3D printed negative stiffness meta-structures with superior energy absorption and super-elastic shape-recovery features

The aim of this paper is to create novel 3D cubic negative stiffness (NS) structures (NSSs) with superior mechanical performances such as high energy absorption, shape recovery, super-elasticity, and reversibility. The conceptual design is based on an understanding of geometrical influences, non-linear buckling-type instability, snap-through mechanism, elasto-plastic deformation growth and plastic hinges. A finite element (FE) based computational model with an elasto-plastic material behavior is developed to design and analyze NSSs, saving time, material, and energy consumption. Material samples and meta-structures are 3D printed by selective laser sintering printing method. Material properties are determined via mechanical testing revealing that the printing process does not introduce much anisotropy into the fabricated parts. Experimental tests are then conducted to study the behavior of novel designs under loading–unloading cycles verifying the accuracy of the computational model. A good correlation is observed between experimental and numerical data revealing the high accuracy of the FE modeling. The structural model is then implemented to digitally design and test NSSs. Effects of the geometrical parameters of the negative stiffness members under three cyclic loading are investigated, and their implications on the non-linear mechanical behavior of NSSs under cyclic loading are put into evidence, and pertinent conclusions are outlined. In addition, the dissipated energy and loss factor values of the designed structures are studied and the proposed unit cell is presented for the energy absorbing systems. The results show that the structural and geometry of energy absorbers are key parameters to improve the energy absorption capability of the designed structures. This paper is likely to fill a gap in the state-of-the-art NS meta-structures and provide guidelines that would be instrumental in the design of NSS with superior energy absorption, super-elasticity and reversibility features.

The aim of this paper is to create novel 3D cubic negative stiffness (NS) structures (NSSs) with superior mechanical performances such as high energy absorption, shape recovery, super-elasticity, and reversibility. The conceptual design is based on an understanding of geometrical influences, non-linear buckling-type instability, snap-through mechanism, elasto-plastic deformation growth and plastic hinges. A finite element (FE) based computational model with an elasto-plastic material behavior is developed to design and analyze NSSs, saving time, material, and energy consumption. Material samples and meta-structures are 3D printed by selective laser sintering printing method. Material properties are determined via mechanical testing revealing that the printing process does not introduce much anisotropy into the fabricated parts. Experimental tests are then conducted to study the behavior of novel designs under loading-unloading cycles verifying the accuracy of the computational model. A good correlation is observed between experimental and numerical data revealing the high accuracy of the FE modeling. The structural model is then implemented to digitally design and test NSSs. Effects of the geometrical parameters of the negative stiffness members under three cyclic loading are investigated, and their implications on the non-linear mechanical behavior of NSSs under cyclic loading are put into evidence, and pertinent conclusions are outlined. In addition, the dissipated energy and loss factor values of the designed structures are studied and the proposed unit cell is presented for the energy absorbing systems. The results show that the structural and geometry of energy absorbers are key parameters to improve the energy absorption capability of the designed structures. This paper is likely to fill a gap in the state-of-the-art NS meta-structures and provide guidelines that would be instrumental in the design of NSS with superior energy absorption, super-elasticity and reversibility features. * Author to whom any correspondence should be addressed.
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Introduction
The ability to absorb and dissipate kinetic energy by different static and dynamic loadings or vibrations is an essential requirement for many engineering applications. Energy absorbers like car bumpers and bicycle helmets absorb mechanical energies via permanent deformations of foams or cellular materials mostly made of plastics [1]. In general, the mechanical energy can be dissipated through non-recoverable methods, such as auxetic structures with permanent deformation in the plastic domain that unlike traditional structures, auxetic meta-materials show lateral shrinkage under compressive loading or lateral expansion under tensile loading [2][3][4][5][6][7] or recoverable methods such as non-linear instability [8,9] and viscoelastic mechanisms [10,11]. Practically, energy absorbing structures should be lightweight and capable of energy absorption in a reversible manner.
Negative stiffness (NS) structure (NSS) is a class of metamaterial structures designed to operate based on magnetic interactions [12] and the buckling mechanism in its members to absorb energy, maintain their original configuration, and prevent vibration of structures through cyclic loadings [13]. During buckling in a member of the NSS, with an increase in the applied compressive loading, a force-drop occurs, a snap-through mechanism is activated in the structure, and it reveals an NS behavior [14]. Such structures have been used as actuators [15,16], bumpers [17], energy absorbers [18][19][20][21][22], and vibration absorbers [23,24]. While other energy absorber structures such as auxetics are disposable, NSSs have the advantage of multiple-use applications [25][26][27][28]. Furthermore, the load bearing ability of viscoelastic materials is under question [14,15].
