Delamination buckling growth in laminated …webpages.iust.ac.ir/bijan_mohammadi/Bijan_Mohammadi...Delamination buckling growth in laminated composites using layerwise-interface element

Composite Structures xxx (2010) xxx–xxx

ARTICLE IN PRESS

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Delamination buckling growth in laminated composites usinglayerwise-interface element

Hossein Hosseini-Toudeshky a,*, Samira Hosseini a, Bijan Mohammadi b

a Aerospace Engineering Department, Amirkabir University of Technology, Hafez Ave., Tehran, Iranb School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran

a r t i c l e i n f o

Article history:Available online xxxx

Keywords:Delamination buckling growthLayerwiseDecohesive lawCompositeFEM

0263-8223/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compstruct.2010.01.013

* Corresponding author. Tel.: +98 21 64543224; faxE-mail addresses: [email protected] (H. H

[email protected] (B. Mohammadi).

Please cite this article in press as: Hosseini-ToudCompos Struct (2010), doi:10.1016/j.compstruc

a b s t r a c t

In this paper, a numerical investigation on the buckling of composite laminates containing delamination,under in-plane compressive loads, is presented. For this purpose, delamination propagation is modeledusing the softening behavior of interface elements. The full layerwise plate theory is applied for approx-imating the displacement field of laminates and the interface elements are considered as a numericallayer between any two adjacent layers where the delamination is expected to propagate. A non-linearcomputer code was developed to handle the numerical procedure of delamination buckling growth incomposite laminates using layerwise-interface elements. The load/displacement behavior and the con-tours of embedded and through-the-width delamination propagation for composite laminates are pre-sented. It is shown that delamination growth can be well predicted using this layerwise-interfaceelements with decohesive law.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Delamination is a common failure mode in layered compositematerials which may result from the manufacturing imperfections,lay-up geometries, edge effects and various loadings. The presenceof the delamination can cause significant reduction in stiffness andstrength of the laminate under compressive loads; hence, a clearunderstanding of the compressive failure behavior of the laminateis extremely important. Many investigations were already per-formed on the buckling induced delamination propagation of com-posite laminates.

Whitcomb and Shivakumar [1] studied the delaminationgrowth due to the local buckling of a composite plate with squareand rectangular embedded delaminations. They applied the frac-ture mechanics approach for predicting the delamination growthusing a virtual crack closure technique (VCCT) in order to calculatethe total strain energy release rate. They showed that, whether adelamination grows in the load direction or perpendicular to theload direction, it depends on the laminate aspect ratio, the strainlevel and the absolute size of the delamination. Nilsson et al. [2]studied delamination buckling and growth of slender compositepanels using both numerical and experimental methods. Theinvestigated panels were cross-ply laminates under compressionloading and containing embedded delaminations in different

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: +98 21 66959020.osseini-Toudeshky), Bijan_

eshky H et al. Delamination but.2010.01.013

depths. For all delamination depths, the analysis showed a drasticincrease in the energy release rate when global buckling takesplace. Riccio et al. [3] investigated the compressive behavior of car-bon fiber/epoxy laminated composite panels containing through-the-width and embedded delaminations. They also used a modifiedVCCT for predicting the delamination growth. They studied the ef-fect of trough-the-thickness location of delamination on the stabil-ity or instability of delamination growth. Lachaud et al. [4] alsoused the VCC integral to simulate the propagation of delaminationcaused by the local buckling, on thermoset and thermoplastic car-bon/fiber composite laminates having embedded delaminations.They also performed experiments to verify the obtained results.

Tafreshi and Oswald [5] developed finite element models tostudy the global, local and mixed-mode buckling behavior of com-posite plates with embedded delaminations. They applied a dis-placement field to the laminate that arises from the Mindlin/Kirchhoff plate theory and they also used the modified crack clo-sure technique to calculate the strain energy release rate in orderto predict the delamination growth. In the case of global buckling,they performed a parametric study to investigate the influence ofthe delamination size, shape and stacking sequence on the buck-ling load. It is worth to mention that delamination growth analysesusing fracture mechanics approaches are very sensitive to the sizeand shape of the elements around the delamination front and theyneed special considerations of boundary movement problems inthe step by step crack propagation process.

Hwang and Liu [6] performed experiments to study the buck-ling loads, buckling modes, post-buckling behavior and delamina-

ckling growth in laminated composites using layerwise-interface element.

http://dx.doi.org/10.1016/j.compstruct.2010.01.013

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Fig. 1. Multilayer element with quadratic variation through each numerical layer.

