adhesive bonded composite joint analysis with ed surface ply

Adhesive-Bonded Composite Joint Analysis with DelaminatedSurface Ply Using Strain-Energy Release Rate

Alireza Chadegani∗ and Chihdar Yang†

Wichita State University, Wichita, Kansas 67260

and

Stanley S. Smeltzer III‡

NASA Langley Research Center, Hampton, Virginia 23681

DOI: 10.2514/1.C031516

This paper presents an analyticalmodel to determine the strain energy release rate due to an interlaminar crack of

the surface ply in adhesively bonded composite joints subjected to axial tension. Single-lap shear-joint standard test-

specimen geometry with thick bondline is followed for model development. The field equations are formulated by

using the first-order shear-deformation theory in laminated plates together with kinematics relations and force

equilibriumconditions. The stress distributions for the adherends and adhesive are determined after the appropriate

boundary and loading conditions are applied and the equations for the field displacements are solved. The system of

second-order differential equations is solved to using the symbolic computation tool Maple 9.52 to provide

displacements fields. The equivalent forces at the tip of the prescribed interlaminar crack are obtained based on

interlaminar stress distributions. The strain energy release rate of the crack is then determined by using the crack

closure method. Finite element analyses using the J integral as well as the crack closure method are performed to

verify the developed analytical model. It has been shown that the results using the analytical method correlate well

with the results from the finite element analyses. An attempt is made to predict the failure loads of the joints based on

limited test data from the literature. The effectiveness of the inclusion of bondline thickness is justified when

compared with the results obtained from the previous model in which a thin bondline and uniform adhesive stresses

through the bondline thickness are assumed.

Nomenclature

A11, AU11, A

L11 = in-plane modulus per unit width, N=m

A55, AU55, A

L55 = transverse modulus per unit width, N=m

a = prescribed crack length, mB11, B

U11, B

L11 = coupling modulus per unit width, N

b = virtual crack extension length, mD11, D

U11, D

L11 = flexural modulus per unit width, N:m

Ea = adhesive Young’s modulus, PaFx, Fy = reaction force per unit width, N=mGa = adhesive shear modulus, PaGT = total strain energy release rate, J=m2

GTc = critical total strain energy release rate, J=m2

Gxz = adherend x-z plane shear modulus, PaGI = mode I strain energy release rate, J=m2

GIc = critical mode I strain energy release rate, J=m2

GII = mode II strain energy release rate, J=m2

GIIc = critical mode II strain energy release rate, J=m2

h2 = adherend ply thickness, mhU, hL, h = adherend thickness, mJ = J integral value, J=m2

ks, kUs , k

Ls = shear correction factor

ln = notch length of the specimen, mlo = overlap length before crack is initiated, mMC = equivalent crack-tip moment, NMy,M

Uy ,M

Ly = bending moment per unit width, N

NC = equivalent crack-tip force, N=mNx, N

Ux , N

Lx = normal stress resultants per unit width, N=m

P = applied tensile force per unit width, N=mQC = equivalent crack-tip transverse force, N=mQz, Q

Uz , Q

Lz = transverse shear stress resultant per unit width,

N=mqa = adhesive peel stress, Paqi = interlaminar peel stress, PaTi = (i� 1; 2; 3), traction components, N=m2

U = strain energy per unit width, J=mU� = strain energy density, J=m3

u, uU, uL = x directional displacement, mua = adhesive x-directional displacement, mui = (i� 1; 2; 3), displacement components, muo, uoU, uoL = midplane x directional displacement, muo8 , w

o8 = displacement components at midplane of

adherend portion 8, mu� = bottom-surface x-directional displacement, mu�8 , w

�8 = displacement components at the bottom

surface of adherend portion 8, m�a = adhesive Poisson’s ratioW = work required to close virtual crack per unit

width, Nw, wU, wL = z-directional displacement, mwa = adhesive z-directional displacement, mw� = bottom-surface z-directional displacement, m"a, �a = adhesive strains"x, "z, �xz = adherend strains� = adhesive thickness, m�x = adhesive normal stress, Pa�a = adhesive shear stress, Pa�i = interlaminar shear stress, Pa , U, L = bending slope, rad

I. Introduction

A DVANCED composite materials and adhesive-bondingtechnology have been widely applied in aerospace/aircraft

Received 23 May 2011; accepted for publication 8 August 2011.Copyright ©2011 by theAmerican Institute ofAeronautics andAstronautics,Inc. All rights reserved Copies of this paper may be made for personal orinternal use, on condition that the copier pay the $10.00 per-copy fee to theCopyright ClearanceCenter, Inc., 222RosewoodDrive,Danvers,MA01923;include the code 0021-8669/12 and $10.00 in correspondence with the CCC.

∗Graduate Research Assistant, Department of Aerospace Engineering;currently Ph.D. Candidate and Research Assistant, Department ofEngineering Science and Mechanics, Virginia Polytechnic Institute andState University; [email protected].

†Professor and Airbus Fellow, Department of Aerospace Engineering;[email protected].

‡Deputy Manager, Ares Project Office; [email protected].

JOURNAL OF AIRCRAFT

Vol. 49, No. 2, March–April 2012

503

http://dx.doi.org/10.2514/1.C031516

structures thanks to their high strength-to-weight ratios and excellentresistance to corrosion. Inmany applications, bolted joints have beenreplaced by adhesively bonded joints because of the weight penaltyand corrosion problems associated with bolted joints. One of themajor issues in applications of adhesively bonded composite joints isthe prediction of damage and failure mechanisms. A completeinvestigation of the failure modes leads to an efficient and durablejoint design.

Delamination always refers to interlaminar failure, which initiatesby a crack in thematrix andmay lead to fiber separation. Therefore, amatrix crack occurs within laminates where the fibers are parallel tothe load direction and these fibers are placed adjacent to an adhesivebond. Based on the direction of crack propagation, edge or localdelaminations have been categorized. Edge delamination occurs atthe load-free edges of the laminate, while local delamination initiatesfrom a transversematrix crack. It has been observed in simple tensiontests of a uniform rectangular-cross-section specimen that delami-nations begin along the load-free edges and propagate normal to theload direction [1]. Many research papers have been devoted to thestudy of matrix cracking in laminated composite plates, althoughmost have developed finite element approaches. Moreover, severalresearchers have studied the surface-ply delamination failure oflaminated composite plates analytically, while surface-ply failure inbonded laminated composite joints rarely has been discussed.Among researchers, Tong et al. [2] and Tong [3] have shown thefailure of adhesively bonded composite joints with embeddedinterlaminar cracks based on Williams’s mode partitioning model[4]. Recently, Chadegani et al. [5] have developed an analyticalmodel based on the fracture-mechanics approach for calculatingstrain energy release rate of adhesively bonded composite joints withinterlaminar crack for thin bondlines.

An accurate prediction of a delamination failure mechanism andsubstrate failure load are essential in understanding the failureprocess aswell as reliability of bonded laminated composite joints. Ina bonded composite joint, failure typically occurs in the surface plyof the adherend near the stress singularity [1].

Delaminations near free edges, holes, ply drops, and notches havebeen noticed in laminated composites in service. In practicalcomposite structures, delamination is a mixed-mode fractureprocess, which includes mode I (opening mode), mode II (shearingmode), and mode III (tearing mode). The growth process of edgedelaminations and local delamination is often modeled by a fracture-mechanics-based approach leading to the calculation of the strainenergy release rate, which can be considered for various geometriesand loading conditions. In combination with an appropriate failurecriterion, the strain energy release rate can be used as a means topredict the failure load of the structure. In a research study conductedbyYang et al. [6] on adhesively bonded joints, it was concluded that afracture-mechanics-based approach would be an effective methodfor predicting the load-carrying capacity of bonded joints.

