- Email: [email protected]
strength and density
Materials and Design 160 (2018) 496–502
Contents lists available at ScienceDirect
Materials and Design journal homepage: www.elsevier.com/locate/matdes
A nanolattice-plate hybrid structure to achieve a nearly linear relation between stiffness/strength and density Zhigang Liu a, Ping Liu a,⁎, Wei Huang b, Wei Hin Wong a, Athanasius Louis Commillus a, Yong-Wei Zhang a a b
Institute of High Performance Computing, A*STAR, 138632, Singapore School of Aeronautics, Northwestern Polytechnical University, 710072, People's Republic of China
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
• Hybrid structures consisting of a spacefilling three-dimensional octet-truss alumina nanolattice and stretchingdominant plates are proposed in this study. • The structures are found to exhibit a nearly linear scaling of stiffness and strength with density when the hybrid structures fail via intrinsic material failure mechanism. • The stiffness, failure strength and failure mode of these hybrid structures can be tuned by changing their geometrical parameters. • This structure thus provide a much larger design space in terms of stiffness, strength and density than that of pure nanolattice structures.
a r t i c l e
i n f o
Article history: Received 7 June 2018 Received in revised form 19 September 2018 Accepted 22 September 2018 Available online 24 September 2018 Keywords: Nanolattice hybrid structure Stiffness Strength Failure Linear scaling
a b s t r a c t A great deal of effort has been made on the design and fabrication of materials or structures that simultaneously possess ultra-high stiffness, ultra-high strength, and yet ultra-low density. Here, using finite element simulations, we design hybrid structures comprising a space-filling nanolattice and stretching-dominated plates and study how the stiffness, failure strength and failure mode of such hybrid structures depend on the geometrical parameters of the nanolattice. It is found that the stiffness, failure strength and failure mode of these hybrid structures can be tuned by changing the geometrical parameters. In particular, we show that a nearly linear scaling can be achieved between the stiffness/failure strength and the density if intrinsic material failure occurs. Hence, such hybrid structures are able to expand the design space of ultra-light and strong materials for wide structural applications. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction
⁎ Corresponding author. E-mail address: [email protected] (P. Liu).
https://doi.org/10.1016/j.matdes.2018.09.039 0264-1275/© 2018 Elsevier Ltd. All rights reserved.
Design and fabrication of materials or structures that simultaneously possess ultra-high stiffness, ultra-high strength, and ultra-low density has attracted a great deal of attention due to their potential applications in acoustic insulation, vibration-damping, shock energy absorption,
Z. Liu et al. / Materials and Design 160 (2018) 496–502
thermal insulation, and also weight reduction in transport, construction and aerospace structures [1–9]. Many monolithic materials that exhibit an ultra-high strength and stiffness are not suitable for many important applications due to their high density [10–13]. It is well-known that the mechanical response of a material is closely correlated with its internal architecture and density. In general, the strength and stiffness of materials decrease rapidly with decreasing their density. Many natural materials with cellular structures, such as trabecular bone [14], plant parenchyma [15], and sponge [16], are found to simultaneously exhibit high stiffness, high strength and high toughness and yet low density. Inspired by these natural cellular structures, and made available by recent advances in fabrication of metamaterials with complicated geometries, various lightweight cellular materials have been designed and fabricated to achieve outstanding specific stiffness and strength (that is, high stiffness vs density ratio and high failure strength vs density ratio). Recently, a new class of metamaterials in the form of microlattices and nanolattices has been designed, fabricated and tested [17–19]. For example, Zheng et al. [20] fabricated microscale cellular materials and found that these materials are able to maintain a nearly constant stiffness per unit mass density even at an ultra-low density. Meza et al. [21] designed and created three-dimensional (3D) hierarchical nanolattices consisting of multiple self-similar unit cells spanning length scales over four orders of magnitude in fractal-like geometries, and showed that these nanolattices exhibit a unique combination of ultra-lightweight, deformation recoverability, and a near-linear scaling of stiffness and strength with density. Bauer et al. [22] created ultra-strong glassy carbon nanolattices with single strut length shorter than 1 μm and diameter as small as 200 nm, and found that their material strengths can be up to 3 GPa, and their strength-to-density ratio can be six times higher than those of previously reported microlattices [23,24]. Berger et al. [25] studied mechanical metamaterials with relatively simple cubic + octet foam geometry, and found that the resulting low-density metamaterials have many advantageous mechanical properties, arising from the ordered hierarchical structure. These studies have shown that structures with microlattice or nanolattice are of fascinating mechanical properties, and promising for many novel applications [26–31]. Previous studies have shown that the elastic stiffness and failure strength of a cellular material follow power laws with its density [32,33]. The power law exponents are closely related to the architecture and deformation mode of the materials. For pure stretching deformation, the exponent is 1; while for a pure bending deformation, the exponent is 3. In general, for a mixed mode, the exponent is in between 1 and 3 [34]. Hence how to design a cellular structure with stretching dominance is of great importance to achieve a metamaterial with high stiffness vs density and strength vs density ratios. Here, we designed hybrid structures consisting of a space-filling octahedral truss alumina nanolattice and stretching-dominated solid plates. Using finite element (FE) analysis, we obtained the stiffness, failure strength and failure mode of these hybrid structures under uniaxial tensile loading. By comparing their stiffness and strength with those of pure alumina nanolattice and pure alumina solid plate, we found that these hybrid structures show interesting scaling relations between the stiffness/strength and density. In particular, we found that such a combination of space-filling nanolattice and stretching-dominated solid sheets is able to achieve a nearly linear scaling relation of material strength and stiffness with density if intrinsic material failure occurs. 2. Structure model, material model, boundary and loading conditions Fig. 1 shows the CAD design of the hybrid structure (Fig. 1A) comprising a space-filling octahedral truss alumina (Al2O3) nanolattice sandwiched between two alumina plates, and the definitions of the geometrical parameters are also given. The unit cell (Fig. 1B) of the nanolattice consists of hollow tubes (Fig. 1C). We chose the octahedral
497
Fig. 1. (A) The CAD design and geometry of the hybrid structure comprising a space-filling octahedral truss alumina nanolattice sandwiched between two alumina plates. (B) The unit cell comprising hollow tubes and exhibiting a cubic symmetry with a nodal connectivity of 12. (C) The hollow tube. (D) Illustration of the connections between the nanolattice and the plate.
