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Acta Materialia 58 (2010) 2822–2835

The compressive and shear response of titanium matrixcomposite lattice structures

P. Moongkhamklang a, V.S. Deshpande b, H.N.G. Wadley a,*

a Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904-4745, United Statesb Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, United Kingdom

Received 14 August 2009; received in revised form 5 January 2010; accepted 5 January 2010Available online 22 February 2010

Abstract

A method has been developed for fabricating millimeter cell size cellular lattice structures with square and diamond collinear trusstopologies from 240 lm diameter Ti–6Al–4V-coated SiC monofilaments (titanium matrix composite (TMC) monofilaments). Latticeswith relative densities in the range 10–20% were manufactured and tested in both compression and shear. Because of the very highstrength of the TMC monofilaments, the compressive strengths of both topology lattices were dominated by elastic buckling of the con-stituent struts. However, under shear loading, some of the struts are subjected to tensile stresses and failure is then set by tensile fractureof the monofilaments. Analytical expressions are derived for the elastic moduli and strength of both lattice topologies and the predictionsare compared with measurements over the range of relative densities investigated in this study. Excellent agreement between the mea-surements and predictions is observed. The specific shear strength of the TMC lattices is superior to all other cellular materials investi-gated to date, including carbon fiber-reinforced polymers (CFRP) honeycombs. Their compressive properties are comparable to CFRPhoneycombs. The TMC lattices have a brittle response and undergo catastrophic failure at their peak load. They appear most promisingas candidates for the cores in sandwich structures intended for elevated temperature and multifunctional applications where their limitedductility is not a significant constraint.� 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Cellular materials; Metal–matrix composites (MMCs); Mechanical properties

1. Introduction

Lightweight sandwich panels with lattice truss cores arebeing developed for structures that require high specificstiffness and/or strength [1–8]. The truss members inlattice materials can be topologically configured to experi-ence predominantly axial stresses (i.e. tension or compres-sion) when the sandwich panels are loaded in bending (i.e.the cores are stretch dominated) [1]. This results in amechanical performance that is superior to panels thathave cores made of stochastic foams whose ligamentsdeform by bending [9]. In addition, the interconnected

1359-6454/$36.00 � 2010 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2010.01.004

* Corresponding author. Tel.: +1 434 924 3032.E-mail address: haydn@virginia.edu (H.N.G. Wadley).

void spaces within lattice cores provides numerous multi-functional opportunities such as cross-flow heat exchange[10–12], dynamic load protection [13–15] and acousticdamping [16].

The compressive and shear stiffnesses and strengths of allcellular structures depend on the intrinsic properties of thesolids from which they are made. Hence, their stiffness andstrength can be significantly increased by the use of stiffer,higher strength materials. Recently, carbon fiber-reinforcedpolymers (CFRP) have been used to make sandwich struc-tures with both pyramidal lattice [17] and square honeycomb[18] topology cores. To date, these CFRP sandwich struc-tures exhibit the highest specific stiffnesses and strengths,but since the polymers in CFRP decompose at low tempera-tures, these structures cannot be used in elevated tempera-ture applications.

rights reserved.

Fig. 1. Sketches of the: (a) diamond collinear and (b) square collinearcellular sandwich cores. The geometrical parameters used to characterizethe specimens and the core topologies are marked along with thecoordinate system employed.

P. Moongkhamklang et al. / Acta Materialia 58 (2010) 2822–2835 2823

A method for fabricating millimeter cell size cellular lat-tice structures with a square or diamond collinear truss

Fig. 2. Schematic of the procedure used to fabricate the TMC collinear latticevacuum diffusion bonding of the core, (c) sandwich panel lay-out, and (d) bra

topology (see Fig. 1) from 240 lm diameter Ti–6Al–4V-coated SiC monofilaments has recently been developed[19]. The monofilaments used to make this structures havean elastic modulus of 195 GPa, a tensile strength in excessof 800 MPa and a density of only 3.93 Mg m�3. The specificcompressive stiffness and strength of titanium matrix com-posite (TMC) lattices made from these filaments wasbetween 2 and 10 times that of other cellular structures ofsimilar density. Since titanium composites retain gooddimensional stability at temperatures up to 500 �C [20], theselattices appear to be promising candidates for multifunc-tional applications at elevated temperature.

