Experimental and numerical characterization of honeycomb
sandwichcomposite panelsAhmed Abbadia, Y. Koutsawaa, A. Carmasolb,
S. Belouettara, Z. Azarib,*aCentre de Recherche Public Henri Tudor,
29, Avenue John F. Kennedy, L-1855, LuxembourgbLaBPS (Laboratoire
de mcanique, Biomcanique, Polymres, Structures), Ecole Nationale
dIngnieurs de Metz, Universit Paul Verlaine-Metz,Ile du Saulcy,
F-57045 Metz, Francearti cle i nfoArticle history:Received 3 June
2008Received in revised form 30 April 2009Accepted 8 May
2009Available online 21 June
2009Keywords:SandwichHoneycombMechanical propertiesModellingPlate
theoryabstractIn this paper, an experimental investigation, an
analytical analysis and a numerical
modelofatypicalfour-pointbendingtestonahoneycombsandwichpanelareproposed.
Thehoneycomb core is modelled as a single solid layer of equivalent
material properties. Ana-lytical and numerical (nite element)
homogenization approaches are used to compute theeffective
properties of the honeycomb core. A general kinematic model (unied
formula-tion) has been adopted and used for the modelling of
honeycomb sandwich panel submit-ted to the bending test. A
comparative study of major classes of representative theories
hasbeenconsidered.
Qualitativeandquantitativeassessmentsofdisplacement, stresshavebeen
presented and discussed. 2009 Elsevier B.V. All rights reserved.1.
IntroductionHoneycomb (HC) sandwich structures consist of a thick
layer (core) intercalated between thin-stiff layers (skins) (Fig.
1).They are produced by bonding metal or composite laminate skins
to a honeycomb core. These layered-like materials arecharacterized
by lightweight, high exural stiffness and can support classical
loadings like tension and bending. The manyadvantages of honeycomb
sandwich constructions, the development of new materials and the
industrial needs for high per-formance and low-weight structures
ensure that honeycomb sandwich construction will continue to be in
demand. HC com-posites are increasingly being used to replace
traditional materials in highly loaded applications [1,2].
Honeycomb cores aredescribed as cellular solids [2,4], that make
use of voids to decrease mass, whilst maintaining qualities of
stiffness and energyabsorption. This improvement, at relatively
little expense, in terms of mass, is of great interest in
aerospace, automotive andmany other applications [2]. In order to
use these materials in different applications, the knowledge of
their mechanicalbehaviour is required. This calls for the
development of rigorous mathematical and experimental methods
capable of char-acterizing, modelling, designing and optimising of
the composite under any given set of conditions. Numerical
simulation ofthese structures requires, rstly, a proper
experimental identication of the core and the skins material
behaviors, and sec-ondly an adequate kinematic model to obtain a
reasonable computational cost. In the present paper, we propose to
exper-imentally investigate and to identify the basic mechanical
properties of the honeycomb sandwich panels and to
analyticallydescribe the response of HC sandwich panel submitted to
a four-point bending test. The experimental investigations
carriedout consist infour-pointsbendingstatictestsontwotypesof
HC(AramideFibre, Aluminium) sandwichpanels. The1569-190X/$ - see
front matter 2009 Elsevier B.V. All rights
reserved.doi:10.1016/j.simpat.2009.05.008*Corresponding author.
