Manufacturing of 3D Auxetic Structures Through Perforations of Corrugated Systems

Abstract

Fabrication of auxetic structures has always been a limiting factor in their availability. Their complex shape, a requirement originating from the deformation mechanism that leads to a negative Poisson’s ratio, has also limited their manufacturability. In the case of auxetic systems that deform through the rotating semi-rigid mechanism—which allows for the concurrent deformation and rotation of their constituent element—the situation is even more complicated. Relatively few examples of these types of structures are known, with most work on them being largely theoretical. This includes their use in explaining the auxetic mechanism in certain molecules. Nevertheless, these systems can, in principle, offer added functionalities, as they undergo a shape change while still exhibiting a negative Poisson’s ratio. To this end, this work presents a practical scheme for the manufacturing of 3D rotating semi-rigid units, whereby these are produced through perforations of corrugated sheets. For the purpose of this investigation, diamond-shaped perforations were chosen, and the side profile of the corrugated sheet consisted of successive semicircles that alternate in orientation. Analysis of the system indicated that a 3D negative Poisson’s ratio can be obtained while allowing the distance between the hinges to change during deformation.

1. Introduction

Most commonly found materials contract laterally when uniaxially loaded in tension. However, it is also possible to find materials that expand laterally when pulled. These are characterized by a negative Poisson’s ratio and are commonly referred to as auxetics [1]. Their behavior is generally related to the geometry and the way they deform during loading. Various mechanisms have been identified leading to such behaviors. These include, amongst others, the rotating rigid units [2,3,4,5,6,7,8,9,10], chiral structures [11,12,13,14,15,16,17,18,19,20,21], re-entrant cells [22,23,24,25,26,27,28,29], the collective behavior of interacting particles [11,16,30,31,32,33,34,35], and triply periodic minimal surface lattice systems [36,37,38].

Interest in these materials has surged since the end of the 1980s, following the pioneering work of the likes of Lakes [39], Wojciechowski [11,16] and Evans [40]. It was spurred, at least in part, by the advantages auxetics have over conventional materials. These include improved shear stiffness [39,41], indentation resistance [39,42,43], fracture toughness [39,44], and energy absorption [45] properties. For this reason, auxetics have been proposed for a number of applications for which their behavior can result in improved functionalities. Amongst the suggested potential uses, there are dilators for medical applications [46], fastening devices [47,48], impact loading devices [49,50,51,52,53,54], smart particle filters [41,55], and textiles [56,57,58,59,60,61].

One of the earliest and most well-known auxetic mechanisms is that of the “rotating rigid units”. It is based on rigid elements that are joined at their corners or edges. The connection points serve as pivots (hinges) that allow the elements to rotate relative to one another without changing shape themselves. The initial geometries considered were rotating squares [2,4,5] and equilateral triangles [62], these being the only regular polygons that can be hinged at their edges so as to act as rotating units and, at the same time, still be space-filling [63]. While these naturally acted as an initial starting point, irregular polygons such as rectangles [63,64], parallelograms [3], and scalene triangles [65] soon followed. More recently fragmentation-reconstitution [8,9,10] was introduced, extending how rotating rigid units can operate. Further relaxation on the shape of polygons considered allowed the design of systems where the elements can have different shapes [66,67,68,69].

The concept of rotating rigid units was also extended to 3D. For this purpose, polyhedra were mostly considered. One of the earliest examples is the one presented by Alderson and Evans [70] who considered connected tetrahedra. Other geometries quickly ensued, including cuboids [6] and connected prisms [71,72,73,74,75,76], as well as more general polyhedra [7].

Relaxing the rigidity requirement slightly leads to a class of structures, known as rotating semi-rigid units, that can also exhibit auxetic behavior. This concept was developed for systems including the rotating square [77], triangles [78], and tetrahedra [70] ones. While this mechanism was fruitfully used to explain the auxetic behavior of interacting atoms, practical implementations appear to have been very limited. One of the few examples can be considered to be the “cross” system [79,80], where regions of the parameter space exist allowing for the rotating rigid units and flexing of the ligaments to be equally important as a deformation mechanism. Another is represented by the work of Mrozek and Strek [81], who studied the behavior under both static and dynamic loading of what can be considered to be connected square frames. Their findings highlighted clear differences between conventional and auxetic configurations in terms of both stiffness-related behavior and vibration response. The investigation also identified the frequency intervals for which the mechanical impedance and transmissibility are appropriate.

