buckling behaviour of carbon–epoxy adhesively...

Journal of Adhesion Science and Technology 23 (2009) 1493–1513www.brill.nl/jast

Buckling Behaviour of Carbon–EpoxyAdhesively-Bonded Scarf Repairs

R. D. S. G. Campilho a,∗, M. F. S. F. de Moura a, D. A. Ramantani a, J. J. L. Morais b

and J. J. M. S. Domingues c

a Departamento de Engenharia Mecânica e Gestão Industrial, Faculdade de Engenhariada Universidade do Porto, Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal

b CITAB/UTAD, Departamento de Engenharias, Quinta de Prados, 5001-801 Vila Real, Portugalc Departamento de Engenharia Mecânica, Instituto Superior de Engenharia do Porto Rua Dr. António

Bernardino de Almeida 431, 4200-072 Porto, Portugal

AbstractThe present work is dedicated to the experimental and numerical study of the buckling behaviour under purecompression of carbon–epoxy adhesively-bonded scarf repairs, with scarf angles varying from 2 to 45◦.The experimental results were used to validate a numerical methodology using the Finite Element Methodand a mixed-mode cohesive damage model implemented in the ABAQUS® software. The adhesive layerwas simulated using cohesive elements with trapezoidal traction–separation laws in pure modes I and II toaccount for the ductility of the adhesive used. The cohesive laws in pure modes I and II were determinedwith Double Cantilever Beam and End-Notched Flexure tests, respectively, using an inverse method. Sincein the experiments interlaminar and transverse intralaminar failures also occurred, cohesive laws to simulatethese failure modes were also obtained experimentally following a similar procedure. Good correlationswere found between the numerical predictions and experimental results for the elastic stiffness, maximumload and the corresponding displacement, plateau displacement and failure mode of the repairs.© Koninklijke Brill NV, Leiden, 2009

KeywordsFracture, finite element method, cohesive damage model, repair

1. Introduction

Carbon-Fibre Reinforced Plastic (CFRP) composites are quite appealing for struc-tures requiring high specific strength and stiffness. However, these layered materialsare highly susceptible to suffer delamination damage, due to their low interlaminarstrength. Delamination can originate from low velocity impact events. Consideringan airplane wing, impact may occur during regular operation (e.g., bird strike) or

* To whom correspondence should be addressed. Tel.: +351939526892; Fax: +351225081584; e-mail:[email protected]

© Koninklijke Brill NV, Leiden, 2009 DOI:10.1163/156856109X433045

1494 R. D. S. G. Campilho et al.

Journal of Adhesion Science and Technology 23 (2009) 1493–1513

by an accident (e.g., tool impact during maintenance). Since this kind of damagesignificantly reduces the structures’ strength, replacement or repair must be fol-lowed. Repair of these structures is more efficient from economical and ecologicalpoints of view, since composite materials are difficult to recycle. Repair by adhe-sive bonding is a valid option due to its numerous advantages over the conventionalbolting or riveting methods, e.g., more uniform stress distributions, reduced weightpenalty, minimal aerodynamic disturbance, and fluid sealing characteristics. If afull or significant strength recovery is required, or if a repair without aerodynamicperturbation is needed, a scarf repair should be used. The higher efficiency of thisrepair method, compared with the easy-execution strap repairs, is due to the largerbond areas and the reduction of stress concentrations at the bond edges due to theadherend tapering effect.

