Finite Elements in Analysis and Design 74 (2013) 67–75

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Finite Elements in Analysis and Design

0168-87http://d

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

Investigation of effective parameters on composite patch debondingunder static and cyclic loading using cohesive elements

Hossein Hosseini-Toudeshky a,n, Ali Jasemzadeh a, Bijan Mohammadi b,1

a Aerospace Engineering Department, Amirkabir University of Technology, 424 Hafez Avenue, Tehran, Iranb School of Mechanical Engineering, Iran University of Science & Technology, Narmak, Tehran, Iran

a r t i c l e i n f o

Article history:Received 29 July 2012Received in revised form3 March 2013Accepted 8 June 2013Available online 5 July 2013

Keywords:Bonded repairInterface elementCohesive lawComposite patchAluminum panel

4X/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.finel.2013.06.003

esponding author. Tel.: +98 21 6405032/6454ail addresses: Hosseini@aut.ac.ir (H. Hosseini-deh@aut.ac.ir (A. Jasemzadeh),ohammadi@iust.ac.ir (B. Mohammadi).l.: +98 21 77240208.

a b s t r a c t

The major addressed issue in this paper is investigation of impressive parameters on initiation andpropagation of debonding in the adhesive layer when it occurs concurrently with the growth of an initialcrack in a single-side repaired aluminum panels by composite patches under cyclic loading. Using thesoftening behavior of thin layer solid like interface elements, debonding is modeled between thecomposite patch layer adjacent to the aluminum panel as a function of loading condition and stress field.A user element routine and two damage model routines were developed to include the interface elementand to simulate the distribution of damage in adhesive layer under static and cyclic loading. Fatigue crackgrowth in aluminum panel was also modeled using a simple approach developed by the authors. It isshown that, it is possible to decrease the debonding propagation by implementing appropriatecomposite patch and adhesive dimensional and material properties.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Extending the service life of cracked components with compo-site patch has generated a lot of controversy in recent years. Thistechnique is more structurally efficient, formable on curved panelswith fewer damages and side effects on the structure than otherssuch as metallic patches. Repair of aluminum structures wasinitiated by Baker in the early 1970s [1]. Different numericalmodels have been already used for calculating the stress field ofsingle-side repairs and optimizing the patch geometry to reducethe stress intensity factor (SIF). Chung and Young [2] performedfatigue crack growth tests for the repaired thick plates containingan inclined edge crack. They used Al 6061-T6 alloy for the basematerial and HT145/RS1222 for the single-side composite patch.Hosseini-Toudeshky et al. [3–7] performed 3D finite elementsfatigue crack growth analysis of single-side repaired panels tosimulate the actual crack-front shape and crack growth life forvarious panels with dominant mode-I and mixed mode conditioncracks. They also examine their observation in the finite elementanalyses by performing experiments on single-side repairedpanels using different composite patches in various fracture modeconditions [3,8–10]. Jones and Chiu [11] also performed

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3224; fax: +98 21 66959020.Toudeshky),

experimental and numerical studies on the repair of crackedstructural components. They observed that the crack propagateswith a curved front shape along the panel thickness.

In contrast to all the advantages mentioned, fatigue lifeprediction of repaired component by composite patch seems tobe complicated due to several possible failure modes, such asgrowth of initial crack in the aluminum panel, patch debondingand new fatigue cracks in the skin (panel) at the patch edges. Thefocus of this paper is prediction of possible patch debonding dueto the fatigue loading concurrent with the possible growth of theinitial crack in the repaired panels.

Whether debonding of patch could significantly affect therepaired panel life or not, further questions may arise such ashow can it be modeled trustworthily, and which parameters affectthe propagation of debonding? Papanikos et al. [12] used finiteelement-based progressive failure model (elastic property degrada-tion) to investigate the geometrical effects on patch debondinginitiation and propagation induced by monotonic loading. It wasfound that, depending on the patch thickness, debonding mayinitiate either at the upper patch edge (being catastrophic for therepair) or at the crack faces (being not catastrophic). Denney andMall [13] and Naboulsi et al. [12] used experiments and the three-layer technique to investigate the effects of pre-existing debondingof various sizes and in different locations on the fatigue crackgrowth and life of repaired structures. Megueni et al. [14] used thefinite element method to compute the stress intensity factor forrepairing cracks with bonded composite patch taking debondinginto account. The previous studies did not consider principal

