Composite Structures 100 (2013) 79–88

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Composite Structures

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Analytical modeling of indentation of composite sandwich beam

P. Navarro a,⇑, S. Abrate b, J. Aubry a, S. Marguet a, J.-F. Ferrero a

a Université de Toulouse, Institut Clément Ader (ICA), UPS, INSA, Mines Albi, ISAE -118 Rte de Narbonne, 31062 Toulouse, Franceb Department of Technology, College of Engineering, Southern Illinois University, Carbondale, IL 62901-6603, USA

a r t i c l e i n f o a b s t r a c t

Article history:Available online 4 January 2013

Keywords:Sandwich structuresAnalytical modelingNon-linear behaviourIndentation

0263-8223/$ - see front matter � 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruct.2012.12.017

⇑ Corresponding author. Tel.: +33 5 6133 9570; faxE-mail address: Pablo.Navarro@isae.fr (P. Navarro)

This paper deals with static indentation of sandwich beams with a foam core. An analytical model is pre-sented assuming that the specific response of foams in compression can be assimilated to an elastic–per-fectly plastic behavior. The elastic part is represented using Vlasov’s model. The displacements and stresscalculated with this two parameters model are compared with results given by Winkler’s theory andFinite Element Method. Vlasov’s idealization gives more accurate results that Winkler’s model. A com-plete study of the influence of the parameters of Vlasov’s model is performed. Then, plasticity is addedto the model to represent the non-linear response of the core. The load–displacement response of sand-wich beams under static indentation is calculated and compared to experimental and Finite Elementresults. A good correlation is found. The size of the area where the foam is crushed given by the developedmodel is in good agreement with Finite Element analysis.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

This article deals with the analytical modeling of indentation ofcomposite sandwich beam with composite face sheets and foamcores. Composite sandwich structures are widely used in the fieldof transportation (helicopter blades, ship’s hull, etc.) for their lowweight and high in plane and flexural stiffness. During their life,these structures can be subjected to localized loadings, like im-pacts of birds or hailstones. Even though a visual examination ofthe loaded surface may reveal little damage, significant damagemight exist and cause important reductions in the strength of thestructure. That is why significant research efforts have been fo-cused on the study of impacts on sandwich structures and thestudy of the local deformations in the contact zone is an importantcomponent of it.

A comprehensive review article of Abrate [1] discusses impactson composite sandwich structures. The contact laws for sandwichstructures differ significantly from that for a monolithic laminates.The local deformation in the contact zone consists of the localindentation of the top face sheet and in a large part of the deforma-tion of the core material under that face sheet. Many experimentalstudies have been carried out [2–7] to understand the behavior ofsandwich beams under quasi-static and dynamic loadings. Failuremodes have been identified. For low impact energy levels, thebeams deform within the elastic range. For higher energy levels,

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four basic modes are identified: core crushing, facesheet buckling,delamination within the facesheet and debonding between thefacesheet and the core. The contact force-indentation relation forquasi-static and low velocity impacts is non-linear, owing to thespecific response of the foam core or honeycomb core in compres-sion [8,9].

In order to predict the overall response of the sandwich struc-tures, several models have been proposed. The Finite ElementMethod is often used [10–13] to study the static and dynamicindentation of sandwich structures. The developed models are ableto well represent the static and dynamic indentation response ofsandwich panels. The computed failure modes match experimentalobservations for several impact energies, impact angles andspecimen configurations. However, setting up such models iscomplicated and calculation times are high.

Many analytical models have been developed to predict inden-tation failure in sandwich structures. The comprehensive review ofWang et al. [14] discusses on the various idealizations developed tomodel beams and plates on elastic foundations. The modeling ofthe core generally follows one of two main strategies. The firstone consists of modeling the core using linear elastic springs. Thesesprings are independent so shearing effects are not taken intoaccount [15,16]. This model has been modified to introduce inter-actions among the spring elements by incorporating a thin elasticmembrane subjected to a constant tension (Filonenko-Borodichmodel [17]); an elastic beam or plate (Heteneyi model [18]); or abeam or a plate that only undergoes transverse shear deformations(Pasternak model [19]). One of the major disadvantages of thesemodels is that the identification of the parameters of the

x

z

P

k

Fig. 1. Beam resting on one parameter Winkler elastic foundation.