Qiu et al [29] proposed the possibility of using curved beams to create the NS behavior. They developed analytical equations to characterize the behavior of curved beams. Izard et al [30] designed an NSS based on a curved beam. Their results showed that using a buckling beam as a member with the NS property improved the energy absorption in comparison with other conventional structures. Tan et al [31,32] studied the mechanical behavior of NSSs, including V-shaped, curved and arched beams with different configurations, through analytical, finite element (FE) simulation, and experimental tests. They concluded that the use of the curved and V-shaped beams in NSSs exhibited better NS behavior than the arched beams. Virk et al [33] designed cellular structures with zero stiffness and NS effects. Their cellular structures demonstrated a large deformation capability with zero stiffness and NS regimes for a special range of strain rates. Yoon and Mo [34] designed a new structure with a bidirectional NS mechanism and conducted an experimental test to determine the effect of the snap-through mechanism on its performance. Zhang et al [26,35] improved the energy dissipation capability of NSSs by increasing the bending stiffness of a curved beam and maintaining a constant volume to mass ratio. They designed two types of lattices and hollow cross-sections as building blocks for multi-stable structures. Recently, Ma et al [36] proposed a multiple-use and self-recoverable NSS with tetrahedral motif unit cells (UCs) and curved beams with an ability to dissipate energy in three directions.
The 3D printing technology is one of the common highprecision methods to manufacture conventional [37,38] and meta-material [38] structures. NSSs can be manufactured through various 3D printing technologies such as fused deposition modeling (FDM) [38][39][40], selective laser sintering (SLS) thin, stereolithography (SLA) [41,42], and multi-jet fusion [43]. For instance, Ha et al [44] presented a 3D cubic NS lattice structure including multiple UCs with a high energy absorption capacity and shape recovery in the elastic domain with 20% strain. They used SLS printing technology to manufacture the structure and verified the results with FE method (FEM). Ha et al [45] designed and printed an NSS including multiple tetra-beam-plate UCs and investigated the mechanical behavior of the structure through numerical and experimental analyses. They concluded that the snap-through mechanism of the NSS is due to the nonlinear geometry of the elastic beam members. Tan et al [46] designed disk elements on the face sides of cubic UCs and introduced a 3D NS disk structure. They fabricated the structure using SLA printing method, evaluated its mechanical behavior and energy absorption values under compressive loading, and compared their results with those from FEM. Gao et al [47] introduced negative stiffness UCs manufactured by SLS printing technique with the ability to absorb energy and recover their original configuration in two loading directions. They investigated the mechanical behavior of the UC and 3D cubic NS lattice structures under compressive and tensile loadings through theoretical, FE simulation, and experimental studies. Gholikord et al [48], assembled the positive and negative stiffness members manufactured by FDM method as new NSSs and evaluated their energy absorption capabilities under cyclic loadings and verified the energy absorption values with FEM.
The literature review reveals that although NSSs have been investigated in the recent years, their capabilities for energy dissipation, super-elasticity, shape recovery and reusability have not yet been discovered. This paper aims at presenting novel designs of 3D cubic NS UCs with new features like high energy absorption, shape recovery, superelasticity, and reversibility. The design approach is based on an understanding of geometrical effects, non-linear bucklingtype instability and elasto-plastic deformation mechanism. A FEM based on the elasto-plastic model is established as a digital tool to study geometrical effects on the structural behaviors reducing material waste and experimental efforts. Several dog-bone samples and NSSs are 3D printed by SLS printing technology. Experiments are first conducted to determine mechanical material properties of the fabricated part revealing that the printing process does not introduce much anisotropy into the printed parts. Experimental studies are then carried out to investigate novel designs' energy absorption in response to loading-unloading cycles. A good correlation between experimental and numerical results is seen confirming high accuracy of the FEM. FE solution is then applied to digitally design and test NSSs mechanically. A detailed analysis of the influence of geometrical parameters on the mechanical behaviors of NSSs under cyclic loading is carried out. Due to the absence of similar design and results in the specialized literature, it is expected that the results of this research will contribute to a better understanding of the mechanical behavior of energy absorbing NS meta-structures with super-elasticity and reversibility features.

