Accepted Manuscript Compressive deformation mechanism of honeycomb-like graphene aerogels Jun–Jun Shang, Qing-Sheng Yang, Xia Liu, Chao Wang PII:
S0008-6223(18)30358-0
DOI:
10.1016/j.carbon.2018.04.013
Reference:
CARBON 13050
To appear in:
Carbon
Received Date: 14 January 2018 Revised Date:
4 April 2018
Accepted Date: 6 April 2018
Please cite this article as: Jun–Jun. Shang, Q.-S. Yang, X. Liu, C. Wang, Compressive deformation mechanism of honeycomb-like graphene aerogels, Carbon (2018), doi: 10.1016/j.carbon.2018.04.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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ACCEPTED MANUSCRIPT Compressive Deformation Mechanism of Honeycomb-like Graphene Aerogels Jun-Jun Shang a, Qing-Sheng Yang a, ∗, Xia Liu a, ∗, Chao Wang b, ∗ a
Department of Engineering Mechanics, Beijing University of Technology, Beijing 100124,
b
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China LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China
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Abstract: Graphene aerogels (GAs) are a kind of advanced graphene assemblies, which are
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quite elastic and can easily restore their original form after compression. The macroscale compression behavior of the GAs is the result of microstructural evolution of the graphene sheets. In this paper, the microstructural compression response of honeycomb-like GAs is investigated via coarse-grained molecular dynamics (CGMD) method. Based on the
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coarse-grained (CG) model of a single-layer graphene, honeycomb-like structures can be built and arranged layer-by-layer to form the complete GA model. CGMD simulations presented in this work are carried out with a TersoffCG potential, which was developed specifically for
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single-layer graphene. The effects of layer number, size of the graphene sheets and the
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interlaminar overlap pattern on the compressive behavior of the GAs are studied. Moreover, the stress distribution in graphene networks of the GAs during the compression process is analyzed. Different microscale strain concentrations and stress localizations are discovered in the GAs with different network patterns. Furthermore, a negative Poisson’s ratio of the models within a certain range of compressive strain is found, which is related to both the ∗ ∗ ∗
Corresponding author. E-mail address:
[email protected] (Q. Yang). Corresponding author. E-mail address:
[email protected] (X. Liu). Corresponding author. E-mail address:
[email protected] (C. Wang). 1
ACCEPTED MANUSCRIPT microstructural arrangement and the deformation of graphene sheet.
1. Introduction
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Graphene is a one-atom thick nanomaterial with extraordinary mechanical [1], electrical [2] and thermal [3] properties. Extensive studies have been made to exploit its unusual performance from nanoscale to macroscopic applications by assembling it into
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three-dimensional (3D) bulk materials [4, 5]. Graphene aerogels (GAs) are a kind of 3D
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macroscale assemblies of graphene, which possess extremely high porosity [6, 7], ultralight weight [8], excellent compressibility [9, 10] and electrical conductivity [11]. The microstructures of GAs can be well-controlled by various methods, including template growth [11-13], self-assembly [14-16], cross-linking [17, 18] and 3D printing [19, 20]. According to
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the assembly pattern of graphene sheets, the networks in GA can be categorized as core-shell-like hierarchical structures [21], honeycomb-like structures [22] and structures with randomly oriented graphene sheets [6].
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Since the superior elasticity of GAs arises from their microstructure, great efforts have been
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made to determine the relation of the network to graphene sheets. Nieto et al. [23] evaluated the intrinsic deformation mechanisms of a freestanding 3D graphene foam utilizing in situ tensile testing inside a scanning electron microscope (SEM). They found that the graphene branch alignment enables branches to bear load along the high-strength in-plane direction of graphene, causing the graphene foam to have high elastic modulus during tensile loading. The deformation mechanisms of compression, graphene branch bending and branch wall elastic depression, were also investigated by using the nano-indentation method [23]. By using 2
ACCEPTED MANUSCRIPT full-atom molecular dynamics (MD) simulations, Baimova et al. [24] discovered that the graphene bulk material consisting of crumpled graphene is extremely resistant to diamondization when subjected to hydrostatic compression at room temperature. Their work
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also shows that severe plastic shear deformation can alter the structures and mechanical properties of graphene bulk materials. Mimicking the synthesis of the porous material, Qin et al. [25] built full atomic models of a 3D graphene assembly using MD simulations and
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calculated the critical densities below which the 3D graphene assembly starts to lose its
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mechanical advantage over most polymeric cellular materials.
Comparing to the full-atom MD simulations, coarse-grained MD (CGMD) simulations on large-scale models like GAs are more efficient and better represent real experiments. Wang et al. [26] investigated the microscopic deformation mechanism of 3D graphene foam materials
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with random arranged graphene sheets under uniaxial compression, using CGMD method developed by Cranford et al. [27]. They achieved rubber-like behavior and a near-zero Poisson’s ratio of 3D graphene foam under uniaxial compressive loading and showed that
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these compelling phenomena can be attributed to microstructural deformation, rearrangement
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and compaction in three stages. The same CGMD method was used to evaluate the uniaxial tension behavior and fracture mode of 3D graphene foam by utilizing physical cross-links and van der Waals forces acting among different graphene flakes [28]. Honeycomb structures, inspired from bee honeycombs, have found excellent mechanical properties and widespread applications [29]. Krainyukova et al. [30] obtained exceptionally stable 3D carbon honeycomb (C-honeycomb) with high level of physical absorption of various gases by deposition of vacuum-sublimated graphite. The structural variation, 3
ACCEPTED MANUSCRIPT electronic band structure, elastic properties, and localized compression of the C-honeycomb were studied using density-functional theory (DFT) and MD simulations by Zhang et al. [31]. Pang et al. [32] showed superb specific strength of C-honeycomb by DFT calculations and
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high thermal conductivity by equilibrium MD simulation. Gu et al. [33] investigated the influence of junction structures on the mechanical and thermal properties of C-honeycomb further. Yi et al. [34] presented giant energy absorption capacity of C-honeycomb in all three
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directions. However, the above comprehensive studies mainly focus on 3D graphene
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honeycomb with sidewall size of about 5 Å to 25 Å, which is much less comparing with those constituting GA. Moreover, previous work described properties of C-honeycomb consisting of monolayer graphene sheet. For the real materials, multi-layer graphene sheets may exist [11, 23, 35]. Besides, the DFT calculation and full-atom MD simulation are also time-consuming.
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Although Qiu et al. [22] fabricated GA with a honeycomb structure, which exhibits a combination of ultralow density, superelasticity, good electrical conductivity and high efficiency of energy absorption, there is no model could simulate the realistic structure of
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honeycomb-like GA at present. The microscopic deformation mechanism of honeycomb-like
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GA has not been fully uncovered. The influence of the microstructure including the layer number and size of graphene sheet on mechanical behavior of honeycomb-like GA is unknown. What’s more, the hexagon holes do not usually run through the whole structure [22], which means stagger exists between honeycomb structures. In addition, some functional system may be designed by assembling honeycomb structures to specific shape. There is still a lack of study on the effect of staggered displacement. In this work, uniaxial compressive and recoverable behavior of GA with staggered 4
ACCEPTED MANUSCRIPT honeycomb structure of graphene sheets is investigated by employing the CGMD method [36] based on Tersoff potential [37, 38]. The method improves the computational efficiency and highly simplifies the modeling process of graphene as it avoids the necessity to define
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bonds/angles/dihedrals connectivity. Meanwhile, the model largely represents the realistic structure of honeycomb-like GA. Various layer numbers, sizes and arrangements of graphene sheets are considered. The microstructural evolution and stress distribution of GA are shown.
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Furthermore, the Poisson’s ratio as a function of compressive strain is calculated. The rest of
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this paper is organized as follows: Firstly, the CGMD method and the simulation model of honeycomb-like GA are introduced. Then, the uniaxial compressive and recoverable behavior of GA are reproduced. Subsequently, the influence of microstructures on compressive response are investigated, including the layer number and size of graphene sheet, as well as
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the staggered displacement of honeycomb structures. Finally, the Poisson’s ratio depending on the microstructures of GA and the applied strain is discussed. The conclusion and
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acknowledgements are given at the end of this paper.
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2. Model and methodology
The CG model of graphene adopts a 4-to-1 mapping scheme, in which four atoms are substituted by a single bead, as shown in Fig. 1(a). The hexagonal structure is maintained in the CG model and the distance between the adjacent beads is twice of the original atomic distance (Fig. 1(b)). Since the TersoffCG potential [36] is calculated according to the atomic Tersoff potential [37, 38], it inherits its three-body properties. The Tersoff potential has the form [38] 5
ACCEPTED MANUSCRIPT Vij ( r ) = fC ( rij ) fR ( rij ) + bij fA ( rij )
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where fR represents a repulsive pair potential, fA represents an attractive pair potential, and fC is merely the smooth cutoff function. bij represents a measure of the bond order. The functions
fR ( r ) = A exp ( −λ1r ) fA ( r ) = −B exp ( −λ2 r )
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of fR and fA are expressed as (2) (3)
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For TersoffCG potential, the coefficients relating to repulsive and attractive pair potentials (A
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and B) are obtained by multiplying the ones in Tersoff potential by four. The indexes (λ1 and λ2) of the repulsive and attractive pair potentials in TersoffCG potential are set to half of their original values. The cutoff parameters R and D are determined by fitting the simulation results of the CG model to that of the full-atom model, and the values of R = 4.1 Å and D = 0.6 Å are