2 H. Hosseini-Toudeshky et al. / Composite Structures xxx (2010) xxx–xxx

ARTICLE IN PRESS

tion growth of the delaminated unidirectional carbon/epoxy lami-nates, containing strip shaped delaminations. They investigatedthe effect of delamination length and its through-the-thicknesslocation on the buckling load. Zhang and Wang [7,8] presented alayerwise B-spline finite strip method with consideration ofdelamination kinematics to study the buckling and post-bucklingbehavior of delaminated composite laminates. For the predictionof delamination propagation, they employed a fracture mechanicsapproach which is based on the energy release rate criterion. Aslanand Sahin [9] studied the effects of delamination size and through-the-thickness location, on the critical buckling load and compres-sive failure load of the E-glass/epoxy composite laminates contain-ing multiple through-the-width delaminations. The experimentaland numerical study performed for the [0/90/90/0]s cross-ply lam-inates with and without delaminations. The lengths of delamina-tions were different for different depths. Suemasu et al. [10]presented a numerical study to investigate the compressive behav-ior of carbon fiber reinforced plastic laminated plates containingmultiple embedded delaminations. They used 20 node brick ele-ments for discretizing the composite layers and incorporated cohe-sive elements for modeling the delamination. They investigatedthe effects of interlaminar toughness on the buckling mode anddelamination growth stability.

Kyoung and Kim [11] presented an analytical method to deter-mine the delamination buckling and growth of one-dimensionalbeam-plate. They also investigated the effects of delaminationlength and location on the buckling load and delamination growth.De Borst and Remmers [12] used mezo level approach to study thelocal buckling growth of a delaminated layer. Wagner et al. [13]proposed a finite element method to simulate the delaminationpropagation in plate strips and plates with circular embeddeddelamination. They used interface elements in the regions thatthe delamination is expected to propagate. The used interface ele-ment formulation was based on the plasticity theory.

The objective of this paper is to employ an interface elementincorporating with layerwise theory to study the delaminationbuckling and growth of laminated composites containing initialthrough-the-width or embedded delamination. The fracturemechanics approaches can not easily predict the simultaneousdelamination initiation and propagation, therefore, the interfaceelement concept in conjunction with multilayer elements (layer-wise theory of laminates) is used to determine the delaminationgrowth and the fracture mechanics concept is employed indirectlyin this approach. For this purpose, a finite thickness interface ele-ment model is used to simulate the progressive delamination. Abilinear constitutive equation relates the stress and strains of theelement. The initiation and propagation of the delamination ismodeled through the softening behavior of the interface layer.The present investigation is based on the formulation of the fulllayerwise plate theory to approximate the displacement field ofthe layers and the interface element is considered as a thin filmnumerical layer between any two adjacent layers. A non-linear lay-erwise finite element code was also developed to handle thenumerical procedure of delamination buckling growth in compos-ite laminates.

2. Layerwise finite element formulation

In this study the composite laminate is discretized by 8-nodedmultilayer elements as shown in Fig. 1, where the full layerwiseplate theory is used to approximate the displacement of each nodeand each material layer. Each node of the element has three de-grees of freedom. In contrast with the 3-D solid elements, the lay-erwise elements provide the capability of discretizing the structureby a 2-D mesh. Therefore, the amount of input data is reduced and

Please cite this article in press as: Hosseini-Toudeshky H et al. Delamination buCompos Struct (2010), doi:10.1016/j.compstruct.2010.01.013

the number of operations for constructing the element stiffnessmatrix is decreased. The interface element concept in the layerwisefinite element method is considered as a numerical layer of themodel inserted between any two adjacent layers, which the delam-ination is expected to propagate as typically shown in Fig. 3. Thedisplacement field of the laminate can be approximated using 2-D interpolation functions multiplying by through-the-thicknessinterpolation functions and is written as [16,19]:

uðx; y; zÞ ¼Xn

j¼1

Ujðx; yÞHjðzÞ

vðx; y; zÞ ¼Xn

j¼1

Vjðx; yÞHjðzÞ

wðx; y; zÞ ¼Xn

j¼1

Wjðx; yÞHjðzÞ

ð1Þ

In this equation, n is the number of nodes in the thickness direc-tion and Uj, Vj and Wj are the in-plane approximate of the nodalvalues of displacement components, u, v and w respectively. Forquadratic variation of the displacement, three nodes are consid-ered through-the-thickness of each numerical layer, Ne. The num-ber of numerical layers can be greater than, equal to or less thanthe material layers of the composite laminate. Hj is through-the-thickness quadratic interpolation function of the jth node and isdefined by Van Hoa and Feng [18]:

