Analysis of Mesostructure Unit Cells Comprised of Octet-truss Structures

Transcription

1 Analyi of Meotructure Unit Cell Compried of Octet-tru Structure Scott R. Johnton *, Marque Reed *, Hongqing V. Wang, and David W. Roen * * The George W. Woodruff School of Mechanical Engineering, Georgia Intitute of Technology, Atlanta, GA R&D Group, IronCAD, 7 Galleria Pkwy, Atlanta, GA 9 Abtract: A unit tru finite element analyi method allowing non-linear deformation i employed to analyze a unit cell compried of n octet-tru tructure for their tiffne and diplacement compared to their relative denity under loading. Axial, bending, hearing, and torion effect are included in the analyi for each trut in the octet-tru tructure which i then related to the meotructure level (unit cell). The veratility of additive manufacturing allow for the fabrication of thee complex unit cell tru tructure which can be ued a building block for macro-cale geometrie. The finite element calculation are compared to experimental reult for ample manufactured on a Stereolithography Apparatu (SLA) out of a tandard rein. Introduction Lightweight compliant tructure are becoming more deirable in many area of indutry due to the advancement in manufacturing method that are now capable of fabricating their complex geometrie. Component with an internal tru tructure deigned for pecific loading application i an example of a lightweight tructure. The automotive and aeropace indutrie are very intereted in lightweight tructure becaue lightweight tructure can improve their product performance. Tru tructure application have been limited in the pat becaue of the large cot required for fabrication, or the tructure wa impoible to fabricate uing traditional manufacturing method. Additive manufacturing i now capable of fabricating lightweight tru tructure, poibly with imilar mechanical propertie to conventional manufactured component with the ame exterior component deign. The initial tep for introduction of tru tructure in commercial application require mechanical analyi of the capabilitie of the tru tructure to determine their mechanical propertie. Thi will allow deigner to chooe tru deign with knowledge of their trength, tiffne, and weight. Since tru tructure are a erie of trut and node, they behave differently from olid component. The development of fat computing ytem ha allowed Finite Element Analyi (FEA) to become a tandard method for component deign analyi, but mot FEA code uually ue element dicretely in the form of beam, rod, plate, or olid element. The node of the tru tructure phyically poe the loading contraint (tenile, compreion, torion, elongation, and buckling) of many different type of the aforementioned element type, but an element poeing all of the unique loading condition which a tru tructure node experience doe not currently exit. A unit tru approach FEA program ha been developed at Georgia Tech [] to pecifically analyze complex tru tructure (decribed in more detail in Section.4) which include tenile, compreive, torion, and buckling effect due to tructural loading. The unit tru FEA analyi i compared to experimental reult for tiffne and diplacement for a pecified relative denity of a unit cell. Each unit cell contain a different number of octet tru tructure [] poeing 4

2 the ame relative denity of.5. A comparion of experimental and unit tru FEA approach are preented in Section.4. Meo-tructure deign analyi of component Component can be geometrically decompoed into maller element for analytical purpoe which i preciely the analytical proce employed by Finite Element Analyi (FEA). The ame idea can be ued to ucceively decompoe a component (macro-cale) into medium ized element (meo-cale) which i further ubdivided into maller building block (micro-cale). Thi decompoition ideology i the bai for the unit cell deign and analyi approach to lightweight component by uing tru tructure within the unit cell.. Unit cell approach for component deign A multi-cale deign approach i employed to create an appropriate tru tructure for pecific component geometrie. The geometric deign of the component can be decompoed into meotructure unit cell (tructure poeing feature ize between micro and macro-cale a diplayed in Figure a) uch that the meotructure are ued a the baic building block for the component geometry. Each meotructure unit cell i further decompoed into maller tru tructure (Figure c) where the octet-tru tructure (Figure b) ha been choen to be the building block for each unit cell [] becaue the mechanical behavior of the trut (lattice tructure) can be predicted more readily than material with random void, uch a foam. Performing thi decompoition create a tranition from a macro cale (the component level) to a meo-cale (unit cell) and finally to the micro-cale (octet-tru). An example of thi decompoition i diplayed in Figure to illutrate the multi-cale analyi of the component (a) (b) (c) Figure. (a) Component geometry decompoed into unit cell, (b) octet-tru tructure, and (c) a unit cell containing a xx octet-tru tructure. There are many reaon for employing the unit cell deign methodology. Firt, by uing unit cell with known mechanical behavior, the unit cell can be ued a an element type in a Finite Element Analyi (FEA) which reduce the computational intenity of the FEA calculation compared to the current FEA definition of numerou node and trut compriing the tru tructure. A econd advantage in uing thi meotructure approach provide the deigner local control of the mechanical behavior at the component level by altering the compoition of the unit cell at the meotructure level. Thi advantage alo allow for a maximum component performance with repect to weight under defined loading condition becaue le dene unit cell may be ued in location that carry le of the applied load. The following invetigation of unit cell compried of array of octet-tru tructure will expand upon previou reearch of the octet-tru tructure by: () including bending, hearing, and torion effect in addition to the tenile effect and () analyi of array of octet-tru 4

