Behavior of laminated shell composite with imperfect contact between the layers

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Behavior of laminated shell composite with imperfect contact between the layers

David Guinovart-Sanjuán, Raffaella Rizzoni, Reinaldo Rodríguez-Ramos, Raúl Guinovart-Díaz, Julian Bravo-Castillero, Ransés Alfonso-Rodríguez, Frédéric

Lebon, Serge Dumont, Igor Sevostianov, Federico Sabina

To cite this version:

David Guinovart-Sanjuán, Raffaella Rizzoni, Reinaldo Rodríguez-Ramos, Raúl Guinovart-Díaz, Julian Bravo-Castillero, et al.. Behavior of laminated shell composite with imperfect contact between the layers. Composite Structures, Elsevier, 2017, 176, pp.539-546. �10.1016/j.compstruct.2017.05.058�.

�hal-01694039�

Behavior of laminated shell composite with imperfect contact between the layers

David Guinovart-Sanjuán^a,⇑, Raffaella Rizzoni^b, Reinaldo Rodríguez-Ramos^c, Raúl Guinovart-Díaz^c, Julián Bravo-Castillero^c, Ransés Alfonso-Rodríguez^a, Frederic Lebon^d, Serge Dumont^e, Igor Sevostianov^f, Federico J. Sabina^g

aDepartment of Mathematics, University of Central Florida, 4393 Andromeda Loop N, Orlando, FL 32816, USA

bDipartimento di Ingegneria, Università di Ferrara, Via Saragat 1, 44122 Ferrara, Italy

cDepartamento de Matemática, Universidad de La Habana, San Lazaro y L, 10400 La Habana, Cuba

dAix-Marseille University, CNRS, Centrale Marseille, LMA, 4 Impasse Nikola Tesla, CS 40006, 13453 Marseille Cedex 13, France

eUniversity of Nîmes, Place Gabriel Péri, 30000 Nîmes, France

fDepartment of Mechanical and Aerospace Engineering, New Mexico State University, P.O. Box 30001, Las Cruces, NM 88003, USA

gInstituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Delegación Álvaro Obregón, Apartado Postal 20-726, 01000 CDMX D.F., Mexico

The paper focuses on the calculation of the effective elastic properties of a laminated composite shell with imperfect contact between the layers. To achieve this goal, first the two-scale asymptotic homogenization method (AHM) is applied to derive the solutions for the local problems and to obtain the effective elastic properties of a two-layer spherical shell with imperfect contact between the layers. The results are com- pared with the numerical solution obtained by finite elements method (FEM). The limit case of a laminate shell composite with perfect contact at the interface is recovered. Second, the elastic properties of a spherical heterogeneous structure with isotropic periodic microstructure and imperfect contact is ana- lyzed with the spherical assemblage model (SAM). The homogenized equilibrium equation for a spherical composite is solved using AHM and the results are compared with the exact analytical solution obtained with SAM.

1. Introduction

Composite materials have emerged as the materials of choice in various branches of industry – aerospace, automotive, sport, etc. – for increasing the performance and reducing the weight and cost.

However, defects induced during the manufacturing process or accumulated due to environmental and operational loads lead to the reduction in the mechanical performance and material strength and are recognized as a general problem in this type of composites,[1]. Most typically, such defects can be found at the interfaces between the layers creating an imperfect contact condi- tion,[2,3].

The effect of the contact imperfectness on elastic properties of composites attracted the attention of researchers from 1970’s, [4,5]. In[6–9], the authors obtained analytical expressions for the effective elastic properties of rectangular fibrous composites with

imperfect contact between the matrix and the reinforcement. On the other hand, the multilayered curvilinear shell structures have received special attention in the last years. In [10–13] several mathematical methods have been used to derive analytical expres- sion for the elastic properties of laminated shell composites. As a particular case, in[14], the expression of the effective coefficients for a curvilinear shell composite with perfect contact at the inter- face is obtained.