Another system that has received some attention in the context of auxetics is corrugated sheets. These typically consist of an undulating pattern of crests and troughs. For example, Zhang et al. [82] and Li et al. [83] combined corrugated sheets with tubes to obtain a system exhibiting a negative Poisson’s ratio, while Grima et al. [84] modified graphene by introducing patterned “defects”, with the resultant attaining a wave-like shape and exhibiting auxeticity. It is also possible to use corrugated sheets to design structures mimicking known mechanisms, such as in the work of Li et al. [85], who used them to create a system based on the double arrow pattern. An important consideration when studying these systems is that the shear correction factor is not constant, but strongly depends on the structural configuration of the corrugated panels, especially the fluting geometry [86].

In view of these considerations, this work presents a simple and effective manufacturing methodology that can be used to potentially produce an infinite variety of structures that deform through the semi-rigid rotating unit mechanism. It consists primarily of making perforations in corrugated plates. The concept can, to some extent, be considered as a practical implementation at the macro scale of the modified graphene discussed by Grima et al. [84]. However, in the case considered here, the corrugations are not being induced by the holes, as occurs for the modified graphene, but can be chosen arbitrarily. The same applies to the shape of the holes.

For the purpose of the investigation, a corrugated sheet with a side profile consisting of semicircles alternating in orientation was selected, and diamond-shaped perforations were adopted. The system was studied using numerical and experimental techniques. Analysis of the results indicates that such structures can indeed deform through a semi-rigid units type of mechanism exhibiting both a negative Poisson’s ratio along multiple planes and a change in shape. Considering the already widely available off-the-shelf variety of corrugated plates, the methodology could easily allow for extended industrial adoption and use.

2. Materials and Methods

2.1. Design of the Structure

The usual rotating square system can be obtained through diamond-shaped perforations of a planar plate. Adjacent perforations are spaced by a distance s between the tips of the diamonds. Furthermore, successive diamond shapes are rotated by 90° with respect to each other. The end result is illustrated in Figure 1a. Apart from the distance between the corners of the perforations s and the thickness t in the out-of-plane direction, the system is characterized by the side length of the diamonds l and an internal angle, which can be taken as that between a side and the minor diagonal, namely θ, as shown in the figure. Based on these parameters, the unit cell of the flat rotating squares system has dimensions along the x, y, and z directions given, respectively, by

$$L_{\mathrm{F},x}=2\left[l\cos \left(\theta \right)+l\sin \left(\theta \right)+s\right];$$

(1)

$$L_{\mathrm{F},y}=t;$$

(2)

$$L_{\mathrm{F},z}=2\left[l\cos \left(\theta \right)+l\sin \left(\theta \right)+s\right].$$

(3)

This work extends the concept of the rotating square system to corrugated plates. For the purpose of the investigation, the profile along one of the sides of the plate was chosen to consist of consecutive semi-circular arcs having equal dimensions but rotated by 180° with respect to the adjacent ones. As illustrated in Figure 1b, these have external and internal radii denoted by r₁ and r_2, respectively. The thickness of the plate is thus given by t = r₁ − r₂.

Even though the resultant plate does not have a flat side profile, diamond-shaped perforations can still be performed on it in the same fashion as those of the usual rotating square system. As viewed from above, the system looks like the rotating square one. It is only the lateral view that is different, as illustrated in Figure 1b.

Before proceeding, it is important to note that this work will only consider cases where the four squares shown in Figure 1a can be fitted exactly on two semi-circular arcs. With this restriction, l, s, t, and θ are not independent of r₁ and r₂ (and vice versa). In fact, due to geometric constraints, these variables are related through the equation:

$$r=l\left[\cos \left(\theta \right)+\sin \left(\theta \right)\right]+s,$$

(4)

where r is the radius of curvature associated with this system, defined as the average of r₁ and r₂ (i.e., the radius of the surface corresponding to half the thickness t). This may also be written in terms of r₁ and r₂ as follows:

$$r_1={\displaystyle \frac{l\left[\cos \left(\theta \right)+\sin \left(\theta \right)\right]+s+t}{2}},$$