3 tructure that include the interaction between neighboring octet. Thi additional information about unit cell octet-tru tructure provide neceary information to create a unit cell library to contruct macro level component geometrie.. Octet-tru tructure analyi The octet-tru tructure (diplayed in Figure b) wa choen a the building block for the unit cell becaue of a previou analyi of the tructure [] and it good mechanical propertie under mot type of loading condition. The initial analyi of the octet-tru tructure wa performed by Dehpande, Fleck, and Ahby [] where their analyi rigorouly tudied the octahedral portion (Figure a) of the octet-tru tructure for tiffne, compliance, and buckling. They tate that the octahedral portion of the octet-tru tructure will contribute the majority of the tiffne of the tructure under compreive loading. Therefore their analyi i retricted to the octahedral ection of the octet-tru tructure and doe not include the tetrahedral ection of the tructure (black ection diplayed in Figure b). Herein, a further analyi of the octet-tru tructure i preented that include the eight tetrahedron of the octet tructure for compliance. Both of thee algebraic preentation conider each trut to be pinned at their repective node. An algebraic formulation for trut that are fully contrained at each node become too complex when all of the loading condition (tenile, bending, hear, and torion) are included and will not be preented. (a) (b) Figure. (a) Octahedral ection of the octet-tru tructure (red inner ection) and (b) octettru tructure with highlighted a corner tetrahedron ection (black ection). The relative denity of a unit cell will be ued a a metric for characterization of the mechanical behavioral propertie of the cell. The unit cell relative denity i determined by the thickne of each octet-tru trut and number of octet-tru tructure located within the cell defined by a 6 () l where a i the trut diameter and l i the length of each trut. Equation repreent the material volume of the incribed unit cell containing an octet-tru tructure, pecifically, only half of the volume of each trut on each of the ix face i included becaue the remaining volume lie in the volume for adjacent cell. Each unit cell i analyzed for it tiffne (compliance) with repect to 4

4 the relative denity. Unit cell containing n x n x n array of octet-tru tructure are alo be analyzed (for n =,,). Previou invetigation of the octet-tru tructure by Dehpande, Fleck, and Ahby [] aumed that each node of the tructure wa pinned and free to rotate about the node, thu auming only tenile condition are preent within the trut (trut are either in tenion or compreion). Additionally, their analyi primarily involved invetigation of failure of the octettru tructure due to elatic buckling or platic collape. Thu, bending moment, hearing force, and torion effect were not included within their analyi... Octahedral tructure analyi Dehpande, Fleck, and Ahby [] have determined that the octahedral ection of the octettru tructure (diplayed in Figure a) determine the tiffne for the entire tru tructure. For their analyi, they aume that all of the joint are pinned and only axial force are preent within the trut of the octahedral ection. The compliance matrix for only the octahedral tructure i iotropic [4] and i determined algebraically to be xy xz yz z y x xy xz yz z y x ym () and expreed in matrix form C () where E E a l 9 4 (4a) E E a l 4 (4b) and E E a l 6 (4c) where E i the elatic modulu for the olid trut material. (Note that in Equation 4c i different than printed in [], we believe that there i a typographical error in the publication.) The compliance matrix [C] tated in Equation i for a pinned node octahedral tructure (Figure a) and not the entire octet-tru tructure (Figure b). Dehpande analyi ue the compliance matrix [C] to determine the tiffne of the octet-tru and then make a comparion to FEA reult. The eight tetrahedral tructure that are required in conjunction with the octahedral tructure to create the octet-tru tructure are not included in thee calculation and are not repreented in Equation

