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8/18/2019 Analytical, numerical and experimental study of the transverse shear behavior of a 3D reinforced sandwich struct…
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Analytical, numerical and experimental study of the transverse shear
behavior of a 3D reinforced sandwich structure
Cyril Laine a , Philippe Le Grognec a , *, Stephane Panier a , Christophe Binetruy a , b
a Mines Douai, Polymers and Composites Technology & Mechanical Engineering Department, 941 rue Charles Bourseul, CS 10838,
59508 Douai Cedex, Franceb LUNAM Universit e, Ecole Centrale de Nantes, Research Institute in Civil Engineering and Mechanics (GeM), 1 rue de la No€e, BP 92101,
44321 Nantes Cedex 3, France
a r t i c l e i n f o
Article history:
Received 7 November 2013
Accepted 18 April 2014
Available online 9 May 2014
Keywords:
Transverse shear stiffness
Reinforced sandwich
Unit cell model
a b s t r a c t
Sandwich structures are known to be very sensitive to transverse shear effects when submitted to out-of-
plane loads. The use of a MindlineReissner type equivalent plate model is then certainly the simplest
way to take into account these transverse shear strains that strongly inuence the global deection in
simple bending. Such a model requires the estimation of the transverse shear stiffness or of the so-called
shear correction factor. In the case of a traditional sandwich (with homogeneous foam core), this shear
correction factor is set to unity, so that the equivalent transverse shear modulus coincides with the shear
modulus of the foam core, which is fatally insubstantial. In order to improve the through-thickness
properties of sandwiches, which are governed by the core layer, use is made of thin-walled core ma-
terials or reinforcements. In these more complicated cases, the equivalent shear modulus of the core
material (in a 3D framework) highly depends on the geometry of the reinforcements and may only be
calculated numerically. Moreover, the use of this homogenized shear modulus for the heterogeneous
core layer and of a shear correction factor of unity does not generally convey to the proper value of the
transverse shear stiffness, due to the possible interactions between the reinforcements and the skins.
This paper particularly deals with sandwich structures manufactured with polymeric foam core rein-forced thanks to the Napco® technology (which is based on transverse needle punching) and is devoted
to obtaining their transverse shear stiffness. Bearing in mind the remarks made earlier, a one-step ho-
mogenization procedure is employed, involving simultaneously the contribution of the reinforcements to
the equivalent shear modulus of the reinforced foam core and the interactions between reinforcements
and skins. An analytical (respectively numerical) solution is derived, considering a 2D (respectively 3D)
unit cell and using the basic principle of energy equivalence. The transverse shear stiffnesses obtained by
these two simplied methods are then compared to the one obtained by a nite element numerical
computation on a whole beam-like structure for validation purposes, and nally confronted to the
experimental values resulting from 3-point bending tests performed with various volume fractions of
reinforcements.
© 2014 Elsevier Masson SAS. All rights reserved.
1. Introduction
Sandwich materials are commonly used in many applications of
aerospace, marine or transportation industries, among others, due
to the attractive combination of a lightweight and strong me-
chanical properties. The exural stiffness of sandwiches is indeed
particularly signicant, thanks to the high strength of the skins andtheir distance from the middle-surface of the structure. When
dealing with bending beams (respectively plates), transverse shear
effects can be neglectedif the structure is almost homogeneous and
suf ciently slender (respectively thin). In this case, the so-called
EulereBernoulli (respectively LoveeKirchhoff) hypotheses apply,
and only the exural stiffnesses are involved in the bending
response. Since one considersa thicker structure and/ora sandwich
composite material, these transverse shear effects can no longer be
neglected and transverse shear stiffnesses may be introduced in the
context of a Timoshenko (respectively MindlineReissner) model,
for example (Reissner, 1945; Mindlin, 1951).
* Corresponding author.
E-mail addresses: [email protected] (C. Laine), philippe.le.grognec@
mines-douai.fr (P. Le Grognec), [email protected] (S. Panier),
[email protected] (C. Binetruy).
Contents lists available at ScienceDirect
European Journal of Mechanics A/Solids
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . co m / l o c a t e / e j m s o l
http://dx.doi.org/10.1016/j.euromechsol.2014.04.006
0997-7538/©
2014 Elsevier Masson SAS. All rights reserved.
European Journal of Mechanics A/Solids 47 (2014) 231e245
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://www.sciencedirect.com/science/journal/09977538http://www.elsevier.com/locate/ejmsolhttp://dx.doi.org/10.1016/j.euromechsol.2014.04.006http://dx.doi.org/10.1016/j.euromechsol.2014.04.006http://dx.doi.org/10.1016/j.euromechsol.2014.04.006http://dx.doi.org/10.1016/j.euromechsol.2014.04.006http://dx.doi.org/10.1016/j.euromechsol.2014.04.006http://dx.doi.org/10.1016/j.euromechsol.2014.04.006http://www.elsevier.com/locate/ejmsolhttp://www.sciencedirect.com/science/journal/09977538http://crossmark.crossref.org/dialog/?doi=10.1016/j.euromechsol.2014.04.006&domain=pdfmailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]
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The transverse shear behavior of a sandwich structure (as well
as any other out-of-plane behavior such as the through-thickness
compression) is one of the principal weaknesses of classical sand-
wiches, given that the corresponding stiffness is directly related to
the low mechanical properties of the soft core material. When a
sandwich structure is subjected to simple bending, the out-of-
plane loads cause transverse shear strains in the core layer, that
give rise to a supplementary non-negligible deection in addition
to the classical deection associated with theexural behavior. As a
matter of fact, in the case of a homogeneous core with a very low
modulus compared to the skin one, the deection due to the
transverse shear effects may even become predominant.
In order to improve the load carrying capacity of sandwiches,
especially in the transverse shear behavior (by increasing the
equivalent transverse shear stiffnesses), without being detrimental
to lightness, the low density core layer is usually strengthened by
appropriate reinforcements. One of the simplest ways to proceed
comes down to add orthogonal reinforcements embedded in the
upper andlower skins.Amongthe existingmethods,such as tufting,
Z-pinning and stitching (Lascoup et al., 2006), the patented Napco®
technology, which is based on transverse needling, allows one to
produce tailored sandwich structures in a continuous way, while
preserving a high production ef ciency and a relatively low cost.The overall purpose of this study is to analyze the mechanical
behavior of such Napco® sandwich structures submitted to out-of-
plane loads. The through-thickness compression has already been
analyzed by some of the authors in Laine et al. (2013). Here, our
attention focuses on the transverse shear behavior involved in
simple bending loading conditions. For this purpose, experimental
3-point bending tests are rst performed for various volume frac-
tions of reinforcements (including the case of a non-reinforced
sandwich). In all cases, the signicance of the transverse shear
deection in relation to the pure exural one is emphasized, and
the increase of the equivalent transverse shear stiffness due to the
presence of reinforcements is assessed. The main objective is then
to develop analytical and numerical approaches for the determi-
nation of the effective transverse shear modulus of Napco®
sand-wiches. Such ef cient predictive tools would further be employed
to optimize the size and volume fraction of reinforcements in
relation to the transverse shear behavior.
