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Acta Materialia 60 (2012) 1455–1466

Shear and bending performance of carbon fiber compositesandwich panels with pyramidal truss cores

J. Xiong a,b, L. Ma a, S. Pan a, L. Wu a,⇑, J. Papadopoulos b, A. Vaziri b

a Center for Composite Materials and Structures, Harbin Institute of Technology, Harbin 150001, Chinab Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA

Received 18 May 2011; received in revised form 11 November 2011; accepted 15 November 2011

Abstract

Structural performance in direct (pure) shear and three-point bending was investigated for sandwich panels with a carbon fiber pyra-midal truss core. Analytical estimates for sandwich panel strength for each loading condition were presented for possible competing fail-ure modes. In the experimental part of the study, pyramidal truss cores were made using the hot press molding technique and thenattached to flat carbon fiber composite face sheets to build all-composite sandwich panels. Panels with different configurations (e.g., corerelative density and face sheet thickness) were tested to probe different failure modes and investigate the mechanical properties. In gen-eral, measured failure loads showed good agreement with the analytical predictions. Failure mechanism maps illustrate the controllingfailure mechanisms in various regions of parameter space.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Composites; Pyramidal truss cores; Shear; Bending; Failure mechanisms

1. Introduction

Fiber reinforced plastic (FRP) composite sandwich pan-els with lattice core construction are attractive for buildinglightweight multifunctional structures, especially for aero-space and marine systems [1–12]. These panels are com-monly subjected to lateral loading in service, so there is adriving need to study their bending behavior, includingstiffness, strength and failure mechanisms. Recently, therehave been significant research efforts to understand thebehavior of metal panels with low density core construc-tion under out of plane loading (e.g., see Refs. [13–20]).Mohr [21] indicated that the mechanical behavior of ideallattice truss materials is controlled by the axial strainingof individual truss members. Doyoyo and Hu [22] pointedout that the strut parameters are very important for under-standing core failure. The insights provided by these stud-

1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2011.11.028

⇑ Corresponding author. Tel.: +86 451 86412549; fax: +86 45186402386.

E-mail addresses: xiongjian0309@126.com (J. Xiong), wlz@hit.edu.cn(L. Wu).

ies could help to explain the performance of similarsandwich panels made of FRP. However, account mustalso be taken of the relatively weaker core-to-face-sheetbonds, and possibly also by the strongly anisotropic FRPmaterial behavior and more complex failure possibilitiesfor composites.

In this work, a series of analytical and experimentalinvestigations were performed into both shear and three-point-bending responses of composite sandwich panelswith pyramidal truss cores. Sample cores were fabricatedfrom 2–16 layers of unidirectional carbon fiber/epoxy pre-preg sheets of thickness 0.15 mm (T700/epoxy composite,Beijing Institute of Aeronautical Materials, China). Themeasured large-sample properties of the unidirectional pre-preg used in the experiments are provided in Table 1.

Fig. 1 shows the method used for fabrication of the trusscores by the hot press molding technique [9]. First, themold parts were assembled together, and all the expansiontooling was laid into the gaps between the bottom framesand the base tooling. The slender prepreg strips wereinserted into the truss form and subjected to a pressure

rights reserved.

Table 1Properties of unidirectional laminate (T700/epoxy composites).

Properties Value

0� Tensile strength (MPa) 14000� Tensile modulus (GPa) 1230� Compression strength (MPa) 8500� Compression modulus (GPa) 10090� Tensile strength (MPa) 1890� Tensile modulus (GPa) 8.390� Compression strength (MPa) 9690� Compression modulus (GPa) 8.4In-plane shear strength (MPa) 16.0In-plane shear modulus (GPa) 4.8Interlayer shear strength (MPa) 60Major Poisson’s ratio 0.3Volume fraction of fibers 57 ± 3%

(i) Assembling the bottom parts and laying up the prepreg

(ii) Applying pressure and heat

Top frames

Bottom frames Blocks

Strut compaction area

Composite pyramidal truss lattice core preforms

Top frames

Bottom frames

Fig. 1. Schematic of the manufacturing process of carbon fiber compositepyramidal truss cores.

1456 J. Xiong et al. / Acta Materialia 60 (2012) 1455–1466

of 0.5 MPa (70 psi) by pressing the top frame against thebottom frames. The composite trusses were heated andcured at a constant temperature of 125 �C for 1.5 h.Fig. 2a shows the schematic of a unit cell of a pyramidaltruss core with strut thickness t, strut width d, core heighth, angle from horizontal x, and bonding length b. The rel-ative density of the core �q is the core relative density calcu-lated accurately as

�q ¼2 2bþ ðh� tÞ= sin x� t sin x

2� d

2

� �dt

2bþ ðh� tÞ= tan x� t sin x2

� �2h

ð1Þ

A simpler expression for the density is [17]

�q ¼ 2 2bþ ðh� 2tÞ= sin x½ �dt

ð2bþ h= tan xÞ2hð2Þ

which is less accurate for thick struts. In the present sam-ples, in order to keep consistency with previous papers,core relative densities were always calculated analytically

by Eq. (2), and ranged from 0.64% to 4.07%. These resultsderive from t varying between 0.3 and 2.4 mm, while theother parameters were fixed at the values d = 3 mm,b = 4 mm, h = 15 mm, x = 45�.