80 P. Navarro et al. / Composite Structures 100 (2013) 79–88

foundation is very difficult. Indeed, as shown by Gdoutos et al. in[15], the stiffness value of the foundation has to be deduced fromexperimental results. In addition, Heteneyi [18] shows that thesprings’ stiffness expression depends on the loading conditions.

The second modeling strategy consists of idealizing the core as athree-dimensional continuous elastic solid. From this approach,Vlasov [20] developed a two parameter model of the core assumingthat there is no horizontal displacements and that vertical dis-placements can be represented by a single shape function. Themain advantage of this method is that, contrary to one parameteridealizations, the stiffness parameters can be easily deduced frommaterial data, regardless of the loading conditions. Moreover, nospecific experimental identification is necessary. This idealizationhas been used to model indentation on sandwich beams [21] andon sandwich panels [22], but only within the elastic range.

To reproduce the non-linear behavior of a sandwich beam, Zenk-ert et al. [23] presents a modified Winkler model in which the coreis represented by a perfect elastic plastic material law. Experimen-tal load–displacement curves for indentation tests are matched bythe model, but the model contains rather rough approximations.Firstly, transverse shear stresses, that play an important role inthe damage of sandwich structures, are not accounted for. Secondly,the foundation modulus is geometry dependant and can only be ex-tracted by quite complex experimental procedures.

In the present paper, the core is modeled using Vlasov’s theory,by a two parameter elastic–perfectly plastic foundation. This mod-eling strategy presents three main advantages. Firstly, the model isable to reproduce the linear and non-linear response of the sand-wich beam. Secondly, transverse shear in the foam is taken into ac-count for a better accuracy of the stress state in the sandwichbeam. Thirdly, the parameters of the model can be found only frommaterial data: no specific experimental identification is necessary.

The calculated behavior of the beam before the crushing of thecore is compared with results given by the Winkler theory and Fi-nite Element Method. Contrary to Winkler’s model, the deflectionof the skin and the stresses in the beam calculated with Vlasov’smodel correlate well with FEM results. The influence of the param-eters of the Vlasov modeling on the response of a sandwich beam isstudied. Then, the non-linear behavior of the core is taken into ac-count to predict the size of the crushed zone and the contact forceresponse. Results are compared to Finite Element and experimen-tal results. A good correlation is found for the indentation load–deflection curve and for the size of the area where the foam iscrushed.

2. Linear elastic response

Local loading on sandwich beams can be studied analytically byconsidering the core as an elastic foundation. In this section we re-call the results obtained with Winkler’s model and develop a twoparameter model for the foundation using Vlasov’s approach.

2.1. Winkler’s model

The one parameter Winkler foundation model is widely used tostudy local indentation of sandwich beams. For the sake of com-pleteness, Winkler’s formulation is presented here. This model rep-resents the core as a system of independent and linear elasticsprings (Fig. 1).

The pressure–deflection relation is given by:

p ¼ kw ð1Þ

where p is the pressure response of the core, k the stiffness of thesprings and w the deflection of the facesheet. For a localized loadingthe governing equation of the skin is:

EsIsd4w

dx4 þ kw ¼ 0 ð2Þ

where EsIs is the flexural rigidity of the skin. The parameter k is theonly parameter that represents the behavior of the whole core. Itsvalue must be determined with care. In the models described in[23], k is roughly estimated as:

k ¼ Ecbtc

ð3Þ

where Ec is the core Young’s modulus, b the width of the beam andtc the core thickness. The stiffness due to shearing effects is ne-glected and this estimation may be inappropriate in some cases.

The value of k can also be identified experimentally. Gdoutoset al. [15] gives an estimated expression of k, function of the skinand core moduli (respectively Es and Ec) and of the facing thick-ness ts:

k ¼ 0:64Ec

ts

Ec

Es

� �1=3

ð4Þ

The solution of Eq. (2) is:

wðxÞ ¼ e�kxðA sinðkxÞ þ B cosðkxÞÞ þ ekxðC sin ðkxÞ þ D cos ðkxÞÞ

with k ¼ k4EsIs

� �1=4

ð5Þ

The constants A, B, C and D are determined from boundaryconditions:

� the skin remains bounded at infinity.

limx!1

wðxÞ ¼ 0 and limx!1

dwdxðxÞ ¼ 0 ð6Þ

� the slope at the origin is zero (symmetry).

dwdxð0Þ ¼ 0 ð7Þ

� the shear force at the origin is half the force P applied.

