In-plane elastic properties of hierarchical cellular solids

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1 Available online at Phyic ngineering 0 (20) th International Conference on the Mechanical Behavior of Material In-plane elatic propertie of hierarchical cellular olid Nicola M. Pugno a *, Qiang Chen a a Laboratory of Bio-Inpired Nanomechanic "Giueppe Maria Pugno",Department of Structural ngineering and Geotechnic, Politecnico di Torino, Torino, 029, Italy. Abtract In thi paper, we analytically calculate the in-plane linear-elatic propertie of a new cla of bio-inpired nano-honeycomb material poeing a hierarchical architecture, which i often oberved in natural material. Incorporating the urface tenion, peculiar of the nano-cale, modification of the claical reult for macrocopic and nonhierarchical honeycomb are propoed. A parametrical analyi reveal the influence of relative denity and of two key geometrical parameter on the overall elatic propertie. We dicover optimal value for ome of the mechanical propertie, e.g. tiffne-to-denity ratio. The developed theory allow u to deign a new cla of material with tailored elatic propertie at each hierarchical level and could be ueful for many application. 200 Publihed by levier Ltd. Keyword: Hierarchical; Nano-honeycomb; Orthotropic Material; latic contant.. Introduction Honeycomb material, due to their pecial tructure and propertie, are very promiing for tructural, mechanical and material deign, e.g. they are ued a a core material in andwich tructure for energy aborption []. One of the important iue in material cience i to characterize and model the in-plane and out-plane mechanical behavior [2-5] of honeycomb tructure. For the in-plane deformation mechanim, the tre-train curve [3-6] i decribed by three regime (the linear elatic, plateau and denification region). If the honeycomb tructure i nano-ized, the urface effect hould be taken into account becaue of the high urface-to-volume ratio. xtenive work [7-] tudied the influence of the * Correponding author. Tel.: ; fax: mail addre: nicola.pugno@polito.it Publihed by levier Ltd. doi:0.06/j.proeng

2 Nicola M. Pugno and Qiang Chen / Phyic ngineering 0 (20) urface effect on the linear elatic behavior of nanowire, ince nano-wire or nano-rod hold a promie for nano-device application. Fig. SM image of pure apen wood [5]. Fig. 2 Hierarchical nano-honeycomb. Fig. 3 An inclined orthotropic beam. In thi paper, we contruct a hierarchical nano-honeycomb tructure [2-4], inpired by biological material (Fig. ) [5], and tudy it in-plane elatic propertie (Fig. 2). Here, tarting from an orthotropic contituent material and conidering the influence of the urface effect, we derived the effective longitudinal Young modulu of the n-level tructure thank to a claical iterative approach [6]. Beide, we alo find the expreion for the tiffne-to-denity ratio. Finally, we perform a parametric analyi to invetigate the influence of the relative denity and the geometrical parameter on the overall elatic behavior. 2. Deflection of an orthotropic beam with urface effect Auming the conervation of plane ection, for the elatic line of an orthotropic beam with principal direction coincident with the beam axi (Fig. 3). The claical expreion for the deflection of the uler beam i found [6,7,8]: Fl 3 2 max co () 2I where, i the vertical diplacement of the guided end of the orthotropic beam, F i the concentrated max force acting on the guided end, l i the beam length, I i the flexural rigidity, and i the inclined angle between beam and horizontal line. If the beam i nano-ized, the modification of the influence of the urface effect hould be conidered, finding the maximum diplacement a [9]: Fl3 2 max co eq (2) 2 I with eq I bt bt t (3) eq where, I i the equivalent flexural rigidity; b, t are width and thickne of the beam, repectively;, which depend on the crytal orientation [9], i the urface Young modulu. 3. Linear-elatic propertie of hierarchical nano-honeycomb

