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Modelling of Single-Lap Joints UsingCohesive Zone Models: Effect of theCohesive Parameters on the Output ofthe SimulationsR. D. S. G. Campilho a b , M. D. Banea c , J. A. B. P. Neto c & L. F. M.da Silva ca Faculdade de Economia e Gestão, Universidade Lusófona do Porto,Porto, Portugalb Departamento de Engenharia Mecânica, Instituto Superior deEngenharia do Porto, Porto, Portugalc Departamento de Engenharia Mecânica, Faculdade de Engenhariada Universidade do Porto, Porto, Portugal

Available online: 25 Apr 2012

To cite this article: R. D. S. G. Campilho, M. D. Banea, J. A. B. P. Neto & L. F. M. da Silva (2012):Modelling of Single-Lap Joints Using Cohesive Zone Models: Effect of the Cohesive Parameters on theOutput of the Simulations, The Journal of Adhesion, 88:4-6, 513-533

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Modelling of Single-Lap Joints UsingCohesive Zone Models: Effect ofthe Cohesive Parameters on the

Output of the Simulations

R. D. S. G. CAMPILHO1,2, M. D. BANEA3,J. A. B. P. NETO3, and L. F. M. DA SILVA3

1Faculdade de Economia e Gestao, Universidade Lusofona do Porto, Porto, Portugal2Departamento de Engenharia Mecanica, Instituto Superior de Engenharia

do Porto, Porto, Portugal3Departamento de Engenharia Mecanica, Faculdade de Engenharia

da Universidade do Porto, Porto, Portugal

The available techniques for strength prediction of bonded jointshave improved over the years. Cohesive zone models (CZM) coupledto finite element method (FEM) analyses surpass the limitations ofstress=strain and fracture criteria, and simulate damage growth.CZMs require the instantaneous energy release rates in tension(Gn) and shear (Gs) along the fracture paths and respective fractureenergies in tension (Gn

c) and shear (Gsc), and crack growth is ruled

by traction-separation laws that are established at the failure paths.Additionally, the cohesive strengths must be defined (tn

0 for tensionand ts

0 for shear) relating to the onset of damage. A few techniquesare available for the estimation of these parameters (e.g., the pro-perty identification technique, the direct method and the inversemethod) that differ in complexity and expected accuracy of theresults. In this work, the influence of the cohesive law parametersof a triangular CZM used to model a thin adhesive layer in bondedjoints is studied, to estimate their effect on the predictions. Someconclusions were established to provide important data for the

Received 7 July 2011; in final form 26 September 2011.Presented in part at the 1st International Conference on Structural Adhesive Bonding

(AB2011), Porto, Portugal, 7–8 July 2011.Address correspondence to R. D. S. G. Campilho, Departamento de Engenharia Mecanica,

Instituto Superior de Engenharia do Porto, Rua Dr. Antonio Bernardino de Almeida, 431, Porto4200-072, Portugal. E-mail: raulcampilho@gmail.com

The Journal of Adhesion, 88:513–533, 2012

Copyright # Taylor & Francis Group, LLC

ISSN: 0021-8464 print=1545-5823 online

DOI: 10.1080/00218464.2012.660834

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proper selection of the estimation technique and expected accuracyof the simulation results.

KEYWORDS Bonded joints; Cohesive zone models; Finite elementmethod; Structural adhesive

1. INTRODUCTION

Adhesively bonded joints are largely employed in various fields of engineer-ing, such as automotive, aeronautical, and design of special structures, as a fastand effective process to join components [1,2]. Bonded joints also provideother benefits, such as more uniform stress fields at the bonding region thanconventional techniques such as fastening or riveting, capability of fluid seal-ing, high fatigue resistance, and possibility to keep the integrity of the parentmaterials and to join different materials, as the adhesive prevents corrosionand it compensates different thermal expansion of the adherends [3]. On theother hand, the available analytical and numerical techniques for the strengthprediction of bonded joints have also been improved over the years. The firstpublished studies proposed theoretical (mainly closed-form) solutions forstress distributions along the bondline of simple geometries such as the singleor double-lap joint. Strength prediction was typically accomplished by com-parison of the maximum stresses with the material strengths, and the accuracyof the predictions mainly relied on the amount of simplification hypotheses,which as a rule were considerable and limitative to the real problem to besimulated [4]. The FEM was subsequently used with the same purpose,traditionally by using stress=strain or fracture mechanics criteria for strengthprediction. The adhesive layer was modelled by finite elements and it wasassumed to be perfectly bonded to the adherends [5]. These techniquesshowed acceptable results, but they suffered from different intrinsic limita-tions: stress=strain methods are mesh-dependent, while fracture criteria suchas the virtual crack closure technique (VCCT) are restricted to linear elasticfracture mechanics (LEFM) analyses and rely on the existence of an initialcrack, although being relatively insensitive to large stress gradients. Surpass-ing of these limitations, added to the possibility of damage growth modelling,was accomplished by CZM [6]. Equally to the VCCT, CZM also require theknowledge of Gn and Gs along the fracture paths and respective materialproperties Gn

