Theoretical study and simulation analysis on re-entrant square cellular material’s mechanical properties

Abstract

With the development of three-dimensional printing technologies, so-called “designer materials” including re-entrant, chiral, and rotating rigid structures have attracted much attention due to unusual properties. However, most studies ignore theoretical analysis and pay much attention to numerical simulation. In this article, a novel re-entrant cellular material is proposed and taken as an example to introduce the methods of theoretical study on relations between mechanical properties and geometric parameters of this “designer material.” This work starts from developing theoretical models of relative density using two-dimensional unit cell in terms of three geometric parameters: edge length, re-entrant distance, and cellular thickness. Then, linear-elastic mechanical properties including elastic modulus and shear modulus, and nonlinear mechanical properties are explored. Based on theoretical models, predominant compression failure mode of this material is plastic collapse. Finite element method is adopted to identify accuracy of static mechanical models. On this basis, dynamic mechanical model is put forward, the relationship of dynamic plateau stress and impact velocity, and structural parameters are established by empirical formula. Moreover, the relationship of unit volume energy absorption and geometric features, impact velocity are also established. Eventually, finite element method is utilized to prove that simulation results show a well coincidence to theoretical results.

Keywords

Re-entrant cellular material geometric features mechanical properties theoretical model unit volume energy absorption

Introduction

Cellular materials with different shapes have emerged during the last decades as an exciting paradigm for developments of materials and have been studied further, no matter in mechanics or potential applications. For the cellular materials, geometric sensitivity is a vital important feature. Its mechanical properties are not only related to material itself but also determined by structural characteristics. Thus, establishing the relations of geometry and performance is always the research hot topics.¹

To date, substantial efforts have been devoted to study on simulation and experiment analyses of “designer materials.” Alderson et al.² presented that finite element (FE) models of conventional and re-entrant honeycombs show excellent agreement with the experimental values. Mizzi et al.³ used a fairly new, but promising perforation method to emulate the re-entrant honeycomb mechanism. Based on the aforementioned research, Rad et al.⁴ presented a successful analytical and FE approach for the modeling of three-dimensional (3D) re-entrant structures, in order to extend models’ application in various loading and material cases. Yang et al.⁵ successfully established a novel analytical model, which was verified by FE modeling and experiments to be relatively accurate at predicting the performance of cellular structure. In addition, others used soft lithography for the microproduction of re-entrant hexagonal honeycombs.⁶ Lakes and Elms⁷ pointed out that concave foams showed better properties in yield strengths and energy absorptions than conventional foams of identical original density, a nonlinear auxetic novel vibration damper was proposed by FY Scarpa et al.⁸ Several experiments have been performed to examine the properties of various foam specimens.^9–17

Unlike most traditional studies on “designer materials” only involving simulation/experiment analysis and applications, this study takes re-entrant cellular materials as an example to introduce the methods of theoretical study on the static and dynamic mechanics of this “designer material” in elastic stage and plateau region. Owing to their rationally designed structural topology, re-entrant square cellular material and other re-entrant architected cellular materials exhibit unusual properties not usually found in nature.^18,19 These novel mechanical properties or physical properties could be exploited for the development of materials with advanced functionalities, with applications in the developments of novel fields, such as soft robotics, biomedicine, soft electronics, and acoustic cloaking.^20–22 This is not only for their unusual mechanical response found to originate from its geometry or microstructure,²³ but offers a route to attaining extreme values of other material properties, like a higher indentation resistance, shear resistance, energy absorption, hardness, and fracture toughness.^10,23,24 These methods of theoretical study provide theoretical basis and data support for preliminary design and applications of cellular materials in the near future. In what follows, this article starts from geometry-relative density relationship. Subsequently, static mechanical properties, including linear-elastic property, nonlinear-elastic property, and plastic behavior, are analyzed. According to theoretical models in plateau, the critical stress formulae of plastic collapse and elastic buckling are acquired, and finally, plastic collapse proves to be the predominant failure mechanism. Subsequently, the overall mechanical properties of the cellular material are systematically analyzed using FE method and compared with theoretical models. Then, dynamic mechanics is proposed, the effects of impact velocity on dynamic plateau stress of this material are analyzed, and empirical formulae are given to estimate the dynamic plateau stresses. Meanwhile, the relationship between unit volume energy absorption and geometric features, impact velocity are also established. Eventually, FE method is utilized to prove that simulation results show a well coincidence to that of theoretical values, that is, the agreement between theory and simulation is good and the equations derived herein could predict measured properties well.

Relations between relative density and geometric parameters

Theoretical models of relative density

The re-entrant cellular material proposed is equipped with anisotropy. To deeply understand structural characteristics and deformation mechanism, the relations between three geometric parameters, including cellular thickness $t$ , edge length $L$ , re-entrant distance $h$ , and relative density $ρ_{RD}$ are explored. Figure 1 schematically illustrates the geometric parameters.

Figure 1.

Re-entrant cellular material with three geometric parameters.

Relative density $ρ_{RD}$ defines the ratio of cellular structural equivalent density to the density of material, that is

ρ_{RD} = \frac{ρ_{c}}{ρ_{s}}

(1)

where $ρ_{c}$ denotes equivalent density of cellular structure and $ρ_{s}$ represents the material density. Figure 2 reveals relative density of two-dimensional (2D) re-entrant cells.

Figure 2.

Relative density of 2D re-entrant cells.