Earlier studies on adhesively bonded joints were directed byGoland and Reissner [7] Erdogan and Ratwani [8], Hart-Smith [9],and Adams and Peppiatt [10]. Enormous literature on the analysis ofadhesively bonded joints-can be found, among others, in extensivereviews by Kutscha [11], Kutscha and Hofer [12], Matthews et al.[13], Vinson [14], da Silva et al. [15], and Zhao et al. [16]. Tsai andMorton [17] compared results from a two-dimensional (2-D) geo-metrically nonlinear finite element analysis with those from theanalytical solutions. Yang and Pang [18] derived an analytical modelthat provided the stress distributions of adhesively bonded single-lapcomposite joints subjected to axial tension. Huang et al. [19] andYang et al. [20] also derived an elastic–plastic model for adhesively

bonded single-lap composite joints. Their approaches have includedimportant capabilities such as the asymmetry of the adherendlaminates and effects due to transverse shear deformation. Anexisting crack is usually assumed to be in a joint when conducting afracture analysis. Several methods in the literature are available forcalculating the strain energy release rate: finite element analysis andthe complex variable stress potential approach [1]. In a reportpublished in 2002, Krueger [21] described the virtual crack closuretechnique (VCCT), including its history, approach, and applicationsin conjunction with finite element analysis. Davidson et al. [22]published a series of papers that employed the classical plate theoryversion of the VCCT to predict the strain energy release rate ofmixed-mode delamination in composite laminates. A crack–tip forcemethod was derived by Park and Sankar [23] to compute the strainenergy release rate in delaminated beams and plates. Kim and Kong[24] proposed a simplified method for determining the strain energyrelease rate of free-edge delamination in composites using theclassical laminated plate theory. Davidson [25] andYu andDavidson[26] used three-dimensional (3-D) crack-tip elements to analyticallydetermine strain energy release rate and mode-mixity for differenttypes of laminated plates containing delamination. Their results werecomparedwith 3-Dfinite element analyses and theVCCT.Moreover,Rybicki and Kanninen [27] and Raju [28] used finite elementmethods to calculate the strain energy release rate.

Finite element methods play a significant role in structuralanalysis and have been widely used to study the adhesively bondedcomposite joint. Wang et al. [29] applied the VCCT to calculate thestrain energy release rate of cracked composite panels with nonlineardeformation. Wei et al. [30] presented an improved VCCT todetermine the energy release rate using a three-step analysis that wasused also by Murthy and Chamis [31]. Contour integrals to computethe strain energy release rate have been employed, among others, byFernlund et al. [32], Yang et al. [33,34], and Chadegani et al. [5].Yang et al. [33,34] developed finite element models using the finiteelement software ABAQUS® [35] to estimate the J integral of anadhesively bonded joint with a crack. Although finite elementmethods are capable of solving problems with various types ofmaterials and complicated geometrical configurations, analyticalmethods offer advantageous performance and solutions, especiallywith parametric analyses and optimization. Interlaminar fracture incomposites using the sublaminate approach has been studied, amongothers, by Armanios and Rehfield [36–38] and Rehfield et al. [39], inspite of simplifications related to their case studies. A comparativestudy of the analytical models can be found in a review paper by daSilva et al. [40].

As described earlier, the fracture-mechanics-based analysis isbelieved to be effective in predicting the load-carrying capacity of anadhesively–bonded composite joint [6]. It should be emphasize thatthe advantage of a simple and robust analytical model motivates thecurrent study. The objective of the present paper is to develop ananalytical fracture-mechanics-based model that can be used todetermine the strain energy release rate and failure load in anadhesively bonded single-lap composite joint due to surface-plydelamination failure. Because of the availability of test data, themodel derivation and failure analysis are based on the ASTMD3165[41] specimen geometry shown in Fig. 1. Cracks usually start atlocations of high stress concentration. For the ASTM D3165specimen configuration the critical area is located at the corner of theadhesive layer, where a continued adherend and a discontinuedadhesive are present. Often times, either the adhesive/adherend inter-face starts to fail, as in the case of metal adherends, or the adherendstarts to fail, as in the case of composite adherends. Failure of such

Fig. 1 ASTM D3165 [41] specimen geometry and dimensions.

504 CHADEGANI, YANG, AND SMELTZER

joints at the adherend/adhesive interface with either thin or thickbondlines has been studied by Yang et al. [33,34] and Chadeganiet al. [5] Because focus of the current study is on the surface-plyfailure/delamination of the adherend, the failure is assumed to initiateat the surface ply adjacent to the adhesive, as shown in Fig. 2.Thereafter, the crack propagates along the length direction of thejoint until the entire joint unzips.

Linear–elastic material properties as well as small displacementsare assumed for both the adhesive and adherends in order tomake theanalytical approach feasible. In the following sections, step-by-stepprocedure and formulation of the problem using laminatedorthotropic plate theory for the adherend to calculate interlaminarstress distributions are discussed. The analytical solutions are deter-mined using the symbolic computation toolMaple 9.52 [42]. Resultsfrom the analyticalmodel are verified byfinite element analysis usingABAQUS 6.8-1 [35].

II. Stress and Displacement Model Development

Details of the analytical method for determining the strain energyrelease rate of adhesively bonded single-lap composite joints with aprescribed interlaminar crack is based on the laminated plate theoryversion of Irwin’s virtual crack method [43]. The strain energyrelease rate is derived in terms of the forces and moment at the cracktip: NC, QC, and MC. These forces and moment at the crack tip aredetermined from the linear–elastic shear and peel-stress distributionsat the interface between the first and second plies of the adherendadjacent to the adhesive within the overlap area. Therefore, adescription of the stress and displacement states in the pre- andpostpropagation specimen geometry is required before an estimate ofthe strain energy release rate can be obtained. A summary of themethodology used to derive the equations for determining therequired stress and displacement fields in an adhesively bonded jointis presented in this section.

An adhesively bonded single-lap joint with the standard geometryof an ASTMD3165 [41] specimen and an applied tensile load P perunit width is shown in Fig. 2. The joint is divided into six regions inthe x direction for the convenience in the model development, whereregions 1, 4, and 6 consist of two adherends and a thick adhesivelayer, regions 2 and 5 represent two notches, and region 3 is the areawhere surface-ply delamination failure of the laminate is located, as

shown in Fig. 2. Region 4 is the bonded-joint overlap area where theapplied mechanical loads are transferred from one adherend to theother; this is also the area on which joint strength is typically based.

Overall procedure of the model development can be summarizedas follows:

1) Divide the model into adherend portions according to theASTM D3165 [41] specimen geometry, and derive the kinematicsrelations for each adherend portion.

2) Derive force and moment equilibriums for each adherendportion according to sign convention.

3) Derive equations for the adhesive and interlaminar stressesusing the kinematics of adherends.

4) Specify boundary conditions according to the loading status anddisplacement constraints.

5) Find the equivalent crack-tip forces andmoment, and propagatethe crack to find displacement distributions.

6) Apply the crack closure method to calculate the strain energyrelease rate.

A. Adherend Formulation

The specimen is divided into 12 adherend portions from left toright, as shown in Fig. 3. The general formulas for the adherendportions are the same for all, except for adherend portions 5 and 8.The displacement fields of the adherend are described by thelaminated orthotropic plate theory.

1. Adherend Portions 1–4, 6, 7, and 9–12

Based on the first-order laminated plate theory, the displacementfields for adherend portions 1–4, 6, 7, and 9–12 can be written as

u� uo�x� � z �x� (1)

a) The entire joint

b) The overlap areaFig. 2 Regions used in model derivation.

Fig. 3 Discretized model with 12 adherend portions.