The transverse shear behavior of beams and plates has been
studied for many years, both theoretically and experimentally. The
theoretical developments mainly concern composite laminates or
sandwiches which are most affected by transverse shear effects.
The general purpose of these works is to represent at best the
transverse shear response of such composite materials. The most
straightforward solution comes down to use a rst-order shear
deformation theory (namely a Timoshenko model for beams or a
MindlineReissner model for plates, as an example) and to derive
the corresponding equivalent transverse shear stiffness(es) of the
composite beam or plate in the context of the so-called laminate orsandwich theory. The strong simplifying assumptions in the
transverse shear strain and stress distributions, which are sup-
posed to be uniform in all layers throughout the thickness of the
composite structure, require the introduction of shear correction
factors that are often dif cult to assess. As an alternative, higher-
order shear deformation theories have gradually emerged.
Higher-orderbeam/platetheories arereferred to as such precisely
because they involve strain and stress distributions in the section/
thickness of higher order. The objective of higher-order shear
deformation theories is to better represent the transverse shear
strain and stress elds in the beams or plates in hand in order to
better estimate the bending response of the structure without the
use of a shear correction factor. Lots of models can be found in the
literature, regarding equivalent single layer theoriesand
rst applied
to homogeneous structures, which result most of the time from the
choice of specic kinematic hypotheses. Non-linear displacement
elds are introduced, especially in the thickness direction, using
appropriateshape functions for shear (polynomial or sinusoidal). For
instance, Barut et al. (2002) developed a higher-order plate theory
using quadratic and cubic expansions for the out-of-plane and in-
plane displacements, respectively. Mantari et al. (2012) recently re-
ported many shape functions (mostly polynomial) that have been
proposed in the literature during the last century and dened a new
trigonometric one that guarantees the stress free boundary condi-
tions on the top and bottom surfaces of the structure. As far as
laminated or sandwich structures are concerned, all these shape
functions may still be used for the overall structure in the context of
equivalent single layer theories but also in the framework of layer-
wise theories. In the latter case, similar shape functions are dened
for each layer of the composite material, giving rise to a global
polynomial or sinusoidal piecewise function. In both cases of rst-
order and higher-order shear deformation theories, a zig-zag func-
tion can be addedin orderto introduce theadequate discontinuityin
the rst derivative of the displacement eld, which is called the zig-
zageffect.The resulting zig-zag modelcanalsobe seenas a piecewise
layer approach (see Brischetto et al. (2009) for an application of the
zig-zag theory to sandwich plates and Carrera (2003) for a generalreview on the use of zig-zag functions for multi-layered plates).
Numerous other models have been developed for a more ac-
curate determination of the transverse shear response of various
structures. Without being comprehensive, one can mention Yu
et al. (2003) who dened a 2D plate theory applicable to lami-
nated plates based on an asymptotic analysis. Their 3D formulation
enables them to recover the 3D displacements, strains and stresses
with a very good accuracy. Nguyen et al. (2005) also proposed a 3D
approached model for thick laminates and sandwich plates. More
recently, Lebee and Sab (2011) presented an original Bending-
Gradient plate model which appears to be an extension of the
well-known MindlineReissner model for heterogeneous plates.
This model was successfully applied to multi-layered plates and
even complex sandwich structures with cellular cores (Lebee andSab, 2012). Moreover, such a model allows one to derive the
transverse shear stiffnesses of heterogeneous plates (including
sandwiches) using a direct homogenization procedure. Lastly,
Buannic and Cartraud (2001) made useof the asymptotic expansion
method in the context of periodic heterogeneous beams. The
consideration of higher-order terms in the developments of
displacement, strain and stress elds enabled them to improve the
classical solution corresponding to the EulereBernoulli theory
without a priori new hypotheses, unlike in standard rened beam
models.
All the higher-order beam or plate models presented above are
likely to provide the equivalent transverse shear stiffness of com-
posite structures but also accurate information about the local
behavior at the heterogeneity scale. However, such models involvemany more degrees of freedom and give rise to increased compu-
tational costs. In this paper, the intention is not to develop or even
use a higher-order model, but rather to estimate at best the
transverse shear stiffness of a non-conventional 3D reinforced
sandwich structure, for practical use in a Timoshenko or Mind-
lineReissner type model.
Regardless of which model is employed, lots of studies have
been carried out with the primary purpose of nding the transverse
shear stiffness of various composite structures. In the context of the
rst-order shear deformation theory, several methods have been
proposed in the literature for the determination of the shear
correction factor. These are typically homogenization or averaging
methods, based on comparisons of forces and moments between
dual problems (Altenbach, 2000) or, most often, on an energy
C. Laine et al. / European Journal of Mechanics A/Solids 47 (2014) 231e 245232
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equivalence. As concerns sandwich panels, Kelsey et al. (1958)
derived rst lower and upper bounds for the transverse shear
stiffness, by applying uniform in-plane displacements or forces on
the top and bottom surfaces of the core layer, focusing on the case
of honeycomb cores. Shi and Tong (1995) also got interested in the
equivalent transverse shear stiffness of honeycomb structures us-
ing a 2D periodic unit cell. Isaksson et al. (2007) investigated the
equivalent transverse shear behavior of corrugated core structures.
Lascoup et al. (2012) derived simplied analytical expressions for
the equivalent shear moduli of stitched sandwich structures
(featuring various stitching angles), but the inuence of the foam
surrounding the stitch yarns was neglected. Also considering
structures similar to Napco® sandwiches, Liu et al. (2008) analyti-
cally expressed the global stiffness tensor of pin-reinforced foam
cores (with various arrangements of pins), including notably the
transverse shear moduli. In their approach, the through-thickness
reinforcements are modeled by simply-supported beams and the
continuous core material is replaced by the superimposition of
horizontal and vertical elastic spring distributions. Lastly, Lebee
and Sab (2010) applied the approach from Kelsey et al. (1958) to
chevron folded cores. The same authors thoroughly discussed of
the principal questions raised by the determination of the trans-
verse shear stiffness of sandwiches in both cases of homogeneousor cellular cores (Lebee and Sab, 2012). They stated that the bounds
obtained by Kelsey et al. (1958) or any other estimations derived by
the above-mentioned approaches are not satisfactory, since the
skin effects are not (or inadequately) considered.