Through-thickness compressive compliance andstrength were studied previously for pyramidal truss corethat was either snap-assembled from small water jet-cutstruts [3] or was molded complete [9]. Truss core typicallyexhibits a crushing strength greater than most high-void-content (i.e., low relative density) materials with a density<100 kg m�3. In contrast, the flexural stiffness and strengthof the sandwich panels is governed by the truss core shearcompliance and strength [23]. Thus, studying the shearbehavior of the composite core is a key step towards under-standing the flexural responses of these panels.

In Section 2, the out-of-plane shear behavior of pyrami-dal truss cores manufactured is studied, using a hot-pressmolding technique. The sandwich panels were fabricatedby attaching the pyramidal truss structures to flat carbonfiber reinforced face sheets with an epoxy adhesive (08-57, Heilongjiang Institute of Petrochemistry) (seeFig. 2b). In Section 3, the three-point bending behaviorof composite sandwich panels is studied, using both theo-retical models and experimental observations. This buildson previous analyses of metallic sandwich panels with trusscores [13,14,24,25] and sandwich panels made of glass fiberreinforced epoxy face sheets with polymer foam cores[26,27]. Section 4 expands the results to provide predictivefailure maps for composite sandwich panels under three-point bending. The conclusions are discussed in Section 5.

2. Core shear response

In this section, an analytical expression is derived for the“effective” shear stiffness of a composite pyramidal trusscore by analyzing the elastic deformation of a single “pyr-amid“ or unit cell of the truss core. Analytical estimates arethen provided for the failure load of a pyramidal coreunder shear for three different potential collapse scenarios:(i) Euler buckling of the struts; (ii) compressive fracture ofthe struts; and (iii) core-to-face-sheet bond failure. Sheartests on pyramidal truss cores with four different relativedensities were conducted to measure the shear responseand to probe different failure modes.

2.1. Theoretical investigations

Fig. 3a shows a unit cell of a pyramidal truss core withaxes oriented along the cell diagonal, and subjected to xz

shear. (Note that the most common loading and test direc-tion is at 45� to x and y.) Although Lebee and Sab [28] indi-cated that high-accuracy shear-modulus bounds shouldtake core/face-sheet interactions into account, in this sim-ple treatment the face sheets are treated as rigid. It isassumed that each core member is fully bonded to the facesheets which experience no slope change, so that the endsof the core member may be treated as clamped.

30mm

(b) (a)

Fig. 2. (a) Sketch of the unit cell of the pyramidal core. (b) A CFRP composite pyramidal truss core sandwich panel. A portion of the top face sheet wasremoved to reveal the interior construction of the panel, each strut of the pyramidal truss was made of two slender laps and has thickness t = 0.3 mm. Thepyramidal truss core relative density is �q ¼ 0:644%.

Fig. 3. (a) Views of a pyramidal truss core unit cell subjected to shearloading. (b) Top views of the core structure deformation under shear. (c)Schematic of a single strut deformation under shear and its free-bodydiagram.

J. Xiong et al. / Acta Materialia 60 (2012) 1455–1466 1457

Deflecting the pyramid apex a distance dx in directionx gives rise to a shear angle cxz and requires an appliedforce Fx. Using beam theory, the authors develop anexpression for the strain energy U, from which it is easyto calculate Fx = dU/dx. The axial extension dA of a strutcreates strain energy d2

A=2� �

ðEAc=LÞ, where E is the axial

elastic modulus, Ac is the cross-sectional area equal totd, and L is approximated as (h�2t)/sin x. It should beemphasized that an expression for L which accuratelycaptures its end compliance is not known very precisely,since the angled strut blends into horizontal bonding legof the same thickness, but lower modulus in the strutdirection. Ignoring shear strain, transverse flexure dS ofa strut with no slope change at either end creates bendingstrain energy d2

S=2� �

ð12EI=L3Þ, where E and L are asbefore, and I (second moment of area) depends on thedirection of flexure: flexure perpendicular to the d dimen-sion requires It = dt3/12, and flexure perpendicular to thet dimension requires Id = td3/12. For the prescribed hori-zontal apex deflection dx, the two struts oriented in the xz

plane undergo axial deflection dA = dx cos(x) and trans-verse deflection dS = dx sin(x). The two struts orientedin the yz plane undergo transverse deflection dS = dx.The total energy for the four struts making up a pyramidis

U ¼ 2 d2x=2

� �½cos2ðxÞðEA=LÞ þ sin2ðxÞð12EIt=L3Þ

þ ð12EId=L3Þ� ð3ÞNote that the three terms in square brackets are of very dif-ferent magnitudes. For typical dimensions, the second(weak-direction flexure) is outweighed by the third(strong-direction flexure), which is greater by the factor[d/(t sin x)]2. And the third is far outweighed by the first,i.e., the axial stiffness, which is greater by the factor(Lcos x/d)2.

