www.elsevier.com/locate/compstruct

Composite Structures 78 (2007) 551–559

The behavior of angle-ply laminated cylindrical shells withviscoelastic interfaces in cylindrical bending

Wei Yan a,b,*, J. Ying c, W.Q. Chen a,b

a Department of Civil Engineering, Zhejiang University, Hangzhou 310027, PR Chinab State Key Lab of CAD & CG, Zhejiang University, Hangzhou 310027, PR China

c Department of Mechanical Engineering, Zhejiang University, Hangzhou 310027, PR China

Available online 27 December 2005

Abstract

The behavior of a simply supported angle-ply laminated cylindrical shell in cylindrical bending with viscoelastic interfaces is studied.The Kelvin–Voigt model is adopted to represent the character of interfaces. State-space formulations are developed based on the exactelasticity equations, and in particular, a variable substitution technique is used to derive the state equations with constant coefficients.Since the behavior of this structure under static loading is time-dependent, the power series expansion technique is used to approximatethe variations of physical variables with time. The response for a laminated shell with viscous interfaces is also investigated as a particularcase. Results show that the displacements as well as the maximum stresses in the panel increase rapidly with time. Thus, the imperfectbonding properties should be considered carefully in the design of laminated structures.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Cylindrical shell; Kelvin–Voigt model; State-space method; Variable substitution technique; Power series expansion

1. Introduction

In recent years, fiber-reinforced laminated compositestructures have been widely used in aerospace, marine,automobile and other engineering industries. These typesof composite systems can exhibit many favorable charac-teristics such as high specific modulus and strength, resis-tance to fatigue and damage, low specific density, anddirectional properties [1]. In particular, laminated shellsare very prominent in bearing various loads and have beenwidely used in engineering structures. So many researchershave paid their much attention to developing different shelltheories to model and analyze laminated shell structures. Aclassical shell theory based on the Kirchhoff–Love kine-matics hypothesis may not be acceptable for the analysis

0263-8223/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2005.11.017

* Corresponding author. Address: Department of Civil Engineering,Zhejiang University, Hangzhou 310027, PR China. Tel.: +86 571 87952284; fax: +86 571 879 52165.

E-mail address: yanw@zju.edu.cn (W. Yan).

of composite structures [2,3]. This is because the effect oftransverse shear deformation becomes significant in thesestructures. Applications of such classical theories to aniso-tropic layered shells could lead to as much as 30% or evenmore errors in the global response parameters such asdeflections and stresses, etc. [4]. Thus, a number of moreaccurate theories for anisotropic layered shells have beenproposed in the literature. Zenkour and Fares [4] studiedthe bending, buckling and free vibration of non-homoge-neous composite laminated cylindrical shells using a refinedfirst-order theory. To overcome the deficiency of the first-order theory, a higher-order shell theory with varyingdegree of refinements of the kinematics of deformation isadopted to analyze the response of composite shells [2,3].The layerwise shell theory, which reduces a 3-D problemto a 2-D problem, is also used to model discretely cylindri-cal shells for stress, vibration, pre-buckling and post-buck-ling analysis [5]. For the problem of perfectly bondinglaminated cylindrical shells in cylindrical bending, it is wellknown that the exact elasticity solutions have been derived[6,7]. However, the conventional exact elasticity analysis

0

rnrk

r0

1

k

n

h

hk

rz

θ0

θ

Fig. 1. Geometry and coordinates of laminated cylindrical shell incylindrical bending.

552 W. Yan et al. / Composite Structures 78 (2007) 551–559

following Pagano’s method [8–10] becomes computation-ally expensive when the number of layers in the laminateincreases, because of a large number of integral constantsthat are involved. On the other hand, the state-spacemethod has been developed and proven to be an effectivetool for solving problems of laminated or FGM structures[11,12].

In the case of a laminated cylindrical shell in cylindricalbending, a state equation can be obtained with variablecoefficients in the cylindrical coordinate system, whosesolution is generally difficult to obtain directly. Under thisconsideration, Fan and Zhang [13] solved a set of differen-tial equations with variable coefficients using transfermatrix method along with the layerwise approximation.Cai et al. [14] utilized two different numerical methods tostudy the mechanical behavior of angle-ply laminatedcylindrical panels with imperfect interfaces. In contraryto the above mentioned techniques, Chen and Ding [15]derived the state equations with constant coefficients andgot the exact elasticity solution for a multilayered sphericalshell with spherical isotropy using a variable substitutiontechnique. In this paper, the above variable substitutiontechnique is extended to treat problems pertinent to themechanical response of angle-ply laminated cylindricalshell in cylindrical bending with viscoelastic interfaces.