� P2¼ Vð0Þ ¼ �EsIs

d3w

dx3 ð0Þ ð8Þ

Thus, the elastic solution can be written as:

wðxÞ ¼ P

8k3EsIs

e�kxðsin ðkxÞ þ cos ðkxÞÞ ð9Þ

2.2. Vlasov’s model

The most important weakness of Winkler idealization is theindependence of the springs. A first estimation of their stiffnessmust be done experimentally and the applied loads get localized.

Table 1Material and geometric characteristics of the modeled beam.

Es (GPa) 100 ts (mm) 1Ec (MPa) 200 tc (mm) 30tc 0.15 b (mm) 1

P. Navarro et al. / Composite Structures 100 (2013) 79–88 81

To take into account the vertical shearing of the core, Vlasov devel-oped a two parameter foundation model derived from a continuumapproach and based on the variational principle. The foundation isconsidered to be elastic, isotropic and homogeneous. The totalstrain energy of the skin/core system is:

P¼ 12

Z L=2

�L=2EsIs

d2w

dx2

!2

dxþ 12

Z þ1

�1

Z 0

�tc

bðrxex þrzez þ sxzcxzÞdzdx

ð10Þ

where L is the length of the sandwich beam; rx, rz, sxz, ex, ez and cxz

are the stresses and the corresponding strains in the core.The vertical displacement in the core is assumed to be the prod-

uct of w(x), the displacement of the facesheet, by an appropriatemode shape /ðzÞ. The horizontal displacements of the core areneglected:

vðx; zÞ ¼ wðxÞ/ðzÞ with /ð0Þ ¼ 1 and /ð�tcÞ ¼ 0uðx; zÞ ¼ 0

�ð11Þ

Thus:

ex ¼@u@x¼ 0; ez ¼

@v@z¼ wðxÞ @/ðzÞ

@z; cxz ¼

@u@zþ @v@x¼ @w@x

/ðzÞ

ð12Þ

Using the constitutive relations of plane elasticity and thestrains given by Eq. (12), Eq. (10) yields:

P ¼ 12

Z L=2

�L=2EsIs

d2w

dx2

!2

dxþ 12

Z þ1

�1

Z 0

�tc

bEcð1� tcÞ

ð1þ tcÞð1� 2tcÞw2ðxÞ

�

� d/dz

� �2

þ Ec

2ð1þ tcÞdwdx

� �2

/2ðzÞ!

dzdx ð13Þ

Minimizing the function of P with respect to w, the governingequation for the skin/core system is obtained:

EsIsd4w

dx4 � k1d2w

dx2 þ k2w ¼ 0 withk1 ¼ Ec bð1�tc Þ

ð1þtcÞð1�2tcÞR 0�tc

d/dz

� �2dz

k2 ¼ Ec b2ð1þtc Þ

R 0�tc

/2dz

8><>:

ð14Þ

where tc represents the Poisson ratio of the core. k1 is the shearfoundation parameter and k2 is the compression foundation param-eter. According to Vallabhan [24], the mode shape /ðzÞ is found byminimizing P with respect to /:

/ðzÞ ¼ sinh cð1� z=HÞð ÞsinhðcÞ with c2 ¼ tc

1� 2tcð ÞRþ1�1 ðdw

dx Þ2dx

2ð1� tcÞRþ1�1 w2dx

ð15Þ

where c is the core deformation parameter. It is characteristic of thesandwich structure and it can be determined using an iterative pro-cedure [25]. One initial value of c is chosen and the parameters k1

and k2 are calculated (Eq. (14)). Then, the value of c is calculatedfrom Eq. (15) and from the deflection found in the previous step.With this new value of c, k1 and k2 are reestimated from Eq. (14).The iteration process is stopped when the difference between twosuccessive values of c is lower than a given tolerance (10�3 forinstance).

The general solution of Eq. (14) is:

wðxÞ ¼ Aeffiffiffis1p

x þ Beffiffiffis2p

x þ Ce�ffiffiffis1p

x þ De�ffiffiffis2p

x

withs1 ¼ k1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

1�4k2EsIs

p2EsIs

s2 ¼ k1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

1�4k2EsIs

p2EsIs

8><>: if k2

1 � 4k2EsIs > 0;

wðxÞ ¼ eaxðA cosðbxÞ þ B sinðbxÞÞ þ e�axðC cosðbxÞ þ D sinðbxÞÞ

witha ¼ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1þ2

ffiffiffiffiffiffiffiffiffik2EsIs

pEsIs

r

b ¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�k1þ2

ffiffiffiffiffiffiffiffiffik2EsIs

pEsIs

r8>>><>>>:

