European Journal of Mechanics A/Solids

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1 European Journal of Mechanic A/Solid 7 (0) 8e Content lit available at SciVere ScienceDirect European Journal of Mechanic A/Solid journal homepage: www. elevier. com/ locate/ ejmol In-plane elatic propertie of hierarchical nano-honeycomb: The role of the urface effect Qiang Chen a, Nicola M. Pugno b,c,d, * a School of Biological Science and Medical Engineering, Southeat Univerity, 0096 Nanjing, P.R. China b Laboratory of Bio-inpired Nanomechanic, Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Coro Duca degli Abruzzi,, 09 Torino, Italy c National Intitute of Nuclear Phyic, National Laboratorie of Fracati, Via E. Fermi 0, 000 Fracati, Italy d National Intitute of Metrological Reearch, Strada delle Cacce 9, I-0 Torino, Italy a r t i c l e i n f o a b t r a c t Article hitory: Received 7 January 0 Accepted June 0 Available online 9 June 0 Keyword: Hierarchical Nano-honeycomb Elaticity Buckling Strength In thi paper, we analytically calculate the in-plane elatic propertie (linear-elaticity and elatic buckling) of a new cla of bio-inpired nano-honeycomb material poeing a hierarchical architecture. Incorporating the urface effect, modification to the claical reult for macrocopic and nonhierarchical honeycomb are propoed, and the reult are compared with thoe in the literature. A parametrical analyi reveal the influence of two key geometrical parameter on the overall elatic propertie. We dicu the relevant mechanical propertie, e.g. tiffne efficiency (tiffne-to-denity ratio) and trength efficiency (trength-to-denity ratio), which are indice reflecting the mechanical efficiency of material, and dicover that the tructural trength can be optimized. The developed theory allow u to deign a new cla of nano material with tailored mechanical propertie at each hierarchical level and could be ueful for many application. Ó 0 Publihed by Elevier Maon SAS.. Introduction Honeycomb-like tructure can be often found in Nature, for intance, the armadillo hell (Rhee et al., 0), the beak of Tucan bird (Seki et al., 00), and the widely tudied lobter claw of lobter (Raabe et al., 00): the tructure i low-weight but trong and tough, that i to ay, it i more efficient (Karam and Gibon, 99). So far, it i well accepted that the tructure of the natural material i an optimized reult by ambient environment in the evolutionary proce (e.g. the armadillo hell can protect internal organ from being attacked by predator). Thu, nature eem to grant u a bet olution to deign more efficient material. Therefore, inpired by nature, honeycomb material have been extenively invetigated for tructural, mechanical and material deign. For example, in the field of material cience, they are ued a a core material in andwich tructure for energy aborption (Wang, 009; Wang et al., 009); alo, honeycomb caffold with mechanical tability, biocompatibility and biodegradability are ued for tiue regeneration (George et al., 008). With uch extenive application, from the mechanical point of view, it i * Correponding author. Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Coro Duca degli Abruzzi,, 09 Torino, Italy. Tel.: þ9 (0)690. addre: nicola.pugno@polito.it (N.M. Pugno). important for material cientit and engineer to characterize and model the in-plane and out-of-plane mechanical behavior (Gibon et al., 98; Warren and Kraynik, 987; Zhang and Ahby, 99; Papka and Kyriakide, 99) of honeycomb tructure. For it contitutive behavior, the treetrain curve (Papka and Kyriakide, 99; Gibon and Ahby, 997) are typically decribed by three regime (the linear elatic, peudo platic plateau and peudo hyper-elatic denification region). Gibon and Ahby (997) ummarized mot of the work on the tructural and mechanical propertie for D and D cellular olid, and ytematically invetigated the tructure-olid mechanic. In recent year, a variety of topological honeycomb i being tudied for multifunctional application (Wadley, 006), e.g. the thermal conductivity propertie of a rectangular-hexagonal honeycomb tructure (Bezazi et al., 008). Even though many tudie in thi field provide ome method to deign new porou material, and enrich the exiting one, there i a lack of detailed invetigation on natural honeycomb. So, in order to mimic the natural honeycomb, Zhang et al. (00) revealed the ophiticated and hierarchical tructure of honeybee comb, and howed that the cell wall of the natural honeycomb i a multi-layered tructure, which i continuouly contructed by adding wax layer reinforced by ilk a time increae; the tructure i trengthened and avoid the fragility, and thu provide a mechanically afe place for toring honey and brooding /$ e ee front matter Ó 0 Publihed by Elevier Maon SAS.