CHADEGANI, YANG, AND SMELTZER 505

w�w�x� (2)

where the superscript o represents the midplane displacement, z ismeasured from the midplane of each adherend portion, and is thecorresponding bending slope. Considering the given displacementfunctions and the strain-displacement relations yields the normalstrain "x and shear strain �xz for each adherend portion as

"x �@u

@x� duo�x�

dx� z d �x�

dx(3)

�xz �@w

@x� @u@z� dw�x�

dx� �x� (4)

Therefore, based on the classical laminate theory, the relationshipsamong stress resultants, strain, and bending slope can be formulated.Furthermore, using the extensional stiffness �A�, the couplingstiffness �B�, and the bending stiffness �D� for orthotropic laminates,the normal stress resultant Nx, bending moment for unit width My,and transverse shear stress resultant Qz can be obtained as

Nx � A11

duo�x�dx� B11

d �x�dx

(5)

My � B11

duo�x�dx�D11

d �x�dx

(6)

Qz � ksA55

� �x� � dw�x�

dx

�(7)

where ks is the shear correction factor, andAij,Bij, andDij are termstaken from the common extensional, bending, and extensional-bending coupling stiffness matrices from the laminated plate theoryas

�A11; B11; D11� �Zh=2

�h=2Q�i�11 � 1; z; z2 � dz (8)

A55 �Zh=2

�h=2Q�i�55 dz (9)

where Q�i�11 and Q�i�55 represent the normal and transverse shearstiffness, respectively, of the ith ply, and h is the thickness of theadherend.

2. Adherend Portion 5

As shown in Fig. 3, adherend portion 5 is the delaminated surfaceply within the overlap area. Because of the small length-to-thicknessratio of adherend portion 5, which is a single ply, displacementdistributions u and w in the x and z directions, respectively, areassumed to be independent of z, uo5 , and w

o5 (see Fig. 4), i.e., no

bending slope contributes to u within this portion. This is areasonable assumption because of the thickness in this adherendportion, which is a single ply, small, and traction-free at the topsurface. The normal and shear stress resultants N5x and Q5z and thebending momentM5y per unit width are related to the strains withinthis adherend portion and the constitutive relations of the adhesivematerial. Because of small thickness, M5y is assumed to benegligible:

N5x �Zh2=2

�h2=2Q11

duo5�x�dx

dz (10)

M5y �Zh2=2

�h2=2Q11

duo5�x�dx

z dz 0 (11)

Q5z �Zh2=2

�h2=2ksQ55

�dwo5�x�dx

�dz (12)

where h2 is the thickness of adherend portion 5.

3. Adherend Portion 8

Displacement distributions u and w in the x and z directions,respectively, within adherend portion 8, which is also a single ply, areassumed to follow a similar approach as discussed for adherendportion 5. The normal and shear stress resultants and bendingmoment per unit width can be related to the displacement fields asspecified in Eqs. (10–12).

To correlate the shear and peel stresses at the adherend/adhesiveinterface at the bottom of adherend portion 8, the displacements uo8and wo8 at the midplane, and the displacements u�8 and w�8 at thebottom surface of adherend portion 8, as shown in Fig. 5, are used byassuming that �z and �xz are uniform within the lower half ofadherend portion 8. Therefore, as shown in Fig. 5, the shear stress andpeel stresses ��a and q

�a are the same as �xz and �z, respectively, within

the lower half of adherend portion 8 and can be represented in termsof the midplane displacements uo8 and wo8 and the bottom-surfacedisplacements u�8 and w

�8 as

��a � �xzjz��h2=2 ��Gxz�2

h2�uo8 � u�8� �

dwo8dx

�(13)

q�a � �zjz��h2=2 ��xz � �xy�yzExEy�

�duo8dx

� 2

�1 � �xy�yxExEy�

��wo8 � w�8�

h2(14)

where �xy and �yx are the major and minor Poisson’s ratios in the x-yplane;�xz is themajor Poisson’s ratio in the x-z plane;�yz is themajorPoisson’s ratio in the y-z plane; Ex, Ey, and Gxz are the moduli ofelasticity in the x and y directions and shear modulus in the x-z plane,respectively; and � can be given as

��1 � �xy�yx � �yz�zy � �zx�xz � 2�xy�yz�zx

ExEyEz(15)

where �zy and �zx are minor Poisson’s ratios in the y-z plane and thex-z plane, respectively, and Ez is the modulus of elasticity in the zdirection.

The interlaminar shear and peel stresses �i and qi at the uppersurface of adherend portion 8, as shown in Fig. 5, which are thestresses between adherend portions 7 and 8, are to be used for theequivalent crack-tip forces calculation and, later on, for the strainenergy release rate determination. Assuming uniform �z and �xzwithin the upper half of adherend portion 8 and a perfect bondbetween adherend portions 7 and 8, based on the kinematics and

Fig. 4 Free-body diagram of adherend portion 5. Fig. 5 Free-body diagram of adherend portion 8.


constitutive relations, �i and qi are related to the displacementfunctions of adherend portions 7 and 8 as

qi ��xz � �xy�yzExEy�

�duo8dx� 2

h2

�1 � �xy�yxExEy�

��w7 � wo8� (16)

�i ��Gxz�2

h2

�uo7 �

h12 7 � uo8

�� dwo8

dx

�(17)

where h1 is the thickness of adherend portion 7.The bottom surfaces u�8 and w�8 , shown in Fig. 5, will be solved

later by the equilibrium equations of adherend portions 8 and 9.

B. Adhesive Formulation

The adhesive is assumed to behave as an elastic–isotropicmaterial.The adhesive displacements in the x direction, ua, and the z direction,wa, are assumed to be polynomials of both x and z. To ensure thecontinuity of the displacement fields of the adhesive and adherends atthe interfaces (z�h=2), ua and wa are written as

ua � uo�x; z� �Xnm�1

amum�x; z� � uo�x; z�

��z� �

2

��z � �

2

��a1 � a2x� a3z� � � �� (18)

wa � wo�x; z� �Xnm�1

bmwm�x; z� �wo�x; z�

��z� �

2

��z � �

2

��b1 � b2x� b3z� � � �� (19)

where � is the adhesive thickness, and uo�x; z� and wo�x; z� aredifferent for each region and will be described in the followingsections.

The assumption of terms um andwm in the adhesive displacementfields, which are zero at both upper and lower adhesive/adherendinterfaces, z� �=2 and z��=2, can be regarded as the deviationof the displacement field from a linear function of z.

It is assumed that the adhesive normal stress in the x direction isnegligible; only peel stress �a;z and shear stress �a;xz exist in theadhesive. Under the assumption of plane-strain condition, theadhesive peel and shear stresses �a;z and �a;xz can be formulated interms of the adhesive strains "a;x, "a;z, and �a;xz as

�a;z �Ea

�1� �a��1 � 2�a��a"a;x � �1 � �a�"a;z�

� Ea�1 � �a�2

"a;z �Ea

�1 � �a�2@wa@z

(20)

�a;xz �Ga�a;xz �Ga�@ua@z� @wa@x

�(21)

where Ea and Ga are the adhesive Young’s and shear moduli,respectively, and �a is the Poisson’s ratio of the adhesive.