if k21 � 4k2EsIs < 0 or ð16Þ

wðzÞ ¼ effiffikp

xðAxþ BÞ þ e�ffiffikp

xðCxþ DÞ with

k ¼ k1

2EsIsif k2

1 � 4k2EsIs ¼ 0

A, B, C and D are four unknowns determined from the boundaryconditions (Eqs. (6)–(8)); so the solution is:

wðxÞ ¼ P2EsIsðs1 � s2Þ

e�ffiffiffis2p

xffiffiffiffiffis2p � e�

ffiffiffis1p

xffiffiffiffiffis1p

� �with

s1 ¼ k1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

1�4k2EsIs

p2EsIs

s2 ¼ k1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

1�4k2EsIs

p2EsIs

8><>: if k2

1 � 4k2EsIs > 0;

wðxÞ ¼ P

4EsIsða2 � b2Þe�ax cosðbxÞ

aþ sinðbxÞ

b

� �with

a ¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1þ2

ffiffiffiffiffiffiffiffiffik2EsIs

pEsIs

r

b ¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�k1þ2

ffiffiffiffiffiffiffiffiffik2EsIs

pEsIs

r8>>><>>>:

if k21 � 4k2EsIs < 0 or ð17Þ

wðzÞ ¼ P2k1

e�ffiffikp

x xþ 1ffiffiffikp

� �with k ¼ k1

2EsIsif k2

1 � 4k2EsIs ¼ 0

2.3. Example

The indentation of a sandwich beam is modeled using Winkler’smodel, Vlasov’s model and Finite Element Method. The materialand geometric characteristics of the beam are given Table 1. The gi-ven values are typical values for a unidirectional composite skinand a polyurethane foam core. The foam and skin thicknesses aretypical for structures like helicopter blades.

The software Patran/Nastran is used for the finite element linearcalculation. A planar modeling of the sandwich beam is performed.The skin is modeled with unidirectional beam elements and thefoam core is represented with two-dimensional membrane ele-ments. The material behavior is linear elastic for the facesheetand for the core. The mesh size is 0.5 mm. The indentation load Pis applied at the middle, on one node of the skin. The bottom nodesare embedded. This modeling contains about 400 beam elementsand 8500 membrane elements.

The applied load value is P = 1000 N. The deflection of the skinand the stresses in the skin/core system are given Fig. 2.

The deflection of the skin and the stress in the skin calculatedwith Vlasov’s model correlate well with the FEM results. The max-imal deflection is 8.7% smaller than the FEM results (Fig. 2a). Thedeflection shape of the skin is not correctly predicted by Winkler’smodel: the maximal deflection is 20.8% smaller than the FEM re-sults and the skin presents a non-realistic positive deflection forx > 16 mm. The maximal flexural stress in the skin calculated withVlasov’s model is 9.4% smaller than the FEM calculation (Fig. 2b).

(b)(a)

(d)(c)

Fig. 2. (a) Deflection of the skin; (b) flexural stress in the skin; (c) shear stress in the foam; (d) compressive stress in the foam.

82 P. Navarro et al. / Composite Structures 100 (2013) 79–88

Shear stresses and compressive stresses in the core are obtainedfrom Eqs. (12), (15), and (17). The shear stress in the foam is max-imal at 7 mm from the loading point (Fig. 2c). The value calculatedwith Vlasov’s model is 6.7% greater than the FEM result. Shearstresses in the foam cannot be calculated with Winkleridealization.

The compressive stress in the foam calculated with Vlasov’smodel is more accurate than that calculated with Winkler’s model(Fig. 2d). Nevertheless, the maximal compressive stress is 33%smaller than that obtained using FEM. This difference can be ex-plained by the fact that compressive strains and stresses are calcu-lated from the derivative of the shape function /ðzÞ (Eq. (12)), thatmay not be accurate enough near the loading point.

Winkler’s idealization is not able to predict correctly the deflec-tion shape and the stresses of the foam. Independent springs do notrepresent well the behavior of the core far from the loading point.Vlasov’s idealization is more accurate and no experimental identi-fication of the parameters k1 and k2 is necessary. Thus, for the pres-ent study, this two parameter foundation modeling is used.

3. Parametric study of the Vlasov’s model

A parametric study of the Vlasov’s model is carried out. Thebehavior of the skin/foam system is studied using a dimensionlessmethod. The influence of the geometric and material parameters isobserved.

