MECHANICAL PROPERTIES OF 3D RE-ENTRANT AUXETIC CELLULAR STRUCTURES

Transcription

1 21 t International Conference on Compoite Material i an, th Augut 2017 MECHANICAL PROPERTIES OF D RE-ENTRANT AUETIC CELLULAR STRUCTURES in-tao Wang, Bing Wang, iao-wen Li, Li Ma* * Center for Compoite Material, Harbin Intitute of Technology, Harbin , PR China addre: mali@hit.edu.cn (Li Ma), webite: Keyword: Auxetic tructure, Cellular olid, Energy method ABSTRACT In thi work, an analytical model of a D re-entrant auxetic cellular tructure ha been etablihed baed on energy method. Analytical olution for the modulu and Poion ratio of the cellular tructure in all principal direction were deduced. The reult how that when the trut are relatie tubby, the oerlapping of the trut a well a axial extenion or compreion hould be taken into conideration for modelling. 1 INTRODUCTION Auxetic wa firtly coined by Ean to define material with a negatie Poion ratio for conenience [1]. It ha been uggeted that the negatie Poion ratio character make auxetic material exhibit excellent mechanical propertie uch a increaed hear modulu [2], uperior indentation reitance [], improed fracture toughne [4], higher energy aborption ability [5], poroity/permeability ariation with train [6], unique ynclatic curature [7]. Due to thee excellent propertie, auxetic material hae great potential for many application field [8]. A promiing metamaterial, auxetic material hae attracted great interet. Figure 1 : (a) 2D re-entrant auxetic tructure. (b) D re-entrant auxetic tructure. (c) Unit cell of the D auxetic tructure and it dimenion. (d) Geometry parameter. The mechanical property of a D re-entrant auxetic cellular tructure that exhibit auxetic behaior in all three principal direction i of concern in thi paper, a hown in Figure 1(b). The D tructure i extended from the 2D re-entrant hexagonal honeycomb tructure [9], a hown in Figure 1(a). In thi paper, the D re-entrant auxetic cellular tructure wa theoretical analyzed baed on energy method. Analytical olution for the modulu and Poion ratio of the cellular tructure in all principal direction were deduced.

2 in-tao Wang, Bing Wang, iao-wen Li, Li Ma 2 STRUCTURAL DESIGN AND SIMPLIFICATION The unit cell of the tructure i hown in Figure 1(c). The mechanical propertie along and Y axe are identical due to the geometry ymmetry. For implicity, all the trut were aumed to hae the ame quare cro ection. Then the geometry of tructure can be decribed by four parameter: length of ertical trut ( h ), length of oblique trut ( l ), re-entrant angle between the ertical trut and the oblique trut (θ ), and the ide length of the trut cro ection ( t ), a hown in Figure 1(d). For the modeling, ome implification were made. Firtly, the boundary effect need not to be conidered. Secondly, deformation mechanim of the tructure conit of flexure, tretching and hearing. Thirdly, train are mall enough that gro change in geometry do not occur. Fourthly, all of the joint in the tructure are conidered to be rigid. Becaue of the oerlapping a hown in Figure 2, the effectie length of both type of trut are horter than the deign alue. Figure 2: (a) Etimation of the length reduction of the ertical trut. (b) Etimation of the length reduction of the oblique trut. A hown in Figure 2(a), the length reduction of ertical trut at one end can be etimated a t 2 h = (1) tan( θ 2) A hown in Figure 2(b), the length reduction of oblique trut at one end can be etimated a where l = ϕ l + (1 ϕ) l 0 ϕ 1 (2) 1 2 t 2 inθ l1 = and l2 t 2 = () tan( θ 2) and the coefficient ϕ i an experiential parameter. It wa found that ϕ = 0.6 can make the analytical reult agree well with the numerical reult. MODELING.1 Uniaxial compreion in direction Due to the ymmetry of the D tructure in direction and in Y direction, the 2D re-entrant tructure can be ued to analyze the D tructure under compreion loading along direction. And the repreentatie element RQR' Q' wa ued for analyi, a hown in Figure (b).

3 21 t International Conference on Compoite Material i an, th Augut 2017 Figure : (a) The 2D re-entrant tructure and the arrow element ued to analyze the D tructure. (b) Loading condition of the arrow element RQR' Q '. where and Then, Poion ratio in direction under compreie tre can be obtained a ε 2l inθ h lcoθ = = = ε 2l inθ h lcoθ The effectie modulu of the re-entrant cellular tructure can be determined a 4F1 2 (2lin θ ) F1 h E = = = 2 2 ε l in θ h lcoθ ( lcoθ ) ( ) ( ) ( ) F l 2 l inθcoθ F l 2 l inθcoθ 6F l 2 l inθcoθ = + (6) 12EI ( 2 ) in 2 θ ( 2 ) co 2 θ ( 2 ) in 2 θ ( 2 ) F l l F l l F l l F h h = (7) 24EI 2 E and G are the Young' modulu and hear modulu of the olid material eparately, A i croection area, I i econd moment of area. (4) (5).2 Uniaxial compreion in direction Due to ymmetry, the ertical trut do not experience any effectie load when the tructure i uniaxial compreed in direction. Therefore, the deformation of the tructure wa olely determined by the oblique trut. It hould be note that re-entrant trut QR ' (type 1) and Q'' R ' (type 2) are not mechanically ymmetrical, therefore hould be conidered eparately, a hown in Figure 4.