truss nanolattice because of its high predicted fracture toughness [35]. In connecting the nanolattice to the plates, we have made a cut to the nanolattice at a distance of tube wall thickness t from the outmost boundary of the nanolattice to create cross sections (see the shaded areas in Fig. 1D). These shaded areas are then bonded to the surfaces of the plates to form the connections. In the simulations, the length of the hybrid structure is kept at L = 151.04 μm, the width of the hybrid structure at W = 75.52 μm, the plate thickness at h1 = 1 μm and the radius of the tube at r = 491 nm. The length of unit cell l is changed from 2.36 μm to 37.76 μm and the tube wall thickness t from 1 nm to 120 nm. As a result, the lattice structure thickness h2 is changed from 4.72 μm to 37.76 μm, depending on the unit cell length and the number of cell layer n along the thickness of the hybrid structure. The thickness of the hybrid structure is H = 2 × h1 + h2, which is changed from 6.72 μm to 39.76 μm, again depending on the unit cell length and the number of cell layer n along the thickness of the hybrid structure. During uniaxial tensile loading, the bottom surface of the hybrid structure is fixed along Y direction and the middle point of the bottom surface is fixed along X, Y, and Z directions, a loading velocity of v = 500 mm/s along Y direction is prescribed at the top surface of the structure (Fig. 1A), while other boundary surfaces are traction-free. Considering the physical instability arising from material failure and numerical efficiency, we adopted explicit dynamics procedure to model the quasi-static uniaxial tension loading. To do so, a velocity-controlled loading condition was applied to model these structures. To ensure a quasi-static response, the energy balance of the modeling system was constantly monitored such that the kinetic energy
498
Z. Liu et al. / Materials and Design 160 (2018) 496–502
of the system was negligible (less than 1%) compared to the total elastic energy and external work. FE simulations on the alumina solid plate, the hollow octet-truss nanolattice, and their hybrid structures were performed to predict and compare the mechanical properties of these three structures. In the FE models, three-dimensional 4-noded rectangular shell elements with reduced-integration were employed for the plates and 2-noded beam elements were used for the tubes of the nanolattice. Computations were performed within the finite strain setting using the general-purpose finite element program ABAQUS/Explicit Version 6.14.2 [36]. The material properties of alumina Al2O3 were obtained from bulge experiments of Al2O3 thin films achieved by atomic layer deposition [37]. Specifically, the Young's modulus E is in the range of 164–165 GPa and the ultimate tensile strength (UTS) is in the range of 1.57–2.56 GPa. In the present work, E and UTS of alumina Al2O3 were taken as 165 GPa and 1.57 GPa, respectively. The constitutive behavior of this brittle material follows Hooke's law, and its damage can be considered as a macroscopic state variable that affects the macromechanical properties (stiffness degradation) of the material. Physically, the loss of stiffness can be considered a consequence of the propagation of distributed microcracks. In [36], a smeared crack model was used to represent the brittle behavior of Al2O3. This model does not track individual “macro” cracks. Instead, constitutive calculations were performed independently at each material point. The presence of cracks enters into these calculations by the way in which the cracks affect the stress and material stiffness associated with the material point. In the present work, a simple Rankine criterion is used to dictate the crack initiation [36]. This criterion states that a crack forms when the maximum principal tensile stress exceeds the tensile strength of the brittle material. Since the strain-at-failure for Al2O3 is typically in the range of 0.0077–0.0082 [35], the strain-at-failure of Al2O3 (defined as ε0) is taken as 0.008 in this study.
Previous studies [20–25] revealed that the main factors that influence the strength and stiffness of a nanolattice structure are the number of cell layer n along the thickness direction (or equivalently the thickness of the nanolattice h2 in Fig. 1), the tube wall thickness t and the length of unit cell l. Thus, in the present study, the effects of these three parameters on the stiffness, strength and failure mode of the hybrid structures were analyzed in details; while the radius of the tube r, the plate thickness h1, the length of the hybrid structure L and the width of the hybrid structure W were fixed. 3. Results and discussion Fig. 2 shows the stress vs strain curves for hybrid nanolattice structures at different values of n, t and l. In the stress-strain curves, the stress is obtained as the ratio of the sum of the reaction forces at the bottom surface to the area of the bottom surface, while the strain is obtained as the ratio of the length change of the hybrid structure ΔL to the initial length L. It is seen that all the samples initially deform linear-elastically and then fail instantaneously and catastrophically, as expected for a brittle material. The elastic modulus E and ultimate tensile strength UTS (defined as σ) of the hybrid nanolattice structures were collected from the stress-strain curves and shown in Fig. 3 to Fig. 5. Fig. 2A displays the stress vs strain curves of the hybrid structures at different numbers of cell layer n. In the calculations, the lattice structure thickness h2 is changed from 4.72 μm for n = 1 to 37.76 μm for n = 8, and thus the thickness of the hybrid structure H is changed from 6.72 μm for n = 1 to 39.76 μm for n = 8. The length of unit cell l is taken as 4.72 μm, and the tube wall thickness t as 50 nm. One can see that the slope of the stress vs strain curves shifts downwards rapidly and steadily as n increases. All the curves exhibit a brittle failure mode, consistent with the brittle nature of Al2O3. Fig. 3A and B shows the variation of elastic modulus E and ultimate tensile strength σ of the hybrid nanolattice structures with the number of cell layer n. It is seen that
Fig. 2. Plots of stress versus strain curves for (A) number of cell layer n, (B) tube wall thickness t and (C) length of unit cell l.
Z. Liu et al. / Materials and Design 160 (2018) 496–502
499
Fig. 3. Elastic modulus E and ultimate tensile strength σ versus number of cell layer n of the hybrid nanolattice structure. (A) Elastic modulus E against number of cell layer n. (B) Ultimate tensile strength σ against number of cell layer n. (C) Specific stiffness E/ρ against number of cell layer n. (D) Specific strength σ/ρ against number of cell layer n.
both the stiffness E and strength σ exhibit a monotonically decreasing trend as the number of cell layer n increases. When the number of cell layer continues to increase, the stiffness E and strength σ tend to level off and gradually saturate. Fig. 3C and D shows the variation of stiffness-to-density ratio (defined as specific stiffness E/ρ) and strength-to-density ratio (defined as specific strength σ/ρ) of hybrid structures with the number of cell layer n, where ρ is the density of the hybrid nanolattice structure, which is defined as the ratio of the total weight of the hybrid structure to the total volume of the hybrid structure. The total weight of the hybrid structure is the weight of the plates plus the weight of the nanolattice, while the total volume of the hybrid structure is L × W × H. It is seen that the specific stiffness E/ρ and specific strength σ/ρ also exhibit a monotonically decreasing trend as the number of cell layer n increases, similar to the stiffness E and strength σ. Fig. 2B displays the stress vs strain curves of the hybrid structures at different values of tube wall thickness t. In the calculations, the length of unit cell l is taken as 4.72 μm, the number of cell layer n as 2 and the lattice structure thickness h2 as 9.44 μm, so the thickness of the hybrid structure H is 11.44 μm. It is seen that these curves only change slightly and progressively as t increases from 1 nm to 120 nm, which is in contrast to the previous study that the thickness of nanolattice tube showed a significant influence on the stiffness and strength of pure lattice structure [38]. Fig. 4A and B depicts the variation of stiffness E and strength σ of the hybrid nanolattice structures with the tube wall thickness t. It is seen that the stiffness E shows a monotonic increase as t increases, while the strength σ exhibits two stages of increase as t increases. The transition of these two stags occurs at t ≈ 5 nm. Fig. 4C and D shows correspondingly the variation of the specific stiffness E/ρ and specific strength σ/ρ of the hybrid structure with the tube wall thickness t. It is seen that the specific stiffness E/ρ decreases significantly as t increases, while the specific strength σ/ρ initially increases, peaks at t
≈ 5 nm and then decreases, suggesting a change in failure mechanism with the variation of the tube wall thickness t. Our detailed analysis showed that when the value of t is small, Euler buckling failure prevails; while when t is larger than a critical value, the intrinsic material failure prevails. The critical strength-to-density ratio σ/ρ for the transition occurs at t ≈ 5 nm. To better understand the failure transition, we use an Euler slender column to analyze the buckling behavior of the hollow tubes. The Euler buckling criterion can be written as,
F¼
π2 Ec Ic ðKLc Þ2
ð1Þ
where F is the maximum or critical force (vertical load on column), Ec is the elastic modulus of the material, Ic is the smallest area moment of inertia of the column, Lc is the length of the column, and K is a parameter, whose value depends on the condition of the end support to the column. For example, for both ends pinned (hinged, free to rotate), K = 1.0, and for one end fixed and the other end free to move laterally, K = 2.0. For the tube with a hollow cylindrical section, Ic ¼ π4 ðr 4o −r4i Þ, where r o ¼ r þ 2t is the outer radius and r i ¼ r− 2t is the inner radius. Since the area of the cross section is S = 2πrt, so the stress σc = Ecε0 = F/S. From Eq. (1), the critical thickness of the tube is t ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ε0 ðKLc Þ2 −r 2 . In the present work, r = 491 nm, Lc = 6.675 μm, ε0 = π2
0.008. We find that when K = 1.83, t ≈ 5 nm. The high value of K suggests that the support from the nodes to the column is not strong. More specifically, it is clearly weaker than that with both ends pinned (hinged, free to rotate), but slightly stronger than that with one end fixed and the other end free to move laterally.