3.1. Dimensional analysis

The formulation of the skin/core indentation problem is non-dimensionnalized. The equation of motion becomes:

d4 �wd�x4 þ B

d2 �wd�x2 þ �w ¼ 0 ð18Þ

The boundary conditions are:

limx!1

�wð�xÞ ¼ 0; limx!1

d �wd�xð�xÞ ¼ 0;

d �wd�xð0Þ ¼ 0;

d3 �wd�x3 ð0Þ ¼ 1 with

w ¼ C � �w�x ¼ A � x

�and A ¼ k2

EsIs

� �1=4

; B ¼ k1ffiffiffiffiffiffiffiffiffiffiffiffik2EsIs

p !

;

C ¼ P

2k3=42 ðEsIsÞ1=4

!ð19Þ

Thus, general solutions for the indentation of sandwich beamscan be found by varying a single parameter B. Fig. 3 shows thedimensionless deflection and curvature for B varying between 0and 100:

B = 0 in the dimensionless model is equivalent to k1 = 0 inthe initial model which corresponds to Winkler’s model. Thecurvature of the deflection decreases when B increases. Indeed, Bquantifies the importance of the term d2 �w

d�x2 in the simplified equationof motion (Eq. (18)). It represents the part of the bending stiffnessof the skin due to the presence of the core. Coefficient B can beexpressed as:

B ¼ tcðsinhðcÞ � coshðcÞ � cÞ2ts sinhðcÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6Ectcð1� 2tcÞ

Estsð1� t2c ÞðsinhðcÞ coshðcÞ þ cÞ

s

ð20Þ

According to [22], it can be assumed that the parameter c doesnot vary significantly for the materials and configuration studied.Thus Eq. (20) shows that the coefficient B depends on the ratiostcts

, EcEs

and the value of tc. The deformation of the skin is more local-ized as the ratio tc

tsor Ec

Esdecreases. The width of the beam does not

have any influence on the dimensionless deflection shape and onthe curvature.

-0,7

-0,6

-0,5

-0,4

-0,3

-0,2

-0,1

0

0,1

0 2 4 6 8 10 12 14

B=0

B=1B=5

B=10B=20

B=50B=100

x

w

-0,7

-0,6

-0,5

-0,4

-0,3

-0,2

-0,1

0

0,1

0,2

0 2 4 6 8 10 12 14

B=0

B=1B=5

B=10B=20

B=50B=100

x

d²w/dx²

(a)

(b)Fig. 3. (a) Dimensionless deflection of the skin; (b) dimensionless curvature of the skin.

P. Navarro et al. / Composite Structures 100 (2013) 79–88 83

3.2. Influence of the geometric and material parameters

Here, the sandwich beam configuration is that presented inTable 1. The geometric and material parameters are variedindependently to see their influence on the behavior of the skin/core system.

First, the ratio tcts

is varied from 1 to 1000 while material proper-ties are kept constant (Table 1). That corresponds to a variation of Bfrom 0.05 to 1100. The deformed shape and the maximal deflectionof the skin are given Fig. 4:

When tcts< 100; the maximal deflection increases rapidly with tc

ts.

When tcts> 100 the maximal deflection tends to a limit that corre-

sponds to the theoretical maximal deflection for an infinite corethickness (Fig. 4). The maximal stress in the skin and the maximalshearing stress in the core reach a maximum value for tc

ts¼ 20 and

decrease slowly when tcts

become high (Fig. 5).The value of tc is varied from 0 to 0.45. That corresponds to a

variation of B from 6.8 to 2.4. The deformed shape is given Fig. 6.When tc varies from 0 to 0.25, the deflection shape remains thesame. For tc > 0.25, the deflection depth decreases when tc in-creases; it is 50% lower for tc = 0.45 than for tc = 0.

Fig. 7 shows that tc does not have a significant influence on thestress in the skin and that the shear stress in the core slowly de-creases with tc. The typical value of tc for polymer foams is be-tween 0.1 (Rohacell foams) and 0.3 (Divinycell foams). Thus, in

our case of study, the value of tc has not a significant influenceon the response of the sandwich beam.

The value of Es is varied from 1000 MPa (typical value for poly-meric materials) to 200,000 MPa (typical value for the steel). Thatcorresponds to a variation of B from 100 to 5. The deformed shapesare given Fig. 8.