Several mathematical models and techniques have been devel- oped to evaluate the elastic properties of curvilinear laminated shell composites with imperfect contact at the interfaces. In papers as[10,12,15–20], the assemblage model, finite elements method and the two-scale asymptotic homogenization method are used to derive in one way or another the effective behavior of the elastic properties of particular composites with imperfect contact at the interface.

In this paper a spherical shell structure is studied. In[21]the authors considered the effect in the elastic properties of a spherical laminated shell composite under the influence of stress and strain

⇑ Corresponding author.

E-mail address:davidgs1601@gmail.com(D. Guinovart-Sanjuán).

distributions for two composites with perfect and imperfect con- tact at the interface using AHM and SAM. In[21]the imperfect con- tact condition is modeled considering a thin interphase between the layers of the composite, i.e., a three phase composite is used in the analysis. Here the same effect in the spherical shell structure is studied except that the imperfect contact condition is modeled as a linear spring type and the FEM is used to validate the results obtained via AHM and SAM. The purpose of studying this kind of spherical structures obeys to the development and application of mathematical methods for the study of the cornea and other sim- ilar soft tissues.

In the present paper, first the AHM technique is used to evalu- ate the elastic properties of a two-layer laminated shell with imperfect contact of the spring type at the interface. The general analytical expressions of the effective coefficients are derived from the solution of the local problem. We focus on a two-layer spheri- cal shell subjected to internal pressure assuming that the layers are isotropic. To validate the model, the effective coefficients of the spherical structure are compared with FEM calculations. The elas- tic fields (stresses, strains and displacements) are also compared with ones calculated by the method of Bufler[22]for the analysis of a spherical assemblage model (SAM). The approach is based on the transfer matrix method and yields closed form calculation of the equivalent elastic properties of a periodically laminated hollow sphere made of alternating layers of isotropic elastic materials with imperfect contact. The effective displacement, radial and hoop stresses computed via AHM are compared with the elastic fields calculated by FEM and SAM.

2. The linear elastic problem

A curvilinear elastic periodic composite is studied. The geome- try of the structure is described by the curvilinear coordinates sys- tem x¼ ðx1;x2;x3Þ 2XR³, where X¼X1[X2 is the region occupied by the solid, it is bounded by the surface@X^¼R1[R2, whereR1\R2¼£; Xaa^¼1;2 are the elements of the composite, separated by the interfaceC^{e. In}X, the stressr^{and strainare}

related through the Hooke’s law,r^ij¼Cijklkl, whereC^ijkl are the components of the elastic tensorC. For a linear periodic solid struc- ture, the elastic tensor CCðx;yÞis regular with respect to the slow variable xand Y-periodic with respect to the fast variable y¼x=e^{2Y, where 0}<e1 characterizes the periodicity of the composite andYdenotes the periodic cell.

The linear elastic equilibrium equation for a curvilinear lami- nated shell composite with imperfect contact (spring type) at the interface is

r^ij_;j^þCⁱjkr^kjþCjk^jr^ikþfi¼0; in X; ð1Þ subject to boundary conditions,

ui¼u⁰_i on R1; r^ijnj¼Sⁱ on R2; ð2Þ and interface contact conditions,

r^ijnj¼K^ij uj

; r^ijnj

¼0; on C^e; ð3Þ

wherefg_;j¼_@x^@

jfgis the derivative with respect to the slow curvi- linear coordinate,Cⁱjk are the Christoffel’s symbols of second type,

½

½ ¼ ðÞ^ð2Þ ðÞ^ð1Þdenotes the jump at the interfaceC^e,njis the nor- mal vector to the corresponding surface (R2; C^e),Kijare the compo- nents of a matrixK, that characterizes the imperfect contact inC^e and the order ofKisOðe¹Þ. Replacing the Hooke’s law and consid- ering the Cauchy’s formula,^{ij¼ ðu}i;jþu_j;iÞ=2, the equations(1)–(3) can be rewritten for the displacement vector function[14].