5 .. Pinned octet tru tructure analyi The author concur with Dehpande that the majority of the compreive or tenile load are aborbed within the octahedral ection of the of the octet-tru tructure, but the tetrahedral element at the corner alo contribute to the compliance or tiffne of the complete octet tructure. Thi i epecially true when analyzing array of octet-tru tructure, which will be illutrated in the following ection. The direct tiffne method can be ued to determine the compliance matrix and reproduce Equation -4 for a pinned node octet-tru tructure []. The compliance matrix [C ] for the octet-tru tructure ha the ame matrix equation C (5) where the octet compliance matrix [C ] for a pinned node octet-tru tructure (Figure b) in Equation 5 ha the ame matrix tructure a Equation with the i value in Equation 4 replaced by the i value preented in Equation 6a-c which have been algebraically determined and l 6 (6a) a E E l (6b) a E E l 6 (6c) a E E The incluion of the eight tetrahedron at the corner of the octahedral tructure (thu creating the octet tructure) doe effect the compliance matrix [C], pecifically, the and term. The coefficient for decreaed from 9 to (octahedral to octet tructure) and the decreaed from to. 8, both decreaing by ~5%). The decreae in the compliance coefficient implie that the octet tructure i le compliant (or more tiff) than the octahedral tructure (along the principle load axi) and in tranvere direction under loading (ee Figure 6a). The hearing term did not change between the octahedral and octet analyi becaue the twiting effect of the octet tru tructure wa not contrained to remain planar under the applied hearing load (i.e., the corner node were allowed to move out of plane due to the twiting nature of the hearing force). Since thi wa only an analyi of a ingle octet, the hearing effect will become more ignificant when the octet tructure are combined to create array of octet (i.e., unit cell).. Unit tru Finite Element Analyi (FEA) approach Unit cell compried of octet tru tructure are currently being analyzed uing a unit-tru finite element analyi program in MATLAB that ha been developed at Georgia Tech []. The unit-tru finite element program i capable of non-linear analyi, but for the current tage in thi project, only linear elatic behavior i conidered. The FEA program include axial, bending, hearing, and torion effect that are preent in loading of the octet tru tructure where previou analye only included axial effect []. A dicuion of the effective mechanical behavior of the unit cell uing the unit tru FEA program will be dicued. 45

6 In the unit tru approach, a unit tru i ued a a new unit cell for mechanic analyi of cellular tructure, including lightweight tructure, and compliant mechanim. A unit tru conit of a central node and a et of half-trut that are connected to the node. Every two neighboring unit true hare a common trut. An example of unit true i hown in Figure. Unit Tru Unit Tru Unit Tru Figure. Serie of three unit tru tructure that are connected between each node. Unit true can be parameterized, analyzed, patterned, and manufactured to upport the deired deign. In a unit tru, the train and tre around the node (diplayed in Figure 4), are uually complicated due to coniderable inter-trut interaction and large bending moment []. The unit tru i leveraged from the ground tru approach and homogenization method [5, 6]. The contitutive equation of -D and -D unit true are hown in Equation 7-9. The linear elaticity of a unit tru i repreented by K e, while U and F repreent the nodal diplacement and force. Unit true can have any number of incident trut and they are pecial finite element for analyzing large cellular tructure. () u () () u u () u () u () u () u () u () u () u (5) u () u () u (5) u (5) u (4) u (4) u (4) u Figure 4. Unit tru nodal degree of freedom for a unit tru node with five attached trut. Static equilibrium: Ke U F N () i () () ( N) i () () Stiffne: Ke () () N N ( ) ( ) ( N) ( N) (7) (8) 46

7 () () ( N ) Nodal Diplacement: U u u u ~ ~ ~ Nodal force: () () ( N ) F f f f ~ ~ ~ A unit tru can be manufactured, wherea the microtructure for homogenization can not be phyically fabricated becaue homogenization i an artificial repreentation. Uing the unit tru approach, combined with the tangent tiffne method, tru tructure can be analyzed under axial force, bending, torion, nonlinearity, and buckling [7]..4 Unit cell compried of octet tru tructure The unit cell analyi will be applied to three different type of unit cell that are compried of octet-tru tructure, diplayed in Figure 5. The analyi i baed on the approach ued by Dehpande by relating the relative tiffne (E/E ) to the relative denity ( in Equation ) for a fixed ized unit cell. The relative denity for a unit cell geometry i altered by varying the diameter of the trut of the octet-tru tructure. Unit cell compried of a ingle octet, a xx array, and a xx array are analyzed within thi work, but thi analyi can eaily be expanded to arbitrary unit cell and octet array. T T (9) (a) (b) (c) Figure 5. CAD image of unit cell compried of octet-tru tructure, (a) ingle octet, (b) xx array of octet, and (c) xx array of octet..5 Unit tru FEA analyi of unit cell A direct comparion of the relative tiffne of unit cell that are compoed a ingle octet and an array of octet-tru tructure i preented herein. To maintain uniformity between analye of the different meotructure (unit cell contructed from an array of octet), the dimenion of the unit cell i et to a fixed value and the applied compreive load i appropriately ditributed to each node creating the ame uniform applied tre. A comparion between the unit-tru FEA approach (decribed in Section.) of a meotructure unit cell and Dehpande analyi of the octet-tru tructure i detailed below. Only linear elatic deformation have been conidered in thi work, but the unit-tru FEA program i capable of determining non-linear deformation and buckling effect, which will be analyzed in future work. Figure 6 diplay the relative tiffne plotted with repect to the relative denity of the meotructure unit cell. An increae in the relative tiffne i oberved a the relative denity increae for the unit cell. To increae the relative denity of the unit cell i to increae the trut diameter, and a larger diameter trut will poe a greater tiffne value that tranlate to the overall relative tiffne of the tructure. Figure 6a diplay the relative tiffne (E/E ) for an 47

8 octahedral and octet tru tructure (Equation 4 and 6, repectively) at the center node (point p c in Figure b). Note that the incluion of the eight corner tetrahedral ection to the octahedral ection to create the octet tructure only lightly increae the relative tiffne of the tructure a tated earlier in Section.... octahedral. octet.5 E.5. E. E. E..5.5 Single octet relative denity. xx xx Octet (Eq. 6) relative denity (a) (b) Figure 6. Log-log plot of the relative tiffne (E/E ) vere relative denity for (a) the pinned node octahedral (Eq. 4) and octet tructure (Eq. 6), and (b) the pinned node octet tructure and the unit-tru FEA analyi of unit cell (analyi include tenile, hear, bending, and torion). The unit-tru FEA reult of unit cell compried of a ingle, xx, and xx array of octet tru tructure i diplayed in Figure 6b. The general trend of an increae in relative tiffne (E/E ) i oberved, but the unit-tru FEA reult indicate that the rate of change in relative tiffne with repect to relative denity i le than predicted by the algebraically determined relative tiffne (Equation 4 and 6). A comparion between Eq. 6 (valid for only a ingle octet) and the unit-tru FEA reult for a ingle octet indicate that Eq. 6 predict higher value of relative tiffne for., a diplayed in Figure 6b. The unit-tru FEA reult alo indicate that a the number of octet tru tructure increae within a fixed unit cell geometry, the relative tiffne alo increae. Specifically, a unit cell with a fixed relative denity ha a greater relative tiffne with 7 (xx) octet-tru tructure (Figure 5c) when compared to the unit cell containing eight (xx) octet-tru tructure (Figure 5b). Therefore, unit cell compried of octet-tru tructure with a fixed relative denity become more tiff (le compliant) when the number of octet-tru tructure increae. In order to increae the number octet within the unit cell and till maintain the ame relative denity, the trut diameter and length mut both be decreaed. Indicating that the relative tiffne i more enitive to the trut length than the trut diameter (by Equation ). Unfortunately, thi reult wa not oberved experimentally, which i preented in Section. Experimental compreion tet of unit cell compried of octet-tru tructure Unit cell compried of octet-tru tructure were fabricated and ubjected to a mechanical compreion tet to determine their material propertie. The unit cell were compried of a ingle, xx, and xx array of octet-tru tructure, a diplayed in Figure 5. Thee unit cell tet pecimen were created to oberve the effect on the unit cell mechanical propertie with repect to the number of compoing octet-tru tructure. The compreion tet procedure for the unit cell followed the tandard compreion tet method for determining compreive propertie of rigid platic, ASTM D 695-a. 48

9 . Specimen deign and fabrication A relative denity value of.5 wa choen to compare the effect of changing the number of octet-tru tructure in the unit cell. The tet pecimen unit cell wa deigned to be a 5 mm cubic cell, containing n octet-tru tructure (n =,, or ). In order to create geometric condition a cloe to the octahedral analyi conducted by Dehpande [], the octet-tru tructure were arranged uch that there are either, 8, or 7 octahedral tructure in the unit cell (formed by the octet-tru tructure). The geometry of the unit cell are diplayed in Figure 5. The trut length and diameter of the octet-tru tructure are calculated uch that each unit cell poee the ame relative denity,. 5. The average value for the overall dimenion of the three different type of unit cell i lited in Table. The ASTM ample pecimen are right circular faced cylinder that are 5.57 mm tall with a diameter of.67 mm. Table. Average dimenion meaurement of unit cell tet pecimen with. 5 (6 tet ample for each group). Table. Material propertie of RenShape SL 75 rein. Array Strut Dia. Height Width Width XY Projected Material Property Value Size (mm) (mm) (mm) (mm) Area (mm ) Tenile mod. (E ).64 [GPa] Single Tenile Str. ( Y ) 57 [MPa] xx Elongation at break. % xx Denity.8 [g/cm ] The experiment pecimen were manufactured on a D Sytem SLA5 uing RenShape SL 75 rein. The exact build tyle wa ued for pecimen fabrication with a layer thickne of.6 mm (.4 in) and a beam width of 5 m. The ample were ultraonically cleaned in iopropyl alcohol for minute and pot cured in a UV oven for 6 minute. The ample were iolated from UV light and ealed in a platic bag to minimize any environmental effect on the pecimen before teting. The ample were teted approximately 4 day after fabrication.. Empirical compreion meaurement and reult The unit cell pecimen with relative denity of.5 exhibited both brittle and ductile failure repone. The unit cell containing a ingle octet and the xx octet array both reulted in catatrophic failure with little or no audible warning before failure (complete detruction). The tre-train curve (diplayed in Figure 7) clearly how a brittle failure repone for the ingle octet and xx array unit cell. The xx octet array unit cell exhibited a more ductile failure accompanied by audible cracking before yielding. Some of the xx octet array did not catatrophically fail and thee pecimen remained relatively intact after teting. The initial behavior of the xx and xx unit cell were very imilar poeing almot the ame tiffne (.6 and.65 GPa, repectively), but the xx had a brittle failure repone while the xx ample exhibited a more ductile failure repone. The average experimental reult for the unit cell octet array i provided in Table. The amount of train produced in the unit cell increaed a the number of octet-tru tructure increaed, pecifically from x.7 for the ingle to x.889 for the xx unit cell. Ironically, the ame trend can not be tated for the maximum allowed tre and unit cell tiffne. The maximum applied tre and the tiffne for the xx octet array unit cell are le than the value for both the ingle and xx octet array unit cell. The tiffne of the 49

10 xx octet array unit cell (E emp =.65 GPa) wa ignificantly le tiff than the ingle octet unit cell (E emp =.7 GPa) and required approximately half of the applied load (F max ) to achieve 8% more train than the ingle octet unit cell. Table. Average meaured and calculated value of the experimental compreion tet. Octet Array max F max max max E emp E emp /E Size (mm) (kn) (mm/mm) (MPa) (GPa) Single xx xx ASTM Sample N/A Note that the tiffne for the SL 75 rein ha been determined to be E ASTM =.8 GPa by the ASTM ample reult, which i approximately / of the manufacturer tated value of E =.64 GPa (preented in Table ). For ake of implicity, the manufacturer tated modulu value for E will be ued in all the following relative tiffne calculation. Octet-tru Stre (MPa) 5 4 ASTM pecimen Single xx xx Strain (mm/mm) Figure 7. Stre train plot of the compreion tet for the unit cell compried of octet-tru tructure (all poeing a relative denity of.5) and the ASTM pecimen. The ingle octet unit cell poeed the greatet relative tiffne (E/E ) of.45 (or approximately 4% of E ) while the xx poeed approximately half the tiffne of the ingle octet with a relative tiffne of.47 (or.47% of E ). All of the unit cell pecimen experienced total diplacement ranging mm reulting from maximum load ranging from kn.. Unit tru FEA computation for unit cell compried of octet-tru tructure A FEA imulation of the compreion experiment conducted in Section. uing the unit tru FEA program i preented in thi ection for the octet-tru tructure (unit cell). The geometric parameter of the unit cell (preented in Table ) and material propertie for SL75 rein (preented in Table ) are ued in the unit tru FEA program to produce the reult ASTM Stre (MPa) 4

11 diplayed in Table 4. The unit tru FEA program calculated ignificantly le maximum train ( max ) of the unit cell by two order of magnitude. Table 4. Unit tru FEA reult for unit cell uing experiment loading condition. Octet Array Applied max max E calc Size Stre (MPa) (mm) (mm/mm) (MPa) E calc /E Single xx xx The unit tru FEA analyi produce the ame trend in relative tiffne between the unit cell that i oberved experimentally. The FEA analyi ha determined that the xx octet unit cell i the mot compliant (leat tiff) of the octet-tru unit cell and allowing the larget amount of train to be produced..4 Comparion of analytical, experimental reult, and,unit tru FEA computation A comparion of the analyi approache of the octet-tru unit cell preented in Section.,., and. i diplayed in Table 5. The relative tiffne equation (Eq. 4 and 6) are for a ingle tru tructure (octahedral and octet, rep.) and are alo ued for unit cell compried of array of octahedral/octet tru tructure (Dehpande [] alo employed thi type of caling). Table 5. Relative tiffne comparion between the analytical octahedral, octet, and unit-tru FEA compliance to the empirical relative tiffne reult for a unit cell with. 5. (Table ) (Eq. 4) (Eq. 6) (Table 4) Relative error (Err rel ) to empirical Octet Eemp Eoctahed Eoctet EFEA Eoctahed Eoctet EFEA Array Size E E E E E E E Single %.7 % 9.6 % xx % 8.7 %.96 % xx % 66.8 %. % The analytically determined relative tiffne for the octahedral and octet tru tructure mot accurately modeled empirical relative tiffne for the ingle octet unit cell (relative error of.95% and.7%, repectively), but thee analytical model ignificantly looe their accuracy when applied to unit cell compried an array of tru tructure (error increaed to 57-8%). Therefore thee analytical model have a ignificant limitation in their application. The unit-tru FEA wa very conitent in i predictive capability of the relative tiffne with relative error ranging tightly about % for all of the unit cell analyzed. No ignificant accuracy wa lot in changing the number of tru tructure compriing the unit cell. 4 Concluion An analyi of unit cell compried of different ize of octet tru tructure ha been performed. An analytical formulation for the octet-tru tructure ha been preented that can more accurately predict the relative tiffne for a ingle octet tru tructure, which ha been confirmed experimentally. A unit-tru FEA program ha been employed to imulate compreion teting of unit cell, exhibiting an accuracy of % relative error to experimentation 4

12 for the relative tiffne of all of the unit cell analyzed. The unit tru FEA program how promie a an analyi tool for unit cell compried of arbitrary tru tructure. 5 Future reearch tak The analyi of the octet-tru tructure i the beginning of the next few tep for the meotructure deign ynthei. A library of unit cell will be created after determining the behavior of variou n x n x n array of octet under different loading condition (non-uniform load, hearing load, and torion). Thee tak may now progre with the knowledge that the unit-tru FEA program i capable of imulating empirical reult. The library of the meotructure unit cell can then be ued a building block for component geometrie. An optimization program can be applied to deign the component geometry for lightweight by uing le dene unit cell with either greater or le tiffne where applicable due to defined loading on the component. For each loading condition, an optimization function will be ued to reduce internal energy, increae trength, and reduce weight baed on the application of the tructure. The optimization function will accomplih thi through the removal of unneceary element, increaing the cro-ectional area of trut with high order of tre, and reducing the cro-ectional area of thoe with low order of tre. The goal in thi i to create a component compried of unit cell tru tructure that have been deigned according to their required trength, flexibility, toughne, etc. 6 Acknowledgement Financial upport for thi work ha been funded by the National Science Foundation (NSF) grant DMI-58. The author would like to thank the following individual at Georgia Tech for their individual contribution to thi reearch: Robert Cooper for helping with compreion teting, and Jeff Lloyd and Amanda O Rourke for CAD and image generation. The author would alo like to thank Profeor Paul Laboiere at the Univerity of Wahington-Seattle for performing the econd round of compreion tet. Reference [] Wang H. A unit cell approach for lightweight tructure and compliant mechanim. School of Mechanical Engineering, vol. Ph.D. Atlanta, GA: Georgia Intitute of Technology, 5. [] Dehpande VS, Fleck NA, Ahby MF. Effective propertie of the octet-tru lattice material. Journal of the Mechanic and Phyic of Solid ;49:747. [] Fuller RB. Synergetic Building Contruction. In: Patent U, editor. United State of America, 96. [4] Borei AP, Schmidt RJ, Sidebottom OM. Advanced Mechanic of Material: John Wiley & Son, Inc., 99. [5] Bendoe MP, Kikuchi N. Generating optimal topologie in tructural deign uing a homogenization method. Computer Method in Applied Mechanic and Engineering 988;7:97. [6] Frecker M, Ananthaureh GK, Nihiwaki S, Kirkuchi N, Kota S. Topological ythei of compliant mechanim uing multi-criteria optimization. ASME Journal of Mechanical Deign 997;9:8. [7] Fu YB, Ogden RW. Nonlinear elaticity : Theory and Application: Cambridge Univerity Pre,. 4