Conceptual design of novel NSSs
This section discusses the design of five NSSs (namely UC-1, UC-2, UC-3, UC-4, and UC-5) with NS properties. Figures 1(a)-(c) shows UC-1 schematic of the side view, isometric view, and one of the face views, respectively. All UCs consist of 12 frame members, 6 plates, and 24 cross beams, see figure 1(b). Beams are connected from two sides to the corner of the frames and plates and the stalks are embedded on upon the plates. All of them are placed on each side face of the NSSs. In figure 1, the geometrical parameters include the height of the beam (t b ), width of the beam (b b ), inclined angle (α), length of the beam (L b ), inclined height of the beam (h b ), side length of the plate (L p ), thickness of the plate (t p ), side length of the frame (L f ), and the thickness of the frame(t f ). Table 1 also lists the values of geometric parameters of UC-1-5. The inclined angle α is 15 • for all NSSs. The snap-through mechanism in NSSs is expected to be based on the buckling of cross beams.
UC-1 is a basic design and is drawn according to [44] with changing its geometrical parameters. The frame and plates are designed with larger dimensions than the beams leading to more deformation of the beams than the frames and plates to prevent undesired buckling. The maximum displacements of the UCs are 12 mm (ε max = 17.05%), and for displacements greater than 12 mm, the frames would start to buckle and collapse. The importance of maximum displacement loading will be discussed in section 4.3.1.
Subsequently, based on the limitations of UC-1 in terms of energy absorption and energy dissipation capabilities, new NSSs with a high mechanical performance are designed here. Figures 2(a) and (b) show side and isometric views of UC-2, respectively. The difference between UC-2 and UC-1 is that in UC-2, the geometric dimensions of the plates are larger than UC-1 to delay the corner beams connected to the plates from entering into the plastic region. In addition, the ends of the beams are chamfered to decrease the stress concentration at the attached places of the beams and stalks.
UC-3, UC-4, and UC-5 are then designed based on the addition of different energy absorber members to UC-2, see figures 3-5(c). The energy absorbers are proposed to improve the energy absorption and dissipation capability of the UCs and recover their original configurations in the loading direction. They include curved members embedded in the middles of the UCs. Their design is based on the allocation of sufficient space for expansion during the compressive loading. The geometry of the energy absorber members should be designed such that they can restore their original configuration during the unloading process. Furthermore, energy absorbers are designed from curved corners instead of sharp ones to decrease their stress concentration and improve their energy absorption mechanism.
It is expected that the energy absorption and specific energy absorption values of UC-3 to UC-5 are improved compared to UC-2 and UC-1. That is due to the fact that upon the application the compressive load, the applied energy is transmitted from the moving loading plate to the stalks and then to the plates and finally it propagates through the internal and external energy absorbing elements for performing buckling in the elastic and plastic regions. As a result, by having more energy absorbing elements in UC-3, UC-4, and UC-5, the energy absorption values are expected to increase significantly. Furthermore, in UC-1 and UC-2, the plates enter the plastic region sooner than the other members during the compressive loading. With the addition of energy absorbers to the foresaid UCs, the stress concentration is expected to decrease in UC-3, UC-4, and UC-5, and as a result they could properly recover their original configuration. In section 4.2, the values of the energy absorption and specific energy absorption are presented and compared.
Figures 3(a)-(c) shows different views of UC-3. A camberarch as shown in figure 3(c) is added to UC-2, named UC-3, see figure 3(b). The camber-arch, as an energy absorber member, could exhibit a snap-through type buckling under compressive loading and recover its original shape during unloading. Table 1 lists the dimensions of the camber-arch.
UC-4 as illustrated in figure 4 is designed by adding another energy absorber between the top and bottom stalks of UC-2, named rhombus, see figure 4(c). Table 1 presents dimensions of the rhombus members. The geometry of the rhombus is designed in such a way that the UC-4 has enough space for buckling to exhibit NS behaviors.
Finally, UC-5 as demonstrated in figure 5 is innovated by adding another energy absorber with an DNA shape (see figure 5(c)) with different structural geometries compared to UC-3 and UC-4 to investigate the effects of the geometrical parameters on the energy absorption, dissipation and shape recovery capabilities. The advantage of DNA members compared to the camber-arch in UC-3 and rhombus in UC-4 is that by designing small curves in DNA member, the stiffness of the member and, as a result, the energy absorption and shape recovery could increase.

Materials and 3D printing
Polyamide material, PA 12, with an elastic-plastic behavior is selected as a base material to fabricate UCs. Lisa Pro Sinterit 3D printer with SLS technology is utilized to manufacture tensile test samples and the UC-2 and UC-3. PA 12 with a layer thickness of 0.125 mm is used for printing. The method of printing is as follows: after designing a sample in the Solidworks software, it is exported as printable STL files. After that, a thin layer of the powder is spread on a printing bed, and then a laser fuses it in the shape of the designed sample.