The adhesive shear and peel stresses at the upper and loweradherend/adhesive interfaces, �Ua , �

La , q

Ua , and q

La , as shown in Fig. 6,

are

�Ua ��a;xzjz��2 (22)

�La ��a;xzjz��2 (23)

qUa � �a;zjz��2 (24)

qLa � �a;zjz��2 (25)

Strain energy in the adhesive can be written as

U� 1

2

ZZ��a;z"a;z � �a;z�a;z� dx dz

� Ea2�1� �a�

Z�=2

��=2

�Zl

0

�1

1 � �a

�@wa@z

�2

� 1

2

�@ua@z� @wa@x

�2�dx

�dz (26)

where l is the length of the overlap. Using the principle of minimumpotential energy, the displacement ua and wa must satisfy thefollowing equations when the adhesive is under static equilibriumcondition:

@U

@am�I

�Xum dS m� 1; 2; 3; . . . ; n (27)

@U

@bm�I

�Zum dS m� 1; 2; 3; . . . ; n (28)

where �X and �Z represent the surface loads in the x and z directionsapplied on the adhesive, respectively, and dS covers the periphery ofthe adhesive. Because the adhesive has two free surfaces at the left

and right edges in each region, both surface loads �X and �Z are zero atthese two edges. As described previously in Eqs. (18) and (19), umand wm are zero at both the upper and lower adherend/adhesiveinterfaces in order to satisfy the continuity condition. Therefore,Eqs. (27) and (28) become

@U

@am� 0 m� 1; 2; 3; . . . ; n (29)

@U

@bm� 0 m� 1; 2; 3; . . . ; n (30)

1. Adhesive Midplane Displacement-Field Formulation in Regions 1

and 6

Using displacement fields of the upper and lower adherends,adhesive midplane displacement fields can be written as

uo�x; z� �1

2uoU�x� � 1

4hU U�x� � 1

2uoL�x� � 1

4hL L�x�

� z

2��2uoU�x� � hU U�x� � 2uoL�x� � hL L�x�� (31)

Fig. 6 General free-body diagram and sign convention.


wo�x; z� �1

2wU�x� � 1

2wL�x� � z

��wU�x� � wL�x�� (32)

where hU and hL are the thicknesses of the upper and loweradherends, respectively.

2. Adhesive Midplane Displacement-Field Formulation in Region 3

Using displacement fields of adherend portions 5 and 6 on the topand bottom interfaces of the adhesive, respectively, the adhesivemidplane displacement fields can be written as

uo3�x; z� �1

2uo5�x� �

1

2uo6�x� �

1

4hL 6�x� �

z

2��2uo5�x�

� 2uo6�x� � hL 6�x�� (33)

wo3�x; z� �1

2w5�x� �

1

2w6�x� �

z

��w5�x� � w6�x�� (34)

3. Adhesive Midplane Displacement-Field Formulation in region 4

In region 4, the adherend portion above the adhesive (namely,adherend portion 8) is a single ply.While the adherend portion abovethe adhesive is modeled using the approach discussed previously, theadherend portion below the adhesive is modeled using the first-ordershear-deformation theory. In terms of the displacement fields of theadherend portions above and below the adhesive, the adhesivemidplane displacements can be written as

uo4�x; z� �1

2u�8�x� �

1

2uo9�x� �

1

4hL 9�x� �

z

2��2u�8�x�

� 2uo9�x� � hL 9�x�� (35)

wo5�x; z� �1

2w�8�x� �

1

2w9�x� �

z

��w�8�x� � w9�x�� (36)

where u�8 and w�8 are bottom-surface displacements of the adherend

portion 8 in the x and z directions, respectively, as shown in Fig. 5.

C. Adherend Equilibrium Equations

To establish the equations of equilibrium for each adherendportion, a free-body diagram of a differential element from theoverlap regions is illustrated in Fig. 6. The upper adherend portionshown in the figure is the adherend portion immediately above theadhesive and represents adherend portions 1, 5, 8, and 12 in regions1, 3, 4, and 6, respectively.

The lower adherend portion in the figure is the lower adherendunder the adhesive in each region. The general equations for forceand moment equilibrium of the adherend portion above the adhesiveare given as

dNUxdx� �i � �Ua (37)

dMUy

dx�QU

z �hU

2��Ua � �i� (38)

dQUz

dx� qUa � qi (39)

where �U1a and qU1a are the shear and peel stresses on the top surface of

the adhesive in region 1, �i and qi are the interlaminar shear and peelstresses of the upper adherend portion, and hU is the thickness of theupper adherend portion. It should be noted that �i and qi do not existin regions 1 and 6 because the upper adherend portions in these tworegions are modeled as one piece, while the upper adherend portionsin regions 3 and 4 are single-ply, so �i and qi may be introduced.

Three equilibrium equations also can be obtained for the adherendportion below the adhesive in a similar manner, but without anystresses at the bottom surface, as

dNLxdx� �La (40)

dMLy

dx�QL

z �hL

2�La (41)

dQLz

dx��qLa (42)

where �L1a and qL1a are the shear and peel stresses on the bottomsurface of the adhesive in region 1.

Based on the general equilibrium equations for the adherendportions above and below the adhesive, equations of equilibrium foreach adherend portion can be written and are described in detailbelow.

1. Region 1: Adherend Portions 1 and 2

From the free-body diagram in Fig. 6, it can be observed that�i � 0 and qi � 0, because the top surface of adherend portion 1 is atraction-free surface. Hence, Eqs. (37–42) can be rewritten forregion 1 as

dN1x

dx��U1a (43)

dM1y

dx�Q1z �

h

2�U1a (44)

dQ1z

dx� qU1a (45)

dN2x

dx� �L1a (46)

dM2y

dx�Q2z �

h

2�L1a (47)

dQ2z

dx��qL1a (48)

By substituting the stress resultants from Eqs. (5–7) and theadhesive stresses on the top and bottom surfaces from Eqs. (20–25),six coupled second-order ordinary differential equations in terms ofuo1 , u

o2 , 1, 2, w1, and w2 are obtained.

2. Region 2: Adherend Portion 3

Region 2 represents a notch in the overall specimen geometry.Because of the interlaminar failure assumed between adherendportions 4 and 5, where the surface ply of the upper adherendadjacent to the adhesive is supposed to continue from the lower rightend of adherend portion 3 and form adherend portion 5, a crack isinitiated between adherend portions 3 and 5, as shown in Fig. 7, as thefree surface. Because of the free surface at the lower right end ofadherend portion 3, the effective thickness and the tensile, bending,and shear stiffnesses of adherend portion 3 are different from its leftend to its right end. Based on the free-body diagram shown in Fig. 7,the equations of equilibrium can be written as

NL3x � NR3x (49)

QL3z �QR

3z (50)


MR3y �ML

3y � NL3xh22�QL

3zL2 (51)

where superscriptsL andR refer to the left and right ends of adherendportion 3, respectively; h2 is the thickness of a single ply; and L2 isthe entire length of adherend portion 3. To obtainuo3 , 3, andw3 at theright end of adherend portion 3, the following equations are used,assuming a small change in their derivatives:

uo3 jx2�L2� uo3 jx2�0 �

L2

2

�duo3dx

��x2�0� duo3

dx

��x2�L2

�(52)

3jx2�L2� 3jx2�0 �

L2

2

�d 3

dx

��x2�0� d 3

dx

��x2�L2

�(53)

w3jx2�L2� w3jx2�0 �

L2

2

�dw3

dx

��x2�0� dw3

dx

��x2�L2

�(54)

where the first-order derivatives of uo3 , 3, and w3 at the right end,where x2 � L2 can be determined by the following equations, whichare based on equilibrium equations Eqs. (49–51). In Eqs. (55–57),the values of uo3 , 3, w3, and their derivatives at the left end, wherex2 � 0, are obtained from the right end of adherend portion 1 usingthe continuity condition. It should be noted that stiffnesses A11, B11,D11, and A55 have different values at the left and right ends due todelamination of the surface ply at the right end:�

A11

duo3dx

��x2�0��B11

d 3

dx

��x2�0��A11

duo3dx

��x2�L2

��B11

d 3

dx

��x2�L2

(55)

�ksA55

� 3 �

dw3

dx

��x2�0��ksA55

� 3 �

dw3

dx

��x2�L2

(56)

