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Structural performance of metallic sandwich beams with 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 b

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

Received 25 April 2006; accepted 21 July 2006 Available 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 respect to the face sheets. The responses in shear, tension and compression, as well as simply supported and fully clamped bending, are measured for specimens with both core orientations. While the two cores perform equally well in compression, the diamond orientation exhibits higher shear strength but lower stretch resistance. For bend-dominated loadings of the sandwich beams, the core in the diamond orientation is preferred because of its higher shear strength. For stretch-dominated loadings encountered in large-displacement, fully clamped bending, the square orientation is superior. Models of core and beam yielding are used to rationalize these observations. Optimizations are 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, lattice truss and prismatic cores are being developed for structures that require high strength [1–5] and blast resistance [6–9]. Some topological conﬁgurations are amenable to additional functionality. For instance, shape morphing of the 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 typical application, high heat ﬂuxes are deposited onto one of the face sheets of a sandwich panel; the heat is conducted away from the faces via the core elements and subsequently removed by a coolant ﬂowing through the core [13,14]. Any open cell metallic structure that allows coolant ﬂow can be used as a heat exchange medium. Through appropri*

Corresponding author. Tel.: +1 805 893 8699; fax: +1 805 893 8486. E-mail address: [email protected] (F.W. Zok).

ate selection of material and core topology, these structures can be designed to possess high thermal conductivity and eﬃciently transfer heat to the cooling ﬂuid [15,16]. One approach envisioned to further increase the thermal performance of sandwich structures involves the use of heat pipes as the truss members within the core [14,17]. Heat pipes increase the performance by homogenizing the temperature distribution and increasing the average temperature diﬀerence at the interface between the external pipe surface and the coolant. Additionally, because of their high second moment of area, such pipes oﬀer superior buckling resistance relative to solid members of equivalent mass [17,18]. Consequently, the cores can be designed with enhanced structural eﬃciency, especially in the domain of low core relative density. The present study explores the structural characteristics of metallic sandwich beams with hollow truss lattice cores. Using recently proposed methods for the manufacture of hollow truss cores and sandwich panels that utilize them

1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.07.016

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Nomenclature B E F Fs Hc Heq It ‘ ‘0 Lc M Mt r R S tf V a b

beam width elastic modulus tensile force shear force core thickness thickness of a solid plate of equivalent mass second moment of area of hollow tube ratio of bending moment to shear force (‘ ” M/V) characteristic length center-to-center tube spacing bending moment applied to sandwich structure bending moment in hollow tube inner tube radius outer tube radius loading span face sheet thickness shear force applied to sandwich structure fractional change in ﬂow stress following brazing non-dimensional core strength

[17], the article focuses on plastic deformation of simply supported and fully clamped bending beams with orthogonal (‘‘cross-ply’’) core architectures, in ±45 (diamond) and ±0/90 (square) loading orientations. Mechanical performance diﬀerences are rationalized on the basis of the core properties in the two orientations, as well as the prevailing core stresses in the beams. Models are used in parallel with the experiments to glean insights into the connections between core topology, constituent material properties and beam performance. Finally, optimizations are performed to identify strong, lightweight sandwich panel designs with hollow truss cores. Their performance is shown to compare favorably with optimized panels with square honeycomb cores. 2. Experimental procedures 2.1. Sandwich fabrication Hollow truss cores were made from medical grade 304 stainless steel tubes (Vita Needle, Needham, MA), manufactured by a tungsten inert gas welding and plug drawing process. The inner and outer tube radii were r = 0.61 mm and R = 0.74 mm, respectively. In preparation for brazing, each tube was sprayed with a mixture of a polymer-based cement (Nicrobraz Cement 520) and a Ni–25%Cr–10%P– 0.03%C braze powder (Nicrobraz 51), both supplied by Wall Colmonoy (Madison Heights, MI). The phosphorous in this alloy acts as a melting point depressant and enhances the ﬂow and wetting characteristics of the molten