In the mechanics of composite materials, it has been rec-ognized that imperfect interfacial bonding has a significantinfluence on the behavior of laminated composites. Due tothe presence of interfacial damage caused by fatigue andenvironmental effects, or the complexity inhering in theprocess of fabrication of composite laminates themselves,various flaws, such as microcracks, inhomogeneities, andcavities, shall be introduced into the adhesives, which inturn affect the behavior of laminated structures. One ofthe key problems associated with the research of imperfectlaminates is how to depict the interfacial properties. Somesimplified interfacial models have been introduced foraccessing the effect of the imperfect interfaces. The mostpopular one is the linear spring-like model [11,14,16–19].Recently, the dislocation-like model was proposed to math-ematically describe the effect of imperfect interfaces on theload transfer [20,21]. In all these works [11,14,16–21], theresponses of laminates under static loading are indepen-dent of the time variable. Recently, the time-dependentbehavior of laminated elastic plates with viscous interfaceswas investigated, and the results showed that the viscousinterfaces would lose the ability of transferring shear stresstotally when time approaches infinity [22–24]. Morerecently, Yan and Chen [25] and Yan et al. [26] studiedthe time-dependent response of laminated strips with visco-elastic interfaces based on Kelvin–Voigt model, since theKelvin–Voigt constitutive relation has its particular meritto describe the behavior of imperfect interfaces. One is thatthis constitutive law covers two particular cases, i.e. thespring-like model and the viscous model [25–28], and ismore universal than those available in the literature[11,14,16–19,22–24]. Another is that this model is more

appropriate for characterizing the creep and relaxationbehavior of interlaminar bonding material under high tem-perature circumstance, according to Hashin [27] and Fanand Wang [28].

It seems difficult to obtain an exact solution directlybased on the state-space formulations, because a set of dif-ferential equations involving time variable should be solvedwhen the Kelvin–Voigt model is employed to depict thebehavior of viscoelastic interfaces. Note that the power ser-ies expansion technique is a popular and effective numericalmethod, which can transform the differential equations intoa series of algebraic formulations. Thus, the above-men-tioned difficulties can be overcome easily. Recently, thismethod is adopted to approximate the variations of elasticfields with time for laminates with viscoelastic/viscousinterfaces and is proven to be very effective and highlyaccurate [23,24].

In this paper, a hybrid analysis, which combines thestate-space method, the power series expansion techniqueand the variable substitution technique, is developed hereto investigate the mechanical behavior of a simply sup-ported angle-ply laminated cylindrical shell in cylindricalbending with viscoelastic imperfect interfaces under staticloading. Kelvin–Voigt model is employed to characterizethe interfacial sliding, which results in an in-plane relativedisplacement at the interface. For simplicity, we assumethat the deformation is relatively slow so that the inertiaterms can be neglected in the analysis and the layer materi-als are assumed to be linearly elastic. Finally, numericalresults are presented and discussed.

2. State-space equations

We consider an n-layered angle-ply cylindrical shell incylindrical bending, as shown in Fig. 1. The laminatedpanel considered here is infinitely long in the z-directionand is simply supported at h = 0,h0. In the cylindrical coor-dinate system (r,h,z), the three displacement components

W. Yan et al. / Composite Structures 78 (2007) 551–559 553

ur, uh and uz in the r-, h- and z-directions, respectively, areall independent of z. In this paper, we follow the method asproposed in Ref. [15] for the analysis of a multilayeredspherical shell with spherical isotropy. First, the constitu-tive relations [7] can be rewritten as follows:

Rz ¼ rrz ¼ c12

ouh

ohþ ur

� �þ c13r2ur þ c16

ouz

oh;

Rh ¼ rrh ¼ c22

ouh

ohþ ur

� �þ c23r2ur þ c26

ouz

oh;

Rr ¼ rrr ¼ c23ouh

ohþ ur

� �þ c33r2ur þ c36

ouz

oh;

Chz ¼ rshz ¼ c26

ouh

ohþ ur

� �þ c36r2ur þ c66

ouz

oh;

Crh ¼ rsrh ¼ c44

ouh

ohþr2uh � uh

� �þ c45r2uz;

Crz ¼ rsrz ¼ c45

our

ohþr2uh � uh

� �þ c55r2uz;