H. Hosseini-Toudeshky et al. / Finite Elements in Analysis and Design 74 (2013) 67–7568

properties and characteristics of adhesive layers. Moreover, thefracture mechanics approaches cannot easily predict the simulta-neous debonding initiation and propagation; therefore, in thispaper, the cohesive zone modeling concept using interface elementis employed for modeling of adhesive layer and progressivedebonding under static and cyclic loading. Different models werealready presented for interface elements. The recent models havebeen developed by Zeng and Li [15], a novel multi-scale cohesivezone in which the bulk material is modeled as a local quasi-continuum medium and the cohesive force and displacementrelations inside the cohesive zone are governed by a coarse graineddepletion potential. Then [16] they applied their model for compo-site laminates to simulate the failure process. Li and Zeng [17] alsodeveloped a new atomistic-based interphase zone model (AIZM),and employed that for simulation of fractures at small scales.

In this paper, the developed solid like interface element modelby Balzani and Wagner [18] is used, for the analyses of debondingof composite patches from the panels. Besides, a damage modelbased on the cohesive zone concept for the analyses of fatiguedebonding propagation under quasi-static and high cycle fatigueloadings which was developed in [19,20] is added to the consti-tutive law equation. Finally, the explained procedure is used forthe prediction of possible debonding initiation and propagationconcurrently with possible fatigue crack growth in the aluminumpanel, which is analyzed using the previously developed simplemethod by the authors [7].

2. Solid-like interface element formulation

Cohesive interface elements are generally formulated in terms oftractions and relative displacements between two surfaces. In thiswork, we implement a solid-like interface element formulation from[18] (Fig. 1) which tolerates the strain status of material to a damagevariable to define material degradation or damage portion in eachGauss integration point. Such formulation is a standard hexahedralisoparametric solid element but carries only the interlaminar stresses.Therefore, the three-dimensional stress and strain tensors are reducedto interlaminar stress and strain vectors accounting for interlaminarcomponents only and defined by

u¼us

ut

un

264

375; ε¼

γsnγtnεn

264

375¼

us;n þ un;s

ut;n þ un;t

un;n

264

375; r¼

τsn

τtn

sn

264

375 ð1Þ

where s, t and n are the integer subscripts for local coordinatedirections as shown in Fig. 1, u is the real displacement vector,εn; γsn; γtn are the strain components corresponding to modes I, II, andIII, and sn; τsn; γtn are the stress components corresponding to modesI, II, and III of fracture.

The displacement and strain vectors can be approximated asfollows:

u¼ ∑8

I ¼ 1NIuI ; ε¼ ∑

8

I ¼ 1BIuI ð2Þ

Fig. 1. (a) Geometry of a solid-like interface element an

In this equation, NI is the shape function of the node-I and BI isthe shape function derivatives matrix which is defined by

BI ¼ Bi;j ¼NI i¼ j0 i≠j

(ð3Þ

Linearization of principle of virtual work can be written asfollows [18]:

ΔδΠðuÞ ¼ZVδεTCΔεdV ; C¼ ∂r

∂εð4Þ

Using the finite element approximation, Eq. (4) can be writtenas

δΠe ¼ ∑8

I ¼ 1δuT

I

ZVBTI rdV ¼ ∑

8

I ¼ 1δuT

I ReI ;

ΔδΠe ¼ ∑8

I ¼ 1∑8

k ¼ 1δuT

I

ZVBTI CBkdVΔuk ¼ ∑

8

I ¼ 1∑8

k ¼ 1δuT

I KeTIkΔuk ð5Þ

where KeT is the tangent stiffness matrix and Re

I is the residualforce vector and V is the volume of element, δε and r are thevirtual strain and interlaminar stress vectors for the interfaceelement, respectively.