d D ey c P P0 qc R rc core r f r s r rt ry s sp sp sy W W0

bending displacement normalized displacement yield strain shear strain non-dimensional load load capacity at ‘ = ‘0 core relative density normalized load core compressive strength tensile ﬂow strength of core tensile ﬂow strength of heat-treated (not brazed) parent alloy panel tensile strength maximum bending stress in hollow tube yield strength Shear stress fully plastic shear strength non-dimensional fully plastic shear strength non-dimensional shear yield stress non-dimensional weight index weight of sandwich beam at ‘ = ‘0

braze. The braze alloy has a solidus of 880 C and a liquidus of 950 C, with a suggested brazing range of 980–1095 C. Once coated, the tubes were stacked in an orthogonal (‘‘cross-ply’’) pattern to the desired height within an alignment tool (Fig. 1a). Alignment was achieved by a set of uniformly spaced stainless steel dowel pins inserted into holes in a stainless steel base plate. The dowel spacing was selected to produce a center-to-center tube spacing, Lc, of 5 mm. The alignment ﬁxture was spray coated ﬁrst with Nicrobraz Green Stop-Oﬀ (Wall Colmonoy, Madison Heights, MI) and then boron nitride (GE Advanced 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 to 550 C and held at temperature for 20 min to volatilize and remove the polymer binder. Thereafter, it was heated to 1020 C at 104 torr, held for 60 min and cooled to ambient temperature at 25 C/min. After brazing, the cores were removed from the tooling and cut to size (nominally 18 mm thick) using wire electro-discharge machining (Fig. 1(b)). The measured core relative density, qc, was 7.5%. By comparison, the calculated value (neglecting the added weight of the braze alloy) was 7.3% [15]. A second brazing operation was used to attach the lattice cores to 304 stainless steel face sheets to produce sandwich beams (Fig. 1(c)). For most cases, the face sheet thickness, tf = 0.7 ± 0.1 mm. For measurement of the core properties in compression, the face sheets were thicker (tf = 2.5 ± 0.2 mm), to ensure that they remained

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2.2. Mechanical testing The mechanical properties of the cores and the sandwich beams were measured using standard compression, tension, shear and bending tests [4,5,8,17,18]. Pertinent details are presented below. In addition, for comparison with the measured strengths of the sandwich beams and assessment of mechanical models, the uniaxial tensile properties of the 304 stainless steel face sheet material were measured following a heating cycle identical to that used for brazing. Tests were performed in accordance with ASTM E8-01, at a nominal strain rate of 103 s1. 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 heattreated parent alloy, some uncertainty in the in situ material properties in the sandwich structures remains because of the opportunity for chemical interaction between the alloy and the braze materials during fabrication. While previous studies have revealed minimal changes in the initial yield strength, they indicate signiﬁcant elevations in the hardening rate and ﬂow stress of 304 stainless steel after exposure to similar braze alloys [5]. Some insights into the magnitude of such eﬀects in the present sandwich specimens are gleaned from an analysis of the tensile test results, presented in Section 3. 3. Core properties Eﬀects of tube architecture (±45 vs. 0/90) on the core properties were ascertained for three loading modes: inplane tension, transverse (out-of-plane) compression and transverse shear. 3.1. Tension Fig. 1. Schematic of the procedure used to fabricate the ±45 hollow truss core sandwich beams. An analogous process was used for fabricating cores in the 0/90 orientation.

undeformed during transverse compressive testing. Even thicker faces (tf = 12.7 ± 0.2 mm) were attached to lap shear specimens that were used for measuring the shear response of the cores. To enable gripping in the clamped bending and uniaxial tension tests, solid steel inserts were placed between and brazed to the face sheets at the two ends of the beams. The assembly (core, face sheets and inserts) was placed in a clamping ﬁxture. To prevent bonding to the ﬁxture, the ﬁxture surfaces were sprayed with Stop-Oﬀ and boron nitride. In addition, a 3 mm thick layer of high-purity alumina–silica ﬁber paper was placed between the face sheets and the adjacent ﬁxture surfaces. Moderate clamping pressure was used to maintain intimate contact between the faces and the core during brazing. The heating cycle was the same as that used to produce the cores. Examples of representative test specimens in both diamond and square orientations are shown in Fig. 2.