2 Q. Chen, N.M. Pugno / European Journal of Mechanic A/Solid 7 (0) 8e 9 On the other hand, a nanocience and nanotechnology develop, the material deign alo expand toward the nano-cale. If tructure are nano-ized, the urface effect hould be taken into account, due to their high urface-to-volume ratio. Regarding the urface effect, extenive work (Wang and Feng, 009; Shankar and King, 007; Wong et al., 997; Chen and Pugno, 0a) tudied it influence on the linear elatic behavior of nano-wire, ince they hold a promie for nano-device application, e.g. enor and actuator. Thee work emphaized the vital role of the urface effect in determining the mechanic of nano-ytem. Duan et al. (006, 009) invetigated the elatic contant of nanoporou material with unidirectional cylindrical nanochannel by conidering the urface effect, and the reult demontrated that a nanoporou material could be tiffer or ofter than it bulk counterpart. And more, Duan et al. (009) reviewed the elatic theory for nano-cale ytem, epecially for the nanocale inhomogeneitie, dicuing different claical theorie, e.g. Ehelby Formalim, and Levin formula. Thee work are very ueful for tudying the fundamental mechanical and phyical propertie of heterogeneou nanoytem. In thi paper, we contruct a hierarchical nano-honeycomb tructure (Fig. ), uing a claical iterative approach (Lake, 99; Pugno, 006; Pugno et al., 008; Chen and Pugno, 0b,c), and tudy it in-plane elatic propertie. Starting from an orthotropic contituent material and conidering the influence of the urface effect, we derived the effective longitudinal Young modulu and buckling trength at the firt level; then, the reult of the n-level tructure (n ) are obtained thank to the iterative method. Finally, a comparion between different theorie and parametric analye reveal the influence of the geometrical parameter on the overall elatic behavior.. Surface effect Due to the exitence of urface in a olid, there i alway a competition between bulk and urface. In particular, when the ize of the olid come down into dimenion maller than 0 nm, urface/interface effect become an important characteritic in nanotructure, and determine their mechanical propertie. A claical expreion for the urface tre i derived baed on the urface/interface energy (Cammarata, 99), and it i compoed by two part, i.e. urface free energy and urface free energy gradient with repect to the urface train: f ij gd ij þ vg/v ij, where, f ij and ij are the urface tre and train tenor, repectively, g i the urface free energy, d ij i the Kronecker delta. For the elatic theory, according to Gurtin and Murdoch (97), the urface tre i uually expreed a the ummation of the urface reidual tre and urface elaticity (i.e. the two part correponding to the two term on the right-handed ide of the f ij expreion, repectively): 0 þ S i i, where, 0 i the urface reidual tre, S i i i the urface tiffne tenor, i the urface elatic train tenor. Here, we only conider the influence of the urface elaticity (one-dimenion), i.e. the econd term on the right-handed ide of the expreion.. Elatic contant of hierarchical nano-honeycomb.. Deflection of an orthotropic beam with urface elaticity The linear-elatic deformation mechanim of honeycomb i mainly due to the bending of cell wall, and the tandard beam theory i employed here to invetigate their elatic contant. (Fig. ); in particular, if the beam i nano-ized, the modification induced by the urface effect hould be taken into account, and the maximum diplacement of the Euler beam i expreed (Tolf, 98; Roark and Young, 97; Gibon and Ahby, 997; Wang and Feng, 009) a: d max with Fl ðe IÞ eqco q if 0 0 () ðe IÞ eq E bt þ E bt þ 6 E t () where, d max i the vertical diplacement of the guided end of the orthotropic beam, F i the concentrated force acting on the guided end, l i the beam length, q i the inclined angle between beam and horizontal line; (E I) eq i the equivalent flexural rigidity conidering the urface effect; b, t are width and thickne of the beam, repectively; E, depending on the crytal orientation (Shenoy, 00), i the urface Young modulu, which ha the phyical dimenion of a urface tenion.... One-level tructure Firt, we conider a one-level nano-honeycomb made by an orthotropic material (level 0) and introduce a local coordinate ytem (0) (0) related to the global coordinate ytem () (). A hown in Fig., the tructure ha two perpendicular mirror plane, i.e. the one-level tructure remain orthotropic. The deformation i caued by the bending of beam, and the compreion of beam (Fig. (b)) which i the beam parallel to the loading direction, but Fig.. Two-level nano-honeycomb.