From the energy expression is calculated

F x ¼ 2dx½cos2ðxÞðEA=LÞ þ sin2ðxÞð12EIt=L3Þþ ð12EId=L3Þ�

¼ dxKx ð4ÞPerforming a similar calculation for a displacement dy

yields an identical result for Ky, which means that the elas-tic shearing behavior is isotropic, so the subscript isdropped from the shear stiffness K of a unit cell. This cal-culation is finally “homogenized” by representing the result

1458 J. Xiong et al. / Acta Materialia 60 (2012) 1455–1466

as a modulus G: the ratio of shear stress siz to shear strainciz, where subscript i represents any direction in the xy

plane. G is found from K (or F/d) through dividing F by2[2b + (h � t)/tan x]2, the unit cell base area, to get shearstress; and dividing dz by the height h to get shear strain.Dispensing with the second (i.e., the smallest) contributionto the stiffness, the result is

G � Eðhtd=LÞðcos2 xþ d2=L2Þ=ð2bþ ðh� tÞ= tan xÞ2 ð5Þ

To estimate the strength of the truss core under shear,three different failure modes were studied.

2.1.1. Euler buckling of the struts

Core failure by strut buckling is not isotropic. Stressessxz and syz are each individually carried by struts alignedwith them. The critical value for stress aligned with struts(the weakest direction) can be estimated, keeping in mindthat, at 45� to the x axis, which is the test direction, themacroscopic buckling stress will be

ffiffiffi2p

times bigger.The Euler buckling load of an end-clamped strut

subjected to an axial load can be estimated fromF A ¼ 4p2EcI sin2 x

ðh�2tÞ2 according to the elementary theory in refer-ences such as Ref. [29], where Ec is the strut’s compressivemodulus, and (h � 2t)/sin(x) is used to approximate thelength. To determine FA from the shear stress, G is usedto compute the shear strain from the shear stress, multiply-

Fig. 4. Schematic o

Table 2Summary of the geometries employed in shear tests along with the predicted

Specimen �q (%) W

(mm)ls(mm)

h

(mm)t (mm) Analytical r

Analyt. failmode

1 0.64 99 198 11.8 0.26 EBFMDM

2 1.25 100 200 12.1 0.56 EBFMDM

3 1.81 99 198 12.8 0.80 EBFMDM

4 2.83 100 200 13.5 1.30 EBFMDM

EB = Euler buckling mode; FM = fracture mode; DM = debonding mode; DEparameters b = 4 mm, d = 3 mm and x = 45� remain the same regardless of canalytical predictions is found in line (b) of Table 5.

ing by h to get dx, then by cos x to get dA, and finally by theaxial stiffness Ectd/L to get FA. Equating this to the allowedbuckling load, and multiplying by

ffiffiffi2p

to account for thegreater strength of testing at 45� to the axes gives

sE ¼ffiffiffi2p

Gzðp2=3Þðt2 tan xÞ=ðh2 � 2htÞ ð6Þ

2.1.2. Compressive fracture of the strutsThe compressive failure load of a composite strut is

FA = rcfdt, where rcf is the strut’s compressive failurestrength. Following the above development, the theoreticalshear stress associated with the onset of strut fracture sF

can be estimated from

sF ¼ rcf ðGz=EÞð1� 2t=hÞ=ðsin x cos xÞ ð7ÞSimilar to buckling, the weakest direction is along the

struts, and one could write that the actual allowed valueof scf would be the above calculation, divided by cosw,where w is the angle between the test direction and thenearest coordinate axis (hence, 645�). For testing withw = 45�, it suffices to multiply the above expression by

ffiffiffi2p

.

scf ¼ffiffiffi2p

rcf ðGz=EÞð1� 2t=hÞ=ðsin x cos xÞ ð8Þ

2.1.3. Debonding

The shear stress multiplication at a face-sheet-to-corebond is just the area ratio Acell/A

0, where A0 = 4bd � d2 is

f the shear test.

stiffnesses and measured failure loads and collapse modes.

esults Experiment results

. Analyt. strength/stiffness(MPa)

Obs. fail.mode

Obs. strength/stiffness(MPa)

NA/NA EB 0.075/9.531.81/NA0.78/NA

0.39/35.69 DM and DE 0.421/21.592.21/35.690.77/35.69

1.02/49.25 DM 0.519/38.782.94/49.250.78/49.25

4.62/86.64 DM 0.548/58.932.71/86.640.77/86.64

= delamination. Analyt. = analytical; Obs. = observed. The experimentalore density, since they are controlled by the mold. The modulus used in

Fig. 5. (a) Shear response of carbon fiber pyramidal truss cores. Thehighest density cores did not develop their full strength because they failedat the core-to-face-sheet bonds. (b) Euler buckling and debonding ofcarbon fiber pyramidal truss cores during shear loading. (c–e) Delamina-tion and debonding in sandwich panels with different relative densitiesinduced by shear loading.

