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Acta Materialia 54 (2006) 5509–5518

Structural performance of metallic sandwich beamswith hollow truss cores

H.J. Rathbun a, F.W. Zok a,*, S.A. Waltner a, C. Mercer a, A.G. Evans a,D.T. Queheillalt b, H.N.G. Wadley b

a Materials Department, University of California, Santa Barbara, CA 93106, USAb Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904, USA

Received 25 April 2006; accepted 21 July 2006Available online 5 October 2006

Abstract

The article focuses on the structural performance of sandwich beams with hollow truss lattice cores made from a ductile stainless steel.The trusses are arranged in an orthogonal (cross-ply) configuration, in either ±45� (diamond) or 0�/90� (square) orientations with respectto the face sheets. The responses in shear, tension and compression, as well as simply supported and fully clamped bending, are measuredfor specimens with both core orientations. While the two cores perform equally well in compression, the diamond orientation exhibitshigher shear strength but lower stretch resistance. For bend-dominated loadings of the sandwich beams, the core in the diamond orien-tation is preferred because of its higher shear strength. For stretch-dominated loadings encountered in large-displacement, fully clampedbending, the square orientation is superior. Models of core and beam yielding are used to rationalize these observations. Optimizationsare then performed to identify strong lightweight designs and to enable performance comparisons with other sandwich structures.� 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Sandwich beams; Lightweight structures; Hollow tube core

1. Introduction

Metallic sandwich panels with various honeycomb, lat-tice truss and prismatic cores are being developed for struc-tures that require high strength [1–5] and blast resistance[6–9]. Some topological configurations are amenable toadditional functionality. For instance, shape morphing ofthe structure can be achieved via actuation of core members[10–12]. Others can be designed for thermal management,through forced flow of fluids through the core. In a typicalapplication, high heat fluxes are deposited onto one of theface sheets of a sandwich panel; the heat is conducted awayfrom the faces via the core elements and subsequentlyremoved by a coolant flowing through the core [13,14].Any open cell metallic structure that allows coolant flowcan be used as a heat exchange medium. Through appropri-

1359-6454/$30.00 � 2006 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2006.07.016

* Corresponding author. Tel.: +1 805 893 8699; fax: +1 805 893 8486.E-mail address: zok@engineering.ucsb.edu (F.W. Zok).

ate selection of material and core topology, these structurescan be designed to possess high thermal conductivity andefficiently transfer heat to the cooling fluid [15,16].

One approach envisioned to further increase the thermalperformance of sandwich structures involves the use ofheat pipes as the truss members within the core [14,17].Heat pipes increase the performance by homogenizing thetemperature distribution and increasing the average tem-perature difference at the interface between the externalpipe surface and the coolant. Additionally, because of theirhigh second moment of area, such pipes offer superiorbuckling resistance relative to solid members of equivalentmass [17,18]. Consequently, the cores can be designed withenhanced structural efficiency, especially in the domain oflow core relative density.

The present study explores the structural characteristicsof metallic sandwich beams with hollow truss lattice cores.Using recently proposed methods for the manufacture ofhollow truss cores and sandwich panels that utilize them

rights reserved.

Nomenclature

B beam widthE elastic modulusF tensile forceFs shear forceHc core thicknessHeq thickness of a solid plate of equivalent massIt second moment of area of hollow tube‘ ratio of bending moment to shear force

(‘ ” M/V)‘0 characteristic lengthLc center-to-center tube spacingM bending moment applied to sandwich structureMt bending moment in hollow tuber inner tube radiusR outer tube radiusS loading spantf face sheet thicknessV shear force applied to sandwich structurea fractional change in flow stress following

brazingb non-dimensional core strength

d bending displacementD normalized displacementey yield strainc shear strainP non-dimensional loadP0 load capacity at ‘ = ‘0

qc core relative densityR normalized loadrc core compressive strength�rcore tensile flow strength of core�rf tensile flow strength of heat-treated (not brazed)

parent alloy�rs panel tensile strengthrt maximum bending stress in hollow tubery yield strengths Shear stresssp fully plastic shear strength�sp non-dimensional fully plastic shear strength�sy non-dimensional shear yield stressW non-dimensional weight indexW0 weight of sandwich beam at ‘ = ‘0