Sandwich structures are usually used in situations wherethey are subjected to significant bending loads [1–4]. Theflexural stiffness and strength of the panel are then alsodetermined by the compressive and shear responses of thecore. This study explores the out-of-plane compressiveand in-plane shear behaviors of TMC lattice structureswith both square and diamond collinear topologies(Fig. 1). The compressive and shear behaviors of the latticestructures are investigated experimentally and the micro-mechanisms of lattice deformation are identified as a func-tion of lattice relative density. The mechanical propertiesare then compared to analytic estimates based upon theobserved failure modes and with other similarly loaded lat-tice and prismatic topologies. While these materials/struc-tures have limited ductility, they are found to exhibit the

core sandwich panels: (a) assembly sequence to make a TMC lattice, (b)zing of face sheets to the cores.

Fig. 3. Diffusion bonded truss–truss nodes: (a) exterior deformation of the titanium alloy coating at the nodes, (b) cross-section of a pair of nodes, and (c)the diffusion bonded region between a pair of monofilaments.

Fig. 4. Photographs of the as-manufactured �q ¼ 9:6% (L/H = 8): (a)diamond and (b) square collinear TMC lattices. Some of the specimenshad serrations on the inner surfaces of the face sheets to facilitate sheartesting.

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highest specific shear stiffness and strength of any cellularstructure fabricated to date.

2. Experimental protocol

2.1. Specimen fabrication

Collinear lattice structures with square and diamond ori-entations and a relative density, �q, in the range of 9–19%were fabricated from Ti–6Al–4V-coated SCS-6 SiC monofil-aments (FMW Composite Systems, Inc., Clarksburg, WV).Each monofilament was approximately 240 lm in diameter(a = 120 lm) and consisted of a 140 lm diameter SCS-6SiC fiber (aSiC = 70 lm) surrounded by a 50 lm thick phys-ical vapor deposited Ti–6Al–4V coating. The SCS-6 fiberswere made by chemical vapor deposition on 33 lm diametercarbon fiber substrates [21]. The densities of the SCS-6 fiber,Ti–6Al–4V and the composite monofilament were 3.00, 4.43and 3.93 g cm–3, respectively, while the Young’s moduli ofthe Ti and SiC were ETi ¼ 109 GPa and ESiC ¼ 355 GPa,respectively.

The TMC monofilaments were stacked in an orthogonalpattern within an alignment tool (Fig. 2a). Alignment wasachieved by a set of uniformly spaced stainless steel pinsinserted into holes in a stainless steel base plate. The dowelspacing was selected to produce a desired center-to-centertruss spacing l (i.e. unit cell length; see Fig. 1). The align-ment fixture was spray coated with boron nitride (GEAdvanced Ceramics, Lakewood, OH) to prevent sticking.Once assembly of the TMC filament lay-up was complete,a dead weight was used to apply a 7.5 g force to each con-tact (truss–truss node). The assembly was diffusion bondedby placing it in a vacuum furnace (Super VII, Centorr

Table 1The geometric parameters of the three relative densities of the TMClattices investigated in this study.

Center-to-center celllength, l (mm)

After diffusion bonding b �W/Wo

Core width,W (mm)

Relativedensity, �q (%)

1.27 5.80 ± 0.083 18.4 ± 0.27 0.81 ± 0.0122.03 9.07 ± 0.071 11.3 ± 0.09 0.82 ± 0.0062.51 10.30 ± 0.098 9.6 ± 0.09 0.78 ± 0.007

Vacuum Industries, Nashua, NH) at a base pressure of�10�7 Torr. The furnace was heated at 20 �C min–1 to900 �C and then held for 6 h under pressure (Fig. 2b). Priorto diffusion bonding, each truss–truss node formed a smallelastic contact but the contact area increased with diffusionbonding time to approximately 215 � 215 lm. The localcontact pressure applied by the dead weight was therefore

Fig. 5. Measured quasi-static tensile stress vs. strain curves of the asdiffusion bonded/brazed TMC monofilaments. Three representativecurves are included to indicate the level of repeatability of themeasurements.

P. Moongkhamklang et al. / Acta Materialia 58 (2010) 2822–2835 2825

high at the beginning (>735 MPa for a contact areaof < 10 � 10 lm) but was reduced to approximately1.8 MPa at the end of diffusion bonding process. The latticestructures were removed from the tooling and cut to shapeusing wire electrodischarge machining.