ð1Þ

where $2 = ro/or, cij are elastic constants. Second, the equi-librium equations [14], in absence of body forces and withthe neglect of inertia terms, are rewritten as

r2Rr þoCrh

oh� Rh ¼ 0; r2Crh þ

oRh

ohþ Crh ¼ 0;

r2Crz þoChz

oh¼ 0. ð2Þ

Thus, from Eqs. (1) and (2), we can get the followingstate-space equation:

r2

uh

uz

ur

Crh

Crz

Rr

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;¼

1 0 � ooh

c55

a � c45

a 0

0 0 0 � c45

ac44

a 0

� c23

c33

ooh �

c36

c33

ooh � c23

c330 0 1

c33

�b1o2

oh2 �b2o2

oh2 �b1ooh �1 0 � c23

c33

ooh

�b2o2

oh2 �b3o2

oh2 �b2ooh 0 0 � c36

c33

ooh

b1ooh b2

ooh b1 � o

oh 0 c23

c33

266666666664

377777777775

uh

uz

ur

Crh

Crz

Rr

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;;

ð3Þ

and the other three secondary variables

Rh ¼ b1

ouh

ohþ ur

� �þ b2

ouz

ohþ c23

c33

Rr;

Rz ¼ b4

ouh

ohþ ur

� �þ b5

ouz

ohþ c13

c33

Rr;

Chz ¼ b2

ouh

ohþ ur

� �þ b3

ouz

ohþ c36

c33

Rr,

ð4Þ

in which

a¼ c44c55�c245; b1¼ c22� c2

23=c33; b2¼ c26� c23c36=c33;

b3¼ c66� c236=c33; b4¼ c12�c13c23=c33; b5¼ c16� c13c36=c33.

ð5ÞFor the simply supported boundary conditions

rh ¼ shz ¼ ur ¼ 0; at h ¼ 0; h ¼ h0. ð6Þ

we can assume

uh

uz

ur

Crh

Crz

Rr

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;¼

r0uhðn; tÞ cosðnph=h0Þr0uzðn; tÞ cosðnph=h0Þr0urðn; tÞ sinðnph=h0Þ

cð1Þ44 r0Crhðn; tÞ cosðnph=h0Þcð1Þ44 r0Crzðn; tÞ cosðnph=h0Þcð1Þ44 r0Rrðn; tÞ sinðnph=h0Þ

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;; ð7Þ

where n = r/r0 is dimensionless coordinate, cð1Þ11 representsthe elastic constant of the first layer (the bottom layer), nis an integer, and a quantity with overbar indicates thedimensionless one. Note that the deformation as well asthe stress field assumed in Eq. (7) is time-dependent be-cause of the viscoelastic interfaces to be considered later.The substitution of Eq. (7) into Eq. (3) yields

no

onVðn; tÞ ¼ AVðn; tÞ; ð8Þ

where Vðn; tÞ ¼ uhðn; tÞ uzðn; tÞ urðn; tÞ Crhðn; tÞ Crzðn; tÞ�

Rrðn; tÞ�T and

A ¼

1 0 �k1c55cð1Þ

44

a � c45cð1Þ44

a 0

0 0 0 � c45cð1Þ44

a

c44cð1Þ44

a 0

c23

c33k1

c36

c33k1 � c23

c330 0

cð1Þ44

c33

b1

cð1Þ44

k21

b2

cð1Þ44

k21 � b1

cð1Þ44

k1 �1 0 � c23

c33k1

b2

cð1Þ44

k21

b3

cð1Þ44

k21 � b2

cð1Þ44

k1 0 0 � c36

c33k1

� b1

cð1Þ44

k1 � b2

cð1Þ44

k21

b1

cð1Þ44

k1 0 c23

c33

26666666666666664

37777777777777775

;

ð9Þwhere k1 = np/h0. It can be seen that Eq. (8) is a state equa-tion with variable coefficients, whose solution is difficult toobtain directly. So the following variable substitution is ta-ken [15]:

n ¼ nk�1 expðgÞ ðk ¼ 1; 2; . . . ; n; 0 6 g 6 gkÞ; ð10Þin which n0 = 1, nk ¼ rk=r0 ¼ 1þ

Pkj¼1hj=r0, here hk is the

thickness of the kth layer, gk = ln(nk/nk�1). Thus, Eq. (8)can be rewritten as

o

ogVðg; tÞ ¼ AVðg; tÞ. ð11Þ

Eq. (11) is a state equation with constant coefficient,whose solution can be obtained as