5510 H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518

[17], the article focuses on plastic deformation of simplysupported and fully clamped bending beams with orthogo-nal (‘‘cross-ply’’) core architectures, in ±45� (diamond) and±0�/90� (square) loading orientations. Mechanical perfor-mance differences are rationalized on the basis of the coreproperties in the two orientations, as well as the prevailingcore stresses in the beams. Models are used in parallel withthe experiments to glean insights into the connectionsbetween core topology, constituent material propertiesand beam performance. Finally, optimizations are per-formed to identify strong, lightweight sandwich paneldesigns with hollow truss cores. Their performance isshown to compare favorably with optimized panels withsquare honeycomb cores.

2. Experimental procedures

2.1. Sandwich fabrication

Hollow truss cores were made from medical grade 304stainless steel tubes (Vita Needle, Needham, MA), manu-factured by a tungsten inert gas welding and plug drawingprocess. The inner and outer tube radii were r = 0.61 mmand R = 0.74 mm, respectively. In preparation for brazing,each tube was sprayed with a mixture of a polymer-basedcement (Nicrobraz Cement 520) and a Ni–25%Cr–10%P–0.03%C braze powder (Nicrobraz 51), both supplied byWall Colmonoy (Madison Heights, MI). The phosphorousin this alloy acts as a melting point depressant andenhances the flow and wetting characteristics of the molten

braze. The braze alloy has a solidus of 880 �C and aliquidus of 950 �C, with a suggested brazing range of980–1095 �C. Once coated, the tubes were stacked in anorthogonal (‘‘cross-ply’’) pattern to the desired heightwithin an alignment tool (Fig. 1a). Alignment was achievedby a set of uniformly spaced stainless steel dowel pinsinserted into holes in a stainless steel base plate. The dowelspacing was selected to produce a center-to-center tubespacing, Lc, of 5 mm. The alignment fixture was spraycoated first with Nicrobraz Green Stop-Off (Wall Colmo-noy, Madison Heights, MI) and then boron nitride (GEAdvanced Ceramics, Lakewood, OH) [19].

The tube assembly was placed in a vacuum furnace(Super VII, Centorr Vacuum Industries, Nashua, NH)for brazing. The furnace was heated at 10 �C/min to550 �C and held at temperature for 20 min to volatilizeand remove the polymer binder. Thereafter, it was heatedto 1020 �C at �10�4 torr, held for 60 min and cooled toambient temperature at � 25 �C/min. After brazing, thecores were removed from the tooling and cut to size (nom-inally 18 mm thick) using wire electro-discharge machining(Fig. 1(b)). The measured core relative density, qc, was7.5%. By comparison, the calculated value (neglecting theadded weight of the braze alloy) was 7.3% [15].

A second brazing operation was used to attach thelattice cores to 304 stainless steel face sheets to producesandwich beams (Fig. 1(c)). For most cases, the facesheet thickness, tf = 0.7 ± 0.1 mm. For measurement ofthe core properties in compression, the face sheets werethicker (tf = 2.5 ± 0.2 mm), to ensure that they remained

Fig. 1. Schematic of the procedure used to fabricate the ±45� hollow trusscore sandwich beams. An analogous process was used for fabricating coresin the 0�/90� orientation.

H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518 5511

undeformed during transverse compressive testing. Eventhicker faces (tf = 12.7 ± 0.2 mm) were attached to lapshear specimens that were used for measuring the shearresponse of the cores. To enable gripping in the clampedbending and uniaxial tension tests, solid steel inserts wereplaced between and brazed to the face sheets at the twoends of the beams. The assembly (core, face sheets andinserts) was placed in a clamping fixture. To preventbonding to the fixture, the fixture surfaces were sprayedwith Stop-Off and boron nitride. In addition, a 3 mmthick layer of high-purity alumina–silica fiber paper wasplaced between the face sheets and the adjacent fixturesurfaces. Moderate clamping pressure was used to main-tain intimate contact between the faces and the core dur-ing brazing. The heating cycle was the same as that usedto produce the cores. Examples of representative testspecimens in both diamond and square orientations areshown in Fig. 2.