500
Z. Liu et al. / Materials and Design 160 (2018) 496–502
Fig. 4. Elastic modulus E and ultimate tensile strength σ versus tube wall thickness t of the hybrid nanolattice structure. (A) Elastic modulus E against the tube wall thickness t. (B) Ultimate tensile strength σ against the tube wall thickness t. (C) Specific stiffness E/ρ against the tube wall thickness t. (D) Specific strength σ/ρ against the tube wall thickness t.
It is noted that similar phenomena have also been reported in other stretching-dominated lattice structures [20]. The presence of maximal value of σ/ρ with the variation of tube wall thickness (see Fig. 4D) due to the transition from a buckling-dominated failure mode to a fracture-dominated failure mode allows us to control the failure mode of hybrid Al2O3 nanolattice structures by tuning their tube wall thickness. Fig. 2C displays stress vs strain curves of the hybrid structures at various values of length of the unit cell l. In the simulations, the lattice structure thickness h2 is fixed at 37.76 μm, the thickness of the hybrid structure H at 39.76 μm, and the tube wall thickness at t = 50 nm. The number of cell layers n along the thickness of the hybrid structure is changed by varying the length of the unit cell l from n = 1 for l = 37.76 μm, n = 2 for l = 18.88 μm, n = 4 for l = 9.44 μm, n = 8 for l = 4.72 μm, to n = 16 for l = 2.36 μm. It is seen that the slope of the stress vs strain curves initially shifts upwards slowly and then drastically as l decreases. Fig. 5A and B shows the variation of stiffness E and strength σ of the hybrid structures with the length of the unit cell l. It is seen that the stiffness E and strength σ initially decrease drastically and then gradually level off monotonically as l increases. Fig. 5C and D shows correspondingly the variation of specific stiffness E/ρ and specific strength σ/ρ of the hybrid structures with the length of the unit cell l. The specific stiffness E/ρ and specific strength σ/ρ exhibit a similar trend with l, i.e., E/ρ and σ/ρ increase rapidly initially and then level off as l increases. The turning point for E/ρ and σ/ρ occurs at l ≈ 20 μm, after which the stiffness-to-density and strength-to-density remain approximately unchanged. Our simulations showed that all the hybrid structures failed catastrophically, regardless of the failure mechanism, that is, structural buckling or intrinsic material failure. Fig. 6 shows the onset and propagation of failure. One can see clearly that the failure initiates at the nodal connections between the nanolattice and the plate (Fig. 6), and then
propagates into neighboring lattice cells near the plates, suggesting that the nanolattice cell is weaker than the plate. This is not surprising since the stiffness and strength of the nanolattice are far less than that of the Al2O3 plate. The fact that the failure starts at the nodal connections between the nanolattice and the plates suggests that the nodal connections are the weakest link. For all the samples that fail via intrinsic material failure, the strain at failure is at approximately 0.008, an intrinsic failure property of this brittle material. The double logarithmic relation between Young's modulus E and density ρ, that is, log(E) vs log(ρ), and that between the failure strength σ vs the density ρ, that is, log(σ) vs log(ρ), of the hybrid structures, together with the pure Al2O3 plate and pure Al2O3 nanolattice structures, are summarized in Fig. 7A and B, respectively. For comparison, recently reported data for log(E) vs log(ρ) of other lattice materials are also shown in Fig. 7A, including hollow Al2O3 stretching-dominated lattices [20], ultralight metallic microlattices [26], and graphene elastomers [39]. It is seen interestingly that for the hollow Al2O3 nanolattice structures, the relation between log(E) and log(ρ) follows approximately a linear relation with a slope of ~1, indicating that these structures are stretching-dominated ones. For the graphene elastomers, the relation between log(E) and log(ρ) also follows approximately a linear relation with a slope of ~3, suggesting that these structures are dominated by the bending deformation. For these ultralight metallic microlattice structures, the relation between log(E) and log(ρ) also follows approximately a linear relation, however, with a slope of ~2, signifying that these structures are in a mixed stretching and bending deformation. From our computed Young's moduli of the plate, pure nanolattices, and hybrid structures (see Fig. 7A), it is seen interestingly that the slope for the relation between log(E) and log(ρ) from the plate to the pure nanolattice is ~3, signifying a drastic reduction of Young's modulus with density. However, the slope for the relation between log(E) and log (ρ) from the plate to the hybrid structures that fail via intrinsic mode is
Z. Liu et al. / Materials and Design 160 (2018) 496–502
501
Fig. 5. Elastic modulus E and ultimate tensile strength σ versus length of unit cell l. (A) Elastic modulus E against length of unit cell l. (B) Ultimate tensile strength σ against length of unit cell l. (C) Specific stiffness E/ρ against length of unit cell l. (D) Specific strength σ/ρ against length of unit cell l.
~1, suggesting the slowest reduction of Young's modulus with density. However, for those hybrid structures that fail via buckling mode, the slope is ~3, consistent with the mechanism transition from intrinsic material failure to buckling failure. Similarly, from our computed failure strengths of the plate, pure nanolattices, and hybrid structures (see Fig. 7B), it is also seen that the slope for the relation between log(σ) and log(ρ) from the plate to the pure nanolattice is ~3, signifying a drastic reduction of failure strength with density. However, the slope for the relation between log(σ) and log(ρ) from the plate to the hybrid structures with intrinsic material failure is ~1, suggesting the slowest reduction of strength with density. Interestingly, the slope becomes ~3 when the hybrid structures fail via buckling mode. These findings demonstrate that if the hybrid structures are designed in the stretchingdominated deformation mode (via intrinsic material failure), we are able to efficiently utilize materials to achieve high specific performances. 4. Conclusion
Fig. 6. Finite element simulations show that the failure starts from the nodal junction between the nanolattice and plate sheets. Then the failure propagates into the lattice cells next to the plate sheet. The lattice failure is prior to the plate failure. (A) Before failure, ε = 0.0076. (B) Failure at the nodal connection between the plate and nanolattice, ε = 0.0078. (C) Failure propagation along the cells near the plates, ε = 0.0080.
We designed a hybrid structure consisting of a space-filling 3dimensional octet-truss alumina nanolattice and stretching-dominant plates and studied the effect of various parameters on the stiffness and strength of these hybrid structures under uniaxial tensile loading by using finite element simulations. Our simulations showed that depending on the choice of geometrical parameters of the hybrid structures, the strength and the failure mode can be tuned. In particular, the hybrid structures exhibit a nearly linear scaling of stiffness and strength with density when the hybrid structures fail via intrinsic material failure mechanism, which is far more superior than those of the pure nanolattice structures. Furthermore, if we are allowed to change the type and material of space-filling nanolattice and plate, the stiffness
502
Z. Liu et al. / Materials and Design 160 (2018) 496–502
Fig. 7. Ashby charts for (A) the stiffness versus density and (B) the failure strength versus density of the plate, pure lattices and hybrid structures. For the stiffness versus density, three other previously reported structures are also included for comparison.
and failure strength of the hybrid structures can be further tuned and expanded. Since the proposed hybrid structures are able to provide a much larger design space in terms of stiffness, strength and density than pure nanolattice structures, it is expected that these hybrid structures might find a wider range of potential applications in precision engineering structures. Credit authorship contribution statement Liu Ping: Initiated the idea and designed the study, analyzed the simulation results. Liu Zhigang: Did the simulations and analyzed the simulation results. Zhang Yong Wei: Analyzed the simulation results. All of the authors wrote, proofread and revised the manuscript. Acknowledgment The support of this work by the Institute of High Performance Computing, A*STAR is gratefully acknowledged. References [1] D. Jang, L.R. Meza, F. Greer, J.R. Greer, Fabrication and deformation of threedimensional hollow ceramic nanostructures, Nat. Mater. 12 (2013) 893–898. [2] J.R. Greer, J.T.M. De Hosson, Plasticity in small-sized metallic systems: intrinsic versus extrinsic size effect, Prog. Mater. Sci. 56 (2011) 654–724.