H2k�1ðzÞ ¼ GðkÞ1 ðzÞH2kðzÞ ¼ GðkÞ2 ðzÞ; z2k�1 6 z 6 z2kþ1

H2kþ1ðzÞ ¼ GðkÞ3 ðzÞ; k ¼ 1;2; . . . ;Ne

ð2Þ

where

GðkÞ1 ¼ 1��zhk

� �1� 2�z

hk

� �¼ � fð1� fÞ

2

GðkÞ2 ¼ 4�zhk

1��zhk

� �¼ ð1þ fÞð1� fÞ which �z ¼ z� zk

b

GðkÞ3 ¼ ��zhk

1� 2�zhk

� �¼ fð1þ fÞ

2

ð3Þ

and zkb denotes the z-coordinate of bottom of the kth numerical

layer. This equation can be written in the extended form as:

u ¼Xn

j¼1

X8

i¼1

UijNiðn;gÞHjðfÞ

v ¼Xn

j¼1

X8

i¼1

VijNiðn;gÞHjðfÞ

w ¼Xn

j¼1

X8

i¼1

WijNiðn;gÞHjðfÞ

ð4Þ



1

15 13

119

7 5

3u1

w1

v1

x

z

y

X

ZY

O'

Fig. 2. Finite thickness interface element.

H. Hosseini-Toudeshky et al. / Composite Structures xxx (2010) xxx–xxx 3

ARTICLE IN PRESS

In which, Ni is the in-plane shape function of the ith node and n,g and f are local coordinates. According to this layerwise definitionfor the displacement field, the non-linear strain/displacement rela-tions of the von-karman type can be written as:

exx ¼Xn

j¼1

X8

i¼1

UijNi;xHj þ12

Xn

j¼1

X8

i¼1

WijNi;xHj

!2

eyy ¼Xn

j¼1

X8

i¼1

VijNi;yHj þ12

Xn

j¼1

X8

i¼1

WijNi;yHj

!2

ezz ¼Xn

j¼1

X8

i¼1

WijNiHj;z þXn

j¼1

X8

i¼1

WijNi;xHj

! Xn

k¼1

X8

l¼1

WlkNl;yHk

!

cxy ¼Xn

j¼1

X8

i¼1

ðUijNi;yHj þ VijNi;xHjÞ

cyz ¼Xn

j¼1

X8

i¼1

ðVijNiHj;z þWijNi;yHjÞ

cxz ¼Xn

j¼1

X8

i¼1

ðUijNiHj;z þWijNi;xHjÞ

ð5Þ

It is worth to mention that the von-karman non-linear termsare usually sufficient for the modeling of bucking behavior of lam-inates, especially when they are subjected to damage propagationsuch as delamination. The matrix form of this equation is:

feg ¼ ½B�fDg ð6Þ

In which, D is the displacement components vector and B is thematrix of derivatives of the shape functions and is defined by VanHoa and Feng [18]:

½B�fDg ¼ ½b1jb2j . . . b8j�

d1j

d2j

..

.

d8j

8>>>><>>>>:

9>>>>=>>>>;; dij ¼

Uij

Vij

Wij

8><>:

9>=>;; j ¼ 1;2; . . . ; n ð7Þ

and

½B� ¼

Ni;xHj 0 w;xNi;xHj

0 Ni;yHj w;yNi;yHj

0 0 NiHj;z

0 NiHj;z Ni;yHj

NiHj;z 0 Ni;xHj

Ni;yHj Ni;xHj w;xNi;yHj þw;yNi;xHj

266666666664

377777777775

ð8Þ

The potential energy equation is defined as:

Y¼ 1

2

ZVfegT

nl½C�fegnldV � fUgTfFg ð9Þ

In which, [C] is the elastic stiffness matrix of the material layer.The tangent stiffness matrix is defined as the second derivative ofthe potential energy equation:

½K�T ¼ @2Qf@Ug2 ¼

ZV

@feg@fUg

� �T

½C� @feg@fUg

� �dV þ

ZV

@2feg@fUg2

" #T

½C�fegnldV

ð10Þ

In which, the term @feg@fUg

h irepresents the [B] matrix which con-

tains non-linear terms of displacements and can be determined

by Eq. (8). The term @2feg@fUg2

h irepresents the derivative of matrix [B]

with respect to displacement vector and can be defined as:


@2feg@fUg2

" #¼

0 0 Nm;xHpNi;xHj

0 0 Nm;yHpNi;yHj

0 0 00 0 00 0 00 0 Nm;xHpNi;yHj þ Nm;yHpNi;xHj

2666666664

3777777775;

i;m ¼ 1;2; . . . ;8j;p ¼ 1;2; . . . ; n

�ð11Þ

So, the tangent stiffness matrix of each material layer can be ob-tained by:

½K�TL ¼Z

VL

½B�T ½C�½B��

dV þZ

VL

@2feg@fUg2

" #½C�½B�fUg

!dV ð12Þ

3. Implementing the interface layer

The used interface layer is similar to an isoparametric 16-nodehexahedral solid element with a very small thickness, as shown inFig. 2. In this figure (X, Y, Z) denote the global coordinate and the O0

is the central point of the midplane of the element. The thickness ofthe interface layer is about 1/100 of the laminate thickness and it isinserted as a numerical layer between two adjacent laminate lay-ers. The strain vector, e, stress vector, r, and the displacement vec-tor, u, for the interface layer are defined by:

u ¼u

vw

8><>:

9>=>;; e ¼

cx

cy

ez

8><>:

9>=>; ¼

u;zv ;z

w;z

8><>:

9>=>;; r ¼ DIe ð13Þ

In which DI is the interface elastic-damage stiffness matrix. Be-cause the interface layer links the two adjacent physical layers, theinterlaminar components of the strain and stress fields are onlyconsidered for this virtually assumed layer.

In order to avoid Poisson’s locking and for decoupling of theinterlaminar shear stress with the out of plane displacement sim-ilar to solid-like interface element stress filed with the zero-thick-ness interface element, the terms w,x and w,y are assumed to bezero [17]. It means that, the displacement variations in the thick-ness direction are only important in the definition of interfacebehavior.

The displacement and strain vectors can be approximated asfollows:

u ¼X16

i¼1

Niui; e ¼X16

i¼1

Biui ð14Þ

In this equation, Ni is the shape function of the ith node and Bi is theshape function derivatives matrix which is defined by:

Bi ¼Ni;z 0 00 Ni;z 00 0 Ni;z

264

375 ð15Þ



Fig. 3. Implementing of interface layers between material layers.


ARTICLE IN PRESS

Using the finite element approach, the displacement and strainvectors can be approximated from the layerwise formulation withlinear through-the-thickness shape function. In this way in Eq. (4),Ni is the shape function of ith node and Hj is through-the-thicknesslinear interpolation function of the jth node defined by:

H1ðfÞ ¼1� f

2; H2ðfÞ ¼

1þ f2

ð16Þ

Fig. 3 shows a typical composite laminate including four mate-rial layers and three interface layers, which totally divided intonine numerical layers. For a typical laminate containing twomaterial layers connecting with an interface layer (totally threenumerical layers), the global stiffness matrix of the laminate isconstructed by assembling of the stiffness matrices of three singlenumerical layers, as follows:

mε

0mσ

0mε f

mε

mσ

K

Kd )1( −0h

G c

Fig. 4. Mixed mode bilinear constitutive law.

½B�T½D�½B� ¼

BTi¼1DM1Bi¼1 BT

i¼1DM1Bi¼2 BTi¼1DM1Bi¼3 0 0 0

BTi¼2DM1Bi¼1 BT

i¼2DM1Bi¼2 BTi¼2DM1Bi¼3 0 0 0

BTi¼3DM1Bi¼1 BT

i¼3DM1Bi¼2 BTi¼3DM1Bi¼3 þ BT

i¼3DIBi¼3 BTi¼3DIBi¼4 0 0

0 0 BTi¼4DIBi¼3 BT

i¼4DM2Bi¼4 þ BTi¼4DIBi¼4 BT

i¼4DM2Bi¼5 BTi¼4DM2Bi¼6

0 0 0 BTi¼5DM2Bi¼4 BT


0 0 0 BTi¼6DM2Bi¼4 BT


266666666664

377777777775

ð17Þ

In this equation, B is the matrix of shape function derivatives formaterial layers and B is the matrix of shape function derivatives forinterface layers. The superscripts of M1 and M2 denote the materiallayers and I denote the interface layer and i = 1, 2, . . . , n, wheren = 6 is the number of nodes through-the-thickness of thelaminate.

4. Constitutive equation of the interface

For the interface element, an irreversible constitutive law canbe incorporated for modeling of an inelastic softening behavior.In this article, a bilinear constitutive law is chosen which is basedon the zero-thickness cohesive element presented in Ref. [20] andalso is the same as its modified solid-like model presented in Ref.[14]. This constitutive law relates the stresses to strains for anypure fracture or loading mode. In the case of mixed mode loading,the constitutive law relates the effective stresses to the effectivestrains, as shown in Fig. 4. The effective strain is a positive contin-uous quantity and when there is not any compressive normalstrain, it is equal to the norm of strain vector and is defined by:


em ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihezi2 þ c2

y þ c2x

qð18Þ

where the hi operator is: h�i = 0 if � 6 0� if � > 0

�to avoid penetration of

delaminated faces.