Therefore, relative density $ρ_{RD}$ can be expressed as

ρ_{R D} = \frac{S}{{(\frac{L}{2})}^{2}} = \frac{4 S}{L^{2}}

(2)

where

S = S_{1} + S_{2} - S_{3}

(3)

Of this, $S$ stands for one-quarter cellular area, $S_{1}$ is total area of two parallelograms, $S_{2}$ represents the area of square at lower right corner, and $S_{3}$ reflects total area of two triangles in lower small square. Moreover

S_{1} = \frac{2 tm}{\cos α}

(4)

S_{2} = {(\frac{L}{2} - m)}^{2}

(5)

S_{3} = {(\frac{L}{2} - m)}^{2} \tan α

(6)

where $α$ represents intersection angle of cell wall and $X$ direction, giving $\tan α = h / (L / 2) = 2 h / L$ , from which, $h$ denotes re-entrant distance governing re-entrant degree of cellular material, $t$ indicates cell thickness, $L$ signifies cell length, $m$ expresses height of parallelogram, $(L / 2) - m$ is the square edge in the lower right corner. In consideration of geometric relationship

\frac{L}{2} - m = \frac{t}{\cos α} + (\frac{L}{2} - m) \tan α

(7)

Subsequently, $m$ is given by

m = \frac{L}{2} - \frac{t}{\cos α - \sin α}

(8)

Relative density $ρ_{RD}$ can be transformed into

ρ_{RD} = \frac{t \sqrt{4 h^{2} + L^{2}}}{L^{3}} (L - \frac{t \sqrt{4 h^{2} + L^{2}}}{L - 2 h})

(9)

Analyses on relations of relative density and geometric parameters

Since relative density $ρ_{RD}$ ranges from 0 to 1, $ρ_{RD}$ is close to 0 with infinite cellular porosity; otherwise, 1. Besides, $ρ_{c}$ varies in a large range in response to geometric changes.

To thoroughly illustrate the relations, thickness coefficient $a$ and re-entrant degree $b$ are introduced and written as

a = \frac{t}{L}

(10)

b = \frac{h}{L}

(11)

With the restrictions of geometric features, too larger geometric parameters will lead to cell walls’ overlap; whereas, too smaller may eliminate re-entrant characteristics. Based on the conditions, the range of thickness coefficient $a$ and re-entrant degree $b$ are set 0–0.2 and 0–0.2, respectively, simultaneously with the division at four intervals, as exhibited in Table 1.

Table 1.

Value ranges of geometric parameters.

	$L$	$b$	$a$
Figure 3(a)	10	0–0.20	0.05, 0.1, 0.15, 0.20
Figure 3(b)	10	0.05, 0.1, 0.15, 0.20	0–0.20

Figure 3 briefly depicts relevancy of relative density $ρ_{RD}$ and re-entrant degree $b$ , thickness coefficient $a$ , respectively. In Figure 3(a), it can be seen that four curves represent various thickness coefficients $a$ with the introduction of re-entrant degree $b$ , and variation of re-entrant degree $b$ shows slight contribution to relative density $ρ_{RD}$ as thickness coefficient $a$ is less than or equal to 0.1. As thickness coefficient $a$ is over 0.1, relative density $ρ_{RD}$ starts to decline with the increase in re-entrant degree $b$ due to the expansion of voids areas. It can be concluded from Figure 3(b) that relative density $ρ_{RD}$ presents a rising trend with the increment in thickness coefficient $a$ .

Figure 3.

(a) Relationship of relative density $ρ_{RD}$ and re-entrant degree $b$ ; (b) relationship of relative density $ρ_{RD}$ and thickness coefficient $a$ .

Linear-elastic properties

Relations of elastic modulus and geometric features

For re-entrant, cells are arranged in order, as shown in Figure 4. It is very difficult to obtain effective mechanical properties of overall structure directly owing to the complexity of architected cellular structure. Whereas, as for the periodicity of this cellular material, macrostructure can be represented as periodic arrangement of unit cell with complicated microstructure along one or multiple directions. Among this, unit cell is also called as representative volume element (RVE). Based on the arrangement and macroscopic geometric features, this architected cellular structure is equivalent to continuum solid structure. Existing homogenization methods^25–27 have provided effective means for multiscale modeling of continuum solids. The equivalent description, homogenization method, was put forward by Papanicolau et al.²⁸ and Sanchez-Palencia and Zaoui.²⁹ This homogenization method links up microscale analysis of RVE and macroscopic equivalent properties. From a mathematical point of view, the theory of homogenization is a limit theory using asymptotic expansion and assumption of periodicity to substitute differential equations with rapidly oscillating coefficients, with differential equations whose coefficients are constant or slowly varying in such a way that the solutions are close to the initial equations. In this case, the determination of the constitutive behavior can be performed on a mesoscopic level considering a RVE. In any region $x$ within macroscopic design domain, it can be considered as a periodic rearrangement of RVE in the mesoscopic scale. Double diamond structure is confirmed as representative unit described in Figure 5. Moreover, characteristic length $y$ of the RVE is much smaller than the characteristic length $x$ of the macrostructure, that is

x = \frac{y}{u} (0 < u ≪ 1)

(12)

Figure 4.

The arranged re-entrant cellular material.

Figure 5.

Double diamond structure as representative unit.

The macrostructure is regarded as continuous homogeneous solid structure, and its mechanical performance keeps the same everywhere. Thus, mechanical properties of overall structure can be obtained by solving mechanical performance of RVE. Summary of tensor distribution in each RVE is equivalent to surface tensor distribution in the entire macrostructure,³⁰ giving

\int_{Γ_{RVE}^{i}} T_{i} d Γ_{RVE} = \sum_{j = n} T_{j}^{*}

(13)

where $T_{i}$ is the traction vector on the surface of RVE, ${Γ^{i}}_{RVE}$ represents a certain part of its boundary as the part which is orientated parallel to one of the coordinate planes, and $T_{j}^{*}$ stands for the surface tensor of overall macrostructure. Then, the loading conditions of RVE can be acquired from above equation (13).