The deflection depth increases when the modulus of the skindecreases. The size of the zone where the skin is deflected doesnot vary with the value of Es .The maximal strain in the skin andthe maximal shear stress in the core are given Fig. 9. These two val-ues decrease when Es increases.

The value of Ec is varied from 50 MPa to 500 MPa. That corre-sponds to a variation of B from 2.9 to 9. The deformed shape is gi-ven Fig. 10a. The value of the Young modulus of the core has astrong influence on the deflection depth. It increases quickly whenEc decreases.

Fig. 10b shows that the maximal deflection, given by Eq. (17), isproportional to E�3=4

c . This observation is in agreement with theexpression of w(x) given by Winkler’s theory (Eq. (9)). Fig. 11shows also that as the stiffness of the core increases, the stress inthe skin and the shear strain in the core decrease.

In this part, the influence of all the parameters of Vlasov’s modelhas been studied. The thicknesses and the moduli of the core and ofthe skin have a strong influence on the deflection of the skin and onthe stresses in the core and in the skin. On the contrary, the value

-7

-6

-5

-4

-3

-2

-1

0

1

0 5 10 15 20 25 30 35 40 45 50

x (mm)

w (m

m)

tc/ts = 1 tc/ts = 2,5tc/ts = 5 tc/ts = 7,5tc/ts = 10 tc/ts = 20tc/ts = 30 tc/ts = 50tc/ts = 100

-8

-7

-6

-5

-4

-3

-2

-1

00 100 200 300 400 500 600 700 800 1000900

tc/ts

w (m

m)(a) (b)

Fig. 4. (a) Deflection of the skin; (b) maximal deflection versus tc/ts.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 10 20 30 40 50 60 70 80 90 100tc/ts

max

imal

str

ess

in th

e sk

in (M

Pa)

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100

tc/ts

max

imal

she

ar s

tres

s fo

am (M

Pa)

(a) (b)

Fig. 5. (a) Maximal stress in the skin versus tc/ts; (b) maximal shear stress in the foam versus tc/ts.

-6

-5

-4

-3

-2

-1

00 5 10 15 20 25 30 35 40 45 50

x (mm)

w (m

m) Nu=0 Nu=0.05

Nu=0.15 Nu=0.25

Nu=0.30 Nu=0.35

Nu=0.45

Fig. 6. Deflection of the skin.

84 P. Navarro et al. / Composite Structures 100 (2013) 79–88

of the Poisson ratio of the core does not have a significant influencefor the configurations studied here.

4. Non-linear response

4.1. Approach and assumptions

The analytical model presented previously is based on theassumption that the core behavior remains linear elastic. This is

not true for foam cores under high loads. Indeed, the stress–strainbehavior of foam cores in compression can be idealized as consist-ing of an initial linear portion, followed by a plateau correspondingto the crushing of the foam cells and then a strong increase ofstress corresponding to densification (Fig. 12).

Considering that densification happens only at very high strains(50–75%), the assumption is made that the foam core can be char-acterized by an elastic perfectly plastic deformation model undercompressive loading. The foam undergoes a linear elastic deforma-tion up to an elastic stress limit rc characteristic of the material. Asthe indentation load increases, a part of the core with length 2l iscrushed (Fig. 13).

After the elastic limit is exceeded, the stress remains constant.Since crushing corresponds to the failure of the foam at a micro-scopic level, when the foam crushes, shearing is neglected. Thegoverning equation for the skin/crushed-core system is then:

for x < l EsIsd4wc

dx4 þ rcb ¼ 0 ð21Þ

where wc is the deflection of the skin in the crushed zone. With thefollowing boundary conditions:

dwc

dxð0Þ ¼ 0 and

d3wc

dx3 ð0Þ ¼ �P

2EsIsð22Þ

The solution of Eq. (21) is:

wðxÞ ¼ 1EsIs

rcP24

x4 � P12

x3 þ C1

2x2 þ C2

� �ð23Þ

0

5

10

15

20

25

30

35

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45Nu

max

imal

she

ar s

tres

s fo

am (M

Pa)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45Nu

max

imal

str

ess

in th

e sk

in (M

Pa)

(a) (b)

Fig. 7. (a) Maximal stress in the skin versus tc; (b) maximal shear stress in the foam versus tc.

-7

-6

-5

-4

-3

-2

-1

00 5 10 15 20 25 30 35 40 45 50

x (mm)

w (m

m)

Ep=1000Ep=10000Ep=50000Ep=100000Ep=200000

s ssss

Fig. 8. Deflection of the skin.