Tel.: +33 387 31 52 69; fax: +33 387 31 53 03.E-mail address:
[email protected] (Z. Azari).Simulation Modelling Practice and
Theory 17 (2009)
15331547ContentslistsavailableatScienceDirectSimulation Modelling
Practice and Theoryj our nal homepage: www. el sevi er. com/ l ocat
e/ si mpatadditional outcome of the experimental study carried out
is the analysis of the core density and the cell orientation (L and
W)effects on the maximum load as well as on damage processes.The
effective mechanical properties of the honeycomb have been
estimated based on the work of Gibson and Ashby [4],Masters and
Evans [5] and Grdiac [6]. For the modelling of the HC sandwich
panel, an attempt has been made to propose ahigh order unied
kinematical formulation able to describe the local (e.g. shear
deformation) and global (e.g. deection)NomenclatureMxbending
moment,Txtransversal force.D indicates the bending stiffnessEfi (i
= 1; 2) Young modulus of the face itfi (i = 1; 2) thickness of face
iEccore Young modulusTccore thicknessrxstress inside the bottom
skinsxzshear stressB(z) the surface momentS shear stiffnessG shear
modulush thickness of the beamk shear correction factorL length of
specimenL1distance between applied loadsL2 = a distance between the
inner supportsrftensile (or compressive) strength of the
skinsscshear strength of the corex1, x2displacement according to
directionsx3transversal displacementr11, r22, r33, s12stress
componentss13, s23shear stress componentsU displacement eldua, w,
cafunctions of x1, x2f(z) shear functionW+accvirtual work of
acceleration quantitiesW+intvirtual work of interior
forcesW+extvirtual work of exterior forcese+ijvirtual
deformationsNab; Mab; Mab; Qageneralized efforts(Pa; ma; ma)
generalized vectors in the formB, B generalized constitution
matricesq(x, y) sinusoidal functionAluminium
skinNomex/Aluminiumhoneycomb coreAluminium skinCell
SizeTLDirectionWDirectionGlueFig. 1. Schematic detailed description
of the honeycomb sandwich structure.1534 A. Abbadi et al. /
Simulation Modelling Practice and Theory 17 (2009)
15331547responses of a given sandwich panel under a four-points
bending static test. 3D nite element simulations of these
struc-tures, using ANSYS and CASTEM 2003 nite element codes, are
also proposed.2. Materials and experimental method2.1. MaterialsThe
honeycomb sandwich panels have been provided by Euro-Composites
S.A. (Luxembourg) and are intended for theaircraftindustry.
Thegeometrical dimensionsof thespecimenareshowninTable1. Thefacesof
athicknessequal to0.60 mm are made of aluminium (AlMg3) (Table 2),
the core structure is made either from aluminium (ECM) sheets or
fromaramide bres (ECA) [3] folded and glued together (Fig. 1)
forming a hexagonal cell structure. As in the standard layout
forcommercial honeycombs, the assembly of the structure produces
some cell walls with double thickness. In the tested con-guration
these double thickness walls were parallel to the specimen
longitudinal axis. The honeycomb core is an openedcell with various
densities of 55 kg/m3and 82 kg/m3of aluminium core and 48 kg/m3of
aramide bre core, respectively.The cell size is 6.4 and 9.6 mm for
aluminium core and 3.2 mm for aramide bres core. The geometrical
and mechanicalproperties of the panels are depicted in Tables
13.2.2. Experimental methodologyTests were carried out through a
four-point bending testing xture device schematically shown in Fig.
2. The device, de-signed and built expressly for these tests, was
connected to a servo-hydraulic universal testing machine INSTRON
4302 con-trolled by an INSTRON electronic unit. The electronic unit
performs the test control and the data acquisition. A PC
equippedwith a NI acquisition device was used to acquire the load
and stroke signals. Load was measured with a 10 kN strain-gageload
cell directly mounted on the testing machine cross head, while
stroke was measured by means of a LVDT transducerdirectly connected
between the frame of the testing machine and the head of the
hydraulic actuator. The design of the xturedevice allows the inner
supports to rotate around the neutral axis of the specimen. Static
tests were carried out on all con-gurations at room temperature in
stroke control mode at a constant displacement rate of 2 mm mn1in
order to archive aquasi-static loading condition according to the
military standards: MIL-STD-401 DIN 53291. The following condition
was im-posed so that the rupture takes place in the core:where
Prup;face = 2tfbdrf ;max(L2 L1)~ Prup;core = sc;maxbd; (1)with L2
L1 - 2tfrf ;maxsc;max: (2)Table 1Specimen dimensions.L (mm) b (mm)
h (mm) Hc (mm) tf (mm) L2 (mm) L1 (mm) d = hc + tf (mm)500 250 10
8.80 0.60 420 210 9.40Table 2Mechanical properties of faces made of
aluminium.Young modulus (MPa) Strength to failure (MPa) Max.
elongation (%)70,000 367 13Table 3Mechanical properties of the
cores [3].Aluminium core Fibre aramide coreCore ECM ECACell size
6.4 9.6 3.2 3.2Density (kg/m3) 82 55 4.8 144Shear resistance L
(MPa) 2.40 1.48 1.32 3.50Shear modulus L (MPa) 430 253 51 128Shear
resistance W (MPa) 1.40 0.88 0.56 2.20Shear modulus W (MPa) 220 170
49 94Compression resistance (MPa) 1.50 2.75 2.10 15.20A. Abbadi et
al. / Simulation Modelling Practice and Theory 17 (2009) 15331547
1535Four replicate sandwich specimens of different densities and in
two different congurations were tested. During the teststhe
displacement of the inner supports of the four-point bending rig
and the total applied load were acquired.2.3. Experimental resultsA
typical loading maximum displacement curves are shown in Figs. 35.