c and Gsc, but growth is ruled by traction-separation laws that

are established at the failure paths. Additionally, the cohesive strengths, tn0

and ts0, are introduced, relating to the onset of damage, i.e., end of the elastic

regime and beginning of stress reduction. In recent years, many works werepublished regarding the definition of the cohesive parameters (Gn

c, Gsc, tn

0,and ts

0) and a few data reduction techniques are currently available (e.g.,the property identification technique, the direct method, and the inverse

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method) that enclose varying degrees of complexity and expected accuracyof the results. The few of these works that validated with experiments theestimated CZM typically made use of double-cantilever beam (DCB), end-notched flexure (ENF), or single-lap specimens, generally with good results[7–11]. For the property identification technique and the inverse method asimplified parameterized shape is often assumed (e.g., bilinear or trilinear)for the traction-separation laws, depending on the behaviour of the materialto be simulated [12]. On the other hand, in the direct method the precise shapeof the cohesive law is easily defined, since it computes the cohesive law of anadhesive layer from the measured data of fracture characterization tests [13] bydifferentiation of Gn (tension) or Gs (shear) with respect to the relative open-ing (dn for tension or ds for shear). However, for subsequent use by FEMstrength prediction techniques, it is common practice to build a simplifiedapproximation for the simulation. Notwithstanding the parameter identifi-cation method, deviations are expected to occur between the quantitative pre-diction of the cohesive parameters and the real behaviour of the adhesivelayer [14,15]. The property identification method, consisting of the separatedcalculation of each one of the cohesive law parameters by suitable tests,is particularly critical if bulk tests are used, owing to reported deviationsbetween the bulk and thin adhesive layer cohesive properties [16]. This iscaused by the strain constraining effect of the adherends in bonded assem-blies, and also by the typical mixedmode crack propagation in adhesive layers[14,16–18]. Actually, in bulk materials cracks tend to grow perpendicularly tothe direction of maximum principal stress [19]. In thin layers, cracks are forcedto follow the bond length path since, as the adhesive is typically weaker andmore compliant than the components to be joined, failure often occurs cohe-sively within the adhesive layer [20,21]. Contrarily, the inverse and direct meth-ods provide more precise estimations as the adhesive can be characterizedunder identical adherend restraining conditions to real applications, givingaccurate estimations [22]. However, notwithstanding the method, the CZMparameters invariably depend on the adhesive thickness (tA) and adherendthickness (tP), which emphasizes the importance of the tA and tP consistencybetween the fracture tests and the structures to be simulated [14, [16,23–25]. Inthe work of Carlberger and Stigh [26], the cohesive laws in tension and shearwere determined for a thin layer of adhesive using the DCB and ENF testconfigurations, respectively, considering 0.1� tA� 1.6mm. The cohesive lawswere derived by a direct method that used a least squares adaption of aProny-series to the Gn=Gs vs. dn=ds data, to avoid errors in the measured dataresulting from direct differentiation of the experimental results. It was con-cluded that the CZM shapes and respective parameters significantly vary withtA, ranging from a rough triangular shape for the smaller values of tA to a trap-ezoidal shape, i.e., with a plastic region after elasticity, characteristic of biggervalues of tA. Corroboration of the adhesive restraining effects was equallyaccomplished by Ji et al. [27], who studied the influence of tA on tn

0 and Gnc

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for a brittle epoxy adhesive, by using the DCB specimen and the directmethod for parameter estimation. The analysis methodology relied on themeasurement of Gn

c by an analytical technique proposed by Andersson andStigh [16] that required the relative rotation between the adherends of theDCB specimen. Derivation of the tn� dn cohesive laws was quickly obtainedby differentiation of the Gn� dn data [8,28], and clearly showed a reduction oftn0 and increase of Gn

c with bigger values of tA. On the other hand, the influ-ence of the deviation between the parameterized approximations and the realcohesive laws on the output of the simulations is also largely dependent onparameters such as tP or the stiffness of the adherends. Pinto et al. [15] evalu-ated the tensile strength of single-lap joints between adherends with differentvalues of tA and materials, with parameterized trapezoidal shapes being usedin the numerical (CZM) simulations. The load-displacement (P-d) datarevealed that the use of accurate CZM shapes for the adhesive is much moreimportant with stiff adherends, since differential deformation between theadherends is minimized, and the adhesive is evenly loaded along the entirebond length. Oppositely, compliant adherends give rise to substantial shearand peel peak stress gradients, providing relatively insensitive results to theshape of the cohesive law. The results of Ridha et al. [29] also emphasizedon the CZM shape influence, by modelling a thin adhesive layer of a highelongation epoxy adhesive in scarf repairs on composite panels. Linear, expo-nential, and trapezoidal softening was considered to model plasticity of theadhesive, with the use of linear softening resulting on under predictions ofthe actual strengths of the repairs of nearly 20%, because of premature soften-ing at the bond edges after the peak strength that is not consistent with theactual behaviour of the adhesive.

