A novel design of mechanical metamaterial incorporating multiple negative indexes

Mechanical metamaterials that combine multiple negative properties are rare, but have great appeal for applications in multipurpose devices. Herein, a novel metamaterial incorporating negative Poisson’s ratio, negative compressibility, and negative thermal expansion or swelling was proposed. The unit cell was designed by simulating the bond structure of lead monoxide crystals using elastic beams. Analytical models for the mechanical responses of the unit cell were established, and expressions for Young’s modulus, thermal expansion, Poisson’s ratio, and compressibility were derived and confirmed by numerical simulations. The results showed that the metamaterial not only displayed a negative Poisson’s ratio but also had negative linear and area compressibilities as well as negative linear, area, and volume thermal expansions if the geometric parameters were properly adjusted. Furthermore, metamaterials with negative linearity and area swelling were obtained experimentally by replacing the thermal expansion of the material with the water absorption expansion of a hydrogel. Such metamaterials with multiple negative indexes are expected to contribute to the feasibility of advanced multifunctional devices with mechanical, pressure, temperature, and moisture sensitivities.


Introduction
A 'metamaterial' is an artificially designed structural material that has exotic physical properties rarely observed in natural materials [1][2][3][4][5][6][7][8][9]. In the material mechanics field, typical exotic properties include negative Poisson's ratio, negative compressibility, and negative thermal expansion. Negative Poisson's ratio means that the material expands laterally under axial forces [10]. Negative compressibility means that the material expands in the whole volume, a plane, or a certain dimension under uniform pressure, called negative volume, area, or linear compressibility, respectively [11,12]. Negative thermal expansion refers to a material that has dimensional contraction in a volume, plane, or direction induced by a temperature increment, called negative volume, area, and linear thermal expansion, respectively [13,14]. Metamaterials with negative properties have important applications in precision instruments, aerospace technologies, and marine sensors. For example, aerospace satellite antennas with negative thermal expansion reduce the mechanical failure caused by thermal deformation [14]. Wearable devices with negative Poisson's ratios integrate well within the human body [15]. Additionally, materials with negative compressibility can be used to manufacture highly sensitive hydrostatic pressure sensors [16].
Previous studies have primarily focused on mechanical metamaterials with only a single negative property. Recently, considerable efforts have been made to combine multiple negative properties to produce new advanced materials that are multifunctional and multipurpose [16]. Typical examples include metamaterials 2. Simulation methods 2.1. Density functional theory simulations Poisson's ratio and deformation mechanism of periodic PbO crystal were researched through DFT tension simulations in the CASTEP code package [38]. The ultrasoft pseudopotential and local density approximation [39] were used, and the Brillouin zone was sampled using Monkhorst-Pack meshes [40] with a 4 × 4 × 4 k-point set. The stable PbO crystal structure was obtained by geometry optimizations using the Broyden-Fletcher-Goldfarb-Shanno method [41]. The relaxation stopped when the stresses and forces were less than 0.01 GPa and 0.005 eV Å, respectively, with a cutoff energy of 400 eV. To minimize symmetry, the P4/nmm space group was changed to P1. We uniaxially stretched the periodic crystal in multiple steps with a strain of 0.2% for each step and examined the transverse deformations to calculate Poisson's ratio.

Finite element method simulations
Poisson's ratio, compressibility, and thermal expansion of the PbO-mimic metamaterial were calculated through finite element method (FEM) simulations in the commercial software ABAQUS/Standard [42]. In the FEM, the slender filaments with circular section using isotropic materials, including E = 1 GPa and ν = 0.3, were rigidly connected at lattice structure joints. The B31 Timoshenko beam element, which considers the stretching, bending, and shearing deformations of the beam, was used to mesh the filaments. Poisson's ratio and Young's modulus of the material were calculated using a uniaxial tension simulation. A schematic of the 3D FEM models under uniaxial tension was shown in figure 1, where uniform tension displacement fields were specified on the two opposite sample surfaces normal to the stretching direction, while the other degrees of freedom were unconstrained. For the thermal expansion simulation, the thermal expansion coefficient (CTE) of slender filaments was defined during pre-processing, and a predefined variable temperature field was applied to the model. The CTE of the metamaterial was obtained by measuring the deformation of the model during postprocessing.

Auxetic behavior of PbO crystal
The atomic structure of a PbO crystal was depicted in figure 2(a). For convenience, we labeled some Pb atoms by Pb 1 , Pb 2, and Pb 3 . In the figure, y Therefore, the [110] stretch caused the decreases of q 1 and the decreases of q . 2 The crystal then had positive Poisson's ratio when the former dominated and exhibited negative Poisson's ratio as the latter was ascended.