3. Homogenization of two-layer laminated shell composites with imperfect contact

In order to obtain an equivalent problem to(1)–(3)with not fast oscillating coefficients, the two–scales Asymptotic Homogeniza- tion Method (AHM) is used. The general expression of the trun- cated expansion is given by

u^ð_m^eÞ¼v^mþe ^{N^p}mv^pþNlkmvl;k

h i

þoðe^{Þ; ð4Þ}

wherev^mv^{mðxÞ; Nlk}mN^lk_mðx;yÞis the local function for the first order approach, N^lk_ð1Þmðx;yÞ is Y-periodic, where Y¼ ½0;1 and N^^pm¼ C^plkN^lk_m [14]. Substituting the expansion (4) into the Eqs.

(1)–(3)a recurrent family of problem is obtained for different pow- ers of the small parametere^.

Considering a two-layer laminated shell composite with isotro- pic components, i.e.

C^ijkl¼kðyÞg^ijg^klþl^{ðyÞgljgkiþgilgkj; ð5Þ}

where½g^ijis the metric tensor of the coordinatesðx₁;x₂;x₃Þand kðyÞ ¼ k1 y2 ½0;c^Þ

k2 y2 ðc^;1

; l^{ðyÞ ¼} l1 y2 ½0;c^Þ l2 y2 ðc^;1

;

where the layers are transversal to the axisx3, the local problem is obtained fore¹

@=@y C ^i3lkþC^i3m3@N^lkm=@y

¼0 on Y¼ ½0;c^{Þ [ f}c^{g [ ð}c;1; ð6Þ with interface conditions given by the expressions

C^i3lkþC^i3m3@N^lk_ð1Þm=@y

h i

¼ ð1Þ^aþ1K^ijhhN^lk_ð1Þjii

on C^{e¼ fy¼}cg;

ð7Þ C^i3lkþC^i3m3@N^lkð1Þm=@y

h i

h i

¼0 on C^{e¼ fy¼}cg;

ð8Þ where the parametera^¼1;2 denotes the layer.

Substituting(5)into the local problem(6)the following expres- sion is obtained@²N^lk_ð1Þm=@y²¼0. Therefore, the local function has the expression

N^lk_m¼ A^lkð1Þ_m yþB^lkð1Þ_m ; y2 ½0;cÞ;

A^lkð2Þ_m yþB^lkð2Þ_m ; y2 ðc;1:

(

ð9Þ

Considering the periodicity of the functionsN^lk_mand@N^lk_m=@ythe following linear equations system is obtained from Eq.(8)

C^i3lkð1ÞþC^i3m3ð1ÞA_m^lkð1Þ

h i

¼ KimA^lkð1Þ_m ðc^{Þ þAlkð2Þ}m ð1c^{Þ; ð10Þ}

C^i3lkð2ÞþC^i3m3ð2ÞA_m^lkð2Þ

h i

¼ KimA^lkð1Þ_m ðc^{Þ þAlkð2Þ}m ð1c^Þ; ð11Þ where the supraindexða^Þa^¼1;2 refers to each layera. The linear problem(10) and (11)related to the variablesA^lkð_m^aÞcan be solved using classical methods and therefore the local functions are obtained.

Applying the average operator to the coefficient of the parame- tere⁰, the homogenized coefficients are obtained and the general expression is given in the Eqs. (12)–(18) of[14]. The effective coef- ficients for a two-layer laminated shell composite with isotropic layers and imperfect contact condition at the interface have the general analytic expression

bh^ijkl¼DC^ijklE

þV1C^ijm3ð1Þ@N^klð1Þm

@y þV2C^ijm3ð2Þ@N^klð2Þm

@y : ð12Þ

whereV_ais the volume of the layers of the composite and the local functions@N^klð_m^aÞ=@yhave the expression

@N^klð_m^aÞ

@y ¼ C^q3klðaÞK^qnV_bþC^q3n3ðbÞ

þC^q3klðbÞK^qnV_b C^r3m3ð1ÞC^r3n3ð2ÞþC^r3m3ð1ÞK^rnV2þC^r3n3ð2ÞK^rmV1

; ð13Þ

forb¼1;2 andb–a^.