�B11

duo3dx

��x2�L2��D11

d 3

dx

��x2�L2

��B11

duo3dx

��x2�0

��D11

d 3

dx

��x2�0� h2

2

��A11

duo3dx

��x2�0��B11

d 3

dx

��x2�0

�

� L2

�ksA55

� 3 �

dw3

dx

��x2�0

(57)


Because of delamination of the surface ply at the bottom and thetraction-free surface at the top, equations of equilibrium of adherendportion 4 can be written as

dN4x

dx� 0 (58)

dM4y

dx�Q4z (59)

dQ4z

dx� 0 (60)

Substituting the stress resultants as functions uo4 , 4, and w4 intoEqs. (58–60) yields three coupled second-order ordinary differentialequations in terms of uo4 , 4, and w4:

A11

d2uo4�x�dx2

� B11

d2 4�x�dx2

� 0 (61)

B11

d2uo4�x�dx2

�D11

d2 4�x�dx2

� ksA55

� 4�x� �

dw4�x�dx

�(62)

ksA55

�d 4�x�dx

� d2w4�x�dx2

�� 0 (63)


As previously noted, adherend portion 5 contains a single ply andis separated from adherend portion 4 due to the prescribedinterlaminar crack. The equations of equilibrium of adherendportions 5 and 6 are similar to those for adherend portions 1 and 2 butwithout considering the bending moment in adherend portion 5, aspreviously mentioned:

dN5x

dx��U3a (64)

dQ5z

dx� qU3a (65)

dN6x

dx� �L3a (66)

dM6y

dx�Q6z �

h

2�L3a (67)

dQ6z

dx��qL3a (68)

Substituting the stress resultants from Eqs. (10) and (12) for N5x

and Q5z and the stress resultants from Eqs. (5–7) for N6x, M6y, andQ6z, into Eqs. (64–68), five coupled second-order ordinarydifferential equations uo5 , w

o5 , u

o6 , w6, and 6 are obtained.

5. Region 4: Adherend Portions 7, 8, and 9

A similar approach as that used for region 3 for adherend portions5 and 6 can be applied to region 4, with an additional adherendthickness at the top and the addition of the interlaminar stressesbetween adherend portions 7 and 8. Eight equations can be writtenbased on normal stress resultants, bending moment, and shear stressresultants as

dN7x

dx��i (69)

dM7y

dx�Q7z �

h12�i (70)

dQ7z

dx� qi (71)

Fig. 7 Free-body diagram of adherend portion 3.


dN8x

dx� �U4a � �i (72)

dQ8z

dx� qi � qU4a (73)

dN9x

dx� �L4a (74)

dM9y

dx�Q9z �

hL

2�L4a (75)

dQ9z

dx��qL4a (76)

where h1 and hL are thicknesses of adherend portion 7 and adherend

portion 9, respectively. The interlaminar shear and peel stresses �iand qi between adherend portions 7 and 8 are in terms ofdisplacement fields of adherend portions 7 and 8, as shown inEqs. (16) and (17).

Thereby, the displacements at the bottom surface of adherendportion 8, u�8andw

�8 , as shown in Fig. 5, can be solved in terms of all

other variables by equating the shear and peel stresses on the bottomsurface of adherend portion 5, as specified in Eqs. (13) and (14), andtheir corresponding shear and peel stresses from the top surface ofadhesive in region 4:

��a � �U4a (77)

q�a � qU4a (78)


This region represents a notch in the model. The equations ofequilibrium for adherend portion 5 are the same as those given foradherend portion 4 but with different adherend stiffnesses.


Because of the similarities between region 1 and region 6, the sameequations of equilibrium can be derived for region 6 withcorresponding variables.

III. Solution Methodology

The overall system of governing equations, including all sixregions, contains 31 s-order ordinary differential equations with 31unknown variables. A total of 62 boundary conditions, summarizedin Appendix, are obtained at the two ends of each region based oneither continuity or applied force conditions. The symbolic solverMaple 9.52 [42] was used to solve the system of equations and obtainthe displacement, strain, and stress fields. Governing equations foreach region are parametrically solved first and then continuity andboundary conditions are evaluated to solve for constants ofintegrations.

IV. Strain Energy Release Rate Calculation

Once the stress, strain, and displacement fields are known in theadhesively bonded single-lap composite joints, the crack closuremethod is applied to estimate the strain energy release rate of the jointwith a prescribed crack. According to linear–elastic fracturemechanics, the energy released due to an extension/propagation of acrack is equivalent to the work needed to close that extension/propagation [27], and this is the foundation of the crack closuremethod.

A. Analytical Approach

In deriving the expression for the strain energy release rate, a jointis assumed to have an overlap length lo, a notch size L2, and a cracklength a, which is located at the interface of the bottom surface ply ofthe upper adherend, as shown in Fig. 8. The displacement of the cracktipC, after a load is applied, can be determined using the mechanicalmodel previously described with an overlap region length ofL4 � lo � a.

Interlaminar and adhesive stress distributions can be determinedusing the solution from the analytical model described previously.Once the crack propagates a small length b, the previous crack tip Cseparates into two points A and B, as shown in Fig. 9. Before thecrack growth of the additional small lengthb, the surface ply betweenC0 andC adheres to the upper adherend,where interlaminar shear andpeel stresses exist at the interface, as shown on the left side of Fig. 10.

The interlaminar shear and peel stresses between C0 and C vanishafter the crack propagates a small length b. The equivalent crack-tipforces and momentNC,MC, andQC, corresponding to a small crackpropagation b, are related to the shear and peel stresses between C0

andC, as shown on the right side of Fig. 10, and can be calculated as

NC �Zb

0

�i dx4 (79)

MC ��Zb

0

qix4 dx4 (80)

QC ��Zb

0

qi dx4 (81)

where the interlaminar shear stress �i and peel stress qi are obtainedfrom the stress model with an overlap length L4 � lo � a. Note thatonly the bottom surface ply of the upper adherend, adhesive layer,and lower adherend are shown in Fig. 10 and that the positivedirections of NC, MC, and QC are defined to be consistent with thepositive directions of displacements u, , and w, respectively.

To determine the relative displacements between pointsA andB, asubsequent stress analysis is performed using a joint with a centraloverlap length L4 � lo � a � b, which simulates the overlap up tothe new crack tip C0.

Fig. 8 ASTM D3165 [41] specimen with initial crack of length a.

Fig. 9 ASTM D3165 [41] specimen with a virtual crack extension of

length b.


To close the virtual crack propagation (length b), crack-tip forcesare applied on pointsA andB tomove themback to the location of theoriginal crack tip C. Therefore, the total work required to close thesmall crack propagation b is

W � 12�NC�uB � uA� �MC� B � A� �QC�wB � wA�� (82)

For joints with a unit width, the strain energy release rate is definedas the derivative of energy released from the crack propagation withrespect to the length of the crack propagation as

GT �dU

da(83)

where U is the strain energy stored in the body. Based on the crackclosure method, the total energy released from the crack propagationis equivalent to thework needed to close the same crack propagation.The total strain energy release rate GT is a summation of mode I,mode II, and Model III strain energy release rates in a three-dimensional case. However, under the current plane-strain conditionthe mode III fracture does not exist. Therefore,GT can be calculatedas Eq. (84) with a virtual crack propagation of length b, where theequivalent crack-tip momentMC and transverse forceQC contributeto mode I strain energy release rate and the equivalent crack-tip in-plane force NC contributes to mode II strain energy release rate:

GT �GI �GII �dU

da� Wb� 1

2b�MC� B � A�

�QC�wB � wA�� 1

2b�NC�uB � uA�� (84)

B. Finite Element Approach Using Three-Step VCCT

The strain energy release rate due to the small increase in cracklength is equivalent to the energy per created crack surface area that isrequired to close that small crack increment. Therefore, the strainenergy release rate can be computed by finite element models usingthe three-step VCCT described below [33,34]. As shown in Fig. 11,the tip of a crack with an original length a is located at C.