Structural design and theoretical analysis of metamaterials
We designed a unit cell for the metamaterial by mimicking the PbO crystal geometry. The nearest-neighbor Pb-O bonds and second nearest-neighbor Pb-Pb bonds were replaced by slender filaments with circular cross   Dimensions of the unit cell in the x 1 (x 2 ) and x 3 axes were respectively. Since q a and L a are related to q b and L , b the independent parameters used to determine the geometry of the unit cell were q , b L , b R b and R . a Owing to tetragonal symmetry, the mechanical responses of the unit cell in the x 1 and x 2 coordinate axes were identical. Analytical models were formulated based on the theory of small deformations to identify the effects of geometric and material The deformation of the periodic structure was obtained by analyzing the forces of beams AB, DB, and CB, as shown in figures 5(b)-(d). For convenience, some constants and notations were defined as follows: Figure 5(b) shows that beam AB withstood forces F 1 and T , 1 as well as a moment M 1 at point B, and was fixed at point A. No rotation occured between A and B, indicating that Then, the deformations of the beam in x 1 and x 3 axes were derived as Figure 5(c) indicates that the beam DB withstood two forces F 3 and T , 3 as well as a moment M 3 at joint B, and was fixed at joint D. No rotation occured between B and D denoting that Accordingly, the deformations of the beam in the x 2 and x 3 axes were expressed as The force diagram in figure 5(d) shows that beam CB was fixed at C and subjected to forces and bending moments at B. For convenience, a local coordinate system consisting of a unit axial vector t as well as two unit normal vectors n and z was defined.
In the t-z plane, the joint B was subjected to a horizontal force F , 21 a vertical force T 2 and a moment M 21 around the n direction. In the n-t plane, the joint B endured a force F 22 in the n direction and a moment M 22 around the z direction. No rotation occured between B and C, which required The deformations of beam BC in the x , 1 x 2 and x 3 axes were obtained as Based on the balance of forces at joint B, it was derived that By using equations (4), (6), (8), and (9) as well as the relations of the relative displacements d d = 2 , 3 the expressions for the deformations of beams that depend only on the force F x1 were obtained. The deformations of the periodic structure in the x , 1 x , 2 and x 3 directions were denoted as d -2 ,  12 and n n = . 23 13 Next, the mechanical response of the periodic unit cell subjected to uniform tension stress s 3 in the x 3 axis was analyzed. Based on symmetry and periodicity, thex x 1 2 boundary remained planar and no rotation occurred. All joints at thex x 1 2 boundary were subjected to identical vertical forces T . x3 Figure 5(a) shows the forces applied to the - The tensile stress s 3 was related to T x3 by / s = T l l 2 .
x 3 3 12 The deformation of the periodic structure was obtained by considering the forces of beams AB, DB, and CB as The deformations of the beam BD in the x 2 and x 3 axes were derived as The deformations of the beam BC in the x , 1 x 2 and x 3 axes were The conditions for the balance of forces in the x 3 and x 1 directions at joint B were as follows: 0. 14

Compressibilities
Under a uniform hydrostatic pressure P, the linear dimension L, the measured area A, and the whole volume V of a traditional material typically decrease. This phenomenon is often characterized by positive linear, area, and volume compressibilities, which are expressed as respectively. For a tetragonal elastic medium, the linear compressibilities in the x , 1 x 2 and x 3 axes were expressed as The area compressibilities in thex x, The volume compressibility was Accordingly, by substituting the expressions for Poisson's ratios (n , 12 n 13 and n 31 ) and Young's modulus (E 1 and E 3 ) into equations (15)- (17), the linear, area, and volume compressibilities of the metamaterial were obtained.

Thermal expansions
The deformations of the beam BD in the x 2 and x 3 axes were The deformations of the beam BC in the x , 1 x 2 and x 3 axes were  Based on the balance of forces in x 3 and x 1 directions at joint B, it was obtained that The area thermal expansions measured on thex x ,

Results and discussion
The theoretical expressions for Young's modulus, thermal expansion, Poisson's ratio, and compressibility of the novel metamaterial enable the quantitative examination of the effects of geometric and material parameters. The material parameters E , a E , b a a and a b depend on the material selection of the slender beams, and their adjustability is limited compared with the geometric parameters q , b L , b R b and R . a Here, we mainly focused on determining the geometric conditions under which the thermal expansion, compressibility, and Poisson's ratio obtained negative values. Some of the analytical results were verified using numerical simulations. Furthermore, by replacing the thermal expansion of the material with the water-absorption expansion of hydrogel, metamaterials with negative linearity and area swelling were achieved experimentally. To facilitate the analysis and design, the ratio of the cross section / = q R R a b and slenderness ratio / = s L R b b were defined, which, along with the angle q b were used to characterize the geometric features of the metamaterial.  To examine the effects of parameters q and q b on the negative value of n , 12 we calculated the dependences of n 12 on q b at s = 10 and q = 0.1, 0.5, 1.0 and 2.0 as seen in figure 7(a). We observed that, at q = 2.0, n 12 was always positive for any value of q ; b at q = 1.0 and 0.5, n 12 became positive for q q < < 0 b b 0 and negative for q q < <  90 ; b b 0 when q was decreased to 0.1, n 12 was negative for any value of q .

