Compression and energy absorption properties of the truss-like lightweight materials based on symmetric groups

Truss-like lightweight materials (TLMs) have been widely used in aeronautics and astronautics, because of excellent mechanical property and superior energy absorption capability. The design of TLMs’ meso-structures was a critical task to improve its performance. Hence, a structure design method based on the symmetric groups was proposed for TLMs, and a novel hexagonal prism TLM’s meso-structure was deduced by the symmetric and translational operations of the space group P6mm. To investigate the performance of the novel TLM, the mechanical analysis model was established. The predictive equations of compression performance was proposed based on Euler–Bernoulli beam theory. The stress distribution of the novel TLM’s meso-structure under compression load was discussed by the finite element analysis method, and its compression and energy absorption properties were investigated. The simulation results were in agreement with the predictive results. In addition, the common FCC and BCC TLMs were discussed using the symmetry group analysis method, and their compression properties were predicted. The results showed that the proposed novel TLM in this study had better compression property than BCC and FCC TLMs at the same relative density.


Introduction
Truss-like lightweight materials (TLMs) have promising applications in aerospace, aircraft and automotive due to the advantages of low density, high specific strength and specific stiffness, excellent seismic performance and superior energy absorption property [1,2]. Among these essential performance, compression and energy absorption properties have been identified as the most promising features for lightweight industrial applications [3,4]. The meso-structures of TLMs are open-cell structures composed of one or more repeating unit cells in x, y and z directions [5]. It was found that the topology of TLM's representative volume unit (RVU) and the relative density of TLM's meso-structure are the major factors which influence the mechanical properties of TLM significantly [6][7][8]. Thus, the research that focuses on the design of TLM's meso-structures has important significance to obtain lightweight components with superior compression and energy absorption properties.
To further investigate the effect of the topology of TLM's RVU on its compression and energy absorption properties, various types of TLM's RVUs such as Pyramid [9], Kagome [10], Octet-truss [11], body-centered cubic (BCC) [12], face-centered cubic (FCC) [13] and some complex geometric configurations composed of multiple struts [14,15] have been studied in recent years. A series of experimental results showed that the TLMs with different types of RVUs exhibited different compressive behavior and energy absorption capacity [16][17][18][19]. Therefore, the meso-structure design of TLM is the most direct and effective way to improve its performance. Currently, there are two main strategies to design TLMs' meso-structures: (1) the topological optimization design for the TLM's meso-structures composed of single type RVU [20,21]; (2) the hybrid design for the TLM's meso-structures composed of different RVUs [22]. Some optimized TLMs' RVUs were proposed based on the current types according to the method (1), and the finite element analysis (FEA) and experiments were carried out to discuss the compression and energy absorption properties of the optimized TLMs. Zhang et al [23] designed enhanced pyramidal TLM through node optimization design. Ding and Yao et al [24,25] fabricated ARCH TLM by optimizing the center line of BCC strut. Wang et al [13] modified the traditional FCC TLM by adding a cross-rod at the center of its RVU. Zhang et al [26] presented the optimization method for the critical buckling loads of TLMs to obtain the new 3D-Kagome TLMs for energy absorption application.
As for the method (2), the hybrid TLMs that obtained by combining two or more different types of RVUs exhibited superior compression and energy absorption properties compared to TLMs composed of a single type of RVU [22,[27][28][29]. Al-Saedi et al [27] obtained a F2BCC TLM based on the hybrid design method, and its meso-structure was composed of the BCC and FCC RVUs. Zhang et al [22] designed and fabricated two hybrid TLMs based on grid and octet RVUs. Sun et al [28] proposed and examined the novel hybrid TLM which combined the geometrical features of both octet and bending-dominated RVUs. Xiao et al [29] combined octet unit and rhombic dodecahedron unit, and the hybrid TLM exhibited superior energy absorption capacity.
Due to limited innovations of the optimization and the hybrid meso-structures, the improvement of the above obtained TLMs' compression and energy absorption properties is not obvious. The design of RVU's structure is still a challenge to develop high-performance TLMs. As a result of the similar geometric features of TLM's meso-structures and crystal structures, the symmetric groups will be adopted to deduce TLM's mesostructure and a novel structure based on the symmetric operations of the space group P6mm can be determined. Through the mechanical analysis model and finite element simulation, the compression and energy absorption properties of the novel TLM will be studied in detail.