3 3028 Nicola M. Pugno and Qiang Chen / Phyic ngineering 0 (20) latic contant of hierarchical nano-honeycomb Fig. 4 Schematic of the honeycomb at level i. Fig. 4 decribe the i-level of the nano-honeycomb. The tructure at level i i contructed baing on the tructure at level i-. The deformation i caued by the bending of the inclined cell-wall and the compreion of the vertical cell-wall, namely, the beam along which the force i applied, but the compreive deformation i neglected with repect to the bending deflection. Thu, baing on eqn. ()-(3) and employing the claical method [6] and an iterative proce [6], we find the elatic contant of the hierarchical honeycomb. For the longitudinal Young modulu, we find: 3 n n (0) f f4 0 i (4) wherea the tranvere Young modulu and hear modulu are: 2 f2 (0) (0) f (5) G2 f3 (0) (0) f The Poion ratio are: 0.5 f 2 f2 2 with ( h / l in ) f 3 co co f2 2 ( h / l in )in ( h / l in ) f i i i i i ( h / l ) ( 2 h / l )co 3 2co ( h / l in ) f4 ( h / l 2) where, 3 () () 2 () () () ( i) ( i) 2 3 ( i) ( i) t b ( i ) t (8) ( ) and i are the bulk and urface Young moduli in the principal direction ( i ) (zeroth level), repectively; b and t are width and thickne of cro-ection of cell wall, repectively; l (6) (7)

4 Nicola M. Pugno and Qiang Chen / Phyic ngineering 0 (20) and h are length of inclined and vertical beam, repectively; i the included angle made by the inclined beam and the horizontal line (Fig. 4). Note that, the reciprocal theorem hold: (9) qn. (4)-(6) how that the tranvere Young modulu and hear modulu can be derived from the longitudinal Young modulu, and two Poion ratio are only related to the geometry of the n-level tructure. From eqn. (8), we note that the Young moduli and hear modulu are modified by a factor. If t / b 3 (i.e. plate), eqn. (8) can be expreed a: (0) ( i) 6 ( i) ( i) t xpreion (9) coincide with the reult from [20], and it obey the caling law lin / t [2] (0) with l (0) in / and 6.0. Note that, l in i a material intrinic length, under which urface effect play an important role compared to bulk; i a dimenionle contant, which depend on the geometry of the tructural element and their deformation. Beide, we can ee that the urface effect make the tructure tiffer if 0 ; otherwie, it make the tructure ofter The tiffne-to-denity ratio The tiffne-to-denity ratio can be derived eaily baing on eqn. (4) and (5), i.e. 2 (0) n n n 0 f f4 0 i () 2 f2 (0) f G2 f3 (0) f (2) 4. Parametric analyi and dicuion Here, we ue ilver (Ag) a the contituent material and a five-level hierarchical nano-honeycomb tructure. The Young modulu and denity of Ag are 78GPa and 0.94g/cm 3, repectively; the urface elatic modulu i.22n/m on the (00) urface A []; the thickne of the cell wall at the firt level i t () 5nm. For the reaon that we dicued in ection 3, only i treated here. Firt, we invetigated the ( i) ( i) influence of the relative denity (0.2, 0.3, 0.4). Second, under the condition of / 0.3, the influence of h / l and are tudied. The analytic reult of the longitudinal Young modulu and the tiffne-to-denity ratio are reported in Fig. 5 and Fig. 6, repectively. They how that the ( i ) ( i) longitudinal Young modulu increae a /, h / l or increae, but decreae a level n increae: there i no any maximum value (Fig. 5(a), Fig. 6(a) and (b)); in contrary, the tiffne-todenity ratio ha an optimal value at level 2 for / 0.4 (Fig. 5(b)) and at level 3 with the ( i) ( i) variation of h / l (Fig. 6(c)). Fig. 6(d) implie that tiffne-to-denity can alo reach a maximum value with the increae of. Fig. 5 and Fig. 6 indicate that the elatic behaviour can be tuned by changing the relative denity and geometric configuration. Increaing or h / l the Young modulu increae. The reaon i

5 3030 Nicola M. Pugno and Qiang Chen / Phyic ngineering 0 (20) (a) (b) Fig. 5 Longitudinal Young modulu and tiffne-to-denity ratio v level n. (a) Influence of the relative denity on the tiffne; (b) Influence of the relative denity on the tiffne-to-denity ratio (a) (b) (c) (d) Fig. 6 Longitudinal Young modulu and tiffne-to-denity ratio v level n. (a) Influence of h / l on longitudinal Young modulu (for each h / l, 70 0i ); (b) Influence of on longitudinal Young modulu (for each, h / l i ); (c) Influence of h / l on tiffne-to-denity ratio (for each h / l, 70 0i ); (d) Influence of on tiffne-to-denity ratio (for each, h / l i). that their increae lead to the reduction of train and under the ame external force, the tiffne increae. If (or h / l ) increae, while h / l (or ) decreae, then, the two variation of or h / l reult in an invere tendency (i.e., the former make the Young modulu larger; the latter make the