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adopted. The Lennard-Jones 6-12 potential is used to describe the nonbonded interaction. The functional form of the Lennard-Jones 6-12 potential is given by: 12 6 Vnb = 4ε LJ (σ LJ / r ) − (σ LJ / r )
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where εLJ is the depth of the Lennard-Jones potential, and σLJ denotes the Lennard-Jones
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parameter associated with the equilibrium distance between two nonbonded beads. The following parameters are used: εLJ = 0.03559 eV and σLJ = 3.46 Å [39]. A detailed description of the TersoffCG potential model including the design of the functional form and the determination of parameters is referred to our previous publication [36]. Comparing to the bead-spring CGMD method [40-42] which are widely used to investigate properties of carbon materials [26, 27, 43-45], the TersoffCG MD method avoids the necessity to define bonds/angles/dihedrals since it remains the bond order feature from Tersoff potential. The 6
ACCEPTED MANUSCRIPT complex structure of GA can be simulated by only providing the coordinates of the beads. The flakes of graphene represented by beads are connected to form a honeycomb structure, of which a periodic structure is taken as the sub-cell of GA. The honeycomb model was
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established by converting coordinates of beads of one flake. The beads in the junction connect the corresponding neighboring graphene flakes and are shared by these flakes within the honeycomb structure. Two identical honeycomb sub-cells positioned vertically constitute the
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periodic unit cell of GA (Fig. 1(c)). The staggered arrangement of sub-cells is designed to
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reflect the realistic structure of GA, which is not an ideal honeycomb structure with through holes. Fig. 1(d) shows the repeated structure of the unit cell in Fig. 1(c) in three directions. The structure of the GA shown in Figs. 1(c, d) is made of single-layer graphene sheet. For all models, the armchair direction of the cell is placed along the x-axis, while the zigzag direction
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of the cell is placed along z-axis. Therefore, the direction perpendicular to the honeycomb plane is along y-axis. The size of the graphene sheet in GA is 98.24 × 65.23 Å2, where 98.24 Å describes sidewall length lx and 65.23 Å is the length of the sub-cell along y direction ly.
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Unless stated otherwise, the staggered displacement vector of two sub-cells is adopted as
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(0.25 lx, ly+2.84 Å, 0.25lz), where lz is the dimension of one sub-cell in z-axis directions. The microstructure of GA can vary by changing the layer number of graphene flakes, the size of graphene flakes and the staggered displacement vector of sub-cells. Firstly, the walls of the honeycomb structure consisting of one-, three- and five-layer graphene sheets are considered. GA consisting of tri- and five- layer graphene sheets is obtained by replacing the single-layer graphene sheet in Fig. 1(c) into the tri- and five-layer ones, respectively. Furthermore, the influence of sub-cell height, i.e. dimension along the y-axis, is investigated. 7
ACCEPTED MANUSCRIPT Models composed of tri-layer graphene sheets with different ratios of sub-cell height (ly) to sidewall length (lx) are established. The ratios are obtained by changing the value of the sub-cell height and keeping the sidewall length a constant value of 98.24 Å. Compressive
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behavior of GA with ratios of approximate 3: 10, 4: 10 and 6: 10 is discussed in detail. Finally, the staggered displacement vectors of sub-cells in GAs consisting of tri-layer graphene sheets are set to (0.375lx, ly+2.84 Å, 0.375lz), (lx, ly+2.84 Å, 0.25lz) and (0.75lx, ly+2.84 Å, 0.25lz),
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and models of GAs with different microstructural arrangements are obtained. The dependence
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of uniaxial compressive mechanical properties of the GAs on its microstructure is discussed in 3.2.
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ACCEPTED MANUSCRIPT Fig. 1. Schematics of graphene flake and GA. (a,b) CG mapping scheme of graphene. (c) Periodic cell of GA consisting of two sub-cells. (d) Geometry of periodic GA.
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The MD code LAMMPS (large-scale atomic/molecular massively parallel simulator) from Sandia Laboratories [46] is used to perform all simulations in this paper. Simulations of uniaxial compression of different GA models are carried out to investigate the relationship
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between the microstructure evolution and the behavior of the material. The interaction
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between carbon atoms in different layers of one sidewall is described by Lennard-Jones potential, while the other interaction is modelled by TersoffCG potential. The time-step is chosen to be 1 fs and temperature is set to 300 K by the Langevin thermostat. After minimization, the NPT ensemble (with constant number of particles, pressure and temperature)
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is adopted to describe the system with a periodic boundary condition in all directions and with a single barometric pressure controlled by a Berendsen barostat. When the system reaches an equilibrium state at about 1 ns, the ensemble is reset to NVE ensemble (the system has
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constant molarity, volume and energy) with all other conditions constant. The system
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undergoes equilibration process for another 2 ns before reaching final equilibrium state. After equilibration, the displacement-controlled compression in the y direction is performed by deforming the simulation box with a velocity of 0.1 m/s, corresponding to a strain rate of approximately 107 s-1. During the deformation, the pressure is kept constant at one bar in all directions except the loading direction.
3. Results and discussion 9
ACCEPTED MANUSCRIPT 3.1 Uniaxial compressive stress-strain relationship and compression-recovery behavior Uniaxial compression in the y direction is simulated using model of GA consisting of tri-layer graphene sheets, and the compressive stress-strain relation is obtained (shown in Fig.