The design method of TLM's meso-structures based on space group
TLM's meso-structures are constructed by the stacking of RVUs in x, y and z directions, which means that the structures must have three-dimensional periodicity. By simulating the spatial configuration of molecules, researchers had designed a series of TLM's meso-structures such as BCC and FCC. Some TLMs' mesostructures, such as triangular prism, square prism and hexagonal prism, could also be designed by laying constant-section struts along the edges and across faces of solid structures. However, there was no research focused on theoretical deduction of TLMs' meso-structures using mathematical method.
According to the existing TLMs' meso-structures, it was found that all of TLMs' RVUs could be deduced by symmetric operations based on one or several struts and the whole TLMs' meso-structures were obtained by translating these RVUs, as shown in figure 1. Symmetric group theory was a mathematical tool to describe the symmetry of crystal structures, and the theory had been successfully used to describe and design the mesostructure of braided materials [30][31][32]. Compared to braided structures, the symmetry of TLMs' mesostructures have mirror elements. Hence, some space symmetric groups, which contain mirror elements, could be adopted to describe and design TLMs' meso-structures in this study.
In order to investigate TLMs' meso-structures using the symmetric group theory, it is necessary to understand the concept of symmetric groups. The symmetric groups include point groups, translation groups of space lattices and space groups. Point groups are composed of symmetric operations such as rotations (C), mirror reflections (s) and rotation-reflections (S) that could be expressed as: If the position of one or more struts in 3D space could be expressed as the vector = + + r a b c x y z , then the struts of TLMs' RVUs could be obtained by the symmetric operations of point groups and the positions of other struts in RVUs was = ¢ r r W . The initial struts could be expressed as the point group element E, and the corresponding symmetric operation was W E ;the point group elements c n i described that the initial struts were rotated pi n 2 successively around the rotation axis C ; n the point group element / s v h described that the initial struts were reflected along the vertical or horizontal mirror s; the point group elements 3 (n , 1 n 2 and n 3 were integers, a, b and c were translational vectors), and space groups are composed of point and translation groups. TLMs' RVUs obtained by the symmetric operations of point groups could be translated based on the corresponding translation groups, and the whole TLMs' meso-structures could be obtained. Therefore, the meso-structures of TLMs can be described and deduced using space groups. Due to the mirror symmetry of TLMs' meso-structures, only some special symmetric groups can be used to investigate the structures. There are 32 point groups, 14 space lattice and 230 space groups in the symmetric group theory, but the symmetric groups with mirror elements only include 18 point groups, 13 translation groups of space lattice and 160 space groups, and these symmetric groups could be used to investigate TLM's meso-structure. According to the symmetric operations described by the 18 point groups that have mirror elements, some TLM's RVUs could be deduced based on one or several struts, as shown in figure 2.
Taking the space group P6mm that have mirror elements as an example, and the corresponding TLM's meso-structure was deduced in detail. The space group P6mm contained the point group 6mm and space lattice hP. The element E represented the position of the initial strut in the 3D space. When the strut was rotated / p 3, / p 2 3, p, / p 4 3 and / p 5 3 successively around the rotation axis C , 6 the positions of the obtained struts were expressed as the elements c .  The symmetric operations of the point group 6mm could be expressed as follows: 2 , ( Set the rotation axis as z-axis in 3D coordinates o-xyz, and the reflection plane was the yoz plane. The symmetric operations of the point group 6mm and the corresponding TLM's RVU were shown in figure 3(a). The single-layer structure could be formed by translating the RVU along the directions: figure 3(b). The single-layer structure was translated along the vector c, and then the novel hexagonal prism TLM's meso-structure could be designed based on the space group P6mm, as shown in figure 3(c).
The deduction process showed that the symmetric groups could be used to design novel TLM's mesostructure. But TLM's meso-structures that obtained by the symmetric and translational operations of space groups are not fixed, which means that a space group can correspond to multiple TLMs' meso-structures with the same symmetry. The final structures are determined by the position and number of the initial struts, the positions of the rotation axis and reflection plane. Hence, 160 space groups that contained mirror elements corresponded at least 160 types of TLMs' meso-structures.
The current TLM's meso-structures could also be described by these space groups. For example, BCC and square prism structures can be described and deduced by the by the space group P4/m, which contains a rotation axis C 4 and a horizontal mirror plane; the space group P4mm which can be used to describe and deduce FCC structure contains a rotation axis C 4 and a vertical mirror plane; triangular prism structure corresponds to the space group Pm2m that have two mutually perpendicular mirror planes and a rotation axis C ; 2 hexagonal prism structure have a rotation axis C 3 and a horizontal mirror plane, and the structure corresponds to the space group P6. Although some TLMs' meso-structures are similar, such as the hexagonal prism TLM's meso-structure and the novel TLM's meso-structures deduced by the space group P6mm, the two structures have different symmetries and correspond to different space groups. Therefore, the two structures are considered as different TLM's meso-structures in this study. In order to further investigate the mechanical properties of the novel TLM's meso-structures ,the related geometric parameters of the struts in the structure were defined as follows: l was the effective length of strut; d represented the diameter of strut; a was the inclination angle of strut. In this paper, the inclination angle (a) of the strut was set as 45°, then the relative density of the hexagonal prism TLM's meso-structure could be obtained by the following equation: It was found that the relative density of the novel TLM's meso-structure could be increased with the diameterto-length ratio (d/l).