Negative poisson's ratios
b Furthermore, we calculated the variations of n 12 with q b at = q 0.1 and s = 10, 20, 50, and 100, as depicted in figure 7(b). The figure indicates that the negative value of n 12 increased with the increase of ratio s and could attain the limit of n = -1. 12 Moreover, from the figures 7(a) and (b), we found that, when the parameters s and q were fixed, there was an optimal q b to achieve the maximum negative value of n , 12 which could be obtained through theoretical calculation. The schematic diagrams of cell structures with different parameters s, q, and q b were displayed in figure 7(c).

Negative compressibilities
After calculating Poisson's ratio and Young's modulus of the models, the compressibilities were obtained using equations (15)- (17). Figures 8(a)-(c) show the dependences of linear, area, and volume compressibilities on q b  The CTEs of the stainless steels, aluminum  alloy and stainless steels are listed in table 1. By comparing the undeformed and deformed FEM samples, it is known that the metamaterials with λ = 0.1 and λ = 2.0 have negative area thermal expansion and positive thermal expansion, respectively. Moreover, we also examined the effects of the parameters q and s on the negative CTEs of the metamaterial. Figures 11 and 12 indicate the variations of the normalized negative thermal expansions with q, s and q , b at λ = 0.1 and 2.0, respectively. From the figures, we found that the negative values of thermal expansions increased with the increases of s and the decreases of q. When λ = 0.1, the thermal expansions a , L1 a , A12 a A13 and a V could be negative, and when λ = 2.0, the negative thermal expansions a , L3 a A13 and a V were obtained. The calculated results indicate that the material can possess negative linear, area and volume thermal expansions if the parameters s, q, and q b are properly adjusted.

Negative swellings
In addition to thermal expansion, some solid body materials such as hydrogels can expand upon water absorption, corresponding to a positive swelling. This effect is characterized by the strain components e -, x swelling 1 e - x swelling 2 and ex swelling 3 [43]. Here, we constructed the lattice configurations, in which the beams of active and passive materials were used as building blocks. The lattice structures were fabricated using the techniques of multimaterial 3D printing, as shown in figure 13(a). The beam of passive material consisted of an elastomer (TangoBlackPlus, Stratasys; in black), and the beam of active material consisted of a thin encapsulation layer (TangoBlackPlus, Stratasys; in grey) and an active filler (Hydroge SUP705, Stratasys; in blue). To facilitate the inflow and outflow of water, a series of holes were drilled on the sides of the encapsulation layer. All samples had the same geometric parameters q =  50 , The dimensions of the models were = » L L m 0.044 1 2 and » L m 0.027 .
3 Figure 13(b) displays the sample where all beams were composed of active material. When the sample was immersed in water, the hydrogel fillers expanded x swelling x swelling 2 1 When the beams with radius R a changed to passive materials and the beams with radius R b still acted as the active building blocks ( figure 13(c) showing negative area swelling. When the beams with radius R a served as active building blocks and the beams with radius R b acted as the passive building blocks ( figure 13(d)), the sample expanded in thex x showing negative linear swelling. Accordingly, through adjusting the swelling coefficients of the beams with radii R a and R , b the metamaterials can exhibit negative linear or area swelling.

Conclusions
In this study, we found through DFT calculations that Poisson's ratio of PbO crystal changed from positive to negative with the increase of tensile strain. The study on the microscopic deformation characteristics of PbO crystal revealed that the positive and negative Poisson's ratios were attributed to the special crystal geometries composed of the nearest-neighbor Pb-O bonds and second nearest-neighbor Pb-Pb bonds, respectively. By Theoretical expressions for Young's modulus, Poisson's ratios, compressibilities, and thermal expansion of the metamaterial were derived and confirmed by numerical computation. We found that Poisson's ratio, compressibility, and thermal expansion could be simultaneously negative when the geometric parameters were properly adjusted. Furthermore, by replacing the material thermal expansion with the swelling of hydrogel, metamaterials with negative linear and area swellings were achieved experimentally. These findings may contribute to the design of novel metamaterials incorporating negative Poisson's ratio, negative compressibility, and negative  thermal expansion (or negative swelling), which are very attractive for applications in multifunctionality and multipurpose devices with mechanical, pressure, temperature, and moisture sensitivities.

Acknowledgments
Supports from the National Natural Science Foundation of China (Grant No. 12072337) is gratefully acknowledged.

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