In the alternative of all the previous analytical solutions that only
concern relatively simple geometries and/or suffer from strong
simplifying assumptions, several semi-analytical and numerical
methods have been proposed. Xu andQiao (2002) developed a semi-
analytical straightforward approach based on a multi-pass homoge-
nization technique, using a unit cell comprising both skins and core.
Withsucha method, theytookinto accountthe skin-core interactions
and proved their signicance in honeycomb sandwich structures.
Hohe (2003) also suggested a direct homogenization scheme for
sandwiches instead of the classical two-step homogenization pro-cedure where the core layer is rst homogenized separately. Finite
element computations are performed on a Representative Volume
Element of the whole sandwich and the determination of the effec-
tive stiffness matrix rests upon a strain energy equivalence principle.
Finally, Cecchi and Sab (2007) developed a MindlineReissner
equivalent plate model for orthotropic periodic plates, so as to apply
tobrickwork panels.A LoveeKirchhoff model isrst identied thanks
to a numerical periodic homogenization technique, involving a 3D
unit cell. A new similar unit cell problem is then solved, whose
loading is based on the stress elds obtained in the previous calcu-
lations, in order to derive the equivalent transverse shear stiffnesses.
All the previous methods, whenapplied to sandwiches, properlytake
into account the so-called skin effects.
This paper particularly deals with sandwich structures man-ufactured with polymeric foam core reinforced thanks to the
Napco® technology and is devoted to obtaining their transverse
shear stiffness. Bearing in mind the remarks made earlier, a one-
step homogenization procedure is employed, involving simulta-
neously the contribution of the reinforcements to the equivalent
shear modulus of the reinforced foam core and the interactions
between reinforcements and skins. First, an analytical closed-
form solution is sought considering a 2D unit cell model and
using the basic principle of energy equivalence. The foam core is
represented as a continuous medium whereas a beam model is
considered for the reinforcements. The skins are not directly
modeled but their presence is taken into account through
appropriate boundary conditions applied to the reinforced foam
core. Some conventions are specially suggested in order to relate
properly the 2D designed model to the real 3D conguration. A 3D
unit cell nite element model is implemented and operated for
the determination of the transverse shear stiffness based on the
same energy equivalence principle. Such a numerical model al-
lows one to verify the relevance of some simplifying assumptions
made in the 2D analytical approach and extend the scope of the
present approach to more complicated sandwiches (with inclined
reinforcements, for instance). Finally, numerical nite element
computations are performed on complete sandwich beams in
order to validate the previous unit cell models and confront nu-merical reference results to the experimental values derived from
the 3-point bending tests.
2. Experimental data
2.1. Napco® technology
The Napco® technology is a manufacturing process of 3D
sandwich materials based on transverse needling. It consists in
strengthening the foam core of a sandwich structure by adding
orthogonal (or inclined) through-thickness reinforcements in order
to particularly enhance the out-of-plane mechanical properties. It
differs from other technologies such as stitching due to the fact that
the brous reinforcements here come from the skin material, sothat the facing fabrics (mats) and the foam core make up a
monolithic whole (see Fig. 1). In practical terms, a set of needles
regularly penetrates the sandwich structure on both sides, ac-
cording to the desired pattern and density, the needles catching
and carrying yarns from the facings through the core material, as
shown in Fig. 2. Once the 3D sandwich preform is produced, it is
impregnated by a liquid resin. Among the different liquid com-
posite molding techniques, the VARIM process (Vacuum Assisted
Resin Infusion Molding) has been retained for its ef ciency.
The creation of the brous reinforcements and the composite
manufacturing, associated with an experimental campaign of
measurement of geometric and material parameters, lead one to a
realistic and optimal representation of the sandwich architecture
and thus to a proper prediction of the effective mechanical prop-erties when using appropriate analytical or numerical tools.
Fig. 1. Napco
®
sandwiches (foam core is partly removed to show the through-thickness composite beams).
C. Laine et al. / European Journal of Mechanics A/Solids 47 (2014) 231e 245 233
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2.2. Material and geometric data
The 3D sandwich specimens which will be subsequently tested
are made up of a linearly elastic isotropic closed cell polyurethane
foam (whose density is 40 kg.m3). Both facings are made of oneply of chopped strand glass mat and one carbon [0,90] cross-ply
laminate. During the infusion process, use is made of an Epolam
5015 epoxy resin with 20% of 5015 hardener. The material data are
summarized in Table 1.
The skins are supposed to be isotropic, with equivalent Young's
modulus E s and Poisson's ratio ns. The heterogeneous through-
thickness brous reinforcements are composed of aligned
isotropic bers surrounded by resin. The only case of orthogonal
reinforcements will be considered in the sequel. Therefore, suchreinforcements can be viewed as unidirectional composite columns
(UDs) and will further be represented by equivalent homogeneous
cylinders perpendicular to the skins (see Fig. 3). A preliminary
homogenization step, based on advanced mixture laws (Berthelot,
1996), is then rst performed, involving the volume fraction of
the bers within the reinforcements V f (obtained through burn off
tests) and the material properties of both constituents (glass bers
and resin). It gives the following equivalent properties for the
transversely isotropic through-thickness reinforcing composites
(due to the unidirectional arrangement of the bers):
E L ¼ E f V f þ E r
1 V f
nLT ¼
nf V
f þn
r1 V f
GLT ¼ GrGf
1 þ V f
þ Gr
1 V f
Gf
1 V f
þ Gr
1 þ V f
GT ¼ Gr
0BBBBBBBBB@
1 þ V f Gr
Gf Grþ
kr þ 7Gr3
2kr þ 8Gr3
1 V f
1CCCCCCCCCA
K L
¼K r
þ
V f 1
kf kr þGf Gr
3
þ 1
V f
kr þ 4Gr3
E T ¼ 2
1
2K L þ 1
2GTþ 2n
2LT
E L
nT ¼ E T2GT
1
(1)
Fig. 2. Napco® technology (Guilleminot et al., 2008).
Table 1
Material properties.
Polyurethane
foam
Epoxy
resin
Glass
ber
Carbon
ber
Young's modulus (MPa) 6.7 3281 72,400 290,000
Poisson's ratio 0.001 0.35 0.22 0.3
Fig. 3. Fiber reinforcements (Guilleminot et al., 2008).