3 0 Q. Chen, N.M. Pugno / European Journal of Mechanic A/Solid 7 (0) 8e h ðþ =l ðþ þ in q ðþ f ðþ f ðþ f ðþ co q ðþ co q ðþ h ðþ =l ðþ þ in q ðþ in q ðþ h ðþ =l ðþ þ in q ðþ h ðþ =l ðþ þ h ðþ =l ðþ co q ðþ (7) f ðþ co q ðþ h ðþ =l ðþ þ in q ðþ h ðþ =l ðþ þ! the compreive deformation i neglected with repect to the bending deflection. Thu, baing on equation () and () and employing the claical approach (Gibon and Ahby, 997), we find the elatic contant of the one-level nano-honeycomb: E ðþ E ðþ G ðþ m ðþ l ðþ f ðþ f ðþ f ðþ f ðþ f ðþ f ðþ f ðþ with Fig.. An inclined orthotropic beam with one end guided and the other fixed. f ðþ $ EðÞ $ EðÞ! 0: m ðþ! l ðþ þ Eð0Þ t ðþ þ tðþ b! r ðþ () () () (6) where and are the bulk and urface Young moduli in the principal direction (0) (level 0), repectively; b and t () are width and thickne of cro-ection of cell wall, repectively; l () and h () are length of beam and, repectively; q () i the included angle made by beam and horizontal line (Fig. ). From equation ()e(7), we note that compared with the Young moduli and hear modulu derived by Gibon and Ahby (997), here, the Young moduli and hear modulu of the onelevel tructure are modified by a factor l ðþ. If t ðþ =b (plate), equation (6) can be expreed a: l ðþ þ 6 Eð0Þ (8) tðþ Expreion (8) coincide with the reult from Miller and Shenoy (000), and it obey the caling law l ðþ þ al in =t ðþ (Wang et al., 006) with l in = and a 6:0. Note that: l in repreent an intrinic material length, under which urface effect play an important role; a i a dimenionle contant, which depend on the tructural geometry and loading method. Beide, we can ee that the urface effect at level 0 make the tructure tiffer if > 0; otherwie, it make the tructure ofter. Moreover, the geometry of Fig. provide the relative denity: r ðþ h ðþ =l ðþ þ tðþ co q ðþ h ðþ =l ðþ ðþ þ in q l ðþ (9) where, r () and r (0) are denitie of the one-level tructure and it contituent material, repectively. Fig.. Schematic of the firt-level nano-honeycomb. (a) Unit cell; (b) Three pair of cell wall.

