On-demand auxeticity and co-existing pre-tension induced compression stage in a sandwich design with kinematically constrained 3D suture tiles

Abstract

By incorporating concepts from auxeticity, kinematic constraints, pre-tension induced compression (PIC), and suture tessellations, tiled sandwich composites are designed, demonstrating behaviors attributed to the synergy between auxeticity and pre-tension induced contact and compression, simultaneously triggered by a threshold strain. The designs can theoretically achieve on-demand Poisson’s ratio in the widest range (−∞, +∞), and once triggered, the Poisson’s ratio is stable under large deformation. Also, once the overall strain goes beyond the threshold, the designs enter into a PIC stage, ensuring the middle soft layer takes the tensile load, while the tiles are under compression via contact and the 3D articulation of the tooth-channel pairs. In this PIC stage, the tooth-channel pairs provide kinematic constraints via the contact and relative sliding between teeth and channels. The deformation mechanisms and mechanical properties of them are systematically explored via an integrated analytical, numerical, and experimental approach. Mechanical experiments are performed on 3D printed specimens. It is found that the length aspect ratio and the obliqueness of the teeth significantly influence the constraint angle and therefore the auxeticity and strength of the designs.

New tiled sandwich composites demonstrate on-demand auxeticity and a pseudo tensegrity deformation stage, achieving Poisson’s ratios from -∞ to +∞, maintaining stability under large deformation and increasing strength under tension and indentation.

Introduction

In nature, biological composites are often hybrids of soft and hard components, creating synergy to achieve high mechanical performance. Different types of combinations result in different groups of hybrid materials with various enhanced properties and functions. Multilayered^1–3 materials are found to mainly increase toughness, dissipate energy, arrest cracks, and balance flexibility and protection. Outstanding examples include the brick-mortar structure in nacre^4–6, layered structures in crustaceans^7,8 and insects^9,10, and tile-skin armors of ganoids and armadillos^11–18. Tensegrity structures consist of cables and struts experiencing tension and compression respectively, the components of which rely on self-stress between its members for stability^19–21. Biotensegrity^22,23 structures exist in human bodies, from knees, hips, and spines²⁴ to living cells and tissues^25,26, and animals and plants such as the porcupine fish (Fig. 1a) and dandelion puffball²³. The combination of tensioned and compressed components in these arrangements allows for shape stability, efficient load transmission²⁷, large stiffness-to-mass ratios, deployability, and tunability^28–30.

Fig. 1. Conceptual model tiles connected via kinematic constraints, showing widest range of Poisson’s ratio, ν_YX, from −∞ to +∞.

a The inflation and stretching defense mechanism of the porcupine fish (Top-Left), the sutural tessellated surface of the seedcoat of the Portulaca oleracea (Bottom-Left, courtesy of James C. Weaver), and the schematics of the design inspired by the two biological systems. b Schematics of the kinematic constraints (red bars) and undeformed and deformed representative volume elements (RVEs) with both positive and negative Poisson’s ratios. c Representations of the deformations of a set of four kinematically constrained tiles having interior tile angle, 2θ, of 90° and arbitrary tile edge length, L. Three cases are shown for arbitrary constraint angles, β, of less than, equal to, and greater than 45°, illustrating the tuning negative Poisson’s ratio via the constraint angle. d Model prediction (Eq. 1) on the relationship between the effective Poisson’s ratio of the connected tiles vs. the constraint angle, where the blue and orange shaded regions represent the regions of obtainable negative and positive Poisson’s ratio, respectively, for the full range of β. Source data for (d) is provided with the paper.

Interdigitated^31–33 and biological sutures such as those in turtle carapace^34,35, sticklebacks³⁶, human skulls³⁷, and seedcoats³⁸, have also shown enhanced toughness, energy dissipation and balanced flexibility and protection. Recently, it has been shown that the jigsaw puzzle-like suture tessellations on the seedcoats of plant Portulaca oleracea and common millets not only simultaneously amplify strength and toughness, but can also have auxeticity^39,40, inspiring sutural tessellations that were shown though finite element (FE) simulations to have negative Poisson’s ratio when the suture waviness and the stiffness ratio between the hard and soft phases are in certain ranges⁴¹, though the reason behind the auxeticity was unclear. We have attributed this behavior to the Keyed-brick mechanism, named by Evans and Alderson⁴², in which individual geometric units (bricks) are designed to have matching key and channel pairs, providing kinematic constraints to regulate the relative movement between neighboring bricks^{39,40,43–46}. Recently, a double-negative mechanical metamaterials with both negative stiffness and negative Poisson’s ratio was designed by utilizing the keyed-brick mechanism to achieve negative Poisson’s ratio⁴⁷. However, no systematic study on the keyed-brick mechanism was found, hindering its further development.

In this paper, we will systematically investigate the keyed-brick mechanism from the perspective of kinematic constraints, create tiled composites with co-existing auxeticity and pre-tension induced compression (PIC), which is a pseudo-tensegrity deformation stage, and demonstrate and quantify the engineering benefits. We will focus on the following mechanisms and behaviors: (1) the keyed-brick mechanism for auxeticity; (2) PIC resulting from tooth-channel contact; (3) sliding and interlocking of 3D articulated tooth-channel pairs for efficient energy dissipation; and (4) co-existing auxeticity and PIC stage for efficient load transfer and increased stiffness, strength, and failure strain. These deformation mechanisms and behaviors are systematically explored via analytical kinematic analyses, three-dimensional finite element (FE) simulations, and experiments on 3D printed specimens.