If ez 6 0 then; cshear ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

x þ c2y

qð19Þ

The constitutive law contains of three different parts. If theeffective strain is less than e0

m the interface material behaves line-arly elastic and no damage is presented in the element, so thereisn’t any decohesion between layers. When the effective strainreaches the e0

m, the interlaminar damage initiates and after that,the interface stresses decrease linearly. The strain ef

m refers to com-plete decohesion.

In order to complete the description of interface behavior, it isnecessary to specify the strains, corresponding to initiation and

completion of damage. The delamination initiation is predictedby the quadratic failure criterion and the effective strain at delam-ination onset is defined by:



Table 1Material properties of graphite/epoxy.

T300/976 [8] AS-4/3501-6 [14]

E11 (N/mm2) 139,300 138,000E22 (N/mm2) 9720 8960G12 (N/mm2) 5580 7100G23 (N/mm2) 3450 3446m12 0.29 0.3m23 0.4 0.4GIc (N/mm) 0.0876 0.222GIIc (N/mm) 0.3152 0.444ro

n (N/mm2) 44.54 51.7so (N/mm2) 106.9 91.7

0

10

20

30

40

50

60

70

80

0 2 4 6 8

Crack opening-w (mm)

App

lied

load

-P (

N)

experiment [14]

analytical result

present model

Wagner [14]

Fig. 6. Comparison of the obtained load versus crack opening from this study withthe available results for DCB specimen.

Fig. 5. DCB specimen.


ARTICLE IN PRESS

e0m ¼

e0zc0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þb2

ðc0Þ2þ be0zð Þ2

rez > 0

c0 ez 6 0

8><>: ð20Þ

in which, e0n, c0 are the strains at delamination onset corresponding

to mode I and II or III respectively, [14], and b is the mode mixingratio which is defined by:

b ¼ cshear

ezif ez > 0 ð21Þ

The effective critical fracture toughness is equal to the area un-der the effective stress/strain curve and this equivalence is used tocalculate the effective strain at complete decohesion. The effectivefracture toughness is defined by:

Gc ¼Gc

h0ð22Þ

where h0 is the interface element thickness and Gc is the mixedmode critical energy release rate and is calculated by a criterionproposed by Benzeggagh and Kenane [15] (B–K criterion) which isdefined as:

Gc ¼ GIc þ ðGIIc � GIcÞGshear

GT

� �g

ð23Þ

In which GIc, GIIc and GIIIc are the critical strain energy releaserates of mode I, II and III respectively and Gshear = GII + GIII andGT = GI + Gshear and g is an experimental parameter.

By introducing a mixed-mode scalar-valued damage parameter:

d ¼ef

m em � e0m

�emðef

m � e0mÞ

ð24Þ

The interface constitutive law in elastic and elastic-damageconditions can be defined by:

r ¼ DIe ð25Þ

In which:

DI ¼KI em 6 e0

m elastic cond: ðaÞð1� dÞKI e0

m < em < efm elastic—damage cond: ðbÞ

KIc efm 6 em contact cond: ðcÞ

8><>:

ð26Þ

In this equation, K is the penalty stiffness, I is the identity ma-trix, d is the damage parameter and:

Ic ¼0 0 00 0 00 0 h�ezi

�ez

264

375 ð27Þ

To avoid penetration and friction free contact of delaminatedfaces and considering the closure effect in the opening mode, Eq.(26-c) was introduced.

5. Finite element analysis

A non-linear finite element program is developed to handle thenumerical procedure. The laminate model is discretized by 2-Deight-node elements. In the thickness direction of each layer, thedisplacement field is approximated by quadratic shape functions.Each layer of the laminate can be modeled by one or more ele-ments of this type.

The displacement field is approximated by full layerwise lami-nate plate theory and the geometric non-linearity of von-karmantype is applied to the strain/displacement relations. In order to ver-ify the developed layerwise finite element program, the obtained


results are compared with the available numerical and experimen-tal results for a DCB specimen and a laminated beam containing athorough the width pre-delamination.

The geometry and loading of the DCB specimen is shown inFig. 5. The specimen has a length of l = 150 mm, width ofb = 25.4 mm and thickness of 2h = 3.05 mm. The pre-delaminatedregion has a length of ao = 31.75 mm. The AS-4/3501-6 graphite/epoxy material is used which its mechanical properties are listedin Table 1. The thickness of the interface element is consideredas 1/100 of the laminate thickness [14]. The fibers are oriented inthe longitudinal direction of the specimen.

Fig. 6 compares the obtained loads/displacements from thepresent study and reference [14]. This figure shows that, the ob-tained results from this study are in good agreement with theexperimental, analytical, and previously available numerical re-sults. The differences in the linear part of the curve are due tothe considered different penalty stiffness values for the interfaceelements.