Vðg; tÞ ¼ expðAgÞVð0; tÞ ð0 6 g 6 gk; k ¼ 1; 2; . . . ; nÞ.ð12Þ

Setting g = gk in Eq. (12), the relationship between thestate variables at the upper and lower surfaces of the kthlayer can be established

VðkÞ1 ¼MkV

ðkÞ0 ; ð13Þ

554 W. Yan et al. / Composite Structures 78 (2007) 551–559

where VðkÞ1 and V

ðkÞ0 are the state vectors at the upper and

lower surfaces, respectively, of the kth layer, and Mk = ex-p(Agk) is the transfer matrix. Similarly, we get

Vðkþ1Þ1 ¼Mkþ1V

ðkþ1Þ0 . ð14Þ

3. Kelvin–Voigt viscoelastic interfacial model

In the general analysis, it is always assumed that distrib-uted normal pressures p(h) and q(h) are applied on the topand bottom surfaces, respectively. These loads can beexpanded in terms of sine functions as follows

pðhÞ ¼ cð1Þ44

X1n¼1

an sinðnph=h0Þ;

qðhÞ ¼ cð1Þ44

X1n¼1

bn sinðnph=h0Þ; ð15Þ

where ½an; bn� ¼ ½2=cð1Þ44 �R 1

0 ½pðhÞ; qðhÞ� sinðnphÞdh. Here inthis paper, it is assumed that, for the sake of simplicity,only a normal sinusoidal loading pðhÞ ¼ p0 sinðph=h0Þ isapplied on the top surface of the laminated cylindricalshell. So we can get the following boundary conditions:

RðnÞr ðgnÞ ¼ rnp0=ðr0cð1Þ44 Þ;

Rð1Þr ð0Þ ¼ C

ð1Þrh ð0Þ ¼ C

ðnÞrh ðgnÞ ¼ C

ð1Þrz ð0Þ ¼ C

ðnÞrz ðgnÞ ¼ 0.

ð16aÞOn the other hand, at the viscoelastic interface, since the

adhesive layer is very thin, the mechanical forces at the twosides can be regarded to be in a state of equilibrium. Thus,we have

rðkþ1Þr ¼ rðkÞr ; sðkþ1Þ

rh ¼ sðkÞrh ; sðkþ1Þrz ¼ sðkÞrz ; uðkþ1Þ

r ¼ uðkÞr ;

uðkþ1Þh ¼ uðkÞh þ dðkÞh ; uðkþ1Þ

z ¼ uðkÞz þ dðkÞz ; at r ¼ rk;

ð16bÞwhere dðkÞh ; dðkÞz are the relative sliding displacements at thekth interface. In this paper, we assume the shear stressand the sliding obey the Kelvin–Voigt viscoelastic law [28]

sðkÞrh ¼ gðkÞ0h dðkÞh þ gðkÞ1h_dðkÞh ;

sðkÞrz ¼ gðkÞ0z dðkÞz þ gðkÞ1z_dðkÞz ; at r ¼ rk;

ð17Þ

where _dðkÞh ; _d

ðkÞz are the sliding velocities (the dot over a

quantity denotes differentiation with respect to time), and

gðkÞ0h ; gðkÞ0z and gðkÞ1h ; g

ðkÞ1z are the elastic constants and viscous

coefficients in h- and z-directions, respectively. Setting

gðkÞ0h ¼ gðkÞ0z ¼ 0, we get the viscous model employed in Refs.

[22–24]. Setting gðkÞ1h ¼ gðkÞ1z ¼ 0, the constitutive relationdegenerates to the linear spring model [28]. Thus, boththe viscous model and the linear spring model are the par-ticular cases of the Kelvin–Voigt model employed in thispaper. Then we assume

dðkÞh ¼ r0dðkÞh ðg; tÞ cosðnph=h0Þ;

dðkÞz ¼ r0dðkÞz ðg; tÞ cosðnph=h0Þ.

ð18Þ

Thus, Eq. (16b) can be written in a matrix form

Vðkþ1Þ0 ¼ V

ðkÞ1 þQðkÞ; ð19Þ

where QðkÞ ¼ ½ dðkÞh dðkÞz 0 0 0 0 �T. Noticing Eqs. (1),

(7) and (18), Eq. (17) can be rewritten as follows

CðkÞrh ¼ nk gðkÞ0h d

ðkÞh þ gðkÞ1h

ddðkÞh

ds

!;

CðkÞrz ¼ nk gðkÞ0z d

ðkÞz þ gðkÞ1z

ddðkÞz

ds

!; at r ¼ rk;

ð20Þ

where gðkÞ0i ¼ gðkÞ0i r0=cð1Þ44 ði ¼ h; zÞ are the dimensionless elas-

tic constants, s ¼ cð1Þ44 t=ðgð1Þ1h r0Þ is the dimensionless time,

and gðkÞ1i ¼ gðkÞ1i =gð1Þ1h ði ¼ h; zÞ are the viscosity ratios.