Results

Conceptual model of tiles connected via kinematic constraints

Inspired by a combination of the defense mechanism of the porcupine fish, via inflation of its stomach to stretch its body and protrude pointy spines to ward of predators, and by the tessellated surface of the seedcoat of the Portulaca oleracea, this design utilizes the tile continuum with individual tiles that are kinematically constrained to their neighboring tiles, Fig. 1a. Similar to the keyed-brick mechanism⁴², kinematically constrained rigid tiles can achieve negative Poisson’s ratio. To systematically explore this mechanism, model diamond tiles connected via kinematic constraints are schematically shown in Fig. 1b. The red bars between tiles represent the kinematic constraints which define the direction of the relative movement between neighboring tiles at their boundaries. The deformation mechanisms are visualized via the comparison between an undeformed representative volume element (RVE) and the deformed one under vertical tensile load (Fig. 1b). Since the tiles can only have relative displacement along the direction of the kinematic constraints, under vertical tension, the tiles will move either outward or inward, leading to the overall negative or positive Poisson’s ratio, respectively.

As shown in Fig. 1c, the diamond tiles have an interior angle 2θ, and edge length L. The direction of the kinematic constraints is defined by the constraint angle β with respect to the loading direction along the global Y-axis. The constraints on different edges are mirror symmetric about both X and Y axes. Thus, if 0° < β < 90°, the constraint directions of four neighboring tiles form a diamond (dash line in Fig. 1b) with interior angle 2β. Figure 1c schematically shows that for different β angles, the deformed configuration can vary significantly, and the magnitude of the negative Poisson’s ratio can be widely tuned. Based on the length aspect ratio of the RVE and the constraint angle β, for arbitrary β and θ, the Poisson’s ratio can be derived as a scale-independent function (full derivation is in the Supplementary Information):

$${\nu }_{YX}=-\frac{\tan \beta }{\tan \theta }\left(0^{\circ }\le \beta <{90}^{\circ }\mathrm{and}{90}^{\circ }<\beta \le {180}^{\circ }\mathrm{and}{0}^{\circ }<\theta <{90}^{\circ }\right)$$ 1

As shown in Fig. 1d, according to Eq. (1), Poisson’s ratio is plotted as a function of β for three θ values, 30°, 45°, and 60°. As examples, the schematics of square tiles (θ = 45°) connected by kinematic constraints with different β angles are shown in Fig. 1c as well.

Generally, Fig. 1d demonstrates two regions of effective Poisson’s ratio of the tiles: negative Poisson’s ratios, when 0° < β < 90°; and positive Poisson’s ratios, when 90° < β < 180°. There are two extreme cases: when β = 0° and 180°, the Poisson’s ratio is zero; and when β = 90°, Eq. (1) does not apply, because that for this case, rigid tiles are locked with no relative displacement at all, also resulting in a Poisson’s ratio of zero. In reality, approaching this maximum and minimum ranges of Poisson’s ratio results in proportionally increasing forces as well and therefore, makes the case of β = 90° a singularity point. However, theoretically, if β gets very close to 90° from either the <90° or >90° side, Poisson’s ratio will be approaching −∞ and +∞, respectively. For this study, we will focus on the region with negative Poisson’s ratios.

For the above-mentioned analysis, we assumed that the loading direction is along the axis of mirror symmetry of neighboring kinematic constraints. Theoretically, the deformation of the tiles and keys are negligible compared to the relative sliding between them. To ensure this negligible deformation, the stiffness of the tile materials and the keys connecting the tiles need to be large enough so that the local interacting forces between tiles and keys induced by the overall tensile load will only cause negligible deformation. Practically, the assembly techniques are important to ensure these theoretical assumptions. Manually assembling the tiles can be challenging. Additive manufacturing provides effective technology to automatically assemble tiles and keys. In the next sections, we will design and fabricate prototypes via 3D printing to further explain the implementation of this concept.

Sandwich designs with suture tiles

To implement the conceptual model of tiles with kinematic constraints, square tiles with 3D articulated tooth-channel pairs, named suture tiles, are designed. The three-dimensional geometry of the suture tile is shown in Fig. 2a. For simplicity, a sandwich structure (Fig. 2b) is designed here by connecting and mirroring a tile layer across a soft middle continuum layer. The soft middle layer is necessary to fix the tiles to the same plane, allow for flexibility in the structure, and allow for relative motion between tiles. The sandwich structure itself was chosen as its out-of-plane symmetry results in no out-of-plane deformation when subjected to in-plane loading and is closer to the analytical model described in Fig. 1 than a single tile layer would be. The out-of-plane thickness of the tiles is t, and that of the soft layer is t_o. Thus, the neighboring tiles are articulated via each tooth-channel pair stemming from the midpoint of each edge of the tiles. These tooth-channel pairs are the enforcement mechanism for the kinematic constraints, allowing for a tunable Poisson’s ratio. The need for a straight edge to enforce the kinematic constraints during relative sliding left only a rectangular and triangular tooth geometry for possible options, and to both optimize tile space and to avoid jamming between the teeth and channels, the triangular tooth geometry was chosen. As shown in Fig. 2b, there is a gap g between neighboring tiles and the tooth-channel pairs are not in contact at the initial configuration.

Fig. 2. The sandwich design with suture tiles and the analytical prediction on the overall Poisson’s ratio (ν_YX).

a The 3D geometry of a single gray suture tile with 3D teeth and channels (top) with tile edge length (L), interior tile angle (2θ), tooth base length (b), tooth height (h), incline angle (γ), tooth length aspect ratio (ρ), and where the tooth edges form angles φ₁ and φ₂ with respect to the tooth edge. Angled top and bottom views of an arbitrary yellow tile (bottom) are shown. b The sandwich design comprised of the top gray tile layer of thickness t with a gap spacing of g, middle light blue soft layer of thickness t_o, and bottom gray tile layer. The fully assembled sandwich structure is shown on the bottom where one arbitrary tile in the bottom layer is shown in yellow representing the placement of the yellow tile in (a). c The four designs with different geometric parameters, designs B, W, S, and D, are shown in the global X–Y coordinate system. The benchmark design (B) for which the other three are compared back to is shaded light blue. Also shown is the local coordinate x-y system, aligned with the tile edges. d Analytical prediction of ν_YX versus γ for various ρ. The RVE of all designs with height w_y and width w_x are shown in which each tile is indicated by a different color and the constraint shapes, formed by the constraint angle β^*, are shown via the red dashed lines. e The tile design space for negative and positive Poisson’s ratios represented by the gray and white regions respectively. f Images of a 3D printed demo sample of design B showcasing its structure and flexibility before entering into the PIC stage. Source data for (d) and (e) are provided with the paper.

