Mechanical properties of 2D hierarchical re-entrant cellular structures with Voronoi sub-structures

Cellular solids are widely used in engineering applications due to their superior properties such as low mass and high energy absorption capacity [1]. Cellular structures with negative Poisson's ratio are shown to have improved energy absorption capacity compared with the traditional honeycombs [2–6]. In 1987, Lakes invented negative Poisson's ratio foams [7]. Thereafter, many types of 2D [8–13] and 3D [14–17] man-made materials with negative Poisson's ratio property were proposed. Hierarchical honeycomb structures are widely used in engineering designs and architectures by introducing structural hierarchy [18–20]. The conceptualization of hierarchical structure is descriptive: to recognize that structural features occur on different size scales [21]. The hierarchical topology improves the structural efficiency compared with non-hierarchical topologies. For example, a honeycomb with ribs made of self-similar honeycombs can achieve a much higher specific strength than the traditional honeycomb made from solid ribs. Self-similar isotropic hierarchical honeycombs were proposed by replacing each three-edge vertex of a base hexagonal network with a similar but smaller hexagon of the same orientation [22,23]. These hierarchical honeycombs have higher in-plane stiffness than the traditional non-hierarchical honeycombs. The mechanical behavior of hierarchical structures is largely controlled by the level of hierarchy and the stacking method of unit cells at each level of the scale [24,25]. Du [26] introduced random Voronoi sub-structures into honeycombs, and found that the hierarchical honeycombs can be 3 times stiffer than a regular hexagonal honeycomb of identical mass.

In this study, 2D hierarchical re-entrant lattice structures with Voronoi (also known as Dirichlet polygons) sub-structures were proposed. Their in-plane mechanical properties (Poisson's ratio and energy absorption capacity) were studied using the finite-element method (FEM). The effects of geometric parameters, including the degree of irregularity of Voronoi sub-structure, k, the concave angle of the re-entrant structure, θ, and the ratio of wall thickness (t₀) to cell height (H₀), w (i.e., $t_{0}/H_{0}$), on the lattice mechanical properties were also investigated. Results showed that Poisson's ratio first decreased and then increased with an increasing θ when k is less than 0.5. A minimum Poisson's ratio of approximately −0.315 was achieved when $k=0.5$ and $\theta =60^{\circ}$. The hierarchical re-entrant lattice structure attained a maximum energy absorption capacity when $k=0.2$ and $w=1/3$, which is 3.9 times higher than that of the traditional re-entrant structure of identical mass. The design concept proposed here can be used for applications where robust mechanical performance is required.

It is noted that the hierarchical structures studied here are anisotropic in properties, and the results presented in this work are for in-plane deformation only. The out-of-plane properties of these hierarchical structures are of interest and will be the topic of our future studies.

2D hierarchical re-entrant cellular structures with Voronoi sub-structures were built in this work. Voronoi structures have been used to represent a 2D cellular solids with a random microstructure. The Voronoi structure is generated by constructing a perpendicular bisector of line segments between random seed points so that newly generated points within any specific Voronoi cell would have a shorter distance than that of adjacent seed points that define the boundary of that Voronoi cell. To construct a 2D Voronoi structure with N cells in a defined area A₀, a degree of irregularity, k, is defined as

$$\begin{equation} k=1-\frac{\delta }{d_{0}}, \end{equation} \tag{ 1 }$$

where $d_{0} =\sqrt{\frac{2A_{0} }{N\sqrt 3 }}$ is the shortest distance between two adjacent seeds in a regular hexagonal honeycomb with identical numbers (N) of unit cells. The maximum δ (the minimum distance between any two adjacent seed points) in a Voronoi structure should be less than d₀ in order to maintain N cells within the defined area A₀ [27]. For a regular lattice, the irregularity factor equals zero; whereas, for a completely random Voronoi model, the irregularity factor equals 1.

Firstly, a 2D Voronoi model was generated in Matlab as the sub-structures according to the procedure described in ref. [27], with an irregularity factor k ranging from 0 to 0.8. Secondly, we replace the solid ribs of a conventional 2D re-entrant structure with the constructed 2D Voronoi structures to build the 2D hierarchical re-entrant structure, as shown in fig. 1(a)–(d). The geometry of the 2D hierarchical re-entrant unit cell structure is defined as follows: the height is H₀, the inner and outer horizontal lengths are L₁ and L₀, respectively, the re-entrant angle is θ, which determines the degree of re-entrant, the cell wall thickness is t₀, $w=t_{0}/H_{0}$. In this work, we assume that $H_{0}=L_{0}$.

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Aluminum was chosen as the solid material used to the build the hierarchical lattice structures. The density, Young's modulus, Poisson's ratio, yield strength, and tangent modulus are $2700\ \text{kgm}^{-3}$, 69 GPa, 0.3, 70 MPa, 1.13 GPa, respectively. A bilinear strain hardening relationship was employed for the simulation. The cell walls were meshed with Shell 163 elements; the number of elements depends on the edge length of the cell wall, and every edge had at least 4 elements. Poisson's ratio was calculated from the elastic region of the stress-strain diagram obtained from the FE analysis. The energy absorption capabilities of the proposed 2D hierarchical re-entrant structures were analyzed using the ANSYS/LS-DYNA package. The structures were compressed between two rigid platens with the bottom platen fixed, and the top platen moving at a constant velocity of $v=2\text{m/s}$.