Compression and energy absorption properties analysis of TLM
3.1. The prediction of TLM's compression property Due to the symmetry and periodicity of TLM's meso-structures, the mechanical properties of TLM and its RVU could be predicted by analyzing the stress on a group of struts. The novel hexagonal prism TLM's meso-structure was taken as an example to investigate the equivalent elastic modulus in z-axis based on Euler-Bernoulli beam theory. But before the theoretical analysis, the following basic assumptions need to be proposed: (1) the struts of TLM's meso-structures were slender rods; (2) the struts only occurred small deformation under the external load; (3) the axial tension and compression, shear and bending deformation of the struts need to be considered, while the torsion deformation of the struts was ignored; (4) the interaction between RVUs was not considered. According to Maxwell criterion, it can be inferred that the novel hexagonal prism TLM's meso-structure was bending-dominated. The hexagonal prism TLM's RVU was subjected to the external load P, and the load could be equally distributed by each strut due to its symmetry, as shown in figure 4(a). The compressive load of each group struts was P , s and = P P 12 .
s The stress condition of struts AB and BC was shown in figure 4(b). The bending moment caused by P s moving from point A to B was as follows: The section Method and Euler-Bernoulli beam theory were adopted to solve the bending moment of the strut BC under compressive load, respectively.  The rotation angle and deflection at the fixed-point C were 0, and the deflection equation could be expressed as follows: The displacement of point B along z-axis and the moment of inertia for the strut BC were expressed as follows: Since the novel hexagonal prism TLM's meso-structure was bending-dominated, all the struts mainly occurred bending deformation under the external load. The displacement of point B caused by bending moment was much larger than that caused by shear and axial force. Therefore, the effects of shear and axial force were ignored. The displacement of point B could be expressed as follows: TLMs could yield under compression load when the the struts reached the plastic limit bending moment. According to the equation (3), it was found that the maximum bending moment appeared at the joints of struts, and the absolute value of the maximum bending moment could be expressed as follows: The bending moment of the beam when it approached the limit of plastic deformation was as follows: Where s s was the yield strength of matrix material, W was the section modulus in bending of circular crosssection strut. Therefore, the ultimate load when the struts of the hexagonal TLM reached the yield moment was: For struts with circular cross-section, the yield strength was twice the ultimate stress. Therefore, the yield strength of the novel TLM's meso-structure was as follows: Through the above analysis, it was found that the compression properties of the novel TLM was related to the matrix material, the diameter-to-length ratio of strut (d/l). Due to the relationship between these geometric parameters and the structural relative density, the equivalent elastic modulus and yield strength of the hexagonal prism TLM can be considered as increasing with the relative density.