Under the overall tensile load, the middle layer is tensioned, the individual tiles in the tile layers shift until the tooth-channel pairs are in contact. Since after that, the soft layer is mainly under pre-tension and the tiles are under compression due to the contact induced by the layer tension, we call this deformation stage of the design PIC. The PIC is a pseudo-tensegrity stage, because that in this stage, the deformation of the design can be analogous to that in a tensegrity structure with some components under pre-tension and some under compression. It is worth noting the kinematic constraints, and therefore the auxeticity, will only be triggered after the contact in this stage. Before contact, the material will remain compliant and flexible, mainly deforming according to the properties of the softer layer. After contact, it is expected that the overall stiffness will increase in this deformation stage.

The tooth geometry is determined by the tooth base length b, tooth height h, and the incline angle γ (0° < γ < 180°) of the tooth. The incline angle is defined as the angle between the tooth midline and the tile edge (Fig. 2a). An incline angle of 90° represents when the tooth is perpendicular to the tile edge. The tooth length aspect ratio ρ is defined as ρ = h/b. Thus, when ρ approaches zero, the tooth vanishes; and when ρ approaches +∞, the tooth-channel pair becomes a line, i.e. the suture tiles degenerate into the ideal square tiles in Fig. 1b–d.

The channel geometry matches with the tooth geometry. Also, to ensure smooth sliding, the neighboring edges of a channel and a tooth on the same tile edge are along the same straight line. As shown in Fig. 2a, the two edges of the tooth-channel pair form angles φ₁ and φ₂ with respect to the tooth edge, respectively. If γ = 90°, φ₁ = φ₂, otherwise, they are not equal. As shown in Fig. 2a, the 3D geometry of the tiles ensures one side along z direction being flat and square-shaped while the other side has a complicated 3D surface. For both tile layers, the flat side is attached to the soft middle layer, as shown in Fig. 2b, forming a sandwich structure.

Since after contacting with the channel, each tooth can only slide along each channel, due to the 3D tooth geometry, the constraint angle β is different under either tension or compression. Thus, for the suture tile designs, the constraint angle is now β^* and is related to φ₁ or φ₂ under either tension or compression as:

$$\beta *=\left\{\begin{array}{cc}\hfill \theta +{\phi }_1-{90}^{\circ },\hfill & \hfill \mathrm{Tension}\mathrm{along}\mathrm{Y}\mathrm{direction}\hfill \\ \hfill \theta -{\phi }_2+{90}^{\circ },\hfill & \hfill \mathrm{Compression}\mathrm{along}\mathrm{Y}\mathrm{direction}\hfill \end{array}\right),$$ 2

where, the φ₁ and φ₂ angles are determined by the incline angle γ of the teeth and the tooth length aspect ratio ρ as (detailed schematics are provided in the Supplementary Information):

$${\phi }_1=\arccos \left(\frac{\frac{1}{2\rho }-\cos \left(\gamma \right)}{\sqrt{1+\frac{1}{4{\rho }^2}-\frac{1}{\rho }\cos \left(\gamma \right)}}\right),$$ 3

$${\phi }_2=\arccos \left(\frac{\frac{1}{2\rho }+\cos \left(\gamma \right)}{\sqrt{1+\frac{1}{4{\rho }^2}+\frac{1}{\rho }\cos \left(\gamma \right)}}\right).$$ 4

In this study, we will focus on the case of tensile load along Y direction. So, according to Eqs. (1) and (2), the overall Poisson’s ratio is derived as:

$${\nu }_{YX}=-\frac{\tan \left({\beta }^{*}\right)}{\tan \theta }=-\tan \left(\theta +{\phi }_1-{90}^o\right)\frac{w_Y}{w_X},$$ 5

where w_X and w_Y are the width and height of the RVE, respectively. Equation (5) are plotted in Fig. 2d, e.

A demo sample of design B is fabricated via a multi-material 3D printer (Stratasys Connex3). The soft layer is printed as a transparent rubbery layer and the tiles are printed as hard white thermal plastic materials, as shown in Fig. 2f. The images of the demo sample in Fig. 2f show the structure of the sandwich design and its flexibility before tooth channel contact. Supplementary Movie 1 (V1) is also provided to further demonstrate the 3D printed material prototype discussed here.

As the deformation of the continuum is tied to the kinematic constraint, as long as the kinematic constraint angle remains unchanged, any format of kinematic constraints of neighboring tiles will have similar kinematic deformation, although here we use the format of key-channel pairs and each tile have both teeth and channels. For instance, one tile can consist of only channels while each neighboring tile can consist of only teeth, if all other parameters are the same, the design will show the same kinematic deformation. In the current designs, the tiles are designed to have both channel and teeth with a chiral arrangement, simply to utilize the space in tiles more efficiently to avoid overlapping of the channels in the center of the tile.