Consider linear-elastic deformation of re-entrant cellular material shown in Figure 6, subjected to a force $2 F$ parallel to $Y$ axis, equilibrium requires that force parallel to $X$ direction should be zero. Each RVE is modeled with edge length $L$ , cellular thickness $t$ , and re-entrant distance $h$ . As thickness coefficient $a$ is lower and re-entrant degree $b$ reaches higher, bending contributes greatly to the deformation mode. The pioneering work³¹ indicates that as thickness coefficient $a$ is less than 0.25, the beams can be regarded as standard Timoshenko Beam, neglecting shear deformation and the axial extension or compression of the beams. To simplify the analysis, following hypothesis are set that re-entrant distance $h$ should not be too small and cellular thickness $t$ should be small, that is, $b \geq 0.05$ and $a \leq 0.25$ . Thus, this article adopts standard Timoshenko Beam under above assumptions. The calculation only considering dominant bending response of the beam without axial deformation or shear effect is acceptable.

Figure 6.

Diamond structure subjected to force.

In consideration of structural symmetry and kinematic consistency,³² mechanical model, simplified in Figures 7 and 8, shows cell wall deformation under the loading in $Y$ direction.

Figure 7.

Simplified mechanical model under the loading in Y direction.

Figure 8.

Cell wall deformation under the loading in $Y$ direction.

Based on actual stress situation, cell wall deflects by³³

δ = \frac{F \sin θ \cdot l^{3}}{3 EI}

(14)

Of this, $I$ depicts the moment of inertia of an area, $I = (c t^{3} / 12)$ ; $c$ stands for the thickness of cellular material along $Z$ direction; $θ$ and $l$ are exhibited in Figure 5, respectively. A component $δ_{y} = δ \sin θ$ parallel to $Y$ axis, and total deflections $δ_{y, sum}$ is given by

δ_{y, sum} = 2 δ_{y}

(15)

Subsequently, strain $ε_{y}$ , stress $σ_{y}$ , and $\sin θ$ are obtained in the following

σ_{y} = \frac{2 F}{S} = \frac{2 F}{Lc}

(16)

ε_{y} = \frac{δ_{y, sum}}{L}

(17)

\sin θ = \frac{h}{l} = \frac{h}{\sqrt{\frac{L^{2}}{4} + h^{2}}}

(18)

Finally, elastic modulus parallel to $Y$ axis $E'$ are given as

E' = \frac{σ}{ε_{y}} = \frac{t^{3}}{4 h^{2} \sqrt{h^{2} + \frac{L^{2}}{4}}} E

(19)

Relationship of shear modulus and geometric features

As it is subjected to loading in $X$ direction, shear deformation occurs, as illuminated in Figure 9.

Figure 9.

Re-entrant cells under shear force.

Like the calculation of elastic modulus, standard Timoshenko Beam is also utilized to analyze shear modulus of this cellular material, shown in Figure 9. It is concluded that force-deformation mainly attributes to diamond structure’s deformation with vertical shear stress. Similarly, the simplified mechanical model is obtained in Figure 10. Force Analysis of each beam on the left half are conducted in Figure 11.

Figure 10.

Simplified mechanical model under the loading in X direction.

Figure 11.

Force analysis of cell beams: (a) upper left beam and (b) lower left beam.

As presented in Figure 11(a), the deflection $δ_{l, 1}$ is described as³³

δ_{l, 1} = \frac{0.5 F l^{3} \cos θ}{3 EI}

(20)

Hence, the deflection along $X$ direction $σ_{l, 1 x}$ is expressed as follows

δ_{l, 1 x} = δ_{l, 1} \cos θ = \frac{0.5 F l^{3}}{3 EI} \cos^{2} θ

(21)

In Figure 11(b), the moment $M$ , given by

M = \frac{0.5 FL}{2}

(22)

Furthermore, the deflection $δ_{l, 2}$ is written as

δ_{l, 2} = \frac{0.5 F l^{3} \cos θ}{3 EI} + \frac{M l^{2}}{2 EI}

(23)

Among this

I = \frac{c t^{3}}{12}

(24)

\cos θ = \frac{\frac{L}{2}}{l} = \frac{L}{2 l}

(25)

Subsequently

δ_{l, 2 x} = δ_{l, 2} \cos θ = \frac{0.5 F l^{3} \cos^{2} θ}{3 EI} + \frac{M l^{2} \cos θ}{2 EI}

(26)

Thus, total deflection $δ_{lx}$ of the left half can be expressed by

δ_{lx} = δ_{l, 1 x} + δ_{l, 2 x}

(27)

Similarly, total deflection $δ_{lx}$ on the right half is illustrated by

δ_{lx} = δ_{rx}

(28)

That is, the left half exhibits the same deformation with the right, in accord with actual situation

δ_{x} = δ_{lx}

(29)

The shear strain $γ$ is explained as

γ = \frac{δ_{x}}{L}

(30)

And, the shear stress $τ$ is given by the following equation

τ = \frac{F}{Lc}

(31)

Eventually, shear modulus $G'$ is acquired as follows

G' = \frac{τ}{γ} = \frac{4 t^{3}}{7 L^{2} \sqrt{h^{2} + \frac{L^{2}}{4}}} E

(32)

Confirmation of the theoretical model

To confirm the developed theoretical models, FE model is developed. Acrylonitrile Butadiene Styrene (ABS) material is adopted, and its Young’s modulus $E = 1 GPa$ , Poisson’s ratio $ν = 0.39$ , yield strength $σ_{y} = 26.75 MPa$ , and shear modulus $G = 0.319 GPa$ .