Fig. 3 shows nodal regions of the lattice structure afterthe diffusion bonding step. It can be seen that excellentmetallurgical continuity was achieved in the bonded region.Equiaxed a-grains and an intergranular b-phase micro-structure were observed with an average a-grain size of10 lm. During diffusion bonding, the spacing between con-secutive layers, and hence the macroscopic lattice width,decreased as the titanium alloy coating at the contactpoints deformed and interdiffused. The relative density ofthe diffusion bonded collinear lattice structure, �q, is givenby [22]

�q ¼ p2b

al

� �ð1Þ

where a and l are the monofilament radius and the inter-monofilament spacing (i.e. the unit cell length) and b is

Fig. 6. The measured compressive stress r33 vs. strain e33 responses of the:(a) square and (b) diamond lattices. Measurements are shown for the threerelative densities �q investigated in this study.

the diffusion bonding coefficient, defined as b � W =W o,where W and Wo are the lattice widths prior to and afterthe diffusion bonding process (see Figs. 1 and 2b for thedefinition of W and Wo). Here (and subsequently) we shallalways take the half angle of the cells (see Fig. 1) asx = 45�. Numerous samples were fabricated with differentrelative densities by varying the inter-monofilament spac-ing l. Table 1 summarizes the lattice parameters and corre-sponding relative densities of these samples after diffusionbonding.

After cutting to size, the lattices were brazed to Ti–6Al–4V face sheets using a TiCuNi-60� paste braze alloy sup-plied by Lucas-Milhaupt, Inc. (Fig. 2c and d). The sand-wich panel samples were vacuum brazed by heating at20 �C min–1 to 550 �C, holding for 5 min (to volatize andremove the polymer binder) and finally heating to 975 �Cfor 30 min at a base pressure of �10�7 Torr. For specimenstested in shear, corrugations and slots were machined into

Fig. 7. A summary of the measured compressive: (a) modulus Ec and (b)peak strength rc of the square and diamond TMC lattices as functions ofthe relative density �q. The corresponding analytical predictions are alsoincluded as lines.

Fig. 8. (a) Photographs showing the deformation sequence of the�q ¼ 11:3% square lattice compressed at an applied nominal strain rate2 � 10�4 s�1. The corresponding measured stress vs. strain response isincluded in (b), with the strain corresponding to the various imagesmarked as I–IV.

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the inner surface of the face sheets (6.35 mm thick)attached to the diamond and square lattices to ensurerobust bonding at the lattice–face-sheet interface (seeFig. 4). Flat face sheets sufficed for the compression tests.

2.2. Compression tests

For compression tests, the square lattices were cut sothat they were four cells in height H and six cells in lengthL with W � H; see Fig. 1. For the diamond topology, pre-vious studies on stainless steels lattices suggested that theresponse is strongly influenced by the specimen aspect ratioL/H for L/H < 5 but is reasonably independent of the spec-imen aspect ratio for larger values of L/H [22]. All com-pression tests in this study were therefore performed onspecimens of aspect ratio L/H = 5, with three cells alongthe core height direction and W � H.

The sandwich panels were tested in compression follow-ing the guidelines of ASTM STP C-365 for sandwich pan-els. A screw-driven testing machine with a 50 kN load cellwas used to apply a nominal strain rate of 2 � 10�4 s�1.The measured load cell force was used to calculate thenominal stress applied to the lattice truss core while a laserextensometer was used to measure the compressive strains.A high-speed camera (Model Phantom v10, VisionResearch Inc., Wayne, NJ) was used to observe the defor-mation and brittle failure modes of the samples. At leastthree tests were conducted on each structure to confirmthe repeatability of the measurements.

2.3. Shear tests

Experiments were conducted to measure the shear stress(r13) vs. shear strain (c13) response. The ASTM standard tomeasure the shear response of sandwich cores (ASTM STPC-273) requires test specimens with an aspect ratioL=H P 12. However, given the manufacturing limitations,the shear tests were conducted on specimens with an aspectratio L/H = 8, with three cells in the core height H directionand W/H = 1 for both diamond and square topologies.