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2(a)). Figs. 2(b-d) show the deformation and stress distribution of the model overall. Unlike the rubber-like stress-strain response found in uniaxial compression experiments [6, 22, 35, 47] and simulations [26], the typical compressive stress-strain relation can be divided into six
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stages. At stage 1, the stress increases sharply to a relatively stable value and the stress-strain
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curve presents a nearly linear elastic regime. In this stage, the strain is less than 0.01. It is found that the wall slightly bends, with no further microstructural change observed during stage 1. The stress is the highest at the junctions (Fig. 2(e)). At stage 2, a relatively flat stress plateau appears, and elastic buckling occurs at the walls connecting sub-cells (Figs. 2(f, g)),
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which can be taken as bulges. Due to buckling, the stress level decreases at the beginning of this stage. With the increase of the loading, the bulges gradually grow up and stress concentration can be observed. For other walls, rotation is the main way to adapt to the
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deformation, which bears less stress. During this stage, the walls with angles of 60° and 120°
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to x-axis become nearly parallel to z-axis. The stress occurs instability at stage 3 (Fig. 2(h)), which is caused by the connection of walls between adjacent cells, as shown in Figs. 2(l-o). Due to the application of TersoffCG potential, bond interaction can form as the distance between beads reaches the cutoff radius. The walls belonging to different cells are squeezed together and bond interaction yields. Then, the structure recovers to the stable status. At stage 4, another linear regime appears and the deformation mainly takes place at walls connecting different cells where new bond interaction forms. Consequently, all walls present bulges (Fig. 10
ACCEPTED MANUSCRIPT 2(i)), which have similar sizes at the end of stage 4. All bulges become slender shells from saddle at stage 5 (Fig. 2(j)), which results in a relatively stable stress. Due to the microstructural adjustment, the stress fluctuates. At stage 6, the stress increase manifestly
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because the slender shells collapsed. The GAs approach very high strain under compression. The dominant deformation of the GAs are bending and stretching of the graphene sheets. Meanwhile, the connected graphene sheets at the junctions have different bending directions,
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Fig. 2. Uniaxial compression behavior of GA composed of tri-layer graphene sheets in the y direction. (a) Stress-strain curve. The inset is the amplification of the curve in stage 1. (b-d)
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Stress distribution of periodic GA in y direction. (e-k) Stress distribution of local structure in (b) at typical points taken from stress-strain curve in (a). (l-n) Connecting process of adjacent cells corresponding to points taken from stress-strain curve in (a). (o) Position of the
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The uniaxial compression in the x and z directions are also conducted, and the stress-strain
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relation along x-axis is shown in Fig. 3. Unlike stress-strain response of compression in y direction, there is no linear elastic stage and unstable stage in x direction. The stress-strain
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curve can be divided into two stages, including the stress plateau stage and the densification stage. At stage 1, the stress keeps a constant of near 0 as the increase of the strain, indicating that the structure bears little load during the deformation. When the wall-wall distance is smaller than the equilibrium distance, the stress increases fast at stage 2. All beads are squeezed together and the structure of GA fully collapses, which leads the sharp increment of the stress . The results are different from those obtained by Yi et al. [34], which may be 12
ACCEPTED MANUSCRIPT caused by the different sizes of graphene sheets constituting honeycomb structure.
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Fig. 3. Uniaxial compressive stress-strain curve of GA composed of tri-layer graphene sheets in the x direction. The insets show stress distribution in x direction at different strain.
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The uniaxial compressive stress-strain relation along z-axis is also obtained, which is
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similar to that in x direction (Fig. S1). At the stress plateau stage, the structure bears little load, while the stress increases sharply at the densification stage. The compressive behaviors along armchair and zigzag directions are nearly the same, while it shows much difference in the direction perpendicular to the hexagon plane. The compressive behavior of GA in the y direction will be investigated further, since the mechanism of compression in the other two directions seems evident. Besides extraordinary supercompressibility, GA exhibits excellent resilient property after 13
ACCEPTED MANUSCRIPT large compressive strain [9, 19-22, 48]. Fig. 4 shows the compression-recovery behavior of GA consisting of three-layer graphene sheets along y-axis. The structure can recover to the initial status before the unstable stage. The walls with rotation recover first (Fig. 4(c)). Then,
sheets but is localized and has higher values at the junctions.
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the bulges disappear (Fig. 4(d)). Finally, the stress is distributed uniformly at the graphene During the process, there is
little energy dissipation, which suggests that stages 1 and 2 are both elastic. It should be noted
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that stage 2 is a relatively flat stress plateau as a whole, but the stress increases obviously in
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isolation. The stress-strain curve of stage 2 can be divided into slow linear stage, rapid linear stage and fluctuating stage. The bulges occur and grow up gradually at stage 2. The emergence of the bulges result in low stress increment, while the increment of stress becomes larger as the bulges grow. When the size of bulge comes to a certain value, its shape begins to
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change from a saddle to a slender shell, which yields fluctuation of stress. But the bulge still
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keeps a saddle-like shape at the end of stage 2 and it can be taken as a stable bulge stage.