6 Nicola M. Pugno and Qiang Chen / Phyic ngineering 0 (20) Young modulu maller) about the elatic parameter, o, there exit optimal value (Fig. 6 (c)). Thi finding ugget next trategie toward the deign of uper-tiff cellular olid. 5. Concluion We have calculated the in-plane elatic propertie of hierarchical nano-honeycomb. The urface effect modifie the claical reult of non-hierarchical honeycomb and it can tiffen or often the elatic propertie of the tructure. mploying an iterative proce, we derived the tiffne and tiffne-todenity ratio at level n. The parametric analyi reveal the influence of the relative denity and of two key important parameter on the tiffne and tiffne-to-denity ratio, and it how that the elatic propertie can be optimized by tuning thee parameter. The preent theory may have many intereting application. Reference [] Foo CC., Chai GB, Seah LK. Mechanical propertie of Nomex material and Nomex honeycomb tructure. Compo Struct 2007; 80; [2] Gibon LJ, Ahby MF. The Mechanic of Two-Dimenional Cellular Material. Proc R Soc Lond A 982; 382; [3] Papka SD, Kyriakide S. In-plane compreive repone and cruhing of honeycomb. J Mech Phy Solid 994; 42; [4] Papka SD, Kyriakide S. In-plane cruhing of a polycarbonate honeycomb. Int J Solid Structure 998a; 35; [5] Papka SD, Kyriakide S. xperiment and full-cale numerical imulation of in-plane cruhing of a honeycomb. Acta Mater 998b; 46; [6] Gibon LJ, Ahby MF. Cellular olid: tructure and propertie, 2nd ed. Cambridge: Cambridge Univeriry Pre; 997. [7] Cammarata RC. Surface and interface tre effect in thin film. Prog Surf Sci 994; 46; -38. [8] Gurtin M, Murdoch AI. A continuum theory of elatic material urface. Arch Rat Mech Anal 975; 57; [9] Wang GF, Feng XQ. Surface effect on buckling of nanowire under uniaxial compreion. Appl Phy Lett 2009; 94; 493. [0] Shankar MR, King AH. How urface tree lead to ize-dependent mechanic of tenile deformation in nanowire. Appl Phy Lett 2007; 90; 4907 [] Wong W, Sheehan P, Lieber CM. Nanobeam mechanic: elaticity, trength and toughne of nanorod and nanotube. Science 997; 277; [2] Pugno NM. Mimicking nacre with uper-nanotube for producing optimized uper-compoite. Nanotechnol 2006; 7; [3] Pugno NM, Boia F, Carpinteri A. Multicale tochatic imulation for tenile teting of nanotube-baed macrocopic cable. Small 2008; 4; [4] Wang XS, Xia R. Size-dependent effective modulu of hierarchical nanoporou foam. urophy Lett 200; 92; [5] Cai X. Wood modification for valued-added application uing nanotechnology-baed approache. PhD. Thei 2007; Univerité Laval; Canada. [6] Lake R. Material with tructural hierarchy. Nature 993; 36; [7] Tolf G. Saint-Venant Bending of an Orthotropic Beam. Comp Struct 985; 4; -4. [8] Roark RJ, Young WC. Formula for tre and train, 5th ed. New York: McGraw-Hill; 975. [9] Shenoy VB. Atomitic calculation of elatic propertie of metallic fcc crytal urface. Phy Rev B 2005; 7; [20]Miller R, Shenoy VB. Size-dependent elatic propertie of nanoized tructural element. Nanotechnol 2000; ; [2] Wang J, Duan HL, Huang ZP, Karihaloo BL. A caling law for propertie of nano-tructured material. Proc R Soc A 2006; 462;