The samples were shear tested at an applied nominalshear strain rate of 2 � 10�4 s�1 using a single-lap shearconfiguration, as specified in the ASTM standard. Themeasured load cell force was used to calculate the shearstress in the specimen while the relative sliding of the twofaces of the sandwich plates was measured using a laserextensometer. This nominal shear strain in the specimenwas estimated from the sliding displacement. The high-speed camera was again used to observe the deformationand dynamic failure modes of the samples.

2.4. Properties of the parent material

To determine the mechanical properties of the parentTMC monofilaments in their “as-fabricated” condition,tensile tests were performed on fibers subjected to the samethermal cycle used for fabrication of the diffusion bonded

lattice and the brazed sandwich structures. The tensile testswere conducted at a nominal applied strain rate of2 � 10�4 s�1.

The measured true stress r vs. logarithmic strain eresponses of the TMC monofilament samples are shownin Fig. 5 (three representative measurements are plotted).The response is approximately linear until a series of frac-ture events occur in the Ti–6Al–4V coating. The measure-ments suggest that the Young’s modulus of TMCmonofilaments is Ef ¼ 195 GPa while their fracturestrength is rf ¼ 952 � 93 MPa (the variation of�93 MPa is based on 10 separate measurements).

P. Moongkhamklang et al. / Acta Materialia 58 (2010) 2822–2835 2827

3. Summary of the measured responses

3.1. Compressive response

The measured through-thickness compressive stress r33

vs. strain e33 responses of the square and diamond latticesis shown in Fig. 6a and b, respectively, for the three relativedensities (�q = 9.6%, 11.3% and 18.4%) investigated in thisstudy. The responses in all cases were elastic, followed bybrittle failure. The measured compressive modulus (mea-sured from unloading–reloading loops) and the peakstrength of the two lattice configurations are plotted inFig. 7a and b, respectively. Variability can be observedover three repeat measurements. This variability is primar-ily associated with: (i) the inherent statistical nature of theresponse of the TMC monofilaments (see the measuredresponse of the parent material in Fig. 5) and (ii) the man-ufacturing variability between specimens, which results inslight loading misalignments.

The sequence of deformation leading to failure is illus-trated in photographs in Figs. 8 and 9 for the �q ¼ 11:3%

Fig. 9. (a) Photographs showing the deformation sequence of the �q ¼ 11:3% dThe corresponding measured stress vs. strain response is included in (b), with tfailure process of the lattice was very rapid and occurred within a period of a

square lattice and diamond lattices, respectively. Globalbuckling of the vertical struts of the square lattice isclearly observed, with final failure resulting in localizationof the deformation and shearing of the square lattice in asingle layer of cells. By contrast, negligible deformation ofthe struts is observed in the diamond lattice until cata-strophic strut failure was initiated near the edge of thespecimen (photograph III in Fig. 9a). This failure thenpropagates in a crack-like manner at approximately200 m s–1 across the specimen, as evidenced from thehigh-speed photographs.

The failure of the struts always occurs in the vicinity ofthe nodes. Examples of the failure modes of the TMCmonofilaments in the square and diamond lattices areshown in Fig. 10a and b, respectively.

3.2. Shear response

The measured shear stress (r13) vs. strain (c13)responses of the square and diamond lattices are plottedin Fig. 11a and b, respectively. After an initial elastic

iamond lattice compressed at an applied nominal strain rate 2 � 10�4 s�1.he locations of the various images marked on the stress–strain curve. Thebout 140 ls, as shown by the time markers included in (a).

Fig. 10. Photographs showing the failure of the struts near the nodes ofthe �q ¼ 18:4%: (a) square and (b) diamond lattices subjected to uniaxialcompression.

Fig. 11. The measured shear stress r13 vs. strain c13 responses of the: (a)square and (b) diamond lattices. Measurements are shown for the threerelative densities �q investigated in this study.

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response the diamond lattices fail in a brittle manner(except for the �q ¼ 9:6% lattice, which displays an overallductile response due to progressive brittle failure of indi-vidual struts). By contrast, some plastic deformation isseen to occur in the square lattices prior to the peakstress. The measured moduli and peak shear strengthsare summarized as functions of �q in Fig. 12a and b,respectively. Again, variability over three repeated mea-surements can be observed.