2.2. Mechanical testing

The mechanical properties of the cores and the sandwichbeams were measured using standard compression, tension,shear and bending tests [4,5,8,17,18]. Pertinent details arepresented below. In addition, for comparison with the mea-sured strengths of the sandwich beams and assessment ofmechanical models, the uniaxial tensile properties of the304 stainless steel face sheet material were measured fol-lowing a heating cycle identical to that used for brazing.Tests were performed in accordance with ASTM E8-01,at a nominal strain rate of 10�3 s�1. The elastic modulus,E, and yield strength, ry, were 200 GPa and 200 MPa,respectively, with a yield strain ey = ry/E = 0.001.

Despite the measurements on the appropriately heat-treated parent alloy, some uncertainty in the in situ mate-rial properties in the sandwich structures remains becauseof the opportunity for chemical interaction between thealloy and the braze materials during fabrication. While pre-vious studies have revealed minimal changes in the initialyield strength, they indicate significant elevations in thehardening rate and flow stress of 304 stainless steel afterexposure to similar braze alloys [5]. Some insights intothe magnitude of such effects in the present sandwich spec-imens are gleaned from an analysis of the tensile testresults, presented in Section 3.

3. Core properties

Effects of tube architecture (±45� vs. 0�/90�) on the coreproperties were ascertained for three loading modes: in-plane tension, transverse (out-of-plane) compression andtransverse shear.

3.1. Tension

Experimental results from uniaxial tension tests are plot-ted in Fig. 3 for sandwich specimens with both core orien-tations, as well as the heat-treated 304 alloy itself. Therelevant tensile stress is the load, F, divided by the arealdensity, BHeq, where B is the width and Heq is the thicknessof a solid plate of equivalent mass. On this basis, the 304alloy has the greatest stretch resistance, followed by thesandwich beam in the 0�/90� orientation; the weakestresponse occurs in the ±45� orientation.

The tensile flow stress of a sandwich panel can be parti-tioned between the core and its face sheets, each weightedby its respective effective thickness [4]. Making a correctionfor the change in the flow stress of the steel following braz-ing, the resulting panel tensile strength becomes:

�rs ¼�F

BH eq

¼ ð1þ aÞ�rf ð1� bÞ 2tf

H eq

� �þ b

� �ð1aÞ

where b is a non-dimensional parameter characterizingcore strength, defined by

b � �rcore

qc�rf

ð1bÞ

Fig. 2. Representative test specimens with cores in (a) ±45� and (b) 0�/90� orientations.

Fig. 3. Tensile response of both sandwich specimens and monolithic 304stainless steel.

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60

Com

pres

sive

str

ess,

σc/

ρ cσ Y

Compressive plastic strain (%)

±45Þ

0Þ/90Þ

Predicted yield stress

Fig. 4. Compressive response of the hollow truss cores.

5512 H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518

�rcore and �rf represent the tensile flow strengths of the coreand the heat-treated (but not brazed) parent alloy, respec-tively, at a prescribed strain, and a is the fractional changein the flow stress of the steel following brazing. Establishedmechanical models for the two core types yield corestrength values b = 0.5 and b = 0 for the 0�/90� and ±45cores, respectively [2].

Predictions of Eq. (1) are plotted in Fig. 3. When thematerial is assumed to be unaffected by the braze (a = 0),the predicted yield strengths agree well with the measure-ments (using the appropriate value of b). However, at smallplastic strains (up to about 0.5%) the work hardening ratesare underestimated. Thereafter, at larger strains, the hard-ening rates are essentially the same. Conversely, uponselecting a = 0.15, the predicted and measured curves atlarge strains (>0.5%) agree well with one another, but theyield strengths are overestimated. These observations arethe same as the braze effects reported in earlier studies[5]. That is, following brazing, the yield strength of 304stainless steel remains essentially unchanged (relative to

that after an equivalent heat treatment), the work harden-ing rate at small strains is elevated and the flow stress atlarger strains is about 10–30% higher than that of the pris-tine alloy. The inference from these comparisons is that thein situ yield strength of the alloy is only slightly altered bybrazing. This result is utilized in subsequent comparisonsbetween experiment and theory. A second key conclusionis that the stretch resistance of the 0�/90� core (b = 0.5) isindeed much higher than that of the ±45� core (b � 0).