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Fig. 4 Compression-recovery behavior of GA consisting of tri-layer graphene sheets in y-axis direction. The insets are stress distribution and deformation of the cell at typical points taken
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from the stress-strain curve.
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The compression-recovery behaviors of GA with tri-layer graphene sheets in x and z directions are also investigated, as shown in Figs. 5 and S2 respectively. The structure can
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recover to the initial state when the compressive strain is less than 0.33 along x-axis. It can be seen from Fig. 3 that stage 1 of compression along x-axis lasts a long strain span from 0 to 0.71, which is much larger than the recoverable strain. The reason of the phenomenon is that large strain in x direction causes bond interaction formation of beads between sub-cells (Fig. 5(d)), which inhibits the recovery process. However, the compression of almost the whole stage 1 can recover to the initial state in z direction because no bond interaction forms during 15
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Fig. 5 Compression-recovery behavior of GA consisting of tri-layer graphene sheets in x-axis direction. The insets are stress distribution and deformation of the cell at typical points taken
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from the stress-strain curve.
3.2 Influence of microstructures on compressive responses 3.2.1 Influence of layer number of graphene sheets The dependence of compressive properties in y direction of GAs on its microstructure is investigated. Firstly, the walls of the honeycomb structure consisting of one-, three- and 16
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The GAs
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honeycomb-structured networks, which are consisting of continuous graphene sheets. At the intersections of the networks, graphene junctions are formed under the governing of the
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three-body TersoffCG force field. Fig. 6 shows the microstructures of junctions consisting of graphene sheets with different layer numbers. For GAs made of single-layer graphene sheet, the junctions are composed of 6-bead rings (Fig. 6(c)). The atomistic model with the same
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assemble pattern and junction structure has also been proposed by Pang et al. [32]. Besides,
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for the GAs made of tri- and five-layer graphene sheets, there are two kinds of junctions. In the inner layers, two neighboring graphene sheets are connected by 6-bead rings, while in the outer layers, the triple junctions are connected by 8-bead rings. The 8-atom rings for connecting graphene sheets have also been proven by Pang et al. [32]. The size of the
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single-layer graphene sheet is 98.24 × 65.23 Å2. For GA consisting of multi-layer graphene sheets, the outer graphene layers are larger than the inner layers to make sure of their flat structure. For GA made of tri-layer graphene sheets, the size of the graphene sheet in the
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middle layer is 102.26 × 63.96 Å2, and 63.96 Å in the y direction. In addition to the
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mentioned two graphene sheet sizes, another with dimensions of 106.35 × 63.86 Å2 is included in the GA that consists of five-layer graphene sheets, with the length of 63.86 Å in the y direction. The dimension of the graphene sheet in direction perpendicular to the y-axis determines the interlayer distance of multilayer graphene sheet. However, it is not continuous due to the geometric construction of graphene. The selected dimensions result in an interlayer distance of approximately 3.4 Å.
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Fig. 6 Microstructure of junctions of graphene sheets. (a) single-layer graphene sheet, (b)
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tri-layer graphene sheets, (c) five-layer graphene sheets, (d) triple junction for GA consisting of single-layer graphene sheet, (e) junction of two graphene sheets for GA consisting of
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multi-layer graphene sheet, (f) triple junction for GA consisting of multi-layer graphene sheet.
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The compressive behavior of GAs consisting of one-, tri-, and five-layer graphene sheets are obtained as shown in Fig. 7. Similar to the stress-strain response of GA composed of tri-layer graphene sheets, the typical compressive stress-strain relation of model composed of monolayer graphene sheet can also be divided into six stages, while that of model consisting of five-layer graphene sheets can be taken as four stages. The first three stages are similar for all models, which are linear elastic stage, stage before instability and unstable stage. At stage 1, the linear elastic regime grows as the layer number of graphene sheets increases. It should 18
ACCEPTED MANUSCRIPT be noted that the linear relation of GA consisting of monolayer graphene is not as distinct as others, which is caused by the low stiffness of monolayer graphene. At stage 2, bulges occur and grow up at walls connecting sub-cells (Figs. 7(a, d)). The stress of model with five-layer
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graphene sheets increases most obviously due to the largest stiffness. The same phenomenon appears that the stress level decreases at the beginning of this stage because of the emergence of bulges. At the rear of stage 2, there is severe fluctuation in the stress-strain curve of GA
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with five-layer graphene sheets, which is caused by interlaminar dissociation at the junction
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(Fig. 7(e)). Stage 3 is the instability process. For models consisting of single- and tri-layer graphene sheets, there are another three stages, the second linear stage, the second flat stress plateau and densification stage. Bulges appears at all walls at stage 4, which results in the second linear stage. The shape change of bulges from saddle to slender shell make the stress
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become stable (Fig. 7(b)). At last, the slender shells are collapsed leading to sharply increase of stress (Fig. 7(c)). However, for model consisting of five-layer graphene sheets, there is only one stage after the unstable stage, where the interlaminar dissociation expands from the
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junction to walls and the stress increases rapidly (Fig. 7(f)).