4 in-tao Wang, Bing Wang, iao-wen Li, Li Ma Figure 4: (a) Compreie loading of re-entrant auxetic tructure in direction. (b) Loading condition of the re-entrant trut QR '. (c) Loading condition of the re-entrant trut Q'' R '. Then, the diplacement of point R ' along direction QR ' can be obtained a ( ) ( ) ( )( + ) F1 l 2 l coθ βinθ inθ F1 l 2 l inθ βcoθ inθ QR ' = 12EI ( l)( ) 6F1 l 2 coθ βinθ inθ + where β i the ratio of the hear force S 2 to the applied force F 1. The diplacement of point R ' along direction QR ' can be obtained a The diplacement of point ( ) ( ) ( )( + ) F1 l 2 l coθ βinθ coθ F1 l 2 l inθ βcoθ inθ QR ' = + 12EI ( l)( ) 6F1 l 2 coθ βinθ coθ + R ' along Y direction Q'' RY ' can be obtained a (8) (9) ( ) ( ) ( ) βf l 2 l inθcoθ βf l 2 l inθcoθ 6βF l 2 l inθcoθ Q'' RY ' = + (10) 12EI The Poion ratio of the re-entrant auxetic tructure compreed in direction can be readily obtained: n QR ' ε co ' in h l θ QR θ = = = ε QR ' QR ' l inθ ν ε ( hl coθ) (11) Y Q'' RY ' Y = = (12) ε QR ' And the effectie modulu of the D re-entrant auxetic tructure under compreion in direction can be obtained a:

5 21 t International Conference on Compoite Material i an, th Augut 2017 E F1 ( h lcoθ) linθ F1 = = = ε QR ' QR ' l inθ ( h lcoθ) Table 1 how the comparion of the analytical model in preent tudy with the experimental reult and the analytical model in the work of Yang et al. [10]. From the table it can be een that, the analytical model in preent tudy matche much better with the experimental reult than the analytical model in the work of Yang et al. [10]. Deign Y Exp.[Y] Anal.[Y] Anal.[P] Exp.[Y] Anal.[Y] Anal.[P] Exp.[Y] Anal.[Y] Anal.[P] P P Exp.[Y] (abbreiation of experimental reult in reference [10]), Anal.[Y] (abbreiation of analytical reult in reference [10]), Anal.[P] (abbreiation of analytical reult in preent tudy). Table 1: Comparion of the analytical model in preent tudy with the experimental reult and the analytical model in reference [10] 4 CONCLUSIONS In thi paper, D re-entrant auxetic cellular tructure that exhibit auxetic behaior in all three principal direction wa analytical modeled baed on energy method. When the trut are relatie tubby, the oerlapping effect of the trut and axial extenion or compreion hould be taken into conideration. ACKNOWLEDGEMENTS The preent work wa upported by the National Natural Science Foundation of China under grant number REFERENCES [1] Ean K, Nkanah M. Molecular network deign. Nature. 1991;5:124. [2] Lake R. Foam tructure with a negatie Poion' ratio. Science. 1987;25: [] Alderon K, Fitzgerald A, Ean K. The train dependent indentation reilience of auxetic microporou polyethylene. Journal of Material Science. 2000;5: [4] Choi J, Lake R. Fracture toughne of re-entrant foam material with a negatie Poion' ratio: experiment and analyi. International Journal of Fracture. 1996;80:7-8. [5] Scarpa F, Ciffo L, Yate J. Dynamic propertie of high tructural integrity auxetic open cell foam. Smart material and tructure. 2004;1: [6] Alderon A, Raburn J, Ean K. Ma tranport propertie of auxetic (negatie Poion' ratio) foam. phyica tatu olidi (b). 2007;244: [7] Ean K. The deign of doubly cured andwich panel with honeycomb core. Compoite Structure. 1991;17: [8] Liu Y, Hu H. A reiew on auxetic tructure and polymeric material. Sci Re Eay. 2010;5: [9] Ean K, Nkanah M, Hutchinon I. Auxetic foam: modelling negatie Poion' ratio. Acta metallurgica et materialia. 1994;42: (1)

6 in-tao Wang, Bing Wang, iao-wen Li, Li Ma [10] Yang L, Harryon O, Wet H, Cormier D. Mechanical propertie of D re-entrant honeycomb auxetic tructure realized ia additie manufacturing. International Journal of Solid and Structure. 2015;69-70: