International Journal of Mechanical Sciences 53 (2011) 65–73

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

0020-74

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ijmecsci

Mechanical behaviour of tube-woven Kagome truss cores under compression

J.S. Park a, J.H. Joo b, B.C. Lee a, K.J. Kang a,n

a Department of Mechanical Systems Engineering, Chonnam National University, Gwangju, Republic of Koreab Automobile Research Center, Chonnam National University, Gwangju, Republic of Korea

a r t i c l e i n f o

Article history:

Received 19 April 2010

Received in revised form

22 October 2010

Accepted 20 November 2010Available online 25 November 2010

Keywords:

Cellular materials

Lattice truss material

Buckling

Failure map

03/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ijmecsci.2010.11.002

esponding author. Tel.: +82 62 530 1668.

ail address: kjkang@chonnam.ac.kr (K.J. Kang)

a b s t r a c t

Wire-woven bulk Kagome (WBK) has recently been used to fabricate multi-layered truss-type cellular

metals. A tube WBK structure is fabricated of tubes instead of solid wires. In this work, tube WBK

specimens with various combinations of slenderness ratio and inner-to-outer diameter ratio of the

tubular struts were tested under compression to investigate the effects of geometric factors on peak

strength, equivalent Young0s modulus and energy absorption capability. To aid in the physical

interpretation of the results and the development of a design methodology, numerical simulations of

single tubular struts were performed with a wide range of slenderness ratio and inner-to-outer diameter

ratio. The tube WBKs outperformed most cellular metals, but they were inferior to hollow trusses,

especially those with a diamond configuration. However, energy absorption of the tube WBKs was

comparable to that of hollow trusses because of stable deformation of the tube WBKs after initial yielding

or maximum strength.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Periodic cellular metals (PCM) are useful sandwich core materi-als in terms of strength and lightness. PCMs are classified asprismatic or truss-based [1]. The open cell structure of the latteris an advantage because the interior space is accessible foradditional functions such as heat transfer media [2,3] or catalystsupports [4]. To enhance resistance against strut buckling (a mainfailure mechanism of truss PCMs), tubes can be used as rawmaterial because they increase the second moment of inertia ofthe cross-sectional area for a given weight. The ‘‘hollow truss core’’developed by Queheillalt et al. [5–7], is a good example. Using tubeshas additional benefits. Tubes give the flexibility of changingdensity without changing unit cell size and enhance the bondingstrength between the cores and face sheets through large surfaceinterfacial area nodes. A multi-layered hollow truss core can besimply fabricated by aligning tubes in collinear layers with analternating orientation of successive layers to create a lattice trussarchitecture [6]. Despite excellent performance in terms of specificstrength and energy absorption capability, hollow truss cores havehigh anisotropy; that is, the material properties are quite sensitiveto orientation.

Wire-woven bulk Kagome (WBK) is a multi-layered truss PCMfabricated by a three-dimensional (3-D) assembly of wires ratherthan by stacking multiple single-layered truss structures [8].Helically-formed wires are screw-inserted in six evenly distributed

ll rights reserved.

.

directions in space to fabricate a Kagome truss-like structure inwhich the wires cross one another with minimum deflection.Consequently, the strength is only slightly degraded compared toan equivalent ideal configuration composed of straight struts andthe material anisotropy is minimized. Moreover, WBK has goodpotential for mass-production. The effects of geometric factors suchas wire diameter and pitch on the compressive strength of WBK hasbeen investigated [9]. WBK composed of high strength steel andfilled with brass has been developed to attain ultra-high specificstrength [10]. Heat transfer characteristics under forced convection[11] and the methodology for optimal design [12,13] for sandwichpanel cores were reported. In addition, it was recently shown thatWBK could be fabricated using tubular wires and the strength wasas good as that estimated using an analytic solution derived forequivalent Kagome truss PCMs with an ideal configuration com-posed of straight tubular struts [14].

In this work, tube WBK specimens with nine different combina-tions of slenderness ratio and inner-to-outer diameter ratio oftubular struts were tested under compression to investigate theeffects of geometric factors on peak strength, equivalent Young’smodulus and energy absorption capability. To aid in the physicalinterpretation of the results and the development of a designmethodology, numerical simulations for single tubular struts wereperformed using a wide range of slenderness ratio and inner-to-outer diameter ratio. A map was created showing contours of therelative densities and the normalized peak strengths of tube WBKsand the domains of their failure modes plotted as functions of do/cand di/do. Another map was constructed showing relative densitiesand equivalent Young0s moduli of tube WBKs. The measured resultsof the tube WBKs were compared to other cellular metals withrespect to strength and energy absorption capability.

J.S. Park et al. / International Journal of Mechanical Sciences 53 (2011) 65–7366

2. Experimental procedures

2.1. Specimen preparation

WBK specimens were fabricated from medical grade 304stainless steel tubes. In the first step, continuous helical tubeswere formed. Three identical tubes were twisted together untilplastic deformation occurred in the tubes, and then were separatedfrom one another. Residual plastic deformation after twisting madethe helical shape. Next, the helical tubes were assembled. The tubeswere assembled in three directions into a two-dimensional (2-D)Kagome truss layer in a plane. The multiple layers were stackedone-over-one with an interval, and additional tubes were insertedin three out-of-plane directions to create a 3-D Kagome-likestructure. At each cross point three tubes pass each other withminimum deflection. Fig. 1 shows typical examples of tube WBKspecimens. The detailed fabrication procedure is described in theauthors’ previous works [8]. Finally, the tube assemblies weremetallurgically bonded using a brazing technique. An aqueousmixture of Nicrobrazs 51 (BNI-12) filler metal powder and cement(Wall Colmonoy Corp.) was sprayed over the tube assemblies and

Fig. 1. Tube WBK specimens composed of three different tubes (top) and a side view

of the middle one (bottom).

Table 1Geometric parameters of the tube WBK and solid WBK specimens tested under compre

Specimen no. do (mm) di (mm) c (mm) Relative den

Calculated

1 0.82 0.56 9.66 0.64

2 0.82 0.56 6.24 1.55

3 0.82 0.56 4.95 2.44

4 1 0.7 12.1 0.64

5 1 0.7 7.81 1.55

6 1 0.7 5.9 2.44

7 1.27 0.99 12.8 0.64

8 1.27 0.99 8.29 1.55

9 1.27 0.99 6.57 2.44

S1 0.78 0 12.6 0.64

S2 0.78 0 8.1 1.55

S3 0.78 0 6.45 2.44

dried in an oven at 110 1C. Vacuum brazing was carried out for225 min in a furnace at 10�4–10�5 Torr. During first 90 min, thefurnace was heated from room temperature to 930 1C and main-tained at this temperature for 15 min. The temperature was thenincreased to 1040 1C for 15 min and maintained for 15 min. Finally,the furnace was slowly cooled down to room temperature for90 min.

Three kinds of tubes with outer/inner diameters of do/di¼0.82/0.56, 1.0/0.7 and 1.27 mm/0.99 mm were used to fabricate the WBKspecimens. The pitches, which were twice the strut lengths, c,ranged from 2c¼9.9 to 25.6 mm. Nine different types of thespecimens were prepared. Two or three experiments per specimentype were performed. All the helically-formed wires used tofabricate a given specimen have an identical pitch, inner diameterand outer diameter. Therefore, the angles among struts in all thespecimens are constant regardless of the pitch. All the specimenshad two layers (i.e., two half-layers on the top and bottom and onelayer in the middle). Table 1 lists the geometric parameters of thetested specimens. Fig. 2 shows photos of the specimens and theircross sections at the cross points among tubes. The numberslabeled on the photos correspond to those in the leftmost columnof Table 1. The three photos in a row (numbered as 1–9) showspecimens fabricated with identical tubes of a given inner/outerdiameter. The strut lengths, c, were intentionally designed to givethree relative densities, rrel�0.64%, 1.55% and 2.44%, theoreticallycalculated for the specimens composed of tubes of a given inner/outer diameter ratio. In the final specimens, the densities wereincreased by the filler metal supplied at the cross points amongtubes to rrel�0.7%, 1.8% and 2.9% in the fabricated specimens. Twoface sheets were attached to the top and bottom surfaces of a WBKspecimen in preparation for a uniaxial compression test. For theface sheets, SUS 304 plate of 3 mm thickness, which is a materialsimilar to the helical tubes, was selected. Epoxy, AXIA EP-04(Magnolia Plastic Inc.), was used as an adhesive to attach the sheets.

2.2. Uniaxial compression tests

All compression tests were performed on an Instron 880electric–hydraulic material test system. The load and displacementwere recorded by a data acquisition board embedded in a personalcomputer. The measurement and control software was coded in theHP VEE language (Agilent Technologiess). A 250 kN load cell wasused for the test, and all of the specimens were compressedbetween two steel circular compression platens. The diameter ofeach platen, D¼200 mm, was sufficiently larger than the specimensizes. A displacement rate of 0.005 mm/s was applied on thebottom surface of each specimen as an external loading. Allspecimens were loaded up to 60% of the strain to obtain their

ssion.

sity (%) di/do do/c Cross section area (mm2)

Measured

0.73 0.68 0.085 0.282

1.83 0.68 0.131 0.282

2.84 0.68 0.166 0.282

0.70 0.7 0.082 0.401

1.77 0.7 0.128 0.401

2.86 0.7 0.169 0.401

0.73 0.78 0.099 0.497

1.86 0.78 0.153 0.497

2.89 0.78 0.193 0.497

0.72 0 0.062 0.478

1.80 0 0.096 0.478

2.84 0 0.121 0.478

Fig. 2. Photos of the specimens and their cross sections at the cross points among tubes.

J.S. Park et al. / International Journal of Mechanical Sciences 53 (2011) 65–73 67

compressive behaviour. To measure the equivalent Young’s mod-ulus, each specimen was unloaded around the initial yield point byabout 30% of the load level at which the specimen started to beunloaded. The displacements actually occurring in each specimenwere determined by subtracting those caused by the test machineand loading train from the measured total displacements.

Fig. 3. An example of finite element models of the single tubular struts

composing WBK.

3. Numerical simulations

To understand the behaviour of the specimens observed in thecompression tests, numerical simulations could have been per-formed for the entire structure of each of the nine specimens.However, this would give information about the behaviour of onlythe specific specimens and it would require considerable comput-ing time for precision simulation. Instead, we performed numericalsimulations for one of the struts composing tube WBK with a widerange of slenderness ratios and inner-to-outer diameter ratios.WBK has been observed to fail by elastic or inelastic buckling or byyielding of struts positioned in out-of-plane directions undercompression or shear. Therefore, it was expected that for an initialstate of deformation with overall compressive strain of less thanabout 20%, simulations of single tubular struts in one of the out-of-plane directions would provide the information needed to explainthe experimental observations for the test specimens, and to offerinsight on failure mechanism and consequent strength of tubeWBKs over wider range of the geometric parameters. The singletubular struts in WBKs are helically-curved and constrained withfiller metal at both ends. The structural response, especially beyondinitial elastic deformation, changes considerably and unpredictablydepending on the geometric parameters. Therefore, known analyticsolutions do not apply, and numerical simulation was the onlyoption.

Fig. 3 shows a finite element model of a single strut with fillermetal at both ends and the location from which the strut is taken in

the WBK structure. The slenderness ratio had nine values in therange from 0.025 to 0.225. The inner-to-outer diameter ratio of thetubes had 12 values in the range from 0 to 0.98. The highestslenderness ratio is bounded by the intrinsic nature of the geometryof WBK [15]. The inner-to-outer diameter ratio and the slendernessratio of the tubes were controlled by the inner diameter and thelength, respectively, with the outer diameter fixed. All the modelshad a half pitch of helically-formed tubes in the lengths, and bothends were offset from the longitudinal axis by the helical radius, rh,given by the tube outer diameter as rh ¼ do=

ffiffiffi3p

[15]. Microscopic

J.S. Park et al. / International Journal of Mechanical Sciences 53 (2011) 65–7368

observation of the stainless steel WBK specimens brazed throughthe aforementioned standard procedure revealed that the fillermetal at the cross points of three wires normally has an averageheight of 2b¼3.4do [16]. To account for constraint by the fillermetal, one-half of the filler metal part was included at both ends ofeach single strut model (in the length of b¼1.7do).

To generate the 3-D models, the MSC Patran 2005 commercialgraphics code (MSC Software Corp, Santa Ana, CA, USA) was used.The total number of the elements in our models ranged from 11,300to 250,000 depending on the slenderness ratio and the inner/outerdiameter ratio. The meshed models were transferred to the finiteelement solver ABAQUS version 6.6 (Dassault Systems, Lowell, MA,USA). We used 20-node quadratic brick elements with reducedintegration (the ABAQUS C3D20R element). To simulate constraintagainst deformation of the tubular struts by the interference amongthe tubes and the filler metal at the cross points, the boundaries(i.e., both ends of each single strut model) were constrained fromrotation and translation except for translation in the verticaldirection at the upper ends and their vicinities. All nodes on theface (‘‘Face A’’ in Fig. 3) at each end and those on the face (‘‘Face B’’)at the filler metal border at a distance, b¼1.7do, from each end werebound to the centers on the faces using the multi-point constraints(MPC) available in ABAQUS. At each Face A, all the translations androtations at the center point were suppressed except for thetranslation in the vertical direction at the upper end. At each FaceB, the translations perpendicular to the vertical direction (i.e., thedirection from the centers on the A faces to those on the adjacent Bfaces) were suppressed. The simulations were performed bycontrolling the displacement at the upper ends in the verticaldirection, and by applying large deformation theory to investigatethe load–displacement responses and the failure modes in thetubes. Material properties were measured from tensile tests ofstand-alone tubes heat treated together during the thermal cycleexperienced by the WBK specimens during the brazing process.Young’s modulus was E¼200 GPa, the yield strength was so¼278MPa, and Poisson’s ratio was assumed to be v¼0.3. The J2-incremental theory of plasticity was applied.

Fig. 4. Stress versus strain curves measured from compression tests of tube WBKs

and solid WBK specimens labeled in Table 1 as (a) No. 1, 4, 7, and S1, (b) No. 2, 5, 8,

and S2, and (c) No. 3, 6, 9, and S3.

4. Results and discussion

4.1. Compression test results

Fig. 4(a) shows the stress versus strain curves of three tubeWBKs and solid WBK specimens (labeled No. 1, 4, 7, and S1 inTable 1). All the specimens had similar relative densities,rrel�0.7%.The curve of the solid WBK specimen (S1) was taken from Lee andKang [14]. Clear initial peaks were observed for all the specimens.Then, the strength values had minimums in the strain range ofe¼0.15 to 0.2. In comparison to the geometry data listed in Table 1,we note that specimen No. 7, which had the highest slendernessratio, do/c, (i.e., the highest inner/outer diameter ratio, di/do, for agiven relative density) showed the highest peak strength. Speci-mens No. 1 and 4, which had similar medium values of do/c, showedmedium peak strengths, and specimen No. S1, which had thelowest do/c, had the lowest peak strength. Fig. 4(b) shows the stressversus strain curves of three tube WBKs and a solid WBK specimen(labeled Nos. 2, 5, 8, and S2) with similar relative density,rrel�1.8%. After the initial yield points, the strength had plateausuntil e�0.15. Fig. 4(c) shows the responses of specimens No. 3, 6, 9,and S3 with similar relative density, rrel�2.8%. After the initialyield points, the strengths kept increasing at least until e�0.2.Comparison of the three figures revealed that a higher relativedensity always results in a higher initial yield strength and morestable post-yield strength, which agrees with previous work forordinary WBKs [9]. Despite the difference in stress–strain

responses depending on relative density, the strength values ofthe four specimens for a given relative density were always rankedby the slenderness ratio, do/c (i.e., the inner/outer diameter ratio,di/do). WBKs composed of shorter tubular struts with thinner walls

Fig. 5. (a) Peak strengths and (b) equivalent Young’s moduli of tube WBKs estimated

by FEAs on single tubular strut models as a function of the slenderness ratio of the

struts compared to those measured from the tests.

J.S. Park et al. / International Journal of Mechanical Sciences 53 (2011) 65–73 69

demonstrated higher peak strength values and higher energyabsorption.

In Fig. 4(a)–(c), the horizontal dashed lines indicate equivalentstrengths estimated using analytic solutions [14] that were derivedfor inelastic buckling and yielding of the struts for the equivalentideal Kagome trusses composed of straight tubes as follows:

scy inelastic

bucking

¼

ffiffiffiffiffiffiffiffiffi2p3p

128k2Et

do4�di

4

c4

!¼

ffiffiffi2p

8pst

do2�di

2

c2

!,

������ ð1Þ

scy

����yielding

¼

ffiffiffi2p

8pso

do2�di

2

c2

!: ð2Þ

In Eq. (1) for inelastic buckling, Et is the tangential modulus andcan be calculated from the material stress–strain curve obtainedfrom the tensile tests of the tubes. si is the stress required for atubular strut to buckle inelastically, and k is a constant representingthe boundary condition of a strut under a buckling load (assumed tobe equal to 2 in this work). Each of Fig. 4(a)–(c) was plotted for agiven relative density. The relative density, whose detailed equa-tion is shown in the next section, was primarily dependent on ageometric parameter of the tubular cross section, ððd2

o�d2i Þ=c2Þ.

Consequently, the peak strengths estimated using Eqs. (1) or (2)were almost constant among the four specimens in each figure.

In Fig. 4(a)–(c), Eqs. (1) and (2) give fairly good preliminaryestimates for equivalent peak strength and yield strength of thetube or solid WBK specimens, considering their simplicity, unlessthe specimens failed by elastic buckling observed in the specimenNo. S1, which is explained using Fig. 8(a) in the next section.

In Table 1, tube WBK specimens No. 7–9 with di/do¼1.27 mm/0.99 mm had cross-sectional areas of 0.497 mm2, which were similarto the cross-sectional areas of solid WBK specimens No. S1, S2, and S3(0.478 mm2). Also, tube WBK specimens No. 7, 8, 9 had the almostsame strut lengths as those of solid WBK specimens No. S1, S2, S3,respectively. In Fig. 4(a)–(c), the violet and black lines correspond tothe tube WBK specimens and solid WBK specimens, respectively. Thetwo lines in each of the figures clearly demonstrate the effectivenessof using tubes as raw material for WBK: for a given strut length andcross-sectional area (i.e., for a given relative density), the strength ofWBK was significantly enhanced by using tubes.

4.2. Numerical simulation results

Finite element analyses of the single helically-formed tubularstruts were performed to estimate equivalent stress–strainresponses of tube WBKs under axial compression and their failurebehaviour. The equivalent stress and strain were defined as

s¼ 3P

2ffiffiffi3p

c2, ð3Þ

e¼ dffiffiffiffiffiffiffiffiffiffiffiffið2=3Þ

pc

, ð4Þ

where P and d are the load and the displacement applied at the topsof the single tubular strut models in the vertical direction,respectively. The denominators of Eqs. (3) and (4) represent thein-plane area that a tetrahedral structure composed of three of thesingle out-of-plane tubular struts support and the original height ofthe tetrahedral structure, respectively.

To check the accuracy of the finite element analyses of the singletubular struts, equivalent stress–strain curves measured from theexperiments and those estimated from the finite element analyseswere compared. For similar values of combinations of slendernessratio, do/c, and inner/outer diameter ratio, di/do, the two equivalentstress–strain curves were in fairly good agreement until theequivalent strain level reached at least e�0.2. The examples inFig. 5(a) and (b) compare the peak strengths and equivalentYoung’s moduli, respectively, estimated from the finite elementanalyses and measured from the experiments. The peak strengthwas defined as the maximum stress in a range of equivalentcompressive strain, er0.1. Young’s modulus was calculated froman initial linear portion of each equivalent stress–strain curve. InFig. 5(a) and (b), the estimations are in reasonable agreement withthe experimental results, which validates the approach taken inthis study for the finite element analyses.

Depending on the slenderness ratio, do/c, and the inner/outerdiameter ratio, di/do, various shapes of the equivalent stress–strainresponses and failure behaviours were observed. Fig. 6(a) showsthree examples of the stress–strain curves. The stress–strain curvesare from the strut models with di/do¼0.7 and do/c¼0.125;di/do¼0.9 and do/c¼0.125; and di/do¼0.7 and do/c¼0.05. The firstand second models had the same slenderness ratio and differentinner/outer diameter ratio, and the first and third models haddifferent slenderness ratio and the same inner/outer diameterratio. The consequences of the differences in the geometry weresignificant both in the stress–strain responses and in the deformedshapes as shown in Fig. 6(b), where the deformed shapes of each ofthe three models are illustrated at the peak strength points, at thecompressive strain levels of e�0.15, and 0.3. In Fig. 6(a), theequivalent stresses were normalized by the material yield strength,

Fig. 6. (a) Equivalent stress–strain curves of tube WBKs calculated from FEA for

strut models with di/do¼0.7 and do/c¼0.125; di/do¼0.9 and do/c¼0.125; and

di/do¼0.7 and do/c¼0.05. (b) Deformed shapes of each of the three models at the

peak strength points, at compressive strain levels of e�0.15, and at e�0.3.

J.S. Park et al. / International Journal of Mechanical Sciences 53 (2011) 65–7370

so¼278 MPa, and the relative densities, rrel. The relative densitieswere calculated using an equation suggested by Park [16]:

rrel �3ffiffiffi2p

p8

d2o�d2

i

c2

! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2þp2r2

h

qc

þ

bffiffi2p �

4rh

3 sin pbc

� �� �2 ffiffiffi3p

bþffiffiffi3p� p

2

� �d2

ob

2ffiffiffi2p

c3: ð5Þ

The latter part of the first term, and the second term represent

the effects of the helically-curved geometry of the tubular struts

and the volume occupancy by the filler metal brazed at cross pointsamong the tubes, respectively. The model with di/do¼0.7 and do/c¼0.125 (case A ) was the most stable in the curve; i.e., the peakstrength was delayed up to e�0.08 and the post-peak strengthslowly decreased. The deformation was observed to occur uni-formly through the whole body, as shown at the top of Fig. 6(b). Themodel with di/do¼0.9 and do/c¼0.125 (case B ) showed the earlierpeak strength at e�0.03 and the post-peak strength decreasedsomewhat rapidly. Local deformation was observed to occur nearthe upper and lower filler metal borders and in the middle even atthe peak strength point, and local deformation became moreprominent with higher strain. The tubular cross sections in themiddle were flattened at e�0.15, as shown in the middle ofFig. 6(b). The model with di/do¼0.7 and do/c¼0.05 (case L ) showedthe earliest and lowest peak strength and the post-peak strengthdropped very rapidly. However, only slight local flattening wasobserved at a compressive strain level of e�0.15. These threeexamples show three typical failure modes of the tubular struts:global inelastic buckling or yielding, local buckling, and globalelastic buckling.

Fig. 7(a) shows the variation of the peak strength normalized byso and rrel as functions of the slenderness ratio and the inner/outerdiameter ratio. Generally, the peak strength increased with do/c fora given di/do. When the inner/outer diameter ratio was lower thandi/do¼0.7, the normalized peak strength varied consistently withdo/c regardless of di/do. However, above di/do¼0.7, the peakstrength departed from the general trend and the increase rateslowed down with do/c. Closely observing the deformation patternafter the peak in the stress–strain curve of each specimen, we foundthat the departures from the general trend at points A , L , and Kwere responsible for the change in the failure mechanism of thetubular struts; that is, from global inelastic buckling or yielding tolocal buckling. Note that point A has the same coordinates as incase A shown in Fig. 6(a) and (b). With a higher di/do ratio, thedeparture began at a lower do/c value. This means that the thinnerwall of a tube tended to cause to the local buckling at the lower do/cvalue. In addition to the transition of the failure mechanism,volume occupancy of the filler metal was the secondary sourceof the decrease in normalized strength. That is, in a WBK composedof tubes with thinner walls; i.e., tubes with a higher di/do value, thevolume of filler metal took the larger portion in the relative densityof each WBK, which resulted in lower normalized peak strength fora given do/c. Therefore, with the highest inner/outer diameter ratio(di/do¼0.98), tube WBKs demonstrated the lowest normalizedpeak strength. In this specimen the failure mechanism wasdifferent from others: the tubular struts with very thin wallbuckled with circumferential waves on the entire surface, whichis often found in axially-compressed thin cylindrical shells [17].

Fig. 7(b) shows the variation of the equivalent Young’s modulusnormalized by the mother material’s Young’s modulus, Eo, and rrel

as functions of the slenderness ratio and the inner/outer diameterratio. If di/doo0.9, the normalized equivalent Young’s modulussteadily increased as do/c increased for a given di/do, and increasedas di/do increased for a given do/c. This means that tubular strutswith shorter length and thinner wall result in higher stiffness. Theinner-to-outer diameter ratios of the tubes in these models werecontrolled by varying the inner diameter with the outer diameterfixed; thus, tubular struts with a thinner wall saved weight withless loss of their second moment inertia (which governs thebending stiffness of the tubular struts). However, if di/doZ0.9,the normalized equivalent Young’s modulus started to decrease asdi/do increased for a given do/c. In particular, for the models with di/do¼0.98, the normalized equivalent Young’s modulus was lowestfor most do/c values. These low Young’s moduli seem to be due tothe lateral elastic deformation which is often found in axially-compressed thin cylindrical shells [17].

Fig. 7. (a) Normalized peak strengths and (b) normalized equivalent Young’s moduli

of tube WBKs estimated by FEA on single tubular strut models as a function of

slenderness ratio and inner/outer radius ratios.Fig. 8. (a) Failure map with contours of the relative densities, normalized peak

strengths and domains of the failure modes of tube WBKs as functions of do/c and

di/do. (b) Map with contours of the relative densities and the equivalent Young’s

moduli.

J.S. Park et al. / International Journal of Mechanical Sciences 53 (2011) 65–73 71

Fig. 8(a) shows a failure map in which contours of the relativedensities and normalized peak strengths of tube WBKs are plottedas functions of do/c and di/do. The map also shows domains of thefailure modes observed from finite element analyses of the singlestruts. The domains of global buckling (elastic buckling andinelastic buckling or yielding) have clear boundaries with thedomain of local buckling located in the right upper region of themap. That is, if both di/do and do/c are sufficiently high, a tubularstrut fails by local buckling rather than global buckling. In contrast,the transition between the domains of elastic buckling, andinelastic buckling or yielding is governed primarily by the slender-ness ratio. In this study, the transition occurred around do/c¼0.06almost regardless of the inner-to-outer diameters ratio, which weattributed to the high dependence of Euler buckling load on theslenderness ratio rather than on the second moment of inertia ofthe cross section of a slender member via FcrpEI=c2 [18]. On the

map shown in Fig. 8(a), the coordinates of the three examples A , Band C of the single struts shown in Fig. 6(a) and (b) are indicated,and the coordinates of the departure points A , L and K shown inFig. 7(a) are indicated, too. The deformation patterns shown inFig. 6(a) are perfectly matched to the failure modes according tothis map. The departure points shown in Fig. 7(a) are located nearthe boundary between the domains of global buckling and localbuckling. By using this map, not only the equivalent peak strengthand the relative density but also the failure mode of a WBKcomposed of helically-formed tubular struts can be estimated asfunctions of do/c and di/do. Moreover, in comparing the relativedensity contours with the strength contours, we note that acombination of do/c and di/do that gives the maximum peakstrength for a given weight is located on the boundary between

J.S. Park et al. / International Journal of Mechanical Sciences 53 (2011) 65–7372

the domains of global buckling and local buckling. The coordinatesof the twelve specimens listed in Table 1 are also indicated in themap. The specimens No. 1, 2, 4, 5, S2, S3; No S1; and No. 8 and 9 arelocated in the domains of inelastic buckling or yielding, elasticbuckling, and local buckling, respectively. In contrast, the speci-mens No. 3, 6 and 7 are located near the boundary between thedomains of inelastic buckling or yielding and local buckling, whichmeans that these specimens were well-designed with respect tostrength per weight.

Fig. 8(b) shows another map in which contours of the relativedensities and the normalized equivalent Young’s moduli of tubeWBKs are plotted as functions of do/c and di/do. The two families ofcontours are consistently distributed and cross each other except inthe far right region of di/doZ0.9, as shown in Fig. 7(b). Therefore, fora desired equivalent Young’s modulus, one can choose severalcombinations of do/c and di/do depending on the relative density.

Fig. 9. (a) Measured data of the normalized strengths and (b) energy absorption per

unit volume during compression up to strain level of e¼60% of tube WBKs with

respect to relative densities compared to data for other cellular metals.

5. Discussion

Using the maps shown in Fig. 8(a) and (b), the optimal design ofa tube WBK can be determined. For a given relative density, the bestcombinations of do/c and di/do to obtain the highest strength andthe highest stiffness can be determined from Fig. 8(a) and (b),respectively. According to the two maps, in general, higher valuesof do/c and di/do lead to higher strength and higher stiffness for agiven relative density. This demonstrates the effectiveness of usingtubular wires as the raw material for WBK. However, for a relativedensity level lower than rrel�0.04, the risk of local buckling shouldbe taken into account, and a tradeoff between strength and stiffnessshould be considered for the best design. The two maps can also beused to design do/c and di/do to give the minimum weight for a givenstrength and/or stiffness.

Fig. 9(a) shows the test results for the normalized strengths oftube WBKs with respect to relative density in comparison withother cellular metals [6,19,20]. The tube WBKs outperformed solidWBKs, metal foams, egg boxes, and woven textiles, but are stillinferior to hollow trusses, especially those with a diamond con-figuration, although the two materials cannot be directly comparedbecause of the different levels of relative density. This result isattributed to the curved shape of the struts composing the WBKs,whereas hollow trusses are made from straight struts. However, ifthe energy absorption capability of the two materials is taken intoaccount, the gap is narrowed. Fig. 9(b) shows test results of energyabsorption per unit volume during compression up to a strain levelof e¼60% for various cellular metals. The energy absorption of thetube WBKs is comparable to that of the hollow trusses because ofthe stable deformation of the tube WBKs after the initial yielding orpeak strength point, as shown in Fig. 4(a)–(c). The slow decrease oreven increase in the strength level after the initial yielding or peakstrength point seems to be due to the widely spreading plasticdeformation and severe interference among tubular strutsobserved during large deformation of the tube WBKs. Among allavailable cellular metals, tube WBKs are located at the upper limitin energy absorption capability at a low level of relative densityalong with hollow trusses at a high level of relative density.

6. Summary

�

Compression tests with nine different tube WBKs and threesolid WBKs revealed that higher relative density alwaysresulted in higher initial yield strength and more stable post-yield strength. For a given relative density, WBKs composed ofshorter tubular struts with thinner walls resulted in higher peakstrength and higher energy absorption.

�

Finite element analyses of single helically-formed tubular strutswere performed to estimate equivalent stress–strain responsesof tube WBKs under axial compression and their failurebehaviour. Compared to the responses measured from experi-ments, the finite element analyses gave fairly good estimationsof peak strength values and the equivalent Young’s moduli. � Depending on the slenderness ratio, do/c, and the inner/outer

diameter ratio, di/do, three types of equivalent stress–strainresponses and failure behaviours were observed. The WBKscomposed of tubular struts with high di/do and high do/c valuesshowed medium peak strength and a moderate decrease inpost-peak strength, and failed by local buckling near the upperand lower filler metal borders, and in the middle. For a do/c valuelower than about 0.06, WBK showed the lowest peak strengthsand rapid decrease in post-peak strength, and failed by globalelastic buckling. In the rest of the region of do/c and di/do, theWBKs showed delayed high peak strengths and a slow decreasein post-peak strengths, and failed by global inelastic buckling oryielding.

J.S. Park et al. / International Journal of Mechanical Sciences 53 (2011) 65–73 73

�

Generally, peak strength increased with do/c for a given di/do.When the inner/outer diameter ratio was lower than di/do¼0.7,the normalized peak strength varied consistently with do/c

regardless of di/do. However, above do/c¼0.7, the peak strengthdeparted from the general trend and the increase rate with do/c

slowed down. The departure was responsible for the change infailure mechanism of the tube strut; that is, from global inelasticbuckling or yielding to local buckling.

� A map was constructed in which contours of relative densities

and normalized peak strengths of tube WBKs and domains oftheir failure modes were plotted as functions of do/c and di/do.The map shows that a combination of do/c and di/do resulting inmaximum peak strength for a given weight is located on theboundary between the two domains of global buckling and localbuckling. The map demonstrated that results from the tests andfinite element analyses were successfully interpreted. Anothermap with relative densities and equivalent Young’s moduli oftube WBKs was constructed. The two maps can be used todetermine the best combinations of do/c and di/do to produce thehighest strength or stiffness for a given relative density or toproduce the minimum weight for a given strength or stiffness.

� In the measured strength, tube WBKs outperformed solid WBKs,

metal foams, egg boxes, and woven textiles, but they were stillinferior to hollow trusses especially with a diamond configuration.However, the energy absorption of tube WBKs was comparable tothat of hollow trusses because of the stable deformation of thetube WBKs after the initial yielding or peak strength point.

Acknowledgements

This study was supported by the 2006 National ResearchLaboratory program of the Korea Science & Engineering Foundation(R0A-2006-000-10249-0).

References

[1] Wadley HNG. Adv. Eng. Mater. 2002;4:726–33.[2] Kim T, Hodson HP, Lu TJ. Int. J. Heat Mass Tran. 2004;47:1129–40.[3] Tian J, Kim T, Lu TJ, Hodson HP, Queheillalt DT, Sypeck DJ, et al. Int. J. Heat Mass

Tran. 2004;47:3171–86.[4] Choi BC, Jeong JW, Joo JH, Kang KJ. Adv. Eng. Mater. 2009;11:536–40.[5] Queheillalt DT, Wadley HNG. Mater. Sci. Eng. A 2005;397:132–7.[6] Queheillalt DT, Wadley HNG. Acta Mater. 2005;53:303–13.[7] Rathbun HJ, Zok FW, Waltner SA, Mercer C, Evans AG, Queheillalt DT, et al. Acta

Mater. 2006;54:5509–18.[8] Lee YH, Lee BK, Jeon I, Kang KJ. Acta Mater. 2007;55:6084–94.[9] Lee BK, Kang KJ. Compos. Struct. 2009;92:445–53.

[10] Kang KJ. Acta Mater. 2009;57:1865–74.[11] Joo JH, Kang BS, Kang KJ. Exp. Heat Transfer 2009;22:99–116.[12] Lee YH, Kang KJ. Mater. Des. 2009;30:4434–43.[13] Lee YH, Choi JE, Kang KJ. Mater. Des. 2009;30:4459–68.[14] Lee BK, Kang KJ. Scr. Mater. 2009;60:391–4.[15] Hur H.K., Kang K.J., In: Proceedings of the Spring Conference, Korean Society of

Mechanical Engineers, 2008, pp. 82–7.[16] Park J.S., Master Thesis, Chonnam National University, South Korea, 2009.[17] Batterman SC. AIAA J. 1966;3:316–25.[18] Lardner TJ, Archer RR. Mechanics of Solids: An Introduction. Singapore:

MaxGraw-Hill; 1994.[19] Zupan M, Chen C, Fleck NA. Int. J. Mech. Sci. 2003;45:851–71.[20] Andrew E, Sanders W, Gibson LJ. Mater. Sci. Eng. A 1999;270:113–24.