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where the quantities E i, n i, Gi, ki and K i represent Young's moduli,
Poisson's ratios, shear moduli, bulk moduli and transverse bulk
moduli (without longitudinal strain), respectively, and the sub-
scripts f, r, L and T stand for the bers, the resin, the longitudinal
direction and the transverse direction. In order to simplify the
geometric representation, the cylindrical through-thickness re-
inforcements are supposed to have a constant circular section of
radius R.
Four different types of needle pattern have been used to create
different pile yarns densities in the nal sandwich structure. The
reference case of a sandwich material without reinforcements is
also considered for comparison purposes in further experiments
and calculations. The material and geometric parameters of theve
panels under consideration are summarized in Tables 2 and 3,
respectively, where the subscript s stands for the skin parameters.
For information, the specic weight of each sandwich panel has
been indicated. The corresponding variation is only due to the
addition of resin during the infusion process.
2.3. 3-point bending experiments
The 3-point bending tests have been performed on a material-
testing machine (Zwick) mounted with a 10 kN-force cell,
following the NF T 54-606 standard. The samples were simply
supported near both ends and submitted to an enforced transverse
displacement at mid-span with an average speed of 1.15 mm.min1
(see Fig. 4 for the experimental set-up). For each of the ve sand-
wich panels, two different spans were considered, in order to
determine both the exural and transverse shear stiffnesses,
associated with the coupled pure bending and simple shear re-
sponses, respectively. For each span length, seven specimens were
tested with always the same width of 60 mm. According to the
thickness of the sandwich panels (which is approximately the same
for all densities), a total specimen length of 280 mm (respectively
500 mm) was retained for the tests with a short (respectively long)
span length of d1
¼230 mm (respectively d2
¼450 mm). All these
dimensions are suf ciently large so that the specimens containmany reinforcements, even in the case B where the volume fraction
of reinforcements is very low. Although the specimens do not
include a whole number of unit cells, which may act as represen-
tative volume elements, the volume fraction of reinforcements in
the specimens coincides thus pretty much with the theoretical
values of Table 3.
Fig. 5 plots the transverse force applied on the upper skin versus
the mid-span deection measured on the lower skin, for both short
and long specimens and for all the sandwich panels considered. In
each case, despite unavoidable imperfections, the seven curves
obtained for the seven tested specimens are very little scattered, so
much that just one curve is plotted for clarity purposes.
The forceedisplacement curves in Fig. 5 clearly emphasize the
inuence of the volume fraction of reinforcements on the bending
behavior of the sandwich beams. Both the initial stiffness and the
failure load strongly increase with the density of brous re-
inforcements, whereas the failure occurs at about the same
deection, whatever the panels considered, which only depends on
the span length. The response curves of the specimens reinforced
up to 138 r /dm2 and 276 r /dm2, respectively, appear curiously very
close to each other. It is due to the presence of many duplicate
reinforcements that were observed in the panel with an expected
density of 276 r /dm
2
. In concrete terms, the needles arising fromboth sides were not really in perfect alignment during the
manufacturing process of this particular panel, giving rise to more
numerous reinforcements but with a lower radius, whence come
the large discrepancies between the expected results and the ones
nally obtained. Lastly, the curves corresponding to the reference
sandwich panel (without reinforcements) are slightly different
fromthe others, as they present a plateau at the maximum load and
then a small decrease of the force before the sudden collapse. It is
due to the local core crush under the load application point and it is
all the more pronounced that the span length is important.
The exural and transverse shear stiffnesses can be derived from
the forceedisplacement curves, as explained in Dawood et al.
(2010), for instance, in the context of similar 3D glass ber rein-
forced polymer sandwich panels. For each panel, both tests (with
short and long specimens) arerequired for the determination of the
two stiffnesses. In each curve, use is made of a reference point in
the rst linear range. In all cases, the deection w at mid-span can
be viewed as the sum of two deections both depending on the
applied force F . The rst one is related to the exural behavior and
involves the corresponding exural stiffness D, and the second one
is associated to the transverse shear behavior and brings into play
the sought transverse shear stiffness S . The total deection is thus
expressed as follows:
w ¼ Fd3
48Dþ Fd
4S (2)
Using Equation (2) successively for both cases of a short and
long span length, with subscripts 1 and 2 respectively, one candeduce the stiffnesses of the corresponding panel:
D ¼F 2
d32 d2d21
48
w2 w1F 2d2F 1d1
; S ¼F 1d1
1 d21
d22
!
4
w1 w2F 1d
31
F 2d32
! (3)
The exural and transverse shear stiffnesses obtained for the
ve panels considered are listed in Table 4 and depicted in Fig. 6. In
each case, only the mean value x and the standard deviation s are
provided.
In classical sandwich structures, the exural behavior is known
to be governed by the skin behavior. It is therefore expected that
Table 2
Material parameters: through-thickness reinforcements and skins.
Panel A B C D E
Density (r /dm2) 0 69 138 276 415
Through-thickness reinforcements
V f (%) e 4.01 2.63 3 1.93
E L (MPa) e 6052.7 5098.8 5354.6 4615
nLT e 0.3448 0.3466 0.3461 0.3475
GLT (MPa) e 1308.4 1275.5 1284.3 1259.2
E T (MPa) e 3676.9 3558.5 3591.3 3493.4
nT e 0.4228 0.4065 0.4115 0.3957
Skins
E s (MPa) 11,207.5 8463.3 8991.2 9181.7 8310.8
ns 0.364 0.3575 0.3616 0.3658 0.3615
Table 3
Geometric parameters.
Panel A B C D E
Specic weight (kg/m3) 199.8 234.57 277.98 350.45 428.91
Reinforcement
radius (mm)
e 1.1065 1.422 1.002 1.126
Reinforcement volume
fraction (%)
e 2.53 9 11.39 16.52
Foam thickness (mm) 20 20 20 20 20Skin thickness (mm) 1.132 1.465 1.255 1.138 1.101
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the foam core, and by extension the reinforcements, has no
particular inuence on the exural stiffness D. However, experi-
mentally speaking, this exural stiffness does not vary, as expected,
in line with the value of the skin thickness. This uncertain variation
of the
exural stiffness and the associated large scattering are
probably due to the fact that, for such sandwiches, the exural part
of the deection appears negligible against the transverse shear
one, what leads to this degree of uncertainty. Conversely, the
transverse shear behavior is governed by the reinforced foam core
and thus the transverse shear stiffness should highly depend on the
Fig. 4. 3-point bending experimental devices.
Fig. 5. Experimental forceedisplacement curves from 3-point bending tests.
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volume fraction of reinforcements. The present results are consis-
tent with these expectations, since the transverse shear stiffness
regularly increases with the density of the sandwich panel and,
what is more, displays a more reasonable scattering.