(5)

$$r_2={\displaystyle \frac{l\left[\cos \left(\theta \right)+\sin \left(\theta \right)\right]+s-t}{2}}.$$

(6)

Furthermore, since it is understood that r₂ > 0, it then also follows that

$$l>{\displaystyle \frac{t-s}{\cos \left(\theta \right)+\sin \left(\theta \right)}}.$$

(7)

Based on these quantities, the dimensions of the unit cell in the x, y, and z directions can be derived, respectively, as

$$L_{\mathrm{C},x}=2\left[l\cos \left(\theta \right)+l\sin \left(\theta \right)+s\right]=2r;$$

(8)

$$L_{\mathrm{C},y}=4r_1;$$

(9)

$$L_{\mathrm{C},z}=2\left[l\cos \left(\theta \right)+l\sin \left(\theta \right)+s\right]=2r.$$

(10)

Before proceeding, it is important to highlight that the chosen side profile represents only one of the infinite possible designs that can be used. In fact, both sides can have a nonlinear shape, and the two side profiles can be different. Notwithstanding this, after the perforations are performed, as viewed from above, the result will always look like the rotating square system. Different side profiles have the possibility of conferring distinct mechanical properties. However, investigating the effect of different side profiles was considered beyond the scope of the present study and will be explored in future work.

2.2. Finite Element Simulations

The mechanical behavior of the corrugated perforated system described in the previous section was investigated using finite element analysis (FEA) with the commercial software ANSYS^® Academic Research Mechanical APDL Release 13. In order to simulate the structure shown in Figure 1b in a time-efficient way, use was made of the symmetry within the unit cell. This allowed the simulation of just one fourth of the unit cell as representative of the bulk material (see Figure 1b).

For geometry creation, the bulk of the structure was designed using the area primitive functions natively available within the software. Two quarter-ring areas were first created and joined together to form the side profile, which was subsequently extruded along the z direction to generate the three-dimensional volume, hereafter referred to as the main volume. The diamond-shaped perforations were then created in the xz plane by first defining key points, which were used to generate the corresponding areas. Similar area configurations were subsequently produced using the copy and translation functions available within the software. Once completed, these areas were extruded in the y direction to form volumes that passed through the main volume. The newly formed volumes were then grouped together and subtracted from the main volume using Boolean operations. The process was automated using the inbuilt scripting language of ANSYS^® Mechanical APDL.

Once the geometry was prepared, Element SOLID187 under plane-stress conditions was used to mesh the structures [87]. This is a higher-order 3D, 10-node element having a quadratic displacement behavior and three degrees of freedom at each node (translations in the nodal x, y, and z directions). It is also well-suited to model irregular meshes.

Symmetry boundary conditions could then be applied in the x and z directions. On the other hand, in principle, the structure should have been allowed to move freely in the y direction. However, some constraints needed to be applied in order to prevent the system from distorting in the out-of-plane direction. This was attained by applying symmetry boundary conditions along the y direction as well. Justification for this follows from the fact that it is expected that the top part of the corrugation remains in the same plane throughout the loading, at least to a first approximation. A summary of the imposed boundary conditions for tension in the x direction is given below:

• Nodes with the smallest x coordinate were constrained against displacement in the x direction, while remaining free to move in the other two directions.
• Nodes with the largest x coordinate were subjected to a prescribed displacement corresponding to the applied strain, while remaining free to move in the other two directions.
• Nodes with the smallest y (z) coordinate were constrained against displacement in the y (z) direction, while remaining free to move in the other two directions.
• Nodes with the largest y (z) coordinate were prescribed to undergo identical displacements in the y (z) direction, while remaining free to move in the other two directions.

• Analogous boundary conditions were applied for loading along the other directions.

For the simulations, the material properties assigned to the structure were based on those reported for 3D-printed PLA in previous studies [88,89], with a Poisson’s ratio of 0.45 and a Young’s modulus of 1600 MPa. A mesh convergence analysis was then carried out in order to determine the settings that gave results that were within 1% of those obtained with a finer mesh. The minimum mesh density satisfying this requirement was attained by first applying the automated meshing function available within the software using the finest mesh setting. Subsequently, the elements located near the boundary nodes were refined to at least a level of 3 with a refinement depth of 6. This meshing strategy provided an optimal balance between mesh density and computational cost by increasing the mesh resolution only in regions where the majority of the deformation occurred. For example, when l = 20 mm, s = 0.5 mm, t = 5 mm, and θ = 20°, the resulting mesh consisted of 298,218 elements.

