changing arc angles in neighboring cells. Changing the Poisson's ratio within a structure is useful for shape morphing structures with specific localized responses.

The greater elastic deformation region of the s-hinge design also enabled us to develop a latching mechanism to capture and store elastic mechanical energy within the auxetic (Figure 5e, Movie S5), previously a challenging task, due to increasing stress concentrations in the conventional honeycomb. This demonstrated multistable auxetic latch is the first of its kind and yields a number of benefits including its lack of reliance on traditional mechanisms of multistability such as beam buckling or origami folding (Shan et al., 2015a; Waitukaitis et al., 2015), which could lead to plastic deformation, lack of control, and failure. It is also able to store several levels of elastic energy in multiple configurations via its layer-by-layer latching in a larger structure. Upon using a small force to pull the rigid ribs in the transverse direction, the latch is released. The energy held in a single unit cell during the latching process was calculated to be 39 J kg⁻¹ PLA upon latching to a 15% unit cell strain, possible because of the s-hinge structure. While this number is modest, it can be greatly increased by designing much smaller unit cells with thinner ribs, where uniform deformation would lead to a larger fraction of the material at the surface of the ribs being strained near its elastic limit, instead of being constrained to the center of thick beams with little deformation. After optimization, such uniform, directional auxetic cell materials could be used for convenient elastic energy storage. The resulting latch structure can also be applied to other auxetic structures and, given appropriate miniaturization, authors propose using such a structures in multistable auxetic filters where several stable pore sizes could be selected and locked by directional compression.

Such latching and elastic storage mechanisms have wide uses in nature. Prominent examples include not only the Venus flytrap above, but also structures on the tendons of owls, eagles and other birds of prey which produce a fierce ratcheting, yet do not have to continually tense their muscles to keep the grip tight (Einoder and Richardson, 2007). Similar locking of joints in insects enable the extraudinary jumping ability of fleas (Sutton and Burrows, 2011) and froghoppers (Burrows, 2003) to generate forces up to 135 and 414 times their body weight respectively upon unlocking the joints (Burrows, 2003).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.