[3] H. Ebrahimi, D. Mousanezhad, H. Nayeb-Hashemi, J. Norato, A. Vaziri, 3D cellular metamaterials with planar anti-chiral topology, Mater. Des. 145( (2018) 226–231. [4] B.D. Nguyen, J.S. Cho, K. Kang, Optimal design of “shellular”, a micro-architectured material with ultralow density, Mater. Des. 95 (2016) 490–500. [5] H. Ebrahimi, D. Mousanezhad, B. Haghpanah, R. Ghosh, A. Vaziri, Lattice materials with reversible foldability, Adv. Eng. Mater. 19 (2017) 1600646-n/a. [6] B. Haghpanah, L. Salari-Sharif, P. Pourrajab, J. Hopkins, L. Valdevit, Multistable shape-reconfigurable architected materials, Adv. Mater. 28 (2016) 7915–7920. [7] W. Franziska, O. Fuad, A. Lucas, A.L. Matthias, K. Carolin, Fabrication and characterization of a fully auxetic 3D lattice structure via selective electron beam melting, Smart Mater. Struct. 26 (2017), 025013. . [8] X.-T. Wang, B. Wang, X.-W. Li, L. Ma, Mechanical properties of 3D re-entrant auxetic cellular structures, Int. J. Mech. Sci. 131–132 (2017) 396–407. [9] J. Bauer, L.R. Meza, T.A. Schaedler, R. Schwaiger, X. Zheng, L. Valdevit, Nanolattices: an emerging class of mechanical metamaterials, Adv. Mater. 29 (2017) 1701850n/a. [10] L.R. Meza, S. Das, J.R. Greer, Strong, lightweight, and recoverable three-dimensional ceramic nanolattices, Science 335 (2014) 1322–1326. [11] H.M.A. Kolken, A.A. Zadpoor, Auxetic mechanical metamaterials, RSC Adv. 7 (2017) 5111–5129. [12] X. Li, Z. Lu, Z. Yang, C. Yang, Directions dependence of the elastic properties of a 3D augmented re-entrant cellular structure, Mater. Des. 134 (2017) 151–162. [13] Z. Lu, Q. Wang, X. Li, Z. Yang, Elastic properties of two novel auxetic 3D cellular structures, Int. J. Solids Struct. 124 (2017) 46–56. [14] T.M. Ryan, C.N. Shaw, Trabecular bone microstructure scales allometrically in the primate humerus and femur, Proc. R. Soc. B 280 (2013) 20130172. [15] P. Van Liedekerke, P. Ghysels, E. Tijskens, G. Samaey, B. Smeedts, D. Roose, H. Ramon, A particle-based model to simulate the micromechanics of single-plant parenchyma cells and aggregates, Phys. Biol. 7 (2010), 026006. . [16] J.H. Shen, Y.M. Xie, X.D. Huang, S.W. Zhou, D. Ruan, Compressive behavior of luffa sponge material at high strain rate, Key Eng. Mater. 535–536 (2013) 465–468. [17] X. W. Gu, C. N. Loynachan, Z. X. Wu, Y.W. Zhang, D. J. Srolovitz, J. R. Greer, Sizedependent deformation of nanocrystalline Pt nanopillars, Nano Lett., 12(2013) 6385–6392. [18] D.Z. Chen, D. Jang, K.M. Guan, Q. An, W.A. Goddard III, J.R. Greer, Nanometallic glasses: size reduction brings ductility, surface state drives its extent, Nano Lett. 13 (2013) 4462–4468. [19] S.W. Lee, M. Jafary-Zadeh, D.Z. Chen, Y.W. Zhang, J.R. Greer, Size effect suppresses brittle failure in hollow Cu60Zr40 metallic glass nanolattices deformed at cryogenic temperature, Nano Lett. 15 (2015) 5673–5681. [20] X. Zheng, H. Lee, T.H. Weisgraber, M. Shusteff, J. DeOtte, E.B. Duoss, J.D. Kuntz, M.M. Biener, Q. Ge, J.A. Jackson, S.O. Kucheyev, N.X. Fang, C.M. Spadaccini, Ultralight, ultrastiff mechanical metamaterials, Science 344 (2014) 1373–1377. [21] L.R. Meza, A.J. Zelhofer, N. Clarke, A.J. Mateos, D.M. Kochmann, J.R. Greer, Resilient 3D hierarchical architected metamaterials, PNAS 112 (2015) 11502–11507. [22] J. Bauer, A. Schroer, R. Schwaiger, O. Kraft, Approaching theoretical strength in glassy carbon nanolattices, Nat. Mater. 15 (2016) 438–443. [23] J. Bauer, S. Hengsbach, I. Tesari, R. Schwaiger, O. Kraft, High-strength cellular ceramic composites with 3D microarchitecture, PNAS 111 (2014) 2453–2458. [24] L.R. Meza, S. Das, J.R. Greer, Strong, lightweight, and recoverable three-dimensional ceramic nanolattices, Science 345 (2014) 1322–1326. [25] J.B. Berger, H.N.G. Wadley, R.M. McMeeking, Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness, Nature 543 (2017) 533–537. [26] T.A. Schaedler, A.J. Jacobsen, A. Torrents, A.E. Sorensen, J. Lian, J.R. Greer, L. Valdevit, W.B. Carter, Ultralight metallic microlattices, Science 334 (2011) 962–965. [27] L.C. Montemayor, L.R. Meza, J.R. Greer, Design and fabrication of hollow rigid nanolattices via two-photon lithography, Adv. Eng. Mater. 16 (2014) 184–189. [28] A.J. Jacobsen, W.B. Carter, S. Nutt, Micro-scale Truss structures formed from selfpropagating photopolymer waveguides, Adv. Mater. 19 (2007) 3892–3896. [29] X.W. Gu, J.R. Greer, Ultra-strong architected Cu meso-lattices, Extreme Mech. Lett. 2 (2015) 7–14. [30] J. Christensen, M. Kadic, M. Wegener, O. Kraft, Vibrant times for mechanical metamaterials, MRS Commun. 5 (2015) 453–462. [31] L. Salari-Sharif, T.A. Schaedler, L. Valdevit, Energy dissipation mechanisms in hollow metallic microlattices, J. Mater. Res. 29 (2014) 1755–1770. [32] K.R. Mangipudi, S.W. Van Buuren, P.R. Onck, The microstructural origin of strain hardening in two-dimensional open-cell metal foams, Int. J. Solids Struct. 47 (2010) 2081–2096. [33] Y. Conde, R. Doglione, A. Mortensen, Influence of microstructural heterogeneity on the scaling between flow stress and relative density in microcellular Al-4.5% Cu, J. Mater. Sci. 49 (2014) 2403–2414. [34] V. Marcadon, C. Davoine, B. Passilly, D. Boivin, F. Popoff, A. Rafray, S. Kruch, Mechanical behaviour of hollow-tube stackings: experimental characterization and modelling of the role of their constitutive material behaviour, Acta Mater. 60 (2012) 5626–5644. [35] N.A. Fleck, V.S. Deshpande, M.F. Ashby, Micro-architectured materials: past, present and future, Proc. R. Soc. A Math. Phys. Eng. Sci. 466 (2010) 2495–2516. [36] ABAQUS Version 6.14, User's Manual Version 6.14, 2014 Providence, RI, USA. [37] M. Berdova, T. Ylitalo, I. Kassamakov, J. Heino, P.T. Törmä, L. Kilpi, H. Ronkainen, J. Koskinen, E. Hæggström, S. Franssila, Mechanical assessment of suspended ALD thin films by bulge and shaft-loading techniques, Acta Mater. 66 (2014) 370–377. [38] L.C. Montemayor, W.H. Wong, Y.W. Zhang, J.R. Greer, Insensitivity to flaws leads to damage tolerance in brittle architected meta-materials, Sci. Rep. 6 (2016), 20570. . [39] L. Qiu, J.Z. Liu, S.L.Y. Chang, Y. Wu, D. Li, Biomimetic superelastic graphene-based cellular monoliths, Nat. Commun. 3 (2012) 1241.