J. Xiong et al. / Acta Materialia 60 (2012) 1455–1466 1459

the area of the cross-shaped pyramid apex, and Acell =2(2b + (h � t)/tanx � t tan(x/2))2. Bond failure is expectedwhen the shear stress in the bond exceeds the bondadhesion strength denoted by sa. Thus, the effective shearstrength associated with the debonding is scr =sa(2bd � d2/2)/(2b + (h � t)/tan x � t tan(x/2))2. Since lat-tice dimensions can be difficult to measure precisely andare not necessarily uniform (e.g., when squeezed for bond-ing, the pyramids might shorten and spread, reducing bothh and x), an alternate calculation of unit-cell base areainvolves counting cells in a given area, from which

scr ¼ saNð4bd � d2Þ=ðwlsÞ ð9ÞHere, N is the number of unit cells of the test specimen, andw and ls are the specimen overall width and length.

2.2. Experimental results

Shear stiffness and strength tests were performed withthe sample bonded to rigid plates, according to the ASTMC273 test method for sandwich panel cores [30]. The loadwas applied with a screw-driven test machine (INSTRON5569) at an applied macroscopic nominal shear strain rateof 10�3 s�1. The load measured by the load cell of the testmachine was used to define the nominal shear stress, whilea clip gage mounted on the fixture was employed to mea-sure the relative displacement between the two faces ofthe specimens, i.e., the applied shear strain. Fig. 4 showsthe schematic of the test setup. The ASTM standard forshear tests on sandwich cores [30] requires that the testspecimens have ls/h > 12.

Four different core designs were tested in shear, withthree samples of each. Table 2 summarizes the measuredand predicted loads, stiffness and failure modes for differ-ent specimens. Fig. 5 shows the measured shear response,as well as snapshots of the specimens. In all cases, an initiallinear response was observed and followed by a non-linearregime. Subsequently, the stresses decreased with increas-ing strain. For the specimen with �q ¼ 0:64%, the peakshear stress of specimen 1 was 0.075 MPa (11 psi) beforeany visible failure by Euler buckling. Significant bendingdeformation of the core struts was observed, resulting ina softening post-buckling response. The predicted failurestresses and stiffnesses (Table 2) are all higher than themeasured values, presumably due to imperfections in themanufactured specimens, and the peeling stresses devel-oped where tensile struts bond to the face sheets. The shearresponse of specimens 3 and 4 with higher core density wasdominated by the debonding between the face sheet and thepyramidal truss core. The predicted stiffness and failurestress (Table 2) for specimens 3 and 4 are higher than themeasured values (sa = 21.8 MPa).

3. Sandwich panel bending response

3.1. Theoretical investigations

Allen [29] approximates the total deflection d at the mid-point of a sandwich beam loaded by force P in three-pointbending as the sum of deflections due to bending of thecross section (by axial strain of the face sheets) and shearof the core:

d ¼ dB þ dS ¼Pl3

48ðEIÞeq

þ Pl4ðAGÞeq

ð10Þ

Here, (EI)eq is the equivalent flexural rigidity, and sampledimensions are defined in Fig. 7a. The theoretical equiva-lent flexural rigidity of a sandwich panel can be derivedfrom classical composite beam theory:

ðEIÞeq �1

2Ef wtf ðhþ tf Þ2 ð11Þ

(a) Face sheet crushing

(b) Face sheet wrinkling

(c) Core member buckling

(d) Core member crushing

(e) Adhesive joint failure

Fig. 6. Possible failure modes of a carbon fiber composite sandwich panel with pyramidal truss core under three-point bending.

tf

P

h

l

ts

s

(b)

(a)

Fig. 7. (a) Schematic of the three-point bending test. (b) A bending test specimen with the pyramidal truss core relative density �q ¼ 1:25%.

1460 J. Xiong et al. / Acta Materialia 60 (2012) 1455–1466

where w is the panel width and Ef is the elastic modulus inthe direction of specimen. In general, h will be less than theheight of the as-manufactured core owing to deformationand spreading of the struts during the bonding step. (AG)eq

is the equivalent shear rigidity

ðAGÞeq ¼ wðhþ tf ÞGc ð12Þ

Gc is the shear stiffness of pyramidal truss cores, which canbe obtained from shear experiments. For beams that areboth long and solid (i.e., with good shear stiffness), sheardeformations are generally recognized to be negligible.

However, when beams are short, and/or when shearcompliance is increased by a reduced-density core, sheardeformation could be significant or even dominant. Mea-surements show that the low-density core leads to low “Gc”

and therefore reduced three-point bending stiffness.The possible failure modes in three-point bending of a

carbon fiber sandwich panel are illustrated in Fig. 6: (i) facesheet crushing or wrinkling (buckling) from the bendingmoment; (ii) core member crushing (including strut delam-ination or fracture) or buckling under a concentrated load;(iii) shear-induced failure of the core or its bonds to the

Table 3Three-point bending deformation of all specimens along with predicted and measured P/d (N mm�1) and the center displacement d (mm); dB is calculateddisplacement due to panel bending, and dS is calculated displacement due to panel shear.