By performing some mechanical testing, it is observed that after SLS printing of the samples, the mechanical properties of the tensile samples are differed in 0 • , 45 • , and 90 • directions relative to the bed plate [49,50]. To evaluate the mechanical properties of PA 12, twelve tensile samples are fabricated in three directions according to the ASTM D638 standard [51]. Figure 6 shows geometric dimensions of a standard tensile sample. The room-temperature mechanical properties including Young's moduli (E 0 , E 45 , E 90 ), 0.2% offset yield strength (σ 0 , σ 45 , σ 90 ) and Poisson's ratio (v) of PA 12 are determined by the Shimadzu ® tensile testing machine and digital image correlation (DIC) method (to measure axial and transverse strains and hence Poisson's ratio) at loading rate of 2 mm min −1 . Typical engineering stress-strain curves are shown in figure 7 and the respective results are presented in table 2.
Among the designed UCs, UC-2 and UC-3 are manufactured using 3D SLS printing technology to check their performance under two cyclic loading. Three UC-2 and UC-3 are printed to achieve trustable results, and their mean mechanical properties are reported. However, the testing values of fabricated samples exhibit close correlations. Figure 8 (a) and (b) shows the side and isotropic views of UC-2 and UC-3 printed samples.

Mechanical testing procedure
The Shimadzu ® universal machine is used to perform two loading/unloading quasi-static compression tests on UC-2 and UC-3 at the room temperature (25 ± 1 • C). Figure 12 in section 4.1 shows the deformation sequences of UC-3 through the cyclic loading. Furthermore, the experimental movie S1-EXP (see the supplementary section) shows the experimental recording of the UC-3 behavior during two cyclic loadings. In the loading process, the top plate moves down and compresses the UCs to a maximum displacement of 12 mm with a loading velocity of 2 mm min −1 and then moves up with the same velocity to start an unloading process. Furthermore, the bottom plate is fixed during the cyclic loadings. The cyclic loading is repeated twice for both UC-2 and UC-3. During the test, the machine applies the displacement (d). The normal force (F) applied to the structure in all the loading and unloading processes is detected by a 50 kN load-cell. Force-displacement and stress-strain diagrams are obtained at the end.

FE simulation
The ABAQUS/standard FE software package is implemented to investigate the mechanical properties of the designed UCs under quasi-static cyclic loadings. An elastic-plastic material model is selected to determine the mechanical properties of PA 12. The elastic-plastic model is considered to investigate the possibility of entering the UCs into the plastic region.
Since, the Young's modulus and yield stress are approximately the same in three foresaid directions (see table 2), an isotropic material with E = 933 MPa, σ 0 = 28.7 MPa is considered for UCs to simplify the FE simulation; also, Poisson's ratio ν = 0.33 calculated from DIC measurements is assumed.
The top plate and all components of each UC are meshed using ten-node tetrahedral elements (C3D10). Mesh sensitivity analysis is performed according to [4]. For all the UCs to ensure high accuracy for the results. For the mesh sensitivity analysis, an element with the dimension 2.6 mm is selected, and the equivalent von Mises stress of the marked node in UC-5 (see figure 9) is measured. Meanwhile, the element dimension is reduced until the von-Mises stress change is negligible. Figure 10 shows the process of mesh sensitivity analysis for UC-5 structure. Considering figure 10, the element size with a dimension of 1.6 mm is selected as a reference element. The same process is implemented to mesh other designed structures.
In this study, a static-general step is considered to simulate the cyclic loading of the UCs. To apply the boundary conditions as shown in figure 6, a displacement-control loading is applied to the top plate in the z-direction and the other five degrees are constrained. Furthermore, the bottom area of the bottom stalk of the UC is constrained to all six degrees of freedom (three translations and three rotations).