Comparison design group

According to Eq. 5, we can see that in the sandwich design with suture tiles, the tooth length aspect ratio ρ and the incline angle γ are two important non-dimensional parameters governing the overall Poisson’s ratio. To quantify their influences, four designs (Fig. 2c) are created by varying ρ and γ: benchmark design B with ρ = 1.81 and γ = 90°; design W with a wider tooth base and smaller tooth length aspect ratio than design B, ρ = 0.94 and γ = 90°; design S with a shorter tooth height, and also a smaller tooth length aspect ratio than design B, ρ = 1.33 and γ = 90°; and design O with the same tooth length aspect ratio with design B, but with an oblique tooth, ρ = 1.81 and γ = 75°. Figure 2d shows ν_YX versus γ obtained from Eq. (5) for the three ρ values presented in the comparison design group along with ρ = +∞ representing the ideal case. The RVEs of the four designs are shown in Fig. 2d as well, in which the actual direction of the kinematic constraints are shown by the red dash lines. Each design is represented by a symbol and is marked in Fig. 2d.

The design space for tuning the overall Poisson’s ratio ν_YX is plotted in Fig. 2e as a labeled contour as a function of ρ and γ, where the white and shaded domains represent design spaces with positive and negative Poisson’s Ratios, respectively. Again, the symbols are for each of the four designs shown in Fig. 2c and all four designs fall into the negative Poisson’s ratio range.

Both Fig. 2d, e show that as ρ increases, ν_YX decreases, i.e. the design becomes more auxetic. Also, they show that when the design parameters are near the boundary between the two domains with negative and positive ν_YX, the Poisson’s ratio can flip between +∞ and −∞ with the slightest alterations to the ρ and γ.

On-demand auxeticity

3D Finite element models of the RVEs of the four designs are developed in ABAQUS/CAE. As an example, the FE model of design B is shown in Fig. 3a. Periodic boundary conditions are applied at the boundaries of the RVEs. Due to the gap g between tiles (Fig. 2b), the kinematic constraints provided by the contact of the tooth-channel pairs will be triggered only when the overall strain reaches a critical value. This on-demand behavior results in the deformation process of the sandwich designs having three stages (Fig. 3b): Stage I (isolation stage), the deformation of the middle layer dominates, the tiles are simply attached to the layer and no interaction between the tiles, so the designs are as compliant as the soft layer and have overall Poisson’s ratio close to the Poisson’s ratio ν_o of the soft middle layer. This stage only present for non-zero gaps; Stage II (the PIC stage), as shown in Fig. 3b, when the overall strain of the design along the loading direction reaches a threshold ε_Y^*, the tooth-channel pairs start to contact activating the keyed-brick type of kinematic constraints and therefore the auxeticity. The tensioned soft layer and tiles compressed from contact with other tiles are similar to the tensioned cables and compressed struts of conventional tensegrity structures, leading to this stage being the start of the PIC stage; Stage III (sliding stage), when the designs are loaded further, relative sliding between the tooth-channel pairs occurs, resulting in deformation determined by the kinematic constraints, and the designs preserve auxeticity and also stiffen due to the interactions between the tiles. In general, in Stage I, the designs are compliant and there are no tile interactions; in Stages II and III, the initiation of the kinematic constraints due to tension in the soft layer results in these stages both grouped into the PIC stage, where both PIC and auxeticity co-exist and the designs are stiffened due to tile interactions.

Fig. 3. Finite element and analytical predictions of the instantaneous Poisson’s ratio (νʹ_YX).

a 3D FE model of the RVE, by taking the benchmark design B as an example. b The deformation stages as a function of the strain in the Y direction (ε_Y). The red arrows indicate two neighboring teeth positions where in stage I they indicate the initial gap, in stage II they indicate tooth contact at the threshold strain (ε_Y^*), and in stage III they indicate tooth sliding. c Plots of νʹ_YX vs. ε_Y, obtained from FE results and Eq. (8), for the three comparison groups; W and B to observe the effects of the tooth base (b), S and B to observe the effects of the tooth height (h), and O and B to observe the effects of the tooth incline angle (γ), where the blue shade region in the plot for S vs. B represents the stage III. d The apparent Poisson’s Ratio (ν_YX) vs. ε_Y, obtained from FE results and Eq. (7), for each design where the line type, color scheme and symbols are consistent with both (c) and (f). e X displacement (U_X) contours for the least auxetic design, W, and most auxetic design, O, before tooth contact (location 1) and after tooth contact (location 2). f A plot of ε_Y^* vs. the normalized gap (g/L), obtained from Eq. (6), for the four designs discussed for the set normalized offset (κ/L) of 0.013, the same value used for the experimental samples. Source data for (c), (d) and (f) are provided with the paper.

The threshold strain ε_Y* is derived from the geometric parameters of the tiles, the initial Poisson’s ratio ν_o of the soft layer, the height of the RVE w_y, and the gap g as:

$${\epsilon }_Y*=\frac{\frac{g\cos \left({\beta }^{*}-\theta \right)}{\rho \sin \left(\gamma \right)}+4\kappa }{\sin \left({\beta }^{*}\right)\left(1+{\nu }_o\cot \left({\beta }^{*}\right)\right)}\frac{1}{w_Y}=\frac{\left(\frac{g}{L}\right)\frac{\cos \left({\beta }^{*}-\theta \right)}{\rho \sin \left(\gamma \right)}+4\left(\frac{\kappa }{L}\right)}{\sin \left({\beta }^{*}\right)\left(1+{\nu }_o\cot \left({\beta }^{*}\right)\right)}\frac{1}{\frac{1}{\cos \left(\theta \right)}-\left(\frac{g}{L}\right)\frac{2\cos \left(\gamma +\theta \right)}{\sin \left(\gamma \right)}},$$ 6