The analysis of the experimental results of thestatic four-points
bending tests permit use to make the following statements: the
sandwich composite stiffness
increaseswhenincreasingthecoredensityandtheloadtofailureincreaseswithincreasingcoresdensities;themaximumloadsare
higher in the L-direction than in the W-direction for low densities
and almost of the same order of higher core densitiesvalues and the
maximum deection is higher in L-conguration than in the W one for
the same sandwich core. Consideringtogether aramide bres and
aluminium cores, the sandwich panels with aramide bres are almost
more ductile than thosemade of aluminium cores.Ec, GcEfd tctfHbFig.
2. Sketch of the four-point bending test and specimen dimensions.0
10 20 30 40 500123456755kg/m3density 82kg/m3Loads (kN)max.
deflection (mm)(Alu-Alu)(Alu-Alu)Fig. 3. Evolution and comparison
of the deection according to the applied vertical load for the L-
and W-directions. The cores are made of aluminium(55 kg/m3and 82
kg/m3).1536 A. Abbadi et al. / Simulation Modelling Practice and
Theory 17 (2009) 15331547The visual and optical observations (Figs.
6 and 7) made on the damaged honeycomb sandwich panels point out
that allthe specimens failed due to face wrinkling: a local
buckling of the compressed face. We have also observed an
indentationand plastic deformations of faces at the loads
application area as well as cell walls wrinkling in the zone
between the loadapplication zone and the support zone. It appears
from these observations that the failure modes depend essentially
on the0 5 10 15 20 25012345Loads (kN)max. deflection (mm)LwFig. 4.
Comparison of the deection according to the applied vertical load.
The cores are made of aluminium (55 kg/m3) and L-oriented.0 20 40
60 80 1000123456748kg/m3144kg/m3loads (kN)max. deflection
(mm)WWFig. 5. Evolution and comparison of the deection according to
the applied vertical load. The cores are made of aramide bres (48
kg/m3and 144 kg/m3)and W-oriented.A. Abbadi et al. / Simulation
Modelling Practice and Theory 17 (2009) 15331547 1537nature of core
itself: material, density and cells orientation. Indeed, for
honeycomb cores made of aramide bre of a densityof 48 kg/m3and
W-oriented the failure is almost characterized by cell walls
buckling (Fig. 6), small indentation of the faces inthe vicinity of
the loading application and a plastic deformation of the sandwich
skins. On the other hand, for
L-orientedcongurationthefailureisessentiallyduetoasignicant
faceswrinklinginthevicinityof theloadapplicationarea(Figs. 6 and
7). Fig. 7 illustrates the failure modes of L-oriented and
W-oriented aluminium core of a density of 55 kg/m3.These damages
are essentially characterized by cell walls buckling in the zone
between the load application and the xedsupport and by a signicant
skins wrinkling [22].3. Effective properties of the honeycomb
core3.1. Analytical homogenization approachThe development of
constitutive material models for honeycomb materials is complicated
due to the highly anisotropicproperties of the material.
Computationally efcient modelling methods and constitutive laws are
required to reduce CPUFig. 6. Failure modes of sandwich with
aramide bre honeycomb core of 48 kg/m3density.L-Direction
W-DirectionFig. 7. Failure modes of sandwich with aluminium
honeycomb core of 55 kg/m3.1538 A. Abbadi et al. / Simulation
Modelling Practice and Theory 17 (2009) 15331547time and whilst
being accurate enough to realistically represent the overall
structural behaviour. The analytical expressionsused for the
determination of the effective elastic properties of cellular
hexagonal honeycomb core, are based on the impor-tant works of
[410]. The elemental beam theory has been adopted (Fig. 8) for each
component inside the unit-cell to arriveat the different
expressions for effective properties utilizing the strain energy
concept. The length of the diagonal struts, ver-tical struts, and
included angle, as well as the thickness of the struts have been
kept as variable, so various forms of hexag-onal
honeycombcellularstructurescanbeinvestigated.