The majority of the works on the strength of scarf joints or repairs (from thispoint on addressed as assemblies) focus on their tensile behaviour. Analyticalmethods [1, 2], experimental strain-measurement based methods [3–7] and Fi-nite Element Method (FEM) analyses [8–10] are the most common approaches.Whilst some of the analytical methods developed in this area focus only on dis-placement and stress analyses, the strain-measurement methods are often used tomonitor the condition of in-service-bonded structures and also for strength pre-diction. Extensometry can be used to determine the thresholds for the initiationof the first microcracks in these assemblies [5, 6]. Other authors [11] proposed atechnique based on relative strain measurements using Bragg grating sensors toidentify debonding onset in scarf joints. Debonding onset was detected by a differ-ential strain approach, using two sensors whose strain differential increased withthe debond length. The FEM was also extensively used to obtain the stress fieldsand predict failure of these assemblies, using appropriate failure criteria. In ten-sion, experimental and FEM studies showed an exponentially increasing strengthwith the reduction of the scarf angle, due to the corresponding increase in the bondarea [12–15]. As for the failure modes of these assemblies, it was observed that, ingeneral, scarf angles below 2◦ led to laminate tension failures, while larger anglesyielded failures in the adhesive layer [17, 18]. A two-dimensional stress and failureFEM numerical analysis of tensile loaded CFRP scarf repairs was carried out byOdi and Friend [19], for scarf angles varying from 1.1 to 9.2◦. The experimentalstrengths obtained in the work of Adkins and Pipes [17] were used to validate thenumerical strength prediction using the Tsai–Wu and maximum stress failure crite-ria for the adherend and the average shear stress failure criterion for the adhesive.Kumar et al. [18] presented an experimental and FEM numerical study regardingthe tensile strength of CFRP scarf joints. Numerical failure loads as a function ofthe scarf angle were obtained using the FEM and the Hashin–Lee criterion for theadherends, and agreed with the experimental ones.

However, the compression behaviour of these assemblies must also be consid-ered during the design process [20, 21]. In fact, the compressive strength of CFRP

R. D. S. G. Campilho et al. 1495Journal of Adhesion Science and Technology 23 (2009) 1493–1513

laminates is lower than the tensile one [22], implying that bonded assemblies un-der bending are more likely to fail in compression than tension. This also appliesto repaired sandwich structures with composite faces, which would fail more eas-ily within the face under compression. Under a compressive load, different failuremechanisms arise, e.g., fibre microbuckling, especially for structures with buck-ling restraining, or global buckling of the assembly. These mechanisms lead to acompletely different strength behaviour. Kumar et al. [23] studied the compressionbehaviour of adhesively-bonded CFRP scarf joints under uniaxial compression, fo-cusing on the influence of the scarf angle on the joints strength and failure mode.Aluminium tabs were glued on both joint faces to prevent global buckling of thespecimens. Under these conditions and below 3◦ scarf angles, the specimens failedby fibre microbuckling, while for larger scarf angles, failure occurred predomi-nantly by shear in the adhesive layer. A slight increase in the joint failure strengthwas observed with the reduction of the scarf angle. On the other hand, if no restrainton the global buckling of the assembly exists, a completely different behaviouris expected. Finn et al. [24] assessed the effectiveness of different bonded repairconfigurations on the compressive strength of CFRP laminates without global buck-ling restraining. It was observed that, under pure compression, the applied loadincreased until the buckling load, representing the load at which the global lami-nate buckling initiated. A plateau region appeared in the load–displacement (P –δ)curve, corresponding to the development of the laminate buckling, at an approxi-mate constant load. Failure occurred after a certain displacement. Helms et al. [25]developed an analytical model based on the Ritz Method to simulate the bucklingbehaviour of an adhesively-bonded Glass-Fibre Reinforced Plastic (GFRP) scarfjoint under pure compression. The scarf angle and the adhesive layer thickness andstiffness were the parameters studied. Finite element analyses were performed tovalidate the analytical model. The results showed that the buckling strength of thejoints was practically insensitive to the scarf angle for the larger scarf angles. Belowa certain scarf angle, the bucking load decreases abruptly. For these small scarf an-gles, increasing the adhesive layer stiffness or reducing its thickness are both validoptions to minimize this drastic reduction of the buckling load. Recently, Cohe-sive Zone Models (CZMs) have been used to simulate the mechanical behaviourof bonded assemblies [26–29]. This methodology, allowing predicting the strength,also accounts for the progressive damage evolution and identification of the failurepaths until complete failure. Campilho et al. [14] evaluated numerically using theFEM the tensile strength and failure modes of unidirectional CFRP scarf repairsusing a triangular shape CZM, for scarf angles ranging from 2 to 45◦. It was con-cluded that the model successfully predicted the strength and failure modes proneto occur in these repairs.