The homogenized problem is obtained from the Eqs. (19) and (20) of[14].

3.1. Comparison of the effective coefficients for composites with perfect and imperfect contact

In order to illustrate the influence of the imperfect contact on the effective coefficients, a two layer rectangular laminate shell is considered (seeFig. (1)). The layers of the composite are isotropic and the materials are stainless steel (Young’s modulusE1¼206:74 GPa, Poisson ratio,m1¼0:3) with volumeV1, represented byY1on Fig. (1)and aluminum (Young’s modulusE₂¼72:04 GPa, Poisson ratio, m^2¼0:35) with volume V2¼1V1, represented byY2 on Fig. (1). The matrixKcharacterizes the imperfect contact at the interfaceC ^(see Fig. (1)) and it has nonzero components K¹¹¼ K²²¼l=e ^{and K33¼ ð}l^þ2kÞ=e ^where l^¼k¼1 and

e^{¼ ½0}:001;0:01;0:05. The effective coefficients for the imperfect contact case are calculated using the Eq.(12)and they are com- pared with the coefficients obtained usingEq. (26)of[14]for per- fect contact case (see Fig. (2)). The convergence of effective

coefficients for the imperfect contact case can be appreciated as

e^{!0, i.e.Kij! þ1.}

For the case ofV1¼0 orV1¼1, we are in presence of a mono- element multi-layered structure, i.e. it is a structure with several layers but each layer is the same element. It can be studied the per- fect and the imperfect contact between the layers of these struc- tures. For the perfect contact case, the mono-element multi- layered structure are considered as an homogeneous structure.

But a different situation is obtained for the imperfect contact case.

Due to the imperfection, the structures are studied as a heteroge- neous composite.

It can be seen inFig. (2), for the perfect contact case, that the aluminum and stainless steel elastic moduli are reached for V1¼0 orV1¼1, respectively, as a whole homogeneous material.

In particular, forV1¼1 the valueC³³³³¼h³³³³¼278:3038 GPa is attained for stainless steel. In the case of imperfect contact the imperfection is modeled as a linear spring distributed across the common interfaceCbetween the layers. WhenK¹¹takes the values 1000, 100 and 20, the elastic constanth³³³³is equal to 254.6779, 144.3725 and 49.3587 GPa, respectively. Thus, the main effect of the imperfect contact is to degrade the composite’s integrity: the value of the elastic constants diminishes accordingly, so a weak- ened composite is obtained. In other words, the particular case of V1¼0 (V1¼1) reports, for the imperfect contact case, the effective coefficients of the a structureXwhere the unit cell has two layers Y1andY2made by aluminum (stainless steel), but with degrading due to the imperfection. It is very important to remark the fact that the AHM reports different values of the effective coefficients for mono-element multi-layers structures with perfect and imperfect contact at the interface; i.e. due to the imperfection, the effective coefficients of the structures with imperfect contact are weaker.

4. The spherical assemblage model with imperfect contact (SAM)

In this section, a spherical assemblage model consisting ofNdif- ferent thin elastic layers is studied using the transfer-matrix method.

Y

Y Y

Fig. 1.Rectangular structureX, unit cellY, the two elementsY1andY2with the corresponding interfaceC.

Fig. 2.Comparison of the effective coefficientsh¹¹¹¹;h³³³³;h¹¹³³andh¹³¹³of a composite with perfect contact at the interface usingEq. (26)of[14]and the effective coefficients for a composite with imperfect contact using Eq.(12).

The transfer-matrix method is a classic approach[23]. Here, we first review its application to a periodic laminated hollow sphere proposed in[22]and next extend the obtained results to the case of imperfect contact between the layers.