Experimental results from uniaxial tension tests are plotted in Fig. 3 for sandwich specimens with both core orientations, as well as the heat-treated 304 alloy itself. The relevant tensile stress is the load, F, divided by the areal density, BHeq, where B is the width and Heq is the thickness of a solid plate of equivalent mass. On this basis, the 304 alloy has the greatest stretch resistance, followed by the sandwich beam in the 0/90 orientation; the weakest response occurs in the ±45 orientation. The tensile ﬂow stress of a sandwich panel can be partitioned between the core and its face sheets, each weighted by its respective eﬀective thickness [4]. Making a correction for the change in the ﬂow stress of the steel following brazing, the resulting panel tensile strength becomes: F 2tf s ¼ r ¼ ð1 þ aÞ rf ð1 bÞ þb ð1aÞ BH eq H eq where b is a non-dimensional parameter characterizing core strength, deﬁned by b

core r f qc r

ð1bÞ

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

that after an equivalent heat treatment), the work hardening rate at small strains is elevated and the ﬂow stress at larger strains is about 10–30% higher than that of the pristine alloy. The inference from these comparisons is that the in situ yield strength of the alloy is only slightly altered by brazing. This result is utilized in subsequent comparisons between experiment and theory. A second key conclusion is that the stretch resistance of the 0/90 core (b = 0.5) is indeed much higher than that of the ±45 core (b 0). 3.2. Compression

f represent the tensile ﬂow strengths of the core core and r r and the heat-treated (but not brazed) parent alloy, respectively, at a prescribed strain, and a is the fractional change in the ﬂow stress of the steel following brazing. Established mechanical models for the two core types yield core strength values b = 0.5 and b = 0 for the 0/90 and ±45 cores, respectively [2]. Predictions of Eq. (1) are plotted in Fig. 3. When the material is assumed to be unaﬀected by the braze (a = 0), the predicted yield strengths agree well with the measurements (using the appropriate value of b). However, at small plastic strains (up to about 0.5%) the work hardening rates are underestimated. Thereafter, at larger strains, the hardening rates are essentially the same. Conversely, upon selecting a = 0.15, the predicted and measured curves at large strains (>0.5%) agree well with one another, but the yield strengths are overestimated. These observations are the same as the braze eﬀects reported in earlier studies [5]. That is, following brazing, the yield strength of 304 stainless steel remains essentially unchanged (relative to

1.0

Compressive stress, σc/ρcσY

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

The compression test results are presented on Fig. 4. For both orientations, yielding initiates at a normalized stress rc/qc 0.5, the same as that predicted from stress analyses of both cores [2,17]. Following yield, both cores strain harden in the small plastic strain domain, ultimately reaching a peak when the tubes begin to plastically buckle. The ﬂow strength achieves a plateau shortly thereafter, at rc/qcry 0.4 0.5, during which the core members

0.8

0Þ/90Þ ±45Þ

0.6

0.4 Predicted yield stress 0.2

0.0 0

10

20

30

40

50

Compressive plastic strain (%) Fig. 4. Compressive response of the hollow truss cores.

60

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progressively crush. Signiﬁcant core densiﬁcation begins at 40–50% compressive strain, marked by a rapid increase in hardening. In contrast to the core properties under tensile loading, the compressive properties are essentially independent of orientation. 3.3. Shear Representative stress-strain curves from the lap shear specimens are shown in Fig. 5. In the ±45 orientation, yield initiates at a shear stress, sy/qcry 0.5: again consistent with the analytic prediction [3]. Following subsequent strain hardening, the stress drops rapidly as the tubes begin to separate from the face sheets. The low failure strain of the ±45 orientation (c 2%) is associated with node failure at the tube-face sheet interface. Alternate brazes [4,5] and improved node joint designs are expected to produce more robust nodes. In the 0/90 orientation, the response is much softer, with a yield strength, sy/qcry 0.1. The subsequent plastic response is characterized by moderate hardening. The nodes remain intact over the strain range probed by these experiments (c 20%), because the low strength of the core ensures that a correspondingly low stress is transmitted to the face/tube interfaces. To model the shear strength of the 0/90 core, the tubes perpendicular to the loading direction are assumed to act as beams, rigidly aﬃxed to both the faces and the longitudinal tubes. An analysis of a unit cell of the tube array (Fig. 6) yields the shear stress, s, carried by a single tube: s¼