Specimen dB and dS (mm) Analy. total d (mm) Analytical P/d (N mm�1) % Test P/d (N mm�1) Test d (mm) Failure load P (N)

1 0.19 1.02 153.62 18.80 173.79 1.06 Buckling: 156.300.83 81.20

2 0.53 9.36 200.40 5.64 613.02 Initial: 0.37 Initial: 234.858.83 94.36 Peak: 9.95 Peak: 1876.27

3 1.23 2.08 178.38 59.28 260.34 1.82 371.190.85 40.72

4 0.84 3.05 355.01 27.44 693.52 Initial: 0.88 Initial: 531.312.21 72.56 Peak: 8.03 Peak: 1083.42

5 3.09 3.45 392.13 89.56 314.31 Initial: 0.87 Initial: 389.120.36 10.44 Peak: 3.35 Peak: 1353.56

6 0.56 0.82 1683.87 68.61 2548.59 0.56 1377.240.26 31.39

7 0.27 0.64 3568.21 42.00 3694.73 0.67 2289.740.37 58.00

8 0.76 1.30 1836.55 58.63 1901.99 1.62 2384.050.54 41.37

For specimens 5–8, it was assumed that scr = 0.5476 MPa and Gc = 154.13 MPa, as found in shear tests.

J. Xiong et al. / Acta Materialia 60 (2012) 1455–1466 1461

face sheet. The collapse of the panel is generally dictated byone of the competing mechanisms that depend on thegeometry of the panel and the mechanical properties ofthe face and core materials. Here, the analytical modelswere adopted for failure of metal sandwich beams with lat-tice cores presented in Refs. [31,32], to obtain estimates ofthe failure load for each of the modes mentioned above. Inthe analysis, it was assumed that the core alone carries theshear load, while the face sheets carry the applied moment.The results are summarized below for the different failuremodes mentioned above. The sandwich panel strength isdictated by the failure mode with the lowest value ofapplied load P.

3.1.1. Face sheet crushing or wrinkling

The critical loads associated with each of the failureloads can be estimated from:

P ¼ 4rfy tf hwl

ðface sheet crushingÞ ð13Þ

P ¼k1Ef p2t3

f wðhþ tf Þ6l½bþðh� tÞ=tanxÞ� t sinðx=2Þ�2

ðface sheet wrinklingÞ

ð14Þ

where rfy and Ef are the in-plane compressive crushing strengthand compressive Young’s modulus of the face sheets. In theanalytical solution, the exact distance between pyramidalpoints equal to “

ffiffiffi2p½bþ ðh� tÞ= tan x� t sinðx=2Þ�” was

used.

3.1.2. Core member crushing or buckling from concentratedload

The loads associated with each failure mode are

P ¼ 2ffiffiffi2p

rcdtw sin x2bþ ðh� tÞ= tan x� t sinðx=2Þðcore member crushingÞ ð15Þ

P ¼ffiffiffi2p

k2p2Ecdt3w sin3 x

6ðh� 2tÞ2½2bþ ðh� tÞ= tan x� t sinðx=2Þ�ðcore member bucklingÞ ð16Þ

where rc and Ec are the compressive strength and stiffnessof struts. Parameter k2 depends on the end constraint con-dition of the struts. The struts are subject to local bending,so the Euler buckling analysis from [29] provides only avery approximate model. The mutual coupling effect be-tween the core and face sheets was disregarded in the ana-lytical models, and k1 = k2 = 1 was assumed in Eqs. (14)and (16) as a pinned connection.

3.1.3. Shear failure of core or bondSample failure is also expected if the core fails via any of

the shear-related mechanisms given in Section 2.1. This cri-terion can be written

P ¼ 2whsmin ð17Þwhere core debonding occurs when smin = scr. This approx-imation ignores the shear-force contribution of thick facesheets with slope constrained by a boundary condition orsymmetry—a length-dependent calculation which is be-yond the scope of this treatment.

3.2. Experimental results

This section describes three-point bending tests on sand-wich panel specimens with different geometries. Prior to the

Table 4Summary of the geometries employed in three-point bending tests along with the predicted and measured failure loads and collapse modes.

Specimen �q (%) w (mm) l (mm) tf (mm) t (mm) h� (mm) Analytical results Experimental results

Failure mode Failure force (N) Failure mode Failure force (N)

1 0.64 99 231 0.58 0.26 12.16 FW 404.99FC 5173.27 CECE 209.71 FW CE: 156.3CC 7042.41 CDCD 1698.84

2 0.64 99 231 2.00 0.26 15.00 FW 18641.87FC 17838.86 CE CE: 234.85CE 209.71 CD CD: 1876.27CC 7042.41CD 1698.84

3 1.25 100 233 0.58 0.54 12.40 FW 151.69FC 895.34 FWCE 417.23 CE FW/CE: 371.19CC 8650.58 CDCD 946.41

4 1.25 100 233 2.00 0.54 15.20 FW 6984.10FC 3076.39 CL CL: 531.31CE 421.44 CD CD: 1083.42CC 8669.33CD 943.04

5 4.07 107 249 0.58 2.00 14.76 FW 200.57FC 4301.02 FW FW:389.12CE 26525.91 CD CD:1353.56CC 20332.38CD 1594.90

6 4.07 105 175 1.1 2.00 15.60 FW 2007.31FC 11221.97CE 27440.94 CD 1377.24CC 20166.48CD 1542.072

7 4.07 107 178 3.0 2.00 19.60 FW 45452.95FC 31120.35CE 26525.91 CD 2289.74CC 20332.38CD 1594.90

8 4.07 108 249 3.0 2.00 19.60 FW 32796.14FC 22454.59CE 26773.82 CD 2384.05CC 20522.41CD 1609.80

FW = face sheet wrinkling; FC = face sheet crushing; CE = core member buckling; CC = core member crushing; CL = core member delamination;CS = core shear failure; CD = debonding between pyramidal truss cores and face sheet. The experimental parameters b = 4 mm, d = 3 mm and x = 45�are the same for every sample.