where κ is the initial offset between tooth and channel. The detailed schematics of Eq. (6) are shown in the Supplementary Information. Equation (6) shows that the threshold strain ε_Y* is a scale-independent property, determined by scale-independent geometric parameters of g/L, κ/L, β^*, θ, and γ. This means that the on-demand behavior is fully tunable without influencing the auxetic behavior. To precisely quantify the auxeticity of the designs, we need to distinguish the apparent Poisson’s ratio ν_YX and the instantaneous Poisson’s ratio νʹ_YX. The apparent Poisson’s ratio ν_YX is calculated by comparing the current configuration with the initial undeformed configuration. Details of the definition are provided in the Supplementary Information. While the instantaneous Poisson’s ratio νʹ_YX is calculated by comparing the current configuration with the configuration of the previous incremental deformation step. Thus, the instantaneous Poisson’s ratio more accurately reflects the auxeticity at the current configuration. At larger levels of deformation ε_Y ≫ ε_Y*, the instantaneous Poisson’s ratio is close to the apparent Poisson’s ratio. The ν_YX and νʹ_YX for the designs can be derived as

$${\nu }_{YX}=\left\{\begin{array}{cc}\hfill {\nu }_0\frac{w_Y}{w_X},\hfill & \hfill {\epsilon }_Y\le {\epsilon }_Y^{*}\hfill \\ \hfill -\left(\tan {\beta }^{*}-\frac{{\delta }^{*}}{\delta }\left({\nu }_o+\tan {\beta }^{*}\right)\right)\frac{w_Y}{w_X},\hfill & \hfill {\epsilon }_Y\ge {\epsilon }_Y^{*}\hfill \end{array}\right)$$ 7

$${{\nu }^{\prime }}_{YX}=\left\{\begin{array}{cc}\hfill {\nu }_0\frac{w_Y}{w_X},\hfill & \hfill {\epsilon }_Y\le {\epsilon }_Y^{*}\hfill \\ \hfill -\tan {\beta }^{*}\frac{w_Y}{w_X},\hfill & \hfill {\epsilon }_Y\ge {\epsilon }_Y^{*}\hfill \end{array}\right)$$ 8

Equations (7) and (8) show that in Stage I, ν_YX = νʹ_YX and when the overall displacement δ is large, ν_YX = νʹ_YX.

FE vs. analytical results

The analytical results from Eq. (8) and the FE results for νʹ_YX versus the overall strain ε_Y are shown in Fig. 3c. From left to right the plots show comparisons between designs W and B, S and B, and O and B. For the first comparison group, both FE and analytical results are shown for the benchmark design B. Since the analytical and FE results for design B match very well, subsequent plots only show the analytical results of design B.

The apparent Poisson’s ratios of the four designs are compared in Fig. 3d, which shows that for each design, the FE and analytical predictions from Eq. (7) match very well. In Stage I, the apparent Poisson’s ratio is the same as the instantaneous Poisson’s ratio. While in Stage III, different from the instantaneous Poisson’s ratio, the apparent Poisson’s ratio gradually decreases after contact. Design W reaches a zero apparent Poisson’s ratio, designs B, S, and O all reach apparent negative Poisson’s ratio, while all designs have a negative instantaneous Poisson’s ratio.

To demonstrate the deformation mechanisms, the FE contours of X displacement U_X of the designs W and O, i.e. the least and the most auxetic cases, are shown in Fig. 3e at two different levels of strain ε_Y = 0.041 and ε_Y = 0.135, respectively. It can be seen that after contact, due to the constrained sliding between teeth and channels, the neighboring four tiles are pushed away from each other, leading to the overall negative Poisson’s ratio. Figure 3f shows that ε_Y^* can be tuned by varying ρ and γ. Generally, for the same gap g, when ρ increases and/or γ decreases, ε_Y^* decreases. The FE and analytical results correlate very well with each other. For all four designs, the initial Poisson’s ratio is the same, ~0.43. In Stage III, when the overall strain goes beyond ε_Y^*, relative sliding between the teeth and channel occurs and all four designs show negative Poisson’s ratio. Design O has the smallest negative Poisson’s ratio, i.e. being the most auxetic, followed by designs B, S, and W.

As the tiles in this state are under compression from the interactions between neighboring tiles, the concern of buckling may be present. The critical overall strain for buckling is determined by the ratio between the tile thickness t and the tile edge length L. Generally, the critical overall strain to tile buckling increases with t/L. For the current designs, the critical strain to buckling will go beyond 38%. So, the tiles will be safely compressed without buckling under the tensile load before failure.

Uni-axial tension experiments

The four designs are fabricated via the multi-material 3D printer (Stratasys Connex3). For comparison, an additional sample consisting of square tiles of the same edge size but with no tooth-channel pairs, called the just squares (JS) design, is also printed. Uniaxial tension experiments are conducted for each design until failure (Supplementary Movies 2-6 (V2-V6)). The 3D printed specimens of the suture tile designs are shown in Fig. 4a. The experimental effective stress strain curves are compared in Fig. 4b. The FE and experimental curves match well before approaching the peaks of the experimental curves, and then diverge. This is because the FE model did not consider material damage, while in experiments, damage initiates before the peak stress and evolves rapidly after the peak stress. The curves show the trend that for the four designs, the initial stiffnesses in deformation Stage I are the same, while in Stage III, all designs are stiffer than in Stage I due to the contact and the tile interactions. Design O is the stiffest, followed by designs B, S, W, and JS, with the JS design not showing an increase in stiffness as it does not move past Stage I.

Fig. 4. Uni-axial tension experiment vs. FE and analytical results.

a A representative image of the Solidworks design of the benchmark sample along with the close-up images of the 3D printed samples, each with their center RVE’s tile geometry traced with their respective color schemes. b Experimental stress (σ_Y) vs. strain (ε_Y) curves for the four designs, and the additional design JS, c Von Mises stress contours for the tile layer of each design at ε_Y = 0.135 (location 2 from Fig. 3). d Experimental and analytical (Eq. 8) instantaneous Poisson’s ratio (νʹ_YX) vs. ε_Y for the three comparison groups, W and B, S and B, and O and B. e DIC contours of ε_XX for all four samples at ε_Y = 0.103 and close-ups of the least (W) and most (O) auxetic cases. f Experimental images of each design just before total failure occurs. g FE local shear stress contours for the soft layer of each design at ε_Y = 00.135 (location 2 from Fig. 3). h DIC contours of ε_XY for all four samples at ε_Y = 0.103, again with close ups of designs W and O. Source data for (b) and (d) are provided with the paper.