Fs 4Lc R

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

Here, Lc 2R represents the edge-to-edge spacing of the tubes, as illustrated in Fig. 6. Yielding initiates when the maximum bending stress, rt, in the tube reaches the material yield strength, ry, That is, rt ¼

F s ðLc 2RÞ Mt ¼ 2

ð3Þ

sy ¼

Normalized shear stress,τ/σyρc

ð5Þ

This result can be re-expressed in a non-dimensional form as: sy

sY R2 þ r 2 ¼ qc rY 4ðLc 2RÞR

ð6Þ

where the core relative density is given by [17]: qc ¼

s ¼ 0.8 ±45˚ 0.6 Predicted yield 0.4 0°/90°

0.2 Predicted yield 0.02

rY pðR4 r4 Þ 8Lc ðLc 2RÞR2

pðR2 r2 Þ 2Lc R

ð7Þ

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

1.0

0.0 0.00

ð4Þ

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

ð2Þ

where Fs is the applied shear force. The resulting bending moment, Mt, acting on each tube is obtained from static equilibrium, whereupon

M tR 4M t R ¼ ry ¼ It pðR4 r4 Þ

0.04

Predicted limit 0.06

0.08

0.10

Shear strain, γ Fig. 5. Shear response of hollow truss cores.

0.12

sp 2R ¼ qc rY pðLc 2RÞ

ð8Þ

Using dimensions pertinent to the present cores (R = 0.74 mm, r = 0.61 mm, Lc = 5 mm), the normalized strengths are sY ¼ 0:09 and sY ¼ 0:13 for the 0/90 core. These results compare favorably with the experimental measurements (Fig. 5). The slight discrepancy between the predicted fully plastic shear strength and the measured ﬂow strength at large strains can be attributed to strain hardening: a feature neglected in the analytical model. To summarize, while the compressive properties of the two cores are essentially the same, the core in the 0/90 orientation exhibits lower shear strength (by a factor of 5) but signiﬁcant stretch resistance (b = 0.5). These diﬀerences manifest themselves in the properties of the sandwich beams, as detailed below.

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4. Sandwich properties: simply supported bending 4.1. Experiments To assess the mechanical performance of the sandwich beams, measurements have been made using three-point simply supported bend tests. The outer loading points consisted of 25.4 mm diameter hardened steel pins on a ﬂat support base. To inhibit local indentation, the inner loading platen comprised a ﬂat central region, 12.7 mm wide, with adjacent ﬁlets 6 mm radius. The loading span was S = 184 mm. For comparison, tests were also performed on monolithic beams of equivalent areal density. The results are plotted in Fig. 7. The loads and displacements have been normalized by those needed to initiate yielding in the equivalent-weight solid beam; the ensuing normalized load and displacement (deﬁned in the ﬁgure) are denoted R and D, respectively. On this basis, it is evident that the strengths of the sandwich beams are much greater than that of the solid beams by about an order of magnitude. Additionally, in the plastic domain, beams with the ±45 core are also almost twice as strong as those with the 0/90 core. The latter diﬀerences arise from the operative failure modes, elaborated below. With the ±45 core, large deﬂections are obtained without deformation localization (Fig. 8a). In this case, the test was terminated once the outer loading pins had experienced signiﬁcant lateral displacement. The beams with the 0/90 cores exhibited similarly large plastic deﬂection, but ultimately failed by shear of the core beneath the inner loading platen (Fig. 8b). 4.2. Failure model: ±45 core Analytic models for failure initiation of sandwich beams with lattice cores have been developed previously [2,3], using stress analyses of the faces and the core elements subject to a combination of transverse shear, V, and bending

Normalized load, Σ = 3FS/2BHeq 2σy

30

±45˚

moment, M (both per unit width). These models have been adapted here for hollow truss cores. In the current implementation, the core is assumed to carry all of the shear and the faces all of the moment. The key results are summarized below. Failure in the face sheets can occur by either yielding or elastic buckling. The critical loads can be described by a non-dimensional parameter, deﬁned as P ” V2/EM. For beams with ±45 cores [3]: ey t f H c ðface sheet yieldingÞ ‘2 p2 t3f H c P¼ ðface sheet bucklingÞ 24ð1 m2 Þ‘2 L2c