4. Analysis procedure

As mentioned in the Introduction section, it would bedifficult to obtain an exact solution based on the abovestate-space formulations. In this paper, to use the state-space formulations, we adopt an approximate treatmentof interfacial conditions using power series expansions[23,24]. To this end, we divide the time domain into a seriesof equal intervals [0,Ds], [Ds, 2Ds], [2Ds, 3Ds], . . . , each witha small length of Ds. At a typical interval [mDs, (m + 1)D s](m = 0,1,2, . . .), for an arbitrary state variable x, whichmay be dh; dz; uh; uz; ur; Crh; Crz and Rr etc., we have

xðkÞ ¼ xðkÞm;0 þ ðs� mDsÞxðkÞm;1 þ ðs� mDsÞ2xðkÞm;2

þ ðs� mDsÞ3xðkÞm;3 þ � � � ; ð21Þ

and then we can get

dxðkÞ

ds¼ xðkÞm;1 þ 2ðs� mDsÞxðkÞm;2 þ 3ðs� mDsÞ2xðkÞm;3

þ 4ðs� mDsÞ3xðkÞm;4 þ � � � ð22Þ

In view of Eq. (21), by equating coefficients of the sameorder of s � mDs at the two sides of Eq. (19), we obtain

Vðkþ1Þ0;m;i ¼ V

ðkÞ1;m;i þQ

ðkÞm;i ðm; i ¼ 0; 1; 2; . . .Þ; ð23Þ

where QðkÞm;i ¼ ½ d

ðkÞh;m;i d

ðkÞz;m;i 0 0 0 0 �T. Then from Eqs.

(21) and (22), by equating coefficients of the same order ofs � mDs at the two sides of Eq. (20), we obtain

dðkÞhm;i ¼ ðC

ðkÞrhm;i�1=nk � gðkÞ0h d

ðkÞhm;i�1Þ=ðg

ðkÞ1h � iÞ;

dðkÞzm;i ¼ ðC

ðkÞrzm;i�1=nk � gðkÞ0z d

ðkÞhm;i�1Þ=ðg

ðkÞ1z � iÞ ði ¼ 1; 2; 3; . . .Þ.

ð24ÞFrom Eqs. (13), (14), (21) and (23), we get the relations

Vðkþ1Þ1;m;i ¼Mkþ1V

ðkÞ1;m;i þMkþ1Q

ðkÞm;i ðm; i ¼ 0; 1; 2; . . .Þ. ð25Þ

Continuing the above procedure layer by layer, wefinally obtain

VðnÞ1;m;i ¼ TV

ð1Þ0;m;i þ Sm;i ðm; i ¼ 0; 1; 2; . . .Þ; ð26Þ

W. Yan et al. / Composite Structures 78 (2007) 551–559 555

where T ¼Y1

j¼nMj is the global transfer matrix and

Sm;i ¼MnQðn�1Þm;i þMnMn�1Q

ðn�2Þm;i þ � � � þ

Y2

j¼n

MjQð1Þm;i ð27Þ

are the inhomogeneous terms associated with viscoelasticinterfaces, which vanish in the case of a perfectly bondedlaminate. Considering the boundary conditions in Eq.(16a), we have

VðnÞ1 ¼ uðnÞh1 uðnÞz1 uðnÞr1 0 0 rnp0=ðr0cð1Þ44 Þ

h iT

;

Vð1Þ0 ¼ uð1Þh0 uð1Þz0 uð1Þr0 0 0 0

h iT

.

ð28Þ

Substituting this equation in Eq. (26), we obtain

uðnÞh1

uðnÞz1

uðnÞr1

0

0

rnp0=ðr0cð1Þ44 Þ

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

ðnÞ

m;0

¼T

uð1Þh0

uð1Þz0

uð1Þr0

0

0

0

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

ð1Þ

m;0

þSm;0 ðm¼ 0;1;2; . . .Þ;

ð29Þ

uðnÞh1

uðnÞz1

uðnÞr1

0

0

0

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

ðnÞ

m;i

¼T

uð1Þh0

uð1Þz0

uð1Þr0

0

0

0

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

ð1Þ

m;i

þSm;i ðm¼ 0;1;2; . . . ; i¼ 1;2;3; . . .Þ.