From the experiments, the ultimate tensile strengths of the four designs follow the same trend (Fig. 4b): Design O is the strongest, followed by designs B, S, W and JS. Especially, design O shows a ~40% and more increase in strength than the others. Figure 4c shows the FE Von Mises stress contour in the tile layer for each design, illustrating how the kinematic constraints result in the tile layer taking increasing load with increasing auxeticity, causing an increase in stiffness and potentially strength as well. For example, design O shows the highest Von Mises stresses in the tiles at the same overall strain, which is consistent with the experimental results shown in Fig. 4b.

Figure 4d shows the instantaneous Poisson’s ratio νʹ_YX obtained from the experimental digital image correlation (DIC) data. For all four designs, the experimental results show a gradual decrease of the instantaneous Poisson’s ratio, from positive to negative value, and eventually reach the analytical predictions. This is due to the limit in printing resolution and boundary effects, as tooth-channel pairs between different tile pairs do not all come in contact exactly at the same time, and the Poisson’s ratio stays negative throughout the onset of failure.

Figure 4e shows the DIC strain contours of ε_XX at ε_Y = 0.103 on the 3D printed specimens, and close-ups of design W and design O underneath. In the close-up images, outlines of the center RVE tiles are shown to better visualize their deformation. The tiles are in compression at the locations where contact occurs, for example negative ε_XX are observed at those locations, indicating the tiles are the component carrying compression in the sandwich design. Also, the strains measured from DIC at the boundary of the tiles reflect the relative displacements between the tiles. For example, ε_XX at tile boundaries indicates the local relative X displacement between neighboring tiles. The interior of the tiles shows close-to-zero surface strains. Design O shows larger positive ε_XX around tile boundaries compared to Design W, indicating the tiles are expanding outward more and therefore Design O is more auxetic.

As each sample has the similar overall strain at failure, and each fails via the rupture of the thin layer, the failure of the sample can only propagate between the tiles, and where the tooth and channel pairs of neighboring tiles allow for enough deformation of the layer to reach the point of failure. The final instances of each sample during experimentation are shown in Fig. 4f.

The FE local shear stress ε_XY (x-y is a local coordinate system with x and y along the tile edges) contours in the soft layers are shown in Fig. 4g where zones connecting neighboring tiles experience large shear stress while zones directly fixed to the tiles experience virtually little stress. ε_XY in design O is the smallest, followed by designs B, S, and W. This is further illustrated in Fig. 4h where the DIC strain contours of ε_XY for the four designs are shown. The contours at the tile boundaries indicate the local relative sliding between neighboring tiles. Design W shows the largest level of ε_XY at the tile boundaries, indicating the largest relative sliding between neighboring tiles and thus the largest shear stress in the zone connecting the tiles (Fig. 4g). While, Design O has the smallest angle between the sliding direction and horizontal direction and therefore the smallest shear stress in the connecting zone of the soft layer (Fig. 4g). The smallest local shear stress in the soft layer of design O helps delay its tensile failure due to layer rupture. These indicate that after tooth-channel contact (i.e. in the PIC stage) auxeticity and PIC coexist, and designs with larger auxeticity can more efficiently transfer loads to tiles, and get less shear in soft layer, therefore gain larger stiffness and strength.

Indentation experiments with pre-stretch

For tiled armors in nature, they often resist indentation load in a pre-stretched state, such as the inflated puffer fish armor, and the seedcoat under pre-stretch due to the growing seed inside. In this section, we will further explore the benefits of co-existing auxeticity and PIC in resisting indentation loads for designs subjected to an initial pre-stretched state.

In order to apply pre-stretch on the 3D printed specimens, a testing rig is designed and shown schematically in Fig. 5a. Pre-strain, ε_o, can be achieved by stretching the sample, placing it in the rig, and inserting two identical blocks (the dark blocks in Fig. 5a) to achieve self-equilibrium under pre-stretch. By varying the heights of the blocks, different levels of pre-strain are obtained. The pre-stretched sample can be either in the PIC stage or not depending on whether the level of pre-strain ε_o goes beyond the critical strain ε_Y^* for tooth-channel contact. An example pre-stretched specimen is shown in Fig. 5b, and the indentation experimental set-up is schematically shown in Fig. 5c.

Fig. 5. Cyclic indentation experimental results for designs under variable amounts of pre-stretch.

a A representative image of the SolidWorks design of pre-stretched design O and its placement in the pre-stretch indentation rig, where the hard tile phase is dark gray, the soft middle layer is light blue, the indentation rig is light gray, and the blocks used for pre-stretch are black. b An image of pre-stretched design O in the rig, c Solidworks schematic of the cyclic indentation (δ) setup, where the indenter tip is shaded dark blue and the base of the rig is fixed during testing. d Images of samples JS, B, and O under no pre-strain and pre-strain greater than ε_Y^*. The black lines indicate the constraint angle (β^*), the yellow lines and arrows indicate the gap between neighboring tiles, the res lines and arrows represent the gap between neighboring teeth and channels, and the blue arrows indicate the direction of relative sliding. e Cyclic loading results for each design and each pre-strain level for the initial cycle up to 0.5 N, where ε_o = 0% for design O shows the hysteresis energy, ϵ_h, by the blue shaded area between the loading and unloading curves. Note that the color scheme is consistent with both (f) and (g). f Cyclic loading results for each design and each pre-strain level for the cycles up to 1, 2, and 3 N. g Normalized hysteresis energy (ϵ_h/δ_max) for each design, cycle max load and pre-strain level. Source data for (e)–(g) are provided with the paper.