3.2. Compression

The compression test results are presented on Fig. 4. Forboth orientations, yielding initiates at a normalized stressrc/qc � 0.5, the same as that predicted from stress analysesof both cores [2,17]. Following yield, both cores strainharden in the small plastic strain domain, ultimately reach-ing a peak when the tubes begin to plastically buckle. Theflow strength achieves a plateau shortly thereafter, atrc/qcry � 0.4 � 0.5, during which the core members

Fig. 6. Schematic of the 0�/90� tube array and the stress analysis of anindividual tube member under shear loading.

H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518 5513

progressively crush. Significant core densification begins at40–50% compressive strain, marked by a rapid increase inhardening. In contrast to the core properties under tensileloading, the compressive properties are essentially indepen-dent of orientation.

3.3. Shear

Representative stress-strain curves from the lap shearspecimens are shown in Fig. 5. In the ±45� orientation,yield initiates at a shear stress, sy/qcry � 0.5: again consis-tent with the analytic prediction [3]. Following subsequentstrain hardening, the stress drops rapidly as the tubes beginto separate from the face sheets. The low failure strain ofthe ±45� orientation (c � 2%) is associated with node fail-ure at the tube-face sheet interface. Alternate brazes [4,5]and improved node joint designs are expected to producemore robust nodes. In the 0�/90� orientation, the responseis much softer, with a yield strength, sy/qcry � 0.1. Thesubsequent plastic response is characterized by moderatehardening. The nodes remain intact over the strain rangeprobed by these experiments (c � 20%), because the lowstrength of the core ensures that a correspondingly lowstress is transmitted to the face/tube interfaces.

To model the shear strength of the 0�/90� core, the tubesperpendicular to the loading direction are assumed to actas beams, rigidly affixed to both the faces and the longitu-dinal tubes. An analysis of a unit cell of the tube array(Fig. 6) yields the shear stress, s, carried by a single tube:

s ¼ F s

4LcRð2Þ

where Fs is the applied shear force. The resulting bendingmoment, Mt, acting on each tube is obtained from staticequilibrium, whereupon

M t ¼F sðLc � 2RÞ

2ð3Þ

0.00 0.02 0.04 0.06 0.08 0.10 0.120.0

0.2

0.4

0.6

0.8

1.0

Shear strain, γ

Nor

mal

ized

she

ar s

tres

s,τ/

σ yρ c

±45˚

0°/90°

Predicted yield Predicted limit

Predicted yield

Fig. 5. Shear response of hollow truss cores.

Here, Lc � 2R represents the edge-to-edge spacing of thetubes, as illustrated in Fig. 6. Yielding initiates when themaximum bending stress, rt, in the tube reaches the mate-rial yield strength, ry, That is,

rt ¼M tR

I t

¼ 4M tR

pðR4 � r4Þ¼ ry ð4Þ

where It is the second moment of area of the hollow tubeabout the neutral axis. Combining Eqs. (2)–(4) gives theapplied stress at yield initiation:

sy ¼rYpðR4 � r4Þ

8LcðLc � 2RÞR2ð5Þ

This result can be re-expressed in a non-dimensional formas:

�sy �sY

qcrY

¼ R2 þ r2

4ðLc � 2RÞR ð6Þ

where the core relative density is given by [17]:

qc ¼pðR2 � r2Þ

2LcRð7Þ

Following similar procedures, the shear stress sp needed forthe formation of a fully plastic hinge at each node is alsoobtained:

�s ¼ sp

qcrY

¼ 2RpðLc � 2RÞ ð8Þ

Using dimensions pertinent to the present cores (R = 0.74mm, r = 0.61 mm, Lc = 5 mm), the normalized strengthsare �sY ¼ 0:09 and �sY ¼ 0:13 for the 0�/90� core. These re-sults compare favorably with the experimental measure-ments (Fig. 5). The slight discrepancy between thepredicted fully plastic shear strength and the measured flowstrength at large strains can be attributed to strain harden-ing: a feature neglected in the analytical model.