ð30Þ

Table 1A simply supported perfect cross-ply laminated cylindrical shell in cylindrical

S Method w(p/6, r0) �rzðp=6; r0Þ2 Present 1.41653 0.034669

[14] 1.41654 0.034670[6] 1.436 0.0347[16] 1.10533 0.04922

4 Present 0.46291 0.017714[14] 0.46291 0.017714[6] 0.457 0.0177[16] 0.44452 0.01977

10 Present 0.14429 0.0099488[14] 0.14429 0.0099488[6] 0.144 0.0100[16] 0.14291 0.01003

50 Present 0.080828 0.0079822[14] 0.080828 0.0079822[6] 0.0808 0.0080[16] 0.08078 0.00799

100 Present 0.078576 0.0078656[14] 0.078576 0.0078656[6] 0.0787 0.0079[16] 0.07856 0.00787

Obviously, we have dðkÞh0;0 ¼ d

ðkÞz0;0 ¼ 0 because of the zero

initial condition at t = s = 0. So we have S0,0 = 0, andthe state variables at the top and bottom surfaces of theshell can be solved from Eq. (29). Consequently, all physi-cal variables in the shell can be determined from the follow-ing equation as well as Eq. (4):

VðkÞm;iðgÞ ¼ exp½Ag�

Y1

j¼k�1

MjVð1Þ0;m;i þQ

ðk�1Þm;i þMk�1Q

ðk�2Þm;i

þMk�1Mk�2Qðk�3Þm;i þ � � � þ

Y2

j¼k�1

MjQð1Þm;i

!

ð0 6 g 6 gk; k ¼ 1; 2; . . . ; nÞ. ð31Þ

This in turn gives dðkÞh0;1 and d

ðkÞz0;1, i.e. S0,1 by virtue of Eq.

(24). Thus, all variables for i = 1 can be computed fromEq. (30). Continuing this procedure, we eventually obtainall the coefficients in Eq. (21) for m = 0. Setting s = Ds in

Eq. (21) yields dðkÞh1;0 and d

ðkÞz1;0, which give the value of S1,0.

By applying this analysis step by step, all physical variablesat any time can be determined.

5. Numerical computation

First, to validate the effectivity and precision of the pres-ent state-space method and variable substitution technique,we make a comparison with the exact solution developed inRef. [6] and the results in Ref. [14] for the cross-ply lami-nated cylindrical panel with perfect interfaces, as shownin Table 1. The solution derived from the conventionalhigher-order zigzag theory for a perfectly bonded shell inRef. [16] is also listed in Table 1. The orientation of the

bending ([90�/0�/90�], h0 = p/3)

rzðp=6; rnÞ �rhðp=6; r0Þ rhðp=6; rnÞ0.087131 3.46692 2.463100.087131 3.46699 2.463080.0871 3.467 2.4630.02006 4.92227 2.00591

0.029295 1.77140 1.367040.029295 1.77141 1.367040.0293 1.772 1.3670.01357 1.97720 1.35745

0.011472 0.99488 0.897210.011472 0.99488 0.897210.0115 0.995 0.8970.00895 1.00324 0.89452

0.0079311 0.79822 0.783110.0079311 0.79822 0.783110.0079 0.798 0.7820.00783 0.79859 0.78267

0.0078162 0.78656 0.779120.0078162 0.78656 0.779120.0078 0.786 0.7810.00779 0.78675 0.77891

556 W. Yan et al. / Composite Structures 78 (2007) 551–559

unidirectional fibers is denoted by the angle a, measured inthe clockwise direction from the z-axis to the fiber direc-tion. Each layer of the three-layered cross-ply laminatedpanel considered here is of equal thickness and densityratio, and the following material properties, geometry,and normalized parameters are used:

EL=ET¼ 25; GLT=ET¼ 0:5; GTT=ET¼ 0:2;

lLT¼ lTT¼ 0:25; h0¼ p=3; ð32aÞ

rm¼ðrnþ r0Þ=2; S¼ rm=h ðrh;rz;wÞ¼rh

p0S2;

rz

p0S2;10ET

p0hS4ur

� �;

ð32bÞ

where L denotes the direction parallel to the fibers, T thetransverse direction, and l the Poisson’s ratio.