For comparison, designs JS, B,and O are selected and 3D printed. Four levels of pre-strains ε_o = 0%, 2.65%, 5.49%, and 7.89% are applied on each design. Thus, a total of 12 specimens are printed. Experimental results in Fig. 4d show that the three non-zero pre-strains result in ε_o > ε_Y^* for designs B and O, with the pre-strain of 2.65% being just after contact has been made. Thus, all pre-stretched samples B and D are in the PIC stage.

The RVEs of the specimens are shown in Fig. 5d where samples on the left, and samples on the right are in unstretched and pre-stretched stages, respectively. Figure 5d shows where the gap between tiles is indicated by the yellow lines and arrows and the gap between the tooth and channel is marked by the red lines. It should be noted that for ε_o > ε_Y^*, the gap between the tooth and channel vanishes while the gap between the tile grows. The arrows along the teeth edge indicate the direction of relative sliding between the tooth-channel pairs.

To evaluate the energy dissipation capacity of the pre-stretched designs under indentation. Displacement-control (1 mm/min loading rate) cyclic indentation experiments are conducted designs with different levels of pre-stretch and increasing loading levels. For example, each specimen is first cyclically loaded to 0.5 N followed by unloading to zero force, and then load-unload to 1 N, 2 N, and 3 N, sequentially. To prove repeatability, at each level of load, three repeated cycles are conducted before moving to the next load level. The curves from each cycle are almost identical with only ~3% difference on average, so only the curves of the third cycle are shown for each set in Fig. 5e, f.

Figure 5e shows the loading-unloading curves for the cycle of 0.5 N. From the unstretched case to the first level of pre-stretch there is a larger increase in stiffness. The stiffness increases further with increasing pre-stretch after this, although the difference is not as much. The trend of increasing stiffness when ε_o is larger can change due to damage. For example, for designs JS and B, the pre-stretch of ε_o = 7.89% results in some level of damage in pre-stretching, so it appears at a lower stiffness than the ε_o = 5.49% case and is indicated with a dashed line. The loading-unloading curves under other three levels of loads are shown in Fig. 5f where all JS specimens fail before reaching the 3 N. For ε_o = 0%, 2.65%, and 5.49%, specimens reach 2 N, while the specimen with 7.89% pre-strain only reach 1 N. Earlier damages in ε_o = 5.49%, and 7.89% create drastic difference in the stiffness.

A similar trend occurs for designs B and O, with a large increase in stiffness observed from the un-stretched case to the first level of pre-stretch. Greater differences in stiffnesses are found for cases with increasing pre-stretch, attributed to the interaction between the suture tiles in the PIC stage. The highest pre-stretch of ε_o = 7.89% results in damage before indentation occurs and is observed by a lower stiffness. This is shown again in Fig. 5f with cycles up to 1 N, 2 N and 3 N. Curves of design O indicate less damage than in design B at higher level of ε_o.

To quantify the efficiency of energy dissipation under cyclic indentation loading, we first determine both the hysteresis energy ϵ_h, which is equal to the blue shaded area (area under the loading curve minus the area under the unloading curve) illustrated in Fig. 5e for the case of design O with ε_o = 0%, and back deflection δ_max, the maximum indentation displacement. The energy dissipation efficiency (EDE) is defined as the hysteresis energy per unit back deflection ϵ_h/δ_max, which is evaluated and compared for each indentation cycle, as shown in Fig. 5f.

Figure 5f shows that for all levels of pre-stretches, because of the tooth-channel interactions, designs B and O have significantly larger EDE than design JS. Also, when loading level increases, the EDE increases for all designs. The JS design has very little energy dissipation and the pre-stretch barely influences the EDE ϵ_h/δ_max and the very small hysteresis attribute to the viscoelasticity of the soft layer. Generally, design O has less damage than design B at the same level of pre-stretch followed by the same level of indentation load.

To further explore the indentation resistance of the designs, another set of indentation experiments are conducted, in which non-pre-stretched specimen (ε_o = 0%) and pre-stretched (ε_o = 7.89%) specimens of designs JS, B and O are indented until rupture (shown schematically in Fig. 6c).

Fig. 6. Indentation to failure experimental results for designs under pre-stretch amounts of ε_o = 0% and 7.89%.

a Indentation to failure experimental results for ε_o = 0%. b Indentation to failure experimental results for ε_o = 7.89%, where the marked locations 1–4 correspond to deflections of 2 mm, 4.75 mm, 6.0 mm, and 11 mm, respectively, for each design (where applicable). c Schematics of the indentation setup pre and post-failure, where the hard tile phase is dark gray, the soft middle layer is light blue, the indentation rig is light gray, and the indenter tip is dark blue. d RVE’s of designs JS, B, and O where the tiles are colored gray, and the soft layer is light blue is shown to the left of their experimental indentation to failure results grouped by design, with experimental images shown for locations 1–4 from (a) and (d) for the ε_o = 7.89% case. Source data are provided with the paper.

The force–deflection curves are compared in Fig. 6a, b where locations 1–4 correspond to deflections of 2 mm, 4.75 mm, 6.0 mm, and 11 mm respectively. It is shown that generally, the strength and stiffness of the designs increase dramatically from design JS to B and to O. As shown in Fig. 6d, design JS, non-pre-stretched and pre-stretched specimens show no differences in initial stiffness but the pre-stretched specimen shows a significantly lower strength and maximum deflection than the non-pre-stretched one. While for design O, the pre-stretched specimens show larger initial stiffness and larger deflection to complete rupture, but the strength is only slightly lower than the non-pre-stretched specimen. Although all designs show a decrease in the maximum load when pre-stretched, the difference decreases with the design’s auxeticity, and designs with greater auxeticity have the potential to strengthen under pre-stretch compared to no pre-stretch. Experimental images at the marked locations for designs with ε_o = 7.89% are shown in Fig. 6d at the same indentation displacement, depicting the evolution of damage and failure of the designs. For designs B and O, location 3 on the curves corresponds to the falling off of the tile directly under the indenter tip. After this local failure, designs B and O can continue taking load. This is because the interactions between the tooth-channel pairs in the PIC stage enable effective load transfer to other tiles. For additional visualization of the experimental setup, loading and deformation. Supplementary Movies 7–9 (V7-V9) show synchronized experimental videos and curves.