To summarize, while the compressive properties of thetwo cores are essentially the same, the core in the 0�/90�orientation exhibits lower shear strength (by a factor of5) but significant stretch resistance (b = 0.5). These differ-ences manifest themselves in the properties of the sandwichbeams, as detailed below.

5514 H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518

4. Sandwich properties: simply supported bending

4.1. Experiments

To assess the mechanical performance of the sandwichbeams, measurements have been made using three-pointsimply supported bend tests. The outer loading points con-sisted of 25.4 mm diameter hardened steel pins on a flatsupport base. To inhibit local indentation, the inner load-ing platen comprised a flat central region, 12.7 mm wide,with adjacent filets 6 mm radius. The loading span wasS = 184 mm. For comparison, tests were also performedon monolithic beams of equivalent areal density.

The results are plotted in Fig. 7. The loads and displace-ments have been normalized by those needed to initiateyielding in the equivalent-weight solid beam; the ensuingnormalized load and displacement (defined in the figure)are denoted R and D, respectively. On this basis, it is evi-dent that the strengths of the sandwich beams are muchgreater than that of the solid beams by about an order ofmagnitude. Additionally, in the plastic domain, beams withthe ±45� core are also almost twice as strong as those withthe 0�/90� core. The latter differences arise from the opera-tive failure modes, elaborated below. With the ±45� core,large deflections are obtained without deformation locali-zation (Fig. 8a). In this case, the test was terminated oncethe outer loading pins had experienced significant lateraldisplacement. The beams with the 0�/90� cores exhibitedsimilarly large plastic deflection, but ultimately failed byshear of the core beneath the inner loading platen (Fig. 8b).

4.2. Failure model: ±45� core

Analytic models for failure initiation of sandwich beamswith lattice cores have been developed previously [2,3],using stress analyses of the faces and the core elements sub-ject to a combination of transverse shear, V, and bending

0

10

20

30

0 5 10 15

Nor

mal

ized

load

, Σ=

3FS

/2B

Heq

2 σ y

Normalized displacement, Δ = 6δHeq/S2εy

Onset of roller movement

±45˚

0˚/90˚

Onset of coreindentation

Solid sheet

Predicted yield points

Fig. 7. Comparison of simply supported bending response of thesandwich beams with that of the equivalent weight monolithic material.

moment, M (both per unit width). These models have beenadapted here for hollow truss cores. In the current imple-mentation, the core is assumed to carry all of the shearand the faces all of the moment. The key results are sum-marized below.

Failure in the face sheets can occur by either yielding orelastic buckling. The critical loads can be described by anon-dimensional parameter, defined as P ” V2/EM. Forbeams with ±45� cores [3]:

P ¼ eytfH c

‘2ðface sheet yieldingÞ ð9Þ

P ¼ p2t3f H c

24ð1� m2Þ‘2L2c

ðface sheet bucklingÞ ð10Þ

where Hc is the core thickness and ‘ ” M/V. The latterquantity scales with loading span: ‘ = S/2 for simply sup-ported three-point bending and ‘ = S/4 for clamped ends.Similarly, the tubes in the core can fail by either yieldingor buckling. Here thes nodes at which the tube membersintersect are assumed to be pinned such that the adjoiningtube segments are free to rotate about these nodes. Thepertinent failure loads are:

P ¼ peyRH c½1� ðr=RÞ2�4Lc‘

ðcore yieldingÞ ð11Þ

P ¼ p3H cR3½1� ðr=RÞ4�16L3

c‘ðcore bucklingÞ ð12Þ

Beam failure is dictated by the mode with the lowest valueof P.

For three-point bending, wherein ‘ = S/2, the loadparameter P is related to the non-dimensional quantity Rpresented in Fig. 7 via the relation:

R ¼ 3S2

2H 2eqey

!P ð13Þ

For the present experimental configuration, the preced-ing models predict failure by face yielding, at a normalizedload of P = 1.3 · 10�6 or, equivalently, R = 10. This resultcorrelates well with the onset of non-linearity in the exper-imental measurements, shown in Fig. 7.