The spherical assemblage has internal radiusRi, external radius Reand thicknessh¼2t. The inner surfacer¼Riis loaded by a con- stant pressure

r^rrðRiÞ ¼ þp; ð14Þ

whereas the external surfacer¼Reis traction free

r^{rrðReÞ ¼}0: ð15Þ

Thek-th layer comprised between the radiiRk1andR_k, is char- acterized by the thicknessh_kand it is made of linear elastic homo- geneous and isotropic material with Young modulus and Poisson ratio Ek and mk, respectively. According to the transfer-matrix method[22], the radial stressr^rr and displacement ur at radius Rk1 of the laminated sphere can be related to the radial stress and displacement at radius Rk through the field-transfer matrix Tkof the layerk:

rrrðRkÞ EurðRkÞ=h

¼Tk rrrðR_k1Þ EurðR_k1Þ=h

; ð16Þ

Tk:¼ 1akh bkh ckh 1dkh

; ð17Þ

with

ak:¼2 1 m^k

ð1m^kÞ

kk

Ri; ð18Þ

bk:¼ 2 ð1m^kÞEkh

E kk

R²_i; ð19Þ

ck:¼ 1 2m²k

ð1m^kÞ

Ekh

E kk; ð20Þ

dk:¼ 2m^k

ð1mkÞ kk

Ri

: ð21Þ

Here,kk¼hk=his the thickness ratio of thek-th layer, andE and h denote a reference modulus of elasticity and thickness, respectively. Applying(16)N-times for the layered hollow sphere made of layers in perfect contact, we have

r^rrðReÞ

EurðReÞ=h

¼S r^rrðRiÞ

EurðRiÞ=h

; S:¼TNTN1. . .T1: ð22Þ

HereSis thetransfer matrix systemfrom radiusRitoRe, linking the two elastic states at the boundaries of laminated sphere.

Sustituting (16) into (22) and considering only the terms of order zero and one inh, Bufler’s result is obtained:

S¼IhMþoðhÞ; M:¼ a b c d

; ð23Þ

with

a:¼X^N

k¼1

2 1 mk

ð1m^kÞ

kk

Ri; ð24Þ

b:¼X^N

k¼1

2 ð1m^kÞEkh

E kk

R²_i; ð25Þ

c:¼X^N

k¼1

1 2m²k

ð1m^kÞ

EkE

h kk; ð26Þ

d:¼X^N

k¼1

2m^k

ð1m^{kÞkRki: ð27Þ}

To extend these results to a laminated sphere with imperfect contact between the layers, an arrangement of springs is artificially considered at the spherical surface between adjacent layers. In this

case, a jump of the radial displacement has to be taken into consid- eration, [24–27]. In particular, the continuity of radial stress is assumed and the following linear relation between the radial stress and the jump of radial displacement is imposed at the radiusR_k:

r^rrðRþkÞ EurðR^þ_kÞ=h

" #

¼K^_k r^rrðRkÞ EurðR_kÞ=h

; K^_k:¼ 1 0

Ee^k=ðhð2lkþkkÞÞ 1

; ð28Þ where the matrixK^_k characterizes the imperfect contact provided by thek-th layer of springs, withk¼1;2;. . .N1;ek1 is a small length parameter accounting for its thickness and 2lkþkkits elas- ticity coefficient.

Now, the presence of springs are considered, therefore the transfer matrix systemS(cf.(23)) modifies in order to incorporate the matricesK^k,

~S¼TNK^N1TN1. . . ^K1T1: ð29Þ

Substituting(16) and (28)into(29), one obtains the new matrix system for a spherical assemblage made ofNlayers with imperfect contact. It can be shown that