This work addresses the influence of the cohesive law parameters of atriangular cohesive law used to approximate the behaviour of a thin adhesivelayer in bonded joints, to assess the effect of these parameters on the strengthprediction of bonded structures using CZM. Single-lap joints bonded with aductile adhesive were evaluated for varying overlap lengths (Lo), from shortoverlaps and nearly uniform shear stress distributions, to long overlaps, whichgive rise to significant peel and shear stress variations along the overlap [30].Justification of this study is based on issues such as the dissimilar accuracy ofthe available cohesive parameter determination techniques and respectiveassessment of deviations on the simulation results, or clarification on howthe parameter estimations for specific conditions can be extrapolated to differ-ent conditions of tA or tP. Moreover, some applications such as in the automo-tive industry are also characterized by fabrication processes in which it is hardto achieve the design values of tA, or even to obtain a constant value of tA [26].All of these issues emphasize the importance of an accurate knowledge of theinfluence of the CZM parameters and respective deviations on the simulationresults. As a result of this study, some conclusions were established to aidunderstanding of these effects and also to provide bonded joint designers

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with important data for the proper selection of the data reduction techniqueand expected accuracy of the simulation results under different scenarios.

2. EXPERIMENTAL WORK

2.1. Characterization of the Materials

The adherends were fabricated from unidirectional carbon-epoxy pre-preg(SEAL1 Texipreg HS 160 RM; Legnano, Italy) with 0.15mm thickness pliesand [0]16 lay-up. Table 1 specifies the elastic ply properties, modelled as elasticorthotropic [31]. The adhesive Araldite1 2015 (Huntsman Advanced Materials,Basel, Switzerland) was characterized in previous works [32], whose proper-ties, shown in Table 2, were used to construct the triangular cohesive laws(the initial yield stress was calculated for a plastic strain of 0.2%). Bulk testsunder tension were selected to characterize the adhesive in tension and thickadherend shear tests (TAST) were chosen for shear characterization. The bulkspecimens were manufactured by the indications of the NF T 76-142 Frenchstandard, to prevent the creation of voids. Thus, the 2mm thick plates werefabricated in a sealed mould, followed by precision machining to producethe dogbone shape described in the standard. The TAST characterization ofthe adhesive was carried out according to the 11003-2:1999 ISO standard,considering DIN Ck 45 steel adherends. More details about the fabricationand testing procedures can be found in reference [32].

TABLE 1 Elastic Orthotropic Properties of a Unidirectional Carbon-EpoxyPly Aligned in the Fibre Direction (x-direction; y and z are the transverseand through-thickness directions, respectively) [31]

Ex¼ 1.09Eþ 05MPa nxy¼ 0.342 Gxy¼ 4315MPaEy¼ 8819MPa nxz¼ 0.342 Gxz¼ 4315MPaEz¼ 8819MPa nyz¼ 0.380 Gyz¼ 3200MPa

TABLE 2 Properties of the Adhesive Araldite1 2015 [32]

Property

Young’s modulus, E [GPa] 1.85� 0.21Poisson’s ratio, n� 0.33Tensile yield strength, ry [MPa] 12.63� 0.61Tensile failure strength, rf [MPa] 21.63� 1.61Tensile failure strain, ef [%] 4.77� 0.15Shear modulus, G [GPa] 0.56� 0.21Shear yield strength, sy [MPa] 14.6� 1.3Shear failure strength, sf [MPa] 17.9� 1.8Shear failure strain, cf [%] 43.9� 3.4

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2.2. Joint Geometries

Figure 1 represents the joint geometry. The characteristic dimensions weredefined as (in mm): LO¼ 10–80, width b¼ 15, total length between grippingpoints LT¼ 240, tP¼ 2.4, and tA¼ 0.2. The bonding surfaces were preparedby manual abrasion with 220-grit sandpaper, followed by wiping with acet-one. The desired value of tA was achieved with a dummy adherend and a0.2-mm spacer under the upper adherend, jointly with the application ofpressure with grips. Tabs were glued at the specimen edges for a correct align-ment in the testing machine. For proper adhesive curing at room temperature,testing was carried out 1 week after fabrication. Eight different values of LOwere evaluated (10, 20, 30, 40, 50, 60, 70, and 80mm). The joints were testedin an Instron1 4208 (Norwood, MA, USA) electro-mechanical testing machinewith a 100-kN load cell, at room temperature, and under displacement control(0.5mm=min). The testing machine grips displacement was considered tobuild the P-d curves. For each value of LO, six specimens were tested, withat least four valid results.