3. Approximate methods for the determination of the
transverse shear behavior of Napco® sandwiches
3.1. Analytical resolution of the transverse shear stiffness using a 2D
unit cell
First of all, the elementary architecture of such reinforced
sandwich structures with orthogonal reinforcements allows one to
develop an analytical solution for the transverse shear stiffness.
This solution will further be compared with numerical nite
element results and confronted to the previous experimental
values for validation purposes.
3.1.1. Problem de nition
In the subsequent analysis, only the transverse shear behavior is
investigated, disregarding the exural behavior also brought into
play in the simple bending response of the sandwich structure. One
thus focuses on the deformation eld in the reinforced foam core
only, considering the skins as innitely rigid. The following as-
sumptions are then used, in order to be able to derive an explicit
expression for the sought transverse shear stiffness. A 2D repre-
sentation is retained, like in Laine et al. (2013), with a unit cell
model (two half-reinforcements separated by a foam block) which
is supposed to describe the effective behavior of the global sand-
wich structure, once the proper periodicity conditions are pre-
scribed (see Fig. 7). As previously mentioned, only the reinforced
foam core is rst represented, the presence of the skins being
replaced by the proper boundary conditions. Their little inuence
on the transverse shear behavior will be discussed below.
The width (2e) of the composite reinforcements is chosen in
such a way that their second moment-to-area ratio in the 2D model
is equal to the one of the real cylindrical reinforcements in the 3D
material ðe ¼ R ffiffiffi
3p
=2Þ. The same transverse shear behavior wouldthus be obtained in both 2D and 3D congurations in the absence of
foam. The width (2H ) of the foam block is then dened in agree-
ment with the volume fraction of reinforcements (H ¼ e(1 V fr)/V fr). This particular choice will be proven to give satisfactory results.
Finally, the global thickness (2L) is the real foam thickness
measured experimentally.
The homogeneous and isotropic foam core is considered here as
a 2D continuous solid and it is supposed to be linearly elastic (with
Young's modulus E c and Poisson's ratio nc). The 2D model is sup-
posed to reproduce the behavior of a panel with lateral dimensions
much larger than thickness, so that the plane strain hypothesis is
adopted. The transversely isotropic brous reinforcements (UDs)
are assumed to behave like Eulere
Bernoulli beams, with clampedboundary conditions, due to the entanglement of the bers into the
rigid skins. Due to these kinematic hypotheses, only the longitu-
dinal modulus E L will be involved in the sequel among all the elastic
moduli dened in Equation (1), so that the material can also be
considered as isotropic.
3.1.2. General procedure for the calculation of the transverse shear
stiffness of a sandwich structure
The calculation method is based on the classical energy equiva-
lence principle. The unit cell is rst submitted to a prescribed
macroscopic shear strain g in such a way that only pure shear is
involved in the effective mechanical response of the sandwich.
Practically speaking, due to the extreme rigidity of the skins, this
macroscopic shear state can be achieved by enforcing two different
horizontal displacements (in the Y -direction, see Fig. 7) between the
lower and upper boundaries of the unit cell (at the interface be-
tween the foam core and the lower and upper skins, respectively).
The relative displacement between the two skins is denoted by 2 d,
so as to get the relationship g ¼ d/L in small deformations (Fig. 8).This unit cell is then successively considered as being hetero-
geneous (in this case, the foam block and the reinforcements are
provided with their respective moduli) and homogeneous (with an
equivalent homogenized behavior). In both cases, the total strain
energy of the unit cell is estimated, involving the mechanical pa-
rameters of the constituent materials (together with the geometric
ones) and the effective moduli, respectively. By virtue of the energyequivalence, it is possible to express the sought effective properties,
namely the equivalent transverse shear stiffness, as a function of
the material and geometric data.
The so-called “microscopic” strain energy is calculated by inte-
grating the local volume density of strain energy over the unit cell
volume U (a unit depth is retained, for simplicity purposes):
W micro ¼1
2
Z U
s : 3ⅆ U (4)
where s and 3 refer to the local stress and strain tensors,
respectively.
The energy W micro can be viewed as the sum of the strain en-
ergies stored in the reinforcement and the foam block. Due to the
heterogeneities, the macroscopic shear strain applied to the unit
cell may lead to any general stress/strain state at the local level, so
that all the components of tensors s and 3 must be used when
dealing with Equation (4).
Besides, the “macroscopic” strain energy (associated with the
equivalent material) only involves here the transverse shear
behavior, in the absence of any other macroscopic stress/strain
state. For the sake of brevity, one denes the shearing force per unit
of area Q applied to the lower and upper boundaries of the unit cell
(as represented in Fig. 8), which supposedly induces the equivalent
shear strain g. The macroscopic strain energycan then be expressed
as follows:
W macro ¼ 12
Q gð2H þ 2eÞ2L ¼ 2G*ðH þ eÞLg2 (5)
where G* stands for the effective transverse shear modulus (or
transverse shear stiffness per unit of area).
This general procedure is suitable for any sandwich congura-
tion. It is rst applied to the case of a classical sandwich (without
reinforcements) whose results are well-known, for validation
purposes. Next, use will be made of the same method in the more
complicated case of a reinforced sandwich.
3.1.3. Case of a non-reinforced sandwich
3.1.3.1. Calculation of the effective stiffness. The main stumbling
block for the determination of the transverse shear stiffness re-
mains the de
nition of a proper displacement
eld within the unit
Table 4
Experimental values of exural and transverse shear stiffnesses.
Density Flexural stiffness, D (N.mm2) Transverse shear stiffness, S (N)
x s x s
0 r /dm2 1.039 109 1.176 108 4682 14369 r /dm2 3.205 108 2.128 107 10,060 141138 r /dm2 6.171 108 7.753 107 12,025 248276 r /dm2 3.615 108 3.219 107 14,028 458415 r /dm2 4.502
108 2.311
107 22,184 629
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cell. In the case of a non-reinforced foam core (with subscript nr),
the following solution is eligible, in view of the prescribed dis-
placements on the lower and upper boundaries:
U nrð X ; Y Þ ¼ 0V nrð X ; Y Þ ¼ g X (6)
With such a displacement eld, the microscopic strain state is
uniform and the only non-zero strain component happens to be
3
XY ¼ g/2. Thus, the microscopic strain energy comes down to thesingle transverse shear term:
W nr ¼ 12
Z U
Gcg2ⅆ U ¼ 2GcðH þ eÞLg2 (7)
using the same dimensions for the unit cell as in the case of a
reinforced sandwich.