~S¼IhMþoðhÞ; M~ :¼ a b

~c d

; ð30Þ

witha;banddgiven again by Eqs.(24), (25) and (27), respectively, and

~c:¼cþX^N1

k¼1

E h

e^k

ð2lkþkkÞ

¼X^N

k¼1

1 2m²k

ð1mkÞ

EkE h kkþX^N1

k¼1

E h

e^k

ð2lkþkkÞ: ð31Þ The case of a periodic laminate made by repeatingntimes a group ofNlayers is now considered. Asnincreases, the thicknesses hkandekmust decrease withnin order to keep the total thicknessh of the fixed laminate. In particular, it is assumed thathk¼Kkh=n andek¼nkh=n, withKk;k¼1;2;. . .N, the thickness ratio of thek- th layer inside the group, and analogously nk1; k¼1;2;. . .N1, for thek-th layer of springs. The matrix system for the hollow sphere with homogenized properties following Bufler is calculated as

limh!0

1 h~SⁿI

¼lim

h!0

1

hðTNK^N1TN1. . . ^K1T1ÞⁿI

¼ hMi; ð32Þ with

hMi:¼ hai hbi hci hdi

; ð33Þ

hai:¼X^N

k¼1

2 1 m^k

ð1m^kÞ

Kk

Ri; ð34Þ

hbi:¼X^N

k¼1

2 ð1m^kÞEkh

E Kk

R²_i ; ð35Þ

hci:¼X^N

k¼1

1 2m²k

ð1m^kÞ

Ekh E KkþX^N1

k¼1

E h

nk

ð2lkþkkÞ; ð36Þ hdi:¼X^N

k¼1

2m^k

ð1m^{kÞKRki: ð37Þ}

As an example, consider the homogenization of a periodic lam- inate made by repeating a group of two layers under imperfect contact. In this case, the ‘‘unit cell” of the laminate isTþKT, with Tthe transfer matrices of two layers whose elasticity constants are denotedE;m^{, andK^}the matrix characterizes imperfect con- tact. The thickness ratios K are assumed to coincide Kþ¼K¼1=2. Thus,

hai:¼1 Ri

ð12m^Þ

ð1mÞ þð12m^þÞ

ð1mþÞ

; ð38Þ

hbi:¼ h ER²_i

E

ð1mÞþ Eþ

ð1mþÞ

; ð39Þ

hci:¼E h

ð12mÞð1þmÞ

2Eð1m^{Þ þð12}mþÞð1þmþÞ

2Eþð1m^{þÞ þð2}l^nþkÞ

; ð40Þ hdi:¼1

Ri

m

ð1mÞþ m^þ

ð1mþÞ

: ð41Þ

As a final result, note that comparing the latter relations with the matrix system of a transversely isotropic homogeneous elastic sphere (cf.[22, Eqs. (17)–(19)]) one obtains the equivalent material parametersE=ð1m^Þ;E⁰;m⁰of the homogenized sphere consisting of a two–layers laminate with imperfect contact,

E

ð1m^{Þ¼ 2ð1E}mÞþ Eþ

2ð1mþÞ

; ð42Þ

1 E⁰¼

m

ð1mÞþ_ð1^mþ_mþÞ

2

E

ð1mÞþ_ð1^Eþ_mþÞ

þð12m^Þð1þm^Þ

2Eð1mÞ þð12m^þÞð1þm^þÞ

2E_þð1mþÞ þ n

ð2l^{þkÞ; ð43Þ} m⁰

E⁰¼ m ð1mÞþ_ð1^mþ_m

þÞ

E ð1mÞþ_ð1^Eþ_m

þÞ

: ð44Þ

The state of stresses and displacements of the equivalent trans- versely isotropic hollow sphere subjected to the boundary condi- tions(14) and (15)can be obtained by substituting the relations (42)–(44)intoEq. (52)of[22], which are

urðrÞ ¼ pR_e

R_i Re

k11

^R_Re^{i k21} h Ehci

r Re

k2

Rihdi þk2

r Re

k1

Rihdi þk1

2 64

3

75; ð45Þ

r^{rrðrÞ ¼ p}

R_i Re

k11

_R^Ri_e ^k21 r Re

k11

r Re

k21

" #

; ð46Þ

rhhðrÞ ¼ p=2

R_i Re

k11

_R^Ri_e ^k21

ð1þk1Þ r Re

k11

ð1þk2Þ r Re

k21

" #

;

ð47Þ with

k1;2¼1=2 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8C1 p

; ð48Þ

C¼1=2R²_ihbihci ð1RihdiÞRihdi

: ð49Þ

5. The finite element method

In this section, a numerical method based on the finite element is proposed to solve problem(6)–(8). Since this technique is quite standard, it is rapidly outlined here.