3. NUMERICAL ANALYSIS

3.1. Modelling Conditions

The numerical analysis in the FEM package ABAQUS1 (Providence, RI, USA)aimed to check the accuracy of its triangular CZM embedded formulation topredict the strength of adhesively bonded single-lap joints and to evaluatethe impact of cohesive parameter misjudgements on the strength predictions,either caused by intrinsic limitations of the data reduction techniques, or bydifferent restraining scenarios between the characterization tests and thestructures to be simulated (e.g., tA or tP inconsistencies). The triangular CZMformulation was chosen for this analysis because of its simplicity, large usefor investigation purposes, and availability in ABAQUS1 including a mixed-mode formulation, which is absolutely necessary to model the single-lapjoints used in this work. However, other CZM shapes are available, such as

FIGURE 1 Geometry and characteristic dimensions of the single-lap specimens.

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the trapezoidal, which for this particular case would suit more faithfully theductile adhesive behaviour [12]. The numerical analysis considered geometri-cal non-linear effects [12,33] with the elastic orthotropic properties of Table 1to simulate the adherends and the triangular cohesive model presented in thefollowing section for the adhesive. The single-lap joint meshes were builtwithout symmetry conditions (Fig. 1). Figure 2 shows the mesh for theLO¼ 10mm joint. The meshes of the numerical models were built automati-cally considering bias effects with smaller elements near the overlap edges,as these regions are known to be theoretically singular regions with majorpeel and shear peak stresses [34]. The mesh size was adjusted in all modelsfor a similar element size at the overlap edges (edge of � 0.1mm), thusallowing accurate capture of these stress variations [31]. Two-dimensionalnumerical approximations were considered to simulate the joints with four-node plane-strain elements (CPE4 from ABAQUS1) and with COH2D4four-node cohesive elements, compatible with the CPE4 elements [35]. Thejoints were fully restrained (i.e., clamped) at one of the edges to simulate realclamping conditions in the machine grips and the other edge was subjected toa tensile displacement concurrently with transverse restraining [36] (Fig. 1).For the analysis, a single row of cohesive elements was considered for theadhesive [37] incorporating a damage model between each set of pairednodes, as defined further in Section 3.2. The proposed modelling techniqueis currently implemented within ABAQUS1 CAE suite and will be brieflydescribed in the following.

3.2. Cohesive Zone Modelling

CZM model the elastic loading, initiation of damage, and further propagationdue to local failure within a material. CZM are based on a relationshipbetween stresses and relative displacements (in tension or shear) connectingpaired nodes of the cohesive elements (Fig. 3), to simulate the elastic behav-iour up to tn

0 (tension) or ts0 (shear) and subsequent softening, to account for

the gradual degradation of material properties up to complete failure [38].Generically speaking, the shape of the cohesive laws can be adjusted toconform to the behaviour of the material or interface they are simulating[12]. The values of Gn and Gs, representing the areas under the traction-separation laws in tension or shear, respectively, are equalled to Gn

c for ten-sion or Gs

c for shear, allowing one to define the normal or shear maximum

FIGURE 2 Detail of the mesh for the LO¼ 10mm model.

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relative displacements (dnf and ds

f, respectively). Under pure tension or shear,damage propagation occurs at a specific pair of nodes when the stresses arereleased in the respective traction-separation law. Under mixed mode, stressand energetic criteria are often used to combine tension and shear [39]. In thiswork, a continuum-based approach, i.e., using the cohesive elements tomodel solids rather than interfaces, was considered to model the finite valueof tA. The cohesive layer is assumed to be under one direct componentof strain (through-thickness) and one transverse shear strain, which are com-puted directly from the element kinematics. The membrane strains areassumed as zero, which is appropriate for thin and compliant layers betweenstiff adherends. The traction-separation laws assume an initial linear elasticbehaviour followed by linear evolution of damage (Fig. 3). Elasticity isdefined by an elastic constitutive matrix relating the current stressesand strains in tension and shear across the interface (subscripts n and s,respectively) [40]:

t ¼ tnts

� �¼ Knn Kns

Kns Kss

� �:

enes

� �¼ Ke: ð1Þ

The matrix K contains the stiffness parameters of the adhesive layer,given by the relevant elastic moduli. A suitable approximation for thinadhesive layers is provided with Knn¼ E, Kss¼G, Kns¼ 0 [39,41]; E and Gare the tensile and shear elastic moduli. Damage initiation can be specifiedby different criteria. In this work, the quadratic nominal stress criterion wasselected for the initiation of damage, previously tested for accuracy [12],expressed as [40]:

tnh it0n

� �2

þ tst0s

� �2

¼ 1: ð2Þ

FIGURE 3 Traction-separation law with linear softening law available in ABAQUS1.