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Fig. 7. Comparison of compressive stress-strain dependence in y–axis direction for GA composing of graphene network with different layers. The insets are stress distribution and
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deformation of local structure at typical points taken from stress-strain curves.
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It can be seen from Fig. 7 that the increment of layer number makes the stage 2 become slow stress growing stage from flat stress plateau. What’s more, the second stress plateau span
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decreases as the layer number rises. The stresses at the end of stage 2 (corresponding to strain of approximately 0.48) of the GAs consisting of single-, tri- and five-layer graphene sheets are 122.88 MPa, 475.10 MPa and 816.59 MPa, respectively. The stress of GA with tri-layer graphene sheets is 3.87 times of that of GA with single-layer graphene sheet, while the stress of GA with five-layer graphene sheets is 6.65 times of that of GA with single-layer graphene sheet. It can be seen that the enhancement of GAs is increased with the graphene layer, which 20
ACCEPTED MANUSCRIPT demonstrates that van der Waals interactions between graphene sheets also play an important role. The MD simulation results help explain the compressive deformation mechanism of the GAs and their property-structure relations from the viewpoints of micromechanics. However,
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the phenomenon can also be explained by continuum mechanics in a superficial way. For the GAs made of single-layer graphene sheet, the wall thickness of the network is very thin, resulting in easy bending of the walls that are subjected to the compressive load. So, its
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load-carry capability is also small. Then, for the GAs made of multi-layer graphene sheets, the
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bending resistance of the network walls are greatly increased. But, the role of the van der Waals interactions between stacked graphene sheets cannot be described by continuum mechanics.
3.2.2 Influence of dimension of graphene sheets
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Furthermore, the influence of sub-cell height on compressive stress-strain relation (Fig. 8) for models of GA that simulate tri-layer graphene sheets is also analyzed. The ratio of sub-cell height along y-direction (ly) and the sidewall length in the x-z plane (lx) is used to define the
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sub-cell structure. The ratios are obtained by changing the value of the sub-cell height and
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keeping the sidewall length a constant value of 98.24 Å. In order to investigate the influence of dimension of graphene sheets on compressive response of the proposed model, the authors adopted various ratios of approximate 2: 10, 3: 10, 4: 10, 5: 10, 6: 10, 8: 10 and 10: 10. Three types of compressive behavior are observed. For clarity, stress-strain curves of three ratios of approximate 3: 10, 4: 10 and 6: 10 are adopted to discuss in detail, which can represent the typical compressive behavior of GA, including ribbon-like behavior corresponding to ratio of 3: 10, near ribbon-like behavior corresponding to ratio of 4: 10 and typical behavior with 21
ACCEPTED MANUSCRIPT instability corresponding to ratio of 6: 10. The stress-strain curves of GA with ratios of 3: 10 and 4: 10 can be divided into three stages, the linear elastic stage, the flat stress plateau and the densification stage. However, their microstructure deformation differs greatly. For GA
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with ratio of 3: 10, the stress is much higher than that in other cases. Graphene sheets with low ratios behave similar to graphene ribbons, and no bulge occurs (Fig. 8(a)). During the compression, the walls rotate and compression applied perpendicular to the thickness
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direction is born by the slope of graphene sheets (Fig. 8(b)), which can be taken as structural
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instability. The walls are squeezed together and the structure becomes flat gradually (Fig. 8(c)). For GA with ratio of 4: 10, the compressive behavior exhibits characteristics of both GAs with ratios of 3: 10 and 6: 10. At the beginning of the compression, the structure is relatively stable and bulges appears at walls connecting sub-cells, which is the same with that
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of GA with ratio of 6:10. When the strain is near to 0.4, the deformation changes greatly. The walls rotate and behave like that of GA with ratio of 3:10 (Figs. 8(e-g)). Because of the rotation of walls, there is no formation of bond interaction between adjacent cells.
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Consequently, the stress-strain curves of GA with ratios of 3: 10 and 4: 10 show no sharp
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decrease. However, structural instability easily occurs for GA with low ratio. GA with high ratio has more room to accommodate the deformation of the entire structure, which can be controlled more easily.
22
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Stress (eV) -1e7
(c)
(c)
1e7
3:10
(b) (a)
6:10
z
1000
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Stress (MPa)
2000
4:10 (g)
x
(f)
(b) (a)
(f)
0
(e)
(d) (e)
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(d)
(g)
0.0
0.1
0.2
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-1000 0.3
0.4 Strain
0.5
0.6
0.7
Fig. 8. Uniaxial compressive stress-strain dependence in the y direction for GAs composed of tri-layer graphene sheets with different sizes. The insets are stress distribution and
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deformation of local structure at typical points taken from stress-strain curves.