Finite element analysis
FEA method was adopted to further discuss the compression and energy absorption properties of the novel hexagonal prism TLM, and a group of TLMs with different geometric parameters was designed to verify that the theoretical equations could be used to predict TLM's compression properties. The geometric parameters were shown in table 1.
The matrix material selected for the novel hexagonal prism TLMs was A1Si10Mg alloy in this simulation, and all the FEA models were given the input material properties based on Liu's study [17]: the density of the matrix material was 2.67 g cm −3 ; the elastic modulus was 60 GPa; the Poisson's ratio was 0.3; the yield strength was 200 MPa. The bilinear isotropic hardening model was chosen for the plastic behavior of the matrix material, and the tangent modulus of the material was set to 0, which meant the matrix material was ideal elastic-plastic material. According to the symmetric operations of the space group P6mm, FEA models with different diameterto-length ratios were established using SolidWorks software. All of TLM's meso-structures in the simulation had 3 layers, and each layer contained 7 RVUs. The height of TLM's meso-structures was 22 mm. Two circular plates were respectively placed on the top and bottom sides of TLM's meso-structures and the thickness of each circular plate was 2 mm, as shown in figure 5.
The simulations were performed by the commercial finite-element software ANSYS Workbench. The element type of TLM's meso-structures were Solid186, and the meso-structures were meshed using tetrahedron mesh based on the Patch Conforming algorithm. By adjusting the mesh element size, the average mesh element qualities of TLM's meso-structures with different diameter-length ratios were kept around 0.88 and the most mesh element qualities were above 0.8. The top and bottom circular plates, which could simulate the indenter and fixed platform in compression test, were defined as rigid bodies, and the two rigid circular plates established frictionless contact conditions with the top and bottom surfaces of TLM's meso-structure respectively. The boundary conditions reproduced the compression test, in which the top rigid circular plate was moved 8 mm in z-axis at a constant velocity and fixed in other directions, whereas the bottom rigid circular plate was completely fixed in all degrees of freedom. The computational time was about 0.5 h.
According to the simulation results, with the increased of strain, the stress of the four types of TLM's mesostructures also increased gradually. When the strain was 0.01, the maximum Mises stress in the four hexagonal prism TLM's meso-structures both exceeds the yield strength of the matrix material. The Mises stress distribution in the hexagonal prism TLM's meso-structures was shown in figure 6. It was found that the results of Mises stress distribution were the same as those of theoretical analysis, that was, stress concentration occurred at the joints of the struts, while the stress in the middle of the struts was the smallest. With the increase of strut diameter, the area of high Mises stress area was larger. Figure 7 showed the compressive stress-strain curves of the novel TLM's meso-structures from FEA. As a result, the loading capacity of the novel TLM could be improved with the increase of strut diameter. The stressstrain curves could be divided into two stages: elastic deformation and plastic yield. The stress increased linearly with the strain in the elastic deformation stage, which satisfied Hooke's law, and the equivalent elastic modulus  of the structure could be obtained based on the slope of the curve, as shown in figure 6(a). The meso-structures of TLM can be restored to the initial state when the external force was unloaded. The plastic deformation mainly occurred in the plastic yield stage. The stress value stayed at a stage with the increase of strain, and the growth  trend was slow, as shown in figure 6(b). Therefore, the hexagonal prism TLM had a large plastic deformation plateau period, which indicated that it exhibited excellent energy absorption capacity. The equivalent elastic modulus and yield strength of the hexagonal prism TLMs with different geometric parameters could be obtained by the theoretical analysis and FEA, as shown in table 2. According to the data comparison in table 2, there were some errors between the prediction and simulation results. The reasons for these errors may be caused by the following: (1) with the increase of the strut diameter, the deformation caused by shear force and axial force will become a factor that cannot be ignored in the theoretical analysis, and will affect the theoretical calculation results; (2) the failure to consider the interaction of TLM's RVUs is also an important factor causing errors between the theoretical predictions and the simulation results.