4.3. Failure model: 0�/90� core

The most significant modification to the precedingresults (Eqs. (9)–(12)) for beams with 0�/90� cores pertainsto the stress needed for core yielding. Utilizing the modeldeveloped in Section 3 for the core shear strength, the crit-ical load becomes:

P ¼ peyH cðR4 � r4Þ8‘LcðLc � 2RÞR2

ð14Þ

This mode dominates in the present experiments, the loadsfor face failure and core buckling being significantly higher.The resulting critical load is R = 8. This result also agreesfavorably with the onset of non-linearity (Fig. 7).

Fig. 8. Deformation of sandwich beams in simply supported bending.

H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518 5515

5. Sandwich properties: clamped bending

Additional bend tests have been performed using fullyclamped end conditions. Apart from the boundary condi-tions, all testing parameters were the same. The resultingload/displacement curves are summarized in Fig. 9 andimages of the deformed specimens are in Fig. 10. At smalldisplacements (d/S < 0.05), wherein the response is bend-dominated, the curves mimic those obtained under simplysupported conditions (Fig. 7). That is, the beam with the±45� core outperforms that with the 0�/90� core. Con-versely, at larger displacements, membrane stresses ariseand the response becomes stretch-dominated. In thisdomain, the strength rankings reverse; the 0�/90� coreexhibits a higher flow stress than the ±45� core because

Fig. 9. Results of clamped bending experiments on both sandwich beamsand a solid sheet of equivalent mass.

of its superior stretch resistance (Fig. 4). Moreover, whenstretch-dominated, the monolithic sheet becomes the stron-gest, also consistent with the results in Fig. 4.

Also shown in Fig. 9 are rudimentary predictions of theresponse of the sandwich beams in the stretch-dominateddomain. These are obtained by scaling the clamped bend-ing results of the monolithic sheet in accordance with Eq.(1). The core strengths are again taken to b = 0 andb = 0.5 for the ±45� and 0�/90� cores, respectively, anda = 0.15. The resulting curves agree reasonably well withthe measurements in the domain d/S > 0.1. This correlationreaffirms that the change in strength ranking of the twocores is a result of their respective stretch resistance.

6. Optimal designs for bending

Further assessment of the hollow truss core sandwichbeams has been made through comparison of their bendingstrength with those that have been optimally designed.Among the two hollow truss cores, only that in the ±45�orientation is considered, because of its high shearstrength. Comparisons are also made with optimal honey-comb core sandwich panels [20].

The objective of the optimization is to find the geometricparameters of sandwich beams that can support a pre-scribed bending load, P, at minimum weight. The pertinentnon-dimensional weight index is [1]:

W ¼ Wq‘

ð15Þ

where W is the weight per unit area and q the mass densityof the solid material. From geometry, this index can be ex-pressed as:

Fig. 11. Comparisons of the bend strengths of the present sandwichbeams with ±45� cores with both solid and optimized sandwich beams.

Fig. 10. Deformation of sandwich beams under clamped bending. Arrows indicate locations of core separation from face sheets.

5516 H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518

W ¼ 2tf

‘þ pRH cð1� ðr=RÞ2Þ

2Lc‘ð16Þ

Each of the critical loads for the four possible failuremodes (Eqs. (9)–(12)) represents an optimization con-straint. Additionally, to ensure that the designs reside inthe domain of thin beams, the core thickness is restrictedto remain below a critical value: Hc/‘ 6 0.2.

Solutions for the optimal designs have been obtainedusing two complementary methods. In the first, the designobjective and the constraints were coded in an IMSL sub-routine that performs the optimization numerically [1,2].Results from the calculations show that the optima occurat the confluence of either two or three constraints. Withknowledge of the active constraints, analytic solutions havebeen derived using standard mathematical procedures.These procedures and the resulting solutions for the hollowtruss core sandwich beams are presented in the Appendix.Post-yield and post-buckling behaviors are not consideredhere.