Thepresentedanalyticalapproachissimpleandcomputestheeffective
properties in a fraction of the time that is required for FE
analysis with a minimum change in the input le.The proper
implementation of this method embedded in large quasi-static or
dynamic simulations (where part of thestructure could be modelled
with detailed nite element mesh and the rest could be modelled with
a single solid layer ofequivalent material properties) would give
high computational advantage, which is essential in large-scale
modelling andsimulation environment.3.2. Numerical homogenization
approachIt is well known that the previous analytical results are
restricted to the extreme density limits (very low and very
highdensities) because analytical expressions are more difcult to
obtain at intermediate densities. In addition, the
well-knownHashinShtrikman bounds provide analytic information on
the possible range of the effective properties. In parallel, very
re-cently Balawi and Abot [11] have shown experimentally that the
effective elastic moduli for honeycombs with low relativedensities
are not similar in the two in-plane directions as predicted by
previous studies. Thus, conventional nite elementapproaches and
numerical homogenization methods are proposed to assess, and update
if needed, the analytical effectiveproperties. The Representative
Volume Element (RVE) consists in 40 cells meshed with plate nite
elements with 4 nodesand 6 degrees of freedom per node. Every foil
contains 12 elements: 4 according to the height and 3 according to
the length.To estimate the various elastic moduli, a displacement
is imposed on the face of the RVE in a given direction while the
oppo-site face is being xed. Symmetries are taken into account by
using the appropriate boundary conditions. Nine simulationsare
necessary to determine the nine elastic moduli of the honeycomb.
Once the honeycomb core is homogenized, the wholesandwich panel is
likened to a beam constituted of three elastic layers:
isotropic/orthotropic/isotropic, that will be used innumerical and
analytical models described below. The FE results are depicted in
Table 4. The values of the plane stresses inthe skins, shear
stresses in the core, deformations and displacements in the
structure are represented, Table 5.4. Analytical modelling of the
four-points bending testHoneycomb sandwich is basically a layered
composite. The most important feature of HC sandwich construction
is thatthese materials are relatively weak in shear due to their
low shear modulus compared to that of extensional rigidity.
ThemodellingofHCsandwichmaterialsisseentofollowthesamepathoflaminatedcomposites.
Comprehensivereviewsandassessmentsofthemodellingofmulti-layered
andsandwich compositescanbefoundinthesurveypaperbyNooret al. [12]
and in Reddy [13] and in the very recent paper of Hu et al. [14].
In [15], Carrera and co-authors uses a unied for-mulation to
compare about 40 theories for multi-layered, composites and
sandwich plates which are loaded by transversepressure with various
in-plane distributions (harmonic,constant, triangular and
tent-like). Also classical laminate theoryt'aX1X2b1111PPM1M1Fig. 8.
Analytical the homogenization procedure.A. Abbadi et al. /
Simulation Modelling Practice and Theory 17 (2009) 15331547
1539(CLT) and First-order shear deformation theory (FSDT) based
models fail to yield accurate results for thick cores and
severeanisotropy. On the other hand, several studies published in
the literature ([1214,16,17] to cite only few) have shown thatmost
of the higher models (HOT) (e.g. Reddy [13] or Touratier [21]),
while adding more effort in the analysis, do not result inhigher
accuracy than the rst-order shear deformation model, used in
conjunction with a post-processing approach based on3D equations.
To overcome these limitations, we adopt in the present study a
unied formulation, similar to the one pro-posed by Hu et al. [14],
where major existing models (CLT, FSDT, HSDT) could be represented
only by choosing the appro-priate mathematical form of the shear
function f(z).4.1. Formulation and kinematicsIn the following, the
sandwich structure is considered to be plane, three-dimensional
with an orthotropic elastic core andelastic isotropic skins. hs and
hc are, respectively, the thicknesses of the skins and the core.
The coordinates system is chosenso that (x1, x2) is the middle
plane. The derivation of the general governing equations is based
on the following restrictiveassumptions common to many authors (we
cite for instance [14,16]):The core thickness is much higher than
the skins thickness (hchs).The (x1 and x2) components of the
displacements eldin the core are linear functions of the z
coordinate.The transversal displacement is independent of the (x3 =
z) coordinate. Thus e33 can be neglected.The stress components r11,
r22, r33, s12 are neglected in the core.The transversal shear
stress components s13 ands23 are neglected in the skins.In addition
totheprevious assumptions, wesuppose that thethickness of thebeam
(H) is uniform and very small withregard to the other dimensions.