Assuming a virtual crack propagation of length b, the new cracktip becomes C0, and the original crack tip becomes two separatenodes A and B. If nodes A and B are restrained at the original crack-tip location, this virtual crack of length b is closed, and the work toclose this virtual crack can be calculated by multiplying the reactionforces at nodes A and B by the relative displacements of these twoseparate nodes to the original crack tip C. The three-step proceduresto calculate the strain energy release rate using finite elementsoftware are described as follows:

1) Build the finite element model with an original crack of lengtha, and determine the displacements of crack tipC, uc, andwc in the xand z directions, respectively.

2) Propagate the crack with a small length b (usually one elementsize); then the original crack tip C becomes two separate nodes.Record the displacements of both nodes uA, wA, uB, and wB.

3) Constrain the two separate nodes so that they have the samedisplacements as the original crack tip C, and obtain the reactionforces FxA, FyA, FxB, and FyB.

4) The work needed to close the virtual crack is

W � 12�FxA�uc � uA� � FyA�wc � wA�� 1

2�FxB�uc � uB�

� FyB�wc � wB�� (85)

The total strain energy release rate is then obtained by

GT �W

b(86)

In the present study, the commercial finite element softwareABAQUS was used following the three-step procedures describedabove. Two-dimensional four-node bilinear quadrilateral plane-strain elements with reduced integration and hourglass control(CPE4R) were used in the finite element model for the three-stepVCCTapplication. The convergence of the strain energy release ratesolution using VCCT was studied by sequential refinement of thefinite element mesh, as shown in Fig. 12. Hence, the finite elementmodels with a total of 18,000 elements, which corresponds to 20-eight elements through the adherend thickness, i.e., four elements perply, were used to analyze the joints of seven-ply laminates of T300/E765 3KPW plain-weave carbon/epoxy (C/Ep) and 7781/E765satin-weave glass/epoxy (Gl/Ep), both with the same layup sequenceof [0=60= � 60=0= � 60=60=0]. Detailed view of the deformedmesh around the crack tip is shown in Fig. 13. Dimensions of theelements around the crack are 0:0635 0:0635 mm2.

Solution of the near crack-tip field indicates oscillation of thestresses in the immediate vicinity of the crack tip for cracks located atthe bimaterial interface [21,44]. As a result, the mode-mixity is notdeterministic as crack extension (length b) goes to zero. Thereafter,strain energy release rate was calculated for various choice of b, asshown in Fig. 14, to accommodate this issue and b=a� 0:2 (orb� 0:254 mm), the minimum value for stable G values, wasselected and was used hereinafter.

C. Finite Element Approach Using J Integral

Finite element models with the J integral calculation wereconstructed using ABAQUS to verify the present analytical model.Two-dimensional eight-node biquadratic plane–strain elements with

Fig. 10 Equivalent forces at the crack tip.

Fig. 11 Finite element approach using VCCT.

Fig. 12 Convergence of strain energy release rate vs finite element

mesh refinement.


reduced integration and hourglass control (CPE8R) were used in thefinite element model to calculate J integral. It should be noted thatmidside nodes of those elements sharing the crack tip moved to thequarter-points of the each element as described in [25].

The J integral is usually used in quasi-static fracture analysis tocharacterize the energy release associated with crack propagation.It is equivalent to the strain energy release rate if the materialresponse is linear–elastic. Considering an arbitrary counter-clockwise path � around the crack tip, as illustrated in Fig. 15, theJ integral is defined as

J �Z�

�U� dy � Ti

@ui@x

ds

�(87)

where repeated index implies summation, U� is the strain energydensity, ui (i� 1; 2; 3) are the components of the displacementvector, ds is the incremental length along the contour �, and Ti(i� 1; 2; 3) are components of the traction vector. The traction is astress vector normal to the contour. In other words, Ti are thecomponents of the normal stresses acting at the boundary if a free-body diagram on the material inside of the contour is constructed.

Several contour integral evaluations are possible at each locationalong the crack front. In a finite element model, each evaluation canbe thought of as the virtual motion of a block of material surroundingthe crack tip. Each block is defined by contours, and each contour is a

ring of elements completely surrounding the crack tip or crack frontfrom one crack face to the opposite crack face. These rings ofelements are defined recursively to surround all previous contours.ABAQUS/Standard automatically finds the elements that form eachring from the node sets given as the crack-tip or crack-frontdefinition. Each contour provides an evaluation of the contourintegral [27].

One should carefully apply Eq. (87) to calculate J integral sincethe crack is located at the interface of two different linear materials.Smelser and Gurtin [45] and Chou [46] have extended this quantityfor bimaterial bodies and have recommended this equation, without achange, to be applied only when the bondline is straight. Referring tothe deformed configurations of specimen shown in Fig. 16, it isobvious that bondline is straight; therefore, the extension of theEq. (87) can be used.

Theoretically, the J integral should be independent of the domainused, but the J integral estimated from different rings may varybecause of the approximate nature of the finite element solution.Strong variation in these estimates, commonly called domain-dependent or contour-dependent, indicates a need for mesh refine-ment (provided that the problem is suitable for contour integrals).Numerical tests suggest that the estimate from the first ring ofelements abutting the crack front does not provide a high-accuracyresult, so at least two contours are recommended. In the present study10 contours were used, and the average of last five contours wastaken as the final J integral value. This method is quite robust in thesense that accurate contour integral estimates are usually obtainedeven with quite-coarse meshes.

Sharp cracks, where the crack faces lie on top of one another in theundeformed configuration, are usually modeled using small-strainassumptions. Focused meshes for the J integral calculation shouldnormally be used for small-strain fracture-mechanics evaluations.

For linear–elastic materials, the linear–elastic fracture-mechanicsapproach predicts an r�1=2 singularity near the crack tip, where r is

Fig. 13 Detail view of the deformed mesh around the crack tip for

VCCT calculation.

Fig. 14 Convergence of strain energy release rate as a function of

normalized virtual crack extension length.

Fig. 15 Arbitrary contour around crack tip.

Fig. 16 Detail view of the deformed mesh around the crack tip for J

integral calculation.


the distance from the crack tip. In finite element analyses, forcing theelements at the crack tip to exhibit an r�1=2 strain singularity greatlyimproves accuracy and reduces the need for a high degree of meshrefinement at the crack tip [47]. This r�1=2 singularity can beproduced using an eight-node quadrilateral element by moving themidside nodes to the quarter-points, as noted by Barsoum [48] andHenshell and Shaw [49]. Yang et al. [33,34] have demonstrated acomplete step-by-step procedure calculating the J integral usingABAQUS. Even though the singularity at the crack tip of twodissimilar materials or orthotropic plates is known to be of the form�1=2 i� for two-dimensional problems, the use of quarter-pointsingularity elements that produce the classical square-root singu-larity, r�1=2, at the crack tip has been shown to yield converged resultsfor the total strain energy release rate GT calculation and a slightlynonconverged results for the individual modes [50] as describedpreviously. In the current study, two-dimensional eight-nodenonlinear quadrilateral plane-strain elements were used in the finiteelement model for the J integral application.

V. Material Properties, Layup, Dimensions,and Applied Load

To demonstrate the application of the developed model ASTMD3165 [41] specimens usingT300/E765 3KPWplain-weave carbon/epoxy (C/Ep) and 7781/E765 satin-weave glass/epoxy (Gl/Ep) witha ply thickness of 0.25 mm and a quasi-isotropic layup sequence of[60= � 60=0= � 60=60=0] as adherends were used. An interlaminarcrack was assumed at one edge of the overlap as described in themodel development section. The engineering constants for C/Ep andGl/Ep lamina were as follows.

