Analytical, numerical and experimental study of the transverse shear behavior of a 3D reinforced...

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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



6/15

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.



7/15

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



8/15

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.



9/15

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)



10/15

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.



11/15

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.



12/15

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.



13/15

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.



14/15

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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