The energy equivalence principle leads to the following
expression of the effective transverse shear modulus:
G*nr ¼ Gc (8)It turns out that the effective modulus strictly coincides with the
shear modulus of the foam core. In the framework of the
rst-order
Fig. 6. Flexural and transverse shear stiffnesses for various volume fractions of reinforcements.
Fig. 7. Two-dimensional model for the analytical prediction of the transverse shear stiffness.
Fig. 8. Description of the parameters used for the calculation of the macroscopic strain
energy.
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shear deformation theory, it means that the so-called shear
correction factor is unity, which is a classical value for sandwich
structures.
3.1.3.2. In uence of the skin thickness. The previous result (Equa-
tion (8)) shall be valid only for innitely rigid and thin faces. In the
case of Napco® sandwiches, the skin modulus does not alter the
equivalent transverse shear stiffness, as it is suf
ciently high so thatthe skins do not deform under transverse shear. Conversely, the
skin thickness t is not small enough, when compared to the core
thickness, to be neglected in the expression of the transverse shear
stiffness, whence the need for a new expression incorporating the
inuence of the skin thickness. Several corrected estimates for the
transverse shear stiffness, taking into account the thickness of the
faces, have been proposed in the literature. All of them lead to
similar numerical values. Among these, the probably most natural
solution is retained here. The basic idea is depicted in Fig. 9.
At this stage, the prescribed displacements which are respon-
sible for the shear state in the unit cell have been enforced at the
interface between the foam core and the faces. If one considers now
the whole sandwich, including the rigid skins, the same displace-
ments applied to the external skin boundaries may lead to the same
stress/strain distribution in the foam core and consequently to the
same microscopic strain energy. Owing to the energy equivalence
principle, the macroscopic strain energy also remains unchanged.
However, whether you include the skins in the model or not, you
may dene the macroscopic shear strain in two distinct ways, due
to the non-negligible skin thickness (Nordstrand and Carlsson,
1997). In Fig. 9, gc (previously denoted by g) corresponds to the
case where the skin effects are neglected, and gs is the new shear
strain measure dened with the skins included. Thanks to simple
geometric considerations, one obtains the following relationship:
gs ¼ L
L þ t gc (9)
Combining the two expressions of the macroscopic strain en-
ergy (with and without the skins) leads to the following equation:
2G*ðH þ eÞLg2c ¼ 2G*corðH þ eÞðL þ t Þg2s (10)
since the integration volume differs between the two cases. The
new corrected expression of the equivalent transverse shear
modulus writes then:
G*cor ¼ L þ t L
G* (11)
In the case of a non-reinforced sandwich, one simply gets:
G*cornr ¼ L þ t L Gc (12)
which is consistent with the suggestion from Kelsey et al. (1958).
3.1.4. Case of a reinforced sandwich
The same procedure is henceforth applied for the determina-
tion of the equivalent transverse shear stiffness of a reinforced
sandwich. One has to nd again an eligible displacement eld in
the reinforced foam core, which is consistent with the prescribed
boundary conditions. Owing to the high similarity between the
numerical deformed shapes of Napco® sandwiches observed for
both through-thickness compression and transverse shear behav-
iors, the sought displacement eld here will be inspired by the
buckling mode response obtained in Laine et al. (2013) for the
same materials. Indeed, in both cases, the lower and upper skins
support an analogous relative horizontal displacement and the
other lateral boundaries must satisfy the same periodicity
conditions.
On one hand, along with the buckling problem, a sinusoidal
deformed shape is retained for the reinforcement. The same elds
can be used for both half-reinforcements in the unit cell, due to
the enforced periodicity conditions, so that the two half-beams
can be identied as a single entire beam. According to the
EulereBernoulli kinematics, the two following displacement elds
U r and V r are presupposed, standing respectively for the longitu-
dinal and transverse displacement components on the neutral
axis:
8<:
U rð X Þ ¼ 0
V rð X Þ ¼ Lg sinp X 2L
(13)
The macroscopic shear strain value g appropriately appears in
the expression of the transverse displacement V r so as to comply
with the relative horizontal displacement prescribed between the
two skins, namely 2d ¼ 2Lg. Based on this displacement eld, onecan dene the deformation eld (and therefore the stress distri-
bution) within the reinforcement. Owing to the kinematics, the
only non-zero strain (and stress) is the axial component:
3 XX ¼ YV r; XX (14)
Then, the corresponding strain energy writes as follows:
W r ¼ 12
Z LL
Z ee
E L 32
XX ⅆ Y ⅆ X ¼E Lp
4g2e3
48L (15)
still considering a unit depth.
On the other hand, solving the equilibrium equations for the
foam block with the proper boundary and continuity conditions
leads to the same displacement eld for the foam core as obtainedin the buckling analysis (Laine et al., 2013), except for the amplitude
which is here consistent with the displacement eld in the rein-
forcement:
8>>>>>>><>>>>>>>:
U cð X ; Y Þ ¼ LgK 1 cosh
pH
2L þ K 2H sinh
pH
2L
K 3 sinh
pY
2L þ K 2Y cosh
pY
2L
cos
p X
2L
V cð X ; Y Þ ¼ LgK 1 cosh
pH
2L þ K 2H sinh
pH
2L
K 1 cosh
pY
2L þ K 2Y sinh
pY
2L
sin
p X
2L
(16)
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with:
K 1 ¼ 2LpH coshpH
2L þ 4L2ð3 4ncÞ eH p2sinh
pH
2L
K 2 ¼ ep2 coshpH
2L 2Lp sinhpH
2L
K 3 ¼ 2LpðH þ 3e 4encÞcoshpH
2L eH p2 sinhpH
2L
(17)
This solution has been obtained with zero stress boundary
conditions on the lower and upper boundaries of the foam block (at
the interface with the two skins, respectively), instead of
displacement boundary conditions (namely, uniform horizontal
prescribed displacements), that would have naturally better rep-
resented the presence of the skins. In Laine et al. (2013), the present
choice allowed us to derive an explicit expression for the critical
load under through-thickness compression with a good accuracy, at
least for suf ciently high volume fractions
ðV fr
10%
Þ. Here, such a
simplifying assumption also makes it possible to obtain ananalytical solution for the transverse shear stiffness. It will also
most likely limit the validity domain to rather high volume frac-
tions of reinforcements.
In-plane strains and stresses may be then deduced from the
previous expressions of the displacement eld, eventually resulting
in the strain energy within the foam block:
W c ¼ 12
Z LL
Z H H
hðlc þ 2mcÞ
32
XX þ 32YY þ 2lc 3 XX 3YY
þ 4mc 32 XY iⅆ Y ⅆ X (18)where lc and mc are the Lame coef cients of the core material.