The medium plane X is a domain of R R limited by a smooth contour U
and that the Carte-sian reference is associated to the volume V of
the sandwich. The displacement eld is expressed by the following
uniedformulation:U =Ua = ua zw;a f (z)ca;U3 = w;_a 1; 2; (3)where
ua, w, ca could be functions of x1, x2 and t (time); w,a = ow/ oxa,
ca = xa + w,a. ua represents the membrane displace-ment, cais
thetransversal shear strain in themiddle plane. f(z) isthe shear
function that candescribe by thefollowingexpressions depending on
the considered theory [23]:Table 4Mechanical properties of a
honeycomb core made of aluminum 82 kg/m3, L-direction.Aluminum core
82 kg/m3L-direction FE code (Ansys) GibsonE1 (MPa) 1.304 1.332E2
(MPa) 1.334 1.332E3 (MPa) 1733.24 1617.2c121 1c130.0002 0.0002c211
1c230.0002 0.0002c310.33 0.33c320.33 0.33G12 (MPa) 0.346 0.333G21
(MPa) 0.178G32 (MPa) 309.22 308.99G32 (MPa) 23G13 (MPa)
472.72G13-min (MPa) 462.59G13-max (MPa) 513.99Table 5Comparison
between the analytical, nite element and experimental results
obtained for three honeycomb cores densities.Materials Fmax (kN)
rp,max (MPa) ep,max (%) sc;max(MPa) cc,max (%) Analytical (mm)
Experimental (mm) FE (mm)AluAlu 82 kg/m36.299 469.14 0.714 1.34
1.20 11.10 12.36 14.02AluAlu 55 kg/m34.307 320.72 0.488 0.916 1.07
7.58 10 12.30AluFibber 48 kg/m33.278 244.1 0.371 0.697 2.80 6.79
8.68 10.681540 A. Abbadi et al. / Simulation Modelling Practice and
Theory 17 (2009) 15331547Model-1: f (z) = 0, KirchhoffLove Theory
[18];Model-2: f (z) = z, Mindlin rst-order theory [20];Model-3: f
(z) = z1 43zH_ _2_ _, Reddy high order theory [13];Model-4: f (z)
=HpsinpzH_ _, Touratier high order theory [21].4.2. Problem
formulationThe principle of virtual works (PVW) is used to
formulate the problem in order to investigate the ability of the
variousmodels to analyse properly the behaviour of sandwich panels
when subjected to a bending load. For this purpose and con-sidering
the previous introduced unied formulation, a virtual displacement
eld is introduced as:U+ =U+a = u+a zw+;a f (z)c+a;U+3 = w+;_a 1; 2:
(4)The principle of virtual works is written as:W+acc = W+int
W+ext(5)where W+acc: the virtual work of the acceleration
quantities, W+int: the virtual work of the interior forces W+ext:
the virtual workof the exterior forces.The mathematical details of
the analytical modelling are presented in the Annex I. Referring to
the development pre-sented in Annex I, the investigated problem
could be solved entirely by solving the following PDE system:\u+a;
c+aetw+:0 = E11w;111 (E12 2E66)w;122 D11c1;11 (D12 D66)c2;12
D66c1;22 A55c1;0 = (E12 2E66)w;112 E22w;222 (D12 D66)c1;12 D66c2;11
D22c2;22 A44c2;0 = D11w;1111 2(D12 2D66)w;1122 D22w;2222
E11c1;111(E12 2E66)(c1;122 c2;112) E22c2;222 q(6)The boundary
conditions at the panel edges y = b/2 and y = b/2:0 = 2E66w;12
D66c1;2 D66c2;1 C1;0 = E12w;11 E22w;22 D12c1;1 D22c2;2 C2;0 = (D12
4D66)w;112 D22w;222 (E12 2E66)c1;12 2E66c2;11 E22c2;22 Tz C1;1;0 =
D12w;11 D22w;22 E22c2;2 E12c1;1 C2(7)Considering a four-point
bending test conguration, the sandwich panel is simply supported at
the end edges (x1 = 0 andx1 = a). This condition could be expressed
mathematically by:\x; C1 = C2 = Tz = C2 = 0 and y = b=2:For solving
this PDE system, we consider a load given by the following
sinusoidal function:q(x; y) = Qm sinpa x_ _where Qm = 8q0psinpla_
_sinpe2a__ and q0 =P2be: (8)The following close form, similar to
the one proposed by Idlbi et al. [23], must satisfy the load
balance equations and theboundary conditions.w(x; y) = W(y) sinpa
x_ _; c1(x; y) = C1(y) cospa x_ _; c2(x; y) = C2(y) sinpa x_ _:
(9)This complex PDE system is solved using Mathematica 5.1. The
response in terms of displacement eld has permitted tocharacterize
the entire response of the sandwich panel. The results are
presented in Tables 6 and 7.5. Finite element modelling and
analysisFor the purpose of comparison, FE calculations are also
performed on CASTEM 2003. The HC sandwich structure is mod-elled
using 3D solid (eight nodes and six DOFs per node) elements. For
symmetry reasons, only a quarter of the sandwichpanel (Fig. 9) is
considered in the present analysis. The applied boundary conditions
are as follows: at the level of the sup-A. Abbadi et al. /
Simulation Modelling Practice and Theory 17 (2009) 15331547
1541port, the transversal displacement Uz is xed to zero; at the
symmetry level on the face 1, the in-plane displacementUx andthe
rotations hy and hz are xed to zero; then on the face 2, the
in-plane displacement Uy and the rotation hx and hz are
alsozero.Table 6The displacement (U1, U2, U3) eld obtained using
the different analytical models and the FE model (CASTEM
2003).AluAlu 6.482 (load = 1000 N)Displacement: U1(3a/4,0,H/2),
U2(3a/4,b/2,H/2), U3(a/2,0,0)L-Conguration W-CongurationU1 (lm)
Error (%) U2 (lm) Error (%) U3 (mm) Error (%) U1 (lm) Error (%) U2
(lm) Error (%) U3 (mm) Error (%)FE Castem 58.30 0.604 15.03 3.726
2.31 3.91 59.03 1.653 15.06 2.59 2.35 4.864Kirchhoff 57.48 0.811
13.45 7.18 2.174 2.204 57.48 1.016 13.45 8.379 2.174 2.989Reddy
57.94 0.017 14.48 0.069 2.222 0.045 57.94 0.224 14.48 1.362 2.222
0.848Touratier 57.95 Ref. 14.49 Ref. 2.223 Ref. 58.07 Ref. 14.68
Ref. 2.241 Ref.Table 7Comparison of stress values obtained using
the different analytical models and the FE model (CASTEM
2003).Models Alu-Alu 6.4-82 (load = 1000 N)r11(a/2, 0, H/2),
r22(3a/4, 0, H/2), s13 (3a/4, b/2, hc/2)L-Conguration
W-Congurationr11 (MPa) Error (%) r22 (MPa) Error (%) s13 (MPa)
Error (%) r11 (MPa) Error (%) r22 (MPa) Error (%) s13 (MPa) Error
(%)Castem 44.70 0.134 3.85 2.729 5.07 2.238 45.46 1.473 3.70 3.971
5.70 3.012Kirchhoff 44.66 0.223 4.487 13.365 44.66 0.3125 4.487
16.455 Reddy 44.75 0.0223 3.965 0.177 5.204 13.254 44.75 0.1116
3.965 2.907 5.204 11.451Touratier 44.76 Ref. 3.958 Ref. 4.595 Ref.
44.80 Ref. 3.853 Ref. 5.877 Ref.Fig. 9. Sketch of a four-point
bending test modelling and notations.1542 A. Abbadi et al. /
Simulation Modelling Practice and Theory 17 (2009) 15331547Prior to
initiating the evaluation study, an analysis of mesh convergence is
carried out to ensure the accuracy of the pro-posed nite element
solution since it is considered in the present study as the
reference. The convergence was achieved with9000 elements: 50
elements following the x-axis, 12 elements in the thickness of the
core, 3 elements in the thickness ofeach skin and 10 elements
following the y-axis.6. Results and discussionSome of the obtained
results obtained using the various theories as well as using FE
modelling are depicted in Tables 6and 7. In the proposed comparison
effort, the 3D FE results are considered as the reference and
serves to validate the ana-lytical results. The curves of (Fig. 10
and 11) give the variation of the displacements U3, U1 and U2 and
for some positions inthe sandwich panel according to the different
models. In Figs. 12 and 13 are presented the in-plane stress and
the shearstress, respectively. Form these results we could observe
that in Model-1 (KirchhoffLove theory based model or CLT
basedmodel) the deformations due to the transverse shear are
neglected and therefore the deection is underestimated (error of3%
on the transversal displacement and from 1% to 8% on U1 and U2).