ARTICLE IN PRESS

Corigliano and Allix [21] proposed the definition of the penaltystiffness of the 2-D cohesive elements, as a function of the interfacevirtual thickness, t, and elastic modules of the interface (E33, G13

and G23) as: K33 = E33/t, K11 = 2G13/t, K22 = 2G23/t and for simplicityit can be assumed that K11 = K22 = K33 = E33/t. For 2-D (zero-thick-ness) interface models, the penalty stiffness parameter is equalto the slope of the stress/relative displacement curve, whereasfor solid-like interface model, this parameter is the slope ofstress/strain curve, so the stiffness of solid-like element is 1/t timesof the stiffness of the zero-thickness element and can be equal toE33.

The second example is a composite laminate containing thor-ough the width pre-delamination. The obtained results for delam-

0

200

400

600

800

1000

1200

1400

1600

-1.5 -0.5 0.5 1.5 2.5 3.5 4.5

Central deflection-w (mm)

Nx

(N/m

m)

Present Study, Upper Sub-laminateWang [8], Upper Sub-laminatePresent study, Base LaminateWang [8], Base Laminate

oscillation behavior, wang et. al. [8]

(a)

0

200

400

600

800

1000

1200

1400

1600

0 0.1 0.2 0.3 0.4 0.5

End shortening (mm)

Nx

(N/m

m)

Wang et. al. [8]Present study

(b)

Fig. 7. Comparison of the results obtained from the present study with Ref. [8] forthe [04/012//04] lay-up: (a) load versus transverse deflection and (b) load versusend-shortening displacement.

60 mm

40 mm

u0 x

yu

w=

0,u=

u 0

Fig. 8. Geometry dimensi


ination buckling in case of non-growing delamination arecompared with those obtained from the ANSYS finite element soft-ware. The geometry of the laminate with [0/45/45//0] lay-up isshown in Fig. 8, where ‘‘//” is the through-the-thickness locationof delamination. The used material properties are also listed in Ta-ble 1. The laminate is loaded symmetrically in the ‘x’ direction byapplying ends shortening. The boundary conditions are clampedfor loaded ends and free for the other ends. The obtained load ver-sus normalized transverse deflection and load versus end-shorten-ing displacement from the present study are compared with theresults obtained from ANSYS commercial code in Fig. 9a and bb.These two figures show that, the obtained results from the devel-oped non-linear procedure are in good agreement with those ob-tained from the ANSYS finite element software.

20m

m

z

x

0 Interface layer

Upper sublaminateBase laminate

h0

2.4 mm

w=

0,u=

u 0

00 45 45

ons of the specimen.

0

50

100

150

200

250

300

350

400

450

-0.7 -0.2 0.3 0.8 1.3

(w/t)

Present Study Base Laminate

Ansys Base Laminate

Present Study Sub-laminate

Ansys Sub-laminate

Lay-up[0/45/45//0]

(a)

Nx

(N/m

m)

0

50

100

150

200

250

300

350

400

450

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

uo (mm)

Nx

(N/m

m)

Present Study

Ansys

Lay-up[0/45/45//0]

Buckling Point

(b)

Fig. 9. Comparison of the results obtained from the present study with Ansyscommercial code for the [0/45/45//0] lay-up: (a) load versus normalized transversedeflection and (b) load versus end-shortening displacement.




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6. Numerical models

The first numerical example is a unidirectional laminated com-posite plate, containing a central through-the-width delaminationexposing to a uniform in-plane end-shortening. The laminate ismade up of T300/976 graphite/epoxy material, with a [04/012//04]lay-up where ‘‘//” symbol indicate the through-the-thickness loca-tion of delamination. The material properties, needed for numeri-cal analysis, are presented in Table 1. Also, details of geometry,dimensions and boundary conditions of the laminate, are availablein Ref. [8]. Some geometrical information are presented in Fig. 7b.In the present model, the penalty stiffness of the interface layer isconsidered to be 106 N/mm. Instead of imperfection, concentratedloads equal to 3 N are applied to the nodes of centerline of theupper and lower layers. The obtained numerical results for the var-iation of normalized compressive load versus central out of planedisplacement and also versus the applied axial end-shortening,are illustrated in Fig. 7a and b and compared with the presented re-sults in Ref. [8] and the results show good accordance. These fig-ures also show that, at a load around 210 N/mm the localbuckling of the delaminated layers occurs but the base laminatestill has very small deflection. By increasing the load, the delamina-tion propagates and the delaminated layers deflect more. At a load

(b)

(a)