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hi are the Macaulay brackets, emphasizing that a purely compressive stressstate does not initiate damage [42]. After the mixed-mode cohesive strengthis attained (tm

0 in Fig. 3, corresponding to a d value of dm0) by the fulfilment

of Eq. (2), the material stiffness initiates a degradation process. Complete sep-aration and the mixed-mode failure displacement (dm

f in Fig. 3) are predictedby a linear power law form of the required energies for failure in the puremodes [40]:

Gn

Gcn

þ Gs

Gcs

¼ 1: ð3Þ

Table 3 shows the values introduced in ABAQUS1 for the adhesivelayer damage laws [12,37]. These properties were established from the dataof Table 2, considering the average values of the experiments.

4. RESULTS AND DISCUSSION

4.1. Stress Analysis

The plots of through-thickness normal (ry) and shear (sxy) stress distributionsat the adhesive mid-thickness as a function of LO are presented in Figs. 4 and5, respectively. Both stresses are normalized by savg, the average value of sxyalong the overlap for each value of LO [43]. The stress profiles are consistentwith the typical distributions documented in the literature [44]. ry stresses(Fig. 4) show singularities caused by the sharp geometry change at the overlapedges and a large gradient at the nearby regions [45]. ry stresses are compress-ive at the inner overlap, although much smaller in magnitude than savg. Theshape of ry stresses is caused by the joint eccentricity [15], leading to theadherends flexure that results in opening at the overlap edges and com-pression in-between. ry peak stresses known to severely reduce the joint’sstrength, since structural are adhesives typically have small values of Gn

c

[31]. The increase of LO showed a concentration of peak stresses in smallernormalized regions. Bigger values of LO also give rise to ry compressive stres-ses close to the singularities. sxy stresses (Fig. 5) show the typical concave

TABLE 3 Properties of the Adhesive Araldite1 2015 forCZM Modelling [12,37]

Property

E [GPa] 1.85G [GPa] 0.56tn0 [MPa] 21.63

ts0 [MPa] 17.9

Gnc [N=mm] 0.43

Gsc [N=mm] 4.70

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shape profiles, with smaller magnitudes at the overlap inner region and peakstresses at the overlap edges [46]. This shape is accredited to the differentialdeformation between the adherends along the bondline, since they areincreasingly loaded from their free overlap edge towards the opposite one[47,48]. sxy stress gradients are negligible for the smaller values of LO, but thesegradually acquire a concave shape with LO, because of increasing adherendlongitudinal strain gradients caused by the larger bonding areas and higherloads [30]. This feature is in the origin of an increase of the maximum load sus-tained by the specimens (Pm) with LO at a decreasing rate, depending mainlyon the adherend’s stiffness and ductility of the adhesive. In fact, for stiff

FIGURE 5 Normalized sxy stress distributions at the adhesive mid-thickness as a function ofx=LO.

FIGURE 4 Normalized ry stress distributions at the adhesive mid-thickness as a functionof x=LO.

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adherends such as the CFRP ones used in this work, it is expected that thisreduction is smaller than for less stiff adherends, since longitudinal defor-mation gradients along the overlap are not so important [15]. Additionally,joints bonded with ductile adhesives such as the Araldite 2015 are equally lessaffected by this intrinsic feature of single-lap joints since they allow the redis-tribution of stresses at the overlap edges when the peak load is attained [49].As a result, depending on the degree of ductility, the strength approaches alinear relationship with the bonding area [30,50]. Eventually, a constant levelof strength can result from using extremely brittle adhesives or modification ofthe failure mechanism, by cross-sectional tensile fracture of the adherends at apredefined load [50,51].

4.2. Joint Strength

All the joints experienced a cohesive failure of the adhesive layer. Figure 6shows a cohesive failure for a specimen with LO¼ 20mm. Figure 7 reportsPm as a function of LO, showing an increase of Pm at a slightly decreasing ratewith LO [52,53], although the Pm-LO plot is nearly linear. As previouslydiscussed, this occurs from the high stiffness of the adherends and ductilityof the adhesive [15,30,50]. The absence of a constant level of strength in thePm-LO curve is justified by the high strength of the CFRP (i.e., the tensilestrength of the laminates was not attained for the tested range of LO values)and by the ductility of the adhesive that allowed a progressively larger redis-tribution of stresses in the adhesive layer up to the largest value of LO, initiat-ing at the loci of peak stresses, i.e., the overlap edges (Figs. 4 and 5; [49,54]). Infact, since fracture was always abrupt, only with a negligible crack growthbefore Pm for the bigger values of LO, it can be concluded that the adhesiveplasticity always held up crack initiation at the overlap edges up to Pm, keep-ing these regions at the peak strength while stresses increased at the innerregions [30].