Contents lists available at ScienceDirect
Materials and Design journal homepage: www.elsevier.com/locate/matdes
A nanolattice-plate hybrid structure to achieve a nearly linear relation between stiffness/strength and density Zhigang Liu a, Ping Liu a,⁎, Wei Huang b, Wei Hin Wong a, Athanasius Louis Commillus a, Yong-Wei Zhang a a b
Institute of High Performance Computing, A*STAR, 138632, Singapore School of Aeronautics, Northwestern Polytechnical University, 710072, People's Republic of China
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
• Hybrid structures consisting of a spacefilling three-dimensional octet-truss alumina nanolattice and stretchingdominant plates are proposed in this study. • The structures are found to exhibit a nearly linear scaling of stiffness and strength with density when the hybrid structures fail via intrinsic material failure mechanism. • The stiffness, failure strength and failure mode of these hybrid structures can be tuned by changing their geometrical parameters. • This structure thus provide a much larger design space in terms of stiffness, strength and density than that of pure nanolattice structures.
a r t i c l e
i n f o
Article history: Received 7 June 2018 Received in revised form 19 September 2018 Accepted 22 September 2018 Available online 24 September 2018 Keywords: Nanolattice hybrid structure Stiffness Strength Failure Linear scaling
a b s t r a c t A great deal of effort has been made on the design and fabrication of materials or structures that simultaneously possess ultra-high stiffness, ultra-high strength, and yet ultra-low density. Here, using finite element simulations, we design hybrid structures comprising a space-filling nanolattice and stretching-dominated plates and study how the stiffness, failure strength and failure mode of such hybrid structures depend on the geometrical parameters of the nanolattice. It is found that the stiffness, failure strength and failure mode of these hybrid structures can be tuned by changing the geometrical parameters. In particular, we show that a nearly linear scaling can be achieved between the stiffness/failure strength and the density if intrinsic material failure occurs. Hence, such hybrid structures are able to expand the design space of ultra-light and strong materials for wide structural applications. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction
⁎ Corresponding author. E-mail address: [email protected] (P. Liu).
https://doi.org/10.1016/j.matdes.2018.09.039 0264-1275/© 2018 Elsevier Ltd. All rights reserved.
Design and fabrication of materials or structures that simultaneously possess ultra-high stiffness, ultra-high strength, and ultra-low density has attracted a great deal of attention due to their potential applications in acoustic insulation, vibration-damping, shock energy absorption,
Z. Liu et al. / Materials and Design 160 (2018) 496–502
thermal insulation, and also weight reduction in transport, construction and aerospace structures [1–9]. Many monolithic materials that exhibit an ultra-high strength and stiffness are not suitable for many important applications due to their high density [10–13]. It is well-known that the mechanical response of a material is closely correlated with its internal architecture and density. In general, the strength and stiffness of materials decrease rapidly with decreasing their density. Many natural materials with cellular structures, such as trabecular bone [14], plant parenchyma [15], and sponge [16], are found to simultaneously exhibit high stiffness, high strength and high toughness and yet low density. Inspired by these natural cellular structures, and made available by recent advances in fabrication of metamaterials with complicated geometries, various lightweight cellular materials have been designed and fabricated to achieve outstanding specific stiffness and strength (that is, high stiffness vs density ratio and high failure strength vs density ratio). Recently, a new class of metamaterials in the form of microlattices and nanolattices has been designed, fabricated and tested [17–19]. For example, Zheng et al. [20] fabricated microscale cellular materials and found that these materials are able to maintain a nearly constant stiffness per unit mass density even at an ultra-low density. Meza et al. [21] designed and created three-dimensional (3D) hierarchical nanolattices consisting of multiple self-similar unit cells spanning length scales over four orders of magnitude in fractal-like geometries, and showed that these nanolattices exhibit a unique combination of ultra-lightweight, deformation recoverability, and a near-linear scaling of stiffness and strength with density. Bauer et al. [22] created ultra-strong glassy carbon nanolattices with single strut length shorter than 1 μm and diameter as small as 200 nm, and found that their material strengths can be up to 3 GPa, and their strength-to-density ratio can be six times higher than those of previously reported microlattices [23,24]. Berger et al. [25] studied mechanical metamaterials with relatively simple cubic + octet foam geometry, and found that the resulting low-density metamaterials have many advantageous mechanical properties, arising from the ordered hierarchical structure. These studies have shown that structures with microlattice or nanolattice are of fascinating mechanical properties, and promising for many novel applications [26–31]. Previous studies have shown that the elastic stiffness and failure strength of a cellular material follow power laws with its density [32,33]. The power law exponents are closely related to the architecture and deformation mode of the materials. For pure stretching deformation, the exponent is 1; while for a pure bending deformation, the exponent is 3. In general, for a mixed mode, the exponent is in between 1 and 3 [34]. Hence how to design a cellular structure with stretching dominance is of great importance to achieve a metamaterial with high stiffness vs density and strength vs density ratios. Here, we designed hybrid structures consisting of a space-filling octahedral truss alumina nanolattice and stretching-dominated solid plates. Using finite element (FE) analysis, we obtained the stiffness, failure strength and failure mode of these hybrid structures under uniaxial tensile loading. By comparing their stiffness and strength with those of pure alumina nanolattice and pure alumina solid plate, we found that these hybrid structures show interesting scaling relations between the stiffness/strength and density. In particular, we found that such a combination of space-filling nanolattice and stretching-dominated solid sheets is able to achieve a nearly linear scaling relation of material strength and stiffness with density if intrinsic material failure occurs. 2. Structure model, material model, boundary and loading conditions Fig. 1 shows the CAD design of the hybrid structure (Fig. 1A) comprising a space-filling octahedral truss alumina (Al2O3) nanolattice sandwiched between two alumina plates, and the definitions of the geometrical parameters are also given. The unit cell (Fig. 1B) of the nanolattice consists of hollow tubes (Fig. 1C). We chose the octahedral
497
Fig. 1. (A) The CAD design and geometry of the hybrid structure comprising a space-filling octahedral truss alumina nanolattice sandwiched between two alumina plates. (B) The unit cell comprising hollow tubes and exhibiting a cubic symmetry with a nodal connectivity of 12. (C) The hollow tube. (D) Illustration of the connections between the nanolattice and the plate.