Variations in the non-dimensional weight W with load Pare plotted in Fig. 11. Results include those for optimizedhollow truss core and honeycomb core beams [20], as wellas a solid beam of the same material. The weight of the lat-ter is given by [3]:

W ¼ 6Pey

� �1=2

ð17Þ

Additionally, results for the present sandwich beams withthe ±45� hollow tube core are shown. Since failure of thesebeams occurs by face yielding, Eqs. (9) and (16) yield sca-lings of the forms:

W ¼ W0‘0

‘

� �ð18Þ

P ¼ P0

‘0

‘

� �2

ð19Þ

where W0 and P0 are the weight and the load capacity ofthe sandwich beam at a characteristic length, ‘ = ‘0. Thelatter is selected to correspond to the simply supportedthree-point bend test conditions (with ‘0 = S/2 = 92 mm),whereupon W0 = 0.029 and P0 = 1.34 · 10�6. CombiningEqs. (18) and (19) (to eliminate ‘) yields:

W ¼ W0

PP0

� �1=2

ð20Þ

Numerical values have been calculated for 0.009 6 W 60.03. This weight range was obtained by varying the char-acteristic length over 90 mm 6‘6 300 mm. Since the abso-lute core thickness remains fixed (Hc = 17 mm), itsnormalized value falls in the range 0.05 6 Hc/‘ 6 0.2. Faceyielding dominates throughout.

H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518 5517

The comparisons in Fig. 11 reveal that the present sand-wich beams are far superior to the solid beams: theirstrengths differing by an order of magnitude at prescribedweight. Additionally, the beam strengths are within a fac-tor of 2 of those with the optimal hollow truss core designsat high loads, with the difference increasing as the loaddecreases. The geometric parameters needed to achievehigher strengths can be obtained from the solutions inthe Appendix.

The results also demonstrate that the hollow trusscore sandwiches can be designed to be stronger thanthose with square honeycomb cores. However, the pres-ent truss designs only support load effectively in oneplane whereas the honeycomb cores are almost isotropic.This deficiency could be mediated by using alternatetruss configurations, e.g. pyramidal or tetrahedral [1,2].Indeed, methods for fabricating such cores have recentlybeen demonstrated [18]. More importantly, the hollowtruss structures can be used as both heat pipes and heatexchange media within a sandwich panel. The closed cellstructure of the honeycomb core precludes its use inthese applications.

7. Summary

Metallic sandwich beams with hollow tube cores havebeen fabricated in a two-step brazing process. Theresponses in shear, tension, compression, as well as simplysupported and fully clamped bending, have been measuredfor both diamond (±45�) and square (0�/90�) orientations.Among the two cores, the diamond orientation exhibits thehigher shear strength but lower stretch resistance. The twocores perform equally well in compression. For bend-dom-inated loadings, wherein substantial core shear arises, thediamond orientation is preferred. For stretch-dominatedbehavior occurring at large displacements in fully clampedtests, the square orientation is superior.

Optimizations have been performed to identify stronglightweight sandwich designs with both hollow truss andsquare honeycomb cores. Although somewhat sub-optimal, the present hollow truss sandwich beams are farsuperior to solid beams of equivalent weight. With somemodifications in truss topology and dimensions, excep-tional strengths could be achieved. Combined with theiradaptability for efficient heat exchange, the hollow trusscore sandwiches offer outstanding potential for use as light-weight thermostructural components.

Acknowledgements

This work was supported by the Office of Naval Re-search through a contract to the University of Virginia(N0014-01-1-1051) and a sub-contract to the Universityof California, Santa Barbara (GG10376-114969), moni-tored by Dr Steve Fishman. The authors are grateful toProfessors Vikram Deshpande and Norman Fleck (Cam-bridge University, UK) for helpful discussions.

Appendix A. Optimization of hollow truss core sandwich

beams

A.1. Preliminaries

Numerical results from the IMSL subroutine revealthree solution domains, distinguished by the load param-eter, P. Henceforth, the load domains are denoted low,intermediate and high. In each, either two or three designconstraints are active. At low loads, the active constraintsare face yielding (FY) and core buckling (CB). The tran-sition to the intermediate load domain occurs when thecore thickness reaches its limit: Hc/‘ = 0.2. In thisdomain, face yielding and core buckling remain active.At high loads, core yielding replaces core buckling,whereas face yielding and the core thickness limit remain.With knowledge of the active constraints, analytic solu-tions are obtained by combining the constraint functions,as described below.