FIGURE 6 Experimental fracture surfaces for a LO¼ 20mm specimen.

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4.3. Cohesive Parameters Influence

In this section, the influence of percentile variations of Gnc, Gs

c, tn0, ts

0, andtheir combined effect on the value of Pm=Pm

0 of the joints is numericallyassessed (Pm

0 represents Pm for the initial parameters of Table 3). Percentilevariations of the initial properties between �80 to þ100% were considered,whilst the non-mentioned cohesive parameters in all analyses were keptunchanged (value of Table 3). The influence of each parameter on the dam-age laws is depicted in Fig. 8 [(a) forGn

c and the tensile law; (b) forGsc and the

shear law] and Fig. 9 [(a) for tn0 and the tensile law; (b) for ts

0 and the shearlaw]. Varying Gn

c or Gsc changes the slope of the decaying portion of the

respective cohesive law, while tn0 or ts

0 remain identical. The modificationof tn

0 or ts0, by keeping the respective value of Gn

c or Gsc unchanged, greatly

changes the softening behaviour and value of dnf or ds

f, respectively. As willbe discussed in detail in the following section, the fluctuations of Pm=Pm

0 with

FIGURE 7 Experimental and numerical comparison between the Pm values as a function of LO.

FIGURE 8 Cohesive laws for values of (a) Gnc and (b) Gs

c ranging from �80 to þ100% of theinitial ones, in increments of 20%.

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Gnc and Gs

c are justified by the variations of dnf or ds

f of the damage laws bythe modifications of these parameters (Fig. 8) and the actual values of dn or dsalong the entire bondline when Pm is attained (proportional to the stress dis-tributions presented in Figs. 4 and 5), which determine the loads transmittedby the adhesive layer according to the established cohesive laws. On the otherhand, the influence of tn

0 and ts0 on Pm=Pm

0 will mainly depend on the valueof tm

0 (Fig. 3), which is attained when Eq. (2) is fulfilled, and whose value sig-nificantly changes by the modification of tn

0 and ts0 in the same equation.

Actually, by the variation of tm0, Pm=Pm

0 will be affected to an extent thatdepends on the values of dn and ds along the entire adhesive layer at the timeof failure. Ridha et al. [29] addressed the strength of scarf repairs in compositepanels by CZM modelling. An analysis on the cohesive parameters influencewas conducted by testing percentile variations of -50 and 50% of the initialvalues and simultaneous variations of tn

0=ts0 or Gn

s=Gsc. Results showed a

large influence on the strength predictions by reductions of these parameters,whilst positive variations gave small variations on the strength of the repairs.

4.3.1. FRACTURE TOUGHNESS

Figures 10–12 describe the influence of percentile variations of Gnc, Gs

c, andGn

c plus Gsc, respectively, on Pm=Pm

0. Figure 10, relating to Gnc, shows a sig-

nificant under prediction of Pm=Pm0 for reductions of Gn

c (maximum of� 35.5% for an 80% reduction of the initial Gn

c and LO¼ 40mm), occurringby the smaller values of dn

f (Fig. 8a) in the tensile cohesive law, which resultsin the premature failure at the overlap edges where peel stresses peak (Fig. 4).A slight reduction of the reported under prediction is found near LO¼ 80mm,since for bigger values of LO the peak values of dn focus at a smaller normal-ized region at the overlap edges (correspondingly to ry stresses; Fig. 4). Thegradual increase of Pm=Pm

0 from LO¼ 30mm to LO¼ 10mm is accredited tothe smaller values of dn at the overlap edges with the reduction of LO (Fig. 4),

FIGURE 9 Cohesive laws for values of (a) tn0 and (b) ts

0 ranging from �80 to þ100% of theinitial ones, in increments of 20%.

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which leads to smaller actual values of dn in the tensile cohesive lawwhen Pm isattained (Fig. 8a), rendering any under prediction of Gn

c less preponderant.Over predicting Gn

c gives minor improvements of Pm=Pm0 (maximum of

� 4.9% for LO¼ 50mm) [29], equally smaller for shorter overlaps, due to thecorresponding reduction of dn values (Fig. 4). The negligible influence ofthe Gn

c over predictions, when compared with the under predictions, is alsoclosely related to the sole attainment of large dn values (bigger than dn

f forthe initial parameters; Fig. 8a) at the overlap edges, which renders any increaseof Gn

c above its initial value not significant [29]. Figure 11 corresponds to Gsc

and depicts a significant difference from the data of Fig. 10 (Gnc), as Pm=Pm

0

FIGURE 10 Percentile variation of Pm=Pm0 with Gn

c values ranging from �80 to þ100% of theinitial ones.