3. Decohesion element constitutive equation

The notion of this work is to obtain initiation and propagationof debonding under both static and cyclic loading. Therefore, therelated constitutive law should have appropriate response to allpossible loadings including static and cyclic loading. On the otherhand, according to the previous researches [20,21], the damageevolution in a degradation process involving high-cycle fatigueshould be considered as the sum of the damage growth fromquasi-static loads and the one from cyclic loads as follows:

d¼ dstatic þ dcyclic ð6Þ

3.1. Constitutive law under static loading

For the static part of the question a bilinear constitutive law ischosen, which is based on the solid-like interface elementpresented in [18]. In the case of mixed mode loading, it relatesthe effective stress to the effective strain, as shown in Fig. 2 andcontains three parts. In the first part, the interface materialbehaves in a linearly elastic manner. It continues until the effectivestrain is less than the damage onset. In this part, the dissipatedenergy is zero and no damage is presented in the element. In thesecond part, the stress evaluation is defined by a damage variable,which has an amount between 0 and 1. It initiates when theeffective strain reaches the damage onset. In this part, the energydissipation is formulated as follows [19]:

EdisGc

¼ Ad

Ae¼ 1−

εmεm0

ð1−dÞ ð7Þ

d (b) deformed shape of interface element.

Fig. 2. Bi-linear elastic-damage constitutive law for cohesive elements in staticloading.

Table 1Definition of bi-linear constitutive law.

1 Elasticity matrix

c¼KI εmoε0m

ð1−dÞKI ε0moεmoεfm

KIc εfmoεm

8>><>>:

2 Damage parameter ds ¼ εfm ðεm−ε0m Þεm ðεfm−ε0m Þ

3 Effective strain εm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi⟨εn⟩2 þ γ2sn þ γ2tn

q4 Effective strain at delamination onset

ε0m ¼ε0nγ

0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þβ2

ðγ0 Þ2þðβε0n Þ2

rεn40

γ0 εn ≤0

8><>:

5 Effective strain at complete decohesionεfm ¼

2Kh0ε0m

Gc½ � εn40

γfsn εn ≤0

8<:

6 Mixed mode and effective fracturetoughness

Gc ¼ GIc þ ðGIIc−GIcÞ β2

1þβ2

� �η

7 Mode mixing ratioβ¼

ffiffiffiffiffiffiffiffiffiffiffiffiγ2snþγ2tn

pεn

εn40

Fig. 3. Constitutive law for cyclic part of loading in the cohesive zone model. (1)Line 1 is related to the elastic part. (2) Line 2 is related to the degradation of staticpart. (3) Lines 4 and 5 are related to the degradation of cyclic part.

H. Hosseini-Toudeshky et al. / Finite Elements in Analysis and Design 74 (2013) 67–75 69

where d is the damage variable, Ad is the element damage area andAe is the element area. Edis is the dissipated energy. Also, εm is theeffective strain, εm0 is the effective strain at delamination onsetand Gc is the mixed mode critical energy release rate. Note that anon-dimensional parameter defined as the ratio of damaged areawith respect to the element area has been used. Therefore,elements size and area does not influence the damage parameteror fatigue life directly. In the final part, the stress becomes zeroand the energy dissipation reaches the critical energy release rate.This part is the well known complete decohesion.

The required formulations for the interface constitutive lawunder static loading are summarized in Table 1 where εmf is theeffective strain at complete decohesion, K is the penalty stiffness,I is the identity matrix, h0 is the interface element thickness and ηis an experimental parameter.

3.2. Constitutive law under cyclic loading

The deleterious influence of cyclic loads can be formulated as afunction of the number of cycles. Since there is no explicitexpression to define this function, the degradation process whichis based on the high cycle fatigue model presented by Turon [19] isimplemented by incorporating a fatigue damage parameter thatevolves during cyclic loading and is accumulated with theobtained static damage parameter (Fig. 3).

To calculate the evolution of the damage variable, the followingexpression is used to relate the evolution of damage variable to thegrowth rate of damage area in each integration point [19].

∂d∂N

¼ ∂d∂Ad

∂Ad

∂Nð8Þ

where d is damage variable and Ad is the element damage area.