Considering the importance of the transverse stress inthe modelling
of honeycomb sandwich composites, we state that Model-1 is unsuited
for the core modelling. Nevertheless, itcould be adequate for
modelling the faces. Model-2 is a rst-order shear deformation
theory (FSDT) based model. This modelconsiders a linear variation
of the shear stress in the core. Model-3 and Model-4 are higher
order theory based models. Theyare mainly based on hypothesis of
non-linear stress variation through the thickness. The third-order
theory by Reddy [13]250 300 350 400 450-2.5-2.0-1.5-1.0-0.50.0U3
[mm]X [mm] Kirchoof Touratier ReddyCastemFig. 10. Evolution of the
deection U3, x considering the different
theories.-60-40-200204060-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
6U1[m]U2[m]z[mm]LoveReddyTouratierCastem-20-15-10-505101520-6 -4 -2
0 2 4 6z[mm]LoveReddyTouratierCastemFig. 11. Evolution of the
in-plane displacements (U1 and U2) elds versus z coordinate (x =
3a/4 and y = b/2.) considering the different theories.A. Abbadi et
al. / Simulation Modelling Practice and Theory 17 (2009) 15331547
1543(Model-3) is based on the same assumptions as than classical
and rst-order theories, except that the assumption of straight-ness
and normality of a transverse normal after deformation is relaxed
by expanding the displacements as cubic functions ofthe thickness
coordinate. Model-3 satises zero transverse shear stresses on the
bounding planes and the equations of mo-tion are derived from the
principle of virtual displacements. As with the CLT and FSDT, HOT
based model does not satisfy
thecontinuityconditionsoftransverseshearstressesatlayerinterfaces.
InModel-3, Touratier[21]suggestedtrigonometricfunctions instead of
polynomial developments of the transverse coordinate. The proposed
theory recovers the classic thinplate and ReissnerMindlin [19]
theories and satises zero transverse shear stress conditions on the
top and bottom surfaceof plates and avoids shear correction
factor.7. ConclusionExperimental, analytical and numerical
modelling of a typical four-points bending test of honeycomb
sandwich structurehave been investigated. The honeycomb core is
modelled as a single solid layer of equivalent material properties.
Analyticaland numerical (FE) homogenization approaches have been
used to compute the effective properties of the honeycomb
core.Beside the analysis of various in-the-literature models and
theories, 3D governing equations of four-points bending test
havebeenderivedbyusingthevirtualworkprincipleandsolved.
Ageneralkinematicmodel(uniedformulation)hasbeenadopted and used the
static test analysis of honeycomb sandwich panel. Various
kinematics (CLT, FSDT, HOT) have been con-sidered and compared and
the results have been presented. It comes out from this assessment
process that models based onhigh order theories are more accurate
than CLT and FSDT based models and that CLT and FSDT models always
overestimateglobal stiffnesses and do not enable a valid
description of the distribution of the transverse shear stress.-6
-4 -2 0 2 4 6-60-40-200204060 11 [Mpa]Z[mm]Kirchhoff TouratierReddy
Castem-6 -4 -2 0 2 4 6-8-6-4-202468 22 [Mpa]Z [mm]Kirchhoff
TouratierReddy CastemFig. 12. In plane stress evolution (r11,r22)
according to z face for x = a/2 and y = 0.0123456-6 -4 -2 0 2 4
613[Mpa]z[mm]Reddy Touratier CatemFig. 13. Evolution of the shear
stress in the honeycomb sandwich (analytical and FE results).1544
A. Abbadi et al. / Simulation Modelling Practice and Theory 17
(2009) 15331547AcknowledgmentsThe nancial support of the Funds
National de la Recherche is acknowledged (AFR grant of Dr.
Koutsawa). Part of thisresearch work has been achieved in the
framework of Adyma project (FNR/08/01).Appendix A. Expression of
W+accThe virtual work of the acceleration quantities is dened
by:W+acc =_VqU+UdV: (I:1)By using some classical integration
theorems and introducing the following generalized vectors for, n =
0, 1, 2(In; Jn; K) =_H=2H=2q(zn; f (z)n; zf (z))dz;one can write
W+acc as:W+acc =_X((I0ua I1 w;a J1ca)u+a (I1ua;a I2 w;aa Kca;a I0
w)w+ (J1ua K w;a J2ca)c+a)dX_C(I1ua I2 wa Kca)naw+dC: (I:2)A.1.