FIGURE 11 Percentile variation of Pm=Pm0 with Gs

c values ranging from �80 to þ100% of theinitial ones.

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varies nearly proportionally with LO for under predictions ofGsc. This is related

to the more uniform values of ds along the bondline (and correspondingly sxystresses; Fig. 5), compared to dn (Fig. 4). As a result, for small values of LO thevalue of Pm=Pm

0 corresponds to a state of stress in which all the cohesive ele-ments of the adhesive are very close to ts

0 (Figs. 3 and 8b). Thus, any modifi-cation to the shear cohesive law at ds> ds

0 does not reflect by a large amounton Pm=Pm

0. The increase of LO steadily increases the gradients of ds along thebondline, associating Pm=Pm

0 to an increasing portion of the overlap withds> ds

0 (at the overlap edges). As a result, the softening shape of the sheardamage law becomes progressively more preponderant with LO. Themaximum reduction of Pm=Pm

0, of � 34.8%, was found for LO¼ 80mm. Onthe other hand, identically to the Gn

c data, over predicting Gsc only causes a

maximum Pm=Pm0 improvement of � 5.7% (LO¼ 80mm) [29], because of

the occurrence of large values of ds (bigger than dsf for the initial parameters;

Fig. 8b) at a restricted region at the overlap edges (Fig. 5). The combinedmodification of Gn

c and Gsc (Fig. 12) gives a reduced influence on Pm=Pm

0

between �20 and þ100% of the initial values (maximum of � 9.3% forLO¼ 80mm), and large reductions from�40 to�80% that attain its maximum,of � 44.5%, for LO¼ 80mm. The value of Pm=Pm

0 increases from LO¼ 40 toLO¼ 10mm, owing to the combined effect of Gn

c (Fig. 10) and Gsc (Fig. 11).

The bigger deviations of Pm=Pm0, compared with Figs. 10 and 11, also relate

to the joint influence of Gnc and Gs

c on the failure process.

4.3.2. COHESIVE STRENGTH

The influence of tn0 and ts

0 on Pm=Pm0 is shown in Figs. 13–15 for tn

0, ts0, and

tn0 plus ts

0, respectively. The variation of these parameters also affects dnf and

FIGURE 12 Percentile variation of Pm=Pm0 with Gn

c and Gsc values ranging from �80 to

þ100% of the initial ones.

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dsf (Fig. 9) to keep Gn

c or Gsc constant. Figure 13 displays a larger influence of

tn0 on Pm=Pm

0 for the smaller values of LO, for under and over predictions oftn0, due to the concentration of peel, dn, values at a larger normalized region at

the overlap edges. With the increase of LO (Fig. 4), the concentration of peeldn values occurs over a smaller normalized region, giving a less significantinfluence of tn

0 on the global behaviour of the joints. Figure 13 also reportsa much lesser influence on Pm=Pm

0 by over predicting tn0 than under predict-

ing [29]. In both of these scenarios, these variations are closely related to theattainment of tm

0 [Fig. 3 and Eq. (2)]. Actually, the improvement of Pm=Pm0 by

over predicting tn0 (maximum of � 3.2% for LO¼ 20mm) is related to the

FIGURE 13 Percentile variation of Pm=Pm0 with tn

0 values ranging from �80 to þ100% of theinitial ones.

FIGURE 14 Percentile variation of Pm=Pm0 with ts

0 values ranging from �80 to þ100% of theinitial ones.

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smaller influence of ry stresses in Eq. (2). Oppositely, the under prediction oftn0 is largely more preponderant on Pm=Pm

0 (maximum of � 29.1% forLO¼ 20mm), owing to a premature occurrence of tm

0 (Fig. 3) by the largerinfluence of tn on the failure process [Eq. (2)]. Figure 14 depicts a nearly pro-portional percentile reduction of Pm=Pm

0 with ts0 notwithstanding the value of

LO (maximum of � 79.1% for LO¼ 10mm). Actually, the transmission of loadsthrough the adhesive is accomplished mainly by shear [44]. By reducing ts

0,tm

0 diminishes nearly proportionally at almost the entire overlap [Eq. (2)] sincery stresses are only significant at the overlap edges (Fig. 4), and Pm=Pm

0 fol-lows the same tendency. This is valid either for small values of LO, for whichPm=Pm

0 relates to small gradients of ds, and big values of LO, corresponding tolarge ds variations (Fig. 5). On the other hand, the improvement of Pm=Pm

0

with over predictions of ts0 is only close to proportional for LO¼ 10mm

(maximum of � 78.7% for LO¼ 10mm and ts0 increase of þ100%), due to

the evenness of ds values along the overlap that result of a value of Pm=Pm0

almost exclusively depending on ts0. For bigger values of LO, owing to the

enlarging ds gradients along the overlap (Fig. 5), increasing ts0 results in higher

load transfer for dr< dr0 (dr

0 is the shear relative displacement at softeningonset; Fig. 3) and smaller or eventually nil load transfer for dr> dr