Validation of elastic modulus theoretical model

For there exist a wealth of cells in architected cellular structure, it will need extremely extensive computational time and so much work, and often, it is impossible to achieve exact results. While ideal results can be acquired through emulation with the adoption of applying periodic boundary on RVE, it means that mechanical behavior of entire cellular structure can be obtained by just applying appropriate kinematic and kinetic constraints on RVE,^32,34 that is, for periodic structures, their stress–strain fields present periodicity and continuity under external loading conditions. Thus, mechanical analysis model should be constructed under periodic stress or periodic displacement boundary conditions based on periodic unit cell (RVE), with the result of achieving reasonable stress distribution. As for continuous material with periodic unit cell structures, the boundary of unit cell should satisfy continuity conditions including continuous displacement and continuous stress. It should be guaranteed that grid nodes in corresponding axisymmetric planes should be one-to-one correspondence with the adoption of surface mesh offset.

Generally, periodic unit cell boundary planes are paired and axial symmetry, thus displacement field on corresponding axisymmetric boundary planes can be written as³⁵

u_{i}^{j +} = {\bar{ε}}_{ik} x_{k}^{j +} + u_{i}^{*}, u_{i}^{j -} = {\bar{ε}}_{ik} x_{k}^{j -} + u_{i}^{*}

(33)

where ${\bar{ε}}_{ik}$ stands for mean strain of total field, ${\bar{ε}}_{ik} x_{k}$ represents linear distribution displacement field, $j +$ and $j -$ are the $j$ group of axisymmetric planes for unit cell, respectively. $u_{i}^{*}$ is the periodic displacement correction of the boundary planes, and it keeps the same in corresponding points of periodic unit cell corresponding axisymmetric planes, given by

u_{i}^{j +} - u_{i}^{j -} = {\bar{ε}}_{ik} (x_{k}^{j +} - x_{k}^{j -}) = {\bar{ε}}_{ik} Δ x_{k}^{j}

(34)

Of which, $Δ x_{k}^{j}$ is constant for axisymmetric planes of unit cell. For a given ${\bar{ε}}_{ik}$ , displacement difference on the right side of equation (34) can be acquired. Hence, in the FE analysis, periodic displacement constraints of corresponding nodes can be achieved through displacement difference constraint equation (34), and actual displacement boundary values of periodic unit cell can be avoided.

The previous work³⁶ indicates that the uniqueness of the solution and the continuity of the boundary stress can be guaranteed and satisfied through applying periodic displacement constraints based on equation (34), the stress satisfies the following equations

σ_{n}^{j +} - σ_{n}^{j -} = 0, τ_{nt}^{j +} - τ_{nt}^{j -} = 0

(35)

Periodic unit cell is regarded as anisotropic elastomer, and its constitutive equation can be expressed as³⁵

{\bar{ε}}_{i} = S_{ij} {\bar{σ}}_{j}

(36)

Here, $S_{ij}$ denotes equivalent flexibility matrix of unit cell; ${\bar{ε}}_{i}$ and ${\bar{σ}}_{j}$ are mean strain and mean stress, respectively, given by

{\bar{ε}}_{ij} = \frac{1}{V} \int_{V} ε_{ij} dV, {\bar{σ}}_{ij} = \frac{1}{V} \int_{V} σ_{ij} dV

(37)

This article employs Hyperworks/ABAQUS software to emulate deformation behavior of this material. The material’s thickness is 1 mm and meshed with second-order tetrahedron solid elements to prevent “shear locking” and occurrence of “hour glassing mode.”^37,38 In order to apply periodic displacement boundary conditions to unit cell, grid nodes of axisymmetric planes should be kept in one-to-one correspondence. Corresponding grids in relative axisymmetric boundary planes are generated by means of surface mesh translation in Hypermesh software, and the periodic displacement boundary conditions corresponding to equation (34) are applied by writing scripts in ABAQUS software, specific applying methods are illustrated in the literature.^39,40 Subsequently, displacement freedoms of left-lower nodes along $X$ , $Y$ , and $Z$ directions are constrained, displacement freedoms of right-lower nodes along $Y$ direction are limited, and there are no displacements along $Z$ direction for all nodes. In addition, upper nodes are subjected to uniform changing line load $F_{line}$ , and downward displacement velocity of the upper nodes is less than 1 mm/min, which is treated as quasi-static deformation, as revealed in Figure 12.

Figure 12.

FE model for validation of elastic modulus theoretical model.

In order to verify the mesh-independency, the corresponding stress is calculated under various mesh density as strain is 0.035, the variation simulation results of stress with element number is shown in Figure 13. It can be seen that stress tends to be stable as number of mesh elements exceeds 5000. With regards to the mesh-independency verification for shear modulus, number of mesh elements is selected as 7560 to ensure that simulation results are independent with number of mesh elements.

Figure 13.

Various simulation results of direct stress with element number.

First, three groups of geometric parameters are assigned in Table 2. Subsequently the simulation analysis is carried out. Finally, corresponding stress–strain curves are achieved (dashed lines in Figure 14) and elastic modulus $E'$ is also obtained (second column in Table 3). Then, the aforementioned three groups of parameters are brought into theoretical mechanical model (equation (19)), and the theoretical stress–strain curves (solid lines in Figure 14) and elastic modulus $E'$ (third column in Table 3) are achieved.

Table 2.

Three groups of geometric parameters selected in elastic region.

	$L (mm)$	$a$	$h (mm)$
1	9	0.2	0.5
2	12	0.2	1
3	11	0.1	0.5

Figure 14.