1462 J. Xiong et al. / Acta Materialia 60 (2012) 1455–1466

bending experiments, the ends of the panels were firstbonded to a U-shaped steel sheet (ts = 0.5 mm) and filledwith an epoxy resin of length S = 30 mm in order to limitexcessive shear, as shown schematically in Fig. 7a. InFig. 7a, l denotes the beam span, and h and tf are the corethickness and face sheet thickness, respectively. Fig. 7bshows a bending test specimen with �q ¼ 1:25% andtf = 2.0 mm. In the experiments, the sandwich panel wassupported by two 30-mm-diameter hardened steel pinsattached to a flat support base. The indenter had a flat cen-tral region 12.7 mm wide, with adjacent fillets 2 mm inradius and was moved at a constant rate of 0.02 mm s�1.

The applied load was recorded by the INSTRON (5569)during the bending experiments. For each experiment, thetotal deformation of the specimens was calculated fromEq. (10), and the percentage of shear deformation andbending deformation is also calculated (Table 3). In gen-eral, the P/d test shows good agreement with the predictivevalue, except for specimens 2 and 4, which have a verythick face sheet and weak cores compared with the otherspecimens. The first reason is that the percentage of sheardeformation in total deformation is very large, but theshear deflection was restricted by the thick face sheet andboth ends. The second reason is that the shear stiffness of

0 2 4 6 8 10 12 14 160

50

100

150

200

IV

Analytical prediction

III

II

Loa

d (N

)

Displacement (mm)

Specimen 1

I

Fig. 8. Bending response and deformed configurations of specimen 1. Thedeformed configurations are shown at different points of the experimentmarked in the load–displacement response. The specimen failure is mainlydue to core buckling.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

200

400

600

800

1000

1200

1400

1600

1800

Loa

d (N

)

Displacement (mm)

II

I

Analytical prediction (Face wrinkling)

Analytical prediction (Debonding)

Specimen 5

Fig. 9. Bending response and deformed configurations of specimen 5. Theanalytical predictions are shown for both face wrinkling and debonding.The sudden drop in the panel peak strength is mainly due to the coredebonding.

J. Xiong et al. / Acta Materialia 60 (2012) 1455–1466 1463

the thick face sheet was not considered in the analyticalmodels. So the deformation models are only suitable forsandwich panels with a thin face sheet or strong core, inwhich the percentage of shear deformation is <60%. Thebond strength between the truss cores and face sheetappeared to be the limiting factor in several cases.Although beams mostly failed by the predicted mode, othermodes may be manifested at the same time, owing to thecomplex of composite pyramidal truss cores, as summa-rized in Table 4.

3.2.1. Core member buckling

Core member buckling was predicted and observed inspecimens 1 and 2. Fig. 8 shows the measured responseof specimen 1 and the images of the specimen at differentstages of deformation. For this specimen, the loadincreased linearly until core member buckling occurred ata load of �156.3 N. At this point, the truss core struts atthe mid span of the panel buckled, which is the collapsemode predicted by the theory. With increasing appliedload, buckling of the adjacent core struts and face sheetwrinkling occurred at the same time. Combined debondingfrom the face sheets and micro-cracking was also observed

in the struts as the load was increased. For specimen 2 withthe thick face sheet, strut Euler buckling occurred first, butdebonding was the dominant failure mode. The predictedbuckling loads for both specimens are somewhat at oddswith the measurements, and this discrepancy was attrib-uted to imperfections in the manufactured specimens.

3.2.2. Face sheet wrinkling

Face sheet wrinkling was predicted and observed inspecimens 1, 3 and 5, which have very thin face sheets com-pared with the other specimens. The load–displacementresponse and selected deformed configurations of specimen5 are shown in Figs. 8 and 9. Prior to face wrinkling, theobserved response is almost linear. The face wrinklingreduces the stability of the sandwich panels, but does notresult in a sudden drop in the load-carrying capacity ofthe specimen and, thus, is not a catastrophic event. Asthe deflection increases, the debonding between the topface sheet and the pyramidal truss core led to a suddendrop of the load at �1353.6 N. The loads calculated byEq. (14) give only the force associated with face sheet wrin-kling. For specimens 3 and 5, the predicted load associatedwith the wrinkling is somewhat higher than the analyticalprediction. For specimens 3 and 5, the debonding failure

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.50

250

500

750

1000

1250

1500

1750

2000

2250

2500

Loa

d (N

)

Displacement (mm)

IIIIII

Analytical prediction

Specimen 8

Fig. 10. Bending response and deformed configurations of specimen 8.