Energy absorption analysis
According to the analysis of finite element simulations, it was found that the novel hexagonal prism TLM had excellent energy absorption capacity. The impact resistance and energy absorption properties were measured using two important indicators: specific energy absorption (SEA) and ideal energy absorption efficiency [24,25]. The energy absorbed per unit mass was defined as SEA, which could be expressed as follows: Where s was the stress of the hexagonal prism TLM's meso-structures, e was the strain of the structures, r* was the relative density of the structures, and r was the density of the A1Si10Mg alloy. The ideal energy absorption efficiency was the ratio of the absorbed energy to the corresponding compression stress, which could be expressed as follows:  Figure 8 showed the SEA curves and ideal energy absorption efficiency curves of the four hexagonal prism TLMs. The SEA of the novel TLMs increased with the increase of strain, and it could be improved with the increase of strut diameter. This is because the larger the diameter-length ratio of the hexagonal prism TLM's meso-structures is, the larger the proportion of struts in TLM's RVUs is. When the structure generated strain, the structure with larger diameter-length ratio could bear greater stress, so the energy absorbed by the plastic yield of struts was larger. The ideal energy absorption efficiency decreased slightly at the beginning of compression, and then the curve showed the   The compression properties of three TLMs could be discussed using Euler-Bernoulli beam theory, and the analytical models were shown in figure 9.
When TLM's RVUs were subjected to the z-axis compressive loads P, P F and P , B each group of symmetric struts was subjected to the compressive load P ,  Then, based on equations (12), (15) and (16), the equivalent elastic modulus and yield strengths of three TLM's meso-structures could be obtained:   According to equation (17), the relationships between the compression property and relative density of each TLM's meso-structure were shown in figure 10.
Compared with BCC and FCC TLMs, the equivalent elastic modulus of the novel hexagonal prism TLM at the same relative density increased by 466.3% and 54.1%, respectively, and the yield strength of the novel TLM at the same relative density was significantly higher than others. The results indicated that the novel TLM based on the space base P6mm exhibited better compression property with the premise of ensuring lightweight.

Conclusions
A novel hexagonal prism TLM's meso-structure was deduced based on the symmetric and translational operations that described by space group P6mm. The compression property of the novel hexagonal prism TLM's meso-structure was investigated based on Euler-Bernoulli beam theory. The stress distribution, compression behavior and energy absorption property were discussed using FEA method. The compression performance of the hexagonal prism, FCC and BCC TLMs was compared, and the main conclusions could be summarized as follows: (1) TLM's RVU can be deduced by the symmetric operations of point groups, and TLM's meso-structures can be obtained by the symmetric and translational operations that described by space groups.
(2) The equivalent elastic modulus and yield strength of the novel TLM based on the space group P6mm under the z-axis compression load can be predicted by the Euler-Bernoulli beam theory, and the compression property was related to the diameter-to-length ratio of initial strut.
(3) The simulation results were consistent with the theoretical predictions. The novel TLM's meso-structure showed superior compression and energy absorption performance, but there was stress concentration at the joints. The phenomenon could be relieved with the increase of strut diameter.
(4) Compared with the common BCC and FCC TLMs, the novel TLM based on the space group P6mm exhibited better compression property. The equivalent elastic modulus of the novel TLM under the z-axis compression load respectively increased by 466.3% and 54.1% at the same relative density.

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

Competing interests
The authors declare that they have no conflict of interest.

Funding
There is no funding source.