4 Q. Chen, N.M. Pugno / European Journal of Mechanic A/Solid 7 (0) 8e... n-level tructure Fig. decribe an n-level nano-honeycomb. Compared with the one-level tructure, the cell-wall dimenion at the econd or above level are very large, o the urface effect could be neglected, i.e. E ðiþ 0 ði Þ. However, in order to extend the theory to the general cae, we till conider the urface effect at each level, then, iterating equation () we find:! Yn l ðiþ f ðiþ f ðiþ $ rðnþ r i ð0þ (0) natural polymer, natural elatomer, natural cellular material under tenile and flexural load and plotted the o-called Ahby map according to different material indice. Here, according to Ahby (00), we tudy the tiffne-to-denity ratio for the honeycomb tructure under the uniaxially loading condition, which i evaluated by P E/r. For the hierarchical honeycomb tructure, =rðnþ can be derived from equation (9e) a: ð0þ r ðnþ E Y n i! l ðiþ f ðiþ f ðiþ $ rðnþ () G ðnþ $ EðnÞ $ EðnÞ () r ðnþ f G ðnþ r ðnþ ðnþ $ EðnÞ $ EðnÞ () m ðnþ! 0: m ðnþ () where,,, and can be obtained by replacing the upercript () with (n) in equation (7). Note that, the reciprocal theorem hold i.e., mðnþ EðnÞ mðnþ. Equation (0)e() how that the tranvere Young modulu and hear modulu can be derived from the longitudinal Young modulu, and the two Poion ratio are only related to the geometry of the n-level tructure. Therefore, in eential, there are only two independent elatic contant, i.e. the longitudinal Young modulu and one of the Poion ratio... Stiffne efficiency Structural efficiency i baing on the minimum-weight analyi, and it i ued to optimize tructural deign and thu, reduce cot. By conidering the maximum tre and maximum train, Budianky (999) invetigated everal type of compreive tructure which are hollow tube with/without filling foam. Wegt and Ahby (00) ummarized the mechanical efficiency of natural ceramic,. Elatic buckling of hierarchical honeycomb material Different from the linear-elatic deformation of honeycomb, progreive buckling deformation of cell wall dominate the main energy-aborbing mechanim. Therefore, it i ignificant to tudy the buckling behavior of the hierarchical honeycomb in the deign of energy-aborbing material. Regarding the buckling behavior of the conventional honeycomb, Chen and Pugno (0a) tudied the competition between the buckling of the beam (or ) and beam, and they demontrated that the buckling of the beam prevail and there i a poibility for the buckling occurrence of beam (or ) at extreme condition (e.g. the inclination angle approache 90 and the thickne-to-length ratio of the beam (or ) approache zero). Therefore, we only conider the buckling of the beam in the following dicuion... Buckling load of the orthotropic beam with urface effect Euler buckling equation i a imple but efficient way to decribe the buckling behavior for an iotropic beam or column. For an Fig.. Top view of the chematic of the n-level nano-honeycomb.

5 Q. Chen, N.M. Pugno / European Journal of Mechanic A/Solid 7 (0) 8e orthotropic nano-column, due to the tandard beam theory (i.e. Euler beam theory) and the urface effect, the buckling load F cr i obtained (Tolf, 98; Timohenko and Gere, 96): F cr n p ðe IÞ eq l n () where, n i a numerical factor depending on the boundary condition. Equation () i the claical Euler buckling formula with urface effect, in which the Young modulu of the iotropic material i ubtituted by the longitudinal one of the orthotropic beam.... One-level tructure For the one-level tructure (Fig. ), when the external tre act on the tructure (Fig. ), the equivalent concentrated force applied on beam i: F ðþ b ðþ l ðþ co q ðþ (6) Then, if the beam buckle, the force F hould reach the critical load F cr. Subtituting equation (6) into equation (), we find: ðþ cr l ðþ pn ðþ! t ðþ l ðþ! l ðþ h ðþ co q ðþ (7) Here, we define a new function with repect to h () /l () and q () : f ðþ pn ðþ! l ðþ h ðþ co q ðþ (8) where, n () i related to h () /l () and equation (7) i conciely written a: ðþ cr l ðþ f ðþ f ðþ! r ðþ (9)... Two-level tructure For the two-level tructure, we have two object in the analyi, the vertical beam at the firt and the econd level. On one hand, for the econd level, the equivalent concentrated force acting on the beam i: F ðþ b ðþ l ðþ co q ðþ (0) According to equation (0), we find the buckling tre for the beam: ðþ E ðþ l ðþ f ðþ f ðþ! r ðþ r ðþ () E ðþ i calculated by equation (), finally, equation () i rewritten a: ðþ l ðþ l ðþ f ðþ f ðþ f ðþ f ðþ! r ðþ () On the other hand, baed on load tranfer and the equivalent concentrated force acting on the beam at the econd level, the equivalent concentrated force acting on the beam at the firt level can be calculated, which i expreed a: F ðþ! lðþ qðþ tðþco b ðþ l ðþ co q ðþ () then, ubtituting the force into the buckling equation (), the buckling tre i immediately obtained a: ðþ with f ðþ 6 ðþ cr l ðþ f ðþ 6 f ðþ f ðþ h ðþ =l ðþ þ in q ðþ! r ðþ! r ðþ () h ðþ =l ðþ þ () For the buckling tre of the two-level tructure, it i derived a: min ðþ ; ðþ (6)... n-level tructure Like the analyi of the two-level tructure, finding the concentrated force acting on the beam at level i and ubtituting the force into buckling equation (), the dimenionle buckling tre at each level i obtained a: ðiþ Yn jiþ f ðjþ 6 $ Yi k l ðkþ f ðkþ f ðkþ $ f ðiþ f ðiþ! r ðiþ! r ðnþ o, the buckling load of the n-level tructure i obtained a: ðnþ cr (7) min ðiþ (8) Fig.. (a) Unit cell of the honeycomb tructure with cylindrical nano-channel; (b) Unit cell of the honeycomb tructure with hexagonal nano-channel.