Table 5Mechanical properties of carbon fiber composite laminate face-sheets andcarbon fiber composite pyramidal truss core.

No. Stacksequence

Ef

(GPa)rfy ðMPaÞ �q ðkg m�3Þ

(%)Ec

(GPa)rc ðMPaÞ

(a) (0�/90�/90�/0�)s

54.50 473 0.64 NA NA

(b) (90�/90�/90�/0�)s

19.57 80 1.25 20.5 493.76

(c) (0�/45�/�45�/90�)s

22.57 317.22 4.07 21.8 298.25

Ef and rfy are the compressive stiffness and strength of the carbon fibercomposite face sheet. Ec and rc are the compressive stiffness and strengthof the carbon fiber composite struts of pyramidal truss cores. �q is therelative density of pyramidal truss core.

Table 6Stress-based three-dimensional Hashin criteria. XT and XC are the tensileand compressive strength of fibers, YT and YC are the tensile andcompressive strength of matrix, ZT and ZC are the tensile and compressivestrength of unidirectional laminate in the normal direction. rij and Sij arethe components of the stress tensor and shear tensor defined in theclassical form, where x and y are the in-plane axes.

Failure mode Failure criteria

Fiber tensile failure (rxx > 0) ðrxx=X T Þ2 þ ðrxy=S12Þ2 þ ðrxz=S13Þ2 P 1Fiber compressive failure

(rxx < 0)ðrxx=X CÞ2 P 1

Matrix tensile cracking(ryy > 0)

ðryy=Y T Þ2 þ ðrxy=S12Þ2 þ ðryz=S23Þ2 P 1

Matrix compressive cracking(ryy < 0)

ðryy=Y CÞ2 þ ðrxy=S12Þ2 þ ðryz=S23Þ2 P 1

Fiber–matrix shear cracking(rxx < 0)

ðrxx=X CÞ2 þ ðrxy=S12Þ2 þ ðrxz=S13Þ2 P 1

Delamination in tension(rzz > 0)

ðrzz=ZT Þ2 þ ðrxz=S13Þ2 þ ðryz=S23Þ2 P 1

Delamination in compression(rzz < 0)

ðrzz=ZCÞ2 þ ðrxz=S13Þ2 þ ðryz=S23Þ2 P 1

1464 J. Xiong et al. / Acta Materialia 60 (2012) 1455–1466

loads predicted by Eq. (17) are within 10% of the measuredstrengths.

3.2.3. Debonding

Debonding is generally the dominant failure mode in theexperiments for sandwich panels with thick face sheet orthe core �q > 1:25%. A representative load–displacement

curve and the failure modes are shown in Fig. 10. For spec-imen 8, the top face sheet debonds from the core, leading toa sudden drop of the load, while the bottom face sheet isstill attached to the pyramidal truss core. After debonding,the residual loading capacity of specimen 8 is �1250 N, or52% of the peak loading. The predicted failure peak loadfor specimens 4 and 6 is within 30% of the measuredstrength, while the predicted failure loads for specimens 7and 8 are somewhat lower than the measurements.

4. Failure mechanism maps of the sandwich panels

This section provides predictive failure maps for carbonfiber reinforced pyramidal truss composite sandwich panelsbased on dimensions and material properties. The stackingsequence of the face sheets and the relative density of trusscore can have a significant effect on the overall behaviorand failure of composite sandwich panels. Here, the studyis limited to three configurations, based on the sandwichpanels fabricated and tested in the experiment: (a) panelswith core relative density �q = 0.64% and face sheet stackingsequence [0�/90�/90�/0�]s; (b) panels with core relative den-sity �q = 1.25% and face sheet stacking sequence [90�/90�/90�/0�]s; and (c) panels with core relative density�q = 4.07% and face sheet stacking sequence, [0�/45�/�45�/90�]s. The effective material properties of the facesheet for each configuration are listed in Table 5. The effec-tive strength and stiffness were calculated by the compositelaminate theory [33] and Hashin criteria with sudden deg-radation model [34–36] using the ANSYS ParametricDesign Language. The Hashin criteria and associatedparameters are listed in Table 6. In developing the failuremap for each configuration, the core height h and the facesheet thickness tf were varied. The results are plotted inFig. 12 in terms of normalized parameters h/l and tf/l.

In constructing the maps, it was assumed that the panelfailure mode is the lowest failure load based on the

Fig. 11. Failure mechanism maps for carbon fiber composite sandwichpanels with pyramidal truss cores: FW = face sheet wrinkling; FC = facesheet crushing; CE = core member buckling; CC = core member crushing;CD = debonding.

Fig. 12. Failure mechanism map for carbon fiber composite sandwichpanels with different core densities and face sheet thickness.