The total microscopic strain energy of the reinforced foam core
is obtained by simply adding the two strain energies related to the
reinforcement (W r) and the foam block (W c). Based on this strain
energy value, it is possible to deduce the effective transverse shear
modulus G*r by using the same denition of the macroscopic strain
energy as in the previous case of a non-reinforced foam core. This
equivalent transverse shear stiffness may only be used for partic-
ularly thin faces. Otherwise, when the skin thickness is no longer
negligible, it can be upgraded by multiplying by the thickness ratio
(L þ t )/L.For all these quantities, explicit solutions have been obtained
using Maple symbolic calculation software, but they are too
cumbersome to be presented as closed-form expressions.
3.2. Numerical nite element resolution of the transverse shear
stiffness using a 3D unit cell
Three-dimensional numerical nite element computations
have been performed, using Abaqus software, in order to sup-
plement the previous analytical solutions. A hexagonal arrange-
ment of the reinforcements has been retained, according to the
experimental patterns. Thus, the overall mechanical response of
the composite reinforced foam core (and consequently of the
sandwich) is transversely isotropic. A 3D unit cell is only
considered, for ef ciency purposes, but here including the skins
and the full material properties of the transversely isotropic re-
inforcements. The geometry of the plane-parallel unit cell and
the associated nite element mesh (made up of 20-noded hex-
ahedral elements with reduced integration) are depicted in
Fig. 10. The boundary conditions are prescribed in a similar way
than in the previous 2D analysis. Periodicity conditions are
enforced on the lateral faces of the unit cell in both directions.
Lastly, the bottom and top faces of the sandwich cell are sub-
jected to different uniform horizontal displacements (d ¼ 6 mmand d ¼ 6 mm, respectively), so as to produce a pure macroscopicshear state in the unit cell (see Fig. 11 for the loading conditions
and the deformed shape of the 3D unit cell under pure transverseshear).
With such a numerical model involving the same geometric
and material parameters together with the same boundary
conditions as in the previous section, it is possible to re-use the
same procedure for the calculation of the effective transverse
shear stiffness based on the energy equivalence principle.
Whereas the expression for the macroscopic strain energy re-
mains almost the same (W macro ¼ 1/2VG*g2 where V is the unitcell volume without considering the skins), the microscopic
strain energy is hereafter estimated thanks to a numerical
computation (by the way, the contribution of the skins in the
total strain energy is proved to be negligible). This 3D numerical
solution is then expected to be far more accurate than the 2D
analytical one. Indeed, there are not here as many simplifyingassumptions concerning the skins and the corresponding
boundary conditions, the geometry and the full material prop-
erties, and the global kinematics. Finally, the obtained effective
modulus G* is replaced by the corrected value G*cor through the
same thickness ratio as before, if necessary.
Fig. 10. Model for the 3D
nite element computations.
Fig. 9. Correction for the transverse shear stiffness due to the skin thickness.
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4. Numerical and experimental validation
4.1. Numerical computation of the transverse shear stiffness of a
beam under simple bending
Both previous analytical and numerical approaches are based on
several simplifying assumptions and must be therefore validated
using numerical computations on complete structures. In this
section, nite element calculations are performed, involving 3D
heterogeneous beams under simple bending, for validation pur-
poses. The post-processing of the associated numerical resultsnaturally leads to a new estimate for the transverse shear stiffness.
4.1.1. Basic principle
The method used here for the determination of the transverse
shear stiffness from a full 3D numerical simulation has already been
used in many studies (see, forexample, Buannic et al. (2003)).Letus
consider a cantilever sandwich beam, built-in at the left-hand side
and submitted to a transverse force at the right-hand side. In the
present case, the whole beam is represented which consists of
several unit cells placed end to end. A nite element model is
implemented and allows one to determine the mean deection
along the beam. This deection can be viewed as the sum of a
deection due to the bending moment effects (only depending onthe exural stiffness) and another deection due to the transverse
Fig. 11. Transverse shear of the 3D unit cell.
Fig. 12. Three-dimensional beam model for the numerical validation.
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shear effects. The rst deection can be analytically expressed,
using the in-plane effective moduli of the corresponding sandwich
structure, so that the second sought deection can be obtained by
deducting the
rst one from the total de
ection which has beennumerically evaluated. This deection is supposed to be linear with
respect to the longitudinal coordinate along the beam, in such a
way that the transverse shear stiffness can be nally derived from
the associated slope.
4.1.2. Numerical model
The complete beam consists of 10 unit cellsplaced side tosidein
the longitudinal direction. Each unit cell is dened in the same way
as before, in terms of geometry and materials. The common sur-
faces between adjacent unit cells are merged in order to build a
whole structure. The global model is made up of 20-noded hex-
ahedral elements with a coarser mesh as before, for ef ciency
purposes (see Fig.12). The best compromise leads to a minimum of
34,560 elements (461,127 d.o.f.) for the most reinforced case (415 r /
dm2) and 61,440 elements (816,855 d.o.f.) in the opposite case
where the dimensions of the beam are the most important (69 r /
dm2).
The following boundary conditions are then applied. At the left
end of the beam ( x ¼ 0), the whole section is xed. Conversely, atthe right end section ( x ¼ l), the transverse applied force is uni-formly distributed onto the skin surfaces only, with an arbitrary
amplitude. The loading and boundary conditions are depicted in
Fig. 13, together with the deformed shape of the total beam.
4.1.3. Determination of the transverse shear stiffness
The estimation method of the transverse shear stiffness is
illustrated in Fig. 14 and can be summarized as follows.
The previous numerical computation allows us to plot the
average deection along the neutral axis of the beam (wtotal). Thistotal deection can be divided into two parts:
wtotal ¼ wflex þ wshear (19)
The rst part is due to the bending moment and writes analytically
as follows:
wflex ¼ T
D
x3
6 l x
2
2
(20)
where T is the downward transverse force (counted as positive).
The exural stiffness D is also evaluated analytically. The most
general expression for the exural stiffness of a sandwich beam
takes the following form (Zenkert, 1992):
D ¼ 23
E rcbL3 þ 2
3E sb
t 3 þ 3t 2L þ 3tL2
(21)
where b is the width of the beam and E rc stands for the equivalent
longitudinal modulus of the reinforced core.
In practice, the core modulus is substantially below the skin
modulus, even in the case of a reinforced foam core (see
Guilleminot et al. (2008) for more details). In addition, the core
thickness is much higher than the skin thickness, so that the gen-
eral expression in Equation (21) can be simplied in the following
way:
D ¼ 2E sb
t 2L þ tL2
(22)
The values obtained when using Equation (22) are found tobe in
very good agreement with numerical results deriving from a peri-
odic homogenization nite element analysis, for all the sandwich
panels considered.