P¼

ð9Þ ð10Þ

where Hc is the core thickness and ‘ ” M/V. The latter quantity scales with loading span: ‘ = S/2 for simply supported three-point bending and ‘ = S/4 for clamped ends. Similarly, the tubes in the core can fail by either yielding or buckling. Here thes nodes at which the tube members intersect are assumed to be pinned such that the adjoining tube segments are free to rotate about these nodes. The pertinent failure loads are: P¼

pey RH c ½1 ðr=RÞ2 4Lc ‘

P¼

p3 H c R3 ½1 ðr=RÞ 16L3c ‘

ðcore yieldingÞ

ð11Þ

ðcore bucklingÞ

ð12Þ

4

Beam failure is dictated by the mode with the lowest value of P. For three-point bending, wherein ‘ = S/2, the load parameter P is related to the non-dimensional quantity R presented in Fig. 7 via the relation: ! 3S 2 R¼ P ð13Þ 2H 2eq ey For the present experimental conﬁguration, the preceding models predict failure by face yielding, at a normalized load of P = 1.3 · 106 or, equivalently, R = 10. This result correlates well with the onset of non-linearity in the experimental measurements, shown in Fig. 7.

Onset of roller movement 20 0˚/90˚

10

4.3. Failure model: 0/90 core

Onset of core indentation

The most signiﬁcant modiﬁcation to the preceding results (Eqs. (9)–(12)) for beams with 0/90 cores pertains to the stress needed for core yielding. Utilizing the model developed in Section 3 for the core shear strength, the critical load becomes:

Predicted yield points Solid sheet

P¼ 0 0

5

10

15 2

Normalized displacement, Δ = 6δHeq/S εy Fig. 7. Comparison of simply supported bending response of the sandwich beams with that of the equivalent weight monolithic material.

pey H c ðR4 r4 Þ 8‘Lc ðLc 2RÞR2

ð14Þ

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

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

5. Sandwich properties: clamped bending Additional bend tests have been performed using fully clamped end conditions. Apart from the boundary conditions, all testing parameters were the same. The resulting load/displacement curves are summarized in Fig. 9 and images of the deformed specimens are in Fig. 10. At small displacements (d/S < 0.05), wherein the response is benddominated, the curves mimic those obtained under simply supported conditions (Fig. 7). That is, the beam with the ±45 core outperforms that with the 0/90 core. Conversely, at larger displacements, membrane stresses arise and the response becomes stretch-dominated. In this domain, the strength rankings reverse; the 0/90 core exhibits a higher ﬂow stress than the ±45 core because

of its superior stretch resistance (Fig. 4). Moreover, when stretch-dominated, the monolithic sheet becomes the strongest, also consistent with the results in Fig. 4. Also shown in Fig. 9 are rudimentary predictions of the response of the sandwich beams in the stretch-dominated domain. These are obtained by scaling the clamped bending results of the monolithic sheet in accordance with Eq. (1). The core strengths are again taken to b = 0 and b = 0.5 for the ±45 and 0/90 cores, respectively, and a = 0.15. The resulting curves agree reasonably well with the measurements in the domain d/S > 0.1. This correlation reaﬃrms that the change in strength ranking of the two cores is a result of their respective stretch resistance. 6. Optimal designs for bending Further assessment of the hollow truss core sandwich beams has been made through comparison of their bending strength 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 shear strength. Comparisons are also made with optimal honeycomb core sandwich panels [20]. The objective of the optimization is to ﬁnd the geometric parameters of sandwich beams that can support a prescribed bending load, P, at minimum weight. The pertinent non-dimensional weight index is [1]: W¼

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

W q‘

ð15Þ

where W is the weight per unit area and q the mass density of the solid material. From geometry, this index can be expressed as:

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Fig. 10. Deformation of sandwich beams under clamped bending. Arrows indicate locations of core separation from face sheets. 2