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3.2.3 Influence of staggered displacement
Finally, the staggered displacement vectors of sub-cells in GA consisting of tri-layer
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graphene sheets are set to (0.375lx, ly+2.84 Å, 0.375lz), (lx, ly+2.84 Å, 0.25lz) and (0.75lx, ly+2.84 Å, 0.25lz), and models of GAs with different microstructural arrangement are obtained. The models of GA is represented by Patterns 1-4 when the displacement vector is (0.25lx, ly+2.84 Å, 0.25lz), (0.375lx, ly+2.84 Å, 0.375lz), (lx, ly+2.84 Å, 0.25lz) and (0.75lx, ly+2.84 Å, 0.25lz) respectively. As shown in Fig. 9, parts of the wall of different sub-cells in Patterns 1 and 2 are connected by bond interactions, while only the points of different sub-cells in 23
ACCEPTED MANUSCRIPT Patterns 3 and 4 are connected in that way. The connection present in Patterns 1 and 2 is defined as a line contact, while that of Patterns 3 and 4 as a point contact. The stress-strain curves of the 4 patterns are shown in Fig. 9, which can be divided into six stages like that in
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Fig. 2. The deformation processes of four patterns are similar. As detailed description of Fig. 2, the deformation is as follows. Saddle bulges occur and grow up gradually at walls connecting sub-cells. Then bond interaction forms between adjacent cells, resulting in severe decrease of
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stress. When the formation of bond interaction completes, bulges appears at all walls. All
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bulges are compressed from saddle shape to slender shells. At last, the slender shells are collapsed. However, the geometry configurations of the four patterns are different, as shown in Fig. 10. Patterns 1, 3 and 4 have the trend to expand along z-axis but contract along x-axis, while Pattern 2 has the opposite trend. That can be used to adjust the Poisson’s ratio. The
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stress of line contacts is larger than that of point contacts before the unstable stage since line contact is beneficial to stress transfer. Overall, there is little influence of different point contacts on compressive behavior, while line contacts play important role. The geometrical
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configuration of GA consisting of sub-cells connecting by points is more stable than that of
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GA with line contacts.
24
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Point contact
Line contact
2000 0.75lx
1000
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Pattern 3
Pattern 1
-1000
0.25lx Pattern 2
-2000
0.0
0.1
0.5lx Pattern 4
0.2
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0
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Stress (MPa)
0.5lx
0.3
0.4 Strain
0.5
Pattern 1 Pattern 2 Pattern 3 Pattern 4
0.6
0.7
Fig. 9 Uniaxial compressive stress-strain dependence in the y direction for GAs consisting of
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tri-layer graphene sheets in different patterns. The insets are four patterns of the models.
z
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x
(b) Pattern 2
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(a) Pattern 1
Stress (eV) 1e7 -1e7
(c) Pattern 3
(d) Pattern 4 25
ACCEPTED MANUSCRIPT Fig.10 The geometry configurations and stress distribution of GAs consisting of tri-layer graphene sheets in four patterns at the strain of 0.3.
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3.3 Poisson's ratio Fig. 11(a) shows the Poisson’s ratio along x and z axes as a function of compressive strain in the y-axis direction for GA consisting of tri-layer graphene sheets in Pattern 1. The
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Poisson’s ratio depends on the evolution of microstructure. At the beginning of the
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compression corresponding to stage 1 of stress-strain curve in Fig. 2(a), the Poisson’s ratios change dramatically. The Poisson’s ratio along the x-axis decreases sharply, and the values are almost negative, indicating contraction of GA in x-axis direction. Simultaneously, the Poisson’s ratio along the z-axis increases greatly, with the values all above 0, indicating
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expansion of GA. Then bulges are observed at walls with line contact between sub-cells, which makes the model contract in the x-axis direction and expand in the z-axis direction (Fig. 11(b)). As the bulges grow, the angle between the walls with line contact and x-axis becomes
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nearly 90° from 60°. And the walls with angle of 120° to x-axis also rotate near perpendicular
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to x-axis. As a result, the trend of contraction in x-axis direction and expansion in z-axis direction is induced. However, the walls parallel to x-axis occurs rotation leading to contraction in z-axis direction. Consequently, the dimensions of GA increase along x-axis and hold steady along z-axis, as shown in Fig. 11(c). With the increase of the loading, all walls yield bulges and the bulges are compressed from saddle shape to slender shell. The GA keeps contracting in x-axis direction and expanding in z-axis during this process (Fig. 11(d)).
26
ACCEPTED MANUSCRIPT 0.6
vxy
(a)
vzy
0.2 0.0 -0.2 -0.4 -0.6 -0.8 0.6
0.8
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0.4 Strain
(b) Strain = 0.2
(c) Strain = 0.4
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z x
0.2
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0.0
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Poisson's ratio
0.4
(d) Strain = 0.6
Fig. 11. (a) Poisson’s ratio of GA composed of tri-layer graphene sheets in Pattern 1 as a
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function of the compressive strain. (b-d) Comparison of undeformed configuration with
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deformed ones at strain of 0.2, 0.4 and 0.6, respectively.