As shown in Table 1, the results of our method agreewell with the exact ones identically, except for the trans-verse displacement whose relative error is about 1.3% forS = 2 and S = 4. It also can be seen that the solution ofthe present method is nearly identical to that proposedby approximate numerical methods based on state-spaceformulations in Ref. [14]. On the other hand, Table 1clearly shows the relatively large error of the shell theorydeveloped in Ref. [16], especially for small S.

Next, we consider a five-layered cylindrical panel incylindrical bending with a [70�/�30�/45�/�70�/60�] lay-up. The material constants in Eq. (32a) are also adoptedin this numerical example. The mean radius-to-thicknessratio is taken to be S = 4.5 and the central angle ish0 = p/3. We further assume that the first and fourth inter-

faces are viscoelastic with gð1Þ0i ¼ 2gð4Þ0i ¼ 2 and gð4Þ1i ¼ 2gð1Þ1i

ði ¼ h; zÞ, the second is perfect, while the third one is viscous

with gð3Þ1i ¼ gð1Þ1i ði ¼ h; zÞ. A new dimensionless radial vari-able f = (r � r0)/(rn � r0) and the following dimensionlessquantities are used in our calculation for the sake of clarity:

u1 ¼ uhð0; rÞcð1Þ44 =ðp0r0Þ; u2 ¼ uzð0; rÞcð1Þ44 =ðp0r0Þ;u3 ¼ urðp=6; rÞcð1Þ44 =ðp0r0Þ;r1 ¼ srhð0; rÞ=p0; r2 ¼ srzð0; rÞ=p0; r3 ¼ rrðp=6; rÞ=p0.

ð33Þ

Table 2Convergence studya

(M,Ds) u1 u2 u3

s = 2

(2,0.1) 3.844130588 �0.2124243178 12.77692198(3,0.1) 3.844395268 �0.2124552137 12.77777850(3,0.05) 3.797135714 �0.2116446352 12.61461124(4,0.05) 3.797136183 �0.2116445722 12.61461281

s = 10

(2,0.1) 4.423528041 �0.1783484350 14.91936994(3,0.1) 4.423523731 �0.1783408764 14.91937904(3,0.05) 4.428097932 �0.1784490444 14.93506732(4,0.05) 4.428097553 �0.1784490412 14.93506600

a All data are evaluated for f = 0.2 (first layer).

The convergence characteristic of the power seriesexpansion method for the present problem is checked withresults given in Table 2. As in the case of laminates withviscous interfaces [23,24], the results for M = 4 andDs = 0.05 are of highly accuracy. So in the followingnumerical computations, we will assume M = 4 andDs = 0.05. The parameter M in Table 2 represents the num-ber of total terms adopted in the series expansions in Eq.(21) in the calculation.

The through-thickness distributions of the six non-dimensional state variables defined in Eq. (33) are shownin Fig. 2. The response of the shell at t = 0 is the same asa panel with perfectly bonded interfaces. With the occur-rence of interfacial sliding, the significant stress relaxationoccurs, and the six non-dimensional state variables changeremarkably with time. Discontinuities of u1 and u2 at visco-elastic and viscous interfaces, caused by the sliding, areshown in Fig. 2(a) and (b), respectively. The differencebetween u1 (also u2) at the two sides of an interface is justthe sliding displacement, i.e. dðkÞh (or dðkÞz ). We can find thatthe fourth sliding displacement, dð4Þz , at s = 10 is smallerthan that at s = 2 as shown in Fig. 2(b). This interestingand seemingly unreasonable phenomenon can be explainedtheoretically. In general, the sliding velocity at the viscousinterface is larger than that at the viscoelastic interface, fora certain time interval, the dimensionless displacement u2 atthe bottom and top surfaces of the fourth layer increasesmore rapidly than that at the bottom surface of the fifthlayer in this example. So it is logical that dð4Þz at s = 10,which is the difference between u2 at the top surface ofthe fourth layer and that at the bottom surface of the fifthlayer becomes smaller than that at s = 2. It can also be seenfrom Fig. 2(c) that the distribution of the transverse dis-placement seems vertically straight, however, it is just dueto the scale used in this figure. The curve redrawn inFig. 2(d) at s = 10 clearly shows that the value of u3 isno longer a constant along the thickness, although the var-iation is relatively very small. Furthermore, due to theinterfacial sliding, global stiffness of the cylindrical shellis reduced and the displacement components in the threedirections increase significantly with time.

r1 r2 r3

3.735777245 1.198534365 0.26038960903.736044614 1.198611819 0.26046907863.703989126 1.187709420 0.26074158043.703989932 1.187709734 0.2607411697

3.814248091 1.246978622 0.16574139753.814192295 1.246963348 0.16572573413.817538446 1.248086862 0.16575282873.817538211 1.248086780 0.1657528385

Fig. 2. Distributions of field variables along the thickness.