With these settings, a parametric analysis was carried out to determine the dependence of the Poisson’s ratios and Young’s moduli of the system on the geometric dimensions. For this purpose, the values of the parameters listed in

Table 1. The values of the geometric parameters used for the linear simulations.

Parameters	Values
l/mm	10 to 140 in increments of 5
t/mm	2.5, 5, and 10
s/mm	0.5 and 1
θ	5° to 45° in increments of 5°

were adopted. Linear simulations with plane-stress conditions were then carried out to study the behavior of the system under tensile loading.

Using the simulation results, the values of Poisson’s ratios and Young’s modulus were determined by first calculating the original lengths X_i, where i = 1, 2, 3, of the unit cell using the positions of the outermost nodes along each direction. The positions of the same nodes after deformation were then used to calculate the corresponding displacements, δ_i. The applied load was obtained from the reaction force, F_i, acting on the nodes in the loading direction using the inbuilt functions available within the software. Using these quantities, the engineering strains were computed as

$${\epsilon }_i={\displaystyle \frac{{\delta }_i}{X_i}},$$

(11)

allowing the Poisson’s ratios to be determined from

$${\nu }_{ij}=-{\displaystyle \frac{{\epsilon }_j}{{\epsilon }_i}},$$

(12)

where j = 1, 2, 3 and i ≠ j. The Young’s modulus was then calculated using

$$E_i={\displaystyle \frac{1}{{\epsilon }_i}}\left({\displaystyle \frac{F_i}{X_j{X}_k}}\right),$$

(13)

where k = 1, 2, 3 and k ≠ j, i.

For comparison purposes, the flat, rigid, rotating square systems were also simulated using the same parameter space. In this case, the bulk geometry was created using a rectangular area primitive. Once the areas corresponding to the diamond-shaped perforations had been generated as described above, these were subtracted from the bulk area using Boolean operations. The resulting area was then extruded in the y direction. After the geometry was completed, the same element type, boundary conditions, material properties, and dimensions used for the corrugated structures were applied.

Subsequently, nonlinear simulations were carried out on a system having l = 20 mm, s = 0.7 mm, t = 5 mm, and θ = 20°. These dimensions were chosen after taking into consideration the analysis of the results obtained from the linear simulations, as well as the dimensions of the 3D printer that was meant to be used to produce a prototype for mechanical testing. For this purpose, a static simulation that considers the large-deflection effect was chosen. In order to simplify the simulations, it was also assumed that the material deformed in its linear region. This assumption was justified on the basis that a relatively small strain of 7.5% was applied.

In terms of the simulation settings, the line search option was activated, as this facilitates the convergence of the Newton–Raphson solver. Automatic time stepping was employed, with both the initial and minimum number of substeps for the load step set to 100, while the maximum number of substeps was set to 200. All other settings, including those related to mesh generation, were retained from the linear simulations.

2.3. Production of the Prototype and Its Mechanical Testing

In order to further investigate the properties of the curved system, a prototype was produced using additive manufacturing. It consisted of 3 × 3 unit cells having l = 20 mm, s = 0.7 mm, t = 5 mm, and θ = 20°. To facilitate mechanical characterization, rigid T-shaped end sections were incorporated along both sides of the prototype to ensure uniform axial load transfer and secure clamping within the testing machine. Each end section had a bar-like element with dimensions of 13.5 cm (length) × 1.2 cm (height) × 1.3 cm (depth), as shown in Figure 2.