Photographs illustrating the deformation of the�q ¼ 9:6% square lattice are included in Fig. 13: prior tofailure the deformation was reasonably uniform over theentire specimen, with failure occurring abruptly at the lineof nodes nearest to the face sheets. By contrast, failure ismore gradual in the �q ¼ 9:6% diamond lattice, with failureinitiating near the edge of the specimen and progressingthrough the length of the specimen (Fig. 14). However, thisgradual failure is not a feature of all the diamond lattices,with complete failure of the �q ¼ 11:3% diamond latticeoccurring in less than 1 ms, as seen in Fig. 15.

4. Analytical models for the compressive and shear responses

In this section we develop analytical models for the com-pressive and shear responses of the square lattice andbriefly quote the corresponding results for the diamond lat-tice from Zupan et al. [23] and Cote et al. [24].

4.1. Square lattice

Under compression (or tension) in the 3-direction, onlythe vertical struts of the square lattice carry the appliedload. Thus, the effective Young’s modulus Ec of the squarelattice is

Ec ¼1

2Ef �q ð2Þ

with the linear dependence on �q due to the fact that undercompressive or tensile loading in the 3-direction the re-sponse of the square lattice is dominated by the axial com-pression of the constituent struts.For r13 shear loading,the square lattice deforms by bending of the constituentstruts. The effective shear modulus Gc under this loading

P. Moongkhamklang et al. / Acta Materialia 58 (2010) 2822–2835 2829

can be derived as follows. Consider the lattice under shearloading r13, as sketched in Fig. 16a. Using Mohr’s con-struction, this shear loading is equivalent to the biaxialloading of a square lattice rotated through 45�, as shownin Fig. 16b. Consider the unit cell sketched in Fig. 16c andthe corresponding representative strut sketched inFig. 16d. Elementary beam theory dictates that the ap-plied vertical force F is related to the displacement d ofthe tip of the strut by

F ¼12ðEIÞf dðl� 2aÞ3

ð3Þ

where ðEIÞf is the effective flexural rigidity of the TMCmonofilament given by

ðEIÞf ¼ ETiITi þ ESiCISiC ¼ETip a4�a4

SiCð Þ4

þ ESiCpa4SiC

4

¼ 1þ a4SiC

a4ESiC

ETi� 1

� �h iETipa4

4� a ETipa4

4

ð4Þ

Fig. 12. A summary of the measured shear: (a) modulus Gc and (b) peakstrength sc of the square and diamond TMC lattices as functions of therelative density �q. The corresponding analytical predictions are alsoincluded as lines.

The applied force F and corresponding displacement dcan be related to the shear stress r13 and correspondingshear strain c13 in the square lattice via

r13 �Fffiffiffi2p

4albð5Þ

and

c13 �2ffiffiffi2p

dl

ð6Þ

Substituting from the above equations, the shear modu-lus follows as

Gc

ETi

¼ 3ap

8b 1� 2al

� �3

al

� �3

¼ 3ab2�q3

p2 1� 4b�qp

� �3ð7Þ

We proceed to analyze the strength of the square lattice.Under compression in the 3-direction the strength rc is setby cooperative elastic buckling (see Fig. 8). The coopera-tive elastic buckling stress can be obtain by a Rosen anal-ysis [19] as

rc ¼ Gc þP b

4albð8Þ

Here Pb is the elastic buckling load of the vertical strutof the square lattice of height H = nl, where n is the numberof cells along the height of the core. Substituting theexpression for the Euler buckling load Pb for a verticalstrut of height H clamped at both ends in Eq. (8), thestrength rc follows as

rc ¼3

8 1� 2al

� �3þ p2

4n2

" #pETia

bal

� �3

¼ 3

p2 1� 4b�qp

� �3þ 2

n2

" #b2aETi�q

3 ð9Þ

Under shear loading r13 the square lattice deforms bybending at the nodes and thus failure occurs when thestress at the nodes reaches the failure stress rf of theTMC monofilaments. The bending moment at end of thestrut is sketched in Fig. 16d and is given by

M ¼ F ðl� 2aÞffiffiffi2p ð10Þ

The maximum stress in the fiber due to this bendingmoment is

r ¼ MðEIÞf

ðaEÞmin ¼F ðl� 2aÞffiffiffi

2pðEIÞf

ðaEÞmin ð11Þ

where ðaEÞmin ¼ minðaSiC;ESiC; aETiÞ. We use the minimumvalue as this ensures that both the SiC and Ti elements ofthe TMC monofilament have failed. Failure occurs whenthis stress reaches the critical value rf at a critical appliedforce Ff and the corresponding failure shear stress sc ofthe square lattice is given by

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sc

rf¼ F f

2ffiffiffi2p

lab¼ pa3a

8ðl� 2aÞblETi

ðaEÞmin

¼ ab2p

ll� 2a

� ��q2 aETi

ðaEÞmin

ð12Þ

4.2. Diamond lattice

The compressive and shear properties of the diamondlattices are derived in Zupan et al. [23] and Cote et al.