For the sake of simplicity, we denoteY¼ ½0;c^{ÞandYþ¼ ð}c;1.

Then, choosing a test functionv, which can be discontinuous across the interfaceC^e, multiplying the equilibrium Eq.(6) by this test function and integrating amongY, one obtains after integration by parts

Z

Y

C^i3lkþC^i3m3@N^lk_m

@y

@v

@y

!

dyþ C^i3lkþC^i3m3@N^lk_m

@y

!

ðcÞv^{ðcÞ ¼0}

ð50Þ

Z

Yþ

C^i3lkþC^i3m3@N^lk_m

@y

@v

@y

!

dy C^i3lkþC^i3m3@N^lk_m

@y

!

ðc^þÞv^{ðcþÞ ¼0:}

ð51Þ Now, adding these two equations, using the continuity of C^i3lkþC^i3m3@N_@y^lkm across the interface, (see Eqs.(7) and (8)), a weak formulation of the problem can be written as follows

Z

Y

C^i3lkþC^i3m3@N^lkm

@y

@v

@y

!

dyþK^im½½N^lk_ð1Þm½½v ^¼0

Finally, using standard finite element on each sub domain, and a

‘‘flat” finite element onC^e, that have all its nodes onC, the first ones related toYand the other ones toYþ, it is possible to write a rigidity matrix of this problem, that is invertible, with standard error estimation (see[28]or[29]for more details).

Due to finite element discretization, the integrals (see formula (12), for example) for the computation of effective coefficients are substituted by sums over element contributions.

6. Numerical results

6.1. Rectangular shell composite

Here a flat shell structure is considered in order to validate for- mula (12). The unit cell is composed of two layers of isotropic materials like aluminum and reinforced carbon fiber. The proper- ties are Young’s modulus equal to 150 Gpa and Poisson’s ratio 0.3. The volume fraction of both materials are set to 0.5. The matrix Kthat characterizes the imperfect contact takes the same values as given in Section 3.1. In Table 1 the effective coefficients are obtained using formula (12). The same results are also attained using the formulas for perfect contact homogenized constants (1.19) [4, Chap. 5] and (9.9)[30]. As a particular case, from the model used here of imperfect contact, Eq.(12)gives the same val- ues that formula (1.30) of[4, Chap. 5]with perfect contact (i.e. with very largeK¹¹) for a flat laminated structure.

6.2. Spherical shell composite

In contrast to[21]here the imperfect condition is taken as a dis- tribution of linear springs acting at the interface between the two isotropic materials except that now a hollow two layer spherical shell composite is considered with isotropic components.

The inner and outer radius are denoted by Ri¼R0t and Re¼R0þtrespectively, wheret¼R0=10. The spherical coordinate systemðh;u;rÞis used to describe the geometry of the structure, [12]. The layers of the composite are transversal to the coordinate r. The inner surfacer¼R_iof the heterogeneous body is loaded by a constant radial stress, (14), and the external spherical surface r¼Reis traction free. The materials used in the composite have the following elastic properties

l¼10lþ; l^¼exlþ; ð52Þ

m¼0:2; mþ¼0:35; m^¼0:3; ð53Þ wherex2 ½3;3, the index ‘‘” denotes the inner layer, the index ‘‘

+” the outer layer, non-indexed constants are theKparameters. For this particular case, the matrixKis diagonal and has components K¹¹¼K²²¼l=e^{andK33¼ ð}kþ2l^Þ=e^.

To obtain the effective elastic properties of the presented spher- ical shell composite, the two above described approaches, AHM and SAM are used.

As a first step, the local functions@N^lk_m=@y are computed via AHM(13)and FEM. The variational formulation(50) and (51)of