C/Ep:

E11 � 56:50 GPa; E22 � 55:20 GPa

G12 � 3:86 GPa; �12 � 0:076

G1/Ep:

E11 � 26:40 GPa; E22 � 23:60 GPa

G12 � 4:14 GPa; �12 � 0:17

For convenience, other mechanical properties of adherends wereassumed as follows.

C/Ep:

E33 � 8:00 GPa; G13 �G23 � 3:56 GPa

�13 � �23 � 0:35

G1/Ep:

E33 � 3:74 GPa; G13 �G23 � 1:66 GPa

�13 � �23 � 0:16

The shear correction factor ks � 5=6was assumed. Three differentpaste adhesives, Hysol EA9394, PTM&WES6292, andMGSA100/B100, all with distinct moduli, were used in the analyses. Theseadhesives have the following material properties.

Hysol EA9394:

E� 4:24 GPa; �� 0:45

PTM&W ES6292:

E� 1:58 GPa; �� 0:31

MGS A100/B100:

E� 3:51 GPa; �� 0:34

The joint dimensions of the ASTM D3165 [41] specimen includethe central overlap length lo � 30 mm, notch sizeL2 � 1:6 mm, andadherend lengths outside the central overlap L1 � L6 � 78:4 mm.

Load P� 1:78 kN=m as the minimum load without failure wasapplied to calculate the strain energy release rate in this study.

Test data obtained by Tomblin et al. [51] for joint specimens withvarious bondline thickness and distinct adherend and adhesivematerials were analyzed and used for joint-failure load prediction.Results are given in the next section.

VI. Results and Discussion

In this study, ASTM D3165 [41] specimens with an interlaminarcrack were modeled analytically to determine the strain energyrelease rate using the methodology described in the previoussections. The symbolic solver Maple 9.52 [42] was used as themathematical tool. The finite element models for the VCCT and Jintegral were conducted using ABAQUS 6.8-1 [35] to verify theanalytical results.

Once the stress, strain, and displacement fields were obtained, thestrain energy release rates of the joints were estimated, and thesolutions from the analytical models were compared with finiteelement models using the VCCTand the J integral. mode I, mode II,and total strain energy release rates of an initial surface-plydelamination, a, of 1.27 mm under a 1.78 kN tensile load for a C/EpadherendwithHysol EA9394 adhesive of bondline thicknesses from0.127 mm to 3.175 mm are shown in Figs. 17–19, while Figs. 20–22show the results with Gl/Ep adherends. The strain energy release ratevalues for C/Ep and Gl/Ep are plotted vs bondline thickness to depictthe purpose of the present analytical model in dealing with thickbondlines. These figures also show the comparisons of strain energyrelease rate among the present model, the previous model [5] thatuses assumptions of thin bondline and uniform adhesive stressesthrough bondline thickness, and the finite element model using theVCCT. As expected, strain energy release rate values from differentmodels show an excellent match for the cases of a thin bondline,whereas the previous thin-bondline model shows large deviations forthicker bondlines. As can be seen from Figs. 17–22 that thedeveloped new model correlate reasonably well with the finiteelement results using the VCCT and also the J integral for GI, GII,and GT calculations. From these figures, it can be seen that GI, GII,and GT increase with the increase in bondline thickness under thesame applied load. If strain energy release rate is critical for jointfailure, the load-carrying capacity of the joint will be lower when thebondline is thicker. It is also noted that the contribution ofGII toGT isless than GI in all cases.

Even though the purpose of this investigation is to derive ananalytical model to calculate the strain energy release rates of anadhesively bonded composite joint due to a delamination crack at thesurface ply of the adherend, an attempt was made to predict the jointstrength using the limited available data. The test data used foranalysis were obtained by Tomblin et al. [51] of theNational Institutefor Aviation Research at Wichita State University. The specimens

Fig. 17 Mode I strain energy release rate as a function of bondline

thickness with C/Ep adherends and Hysol EA 9394 adhesive.


used for strength analysis are those failed under the same delami-nation mode of the surface ply of the adherend. Ideally, a series ofseparateASTM tests, including the double-cantilever-beam test [52],the end-notch flexure test [53,54], and the mixed-mode bending test[55], should be conducted in order to determine the critical strainenergy release rates, GIc and GIIc, of the composite laminates withvariousmodemixitiesGII=GT . The critical total strain energy releaserateGTc �GIc �GIIc, which is then a function of mode-mixity, canbe used as the failure criterion for the joint specimens analyzed herein this paper. Other criteria, such as power law criterion, exponentialhackle criterion, bilinear criterion, etc., were investigated bydescribed by Reeder [56]. Because of the fact that there are noavailable data on the fracture toughness, GIc, GIIc, etc., of the twocomposite laminates, [60= � 60=0= � 60=60=0] T300/E765 3KPWplain weave and 7781/E765 stain weave, which were used for theASTM D3165 [41] specimens analyzed in this paper, the requiredGIc and GIIc were backcalculated from the experimental data offailure load of ASTMD3165 specimens. This was done by applyingthe experimental failure load of each of the specimens of the samelaminate with a surface-ply delamination of 1.27 mm at the edge ofthe overlap to determine the GI and GII values using the analyticalmodel. The average values of the GI and GII at failure among allspecimens with the same composite adherend, even with differentadhesives, are used as GIc and GIIc. In other words, two sets of GIc

and GIIc were determined, one for T300/E765 (C/Ep) laminate,GIc � 0:95 kJ=m2, GIIc � 0:92 kJ=m2, and one for 7781/E765 (Gl/Ep) laminate, GIc � 0:74 kJ=m2, GIIc � 0:55 kJ=m2, even thoughthe specimens analyzed with T300/E765 (C/Ep) laminate werebonded by three different adhesives. These values are of the same

order as those of similar materials in the literature [57,58]. In thefollowing failure analysis, the predicted failure loads weredetermined based on either GIc, GIIc, or GTc, where GTc�GIc �GIIc. To demonstrate the effectiveness of the present model,the failure loads predicted by the present model were compared withthe failure loads predicted by the previous analytical model [5],which assumes a thin bondline with uniform adhesive stressesthroughout the adhesive thickness. The same method used to

Fig. 18 Mode II strain energy release rate as a function of bondline


Fig. 19 Total strain energy release rate as a function of bondline


Fig. 20 Mode I strain energy release rate as a function of bondline

thickness with Gl/Ep adherends and Hysol EA 9394 adhesive.

Fig. 21 Mode II strain energy release rate as a function of bondlinethickness with Gl/Ep adherends and Hysol EA 9394 adhesive.

Fig. 22 T strain energy release rate as a function of bondline thickness

with Gl/Ep adherends and Hysol EA 9394 adhesive.


determine the GIc and GIIc by the present model was used with theprevious analytical model to obtain the GIc and GIIc used for thefailure load prediction by the previous model, where GIc �0:32 kJ=m2 and GIIc � 0:53 kJ=m2 were used for the T300/E765(C/Ep) laminate andGIc � 0:95 kJ=m2 andGIIc � 0:62 kJ=m2 wereused for the 7781/E765 (Gl/Ep) laminate.

Figures 23–25 compare failure loads calculated from the predictedmethod, based on either GIc, GIIc, or GTc, respectively, using thepresent model and the previous analytical models [5] for the joints ofC/Ep adherends and Hysol EA 9394 adhesive. Similarly, for jointswith Gl/Ep adherends and Hysol EA 9394 adhesive, Figs. 26–28show comparison of the predicted failure loads calculated based oneither GIc, GIIc, or GTc, respectively, using the present and previousanalytical models [5] and experimental data [51]. For specimensmade of C/Ep adherend and other two paste adhesives, PTM&WES6292 and MGS A100/B100, Figs. 29–34 show comparison of theexperimental failure loads and the predicted failure loads calculatedbased on either GIc, GIIc, or GTc using the present and previousanalytical models [5].