truss nanolattice because of its high predicted fracture toughness [35]. In connecting the nanolattice to the plates, we have made a cut to the nanolattice at a distance of tube wall thickness t from the outmost boundary of the nanolattice to create cross sections (see the shaded areas in Fig. 1D). These shaded areas are then bonded to the surfaces of the plates to form the connections. In the simulations, the length of the hybrid structure is kept at L = 151.04 μm, the width of the hybrid structure at W = 75.52 μm, the plate thickness at h1 = 1 μm and the radius of the tube at r = 491 nm. The length of unit cell l is changed from 2.36 μm to 37.76 μm and the tube wall thickness t from 1 nm to 120 nm. As a result, the lattice structure thickness h2 is changed from 4.72 μm to 37.76 μm, depending on the unit cell length and the number of cell layer n along the thickness of the hybrid structure. The thickness of the hybrid structure is H = 2 × h1 + h2, which is changed from 6.72 μm to 39.76 μm, again depending on the unit cell length and the number of cell layer n along the thickness of the hybrid structure. During uniaxial tensile loading, the bottom surface of the hybrid structure is fixed along Y direction and the middle point of the bottom surface is fixed along X, Y, and Z directions, a loading velocity of v = 500 mm/s along Y direction is prescribed at the top surface of the structure (Fig. 1A), while other boundary surfaces are traction-free. Considering the physical instability arising from material failure and numerical efficiency, we adopted explicit dynamics procedure to model the quasi-static uniaxial tension loading. To do so, a velocity-controlled loading condition was applied to model these structures. To ensure a quasi-static response, the energy balance of the modeling system was constantly monitored such that the kinetic energy
498
Z. Liu et al. / Materials and Design 160 (2018) 496–502
of the system was negligible (less than 1%) compared to the total elastic energy and external work. FE simulations on the alumina solid plate, the hollow octet-truss nanolattice, and their hybrid structures were performed to predict and compare the mechanical properties of these three structures. In the FE models, three-dimensional 4-noded rectangular shell elements with reduced-integration were employed for the plates and 2-noded beam elements were used for the tubes of the nanolattice. Computations were performed within the finite strain setting using the general-purpose finite element program ABAQUS/Explicit Version 6.14.2 [36]. The material properties of alumina Al2O3 were obtained from bulge experiments of Al2O3 thin films achieved by atomic layer deposition [37]. Specifically, the Young's modulus E is in the range of 164–165 GPa and the ultimate tensile strength (UTS) is in the range of 1.57–2.56 GPa. In the present work, E and UTS of alumina Al2O3 were taken as 165 GPa and 1.57 GPa, respectively. The constitutive behavior of this brittle material follows Hooke's law, and its damage can be considered as a macroscopic state variable that affects the macromechanical properties (stiffness degradation) of the material. Physically, the loss of stiffness can be considered a consequence of the propagation of distributed microcracks. In [36], a smeared crack model was used to represent the brittle behavior of Al2O3. This model does not track individual “macro” cracks. Instead, constitutive calculations were performed independently at each material point. The presence of cracks enters into these calculations by the way in which the cracks affect the stress and material stiffness associated with the material point. In the present work, a simple Rankine criterion is used to dictate the crack initiation [36]. This criterion states that a crack forms when the maximum principal tensile stress exceeds the tensile strength of the brittle material. Since the strain-at-failure for Al2O3 is typically in the range of 0.0077–0.0082 [35], the strain-at-failure of Al2O3 (defined as ε0) is taken as 0.008 in this study.
Previous studies [20–25] revealed that the main factors that influence the strength and stiffness of a nanolattice structure are the number of cell layer n along the thickness direction (or equivalently the thickness of the nanolattice h2 in Fig. 1), the tube wall thickness t and the length of unit cell l. Thus, in the present study, the effects of these three parameters on the stiffness, strength and failure mode of the hybrid structures were analyzed in details; while the radius of the tube r, the plate thickness h1, the length of the hybrid structure L and the width of the hybrid structure W were fixed. 3. Results and discussion Fig. 2 shows the stress vs strain curves for hybrid nanolattice structures at different values of n, t and l. In the stress-strain curves, the stress is obtained as the ratio of the sum of the reaction forces at the bottom surface to the area of the bottom surface, while the strain is obtained as the ratio of the length change of the hybrid structure ΔL to the initial length L. It is seen that all the samples initially deform linear-elastically and then fail instantaneously and catastrophically, as expected for a brittle material. The elastic modulus E and ultimate tensile strength UTS (defined as σ) of the hybrid nanolattice structures were collected from the stress-strain curves and shown in Fig. 3 to Fig. 5. Fig. 2A displays the stress vs strain curves of the hybrid structures at different numbers of cell layer n. In the calculations, the lattice structure thickness h2 is changed from 4.72 μm for n = 1 to 37.76 μm for n = 8, and thus the thickness of the hybrid structure H is changed from 6.72 μm for n = 1 to 39.76 μm for n = 8. The length of unit cell l is taken as 4.72 μm, and the tube wall thickness t as 50 nm. One can see that the slope of the stress vs strain curves shifts downwards rapidly and steadily as n increases. All the curves exhibit a brittle failure mode, consistent with the brittle nature of Al2O3. Fig. 3A and B shows the variation of elastic modulus E and ultimate tensile strength σ of the hybrid nanolattice structures with the number of cell layer n. It is seen that
Fig. 2. Plots of stress versus strain curves for (A) number of cell layer n, (B) tube wall thickness t and (C) length of unit cell l.
Z. Liu et al. / Materials and Design 160 (2018) 496–502
499
Fig. 3. Elastic modulus E and ultimate tensile strength σ versus number of cell layer n of the hybrid nanolattice structure. (A) Elastic modulus E against number of cell layer n. (B) Ultimate tensile strength σ against number of cell layer n. (C) Specific stiffness E/ρ against number of cell layer n. (D) Specific strength σ/ρ against number of cell layer n.
both the stiffness E and strength σ exhibit a monotonically decreasing trend as the number of cell layer n increases. When the number of cell layer continues to increase, the stiffness E and strength σ tend to level off and gradually saturate. Fig. 3C and D shows the variation of stiffness-to-density ratio (defined as specific stiffness E/ρ) and strength-to-density ratio (defined as specific strength σ/ρ) of hybrid structures with the number of cell layer n, where ρ is the density of the hybrid nanolattice structure, which is defined as the ratio of the total weight of the hybrid structure to the total volume of the hybrid structure. The total weight of the hybrid structure is the weight of the plates plus the weight of the nanolattice, while the total volume of the hybrid structure is L × W × H. It is seen that the specific stiffness E/ρ and specific strength σ/ρ also exhibit a monotonically decreasing trend as the number of cell layer n increases, similar to the stiffness E and strength σ. Fig. 2B displays the stress vs strain curves of the hybrid structures at different values of tube wall thickness t. In the calculations, the length of unit cell l is taken as 4.72 μm, the number of cell layer n as 2 and the lattice structure thickness h2 as 9.44 μm, so the thickness of the hybrid structure H is 11.44 μm. It is seen that these curves only change slightly and progressively as t increases from 1 nm to 120 nm, which is in contrast to the previous study that the thickness of nanolattice tube showed a significant influence on the stiffness and strength of pure lattice structure [38]. Fig. 4A and B depicts the variation of stiffness E and strength σ of the hybrid nanolattice structures with the tube wall thickness t. It is seen that the stiffness E shows a monotonic increase as t increases, while the strength σ exhibits two stages of increase as t increases. The transition of these two stags occurs at t ≈ 5 nm. Fig. 4C and D shows correspondingly the variation of the specific stiffness E/ρ and specific strength σ/ρ of the hybrid structure with the tube wall thickness t. It is seen that the specific stiffness E/ρ decreases significantly as t increases, while the specific strength σ/ρ initially increases, peaks at t
≈ 5 nm and then decreases, suggesting a change in failure mechanism with the variation of the tube wall thickness t. Our detailed analysis showed that when the value of t is small, Euler buckling failure prevails; while when t is larger than a critical value, the intrinsic material failure prevails. The critical strength-to-density ratio σ/ρ for the transition occurs at t ≈ 5 nm. To better understand the failure transition, we use an Euler slender column to analyze the buckling behavior of the hollow tubes. The Euler buckling criterion can be written as,
F¼
π2 Ec Ic ðKLc Þ2
ð1Þ
where F is the maximum or critical force (vertical load on column), Ec is the elastic modulus of the material, Ic is the smallest area moment of inertia of the column, Lc is the length of the column, and K is a parameter, whose value depends on the condition of the end support to the column. For example, for both ends pinned (hinged, free to rotate), K = 1.0, and for one end fixed and the other end free to move laterally, K = 2.0. For the tube with a hollow cylindrical section, Ic ¼ π4 ðr 4o −r4i Þ, where r o ¼ r þ 2t is the outer radius and r i ¼ r− 2t is the inner radius. Since the area of the cross section is S = 2πrt, so the stress σc = Ecε0 = F/S. From Eq. (1), the critical thickness of the tube is t ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ε0 ðKLc Þ2 −r 2 . In the present work, r = 491 nm, Lc = 6.675 μm, ε0 = π2
0.008. We find that when K = 1.83, t ≈ 5 nm. The high value of K suggests that the support from the nodes to the column is not strong. More specifically, it is clearly weaker than that with both ends pinned (hinged, free to rotate), but slightly stronger than that with one end fixed and the other end free to move laterally.