In the framework of fracture mechanics, the term ∂d=∂Ad can becalculated from Eq. (7) as the following expression [19]:

∂d∂Ad

¼ 1Ae

½εmf ð1−dÞ þ dεm0 �2εm0εmf

ð9Þ

Besides, ∂Ad=∂N can be formulated as [19]

∂Ad

∂N¼ Ae

Acz

∂A∂N

ð10Þ

To calculate the mean damage evaluation rate in Eq. (10), thefatigue crack growth law of fracture mechanics, the Paris law, canbe expressed as

∂A∂N

¼ cΔGGc

� �m

ð11Þ

where C and m are the material constants. ΔG is the cyclicvariation in the energy release rate, which can be computed bythe following formulation [20]:

ΔG¼Z

rdε ð12Þ

which represents the area under constitutive law curve in cyclicloading.

3.3. Calculation of cumulative damage

Degradation process in cyclic loading seems to be morecomplicated than static loading. First of all, according to theexplanations in the previous section, damage evolution undercyclic loading results from the loading history and it is obtainedthrough an iterative process in both static and fatigue loadings.Moreover, in a degradation process involving high-cyclic fatigue, acycle-by-cycle analysis becomes computationally intractable and itis time consuming. Therefore, a cycle jump strategy (defining theinterval cycles number and considering the damage portion inone-step) is implemented in the finite element model.

Finally, the damage variable is updated in each cycle jumpcontaining ΔNi fatigue cycles by the following expression.

diþ1 ¼ di þ∂di∂N

ΔNi ð13Þ

where di is the damage parameter in the ith step and diþ1 is that inthe next step.

The obtained damage variable from (13) is the cyclic damageportion of constitutive law and should be added to the static partof damage variable in each step.

It is worth noting that, the developed model used un-loading/re-loading concept in each cycle jump (interval) for taking intoaccount the damage variation as a result of panel crack growthinto the interface layer. For this purpose, when the interface

Fig. 4. Static and fatigue constitutive law for typical loading, unloading andreloading conditions.

H. Hosseini-Toudeshky et al. / Finite Elements in Analysis and Design 74 (2013) 67–7570

elements are under unloading, the previously calculated damagesare saved. In the next step, when a reloading is applied, analysesare repeated according to the new condition in the constitutivemodel as shown in Fig. 4.

It is worth noting that the cyclic degradation should be onlyperformed in the region close to the debonding-front in theconsidered cohesive zone in which interface elements haveexceeded their linear-elastic stress and experience damage [20]would be an oversight.

4. FEM procedure of debonding analysis

In the finite element modeling of progressive debondinganalysis of repaired panels, the following three major parts workcoincidently in each step as illustrated in Fig. 5:

(i)

modeling, meshing, and applying the boundary conditions; (ii) elastic-damage solution, degradation process and damage

evolution calculations under static and cyclic loading formu-lated in the UPF;

(iii)

fatigue crack growth calculation of aluminum panel at eachincrement.

The geometry and typical mesh in the FEM analyses are shownin Fig. 6. The cracked panels are made of aluminum 2024-T3; thecomposite patches are made of Glass/Epoxy, Graphite/Epoxy andBoron/Epoxy. The applied remote stress range is 118 MPa. Toperform the degradation under cyclic loading (cumulativedamage), it is required to restore nodal stress of previous loadstep at each calculation step. Therefore, it is more convenient toun-change the mesh information in every load step. The existingsymmetry conditions are also used to reduce the calculationvolume of the debonding and crack growth modeling ofrepaired panel.

For consideration of the cyclic loading damage in interfaceelements, (Fig. 7), the load is divided to two parts. The first part isstatic loading in which the applied load gradually increases fromzero to the maximum load value. In this part damage variable isdefined by the expression performed in the second row ofTable 1in each Gauss point of the elements. The second part iscyclic loading in which the load amplitude remains constant andfatigue process starts at fatigue zone as shown in Fig. 7. In thispart, the elements damage accumulation or material degradationoccurs which will be considered by Eq. (13). The crack closuretechnique based on the simple method presented by the authorsin [7] is also implemented to compute the fatigue crack growth inthe aluminum panels.