Poisson's ratio and Young's modulus

Poisson's ratio is defined as the ratio of the transverse contraction strain to the longitudinal extension strain in the direction of the stretching force as

$$\begin{equation} \nu =-\frac{{\varepsilon}'}{\varepsilon}, \end{equation} \tag{ 2 }$$

where ν is Poisson's ratio, ${\varepsilon}'$ and ε are the transverse contraction strain and longitudinal extension strain of the sample, respectively. The normalized Young's modulus is defined as the ratio of the elastic modulus of the lattice structure to the elastic modulus of the solid material used to make the lattice structure.

The relationships among Poisson's ratio, normalized Young's modulus and re-entrant angle θ of the proposed 2D hierarchical re-entrant structures were studied for a k value varying from 0 to 0.8, as shown in fig. 2. Simulation results showed that Poisson's ratio first decreased and then increased with an increasing θ except for the case when $k=0.8$, for which Poisson's ratio decreased monotonically with an increasing θ. A minimum Poisson's ratio of approximately −0.315 was achieved when $k=0.5$, and $\theta =60^{\circ}$; whereas a maximum Poisson's ratio of approximately −0.205 was attained when k = 0, and $\theta=55^{\circ}$. The normalized Young's modulus did not have a noticeable change when θ changed from 55° to 60°, but increased and achieved a maximum value of 0.000113 at $\theta =65^{\circ}$ when $k=0.8$. Generally speaking, the degree of irregularity of a Voronoi sub-structure does not have a strong effect on Poisson's ratio and Young's modulus given that other geometry parameters are fixed. The properties of a re-entrant lattice of identical density but without hierarchy are shown for comparison in fig. 2. It can be seen that Poisson's ratio and normalized Young's modulus of non-hierarchical re-entrant structures varied monotonically with increasing θ, and a minimum Poisson's ratio of −1.22 and a maximum normalized Young's modulus of 0.000127 were achieved at $\theta =65^{\circ}$. It is noted that the hierarchical concept with the Voronoi sub-structure significantly reduced the auxetic effect of the hierarchical structure.

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It is worth mentioning that we also studied Poisson's ratio of the proposed hierarchical structures in the thickness direction, and found a value of about 0.4 which has no noticeable change with a varying k value.

In contrast, for a Voronoi structure which is not changed into a hierarchical lattice, its Poisson's ratio is positive; with an increasing degree of irregularity, its Poisson's ratio would gradually increase [6].

Energy absorption capacity

The energy absorption capacities of the proposed 2D hierarchical re-entrant cellular structures with different k and w values were analyzed at a constant compression velocity of 2m/s, and their stress-strain diagrams are shown in fig. 3 and fig. 4. All the lattice structures have an identical relative density.

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The energy absorbed up to densification is commonly used to evaluate the energy absorption capacity of cellular solids [28]. The energy absorption capacity per unit volume (W_v) of a lattice structure up to densification can be calculated via the equation

$$\begin{equation} W_{v} =\int_0^{\varepsilon_{D}} {\sigma (\varepsilon)\text{d}\varepsilon}, \end{equation} \tag{ 3 }$$

where $\sigma (\varepsilon)$ is the flow stress of the structure and $\varepsilon_{D}$ is the densification strain which defines the strain beyond which the structure compacts and the stress rises steeply. The plateau is due to buckling of the ribs, or, if ribs are sufficiently thick, the plastic flow stress of the rib material [28].

The values of the energy absorption per unit volume W_v of the proposed 2D hierarchical re-entrant cellular structures are listed in table 1. We can see from table 1 that W_v has comparable values when $w=1/2$ and 1/3, and is much higher than that when $w=1/4$. The maximum W_v value was achieved when $k=0.2$ and $w=1/3$, which is 3.9 times higher than that of the traditional re-entrant cellular structure without hierarchy topology. From fig. 3, we can see that the plateau stress level of the hierarchical structures had no prominent change with an increasing k, but the plateau region increased when k increases. A similar trend has been reported in our previous work [6].

Table 1:.  Poisson's ratio and energy absorption per unit volume of the 2D hierarchical re-entrant structures of the first order (n = 1) and second order (n = 2) at different k values and at a compression velocity of $v=10\text{m/s}$.

k = 0 $k=0.2$$k=0.5$$k=0.8$

	Wv/MJm−3
$w=1/2$	0.02179	0.01831	0.01379	0.01685
$w=1/3$	0.02034	0.02590	0.01532	0.01507
$w=1/4$	0.00800	0.01009	0.00677	0.00617

The deformation behaviors of the proposed 2D hierarchical re-entrant cellular structures at different compressional strains at various k values are shown in fig. 5. The hierarchical structures shrunk laterally when the lattice structure is being compressed due to the auxetic effect, and no crush bands were observed during compression.

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Collapse of the lattice initiated from the top side of the lattice and then propagated through towards the bottom side of the lattice. A slight tilt was observed in the structures when $k=0.5$ and 0.8 during compression due to uneven loading conditions originated from uneven collapse of irregular Voronoi sub-structure cells.

2D hierarchical re-entrant cellular structures with Voronoi sub-structures were proposed. Their mechanical properties (including Poisson's ratio and energy absorption capacity) were studied using the finite-element method (FEM); the effects of geometry parameters (k, θ and w) on the lattice mechanical properties were investigated. Results showed that Poisson's ratio first decreased and then increased with an increasing θ when k is less than 0.5. A minimum Poisson's ratio of approximately −0.315 was achieved when $k=0.5$ and $\theta =60^{\circ}$. The 2D hierarchical re-entrant cellular structure attained a maximum energy absorption capacity when $k=0.2$ and $w=1/3$, which is 3.9 times higher than that of the traditional re-entrant cellular structure with solid ribs and of an identical relative density. The design concept proposed here can be used for applications where a robust mechanical performance is required.

This work is supported by "The Fundamental Research Funds for the Central Universities (N170504016)".