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558 W. Yan et al. / Composite Structures 78 (2007) 551–559

Fig. 2(e) and (f) display the time dependency of the shearstresses. The variation of the curve is abrupt across the vis-coelastic or viscous interfaces as shown in Fig. 2(e). Thisindicates that a relaxation occurs in the shear stress, andthe two viscoelastic interfaces, i.e. f = 0.2 and f = 0.8, losetheir part capability of transferring shear stress. On thecontrary, we can see that shear stress is nearly zero at theviscous interface (f = 0.6) when s = 10. In fact, they shalleventually vanish when s!1, because the viscous inter-face will lose the capability of transferring shear stress com-pletely. These results are not shown clearly in Fig. 2(f),because the stress relaxation is covered up due to the distri-bution of r2. But we can also see that the shear stress isnearly zero at the viscous interface (f = 0.6) when s = 10.Furthermore, it is observed that the maximum shear stres-ses (r1 and r2) at s = 10 in the cylindrical panel are almostthree times as these at s = 0 and even at the perfect inter-face the shear stresses also become much larger. So it isvery important to consider the effect of imperfect interfacesin the design of laminated structures. Moreover, even ifsome interfaces in a laminated shell (or plate etc.) areperfect, they may also become more dangerous due tothe effect of the other imperfect interfaces in the samelaminate.

Fig. 2(g) shows that, in contrary to those for a laminatedplate [25], the sign of the dimensionless transverse normalstress r3 changes and the maximum value is not on the sur-face. At the top surface, r3 is tensile, but it changes andeventually becomes compressive in the first layer. Alongthe interfaces, an obvious state of non-differentiabilityoccurs and it seems as if the curves are pulled locally tothe right. Such similar instance was also reported in Ref.[7]. The tendency of this pulling is more serious at f = 0.8than elsewhere in the panel. To study this phenomenonmore deeply, we assume three interfacial models assignedto the third interface, while other interfaces keep unaltered.Models 1, 2 and 3 represent the laminated shell with vis-cous, viscoelastic and perfect interfaces at f = 0.6, respec-tively. From Fig. 2(h), we can see that the variation of r3

and the tendency of this pulling become more and moresignificant from Model 3 to Model 2 and to Model 1. Sowe can make a conclusion that the viscous property ofinterface results in a more serious pulling phenomenon.Similar to the distributions of r1 and r2, the maximum nor-mal stress r3 at s = 10 in the cylindrical panel is almost 3.5times as that at s = 0.

6. Conclusions

The response of a simply supported angle-ply laminatedcylindrical shell in cylindrical bending with viscoelasticinterfaces is investigated. As special cases of the viscoelasticKelvin–Voigt model, the response of a laminated shell withviscous interface and spring-like interface can also be stud-ied. Analysis based on state-space formulations is devel-oped for laminates with a relatively large number ofplies. A variable substitution technique is adopted to deal

with the state equation with variable coefficients. Numeri-cal comparison shows that the proposed hybrid methodis very effective and accurate. A special technique usingpower series expansion is further adopted to approximatethe variations of elastic fields with time, and the computa-tion shows that the corresponding numerical solution con-verges rapidly.

Results indicate that the response of a laminated shellwith viscoelastic interfaces (also viscous interface) changesremarkably with time. The response of a laminated panelwith viscoelastic interfaces is found to be greatly differentfrom that of a shell with perfect interfaces or with viscousinterfaces. The most significantly different feature of theviscoelastic interface than the viscous one is that they willkeep part of the capability of transferring shear stresses,even when t!1. On the other hand, the influence of vis-cous interfaces on the laminated structures seems moreremarkable than viscoelastic ones.

Due to the imperfect interfaces, global stiffness of thelaminated structure is reduced and the displacementsincrease significantly with time. Similar to the displace-ments, the maximum stresses in the laminated panel withimperfect interfaces are several times as these in the corre-sponding perfect laminated shell. Furthermore, the behav-ior of transverse normal stress r3 is entirely different fromthat of a laminated plate. The sign of r3 changes in the lam-inated cylindrical shell with the maximum value not beinglocated at the surface. Thus, the effect of imperfect inter-faces should be considered in the design of laminated struc-tures, especially those which are safety critical.

Acknowledgement

This work was supported by the Natural Science Foun-dation of China (No. 10432030 and 50105020).

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