It can be seen from thefigures that the predicted failure loads basedonGIc using the present analyticalmodel in specimensmade of eitherC/Ep or Gl/Ep adherends correlate very well with the experimentaldata [51]. Furthermore, for all the jointsmade of either C/Ep orGl/Epadherends, the predicted failure loads computed based onGIIc using

Fig. 23 Failure load prediction based on critical mode I strain energyrelease rate with C/Ep adherends and Hysol EA 9394 adhesive.

Fig. 24 Failure load prediction based on critical mode II strain energy

release rate with C/Ep adherends and Hysol EA 9394 adhesive.

Fig. 25 Failure load prediction based on critical total strain energy

release rate with C/Ep adherends and Hysol EA 9394 adhesive.

Fig. 26 Failure load prediction based on critical mode I strain energy

release rate with Gl/Ep adherends and Hysol EA 9394 adhesive.







release rate with C/Ep adherends and PTM&W ES 6292 adhesive.






release rate with C/Ep adherends and MGS A100/B100 adhesive.




the current analytical model correlate fairly well with those from thetest data [51], while the present analytical underestimates the failureloads for specimens with C/Ep adherend and Hysol EA 9394adhesive, as shown in Fig. 24.Moreover, the current analyticalmodelgives satisfactory prediction of failure loads based on GTc in allspecimens made of either C/Ep or Gl/Ep adherends. It should benoted that, in all cases, the present analytical model predicts the trendof failure loads vs bondline thickness, where a thicker bondlineresults in a weaker joint. Because the previous analytical modelcannot calculatewell the strain energy release rates for the joints withthick bondlines, it does not provide good failure load estimates eventhough the failure criterion was backcalculated from the test data.

VII. Conclusions

An analytical method was developed to calculate the strain energyrelease rate and failure load specimen due to delamination of thesurface ply of the adherend adjacent to the adhesive. The stress anddisplacement fields of the adhesively bonded single-lap compositejoint were determined based on the laminated orthotropic platetheory. The crack closure methodwas applied effectively in conjunc-tion with the analytical stress and displacement models in deter-mining the strain energy release rate. T300/E765 3KPWplain-weavecarbon/epoxy (C/Ep) and 7781/E765 satin-weave glass/epoxy (Gl/Ep) with a ply thickness of 0.25 mm and a quasi-isotropic layupsequence of [60= � 60=0= � 60=60=0] were used as adherends toevaluate and validate the application of the developed model.

Strain energy release rates obtained from the current analyticalmethod, the previous model, and the finite element models using theVCCTand J integral were compared for joints with C/Ep and Gl/Epadherends and different paste adhesives. Results from the developedmodel correlate well with the finite element models and show greatimprovement comparing to the previous model for the cases withthick bondlines.

Even though the purpose of this investigation is to derive ananalytical model to calculate the strain energy release rates of anadhesively bonded composite joint due to a delamination crack at thesurface ply of the adherend, an attempt was made to predict the jointstrength using the limited available data. Therefore, use of criticalstrain energy release rate as the criterion to predict failure of the jointwas investigated using joints reportedly failed under the samedelamination mode of the surface ply of the adherend. Test data fromthe literature were employed to validate the approach. Because of thefact that therewere not available data on the fracture toughness for thetwo composite laminates studied here, the requiredGIc andGIIc werebackcalculated from the test data. It was shown that the failure loadspredicted based on eitherGIc,GIIc, orGTc using the present analyticalmodel correlate fairly well with those from the test data. Despite thefact that the failure criterion used was backcalculated and averaged

based the experimental data, the previous model does not show asgood a correlation with the experimental data as the present modeldoes. It can be concluded that the current analytical model is able topredict the joint-failure loads reasonably well using eitherGIc orGTcas the failure criterion.

It should be noted that althoughfinite elementmethod is capable ofsolving problems with various types of materials and complicatedgeometrical configurations, analytical methods offer advantageousperformance and solutions, especially with parametric analyses andoptimization. Moreover, it should be emphasized that presenting asimple and robust analytical model was themotivation for the currentstudy.

Appendix: Boundary Conditions

Region 1

uo1 jx1�0 � 0 (A1)

w1jx1�0 � 0 (A2)

M1yjx1�0 � 0 (A3)

N2xjx1�0 �AU11

AU11 � AL11P (A4)

M2yjx1�0 � 0 (A5)

Q2zjx1�0 � 0 (A6)

N2xjx1�L1� 0 (A7)

M2yjx1�L1 � 0 (A8)

Q2zjx1�L1 � 0 (A9)

Region 3

uo3 jx2�L2 � uo4 jx3�0 (A10)

w3jx2�L2 �w4jx3�0 (A11)

3jx2�L2 � 4jx3�0 (A12)

N3xjx2�L2 � N4xjx3�0 (A13)

M3yjx2�L2 �M4yjx3�0 (A14)

Q3zjx2�L2 �Q4zjx3�0 (A15)

N5xjx3�0 � 0 (A16)




M5yjx3�0 � 0 (A17)

Q5zjx3�0 � 0 (A18)

N6xjx3�0 � 0 (A19)

M6yjx3�0 � 0 (A20)

Q6zjx3�0 � 0 (A21)

uo4 jx3�L3 � uo7 jx4�0 (A22)

w4jx3�L3 � w7jx4�0 (A23)

4jx3�L3 � 7jx4�0 (A24)

N4xjx3�L3 � N7xjx4�0 (A25)



uo5 jx3�L3 � uo8 jx4�0 (A28)

w5jx3�L3 � w8jx4�0 (A29)

N5xjx3�L3 � N8xjx4�0 (A30)


uo6 jx3�L3 � uo9 jx4�0 (A32)

w6jx3�L3 � w9jx4�0 (A33)

6jx3�L3 � 9jx4�0 (A34)

N6xjx3�L3 � N9xjx4�0 (A35)


Region 4

N7xjx4�L4� 0 (A37)

M7yjx4�L4 � 0 (A38)

Q7zjx4�L4 � 0 (A39)

N8xjx4�L4� 0 (A40)

Q8zjx4�L4 � 0 (A41)

uo9 jx4�L4 � uo10jx5�0 (A42)

w9jx4�L4� w10jx5�0 (A43)

9jx4�L4� 10jx5�0 (A44)

N9xjx4�L4� N10xjx5�0 (A45)

M9yjx4�L4�M10yjx5�0 (A46)

Q9zjx4�L4�Q10zjx5�0 (A47)

Region 5

uo10jx5�L5 � uo12jx6�0 (A48)

w10jx5�L5 �w12jx6�0 (A49)

10jx5�L5 � 12jx6�0 (A50)

N10xjx5�L5 � N12xjx6�0 (A51)

M10yjx5�L5 �M12yjx6�0 (A52)

Q10zjx5�L5 �Q12zjx6�0 (A53)

Region 6

N11xjx6�0 � 0 (A54)

M11yjx6�0 � 0 (A55)

Q11zjx6�0 � 0 (A56)


N11xjx6�L6 �AU11

AU11 � AL11P (A57)

M11yjx6�L6� 0 (A58)

Q11zjx6�L6 � 0 (A59)

N12xjx6�L6 �AL11

AU11 � AL11P (A60)

M12yjx6�L6� 0 (A61)

Q12zjx6�L6 � 0 (A62)

Acknowledgments

This investigation was partially sponsored by Kansas NASAExperimental Program to Stimulate Competitive Research grantno. NNX07A027A. Support of the National Science Foundationunder grant nos. EIA-0216178 and EPS-0236913, matching supportfrom the State of Kansas, and the Wichita State University HighPerformance Computing Center is also acknowledged.

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adhesive bonded composite joint analysis with ed surface ply

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