Since the exural stiffness is known, it is possible to plot thecorresponding deection wex, which is shown to be small in
comparison with the total deection. The difference between them,
namely the deection wshear due to transverse shear effects, is thus
predominant and appears to be linear, as depicted in Fig. 14. It
proves essential to determine the associated transverse shear
stiffness, using the slope of the linear deection obtained by linear
regression. This slope happens to be the macroscopic shear strain g
and is related to the transverse force T as follows:
g ¼ T S
(23)
so that the transverse shear stiffness writes
S ¼ T g
(24)
and the equivalent transverse shear modulus is
G* ¼ T 2gbðL þ t Þ (25)
4.2. Comparison between analytical, numerical and experimental
results
Finally, the two simplied methods for the calculation of the
transverse shear stiffness, respectively analytical using a 2D unit
cell and numerical using a 3D unit cell, are validated by comparison
Fig. 13. Three-dimensional sandwich beam under simple bending.
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to both numerical results obtained with a complete beam under
simple bending and experimental 3-point bending test results. All
the transverse shear stiffnesses by unit width (expressed in
N.mm1) are plotted in Fig. 15 for the four Napco® sandwichesconsidered as well as the non-reinforced panel.
First, considering the non-reinforced sandwich, the four
methods give rise to very similar values. Thus, the reference
panel (without reinforcements) makes it possible to check the
consistency between the different analytical, numerical and
experimental approaches. The different hypotheses formulated in
each case are conrmed, at the very least in the absence of
reinforcements.
Regarding now the reinforced sandwiches, one can notice the
very good agreement between both numerical approaches (the
relative error has an average value of 2.4% and does not exceed 6%).
The general method based on the energy equivalence principle is
thereby validated. The analytical solution is also shown to be in
good accordance with the reference numerical results, at least for
the three higher densities of reinforcements. In these three cases,
the relative error is around 6% on average. The main difference
between the 2D analytical and 3D numerical unit cell models (other
than the dimension) lies in the consideration of the skins in the
latter. Additional numerical nite element computations have been
performed on 2D unit cells, including the skins. The results almost
Fig. 14. Relative inuence of the exural and transverse shear stiffnesses on the deection of a beam under simple bending.
Fig. 15. Comparisons between transverse shear stiffnesses obtained with different analytical, numerical and experimental methods.
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coincide with the ones obtained with 3D unit cells, what proves the
reliability of the assumptions made when switching from the 3D
conguration to the 2D one.
The inuence of the skin thickness on the transverse shear
stiffness value has already been discussed. Apart from that, the
extreme rigidity of the skins allows one to apply the same
displacement boundary conditions on the core/skins interfaces as
enforced in practice onto the external skin boundaries. Instead,
approximated boundary conditions have been retained in the
analytical approach, that do not match the real conditions in the
presence of skins, in order to simplify the kinematics used in the
analytical resolution and make possible the achievement of a
closed-form expression for the transverse shear stiffness. This
choice of boundary conditions is not so detrimental, as soon as the
volume fraction of reinforcements is about 10% or higher. On the
contrary, with the smallest density, a large discrepancy is noticed
(about47.5% of relative error) between the analytical andnumerical
predictions, what points out the limitations of the present analyt-
ical model. When brous reinforcements are much less numerous
and very distant from each other, the skin effect becomes more and
more apparent in the transverse shear deformation shape and thus
in the corresponding stiffness.
Lastly, the experimental results are confronted to the analyticaland numerical solutions. It is dif cult to quantify and explain the
discrepancies observed between the experimental measurements
and the theoretical predictions, since the analytical and numerical
models are somewhat idealized. A perfect architecture is retained
in the modeling, without any imperfection, and uniform volume
fractions and mechanical properties are considered throughout the
sandwich structure. Furthermore, in the most reinforced case, the
reason why the discrepancy is so high might be the following. Due
to the numerous reinforcements, small cracks may appear in the
foam along the needle track. Then, during the infusion process,
resin may spread into the cracks and strengthen the foam core and
therefore the whole sandwich. Despite all that and independently
of the unavoidable imperfections in practice, the different ap-
proaches presented above provide good estimations for the trans-verse shear stiffnesses of most of the Napco® sandwiches tested in
this study.
5. Conclusions
The Napco® technology is a patented process that transversally
strengthens the foam core of a sandwich structure with ber
yarns taken from facings. In this study, we investigated the po-
tential of such a reinforced sandwich in its transverse shear
behavior, which plays a signicant role in the simple bending
response of a sandwich structure. First, an analytical solution for
the transverse shear stiffness has been proposed. A 2D model was
conveniently dened in which only a unit cell of the reinforced
foam core was considered, due to the material periodicity. Thereinforcements were assumed to behave like EulereBernoulli
beams whereas the foam core was modeled as a 2D continuous
solid, without considering any simplied deformation eld. The
transverse shear stiffness was derived from the energy equiva-
lence principle, by comparing the microscopic strain energy
induced by a macroscopic pure shear loading (using the appro-
priate boundary conditions) and the macroscopic strain energy of
the sought effective material. The reinforcements naturally
strengthen the foam core, especially in its transverse shear
behavior. However, the coupling effects between the re-
inforcements and the skins (due to the manufacturing process)
make the solution here far more complicated than the classical
one based on the equivalent shear modulus of the reinforced core
viewed as a 3D material with an in
nite thickness.
The same method was developed in the context of a 3D unit cell
(including the skins) using nite element calculations. This
approach, though numerical, is an ef cient way to obtain the
transverse shear stiffnesses of sandwich structures, even in more
complicated cases than orthogonal reinforcements (for instance,
with other distributions and/or orientations of the through-
thickness reinforcements), where analytical solutions are no more
available. Numerical computations were also performed on a
complete beam under simple bending, for validation purposes.
Experimental 3-point bending tests have been performed for
ve different sandwich panels (with various densities of re-
inforcements, including a non-reinforced case). Comparisons be-
tween analytical/numerical predictions and experiments were
discussed and clearly showed the accuracy of the theoretical
models. In particular, the expression obtained for the transverse
shear stiffness is proved to be suitable to properly predict the
transverse shear behavior of such sandwiches, as long as the vol-
ume fraction of reinforcements is suf ciently high, say greater than
10%.
Acknowledgments
The authors are indebted to the French Ministry of Economy,
Finance and Industry (NWC-X project, Contract no. 09 2 90 6242)
for its nancial support.
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