0

(Fig. 9b). As a result of these conflicting variations along the overlap whenPm=Pm

0 is attained, the improvement of Pm=Pm0 is limited. The combined

influence of tn0 and ts

0 (Fig. 15) is identical to the sole effect of ts0 (Fig. 14),

but with a slightly bigger impact on Pm=Pm0, especially for the smaller values

of LO (maximum improvement of � 90.0% and reduction of 79.4%, for therespective variations of tn

0 and ts0 equal to þ100% and �80%), because of

the bigger importance of sxy stresses on the joint strength than ry ones [44],and to a larger influence of ts

0 on the results (Fig. 14).

FIGURE 15 Percentile variation of Pm=Pm0 with tn

0 and ts0 values ranging from �80 to þ100%

of the initial ones.

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4.3.3. COMBINATION OF COHESIVE STRENGTH AND FRACTURE TOUGHNESS

Figure 16 reports on the combined influence of similar percentile variationsof Gn

c, Gsc, tn

0, and ts0 on Pm=Pm

0. The results show that the relationship istypically linear for under predictions of the cohesive parameters, followingthe overall tendency of Fig. 15 (tn

0 and ts0 have a higher influence on Pm=Pm

0

than Gnc and Gs

c; Fig. 12). As it regards to the increase of the cohesive para-meters, the relationship is nearly proportional for LO¼ 10mm, but it quicklydiminishes for bigger values of LO (maximum deviation for the LO¼ 80mmjoint: 100% improvement of the cohesive properties gives only a � 69.6%increase of Pm=Pm

0). This trend also resembles the results of Fig. 15, relatingto tn

0 and ts0, but the increase was larger, since Gn

c and Gsc were increased as

well (as previously discussed, the fracture parameters play an important rolefor large values of LO).

5. CONCLUDING REMARKS

This work aimed the evaluation of the cohesive parameters influence of atriangular CZM, used to approximate the behaviour of a thin adhesive layerin bonded joints, on the value of Pm=Pm

0, after validation of the methodologywith experiments. The results allowed a critical perception of the effect ofthese parameters on the numerical predictions. Owing to the range of avail-able techniques for the CZM definition, with dissimilar degrees of complexityand expected accuracy of the quantitative estimations, such a study is highlyimportant for the selection of the most suitable method. A few other practicalscenarios were emphasized for which it is particularly useful to know theinfluence of such parameters’ deviations. The single-lap geometry was chosen

FIGURE 16 Percentile variation of Pm=Pm0 with Gn

c, Gsc, tn

0 and ts0 values ranging from �80

to þ100% of the initial ones.

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covering a wide range of values of LO, allowing testing of different loadingconditions, from a practically even distribution of sxy stresses (small valuesof LO) to a condition of large sxy stress gradients and more localized influenceof ry stresses (large values of LO). The quantitative results presented in thiswork are solely applicable to the particular set of geometric and materialproperties selected, but they can qualitatively be extrapolated for differentbonded geometries and materials. For the conditions tested, under predictingGn

c and=or Gsc is highly detrimental to the accuracy (maximum under predic-

tion of � 44.5% for LO¼ 80mm, by reducing Gnc and Gs

c by �80% of theinitial values), except for extremely small values of LO. On the other hand,the over prediction of Gn

c and=or Gsc only slightly affects the results

(maximum over prediction of � 9.3% for LO¼ 80mm; Gnc and Gs

c improve-ment ofþ 100%). Over predictions of tn

0 are almost inconsequential(maximum of � 3.2% for the LO¼ 20mm joint and þ100% improvement),but moderate variations are expected if this parameter is under predicted,especially for small values of LO (maximum of � 29.1% for LO¼ 20mm and-80% reduction). Opposite to tn

0, ts0 largely influences the results with a nearly

proportional relation between the under prediction of Pm=Pm0 and the per-

centile variation of ts0 (maximum of � 79.1% for LO¼ 10mm and -80%

reduction). For over predictions of ts0, the improvement of Pm=Pm

0 is not sonotorious, especially for large values of LO (maximum of � 78.7% forLO¼ 10mm and ts

0 increase ofþ 100%). The combined effect of ts0 and tn

0

is close to that of ts0 (maximum improvement of � 90.0% and reduction of

79.4%, for the respective variations of tn0 and ts

0 equal toþ 100% and�80%). The simultaneous variation of Gn

c, Gsc, tn

0, and ts0 gives values of

Pm=Pm0 in close proportion to the parameter percentile variations, except

for over predictions and large values of LO. In these circumstances, theimprovement is not so significant, with a maximum deviation for LO¼ 80mm(over prediction of � 69.6% and þ100% properties improvement).

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