500
Z. Liu et al. / Materials and Design 160 (2018) 496–502
Fig. 4. Elastic modulus E and ultimate tensile strength σ versus tube wall thickness t of the hybrid nanolattice structure. (A) Elastic modulus E against the tube wall thickness t. (B) Ultimate tensile strength σ against the tube wall thickness t. (C) Specific stiffness E/ρ against the tube wall thickness t. (D) Specific strength σ/ρ against the tube wall thickness t.
It is noted that similar phenomena have also been reported in other stretching-dominated lattice structures [20]. The presence of maximal value of σ/ρ with the variation of tube wall thickness (see Fig. 4D) due to the transition from a buckling-dominated failure mode to a fracture-dominated failure mode allows us to control the failure mode of hybrid Al2O3 nanolattice structures by tuning their tube wall thickness. Fig. 2C displays stress vs strain curves of the hybrid structures at various values of length of the unit cell l. In the simulations, the lattice structure thickness h2 is fixed at 37.76 μm, the thickness of the hybrid structure H at 39.76 μm, and the tube wall thickness at t = 50 nm. The number of cell layers n along the thickness of the hybrid structure is changed by varying the length of the unit cell l from n = 1 for l = 37.76 μm, n = 2 for l = 18.88 μm, n = 4 for l = 9.44 μm, n = 8 for l = 4.72 μm, to n = 16 for l = 2.36 μm. It is seen that the slope of the stress vs strain curves initially shifts upwards slowly and then drastically as l decreases. Fig. 5A and B shows the variation of stiffness E and strength σ of the hybrid structures with the length of the unit cell l. It is seen that the stiffness E and strength σ initially decrease drastically and then gradually level off monotonically as l increases. Fig. 5C and D shows correspondingly the variation of specific stiffness E/ρ and specific strength σ/ρ of the hybrid structures with the length of the unit cell l. The specific stiffness E/ρ and specific strength σ/ρ exhibit a similar trend with l, i.e., E/ρ and σ/ρ increase rapidly initially and then level off as l increases. The turning point for E/ρ and σ/ρ occurs at l ≈ 20 μm, after which the stiffness-to-density and strength-to-density remain approximately unchanged. Our simulations showed that all the hybrid structures failed catastrophically, regardless of the failure mechanism, that is, structural buckling or intrinsic material failure. Fig. 6 shows the onset and propagation of failure. One can see clearly that the failure initiates at the nodal connections between the nanolattice and the plate (Fig. 6), and then
propagates into neighboring lattice cells near the plates, suggesting that the nanolattice cell is weaker than the plate. This is not surprising since the stiffness and strength of the nanolattice are far less than that of the Al2O3 plate. The fact that the failure starts at the nodal connections between the nanolattice and the plates suggests that the nodal connections are the weakest link. For all the samples that fail via intrinsic material failure, the strain at failure is at approximately 0.008, an intrinsic failure property of this brittle material. The double logarithmic relation between Young's modulus E and density ρ, that is, log(E) vs log(ρ), and that between the failure strength σ vs the density ρ, that is, log(σ) vs log(ρ), of the hybrid structures, together with the pure Al2O3 plate and pure Al2O3 nanolattice structures, are summarized in Fig. 7A and B, respectively. For comparison, recently reported data for log(E) vs log(ρ) of other lattice materials are also shown in Fig. 7A, including hollow Al2O3 stretching-dominated lattices [20], ultralight metallic microlattices [26], and graphene elastomers [39]. It is seen interestingly that for the hollow Al2O3 nanolattice structures, the relation between log(E) and log(ρ) follows approximately a linear relation with a slope of ~1, indicating that these structures are stretching-dominated ones. For the graphene elastomers, the relation between log(E) and log(ρ) also follows approximately a linear relation with a slope of ~3, suggesting that these structures are dominated by the bending deformation. For these ultralight metallic microlattice structures, the relation between log(E) and log(ρ) also follows approximately a linear relation, however, with a slope of ~2, signifying that these structures are in a mixed stretching and bending deformation. From our computed Young's moduli of the plate, pure nanolattices, and hybrid structures (see Fig. 7A), it is seen interestingly that the slope for the relation between log(E) and log(ρ) from the plate to the pure nanolattice is ~3, signifying a drastic reduction of Young's modulus with density. However, the slope for the relation between log(E) and log (ρ) from the plate to the hybrid structures that fail via intrinsic mode is
Z. Liu et al. / Materials and Design 160 (2018) 496–502
501
Fig. 5. Elastic modulus E and ultimate tensile strength σ versus length of unit cell l. (A) Elastic modulus E against length of unit cell l. (B) Ultimate tensile strength σ against length of unit cell l. (C) Specific stiffness E/ρ against length of unit cell l. (D) Specific strength σ/ρ against length of unit cell l.
~1, suggesting the slowest reduction of Young's modulus with density. However, for those hybrid structures that fail via buckling mode, the slope is ~3, consistent with the mechanism transition from intrinsic material failure to buckling failure. Similarly, from our computed failure strengths of the plate, pure nanolattices, and hybrid structures (see Fig. 7B), it is also seen that the slope for the relation between log(σ) and log(ρ) from the plate to the pure nanolattice is ~3, signifying a drastic reduction of failure strength with density. However, the slope for the relation between log(σ) and log(ρ) from the plate to the hybrid structures with intrinsic material failure is ~1, suggesting the slowest reduction of strength with density. Interestingly, the slope becomes ~3 when the hybrid structures fail via buckling mode. These findings demonstrate that if the hybrid structures are designed in the stretchingdominated deformation mode (via intrinsic material failure), we are able to efficiently utilize materials to achieve high specific performances. 4. Conclusion
Fig. 6. Finite element simulations show that the failure starts from the nodal junction between the nanolattice and plate sheets. Then the failure propagates into the lattice cells next to the plate sheet. The lattice failure is prior to the plate failure. (A) Before failure, ε = 0.0076. (B) Failure at the nodal connection between the plate and nanolattice, ε = 0.0078. (C) Failure propagation along the cells near the plates, ε = 0.0080.
We designed a hybrid structure consisting of a space-filling 3dimensional octet-truss alumina nanolattice and stretching-dominant plates and studied the effect of various parameters on the stiffness and strength of these hybrid structures under uniaxial tensile loading by using finite element simulations. Our simulations showed that depending on the choice of geometrical parameters of the hybrid structures, the strength and the failure mode can be tuned. In particular, the hybrid structures exhibit a nearly linear scaling of stiffness and strength with density when the hybrid structures fail via intrinsic material failure mechanism, which is far more superior than those of the pure nanolattice structures. Furthermore, if we are allowed to change the type and material of space-filling nanolattice and plate, the stiffness
502
Z. Liu et al. / Materials and Design 160 (2018) 496–502
Fig. 7. Ashby charts for (A) the stiffness versus density and (B) the failure strength versus density of the plate, pure lattices and hybrid structures. For the stiffness versus density, three other previously reported structures are also included for comparison.
and failure strength of the hybrid structures can be further tuned and expanded. Since the proposed hybrid structures are able to provide a much larger design space in terms of stiffness, strength and density than pure nanolattice structures, it is expected that these hybrid structures might find a wider range of potential applications in precision engineering structures. Credit authorship contribution statement Liu Ping: Initiated the idea and designed the study, analyzed the simulation results. Liu Zhigang: Did the simulations and analyzed the simulation results. Zhang Yong Wei: Analyzed the simulation results. All of the authors wrote, proofread and revised the manuscript. Acknowledgment The support of this work by the Institute of High Performance Computing, A*STAR is gratefully acknowledged. References [1] D. Jang, L.R. Meza, F. Greer, J.R. Greer, Fabrication and deformation of threedimensional hollow ceramic nanostructures, Nat. Mater. 12 (2013) 893–898. [2] J.R. Greer, J.T.M. De Hosson, Plasticity in small-sized metallic systems: intrinsic versus extrinsic size effect, Prog. Mater. Sci. 56 (2011) 654–724.