To plot the force-displacement diagrams for a designed UC, the sum of the element forces (F) and vertical displacement (δ) of a node in the top plate are evaluated, respectively. It should be noted that the top plate nodes have the same vertical displacement, and it is adequate to measure the displacement of only one node. Again, all UCs are subjected to maximum displacement of 12 mm. Then, the applied stress (σ) and strain (ε) of the UCs are calculated by: where A and L are the area of the top plate and initial height of the UCs, respectively. In this paper, the A value is 167.33 mm 2 for all the UCs. After obtaining the stress-strain diagram, the energy absorption (E) values are obtained during the loading through equation (3) as: To conduct weight sensitive analysis of UCs, specific energy absorption (E m ) is investigated. This parameter is defined as the ratio of energy absorption to the mass (m) of the UCs as: Furthermore, the hysteresis loops in stress-strain diagrams which are enclosed during loading/unloading show the values of dissipated energy of the UCs. The values of dissipated energy in a hysteresis loop, ∆E, are calculated using equation (5): where E ′ is the energy release during unloading. Non-zero values of ∆E in the cyclic loading indicate that the NSSs could dissipate energy and have damping capabilities. The loss factor ψ is defined to investigate the ability of the NSSs to dissipate and dampen energy as: Among the designed UCs, the UC with the highest ψ value can dissipate more energy than the others. Figures 11 and 12 show the FE simulation and experimental test of UC-3 for different displacements during one loading/unloading, respectively. Furthermore, the behavior of UC-3 under two loading/unloading cycles is recorded in the S2-FEM movie (see the attached file in the supplementary section). The other UCs show a general behavior to that of UC-3. The detailed behavior analysis of the UCs through FE simulation will be explained in section 4.2. Figures 13  and 14 demonstrate the force-displacement response for two cyclic loadings for UC-2 and UC-3, respectively, obtained from experiments and FE simulation. Furthermore, table 3 presents the force values at the first and second peaks and the error percentages between the FE modeling and experimental results in two cyclic loadings for UC-2 and UC-3. Focusing figures 13 and 14 and table 3 reveal that the main features of NNSs such as softening-hardening-softening behaviors, length of the snap-through instability, hysteresis loop area, peak force value, loading history pattern, irrecoverable residual deformation and shape recovery behaviors are well simulated by the present FE model. The differences between the experimental and FE results could be due to some random defects introduced during SLS printing technology that make the structures imperfect while perfect geometries are assumed in the simulations and also visco-elasticity that is not included in the simulations.

Validation of FE modeling
According to figures 13 and 14 respectively for UC-2 and UC-3, two snap-through mechanical instabilities are observed during two cyclic loadings in both the experiment and the FE simulation. The structural behavior also changes from the first cycle to the second one. It means that UC-2 and UC-3 may experience some inelastic deformations that affect their behavior in the following mechanical loading cycles. A similar behavior would be expected for the other structures designed in this paper. The behavior of UC-3 in the first loading cycle is now analyzed to get a better understanding of the defamation mechanism. Considering the behavior of UC-3 in figures 11, 12 and 14 and S1-EXP-and S2-FEM movies, it is seen that during the first loading cycle, in the deformations less than 4.9 mm, the entire UC-3 first deforms uniformly in an elastic manner. Then, at displacement 4.9 mm, four top cross beams start to buckle, and the first snap-through-like mechanical instability happens and as a result, the first peak in figure 15 is observed. The structure softens and the force drops drastically with an increase in the displacement until displacement of 7.8 mm when the buckling of the top cross beams is completed. Subsequently, with an increase in the applied displacement, the structure hardens and the force increases, and the UC shows the capability to absorb more energy. At displacement of 10 mm, the four bottom cross beams buckle and initiate the second snap-through instability, and the force drops again with an increase of the displacement. The two snapthrough-like mechanical instabilities could be associated to the asymmetrical deformations during the mechanical loading. Upon unloading, the meta-structures start recovering their initial shape while an irrecoverable residual plastic deformation remains into the structure. It is seen that the metastructure experiences a free-stress shape recovery at the end of unloading and the force-displacement curve tracks the horizontal displacement axis towards the origin. That is why, in the second loading cycle, the force-displacement curve starts growing before the first unloading cycle curve. Plastic deformation grows in each load cycle and is captured by the simulation very well. The overall dissipated energy observed in the experiment can be attributed slightly to visco-elasticity and mainly to mechanical instability through compression and plasticity. The mechanical behaviors of UCs and the influence of geometrical parameters on their responses will be studied in the next sections by implementing the developed FE modeling.  (1) and (2), respectively. Furthermore, table 4 presents the values of E, E m , E ′ , ∆E and ψ for every cyclic loading. E and E m are calculated for the first and second loading cycle using equations (3) and (4), respectively. Also, the values of ∆E and ψ for every cyclic loading are evaluated using equations (5) and (6), respectively. According to figure 15 and table 4, the response of the fist loading/unloading cycle does not coincide to the response of the second cycle. That shows the presence of the inelastic deformation in the first cycle that makes the mechanical behavior be cycle dependent. It is seen that all UCs dissipate some energy during the first loading/unloading cycle. By mechanical unloading, mechanical hysteresis, characterized by noncoincident loading-unloading curves, is observed in all cases. UCs recover their original shape partially while some strains remain in the meta-structure due to the plastic deformation. The energy dissipation may arise from mechanical instability and plasticity. As per the second loading/unloading cycle, it is seen that, while UC-2-5 show hysteresis loop and consequently dissipate energy, UC-1 behaves elastically and releases all energy stored during loading upon unloading in the second cycle as ψ value of UC-1 in the second loading cycle becomes zero. UC-1 acts like a springer that releases all the stored energy upon unloading. Considering stress-strain diagrams of second cycle loading in figure 15, it is observed that while UC-2-4 experience some residual plastic deformation at the end of second unloading cycle, interestingly, UC5 does not produce any plastic deformation during the second loading/unlading cycle and dissipates the mechanical energy via the nonlinear mechanical instability. Therefore, it can be concluded that UC-1 and 5 have a super-elastic behavior with a full shape recovery feature in the second loading/unloading cycle. Specifically, UC-1 behaves like non-linear elastic material with no energy dissipation while UC-5 behaves like super-elastic shape memory alloys with a fully shape recovery response. While the energy dissipation mechanism in shape memory alloy (SMA)s is due to the martensitic phase transformation [52,53], solid-state non-linear mechanical instability in NSS UC-5 dissipates the energy. UCs provide the designers with a variety of mechanical behaviors and options to meet different requirements. They can expand the design space for proposing large recoverable energy absorbers and dissipaters with different capacities. The first cycle could also be considered as a training stage to produce shape memory devices where a super-elastic behavior where a large energy absorption and/or dissipation is required. From a numerical point of view, among the designed UCs, UC-1 has the minimum values of E m , ∆E and ψ for both the first and second cycles. UC-2 has more values of E m than UC-1, and therefore it is concluded that changing the plate dimensions of UC-1 to achieve UC-2 is an effective way to improve energy absorption and dissipation capabilities. Also, against UC-1, UC-2 has ψ value in the second loading/unloading cycle, and as a result, chamfering its plate corners causes the original configuration to be properly recovered. According to figures 15(b)-(e) and table 4, it can be observed that, for UC-2 to UC-5, ψ values in the second cyclic loading decreases in comparison with the first load cycle. It implies that plastic deformation growth and mechanical instability reduce by further loading cycles for UC-2-4 and UC-5, respectively.      The results in figure 15 and table 4 reveal that UC-2 has higher E and less E m than UC-3 in the first and second loading cycles. It means that the designing of camber-arch energy absorber should be modified. Therefore, rhombus and DNA members have been designed for UC-4 and UC-5, respectively. Indeed, the higher values of E and E m in UC-4, and UC-5 in comparison with UC-1 and UC-2 show that adding foresaid energy absorbers could increase the E and E m values. Therefore, the energy absorber members in UC-4 and UC-5 can improve the mechanical performance of NSSs. The reason is that the energy absorbers have been designed to decrease the concentration stress in the cross beams and plates and help them recover their original configurations.

Mechanical behaviors of UCs under cyclic loadings
By comparing E and E m values of the UCs with the energy absorber members (UC-3, UC-4, and UC-5) in table 4, it is found that UC-5 has the maximum values of E, E m and ψ. It implies that UC-5 can absorb and dissipate more energy compared to other UCs. For more investigation, the stressstrain response of UC-5 under three cyclic loadings is demonstrated in figure 16. Also, E, E m and ψ for the third cycle of UC5 are calculated and added to table 4. Surprisingly, it is seen that E, E m and ψ remain almost constant for the second and third cycles and stress-strain graphs coincide each other. It is an interesting observation and implies that the plastic deformation growth stops after the first loading/unloading cycle and UC-5 shows a stable shape recovery feature onwards. The energy dissipation mechanism after the first cycle is completely attributed to the non-linear mechanical instability through compression. The stable super-elastic behavior in UC-5 has can be exploited for applications such as strong-yet-flexible instruments. The large recoverable behavior also gives rise to large hysteretic loss during load cycles due to the solid-state energy dissipation. This dissipative response can be exploited to create structural polymeric elements with high energy dissipation for stabilizing sensitive lightweight structures.

Geometric parametric studies
The NS member dimensions are effective parameters in the activation of the snap-through mechanism, improving energy absorption capabilities, and maintaining the original configuration of the NSSs [54]. As mentioned in section 4.2, UC-5 is selected for further investigations. Therefore, the effects of the inclined cross beam angle, α, and DNA member thickness, t D ,

Effect of inclined cross beam angle.
As shown in section 4.1, the cross beams are NS members that could improve the mechanical performance of the designed NSSs through the snap-through mechanism. Therefore, in this section, the effect of the inclined cross beam angle, α, on the snap-through mechanism and energy absorption capability is studied. To examine the effect of α on the UC-5's mechanical stability, the force-displacement responses for the first compressive loading are presented in figure 17 for α = 5 • , 8 • , 12.5 • , and 15 • . It is seen that the force increases monotonically with an increase in the compressive displacement for α = 5 • and 8 • . The mechanical behavior is completely stable during the compression and the material response is hardening.