P. Navarro et al. / Composite Structures 100 (2013) 79–88 85

where C1 and C2 are two unknown constants. A general solution for theskin deflection is obtained by combining the solutions in the crushedarea (Eq. (23)) and in the elastic area (Eq. (16)). Given that the deflec-tion is assumed to be zero far from the loading point, the values of theconstants A and B in Eq. (16) are zero. Four continuity equationsexpressing the continuity of deflections, slopes, bending momentsand shear forces are used to find the four unknowns C, D, C1 and C2:

wcðlÞ ¼ wuðlÞdwcdx ðlÞ ¼

dwudx ðlÞ

d2wc

dx2 ðlÞ ¼ d2wu

dx2 ðlÞd3wc

dx3 ðlÞ ¼ d3wu

dx3 ðlÞ

ð24Þ

where wu is the deflection in the uncrushed zone.

0

0,2

0,4

0,6

0,8

1

1,2

0 20000 40000 60000 80000 100000 120000 140000 160000 180000 200000

Es (MPa)

max

imal

str

ain

in th

e sk

in

(a)

Fig. 9. (a) Maximal strain in the skin versus Es; (b

Finally, an iterative method is used to find the length l of plasticzone. The idea is to respect the continuity of the compressive strainfield in the foam between the crushed and the uncrushed region.First, at the beginning of the iteration n, the deflection of the wholeskin is calculated for an initial value of the length of the plasticzone ln. Then, the compressive strain in the uncrushed regioneuz(x,z) is deduced from (Eq. (12)) for x = ln and z = 0. Finally thisvalue is compared to the strain limit ec of the foam. If |euz(ln,0) � ec|is bigger than a predetermined tolerance value, the size of thecrushed zone is adjusted:

lnþ1 ¼ ln þ bðeuzðln;0Þ � ecÞ ð25Þ

Using this method, the global deflection of the skin and thestresses in the skin/core system are calculated for a given indenta-tion force P.

4.2. Validation of the modified Vlasov’s model

Here, static indentation tests are carried out to control the accu-racy of the developed model. The indented sandwich core is aRohacell A51 polymeric foam. The samples are fabricated withtwo plies of glass–epoxy woven fabrics (7781/913). The sandwichbeams are 200 mm long and their characteristics are given Table 2.They are indented by a 6 mm diameter cylinder. These tests areconducted in a 10 kN INSTRON compression testing machine at adisplacement rate of 1 mm/min (Fig. 14). The vertical displacementof the indentor and the applied load are acquired. The specimen ispositioned on a table in order to prevent the global bending of thesandwich beam. Four tests have been performed. They are veryreproducible.

0

10

20

30

40

50

60

70

80

0 20000 40000 60000 80000 100000 120000 140000 160000 180000 200000

Es (MPa)

max

imal

she

ar s

tres

s fo

am (M

Pa)

(b)

) maximal shear stress in the foam versus Es.

-18

-16

-14

-12

-10

-8

-6

-4

-2

00 5 10 15 20 25 30 35 40 45 50

x (mm)

w (m

m)

Ec=50

Ec=100

Ec=200

Ec=500

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

00 0,01 0,02 0,03 0,04 0,05 0,06

Ec(-3/4)

w (m

m)(a) (b)

Fig. 10. (a) Deflection of the skin; (b) maximal deflection versus Ec-3/4.

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0 50 100 150 200 250 300 350 400 450 500

Ec (MPa)

max

imal

str

ain

in th

e fo

am

0

2000

4000

6000

8000

10000

12000

14000

0 50 100 150 200 250 300 350 400 450 500

Ec (MPa)

max

imal

str

ess

in th

e sk

in (M

Pa)

(a) (b)

Fig. 11. (a) Maximal stress in the skin versus Ec; (b) maximal shear strain in the foam versus Ec.

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8Strain

Stre

ss (M

Pa)

Elasticbehavior

Crushing

Densification

Fig. 12. Typical stress–strain curve for the compression of a polyurethane foam (80 kg/m3).

86 P. Navarro et al. / Composite Structures 100 (2013) 79–88

Results from the present analytical solutions are compared withexperimental results and results from a finite element analysis. Thesoftware RADIOSS is used for this non-linear finite element calcu-lation. The skin is represented by two-dimensional compositeshells elements. The core is modeled with standard 8-node three-

dimensional elements. The mesh size is smaller in the loaded area.The edge length varies from 0.5 mm to 5 mm. The bottom nodesare embedded. The cylindrical indentor is defined as anunderformable body which displacement is imposed. This model-ing contains about 500 composite shell elements and 7500 volume

x

z

PUndamagedfoam

Crushedzone

l

Fig. 13. A schematic of the state of the foam in a sandwich beam during anindentation.