2tf pRH c ð1 ðr=RÞ Þ þ ð16Þ 2Lc ‘ ‘ Each of the critical loads for the four possible failure modes (Eqs. (9)–(12)) represents an optimization constraint. Additionally, to ensure that the designs reside in the domain of thin beams, the core thickness is restricted to remain below a critical value: Hc/‘ 6 0.2. Solutions for the optimal designs have been obtained using two complementary methods. In the ﬁrst, the design objective and the constraints were coded in an IMSL subroutine that performs the optimization numerically [1,2]. Results from the calculations show that the optima occur at the conﬂuence of either two or three constraints. With knowledge of the active constraints, analytic solutions have been derived using standard mathematical procedures. These procedures and the resulting solutions for the hollow truss core sandwich beams are presented in the Appendix. Post-yield and post-buckling behaviors are not considered here. Variations in the non-dimensional weight W with load P are plotted in Fig. 11. Results include those for optimized hollow truss core and honeycomb core beams [20], as well as a solid beam of the same material. The weight of the latter is given by [3]: 1=2 6P W¼ ð17Þ ey

W¼

Additionally, results for the present sandwich beams with the ±45 hollow tube core are shown. Since failure of these beams occurs by face yielding, Eqs. (9) and (16) yield scalings of the forms: ‘0 W ¼ W0 ð18Þ ‘

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

2 ‘0 ð19Þ P ¼ P0 ‘ where W0 and P0 are the weight and the load capacity of the sandwich beam at a characteristic length, ‘ = ‘0. The latter is selected to correspond to the simply supported three-point bend test conditions (with ‘0 = S/2 = 92 mm), whereupon W0 = 0.029 and P0 = 1.34 · 106. Combining Eqs. (18) and (19) (to eliminate ‘) yields: 1=2 P W ¼ W0 ð20Þ P0 Numerical values have been calculated for 0.009 6 W 6 0.03. This weight range was obtained by varying the characteristic length over 90 mm 6‘6 300 mm. Since the absolute core thickness remains ﬁxed (Hc = 17 mm), its normalized value falls in the range 0.05 6 Hc/‘ 6 0.2. Face yielding dominates throughout.

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The comparisons in Fig. 11 reveal that the present sandwich beams are far superior to the solid beams: their strengths diﬀering by an order of magnitude at prescribed weight. Additionally, the beam strengths are within a factor of 2 of those with the optimal hollow truss core designs at high loads, with the diﬀerence increasing as the load decreases. The geometric parameters needed to achieve higher strengths can be obtained from the solutions in the Appendix. The results also demonstrate that the hollow truss core sandwiches can be designed to be stronger than those with square honeycomb cores. However, the present truss designs only support load eﬀectively in one plane whereas the honeycomb cores are almost isotropic. This deﬁciency could be mediated by using alternate truss conﬁgurations, e.g. pyramidal or tetrahedral [1,2]. Indeed, methods for fabricating such cores have recently been demonstrated [18]. More importantly, the hollow truss structures can be used as both heat pipes and heat exchange media within a sandwich panel. The closed cell structure of the honeycomb core precludes its use in these applications. 7. Summary Metallic sandwich beams with hollow tube cores have been fabricated in a two-step brazing process. The responses in shear, tension, compression, as well as simply supported and fully clamped bending, have been measured for both diamond (±45) and square (0/90) orientations. Among the two cores, the diamond orientation exhibits the higher shear strength but lower stretch resistance. The two cores perform equally well in compression. For bend-dominated loadings, wherein substantial core shear arises, the diamond orientation is preferred. For stretch-dominated behavior occurring at large displacements in fully clamped tests, the square orientation is superior. Optimizations have been performed to identify strong lightweight sandwich designs with both hollow truss and square honeycomb cores. Although somewhat suboptimal, the present hollow truss sandwich beams are far superior to solid beams of equivalent weight. With some modiﬁcations in truss topology and dimensions, exceptional strengths could be achieved. Combined with their adaptability for eﬃcient heat exchange, the hollow truss core sandwiches oﬀer outstanding potential for use as lightweight thermostructural components. Acknowledgements This work was supported by the Oﬃce of Naval Research through a contract to the University of Virginia (N0014-01-1-1051) and a sub-contract to the University of California, Santa Barbara (GG10376-114969), monitored by Dr Steve Fishman. The authors are grateful to Professors Vikram Deshpande and Norman Fleck (Cambridge University, UK) for helpful discussions.