J. Xiong et al. / Acta Materialia 60 (2012) 1455–1466 1465

analytical estimates provided in Section 3. The specificexperimental data for each sample are marked by redcircles in Fig. 11 from the tests carried out in Section 3.2(e.g., specimens 1 and 2 for configuration 1). For configu-ration 1 (�q = 0.64% and face sheet stacking sequence [0�/

90�/90�/0�]s), the face crushing (denoted by FC inFig. 11) is only controlling if h=l 6 6:75� 10�4, which cor-responds to a very thin and unrealistic core construction.For this configuration, face wrinkling (denoted by FW) iscontrolling for panels with very thin face sheets(tf =l 6 2:6� 10�3 for the thickest core considered). Hence,neither face crushing nor face wrinkling are active failuremechanisms for sandwich panels with this core densityand face sheet stacking sequence. Core member buckling(denoted by CE) is the only active collapse mechanism inthis map, as was observed in specimens 1 and 2. For con-figuration 2 (�q = 1.25% and face sheet stacking sequence,[90�/90�/90�/0�]s), face crushing is the dominant failuremode for panels with h=l 6 1:15� 10�2, which is againnot a practical scale for sandwich panels with pyramidaltruss cores (typically for these sandwich panels,h=l > 4� 10�2). The face wrinkling and core buckling arethe two dominant failure modes for this sandwich panelconfiguration. In the present experiments, specimen 3exhibited face sheet wrinkling followed by debonding.However, face sheet wrinkling did not cause any consider-able decrease in the load carrying capacity of the panels.For specimen 4, the debonding of the bottom face sheetled to a sudden drop of the load by a combination ofdelamination and micro-cracking so core member bucklingwas not observed.

For the third configuration (�q ¼ 4:07% and face sheetstacking sequence, [0�/45�/�45�/90�]s), the core is muchstronger compared with the previous configurations andthus, core crushing (CC) is the dominant core failure, notthe core member buckling. In addition, the face crushingis expected to be dominant over a wider range of panelgeometry parameters, as quantified in Fig. 11c. In the pres-ent experiments, specimen 5 was subjected to face wrin-kling, which was followed by debonding and thespecimens 7 and 8 failed due to core crushing. However,in specimen 6, debonding occurred prior to any otherfailure modes, leading to significant reduction in the

1466 J. Xiong et al. / Acta Materialia 60 (2012) 1455–1466

load-carrying capacity of the panel. For specimens 7 and 8,debonding was again the dominant mode of failure, andthus core crushing was not observed in the present experi-ments. Fig. 11c also shows the region of the failure mapcorresponding to core debonding (the gray area markedCD). The critical bond adhesion strength was calculatedbased on Eq. (9). This addition allowed the failure modeobserved in specimens 6, 7 and 8 to be predicted.

Fig. 12 presents the failure mechanism map of the sand-wich panel as a function of face sheet thickness tf and corerelative density �q for panels with face sheet stackingsequence [0�/45�/�45�/90�]s and l = 245 mm, which is thebeam span for specimens 5 and 8. In this plot, it is assumedthat the compressive modulus and strength of the core areindependent of the core relative density Ec = 21.8 Mpa andrc = 298.25 Mpa, as measured for core with relative den-sity 4.07% (Table 4). In the figure, the data for specimens6 and 7 are also shown, which have a different beam spanof l = 175 mm. It was checked that the failure maps for thetwo beam spans are not considerably different. The failuremap shows the transition between different failure mecha-nisms: the panels with thin face sheet generally exhibit facewrinkling, while for panels with thick face sheets the failuremechanisms depends on the core density. In general, sinceface sheet wrinkling does not result in significant change inthe load capacity of the panels, the two active mechanismsof panel failure are core member buckling and debonding.The critical value for transitioning to the CD from theother two failure mechanism depends on the bond strengthand quality.

5. Conclusions

Analytical and experimental studies were carried out tomeasure and quantify the structural performance of carbonfiber composite sandwich panels with pyramidal truss coresunder direct shear and three-point bending loads. Failuremechanisms of the panels under each loading conditionwere studied, including face sheet crushing, face sheet wrin-kling, core member buckling and core member crushing forpanels subjected to bending. Eight representative sandwichpanels with carbon fiber pyramidal truss cores were fabri-cated and tested to probe different failure modes. Failuremaps were constructed to predict the response of panelsunder three-point bending. The data could provide insightinto the design of optimized or near-optimized sandwichpanels.

Acknowledgements

The authors thank the anonymous reviewers for valu-able suggestions and Dr. Jim Papadopoulos for fruitful dis-cussions. This work was supported in part by the NationalScience Foundation of China under Grant Nos. 90816024,10872059 and 11002041. The Major State Basic ResearchDevelopment Program of China (973 Program) underGrant No. 2011CB6010303, the Fundamental Research

Funds for the Central Universities Grant No. HIT.NSRIF. 2010069, the Program of Excellent Team in Har-bin Institute of Technology (W.L.) and in part by the USAir Force Office of Scientific Research under AFOSRYIP Grant Award, #FA FA9550-10-1-0145 under the tech-nical supervision of Dr. Joycelyn Harrison (A.V.). L.M.acknowledges the Program for New Century Excellent Tal-ents in University under Grant No. NCET-08-0152. J.X.also gratefully acknowledges the Young Scholar Prizeawarded by MOE of China (AUDQ1010000611) and theChina Scholarship Council (CSC).

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