Discussion

Tiles connected via kinematic constraints can achieve the largest range of Poisson’s ratio, from −∞ to +∞, theoretically. We systematically explored the kinematics of this mechanism and found the overall Poisson’s ratio of the tiled design is determined by two non-dimensional parameters the interior angle 2θ of the tiles, and the constraint angle β. In general, when β changes between 0 and 90 degrees, the overall Poisson’s ratio is negative, and it becomes more auxetic when β increases and θ decreases. To implement this mechanism, sandwich structures with two tile layers and a middle soft layer are designed. Square tiles with tooth-channel pairs with 3D articulation. The tooth-channel pairs function as the kinematic constraints between the tiles. Both analytical and FE analyses are performed and the results match well.

Representative designs are fabricated via a multi-material 3D printer. Uni-axial tension experiments, and indentation experiments of 3D printed specimens are performed. The behaviors and properties that are observed and quantified include the contact-induced on-demand auxetic behavior, widely tunable negative Poisson’s ratio via varying the tooth-channel directions and geometries, and the increased indentation resistance and the efficiency of energy dissipation due to co-existing auxeticity and PIC.

The mechanism and benefits of the co-existing auxeticity and PIC stage can be summarized as the following: under overall tension, the designs can enter into this unique stage, in which the tiles function as the component carrying compressive load through contact, while the layer mainly carrying tensile load. In this stage, due to the kinematic constraints generated by the tooth-channel pairs, relative sliding occurs between contacting teeth and channels, leading to auxeticity and therefore efficient load transfer from the soft layer to the tiles. Also, the relative sliding between teeth and channels can generate an efficient energy dissipation mechanism, which is very valuable for resisting cyclic loading. Because of all these mechanisms, stiffness, strength, final strain to failure, and energy dissipation capacity are increased under both tensile and indentation loads. In addition, due to these benefits, pre-stretch can be applied to increase indentation resistance.

Last but not the least, for many materials, we need different properties and functions under different levels of loading. For example, under relatively small loads (for daily routine activities), flexibility is important, while when the loading level increases (for threatening or accidental loads), increased stiffness and indentation resistance are needed. The current designs with on-demand auxeticity and co-existing PIC provide strategies to meet these different needs under different situations of loading.

Methods

Experiments

Additive manufacturing is used to create the four tensile and three indentation designs using the Objet Connex 3 multi-material 3D printer. A harder material (VeroWhite) is used for the tile material while a softer rubber like material (TangoPlus) is used for the mid-layer. The tensile testing specimens consist of a 35 mm by 61.25 mm by 5 mm testing area with added grips for mounting in the Instron material testing machine. The indentation testing specimens consist of a 40.85 mm by 38.5 mm by 5 mm testing area with added grips for mounting in the pre-stretch indention rig and testing with the Instron material testing machine. Each tile layer is 2 mm thick and the mid-layer is 1.5 mm thick, and the initial gap separation prescribed is 0.15 mm. The tensile specimens are shown in Fig. 4a, while the indentation specimens are shown in Fig. 5a–d. A speckled pattern is used on the tile surface of the tensile specimens and images are taken at even intervals throughout the experiment for digital image correlation (DIC) calculations of strains and displacements. The specimens are statically loaded until failure for tensile testing and undergo static cyclic indentations with a 5 mm radius VeroWhite 3D printed indenter tip.

Finite element simulations

Finite element models are created for each tensile design using ABAQUS to compare with the experimental results and analytical predictions. The models contain the RVE for each design and are subjected to periodic boundary conditions (PBCs) for the exposed X and Y surfaces of the RVEs. PBCs are applied in all boundaries and a prescribed uniaxial strain is applied in the Y direction. FE models (Fig. 3a) of the RVEs of all four designs are created in ABAQUS/CAE. A finer mesh was used for zones with larger deformation and a mesh refinement study was performed.

The RVEs are comprised of two phases, the harder tile pieces, and the softer thin mid-layer. The tile parts are meshed with 3D stress 10-node quadratic tetrahedron elements and are modeled as a linear elastic material with a Young’s Modulus of 2 GPa with a Poisson’s ratio of 0.3. The thin layer parts are meshed with 3D stress 10-node quadratic tetrahedron hybrid elements because the layer uses the hyperplastic Arruda-Boyce material model with an initial shear modulus of 0.2 MPa, a limiting network stretch of 2.5, and an incompressibility parameter of 0.5. Where the two parts are touching, the surfaces in contact are joined with tie constraints.

Surface-to-surface contact interactions are prescribed for each tooth and channel pair, and the interaction properties have defined both normal and tangential behavior with a friction coefficient of 0.1. The surface-to-surface contact is also prescribed to any areas that might touch due to the deformation of the tiles as well. The RVEs are subjected to a uniaxial strain in the Y direction of 0.2.

Author contributions

R.N. and Y.L. designed the experiments and simulations. Y.L. initiated the concept. R.N. conducted the experiments, finite element, and analytical analysis. R.N. and Y.L. interpreted the results. Y.L. supervised the overall research. Both authors contributed to the writing of the manuscript.

Peer review

Peer review information

Nature Communications thanks Andrea Micheletti and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

Source data are provided with this paper.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-50664-8.