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Appendix A. Optimization of hollow truss core sandwich beams A.1. Preliminaries Numerical results from the IMSL subroutine reveal three solution domains, distinguished by the load parameter, P. Henceforth, the load domains are denoted low, intermediate and high. In each, either two or three design constraints are active. At low loads, the active constraints are face yielding (FY) and core buckling (CB). The transition to the intermediate load domain occurs when the core thickness reaches its limit: Hc/‘ = 0.2. In this domain, 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 solutions are obtained by combining the constraint functions, as described below. A.2. Low load domain Upon combining the analytic solutions for the active constraints in this domain (Eqs. (9) and (12)) with that for the weight (Eq. (16)), tf/‘ and Hc/‘ are eliminated, allowing W to be written in terms of two independent geometric variables, notably Lc/R and r/R: 4

W¼

p3 ð1 ðr=RÞ Þ 8ey ðLc =RÞ

3

þ

8ðLc =RÞ

2 2

p2 ð1 þ ðr=RÞ Þ

P

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 to short wavelength buckling. To ensure stability against this mode whilst retaining high resistance to long wavelength buckling, the optimal value of r/R is taken to be 0.9 [21]. Furthermore, setting oW/o(Lc/R) = 0 yields: " #1=5 4 2 Lc 3p5 ð1 ðr=RÞ Þð1 þ ðr=RÞ Þ ¼ P1=5 ðA2Þ 128ey R With Lc/R and r/R known, Hc/‘, tf/‘ and W are readily obtained: " #1=5 Hc 27ð1 þ ðr=RÞ2 Þ3 ¼ P2=5 ðA3Þ 4 2 ‘ 2e3y ð1 ðr=RÞ Þ !1=5 tf 2ð1 ðr=RÞ4 Þ2 ¼ P3=5 ðA4Þ 2 3 ‘ 27e2y ð1 þ ðr=RÞ Þ !1=5 463ð1 ðr=RÞ2 Þ2 P3=5 ðA5Þ W¼ 2 2e2y ð1 þ ðr=RÞ Þ Although W does not depend on Lc/‘, the latter cannot exceed a critical value, dictated by the face buckling constraint. The limiting value is:

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Lc 6 ‘

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p2 24ð1 m2 Þ

4 4

4ð1 ðr=RÞ Þ

2 6

729e9y ð1 þ ðr=RÞ Þ

!1=5

From Eq. (9), the weight is: P6=5

ðA6Þ

A.3. Intermediate load domain The transition to the intermediate load domain occurs when the core thickness reaches its maximum allowable value, Hc/‘ = 0.2. The corresponding load is given by: " # 2 2 1=2 3 5=2 2ey ð1 ðr=RÞ Þ ðA7Þ Ptr ¼ ð0:2Þ 27ð1 þ ðr=RÞ2 Þ With the core thickness ﬁxed, the face sheet thickness is immediately obtained from Eq. (9): tf 5 ¼ P ðA8Þ ey ‘ 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): !1=3 4 Lc p3 ð1 ðr=RÞ Þ ¼ P1=3 ðA9Þ 80 R " # 2 2 1=3 10 2ð1 ðr=RÞ Þ P1=3 ðA10Þ W¼ Pþ 2 ey 25ð1 þ ðr=RÞ Þ An upper allowable limit on Lc/‘ is again obtained (to prevent face buckling), now given by: !1=2 Lc p2 65 P ðA11Þ ‘ 24ð1 m2 Þe3y A.4. High load domain The transition to the high load domain occurs when core yielding replaces core buckling. Throughout, the face yielding constraint and the limiting core thickness Hc/‘ = 0.2 remain active. The transition load is: !1=2 ! e3y 1 ðr=RÞ2 Ptr ¼ ðA12Þ 2 10 ð1 þ ðr=RÞ Þ

W¼

12 P ey

ðA13Þ

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

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