For prediction of possible debonding initiation and propagationconcurrently with possible fatigue crack growth of the preexistingcrack in the repaired aluminum panel, a special procedure is

needed for handling of crack growth in the aluminum panel. Forthis purpose, one may suggest to consider a real crack-front topredict the crack growth length and life in finite element model-ing. But consideration of real crack-front modeling concurrentlywith the use of cohesive elements in the bonding area makes toomuch complexity in the finite element analyses. To overcome suchdifficulty, the simple method performed by the authors [7] can beused. In this simple method, the real crack-front is not consideredin the finite element crack growth modeling (crack-front remainsperpendicular to the panel surfaces during the crack growth), butthe stress intensity factor of a specific point on the thickness isused for life prediction that approximately gives life similar to thereal crack-front modeling.

In each step of crack growth, the mode-I strain energy releaserate is calculated using the following expression [22]:

GI ¼−1

2abFcrack f ronty ðuu

y−udyÞ ð14Þ

where a and b are the length and width of the element respec-tively, and Fy and uy are the nodal forces and displacementsrespectively as shown in Fig. 8(b). Then the stress intensity factorfor a linear elastic fracture condition can be calculated from thefollowing expression:

KI ¼ffiffiffiffiffiffiffiffiEGI

pð15Þ

where E is the modulus of elasticity. To calculate the fatigue life fora known crack length increment or to find a crack growthincrement for a known and limited number of load cycle incre-ments, the well known Paris fatigue crack growth law is used asfollows:

dadN

¼ cðΔKÞm ð16Þ

where c and m are the empirical material constants,ΔK ¼ Kmaxð1−RÞ is the stress intensity factor range in fatigueloading, N is the number of cycles, da is the crack growth lengthand R is the cyclic load ratio. According to the suggestion of thedeveloped simple method [7], the point along the crack-front, Ze,whose stress intensity factor has to be used in Eq. (16) can beobtained from the following equation:

Ze

tp¼ 0:0166

EAlEPatch

� �þ 0:3453 ð17Þ

where Ze is measured from the unpatched surface of the panel asindicated in Fig. 8(a), EAl and EPatch are Young′s modulus ofelasticity of panel and patch materials respectively and tp is thethickness of composite patch.

5. Results and discussion

5.1. Patch debonding analysis under static loading

In this section, a monotonic tension load is applied on therepaired aluminum panel as shown in Fig. 6(b) to predict thedamage parameter values in different loading sub-steps. Theaverage of damage parameter, dAve, in cohesive layer is proposedas follows:

dAve ¼∑e∈czd

eave:Ae

Aczð18Þ

where deave is the average of damage values for an element in theadhesive layer, Ae is the element area and Acz is the adhesive layerarea (cohesive zone area).

Fig. 9 compares the obtained variations of average damageparameter, deave, versus applied stress in static loading for compo-site patches of Glass/Epoxy, Graphite/Epoxy and Boron/Epoxy. It is

Fig. 5. Flowchart of the debonding and crack growth analyses of repaired panels.

H. Hosseini-Toudeshky et al. / Finite Elements in Analysis and Design 74 (2013) 67–75 71

clear that even at the primary steps of loading, damage initiation isidentical, but, as the load increases, damage value and debondingbecome different for various patch materials. Debonding in adhe-sive layer for Graphite/Epoxy patch can be divided into threedistinct loadings: firstly, 0–75 MPa in which damage parameter isalmost zero; secondly, 75–270 MPa in which damage is lower thanone, and finally, loading values of larger than 400 MPa, in whichthe patch experiments the complete decohesion. Also for thispatch material, the gradual rise in deterioration from 0 to approxi-mately 0.1 in the first 250 MPa of loading is followed by a sharperincrease by increasing of applied stress.

Fig. 10 shows the variation of stress intensity factor, KI , versusthe applied remote stress for various patch materials. It also showsan almost linear variation of stress intensity factor by increasingthe applied stress. At the applied load of about 250 MPa, an almostsharp increase of KI value is observed for repaired panels withGraphite/Epoxy and Boron/Epoxy patches which is concurrentwith almost complete debonding of patch from the panel.

Fig. 11 shows the progressive debonding area between theGlass/Epoxy patch and cracked panel (a quarter of the model) atdifferent stages of loading. In this case, debonding initially occursaround the crack region when the applied load becomes close toabout 170 MPa. Then, it grows steadily until the load is below the240 MPa (part 1). Afterward, debonding starts to initiate and grownear the patch edges and propagates by increasing the load valueup to about 320 MPa applied load (part 2). Eventually, it initiatesand grows at the patch upper edges (part 3) and propagates byincreasing the load until the final failure. The final total debondingoccurs at the load of 370 MPa (Fig. 9).