Manufacturing of the structure was undertaken using an Ultimaker S5 3D printer (Ultimaker BV, Utrecht, The Netherlands) loaded with UltiMaker PLA White filament (Ultimaker BV, Utrecht, The Netherlands). The nozzle had a diameter of 0.25 mm, the filament was heated to 200 °C, while the printing bed had a temperature of 60 °C. A printing speed of 70 mm s⁻¹ was selected, the raster angles were set to 60°, and the fan speed to 100%. For the print, a wall thickness of 0.8 mm was chosen, the wall line count was 2, and the layer thickness was 1 mm. The infill was set to 100%. Additionally, supporting material had to be used to ensure that overhanging parts were printed adequately. This was then removed completely once the print was ready. Three specimens were produced to allow repeated readings.

The specimens were then tested using an Instron 5985 universal testing machine (Instron, Milan, Italy) equipped with a 10 kN load cell, in strain control mode at a speed of 1 mm min⁻¹. The experimental setup is illustrated in Figure 3. Three sets of measurements were obtained: one from each specimen. The average longitudinal strain at fixed axial strains was then calculated together with the standard deviation. In order to do this, the axial (x-axis) and transverse (z-axis) strains were determined from the dimensions of a central 1 × 1 unit cell (cf. Figure 1b) in line with the best practice given by Yolcu and Baba [90]. The unit cell, which is shown enclosed by a dashed line in Figure 3, had dimensions L_C,x = L_C,z = 33.5 mm and L_C,y = 46.9 mm. A set of black markers was drawn to delineate it, and their position during the deformation was recorded with the digital camera on the Google Pixel 6a phone (Menlo Park, CA, USA) having a 12.2 MP main sensor (f/1.7) with optical image stabilization and Dual Pixel PDAF technology (Sony Exmor IMX363; Tokyo, Japan), alongside a 12 MP ultra-wide lens (f/2.2) featuring a 114° field of view (Sony Exmor IMX386; Tokyo, Japan). Subsequently, the images were analyzed using the image-processing software ImageJ version 1.54k (National Institutes of Health, Bethesda, MD, USA) to determine the distance between the markers. The values were then used to calculate the engineering strains and Poisson’s ratios according to Equations (11) and (12).

3. Results and Discussion

3.1. Analysis of the Results Obtained from the Linear Simulations

A look through the results obtained from the linear simulations revealed that, in the main, these are very similar for all values of s and t taken. The major difference resides in the range of values attained, with the values of the Young’s moduli increasing with increasing s and/or t while those of the Poisson’s ratios show a more complex relation. For this reason, only the results for s = 0.5 mm and t = 5.0 mm will be discussed in detail, with additional information given in the Supplementary Material. This also includes an investigation of the mass fraction of the structures.

As can be noted from Figure 4, the perforated corrugated system is able to show a negative Poisson’s ratio in both the yz and zx planes, while it is completely non-auxetic in the xy plane, at least for the values of the parameters considered. As a general trend, the Poisson’s ratio of the perforated corrugated structure tends to attain lower values for smaller angles than for angles close to 45°. The trend could be expected since 45° represents the closed position for the rotating square system. On the other hand, the behavior of the system seems to be rather insensitive to the value of l. There are very few exceptions to this behavior, these occurring for small θ and l when loading is in the y direction and for larger θ when loading is in the z direction. The lowest negative Poisson’s ratios occur in the zx plane. This is especially the case when loading in the z direction. The zx plane is the one where the diamond perforations were undertaken. Thus, this behavior can be attributed to a rotating rigid units mechanism. However, the fact that Poisson’s ratio is not constant and close to −1, as is predicted from the rotating squares system, indicates that another mechanism is acting, as will be considered in detail in the analysis of the nonlinear results.

Thus, it can be observed that the behavior of the perforated corrugated system differs in many aspects from that of the regular rotating squares. Comparing the results shown in Figure 4 with those obtained for the corresponding rotating squares system (Figure 5), it can be noted that in the latter case, auxeticity is confined to the zx plane. In the out-of-plane direction, Poisson’s ratio takes values close to zero when loading in the x or z direction and close to 0.45 (the value of the constituent material Poisson’s ratio) when loaded in the out-of-plane direction. At the same time, in the xz plane, Poisson’s ratio takes a value close to −1 for most values of θ and l. The major exceptions to this general behavior occur when loading close to the locked position, i.e., when θ is close or equal to 45°. In this configuration it is hard for the rigid units to rotate relative to one another so that the rotating square model is no longer applicable. It should be noted that the usual rotating square system attains lower and more constant values of the Poisson’s ratio compared to the proposed system. At the same time, the proposed system is able to demonstrate auxeticity in 3D. Thus, the perforated corrugated system allows for the tailoring of a wider range of Poisson’s ratio, extending from negative (close to −1) to positive ones (close to +1).