Fig. 10. (a) Geometry and dimensions and


around 1350 N/mm, the base laminate buckles and deflects down-ward and the delamination area becomes very large. The oscilla-tions in the results for the variation of normalized compressiveload versus central out of plane displacement, presented in Ref.[8], is due to the use of crack closure method with an elastic anal-ysis for delamination growth. However, a continuous variation ofnormalized compressive load versus central out of plane displace-ment was obtained in the present work which is due to the use ofinterface layer with non-linear constitutive law. In addition, thedifferences between the results of the present study with thosein Ref. [8] especially in the last loading steps could be due to theuse of different type of elements and numerical method in whichwe used layerwise-interface elements however reference [8] useddiscontinues layerwise in the finite strip method. The advantageof the present method is the capability to predict the embeddeddelamination growth with any arbitrary shape.

The other series of examples consist of two symmetric cross-plylaminates containing central through-the-width delaminationswith different lay-ups and laminated plates with two types ofstacking sequences containing central rectangular embeddeddelamination. For these laminates, the AS-4/3501-6 graphite/epoxy material with the available mechanical properties and listedin Table 1 is used.

(b) finite element mesh of the model.




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6.1. Through-the-width delamination

The laminate geometry, dimensions and typical mesh of themodels are illustrated in Fig. 10. The plate is symmetrically loadedby applying displacement at two ends. The boundary conditionsare clamped for loaded ends and free for the other edges. An initialtransverse perturbation load is also applied on the midline of themodels with opposite directions for upper and lower layers.

The selected stacking sequences are [05/905/905//05] and [905/05/05//905] where ‘‘//” is the through-the-thickness location ofdelamination. Due to the symmetry condition with respect to they-axis, half of the laminate has been modeled. The penalty stiffnessof the interface elements, K, is considered to be in the order oftransverse modulus of elasticity, E22, and therefore is equal to8960 N/mm2. Variations of reaction load versus normalized trans-verse deflection and also versus the applied axial displacement areobtained and illustrated in Figs. 11 and 12 for both lay-ups.

Variations of load versus normalized transverse deflection inFig. 11a show that for the case of [05/905/905//05] lay-up, the buck-ling mode is of mixed type. The delamination is initially closed,then by increasing the load, the delaminated sub-laminate startsto deflect upward because of the presence of perturbation, butthe base laminate is still unaffected. When the load of the uppersub-laminate, reaches its critical value, which is less than the crit-ical load of the base laminate, the delaminated layers buckle butthey still can carry load in the post-buckling region. The buckling

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-1.8 -1.3 -0.8 -0.3 0.2 0.7 1.2 1.7 2.2

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(I)(II)

(III)

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(b)

Fig. 11. Results for [05/905/905//05] laminate: (a) load versus normalized transversedeflection and (b) load versus end-shortening displacement.


point has been shown in Fig. 11b as well. Fig. 11a also shows thatby further increasing of the load, a mixed-mode buckling phenom-enon occurs, the base laminate becomes critical, and the delamina-tion grows at the same time. Therefore, the laminate cannot standunder further load and the load versus displacement curve isdropped to a softening behavior as shown in Fig. 11b.

Fig. 12a shows the variations of reaction load versus normalizedtransverse deflection for [905/05/05//905] laminate. Because of thelow bending stiffness of delaminated layers (upper sub-laminate)with the fiber direction of 90o, it starts bending upward due tothe perturbation forces at the early stages of loading, and thereisn’t a clear buckling point for this case. This is approved inFig. 12b by illustrating the load versus in-plane end-shorteningdisplacement. In the early loading steps, the base laminate is notaffected significantly. By increasing the applied load, the base lam-inate starts to buckle and at this point, the upper sub-laminate de-flects upward by the constraint of the base laminate, and then thedelamination growth is occurred. Therefore, the laminate can’tstand under further increase of load and the load versus displace-ment curve is dropped to a softening behavior as shown in Fig. 12b.

The contour of damage propagation corresponding to the load-ing steps of (I), (II) and (III) shown in Fig. 11a for [05/905/905//05]laminate, are also illustrated in Fig. 13. In this figure, the dark areasdenote the delaminated parts and the hatched areas correspond tothe initial defined delamination area. These figures show gradualdelamination propagations by increasing the end-shortening atpost-buckling stage of the laminate.

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(b)

Fig. 12. Results for [905/05/05//905] laminate: (a) load versus normalized transversedeflection and (b) load versus end-shortening displacement.



predefined delamination

Delamination growth area

u0= 0.071 mm u0= 0.101 mmu0 = 0.096 mm

(a) (b) (c)

Fig. 13. Damage propagation contours for [05/905/905//05] lay-up at the loading points of I, II and III in Fig. 10a: (a) at point I, (b) at point II, (c) at point III.