[3] H. Ebrahimi, D. Mousanezhad, H. Nayeb-Hashemi, J. Norato, A. Vaziri, 3D cellular metamaterials with planar anti-chiral topology, Mater. Des. 145( (2018) 226–231. [4] B.D. Nguyen, J.S. Cho, K. Kang, Optimal design of “shellular”, a micro-architectured material with ultralow density, Mater. Des. 95 (2016) 490–500. [5] H. Ebrahimi, D. Mousanezhad, B. Haghpanah, R. Ghosh, A. Vaziri, Lattice materials with reversible foldability, Adv. Eng. Mater. 19 (2017) 1600646-n/a. [6] B. Haghpanah, L. Salari-Sharif, P. Pourrajab, J. Hopkins, L. Valdevit, Multistable shape-reconfigurable architected materials, Adv. Mater. 28 (2016) 7915–7920. [7] W. Franziska, O. Fuad, A. Lucas, A.L. Matthias, K. Carolin, Fabrication and characterization of a fully auxetic 3D lattice structure via selective electron beam melting, Smart Mater. Struct. 26 (2017), 025013. . [8] X.-T. Wang, B. Wang, X.-W. Li, L. Ma, Mechanical properties of 3D re-entrant auxetic cellular structures, Int. J. Mech. Sci. 131–132 (2017) 396–407. [9] J. Bauer, L.R. Meza, T.A. Schaedler, R. Schwaiger, X. Zheng, L. Valdevit, Nanolattices: an emerging class of mechanical metamaterials, Adv. Mater. 29 (2017) 1701850n/a. [10] L.R. Meza, S. Das, J.R. Greer, Strong, lightweight, and recoverable three-dimensional ceramic nanolattices, Science 335 (2014) 1322–1326. [11] H.M.A. Kolken, A.A. Zadpoor, Auxetic mechanical metamaterials, RSC Adv. 7 (2017) 5111–5129. [12] X. Li, Z. Lu, Z. Yang, C. Yang, Directions dependence of the elastic properties of a 3D augmented re-entrant cellular structure, Mater. Des. 134 (2017) 151–162. [13] Z. Lu, Q. Wang, X. Li, Z. Yang, Elastic properties of two novel auxetic 3D cellular structures, Int. J. Solids Struct. 124 (2017) 46–56. [14] T.M. Ryan, C.N. Shaw, Trabecular bone microstructure scales allometrically in the primate humerus and femur, Proc. R. Soc. B 280 (2013) 20130172. [15] P. Van Liedekerke, P. Ghysels, E. Tijskens, G. Samaey, B. Smeedts, D. Roose, H. Ramon, A particle-based model to simulate the micromechanics of single-plant parenchyma cells and aggregates, Phys. Biol. 7 (2010), 026006. . [16] J.H. Shen, Y.M. Xie, X.D. Huang, S.W. Zhou, D. Ruan, Compressive behavior of luffa sponge material at high strain rate, Key Eng. Mater. 535–536 (2013) 465–468. [17] X. W. Gu, C. N. Loynachan, Z. X. Wu, Y.W. Zhang, D. J. Srolovitz, J. R. Greer, Sizedependent deformation of nanocrystalline Pt nanopillars, Nano Lett., 12(2013) 6385–6392. [18] D.Z. Chen, D. Jang, K.M. Guan, Q. An, W.A. Goddard III, J.R. Greer, Nanometallic glasses: size reduction brings ductility, surface state drives its extent, Nano Lett. 13 (2013) 4462–4468. [19] S.W. Lee, M. Jafary-Zadeh, D.Z. Chen, Y.W. Zhang, J.R. Greer, Size effect suppresses brittle failure in hollow Cu60Zr40 metallic glass nanolattices deformed at cryogenic temperature, Nano Lett. 15 (2015) 5673–5681. [20] X. Zheng, H. Lee, T.H. Weisgraber, M. Shusteff, J. DeOtte, E.B. Duoss, J.D. Kuntz, M.M. Biener, Q. Ge, J.A. Jackson, S.O. Kucheyev, N.X. Fang, C.M. Spadaccini, Ultralight, ultrastiff mechanical metamaterials, Science 344 (2014) 1373–1377. [21] L.R. Meza, A.J. Zelhofer, N. Clarke, A.J. Mateos, D.M. Kochmann, J.R. Greer, Resilient 3D hierarchical architected metamaterials, PNAS 112 (2015) 11502–11507. [22] J. Bauer, A. Schroer, R. Schwaiger, O. Kraft, Approaching theoretical strength in glassy carbon nanolattices, Nat. Mater. 15 (2016) 438–443. [23] J. Bauer, S. Hengsbach, I. Tesari, R. Schwaiger, O. Kraft, High-strength cellular ceramic composites with 3D microarchitecture, PNAS 111 (2014) 2453–2458. [24] L.R. Meza, S. Das, J.R. Greer, Strong, lightweight, and recoverable three-dimensional ceramic nanolattices, Science 345 (2014) 1322–1326. [25] J.B. Berger, H.N.G. Wadley, R.M. McMeeking, Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness, Nature 543 (2017) 533–537. [26] T.A. Schaedler, A.J. Jacobsen, A. Torrents, A.E. Sorensen, J. Lian, J.R. Greer, L. Valdevit, W.B. Carter, Ultralight metallic microlattices, Science 334 (2011) 962–965. [27] L.C. Montemayor, L.R. Meza, J.R. Greer, Design and fabrication of hollow rigid nanolattices via two-photon lithography, Adv. Eng. Mater. 16 (2014) 184–189. [28] A.J. Jacobsen, W.B. Carter, S. Nutt, Micro-scale Truss structures formed from selfpropagating photopolymer waveguides, Adv. Mater. 19 (2007) 3892–3896. [29] X.W. Gu, J.R. Greer, Ultra-strong architected Cu meso-lattices, Extreme Mech. Lett. 2 (2015) 7–14. [30] J. Christensen, M. Kadic, M. Wegener, O. Kraft, Vibrant times for mechanical metamaterials, MRS Commun. 5 (2015) 453–462. [31] L. Salari-Sharif, T.A. Schaedler, L. Valdevit, Energy dissipation mechanisms in hollow metallic microlattices, J. Mater. Res. 29 (2014) 1755–1770. [32] K.R. Mangipudi, S.W. Van Buuren, P.R. Onck, The microstructural origin of strain hardening in two-dimensional open-cell metal foams, Int. J. Solids Struct. 47 (2010) 2081–2096. [33] Y. Conde, R. Doglione, A. Mortensen, Influence of microstructural heterogeneity on the scaling between flow stress and relative density in microcellular Al-4.5% Cu, J. Mater. Sci. 49 (2014) 2403–2414. [34] V. Marcadon, C. Davoine, B. Passilly, D. Boivin, F. Popoff, A. Rafray, S. Kruch, Mechanical behaviour of hollow-tube stackings: experimental characterization and modelling of the role of their constitutive material behaviour, Acta Mater. 60 (2012) 5626–5644. [35] N.A. Fleck, V.S. Deshpande, M.F. Ashby, Micro-architectured materials: past, present and future, Proc. R. Soc. A Math. Phys. Eng. Sci. 466 (2010) 2495–2516. [36] ABAQUS Version 6.14, User's Manual Version 6.14, 2014 Providence, RI, USA. [37] M. Berdova, T. Ylitalo, I. Kassamakov, J. Heino, P.T. Törmä, L. Kilpi, H. Ronkainen, J. Koskinen, E. Hæggström, S. Franssila, Mechanical assessment of suspended ALD thin films by bulge and shaft-loading techniques, Acta Mater. 66 (2014) 370–377. [38] L.C. Montemayor, W.H. Wong, Y.W. Zhang, J.R. Greer, Insensitivity to flaws leads to damage tolerance in brittle architected meta-materials, Sci. Rep. 6 (2016), 20570. . [39] L. Qiu, J.Z. Liu, S.L.Y. Chang, Y. Wu, D. Li, Biomimetic superelastic graphene-based cellular monoliths, Nat. Commun. 3 (2012) 1241.