Variations of average damage parameters versus applied stressin Fig. 9 for glass/epoxy, graphite/epoxy, and boron/epoxy patchesshow that both initial increase in damage value and debonding(deave¼1.0) of patch from the panel depend considerably on thecomposite patch material. It represents that increasing in theelasticity modulus of patch material causes augmentation in thedebonding initiation and propagation significantly. It is also

Fig. 6. (a) Geometry of cracked panel, composite patch and adhesive and (b) typical finite element mesh for quarter of the repaired panel.

Fig. 7. Schematically loading model.

H. Hosseini-Toudeshky et al. / Finite Elements in Analysis and Design 74 (2013) 67–7572

considerable that it definitely affects the stress intensity factor(SIF) in which, any increase in elastic module of patch materialcould cause the reduction of SIF as shown in Fig. 10.

Fig. 12 shows the variation of average damage parameter versusglass/epoxy patch thickness expressing the initiation and propaga-tion behavior of patches with various thicknesses. This figure showsthat the damage initiation and damage growth into adhesive layerbecomes faster with increase of the patch thickness. It is also noticedthat the damage parameter starts to increase from zero value to alarger value for smaller path thickness earlier than the thick patchthickness. This initial growth of d value is a kind of materialdegradation only. It becomes fully damaged when d approachesunity which the figure shows that it occurs for thick patches earlierthan the thin patches. It is also noted that for smaller patch thickness,the debonding initiates at the crack edge region, while for largerpatch thickness it occurs at the patch edges which is caused bypealing stresses. Therefore, we are faced with different damage areasat patch bonding for thin and thick patches.

5.2. Patch debonding analysis under cyclic loading

In this part, concurrent debonding and crack growth analysesare performed for single-side repaired panels under cyclic loading

by implementing the procedure presented in Fig. 5. Effects of fourimportant parameters, i.e., patch material, bonding properties,adhesive thickness and patch width, on patch debonding initiationand propagation are investigated. The model includes an alumi-num cracked panel (100 mm�50 mm�2.29 mm) with a centralcrack length of 10 mm repaired by a single-side composite patch(40 mm�35 mm�0.72 mm). Geometrical parameters of modelare listed in Table 2.

Variations of average damage parameter versus number of loadcycles for repaired panels with different patch materials and withconsideration of concurrent fatigue crack growth of the panel areshown in Fig. 13. The considered three patch materials are Glass/Epoxy, Graphite/Epoxy and Boron/Epoxy with mechanical proper-ties listed in Table 3. It is evident that, bonding damage initiatesafter about 1300 cycles for all three patches. Therefore, the patchmaterial does not have a serious effect on damage initiation. As itis mentioned in the previous sections, by increasing the number ofload cycles damage propagation in adhesive layer between Boron/Epoxy patch the cracked aluminum panel is larger than those thatoccurred for Graphite/Epoxy and Glass/Epoxy patches. It means, inspite of more decrease of SIF at the crack-front of the aluminumpanel due to the use of composite patches with larger elasticmodule such as Boron/Epoxy, it may cause more damage progressin the adhesive layer.

Fracture toughness of adhesive layers relates to the bondingproperties that appear in the cohesive fatigue constitutive law.Therefore, a sensitivity analysis on the effect of cohesive shear-ing strength on damage progress of adhesive layer becomesvaluable. Fig. 14 shows the damage progress against the numberof load cycles for the considered three different bonding proper-ties with differences in the shear strength as listed in Table 4.This figure shows that by decreasing the shear strength of theadhesive the mean damage value or debonding is significantlyincreased. Therefore, bonding shear strength of interface layercould strongly affect the damage and debonding growth ofpatch from the panel.

Adhesive layer thickness could be another impressive para-meter on damage growth and debonding of composite patch froma cracked panel, which is investigated in this part. For this purposethree typical adhesive thicknesses of 0.01 mm, 0.05 mm and

Fig. 10. Variations of stress intensity factor at crack-front of the panel againstapplied stress for various patch materials.