6 Q. Chen, N.M. Pugno / European Journal of Mechanic A/Solid 7 (0) 8e Fig. 6. (a) The normalized bulk modulu v beam ize; (b) The normalized Young modulu v beam ize... Strength efficiency Like tiffne efficiency, the trength efficiency of the hierarchical tructure i here deduced. And a for uniaxial loading tructure, it i evaluated a P /r. From equation (7), the evaluating criterion, i.e. buckling tre to denity, i expreed a: ðiþ r ðnþ Yn jiþ f ðjþ 6 $ Yi k l ðkþ f ðkþ f ðkþ $ f ðiþ f ðiþ! r ðiþ! thu, the buckling trength to denity i finally obtained a: (9) ðnþ cr min ðiþ rðnþ r ðnþ (0). Comparion between different method In thi ection, we compare our prediction on the one-level tructure (q () 0 and h () /l () ) with other reult preented in the literature (Duan et al., 006, 009). In the literature, the unit cell of the nano-tructure i honeycomb with unidirectional cylindrical nano-channel, thu different from our unidirectional hexagonal nano-channel (Fig. ); the author conidered the cae with poroity 0. and pore diameter from nm to 0 nm. Here, with the condition l l (), we compare the different geometrie impoing the ame pore area fraction. Then, according to a poroity 0. and fixed geometry, ee Fig. a, we find the relationhip between the ide length of the unit cell and pore diameter, i.e. l.89r. Similarly, conidering the equal pore-area, we find l ().89R and t ().R. In order to keep the elf-conitency, we alo employ the data of Al provided Fig. 7. (a) Influence of h/l on tiffne when q 0 ; (b) Influence of h/l on tiffne-to-denity ratio when q 0 ; (c) Influence of q on tiffne when h/l.0; (d) Influence of q on tiffne-to-denity ratio when h/l.0.