Fig. 13. (a) Photographs showing the sequence of deformation of the �q ¼ 9:6%

corresponding measured shear stress vs. strain response is included in (b), wit

[24], respectively. For the sake of completeness, we restatethem for the TMC lattices. Under both compressive andshear loading the diamond lattices deform by stretchingof the constituent struts for aspect ratios L=H P 1. In thisregime, the contribution from the bending of the strutsbecomes negligible. This is the regime of interest in thisstudy and thus all the formulae given in this section onlyinclude the contribution from the stretching of the struts.The effective compressive modulus Ec and shear modulusGc of the diamond lattice are given as

square lattice sheared at an applied nominal strain rate 2 � 10�4 s�1. Theh the various images marked on the stress–strain curve.

Fig. 14. (a) Photographs showing the sequence of deformation of the �q ¼ 9:6% diamond lattice sheared at an applied nominal strain rate 2 � 10�4 s�1.The corresponding measured shear stress vs. strain response is included in (b), with the locations of the various images marked on the stress–strain curve.

P. Moongkhamklang et al. / Acta Materialia 58 (2010) 2822–2835 2831

Ec ¼ Gc ¼Ef �q

41� H

L

� �ð13Þ

while the compressive strength rc and shear strength sc of

the diamond lattice are

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rc ¼ sc ¼rcr�q

21� H

L

� �ð14Þ

where rcr is the failure strength of the strut of length l byeither elastic buckling or tensile failure of the fibers. Undercompressive loading all the struts are in compression andso (neglecting the fiber-crushing mode, which would occurat values of stress typically not achievable for practical cel-lular materials) the operative failure mode is always elasticbuckling. As discussed in Zupan et al. [23], elastic bucklingoccurs by rotation of the nodes of the diamond lattice andhence substituting the Euler buckling load for a pin-endedstrut for rcr, the failure stress rc being given by

rc ¼�q3

21� H

L

� �ab2ETi ð15Þ

By contrast, under shear loading r13, one set of struts isunder tension and the other is under compression. Thus thecritical stress is either the Euler buckling load or the tensilefailure stress of the fibers, i.e.

rcr ¼rf

al >

2p

ffiffiffiffiffiffiffirf

ETia

qp2aETi

4ðal Þ

2 otherwise

8<: ð16Þ

Substituting Eq. (16) in Eq. (14), we get the shearstrength of the diamond lattice in terms of �q as

Fig. 15. (a) Photographs showing the sequence of deformation of the �q ¼ 11:3The corresponding measured stress vs. strain response is included in (b), withfailure process of the lattice was very rapid and occurred over a period of abo

sc ¼�q2ð1� H

LÞrf �q > 1b

ffiffiffiffiffiffiffirf

ETia

q�q3

2ð1� H

LÞab2ETi otherwise

8<: ð17Þ

4.3. Comparison with measurements

The predictions of variation of the effective compressiveYoung’s modulus Ec and compressive strength rc with rel-ative density for both the square and diamond lattices areincluded in Fig. 7a and b, respectively. The correspondingpredictions for Gc and sc are presented in Fig. 12a and b.These predictions employ the formulae developed abovealong with the measured values of Ef ¼ 195 GPa,rf ¼ 952 MPa (Section 2.4) and ETi ¼ 109 GPa, the geo-metric parameters a ¼ 120 lm and aSiC ¼ 70 lm (Section2.1), and a measured value of b = 0.8 (Table 1). In line withthe experiments, the aspect ratio of the diamond latticeswas taken as L/H = 5 and 8 for the compression and shearpredictions, respectively.

