TRUSS WAVINESS EFFECTS ON MECHANICAL BEHAVIORS OF WIRE-WOVEN BULK - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS TRUSS WAVINESS EFFECTS ON MECHANICAL BEHAVIORS OF WIRE-WOVEN BULK KAGOME K. W. Lee 1 , K. J. Kang 1 * 1 School of Mechanical System Engineering, Chonnam National Univ., Gwang-ju, Korea * Corresponding author ( kjkang@chonnam.ac.kr ) Keywords : Periodic Cellular Metal (PCM), WBK (Wire-woven Bulk Kagome), Truss Waviness Effect 20% lower than those of collinear core due to the 1 Introduction waviness effect. Many studies have been conducted for various In this study, to improve the theoretical solutions, cellular mUUaterials, such as truss Periodic Cellular the truss waviness and brazed portion are taken into Metal (PCM). Because they provide not only high account to estimate the strength and stiffness of strength per density, but also make one to use WBK. And the results are compared with those interior space for additional function like heat measured by experiments and estimated by finite element analysis. transfer, catalyst support and storage. The pyramid truss [1], octet truss [2], and woven textile 2 Basic Analytical Solutions topologies [3] have been studied about mechanical Lee at el. [9] derived the analytic solution of performance, optimal designs for specific applications, and fabrication techniques. compressive strength, assuming that WBK has an ideal Kagome truss structure. Fig.2 shows unit cells Kagome truss is a recent addition to lattice truss of the ideal Kagome truss and the WBK. structures. Since the truss elements of the Kagome truss PCM have half the length of those of the Octet,  4 3  d  2   c k E   2 (1) elastic y it has excellent resistance to buckling which is a  c  128 bucking main failure mode of the truss structure, and also has  4 2  d   d  3 2 2  c    k 2 E     (2) high internal space utilization. [4, 5] inelastic y t t  c   c  128 8 bucking Lee et al. [6] introduced a new technique for 2  d  2  c     (3) fabricating multi-layered Kagome truss-like plastic y o  c  8 bucking structures using wires. Helically formed wires were In this equation,   and E are yield strength and systematically assembled in 6 directions evenly Young’s modulus of the wire material, d is the distributed in the 3D space, and then the cross points diameter of a wire and c is the length of a strut. E t among the wires were fixed by brazing to be a defined the slope ( ∂σ / ∂ε ) on stress-strain curve and k robust Kagome truss-like PCM which was named is the constants depending on the boundary WBK after Wire-woven Bulk Kagome in Fig.1. conditions at the ends. They also derived the Since the mechanical strength and stiffness of equivalent Young’s modulus of the ideal Kagome WBK have been theoretically estimated on basis of truss. assumption that WBK is composed of straight struts,   2 d 3 2    [7] the analytic solutions sometimes give substantial E e E   (4)  c  40 errors compared with experimental results. In fact, WBK is assembled with helically-formed wires. 3 Modified Analytical Solutions Consequently, the struts are curved, which resulted 3.1 Maximum bending moment in errors in estimation based on the previous theoretical solution. Recently, Queheillalt et al. [8] Park et al. [10] calculated the maximum bending derived the equation considering waviness effect of moment in a helically formed strut of WBK core strut to predict the mechanical performance about taking account of the constraints at the both ends due the metal textile lattice core. The wires were to brazed filler metal. They applied Prager’s equation [11] for failure strength,  w of a strut modeled to have a sinusoidal shape. They reported that the strength and stiffness of textile core were subjected to bending moment and axial force. The

   equivalent compressive strength is calculated as   2   d 2 d 3 F M ys ys       a   F M (11) , 1 replacing  o with  w to consider the waviness effect     o o F M     4 6 o o of the strut Eq.(3). Here, σ ys is the yield strength of the material We derive the moments considering not only the (200MPa in SUS304), d and A are the diameter and axial force and bending moment but also the shear cross section area of the strut. forces acting one of out-of-plane struts in a Because the force and moments are expressed as tetrahedron consisting WBK truss core in Fig.3(a). functions of the F a in all the above equations, we The axial and shear forces in the strut are related to calculate the critical value of the axial force, F a,cr , to each other by elementary beam theory as follows satisfy Prager criterion by substituting Eq. (10) into [12]; See Fig.3(b). Eq. (11). Then, the critical vertical load, P cr , applied    2 d 3 cos at the top of a tetrahedron consisting WBK in  F   F (5) s   a  c B  4 2 sin Fig.3(b) can be calculated. Finally, the equivalent A helical wire is projected as trigonometric compressive strength follows as: functions on 2-D planes in Fig.3(c). For each of the   2 P  d  3 3 3 2     cr     F F   (12) axes, M y (z), M x (z), T , The following equations are sin cos   c a cr a cr  A  c B  c 2 8 2 2 3   s derived:    z B    ( ) 3.2 Equivalent Young’s modulus        M z F  C z C z a J  ( ) cos sin 1 cos (6) y a  1 2 c  Park et al. applied Castigliano’s second theorem    z B   ( )        to obtain Young’s modulus of the curved strut, E w . M z F  C z C z a J ( ) cos sin 1 sin (7) x a  3 4 c They also calculated the equivalent Young’s    2 d 3 2    modulus by replace E in Eq. (4) with E w . In the    z c B   2 ( 2 )    c B  16 2  similar way, we applied Castigliano’s 2 nd theorem to     z B B  derive the displacement at the top of the helically ( )     T R a sin sin (8)   formed strut, δ a .  c c     Here, a is the radius of the helix, B is the height of F M 1 c  B 1 c  B  2  2   a  x F dz M dz  (13) the brazed portion, c represents the length of the a a  x  AE F E I F  0 0 s a s a element. C 1 , C 2 , C 3 , C 4 are constants calculated by a   M  R 1 c  B 1 c  B  2  2   y  M dz R dz fixed boundary conditions under the influence of y   E I F AG F 0 0 s a s a brazed portions. To simplify the formula by λ , J as a   T 1 c  B  2 substitute, R' is as follows:  T dz   JG F  0   s a F a    2         J   (9) Here, F a axial force acting on the wire, c is an 2 2   EI  c    element length of the tetrahedral configuration and d    B is the diameter of the wire, B is the height of the  a  2 cos   2 d c 3 2    R   F    brazing part, I is the moment of inertia, G s is the a  c  B  c  B   2 8 2   shear modulus, J is the polar moment of inertia, E s is   Finally, the equivalent maximum bending moment Young’s modulus of the wire. And we calculated the equivalent Young’s modulus E e of WBK core can be calculated by the following equation: follows as: 1 1        M M 2 M 2 M 2 M 2 T 2 (10) x y x y   2 2   2 F A F d 3 3 2 1         E s a   (14) 2 sin sin cos   e     h c  c B  6 8 2   3.1 Equivalent compressive strength a 4 Finite Element Analysis Prager’s equation is used to determinate whether the strut fails or not. We also used the equation Finite element analysis was performed using the combined the axial force and the bending moment Periodic Boundary Condition (PBC) in order to on the assumption that perfectly plastic behavior verify the equivalent compressive strength and occurs in the wire. Yong’s modulus. Fig.4 shows the PBC model of the WBK unit cell composed the wires and brazed filler