Stress–strain curves of simulation and theoretical analyses in elastic stage.

Table 3.

Comparison of theoretical results and simulation results.

	Simulation results (GPa)	Theoretical results (GPa)	Relative error (%)
1	1.36	1.29	5.14
2	0.607	0.568	6.43
3	0.264	0.240	9.1

From Figure 14 and Table 3, it can be concluded that simulation results in elastic region is close to that of theoretical, simulation and theoretical results reach a higher agreement. All of these demonstrate the rationality and validity of theoretical models with relatively accurate at describing the performance of re-entrant cellular material.

As shown in Figure 14, the slope of simulation stress–strain curve tends to be smooth. There exists a reason to explain the situation; as the loading stress exceeds plastic collapse stress, plastic hinges will occur at the joints. Moreover, plastic hinges and instability result in the shift deformation of the cellular material. As structures with multiple cells are used to emulate, plastic hinges can be seen more clearly, as presented in Figure 15.

Figure 15.

Shift deformation of cellular material due to plastic hinges and instability.

Validation of shear modulus theoretical model

Similar to validation of elastic modulus theoretical model, the same three groups of geometric parameters are chosen. Figure 16 reveals FE model for verifying shear modulus theoretical model. Periodic displacement boundary conditions based on equation (34) are also applied, displacement freedoms of left-upper nodes along $X$ , $Y$ , and $Z$ directions are limited, displacement freedoms of left-lower nodes along $X$ direction are constrained, and there are no displacements along $Z$ direction for all nodes. In addition, right nodes are subjected to uniform changing line load $F_{line}$ , and upward displacement velocity of the right nodes is less than 1 mm/min, which is treated as quasi-static deformation, as shown in Figure 16.

Figure 16.

FE model for validation of shear modulus theoretical model.

Figure 17 demonstrates various simulation results of shear stress with different number of mesh elements as strain is 0.35. It can be seen that shear stress tends to be stable as number of mesh elements exceeds 7000. Thus, number of mesh elements is selected as 7560 to ensure that simulation results are independent with number of mesh elements.

Figure 17.

Various simulation results of shear stress with element number.

Then, simulation analysis is carried out and corresponding stress–strain curves are achieved (dashed lines in Figure 18), and shear modulus $G'$ can also be obtained (second column in Table 4). Then, the aforementioned three groups of parameters are brought into theoretical mechanical model (equation (32)), and the theoretical stress–strain curves (solid lines in Figure 18) and shear modulus $G'$ (third column in Table 4) are achieved.

Figure 18.

Shear stress–strain curves of simulation and theoretical analyses in elastic stage.

Table 4.

Comparison of theoretical results and simulation results of shear modulus.

	Simulation results (MPa)	Theoretical results (MPa)	Relative error (%)
1	9.31	9.1	2.3
2	9.27	9.0	3.0
3	1.19	1.1	8.18

From above discussion, Figure 18 and Table 4 also illustrate a higher agreement of simulation and theoretical results, proving the rationality and validity of theoretical models.

Analyses

From equations (19) and (32), the conclusion is that both relative elastic modulus $E' / E$ and relative shear modulus $G' / G$ are significantly relevant to geometric parameters. To observe the effects of geometric variables on relative elastic modulus $E' / E$ and relative shear modulus $G' / G$ , thickness coefficient $a$ and re-entrant degree $b$ are also introduced.

Relative elastic modulus

Due to the limitations of geometric features, thickness coefficient $a$ and re-entrant degree $b$ are defined within the range of 0.05–0.2, respectively, with the division of three intervals. Figure 19 exhibits relevancy of relative elastic modulus $E' / E$ and re-entrant degree $b$ , thickness coefficient $a$ , respectively. In Figure 19(a), four curves indicate the relationship of relative elastic modulus $E' / E$ and re-entrant degree $b$ corresponding to four thickness coefficients $a$ . Relative elastic modulus $E' / E$ exhibits a downward trend with the increment in re-entrant degree $b$ . Drop rate of the relative elastic modulus $E' / E$ decreases as re-entrant degree $b$ increases, and slope reaches almost zero eventually. In Figure 19(b), it can be indicated that relative elastic modulus $E' / E$ increases with the increment in thickness coefficient $a$ . Another phenomenon is that as thickness coefficient $a$ is less than or equal to 0.1, variation of re-entrant degree $b$ shows slight contribution to relative elastic modulus $E' / E$ . As thickness coefficient $a$ is over 0.1, change of re-entrant degree $b$ contributes to relative elastic modulus $E' / E$ greatly.

Figure 19.

(a) Relations of relative elastic modulus $E' / E$ and re-entrant degree $b$ ; (b) relations of relative elastic modulus $E' / E$ and thickness coefficient $a$ .

Relative shear modulus

Like relative elastic modulus, Figure 20 exhibits the relevancy of relative shear modulus $G' / G$ and re-entrant degree $b$ , thickness coefficient $a$ , respectively. Figure 20(a) shows the tendency of relative shear modulus $G' / G$ , as indicated in four curves, it can be seen that relative shear modulus $G' / G$ roughly exhibits a linear relation with re-entrant degree $b$ for all thickness coefficient $a$ . The relative shear modulus $G' / G$ increases with the increase in re-entrant degree $b$ . It can be indicated from Figure 20(b) that relative shear modulus $G' / G$ increases with the increment in thickness coefficient $a$ . Another phenomenon is that as thickness coefficient $a$ is less than or equal to 0.1, variation of re-entrant degree $b$ shows little contribution to the change of $G' / G$ . $G' / G$ begins to differ with various re-entrant degree $b$ when thickness coefficient $a$ is over 0.1.

Figure 20.

(a) Relations of relative shear modulus $G' / G$ and re-entrant degree $b$ ; (b) relations between relative shear modulus $G' / G$ and thickness coefficient $a$ .

Nonlinear properties

Nonlinear-static properties

As the subjected stress exceeds the limit stress, deformation of the re-entrant cellular material will be more localized and cellular failures occur due to elastic buckling, plastic collapse, and brittle fracture. Afterward, a longer platform stage will appear in stress–strain curve, and the average stress of platform region is defined as plateau stress.

Higher plateau stress can effectively improve energy absorption abilities. Therefore, it is significantly meaningful to study failure mechanism and yield stress of this cellular material. For the re-entrant cellular material made of elastic–plastic materials, failure modes mainly cover elastic buckling and plastic collapse. Figure 21 describes actual deformation of local re-entrant cellular material in plateau region. This article focuses on constructing mechanical models of axial compression and constructing the relations between yield stress and re-entrant cellular material using single cell as representative unit.

Figure 21.

Actual deformation of local re-entrant cellular material in plateau region.

Elastic buckling

Cellular materials collapse elastically by the cooperative elastic buckling of cell walls, allowing further large deformations at almost constant load. Under the loading of axial compression, cell walls can be simplified as beams with pressure of one end and constraint of the other end. Elastic buckling reveals while the applied load exceeds Euler buckling load. Therefore³³

P_{e} = \frac{n^{2} π^{2} EI}{l^{2}}

(38)

where $n$ describes the coefficient of constraint conditions, with the value range of 0.5–2. As boundary condition is free, $n$ is 0.5; whereas, the value is 2 as boundary condition is fully constrained. $E$ represents cellular material’s elastic modulus, inertia moment of cell wall cross section is symbolized by $I$ , and $l$ stands for beam length.

As elastic buckling happens, the load $F_{e}$ can be expressed as³³

F_{e} = \frac{σ_{e} Lc}{2}

(39)

Besides, the axial force is

P_{e} = \frac{n^{2} π^{2} EI}{l^{2}} = F_{e} \cos θ

(40)

Finally, the limit stress of elastic buckling $σ_{e}$ can be calculated as

σ_{e} = \frac{2 π^{2} E t^{3}}{3 l L^{2}} = \frac{2 π^{2} t^{3}}{3 L^{2} \sqrt{h^{2} + \frac{L^{2}}{4}}} E

(41)

The plastic behavior

As bending moment reaches or beyond the fully plastic moment, the re-entrant cellular material which has a plastic yield point will collapse plastically. Plastic collapse stress $σ_{p}$ can be described as the corresponding stress when complete plastic hinges occur under longitudinal stress. Figure 22 exhibits cell deformation during plastic collapse subjected to $Y - axis$ load.

Figure 22.

Cell deformation during plastic collapse subjected to $Y - axis$ load.

Under the definition of generalized stress, single beam force $F$ follows that³³

F = \frac{σ_{1} Lc}{2}

(42)

As plastic hinges occur, the work done by concentrated force $F$ can be written as

W = \frac{1}{2} FL \sin θ

(43)

where all joints of cell wall rotate through the angle $φ$ .Thus, the work done by plastic hinges can be explained by

W = 2 \times \frac{1}{4} σ_{ys} c t^{2} φ = \frac{1}{2} σ_{ys} c t^{2} φ

(44)

As plastic hinges occur, the work done by concentrated force $F$ is equal to that of plastic hinges. Subsequently, the critical stress $σ$ can be obtained in the following equation. $σ$ also represents limit stress $σ_{p}$ as plastic collapse happens. Among this, $σ_{ys}$ is yield stress

σ = σ_{p} = \frac{2 σ_{ys} t^{2}}{Lh}

(45)

Confirmation of predominant compression failure mode

As for plastic collapse stress $σ_{p}$ and elastic buckling stress $σ_{e}$ , as elastic buckling stress $σ_{e}$ is beyond plastic collapse stress $σ_{p}$

\frac{2 σ_{ys} t^{2}}{Lh} \leq \frac{2 π^{2} t^{3}}{3 L^{2} \sqrt{h^{2} + \frac{L^{2}}{4}}} E

(46)

Simplified as

\frac{a}{\sqrt{1 + {(\frac{L}{2 h})}^{2}}} \geq \frac{σ_{ys}}{3.3 E}

(47)

For conventional elastic materials, like metals and polymers, the magnitude of $σ_{ys} / E$ is 10^-3–10^-2. While value range of thickness coefficient $a$ is 0.05–0.2, and the range of $L / 2 h$ is 2.5–10. Therefore, equation (47) meets the requirements in most cases, and plastic collapse is the main failure mode of the re-entrant cellular material.

Validation of plastic collapse theoretical model

With regard to the validation of plastic collapse theoretical model, meshing and simulation conditions including limitations and loading conditions are the same with that of elastic modulus exhibited as Figure 12.

Three groups of geometric parameters are assigned in Table 5 and then simulation analysis is carried out. The analysis of deformation process in plateau of structures with multiple cells shown in Figure 23 exhibits that plastic hinges emerge in the re-entrant points of unit cell and contact points of adjacent cells. It indicates that the failure mechanism attributes to plastic collapse.

Table 5.

Three groups of geometric parameters assigned in plateau.

	$L (mm)$	$a$	$h (mm)$
First	9	0.2	1.5
Second	11	0.1	0.5
Third	12	0.2	1

Figure 23.

Deformation in plateau with three groups of geometric parameters: (a) first group, (b) second group, (c) third group.

As simulation results are not continuous data, numerical integration method is utilized herein to obtain nominal plastic collapse stress $σ_{ps}$ , given as⁴¹

σ_{ps} = \frac{1}{N} \sum_{i = 1}^{N} σ_{i}

(48)

Of this, $N$ stands for the number of data points in plateau stress region and $σ_{i}$ represents specific plateau stress of each point obtained by simulation.

During the simulation calculation, plateau stress region starts at $ε_{0}$ and ends in $ε_{d}$ (also the beginning of densification strain region). Figure 24 describes $ε_{0}$ and $ε_{d}$ .

Figure 24.

(a) Description of $ε_{0}$ and $ε_{d}$ in stress–strain curve of re-entrant cellular material. (b) Local display of $ε_{0}$ .

$ε_{0}$ can be regarded as the first point when $σ' < 0$ , $σ'$ is the numerical differentiation of stress–strain. This is because stress–strain curve will enter yield stage, namely, plateau region, when loading stress exceeds yield upper limit. Stress will decrease in plateau,⁴¹ as shown in Figure 25(a)

σ' = \frac{σ (i + 1) - σ (i)}{ε (i + 1) - ε (i)}

(49)

Figure 25.

(a) Differential of stress–strain and (b) relation between energy absorption efficiency $E$ and strain.

$ε_{d}$ is regarded as the point corresponding to maximum of energy absorption efficiency $E$ ,⁴¹ as shown in Figure 25(b)

E = \frac{W_{s} (t)}{σ_{s} (t)}

(50)

where $W_{s} (t)$ represents energy absorption of simulation and $σ_{s} (t)$ denotes simulation stress.

Finally, the corresponding stress–strain curves are achieved in Figure 26, and nominal plastic collapse stress $σ_{ps}$ can also be obtained (second column in Table 6). In addition, the above-mentioned geometric parameters are brought into theoretical model (equation (45)), the theoretical stress–strain curves and plastic collapse stress $σ_{p}$ (third column in Table 6) are obtained.

Figure 26.

Stress–strain curves of simulation and theoretical analyses in plateau stage.

Table 6.

Results of theoretical analysis and simulation analysis.

	Simulation results (MPa)	Theoretical results (MPa)	Relative error (%)
1	13.14	12.84	2.28
2	12.34	11.77	4.58
3	27.83	25.68	7.72

From Figure 26 and Table 6, it is seen that simulation fitted results in plateau are close to that of theoretical analysis and maintain a high accuracy. These demonstrate the rationality and validity of theoretical models.

Analyses

Similarly, plastic collapse stress $σ_{p}$ is tightly related to geometric parameters. The relations between plastic collapse stress $σ_{p}$ and re-entrant degree $b$ , thickness coefficient $a$ are described in Figure 27, respectively. It can be concluded from Figure 27(a) that plastic collapse stress $σ_{p}$ displays a downward trend with the increment in re-entrant degree $b$ for all of thickness coefficients $a$ . In Figure 27(b), plastic collapse stress $σ_{p}$ increases with the increase in thickness coefficient $a$ for all of re-entrant degrees $b$ .

Figure 27.

(a) Relations of plastic collapse stress $σ_{p}$ and re-entrant degree $b$ ; (b) relations between plastic collapse stress $σ_{p}$ and thickness coefficient $a$ .

In-plane dynamic crushing behavior

Unlike pure solid materials, re-entrant cellular material owns special mechanical properties owing to re-entrant characteristics, and it shows outstanding energy absorption abilities than conventional ones in terms of unique deformation behaviors.⁴² Energy absorption is characterized by stress–strain curve, a typical nominal stress–strain curve consists of three stages: linear-elastic stage, plateau region, and compression densification stage. The plateau region contributes greatly to energy absorption properties of cellular materials. Hence, it is very meaningful to explore the effects of impact velocity on plateau stress of the cellular materials.^10,42

Theoretical models

In order to evaluate dynamic plateau stress of cellular materials, Reid and Li QM^41,43 gave the relationship between dynamic impact stress and impact velocity according to one-dimensional (1D) shockwave theories, given by

σ_{pd} = σ_{0} + \frac{ρ_{c}}{ε_{d}} v^{2}

(51)

Here, $σ_{0}$ stands for quasi-static plateau stress, $σ_{pd}$ denotes dynamic impact stress, and $ε_{d}$ is related to porosity, approximately given by

ε_{d} = 1 - ρ_{RD}

(52)

Meanwhile, densification strain $ε_{d}$ can be precisely given by the following empirical formula.⁴⁴ Among which, coefficients $A$ , $B$ , and $C$ can be obtained by simulation analysis

ε_{d} = A (1 - B ρ_{RD} + C {ρ_{RD}}^{3})

(53)

As above-mentioned equation (45)

σ_{pd} = \frac{2 σ_{ys} t^{2}}{Lh} + \frac{ρ_{c}}{ε_{d}} v^{2}

(54)

It can be seen from equation (54) that dynamic collapse stress $σ_{pd}$ is related to the square of velocity, thickness coefficient $a$ , and re-entrant degree $b$ .

Validation and analyses of theoretical models

In dynamic impact, shock waves will spread in cellular materials, leading to the complexity of periodic boundary conditions along the direction of dynamic impact.⁴¹ To simplify the calculation, this article adopts larger model with more cells to emulate the dynamic impact performance of materials approximately. Through simulation analysis, it shows that relative elastic modulus and relative platform stress tend to increase with the increment in number of cells, whereas growth rate becomes smaller gradually. As layers of cells reach more than 5, equivalent performances keep stable, as exhibited in Figure 28. Thus, this article employs $5 \times 5$ cells to simulate equivalent performance of this material.

Figure 28.

The relationships of re-entrant cellular materials between relative performances and size.

FE analysis on the dynamic crushing of this cellular material is carried out. Periodic boundary conditions are only applied along $X$ and $Z$ directions, and no periodic boundary condition in $Y$ direction. Figure 29 shows the simulation model for this cellular material. A rigid wall subjected to initial velocity along $Y$ axis impacts the top of the specimen, and the bottom is fixed with all freedoms constrained. In the process of crushing, the change of impact velocity is neglected. Similar to the literature,^45–47 boundary conditions are set that left and right sides are free, and out-plane displacements of all the nodes in the surface are limited to guarantee plane strain state’s deformation.

Figure 29.

Simulation model for re-entrant cellular materials.

Figure 30 shows the trend of dynamic plateau stress $σ_{p}$ with the varying of impact velocity. Calculation method of FE analysis results is the same with that of static mechanical models’ plateau stress (equation (48)), and FE analysis results are marked with cross-shaped, rhombus and snowflake characters in Figure 30. Solid and dashed lines stand for theoretical results. The results display that simulation results show a well coincidence to that of the theoretical in terms of impact velocity. It is also demonstrated that empirical formula could characterize dynamic impact behavior of this cellular materials.

Figure 30.

The trending of dynamic plateau stresses with the variation of impact velocity in terms of various re-entrant degree $b$ and thickness coefficient $a$ .

For all the values of thickness coefficient $a$ and re-entrant degree $b$ , as impact velocity increases, $σ_{p}$ will increase with a rising trend. As the velocity remains a certain value, the higher the thickness coefficient $a$ is and the lower re-entrant degree $b$ reaches, the larger dynamic plateau stress $σ_{p}$ can be expected.

Characterization of energy absorption properties

Under impact loading condition, the surrounding area of stress–strain curve can represent unit volume energy absorption. The total unit volume energy absorption $W_{u}$ is expressed as

W_{u} = W_{u_{1}} + W_{u_{2}}

(55)

Of this, $W_{u_{1}}$ is unit volume energy absorption in elastic and $W_{u_{2}}$ stands for unit volume energy absorption in platform stage. Assuming that elastic region is linear elasticity, as stress lies in elastic stage

σ (ε) = E \cdot ε

(56)

And also, $ε_{0} = σ_{pd} / E$ . Then, described as

W_{u_{1}} = \int_{0}^{ε_{0}} σ (ε) d ε = \frac{σ_{pd}^{2}}{2 E}

(57)

As stress reaches platform region, can be achieved

W_{u_{2}} = \int_{ε_{0}}^{ε_{d}} σ (ε) d ε = (ε_{d} - ε_{0}) σ_{pd}

(58)

According to the simulation model in Figure 29, theoretical and simulation results of unit volume energy absorption are compared with various thickness coefficient $a$ and re-entrant degree $b$ in Figures 31 and 32, respectively. Figure 31 expresses the change of unit volume energy absorption with different re-entrant degree $b$ , where calculated unit volume energy absorption from FE is marked with cross-shaped, rhombus and snowflake characters. It exhibits that simulation results show high agreement with that of theoretical values. For all the values of thickness coefficient $a$ , the unit volume energy absorption decreases with the increment in re-entrant degree $b$ . In Figure 32, calculated unit volume energy absorption from FE is plotted as cross-shaped, rhombus and snowflake symbols. Similarly, simulation results keep high consistent with that of theoretical values in terms of thickness coefficient $a$ . For all the values of re-entrant degree $b$ , the unit volume energy absorption increases with the increase in the thickness coefficient $a$ .

Figure 31.

Theoretical and simulation results of unit volume energy absorption with various re-entrant degree $b$ .

Figure 32.

Theoretical and simulation results of unit volume energy absorption with various thickness coefficients $a$ .

Discussion and conclusion

In this study, a novel re-entrant cellular material is proposed and taken as an example to introduce the methods of theoretical study on relationship between mechanical properties and geometric parameters of this “designer material.” This work initially presents the static mechanical models of the re-entrant cellular material, relative density $ρ_{RD}$ , linear properties including elastic modulus $E'$ , and shear modulus $G'$ , nonlinear properties including elastic buckling stress $σ_{e}$ , and plastic collapse stress $σ_{p}$ are explored. Moreover, their dependence on cell characteristics including edge length $L$ , re-entrant distance $h$ , and cellular thickness $t$ are investigated. According to nonlinear-static mechanical models, failure mechanism of this re-entrant cellular material in plateau region mainly attributes to plastic collapse. In addition, both elastic and plastic properties are tested to identify the accuracy of static mechanical models using FE simulation method, with different values of geometric parameters. It indicates that a higher agreement of simulation and theoretical results proves the rationality and validity of theoretical models. On this basis, dynamic mechanics is proposed, the influences of impact velocity on this material’s dynamic plateau stress are analyzed, and empirical formulae are established to evaluate dynamic plateau stresses. Meanwhile, the relationship between unit volume energy absorption and geometric features, impact velocity are also established. Eventually, FE method is utilized to prove the accuracy of theoretical models, and the results speak for themselves. There can be little question that the theoretical models established adequately describe the static and dynamic properties of the cellular material.

Much of this analysis of the re-entrant cellular material and its simulation confirmation is new and shows great contributions to guiding design and understanding cellular materials. Although this study focuses on 2D cellular material, an extended analysis of 3D re-entrant cellular material can be approached. Also, the results of this study obtained provide theoretical basis and data support for the application of cellular materials.

Footnotes

Handling Editor: Hiroshi Noguchi

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper was funded by the China Scholarship Council (Grant No. 201606170199).

ORCID iD

Ying Zhao

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