Table 2Sandwich configuration.

Es (MPa) 19,400 ts (mm) 0.7Ec (MPa) 20 tc (mm) 20tc 0.15 b (mm) 20rc (MPa) 0.6

Sample

Cylindricalindentor

Fig. 14. Presentation of the static indentation tests on composite sandwich beam.

Fig. 15. Indentation load versus deflection.

Fig. 16. Size of the crushed zone versus indentation load.

Fig. 17. Size of the crushed zone versus P/rc.

P. Navarro et al. / Composite Structures 100 (2013) 79–88 87

elements. An elastic–plastic constitutive law has been used toreproduce the specific behavior of the foam in compression. Whenthe compressive stress is lower than rc the response is linear. Afterthe elastic limit is exceeded, the crushing of the foam is modeledwith a perfectly plastic material law. The densification is assumedto be negligible and is not represented.

The load–deflection responses are given Fig. 15. The response islinear until a critical load Pcr corresponding to the beginning of thecrushing of the foam. Then, the response becomes non-linear: thereaction of the foam core decreases because of crushing. The re-sponse given by Vlasov’s idealization correlates well with the Fi-nite Element results. The load–deflection curve calculated withVlasov’s model presents a discontinuity when the behavior be-comes non-linear. Indeed, when the foam is not crushed, the com-pressive stress given by Vlasov’s model in the loading area isunderestimated. Thus the critical load Pcr is overestimated. Then,when the crushing starts, the compressive stress in the crushedzone is set to the elastic limit rc so the calculated indentation loadand the deflection are well represented. The two models correlatesthe test results in the elastic range. The experimental response inthe non-linear part is 8% lower. Indeed, the two models only takeinto account the crushing of the foam, but other damages (like ma-trix cracking) can occur.

The two calculated sizes of the crushed zone l are compared.Fig. 16 shows the evolution of l function of the indentation load.As the length of the crushed zone could not be measured withaccuracy during the tests, experimental data are not plotted.

In the FEM model, the crushed zone extends with the successivecrushing of the foam elements; that is why the plotted curve pre-sents jumps, corresponding to the discretization of the core. Forboth models, the size of the crushed zone increase linearly withthe indentation load. There is a good correlation between thetwo calculations. The maximal difference between the two crushedzone’s sizes is 9%.

Finally, the size of the crushed zone l calculated from Vlasov’smodel versus P/rc, for rc varying from 0.6 MPa to 2 MPa, is givenFig. 17. The three plotted variations of the size of the crushed zonefit the same straight line. The indentation load P necessary to crushthe core along a length l is proportional to the compressivestrength of the foam rc .

88 P. Navarro et al. / Composite Structures 100 (2013) 79–88

5. Discussion and conclusion

This article presents an analytical approach for the modeling ofa localized loading on a sandwich beam. The specific behavior ofthe foam is represented by an elastic–perfectly plastic model.The elastic response of the foam is based on a two parameters elas-tic foundation model.

For the elastic response to indentation loads, the model is vali-dated by a comparison with a Finite Element analysis. It is thencompared to the widely-used one-parameter Winkler elastic foun-dation model. Contrary to Winkler’s model, Vlasov’s approachadopted here provides more accurate results for the deflection ofthe skin and for the state of stress in the whole beam. For thenon-linear response to indentation loads, the developed analyticalmodel is compared to a Finite Element calculation and to experi-mental results. The calculated load–deflection curve correlateswell with test results. The size of the crushed zone calculated withthe model is validated by the Finite Element results.

The major contributions of this article are firstly a comprehen-sive parametric study of the little-used Vlasov’s foundation modeland secondly the introduction of the non-linear behavior of poly-urethane foams in order to represent the crushing phenomenon.Contrary to existing analytical non-linear models of indentationon sandwich beams, shear stresses in the foam core are accountedfor. This allows having an accurate state of stress within the wholestructure. Moreover, the identification of the parameters of thedeveloped model is simpler than for other existing models: theirvalues are deduced directly from material characteristics and nospecific experimental identification is needed.

Whereas several simplifications used in the model, this methodprovides a simple, time efficient modeling that give very accurateresults.

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