7 Q. Chen, N.M. Pugno / European Journal of Mechanic A/Solid 7 (0) 8e by Miller and Shenoy (000); the bulk Young modulu i E (0) 89.9 GPa, and urface Young modulu on the [00] and [] urface for plate are 7:96 N=m and :8 N=m, repectively. The reult are reported in Fig. 6, in which the Young moduli and bulk modulu of the honeycomb are normalized by thoe of the honeycomb without urface effect. Fig. 6 how, a the beam ize increae, that the influence of the urface effect on the elatic contant decreae. Alo, we find that generally our prediction i comparable with the reult from the literature, even though our reult how a lightly greater influence; beide the different geometrie, the thickne-to-length ratio t () /l () i 0.9 under the condition of the poroity 0., that i to ay, the hear effect hould be taken into account. 6. Parametric analyi and dicuion Here, again, we conider Al a the contituent material and treat different hierarchical nano-honeycomb tructure with hierarchical level number from one to five and identical relative denitie (low relative denity 0. i conidered). We aume the ize of the cell wall i very large compared with that of the firt level, o, the urface effect at level i (i > ) are negligible. The denity of Al i.70 g/cm ; the urface elatic modulu on the [00] urface i 7:96 N=m; the thickne of the cell wall at the firt level i aumed to be t () nm. 6.. Linear-elatic deformation analyi A we dicued in Section, the Young moduli and hear modulu depend on the longitudinal Young modulu, thu, i only analyzed. We conider elf-imilar cae for n-level tructure, i.e. the relative denity r ðiþþ =r ðiþ np ffiffiffiffiffiffiffi 0: and h (i) /l (i) h/l (or q (i) q). The analytic reult of the longitudinal Young modulu and the tiffne-to-denity ratio are reported in Fig. 7. It how that the longitudinal Young modulu increae a h/l or q increae; we can alo ee that a the level number n increae, they increae. 6.. Buckling analyi In thi cae, the parameter are the ame a thoe decribed above. The analytic reult of the buckling trength and trengthto-denity ratio are reported in Fig. 8. Interetingly, it how that increaing h/l or q reult in high trength and trength-to-denity ratio and a the level number n increae, the trength and trength-to-denity ratio reache an optimal value in the threelevel tructure (Fig. 8a and b) or in the four-level tructure for different parameter (Fig. 8c and d). 6.. Dicuion Fig. 7 and 8 indicate that the mechanical behavior of the hierarchical honeycomb can be tuned by changing the geometrical configuration. Increaing h/l or q, the Young modulu, trength and their correponding mechanical efficiency increae; and a the level number n increae, alo, they increae. For the linear-elatic analyi, thi i becaue the tructural deformation decreae when h/l or q increae, then, the Young modulu increae. For the buckling analyi, it can be undertood that increaing h/l or q or level number n reult in a higher tructural tiffne but, a level Fig. 8. (a) Influence of h/l on buckling trength when q 0 ; (b) Influence of h/l on trength-to-denity ratio when q 0 ; (c) Influence of q on buckling trength when h/l.0; (d) Influence of q on trength-to-denity ratio when h/l.0.

8 Q. Chen, N.M. Pugno / European Journal of Mechanic A/Solid 7 (0) 8e number n increae, the relative denity at each hierarchical r (i) /r (0) decreae, thu an optimal value emerge. 7. Concluion We have calculated the in-plane elatic propertie (linear-elatic and buckling propertie) of hierarchical nano-honeycomb. The urface effect modifie the claical reult of a non-hierarchical honeycomb (or conventional honeycomb), which i conidered to be the firt level (or one-level tructure); the prediction are compared with reult from the literature, howing a good agreement. Addreing the hierarchical nano-honeycomb tructure, we have performed parametric analye, and revealed the influence of two key geometric parameter on the tiffne (or tiffne-to-denity ratio) and trength (or trength-to-denity ratio); the reult how that increaing the two geometric parameter can produce increaing mechanical propertie; in particular, an optimal trength or trength efficiency i obtained. The preented theory may have many intereting application, e.g. for deigning biomedical or energy-aborption nano material. Acknowledgment The reearch related to thee reult ha received funding from the European Reearch Council under the European Union Seventh Framework Programme (FP7/007e0)/ERC Grant agreement nu [7998] (ERC StG Idea to NMP on Bio-inpired hierarchical uper nanomaterial ). Reference Ahby, M.F., 00. Material Selection in Mechanical Deign, fourth ed. Butterworth- Heinemann, Burlington, US. Bezazi, A., Remillat, C., Innocenti, P., Scarpa, F., 008. In-plane mechanical and thermal conductivity propertie of a rectangular-hexagonal honeycomb tructure. Compo. Struct. 8, 8e. Budianky, B., 999. On the minimum weight of compreion tructure. Int. J. Solid Struct. 6, 677e708. Cammarata, R.C., 99. Surface and interface tre effect in thin film. Prog. Surf. Sci. 6, e8. Chen, Q., Pugno, N., 0a. Competition between in-plane buckling and bending collape in nanohoneycomb. Europhy. 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