Excellent agreement between the measurements and pre-dictions is observed with one exception: the model overpre-dicts the shear strength of the diamond lattices, especiallyfor �q ¼ 9:6% and 11.3%. This is attributed to the fact thatthese lattices lie at the transition of failure mode from Eulerbuckling of the struts to tensile strut failure. The failure

% diamond lattice sheared at an applied nominal strain rate 2 � 10�4 s�1.the locations of the various images marked on the stress–strain curve. Theut 1 ms, as seen from the time markers included in (a).

P. Moongkhamklang et al. / Acta Materialia 58 (2010) 2822–2835 2833

modes are imperfection sensitive at this transition andhence the measurements are lower than the predictions.

It is worth noting that, other than the shear strength ofthe �q ¼ 18:4% diamond lattice, the strength of the latticesin all other cases experimentally investigated here is setby elastic buckling of the struts. This clearly indicates thatcellular materials made from TMC monofilaments need tohave a higher relative density in order for them tomake optimal use of the high strength of the TMCmonofilaments.

5. Comparison with competing cellular materials

One of the main applications of cellular materials is asthe core material in sandwich structures. Typically, theaim is to maximize the stiffness and/or strength-to-weightratio of the cellular materials so as to enhance the perfor-mance of the sandwich structure in which they areemployed. The compressive and shear stiffness and strength

Fig. 16. Sketches showing: (a) the square lattice subjected to shear loading r13,and (d) the free body diagram of the representative strut used in the analysis.

of the TMC lattices investigated here are compared withother cellular materials in an Ashby [25] style plot in Figs.17 and 18, respectively. In these figures the moduli andstrength are plotted as functions of the density q ¼ �qqs,where qs is the density of the solid truss material. The fig-ures include data for aluminum tetrahedral lattices [26,27],titanium pyramidal lattices [28], and CFRP pyramidal[17,29] and square honeycombs [18]. Data for a Ti textilelattice made by the same process as used here is also shown[30]. It illustrates the significant improvement in strengthand modulus achieved by incorporation of the SiC fiberin the system.

It is clear from these figures that the TMC lattices out-perform all other cellular materials with a density of lessthan about 1 g cm–3 in terms of their shear strength, andthe compressive properties of the TMC lattices are compa-rable to those of CFRP honeycombs. Since titanium com-posites offer useful structural performance at temperaturesup to 500 �C, these diamond lattices also appear to be

(b) the corresponding rotated lattice, (c) the unit cell of the rotated lattice

Fig. 17. An Ashby-style plot of the compressive stiffness: (a) and strength(b) as functions of density for a range of engineering materials. The datafor the TMC square and diamond lattices from this study is included alongwith comparable measurements for other competing cellular materials.

Fig. 18. An Ashby-style plot of the shear stiffness (a) and strength (b) asfunctions of density for a range of engineering materials. The data for theTMC square and diamond lattices from this study are included, along withcomparable measurements for other competing cellular materials.

2834 P. Moongkhamklang et al. / Acta Materialia 58 (2010) 2822–2835

promising candidates for high-temperature, ultralight load-supporting applications, provided their limited ductility isnot a significant constraint.

6. Concluding remarks

Collinear lattices with square and diamond topologieshave been manufactured by diffusion bonding Ti–6Al–4V-coated SiC monofilaments. Three relative densities of

these lattices have been tested in out-of-plane compressionand shear. Given the very high strength of the TMC mono-filaments, the compressive strengths of both the square anddiamond lattices were dominated by elastic buckling of theconstituent struts. However, under shear loading, some ofthe constituent struts are subjected to tensile stresses, andfailure is then set by tensile failure of the TMC monofila-ments. Analytical expressions are derived for the elasticmoduli and strength of the square and diamond TMC

P. Moongkhamklang et al. / Acta Materialia 58 (2010) 2822–2835 2835

lattices, and the predictions compare well with measure-ments over the range of relative densities investigated inthis study.

These TMC lattices have a high specific strength, withtheir specific shear strength exceeding other cellular materi-als investigated to date, including CFRP honeycombs. Thecompressive properties of the TMC lattices are comparableto CFRP honeycombs. However, the TMC lattices havethe drawback of a brittle response and they undergo cata-strophic failure at their peak load. Thus, the TMC latticesappear promising candidate as cores in sandwich structuresfor elevated temperature and multifunctional applications,in situations where their limited ductility is not a significantconstraint.

Acknowledgements

The study was supported by the Office of Navel Re-search (ONR) and monitored by Dr. David Shifler underGrant No. N00014-07-1-0114.

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