Fig. 11. Progressive debonding area with increase in the applied load (showing thecrack and applied load value on typical maps).

Fig. 12. Variation of average damage parameter versus applied stress for differentglass/epoxy patch thicknesses under static loading (damage initiation andpropagation).

Fig. 8. (a) Crack-fronts in different method and (b) definition of parameters used in the crack closure technique.

Fig. 9. Variations of average damage parameter versus applied stress in staticloading for various composite patches.

H. Hosseini-Toudeshky et al. / Finite Elements in Analysis and Design 74 (2013) 67–75 73

0.1 mm are considered for the analyses. The obtained variations ofaverage damage parameter versus number of load cycles areshown in Fig. 15 for the three adhesive thicknesses. This figureshows that the increase in adhesive thickness leads to the increaseof damage values and leads to more debonding potential. It isworth noting that, increasing of this parameter has an inverseeffect on the fracture toughness of adhesive layer and interfacezone which justifies such results.

Lastly, the effects of patch width on damage propagation ofinterface area between the patch and panel are investigated here.For this purpose, three patch width values of 35 mm, 30 mm, and25 mm and the same panel width were considered for theanalyses. The obtained variations of average damage parameterversus number of cycles are shown in Fig. 16 for three different

patch widths. This figure shows that by increasing the patch thedamage values are decreased indicating a positive effect ondebonding phenomenon.

Fig. 13. Variation of average damage parameters versus number of loadcycles for three different patch materials with interface properties ofsmax ¼ 30; τmax ¼ 60 MPa under 118 Mpa.

Table 3Material properties for various composite patches [7].

Glass/epoxy Graphite/epoxy Boron/epoxy

E11 (GPa) 50 134 208E22 (Gpa) 14.5 10.3 25.4E33 (GPa) 14.5 10.3 25.4G12 (Gpa) 2.56 5.5 7.2G13 (GPa) 2.56 5.5 7.2G23 (Gpa) 2.24 3.2 4.9υ12 0.33 0.33 0.1677υ13 0.33 0.33 0.1677υ23 0.33 0.53 0.035

Fig. 14. Variation of average damage parameter versus number of load cycles forthree typical different bonding properties (see Table 4 for bonding types).

Table 4Three typical bonding mechanical properties of adhesive layer.

smax (MPa) τmax (MPa) GIIc (N/mm) GIc (N/mm)

Bonding type(a)30

901.002 0.26Bonding type(b) 60

Bonding type(c) 30

Fig. 15. Variation of average damage parameter versus number of load for threedifferent adhesive thicknesses and properties of smax ¼ 51:7 and τmax ¼ 90.

Fig. 16. Variation of average damage parameter versus number of load cycles forthree different patch widths and properties of smax ¼ 51:7; τmax ¼ 90.

Table 2Dimensions of the panel, adhesive and patch [7].

Aluminum panel Adhesive Patch

L (mm) 100 40 40W (mm) 50 35 35t (mm) 2.29 0.1 0.18 per layer

H. Hosseini-Toudeshky et al. / Finite Elements in Analysis and Design 74 (2013) 67–7574

6. Conclusion

In this paper, a damage model based on the cohesive zoneconcept and finite element procedure was developed for progres-sive damage analyses in the bonding area of composite patchesand cracked panels under quasi-static and high cycle fatigueloadings. This procedure was linked with the concurrent fatiguecrack growth in the aluminum panel using the previously devel-oped simple method by the authors. Effects of patch material,patch width, patch thickness, adhesive thickness, and bondingshear strength on progressive damage of bonding interface wereinvestigated under both static and fatigue loadings. It was shownthat when a patch material with larger elastic modulus, patchthickness and adhesive thickness is used, it leads to more damagein the adhesive layer during the loading (static or cyclic). It wasalso shown that by increasing the shear strength of the adhesiveand patch width, the mean damage value or debonding and crackgrowth rate of the panel are decreased.

Therefore, for the purpose of more strength against thedebonding phenomenon, smaller values for elastic modulus ofpatch, patch thickness and adhesive thickness and also largervalues of adhesive shear strength and patch width are moredesirable.

References

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with composite patches, International Journal of Fatigue 25 (2003) 325–333.

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