Further differences between the behavior of the perforated corrugated plate and the planar one can also be observed from the plots of the Young’s moduli. In the case of the corrugated plate, the maximum values of the Young’s moduli occur prevalently at small values of l, as can be noted from Figure 6. This can be attributed, at least in part, to a leverage effect on the joints, with the induced moment increasing for increasing l. Additionally, the density of the structure increases for a smaller diamond side length. On the other hand, the locking angle, namely 45°, seems to have relatively little effect on the behavior of the Young’s moduli. Its influence appears to be pronounced only when loading in the z direction (Figure 6c). This is also the direction where the highest values of the Young’s moduli are attained, with the maximal values being more than 30 times that attained for loading in the xy plane. The reason why θ = 45° does not lead to a very high Young’s modulus in the perforated corrugated plate is that, for the structure under consideration, this configuration does not really represent a locked position. In fact, the structure is still able to deform with relative ease in the out-of-plane direction.

For the flat rotating square system, as can be observed in Figure 7, the locked position does indeed lead to relatively high Young’s moduli for loading in the xz plane. Yet the highest values of Young’s moduli occur when loading in the out-of-plane (or y) direction when the θ is relatively small. The reason for this is that at this angle, the density of the material is relatively high due to the fact that the cross-sectional area of the perforations is comparatively small. Hence, the stress is being applied on a compact material, leading to a Young’s modulus close to that of the constituent material. At the same time, increasing l results in a decrease in the Young’s modulus when the force is applied in the xz plane. This can be attributed once again to a leverage effect. It is also interesting to note that the minimum value of E_y (the Young’s modulus in the y direction) is more than twice the maximum of E_x and E_z (the Young’s modulus in the x direction and z direction, respectively). Furthermore, the maximal values of the Young’s moduli for the flat system are much more than those obtained for the one with the perforated corrugated plate.

3.2. Analysis of the Results Obtained from the Experimental Testing and Nonlinear Simulations

The results for Poisson’s ratio obtained from the nonlinear simulations are shown in Figure 8. In all cases, the maximum strain in the loading direction was 7.5%. However, the simulation for loading in the y direction failed after ε_y was around 2.85% so that only the results up till this point are shown.

It can be noted that, for all Poisson’s ratios, the observed behavior with increasing axial strain is rather linear. This can also be verified from the values of Pearson’s correlation coefficient r_P that are given in

Table 2. The Poisson’s ratios obtained from the linear analysis are given together with the values obtained through linear regression performed on the results of the nonlinear simulations, including the regression uncertainty and Pearson’s correlation coefficient r_P.

		Nonlinear Regression Analysis
	Linear Value	Nonlinear Value	Uncertainty	rP
νxy	0.390	0.452	0.002	0.9990
νxz	−0.401	−0.351	0.001	0.9997
νyx	1.019	1.010	0.002	1.0000
νyz	−0.218	−0.206	0.001	1.0000
νzx	−0.676	−0.574	0.003	0.9994
νzy	−0.140	−0.137	0.0002	1.0000

. Such a behavior is not common amongst auxetic structures, whereby in many cases the Poisson’s ratio ν tends to increase with axial strain [89]. At the same time, a constant negative Poisson’s ratio can be a desirable feature, for example, in the case of impact loading, as the superior energy absorption abilities conferred by virtue of its auxeticity can be retained throughout the deformation. Hence, this structure can be added to the pool of the few auxetic systems that are able to retain a constant ν.

Of additional interest is the fact that the values of Poisson’s ratios obtained from linear regression are also close to those obtained from a linear simulation (Table 2). This could have been expected given the rather linear variation observed in the behavior. Thus, it is possible to get a good initial idea of the nonlinear value of the Poisson’s ratio from the linear results, potentially reducing the computational time and resources required to investigate and optimize the system for specific applications.