For further discussion, figures 18(a) and (b) exhibits a plastic strain (PE) contour of UC-5 with α = 5 • and 8 • at d max = 12 mm, respectively. Following figures 18(a) and (b) for the deformation of UC-5 with α = 5 • and 8 • , it can be observed that the top and bottom cross beams do not have enough space to buckle, and cross beams do not soften and stay stable during the bending. It is seen that the frame and members connected to the frame, including plates, and cross beams stay in the elastic domain while the DNA member enters into the plastic region for both angles of α = 5 • and 8 • at d max = 12 mm. Plastic hinges are developed in the center, nodes and the end of the beam due to bending moment in the span of the beam and a high level of the stress at these points. For α = 12.5 • , because of the sufficient space for the cross beams, they become unstable and buckle in the elastic region. As shown in figure 17, the force values decrease with an increase in the displacement with a low slope. In addition,  according to figure 18(c) for α = 12.5 • at d max = 12 mm, the frames, plates, and cross beams remain in the elastic region, and the overall PE value of DNA member is lower than that of α = 5 • and 8 • . That is due to the fact that cross beams already absorbed some energy, and a lower level of the energy is transformed to the DNA member reducing the plastic defamation growth. By considering the force-displacement response of UC-5 with α = 15 • in figure 17, it is seen that the force starts to decrease significantly at d = 5.5 mm owing to the activation of the first snap-through mechanism in the top cross beams, and UC-5 with α = 15 • exhibits an NS behavior. The second snap-through instability occurs at d = 10 mm and the bottom cross beams buckle. Figure 18(d) shows the plastic contour of UC-5 with α = 15 • at d max = 12 mm. By comparing PE value and UC-5's response for different α's, it is found that the case of α = 15 • experiences maximum snap-through-like mechanical instability and its DNA member undergoes minimum plastic deformation at hinges. Finally, from an energy absorption and stability points of view, a transient behavior from overall stable bending defamation from α = 5 • to an unstable snap-through-like deformation with a higher anergy absorption for α = 15 • is clearly seen. Figures 19(a)-(d) displays the stress-strain behaviors of UC-5 with different α angles under three cyclic loadings for d max = 12 mm and t D = 2 mm. Furthermore, table 5 lists the values of E, E m and ψ for each cycle. The results presented in figure 19 and table 5 reveal that UC-5 with α = 15 • has the highest values of E and E m in all three cyclic loadings. Also, in a comparison between α = 5 • and 8 • , the values of E and E m in all the three cyclic loadings for α = 5 • are larger than those for α = 8 • . In fact, the stable hardening behavior of UC-5 with α = 5 • leads to absorbing more energy compared to α = 8 • that tends to have a nearly flat plateau.   Before a discussion about ψ values, it is necessary to discuss the maximum allowed applied displacement, d a , for a UC. For d > d a during compressive loading, the frame of the UC is subjected to undesirable buckling which results in the UC instability. According to figure 20, the maximum allowable displacement of the cross beams is 2b. In addition, the following relationship should be satisfied to maintain the stability of the UCs: Considering b = a tan α. in figure 19 leads to: Because the values of a = 18.46 mm and L s = 2.4 mm are constant in this paper, according to equation (8), d a depends on α. Therefore, for UC-5 with α = 5 • , 8 • , 12.5 • , and 15 • , d a values are 8.02 mm, 9.98 mm, 12.90 mm, and 14.68 mm, respectively. It can be concluded that for α = 12.5 • and 15 • , it is possible to maximize the applied compressive displacements (d max ) to more than 12 mm. Therefore, for optimizing UC-5 with α = 12.5 • and 15 • , it is possible to increase d max value, however, to compare the results in an equal compressive displacement, d max = 12 mm is considered here. Also, among allowed applied displacements for different angles, d a for α = 12.5 • is the closest to d max = 12 mm.