A.2. Low load domain

Upon combining the analytic solutions for the activeconstraints in this domain (Eqs. (9) and (12)) with thatfor the weight (Eq. (16)), tf/‘ and Hc/‘ are eliminated,allowing W to be written in terms of two independent geo-metric variables, notably Lc/R and r/R:

W ¼ p3ð1� ðr=RÞ4Þ8eyðLc=RÞ3

þ 8ðLc=RÞ2

p2ð1þ ðr=RÞ2ÞP ðA1Þ

Over the entire physically accessible range 0 6 r/R < 1, oW/o(r/R) < 0 and hence the optimum occurs as r/R! 1.However, in this limit, the walls become susceptible toshort wavelength buckling. To ensure stability against thismode whilst retaining high resistance to long wavelengthbuckling, the optimal value of r/R is taken to be 0.9 [21].Furthermore, setting oW/o(Lc/R) = 0 yields:

Lc

R¼ 3p5ð1� ðr=RÞ4Þð1þ ðr=RÞ2Þ

128ey

" #1=5

P�1=5 ðA2Þ

With Lc/R and r/R known, Hc/‘, tf/‘ and W are readilyobtained:

H c

‘¼ 27ð1þ ðr=RÞ2Þ3

2e3yð1� ðr=RÞ4Þ2

" #1=5

P2=5 ðA3Þ

tf

‘¼ 2ð1� ðr=RÞ4Þ2

27e2yð1þ ðr=RÞ2Þ3

!1=5

P3=5 ðA4Þ

W ¼ 463ð1� ðr=RÞ2Þ2

2e2yð1þ ðr=RÞ2Þ

!1=5

P3=5 ðA5Þ

Although W does not depend on Lc/‘, the latter cannot ex-ceed a critical value, dictated by the face buckling con-straint. The limiting value is:

5518 H.J. Rathbun et al. / Acta Materialia 54 (2006) 5509–5518

Lc

‘6

p2

24ð1� m2Þ

� �4ð1� ðr=RÞ4Þ4

729e9yð1þ ðr=RÞ2Þ6

!1=5

P6=5 ðA6Þ

A.3. Intermediate load domain

The transition to the intermediate load domain occurswhen the core thickness reaches its maximum allowablevalue, Hc/‘ = 0.2. The corresponding load is given by:

Ptr ¼ ð0:2Þ5=2 2e3yð1� ðr=RÞ2Þ2

27ð1þ ðr=RÞ2Þ

" #1=2

ðA7Þ

With the core thickness fixed, the face sheet thickness isimmediately obtained from Eq. (9):

tf

‘¼ 5

ey

� �P ðA8Þ

As in the low load domain, r/R is taken to be 0.9. Then,from Eq. (12) (the core buckling constraint) and Eq. (A1):

Lc

R¼ p3ð1� ðr=RÞ4Þ

80

!1=3

P�1=3 ðA9Þ

W ¼ 10

ey

Pþ 2ð1� ðr=RÞ2Þ2

25ð1þ ðr=RÞ2Þ

" #1=3

P1=3 ðA10Þ

An upper allowable limit on Lc/‘ is again obtained (to pre-vent face buckling), now given by:

Lc

‘6 5

p2

24ð1� m2Þe3y

!1=2

P ðA11Þ

A.4. High load domain

The transition to the high load domain occurs when coreyielding replaces core buckling. Throughout, the face yield-ing constraint and the limiting core thickness Hc/‘ = 0.2remain active. The transition load is:

Ptr ¼e3

y

ð1þ ðr=RÞ2Þ

!1=21� ðr=RÞ2

10

!ðA12Þ

From Eq. (9), the weight is:

W ¼ 12

ey

P ðA13Þ

In this domain, W is independent of Lc/‘, Lc/R and r/R,although the upper limit established on Lc/‘ for the inter-mediate load domain (Eq. (11)) remains valid.

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