Expression of W+intThis virtual work of internal work is given
as:W+int = _Ve+: rdV: (I:3)Considering the assumption of small
perturbations, one can get following relation for the virtual
deformations:e+ij = 12u+i;j u+j;i_ _; i; j = 1; 2; 3;W+(i) =
_V((u+a;b zw+;ab f (z)c+a;b)rab f/(z)c+ara3)dV:(I:4)By introducing
into this relation the following generalized forces and
moments:(Nab; Mab; Mab) =_H=2H=2(1; z; f (z))rabdz;Qa
=_H=2H=2f/(z)ra3dz;and considering the constitutive law:rab =
Qabijeij; a; b; i; j 1; 2; 3and also the generalized constitutive
matrices:(Aabij; Aabij; Babij; Babij; Dabij; Dabij; Eabij)
=_H=2H=2(1; f (z)2; z; f (z); z2; f (z)2; zf (z))Qabijdz:The
virtual work of the interior forces yieldsW+(i) =_X(Aabijui;jb
Babijw;ijb Babijci;jb)_ u+a (Babijui;jba Dabijw;ijba Eabijci;jba)w+
(Babijui;jb Eabij w;ijb Dabijci;jb)c+a Aa3i3cic+a_ dX_C(Aabijui;j
Babijw;ij Babijci;j)_ nbu+a (Babijui;j Dabij w;ij
Eabijci;j)nbnaw+;n ((Babijui;j Dabijw;ij Eabijci;j)nbta);lw+
(Babijui;jb Dabijw;ijb Eabij ci;jb)naw+ (Babijui;j Eabijw;ij
Dabijci;j)nbc+a_ dC:A. Abbadi et al. / Simulation Modelling
Practice and Theory 17 (2009) 15331547 1545A.2. Expression of
W+(e)The virtual work of the external forces is dened by:W+(e)
=_VU+f dV _CfU+F dCf :Using the following generalized vectors:(Pa;
ma; ma) =_H=2H=2(1; z; f (z))fadz; q =_H=2H=2f3dz; Tz =_H=2H=2F3dz
and (Ta; Ca; Ca) =_H=2H=2(1; z; f (z))Fadz;We get from some
integrations by parts the following:W+(e)=_X(Pa u+a(qma;a)w+ ma
c+a)dX_C(Ta u+a(Tz(Ca ta);lma na)w+Ca na w+;nCa c+a)dC:A.3. Balance
equations and natural boundary conditionsBy applying the lemma of
variations calculus to the previous energy equations, one gets the
following equilibrium withtheir associated natural oundary
conditions.I0 ua I1 w;a J1 ca = Aabijui;jb Babijw;ijb Babijci;jb
Pa;J1 ua K w;a J2 ca = Babijui;jb Eabijw;ijb Dabijci;jb Aa3i3ci
ma;I1 ua;a I2 w;aa K ca;a I0 w = Babijui;jab Dabijw;ijba
Eabijci;jba q ma;a:A.4. Boundary conditions0 = (Aabijui;j Babijw;ij
Babijci;j)nb Ta;0 = (Babijui;j Eabijw;ij Dabijci;j)nb Ca;0 = (I1 ua
I2 w;a K ca)na = (Babijui;jb Dabijw;ijb Eabijci;jb)na((Babijui;j
Dabijw;ij Eabijci;j)nbta);l Tz (Cata);l Tz (Cata);l mana;0 =
(Babiju3i;j Dabijw;ij Eabijci;j)nbna Cana:A.5. Application on the
modelling of a sandwich beamThe considered sandwich panels have
mirror symmetry. Thus, the generalized constitution matrices B and
B are zero. Ifone considers that the volume forces are directed
according to the axis z then we have:Pa = ma = ma = 0:By using the
following notation conventionri = Qijej:The equilibrium equations
becomes\u+a; c+aetw+:0 = E11w;111 (E12 2E66)w;122 D11c1;11 (D12
D66)c2;12 D66c1;22 A55c1;0 = (E12 2E66)w;112 E22w;222 (D12
D66)c1;12 D66c2;11 D22c2;22 A44c2;0 = D11w;1111 2(D12 2D66)w;1122
D22w;2222 E11c1;111(E12 2E66)(c1;122 c2;112) E22c2;222 q(38)For
anoblongbeam(thesystemaxes representedFig9), the4boundaryconditions
associatedtotheedges y = b/2andy = b/2are:0 = 2E66w;12 D66c1;2
D66c2;1 C1;0 = E12w;11 E22w;22 D12c1;1 D22c2;2 C2;0 = (D12
4D66)w;112 D22w;222 (E12 2E66)c1;12 2E66c2;11 E22c2;22 Tz C1;1;0 =
D12w;11 D22w;22 E22c2;2 E12c1;1 C2:1546 A. Abbadi et al. /
Simulation Modelling Practice and Theory 17 (2009)
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