Figure 8b also shows the measurements made on the prototypes. As can be noted, the variation in the experimental and numerical results is similar but not exactly equal. In fact, the value of the Poisson’s ratio obtained from linear regression carried out on the experimental measurements was −0.281 ± 0.005 (and r_P = 0.9990), which is more positive than that obtained from the simulations (Table 2). The differences can, to some extent, be explained through imperfections of the physical prototype. For example, if the distance between perforations s was not printed with absolute precision—something that could easily happen—it could easily alter the value of Poisson’s ratio. In this case, the filament nature of the printing meant that the joints cannot be perfectly continuous. Furthermore, the curvilinear profile necessitated the use of supports, thus limiting the precision with which these parts could be printed. Hence, even if there is a difference between the experimental and numerical results, given the limitations of the 3D printing procedure, the agreement can be considered reasonable. At the same time, the discrepancy suggests that deviations can be expected between the Poisson’s ratio values predicted by simulations and those obtained experimentally.

A glimpse into the deformation mechanism acting in this nonlinear deformation can be obtained from Figure 9. As can be noted from the first two panels, the structure with a corrugated side profile becomes distorted so that the side profile starts to be observable when viewing the system from above (see ovals drawn in green). This is unlike the flat system, where only the top part is visible even at the maximum strain applied. Additionally, elongation of the structure with the curvilinear side profile can also be observed, particularly in Figure 9a. The distortion of the structure explains why the value of Poisson’s ratio is not close to −1, as predicted for the ideal flat rotating square system. At the same time, it is possible to envision that the change in shape of the system with a curvilinear side profile could potentially be put to good use if it is possible to control the final form attained. For example, this could be used to create a smart filter where not only the size of the pores can be made to increase when a tensile load is applied but also their shape can be made to attain a desired design.

Thus, the results obtained all indicate that it is possible to extend the rotating square structure to a 3D auxetic system simply by applying perforations on corrugated plates. While the study focused on diamond-shaped perforations, it is to be expected that other systems can be similarly obtained by drilling holes of different shapes. Hence, further investigations are likely to follow on this innovative method.

It should also be mentioned that, while in this study the structure was fabricated using an additive manufacturing technique, it is also possible to produce it using subtractive techniques. This could be particularly advantageous if the perforations are carried out on a flat sheet, before it is corrugated. Furthermore, the use of auxetic corrugated sheets within sandwich panels can provide a lightweight material with enhanced energy absorption capability. The perforation of the sheets can be carried out using already available techniques, providing for a convenient, ready-to-go fabrication methodology.

There are also other important aspects to consider. The first is that corrugation need not be along one single direction. In fact, it is possible to have them along the y and z directions concurrently. Furthermore, these structures can be manufactured using casting. This allows the use of not only traditional materials but also biomaterials, such as mycelia-based composites [91,92]. This method could also have the advantage of combining the impact absorption potential of auxetic systems with the properties of bio-composites made from fungi in order to attain an innovative and sustainable packaging product [93].

4. Conclusions

A practical fabrication methodology for the manufacturing of 3D auxetic structures has been presented in this work. The production process involves the use of perforations on corrugated plates. For the purpose of an in-depth investigation, the structure obtained when diamond perforations are performed on a corrugated plate with a side profile consisting of an alternating pattern of semicircles was considered. As viewed from above, the structure obtained resembles the rotating square system. However, it can differ significantly laterally.

Results show that the structure can exhibit 3D auxeticity. However, its Poisson’s ratio does not attain a constant value of −1 in the plane where the rotating square system can be observed. This was related to the fact that the deformation mechanism does not involve the simple rotation of rigid units. Instead, there is a concurrent deformation of the shape of the structure, leading to a decrease in its auxeticity as compared to a rotating square system. The observed behavior could be used to simultaneously tailor Poisson’s ratio and the final shape attained for specific applications, such as to manufacture smart filters with both pore size and shape changing during loading.

Thus, given the observed mode of behavior, the structure can be considered to deform through a semi-rigid rotating units mechanism, making it one of the first macro-scale systems showing these characteristics. Another innovation from this research is the fact that it is possible to extend 2D auxetic systems obtained through perforations to 3D simply by using a corrugated plate. While in this work, corrugations were only along one side, in practice, both lateral sides can be patterned. This leads to a plethora of possible auxetic structures that can be created, something that can lend itself to an extensive amount of research.