As per energy dissipation, ψ values in table 5 show that α = 12.5 • has the highest value and its value is greater  than the one for 15 • in all three cycles. The reason is that d a = 12.90 mm for α = 12.5 • is closer than d a = 14.68 mm for α = 15 • to the maximum applied displacement (d = 12 mm), and therefore, UC-5 with α = 15 • cannot present its maximum efficiency. Therefore, it is concluded that the highest performance of the NSSs can be achieved by applying  displacements near to their maximum allowable displacements. In the comparison between α = 5 • and other angles, ψ value for the first cycle at α = 5 • is lower than others. Besides, ψ value for α = 5 • becomes zero for the second and third load cycles but it remains constant and non-zero for other cases. As the stress-strain graphs get back to the origin at the end of the second and third unloading cycles, it implies that the plastic hinges do not growth over these cycles. While UC-5 with α = 5 • shows a non-linear super-elastic behavior with no energy dissipation over cycles two and three, other α cases experience a non-linear super-elastic behavior with energy dissipation due to the non-linear mechanical instability. A decreasing-increasing trend of the specific absorbed energy is seen when α value is increased. In a converse manner, an increasing-decreasing trend of the loss factor can be observed as α value is increased. These transient trends can be  associated to the switch in energy absorption and dissipation mechanisms. Our investigation of the further cycles shows that E, E m and ψ remain unchanged for the fourth and fifth cycles results of which are not presented here for keeping the report brief.  to figure 21 and  table 6, with an increase in the DNA member thickness, the stiffness of DNA part increases and the values of E and E m increase for the first cycle. The stiffer the system, the higher the absorbed energy. Also, with an increase in t D , the hysteresis loop area increases in the first cycle, and the energy dissipation capability of UC-5 improves in this cycle. Therefore t D = 3 mm has the maximum value of ψ for the first cycle. From table 6 it is found that, for t D = 1 mm, ψ value becomes zero for the second and third cycles resulting in zero energy dissipation. By following figures 21(b) and (c) and table 6 for the second and third cyclic loadings, it is found that ψ values of t D = 2 mm are a bit greater than those of t D = 3 mm, and t D = 2 mm case results in a higher energy dissipation. In fact, a lower level of the plastic deformation in the case of t D = 2 mm compared to t D = 3 mm lets the meta-structure experience stiffer mechanical instability and consequently results in more energy dissipation. Next, the effects of both α and t D on E m are investigated. Figures 22(a)-(c) shows E m values for the first, second and third loading cycles, respectively. With an increase in α, except for α = 5 • , E m values increase for constant thicknesses. In addition, E m for α = 5 • is greater than that for α = 8 • for all thicknesses in each cyclic loading because as mentioned earlier in section 4.3.2, the DNA member in UC-5 with α = 5 •  Finally, figures 23(a)-(c) shows the ψ values for the first, second and third loading cycles, respectively. According to figure 23(a) for the first cycle, with the increase in t D at a constant α, the ψ values increase significantly. Also, for a constant t D , ψ values become maximum for α = 12.5 • , because, as mentioned earlier, at this angle, d a is closer than other angles to d max . By focusing on figures 23(b) and (c) for the second and third loading cycles, it is seen that the difference between ψ values become considerable for different thicknesses. From these figures, it is seen that loss factors are zero for t D = 1 mm at all inclined cross beam angles. In addition, in the second and third cycles, DNA member with t D = 2 mm and α = 12.5 • results in the maximum loss factor values. Furthermore, for t D = 3 mm, the structure is subjected to a considerable plastic deformation in its first load cycle that affects its energy dissipation performance in the following load cycles.

Conclusion
The aim of this paper was to propose a new class of 3D cubic NSSs that can offer high energy absorption, superior energy dissipation, full/partial shape recovery, super-elasticity, and reversibility in cyclic loadings. The NSSs include stalks, plates, cross beams and a frame. To prevent undesirable buckling of the frames, the thickness of the frames was higher than those of the cross beams. In addition, to absorb high values of energy, energy absorber members with different configurations were designed and assembled inside the UC. Different NSSs were designed exploiting geometrical influences, snap-through instability mechanism, elasto-plastic deformation growth and plastic hinges. A FE modeling via ABAQUS software package assuming an elasto-plastic material behavior was implemented to design and analyze NSSs saving time, material, and energy consumption. SLS 3D printer was used to fabricate material and structural samples. Dog-bone samples were tested and shown that mechanical properties are almost independent of printing direction. The high accuracy of the FE model was verified via the mechanical testing of 3D printed NSSs. The computational model was then employed to further investigate the mechanical behaviors of NSSs. The implications of the effect of inclined cross beam angle, and energy absorbing member thickness on the mechanical response of NSSs were put into evidence via a parametric study, and related conclusions were drawn. It was observed that all NSSs have a high capability of energy absorption and dissipation in the first cycle via plasticity and instability mechanisms. In the further loading/unloading cycles, depending on the geometrical parameters and configurations, NSSs show super-elastic stability/instability with high energy absorption and full/ partial shape recovery. The presented conceptual design and results are expected to contribute to a better understanding of the behavior of NSSs and to be instrumental towards an efficient design of NSS for various applications.
The ways to activate snap-through mechanisms simultaneously in the top and bottom curved beams in the designed NSSs could be investigated in the future research effort. Also, 2 × 2 and 3 × 3 cellular NSSs based on the foresaid designed NSSs could be further investigated to explore new futures.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).