Since microstructure of GA has important effect on the compressive behavior, the influence of layer number of graphene sheets consisting GA and the pattern on Poisson’s ratio is investigated. The relation between Poisson’s ratio along x-axis and the compressive strain along y-axis for GA composed of single-, tri- and five-layer graphene sheets in Pattern 1 is shown in Fig.12. It can be seen that, all the models mainly contract in x-axis direction except 27
ACCEPTED MANUSCRIPT the very beginning of the compression. The contraction can be observed in the insets of Fig. 12, which shows the comparison of the configurations at the initial status and the points taken from curves in Fig. 12. The fluctuation of Poisson’s ratio curves decreases as the layer number
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of graphene sheets increases. However, the layer number of graphene sheets has little effect on the tendency of the deformation. The deformation form can be changed by controlling the staggered displacement. Fig. 13 shows the dependence of Poisson’s ratio along x-axis on
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compressive strain in y-axis direction for GA composed of tri-layer graphene sheets in
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different Patterns, as well as the comparison of deformed configuration with undeformed ones. Contraction in x-axis direction is the main way for GA in Patterns 1, 3 and 4, while obvious expansion in x-axis can be observed for GA in Pattern 2. The patterns of GA can influence the stress transfer between sub-cells, which decides the location of the first bulge occurring. The
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bulges play an important role on the deformation of the whole structure.
28
ACCEPTED MANUSCRIPT (b)
(a)
0.8
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0.0 (b) (a)
-0.4
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Poisson's ratio
0.4
-1.2 0.0
0.2
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-0.8
0.4 Strain
0.6
Single-layer Tri-layer Five-layer 0.8
Fig. 12. Poisson’s ratio along x-axis of GAs composed of single-, tri- and five-layer graphene
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sheets in Pattern 1 as a function of the compressive strain. The insets are Comparison of undeformed configuration with deformed ones at strain of 0.4 for GAs consisting of (a)
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single-layer graphene sheet and (b) five-layer graphene sheets.
29
ACCEPTED MANUSCRIPT (a) (b)
(c)
-0.4 (d)
-0.8 0.2
(c)
0.6
0.8
(d)
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(b)
0.4 Strain
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0.0
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0.0
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Poisson's ratio
0.4
Pattern 1 Pattern 2 Pattern 3 Pattern 4
Fig. 13. Poisson’s ratio along x-axis of GAs composed of tri-layer graphene sheets in Patterns
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1-4 as a function of the compressive strain. (b-d) Comparison of undeformed configuration
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with deformed ones at strain of 0.4 for GAs in Patterns 2-4, respectively.
The Poisson’s ratios along z-axis for GA consisting of single-, tri- and five-layer graphene sheets in Pattern 1, as well as GA composed of tri-layer graphene sheets in Patterns 1-4 as functions of compressive strain in y-axis direction are also investigated, as shown in Figs. S3 and S4 in Supplementary materials. Contraction along z-axis is the main form of deformation for GA consisting of single- and five-layer graphene sheets in Pattern 1. Besides, GA 30
ACCEPTED MANUSCRIPT composed of tri-layer graphene sheets in Pattern 2 most contracts along z-axis. However, GAs in Patterns 3 and 4 expand along z-axis apparently. In general, negative Poisson’s ratio of GA consisting of staggered honeycomb-like structure within a certain range of compressive strain
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is found, and the Poisson’s ratio can be further tuned by designing the microstructure of the GA.
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4. Conclusion
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The CG model of GA with honeycomb-like structure and TersoffCG potential is established to investigate the microscopic uniaxial compressive behavior and resilience after compression release. Models with various microstructures are studied and the microscopic deformation mechanism is revealed. It is found that GA composed of multilayer graphene sheets possesses
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superior stress tolerance and stability. Furthermore, the effect of graphene sheet size on compressive behavior becomes remarkable in various ranges. When the size of the graphene sheet in the load direction is small, the stress during compression increases and the stability of
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the structure deteriorates. Uniform arrangement of a graphene sheet results in more stable
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properties, however, it may cause the stress to decrease. Furthermore, stress focusing at junctions and the buckling location has been observed during the compression process. It was found that the GA exhibits excellent resilience. Moreover, the Poisson’s ratio of the GA can be tuned by adjusting the microstructure parameters, such as the layer number and the arrangement of graphene. Ignoring the stress instability, the stress-strain curves calculated from the present model have good coherence with that from experimental data. The structural evolution during the cyclic deformation process was revealed from the MD simulation results, 31
ACCEPTED MANUSCRIPT as well as the concentration and release of the stress. However, since the defects on the graphene sheets are not considered, the stress calculated from the simulation results is higher than that from the experimental data. Besides, the size of the present simulation model is less
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than that of the actual materials. These results further contribute to the understanding of microscopic deformation mechanisms of GA and its design for various applications.
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Acknowledgements
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This work is supported by the National Natural Science Foundation of China under the Project Numbers 11502007, 11472020, 11632005 and 11602270, the Hong Kong Scholar Program under the Project Number XJ2016021, “Rixin Scientist” of Beijing University of Technology and Opening fund of State Key Laboratory of Nonlinear Mechanics, which are
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gratefully acknowledged.
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