Fig. 14. Geometry and dimensions of embedded delamination specimens.


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6.2. Embedded delamination

The geometry and dimensions of the laminates containing aninitial rectangular embedded delamination are shown in Fig. 14.Two lay-ups of unidirectional ([04/04/04//04]) and quasi-isotropic([0/�45/452/�45/90/02/02/90/�45//452/�45/0]) are consideredfor the laminates. The laminate is loaded by applying displacementat one end as shown in Fig. 14 and the boundary conditions areclamped at the laminate edges. In order to prevent the completefailure of the delaminated layers, and to reduce the computationaltime, a predefined permitted delamination area with the size of40 mm � 30 mm was selected for the laminates. For this purpose,the behavior of the existed interface elements between this areaand the edges of the laminate are forced to be elastic. However,the quasi-isotropic lay-up is un-symmetric about the x- and y-axis,but half of the specimen is only modeled to decrease the computa-tional time. Besides, in order to reduce the computational effort,each group of four material layers is assumed as one numericallayer by means of the laminated element concept.

In the case of unidirectional lay-up, variations of load versusdisplacement results are shown in Fig. 15a and b. The analyses


were performed for two conditions of buckling without delamina-tion propagation (elastic interface elements), and buckling withdelamination propagation. Fig. 15a shows that, before occurrenceof the buckling, the stationary and propagating delaminationbehaviors are coincident. A few steps after the buckling point,the delamination grows unsteadily and leads to an abrupt reduc-tion in the curve slope. After that, delamination grows more stea-dily until it reaches the borders that the propagation has beenprevented intentionally (dashed lines in Fig. 17); in this stage theplate starts buckling in the mixed mode condition.

The obtained variations of load versus displacement for quasi-isotropic laminate are illustrated in Fig. 16a and b for both withand without delamination propagation. Fig. 16a shows that, beforethe point with Nx = 550 N/mm, there is no delamination and thebuckling behaviors without delamination and with delaminationpropagation of the laminate, are the same. After this point, delam-ination starts to propagate and therefore the stiffness of the lami-nate diminishes as shown by decrease in the slope of the curve inFig. 16b as well. It is also worth to mention that, before the prop-agating of delamination, the buckling mode is of global type. But,as the delamination grows and becomes larger, the buckling mode



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Unstable delamination propagation

with Del. Prop.

without Del. Prop.

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without delamination propagation

with delamination propagation

Lay-up[04/04/04//04]s

(b)

Fig. 15. Results of [04/04/04//04] lay-up: (a) load versus normalized transverse deflection and (b) load versus end-shortening displacement.

Fig. 17. Damage propagation contours for [0/�45/452/�45/90/02]s lay-up at the indicated points in Fig. 15a: (a) at point I, (b) at point II, (c) at point III, and (d) at point IV.

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without Del. Prop.

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without delamination propagation

Lay-up[0/-45/452/-45/90/02/02/90/45//-45/452/0]s

(b)

Fig. 16. Results for [0/�45/452/�45/90/02]s lay-up: (a) load versus normalized transverse deflection and (b) load versus end-shortening displacement.


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Please cite this article in press as: Hosseini-Toudeshky H et al. Delamination buckling growth in laminated composites using layerwise-interface element.Compos Struct (2010), doi:10.1016/j.compstruct.2010.01.013



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changes to the mixed mode type. To have a better understanding ofdelamination growth in various loading steps, the contours of thedamage propagation for the loading steps, which are pointed inFig. 16a, are also shown in Fig. 17. In this figure, the dashed linesspecify the permitted delamination growth area and dark regionsdenote the delaminated areas.

7. Conclusion

The buckling delamination growth of composite laminates con-taining initial through-the-width or embedded delamination wasinvestigated in this study. The delamination growth caused bycompressive loads was predicted via considering the softeningbehavior for interface layers which was implemented in the fulllayerwise lamination theory. The geometrical non-linearity of theproblem was made by von-karman non-linear terms. It was shownthat the delamination growth can be well predicted using the lay-erwise-interface elements with decohesive law. It was also shownthat, the buckling mode and delamination growth process, de-pends on the stacking sequence of the laminates. These figures alsoshow the gradual delamination propagations by increasing theend-shortening at post-buckling stage of the laminates containinginitial through-the-width or embedded delamination.

References

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[3] Riccio A, Perugini P, Scaramuzzino F. Modelling compressive behavior ofdelaminated composite panels. J Comput Struct 2000;78:73–81.

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[15] Benzeggagh ML, Kenane M. Measurement of mixed-mode delaminationfracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. J Compos Sci Technol 1996;56:439–49.

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