In this work, an experimental and numerical study is performed on the bucklingbehaviour of CFRP adhesively-bonded scarf repairs under pure compression. Scarfangles varying from 2 to 45◦ are considered. The experimental results were used



to validate a numerical methodology developed using the FEM and a mixed-modecohesive damage model implemented in the ABAQUS® software. The behaviour ofthe adhesive layer was modelled using cohesive elements with trapezoidal traction–separation laws in pure modes I and II. This shape was selected to account for theductility of the adhesive used in this work (Araldite® 2015). The cohesive lawsin pure modes I and II were determined with Double Cantilever Beam (DCB) andEnd-Notched Flexure (ENF) tests, respectively, using an inverse method. Using thisprocedure, the fracture energies in pure modes I and II are obtained from the respec-tive fracture characterization test (DCB or ENF) and the other cohesive parametersare estimated by fitting the experimental and numerical P –δ curves. Since in theexperiments interlaminar and transverse intralaminar failure also occurred, cohe-sive laws to simulate these failure modes were also obtained experimentally with asimilar procedure.

2. Cohesive Damage Model

2.1. Model Description

A mixed-mode (I + II) cohesive damage model implemented within interface fi-nite elements was used to simulate a ductile adhesive layer of Araldite® 2015. Tosimulate the behaviour of ductile adhesives, a trapezoidal law between stresses (σ )

and relative displacements (δr) between homologous points of the interface finiteelements with zero thickness was employed (Fig. 1). These types of laws are con-sidered to accurately reproduce the behaviour of thin adhesive layers in pure mode I[30] and pure mode II [31]. The constitutive relationship before damage onset is

σ = Eδr, (1)

where E is a stiffness diagonal matrix containing the stiffness parameters ei (i =I, II) defined in the next section. Considering the pure-mode model, after δ1,i (thefirst inflexion point in each pure mode, which leads to the plateau region of thetrapezoidal law) the material softens progressively. The softening relationship canbe written as

σ = (I − D)Eδr, (2)

where I is the identity matrix and D is a diagonal matrix containing, in the posi-tion corresponding to mode i (i = I, II), the damage parameter. In general, bondedassemblies are under mixed-mode loading. Therefore, a formulation for interface fi-nite elements should include a mixed-mode damage model (Fig. 1). Damage onsetis predicted using a quadratic stress criterion

(σI

σu,I

)2

+(

σII

σu,II

)2

= 1 if σI > 0,

(3)σII = σu,II if σI � 0,


Figure 1. The trapezoidal softening law for pure-mode and mixed-mode.

where σi (i = I, II) represent the stresses in each mode. It is assumed that nor-mal compressive stresses do not induce damage. Considering equation (1), the firstequation (3) can be rewritten as(

δ1m,I

δ1,I

)2

+(

δ1m,II

δ1,II

)2

= 1, (4)

where δ1,i (i = I, II) are the relative displacements in each pure mode at damageinitiation and δ1m,i (i = I, II) are the corresponding displacements under mixed-mode loading. Stress softening onset under mixed-mode conditions was predictedusing a quadratic relative displacements criterion similar to equation (4), leading to(

δ2m,I

δ2,I

)2

+(

δ2m,II

δ2,II

)2

= 1, (5)

where δ2,i (i = I, II) are the relative displacements in each pure mode at stresssoftening onset and δ2m,i (i = I, II) the corresponding displacements under mixed-mode. Crack growth was simulated by the linear fracture energetic criterion

JI

JIc+ JII

JIIc= 1. (6)

When equation (6) is satisfied damage growth occurs and stresses are completelyreleased, with the exception of normal compressive ones. Using the proposed cri-teria (equations (4)–(6)), it is possible to define the equivalent mixed-mode dis-placements (δ1m, δ2m and δum), with δm corresponding to the current equivalentmixed-mode relative displacement, and to establish the damage parameter (dm) inthe plateau region (δ1m � δm � δ2m) as:

dm = 1 − δ1m

δm(7)



and in the stress softening part of the cohesive law (δ2m � δm � δum) as:

dm = 1 − δ1m(δum − δm)

δm(δum − δ2m). (8)

The damage parameter is introduced in equation (2), thus simulating damage prop-agation. A detailed description of the proposed model is presented in the work ofCampilho et al. [32].

2.2. Cohesive Parameters

In this work, the adhesive layer is modelled numerically using the interface finite el-ements with the trapezoidal shape traction–separation laws described in Section 2.1,instead of the solid finite elements typically employed to this end. Consequently,they include a characteristic length tA (adhesive layer thickness). This parameteris integrated in the interface finite elements formulation in the stiffness matrix (E)components (ei , i = I, II), which simulate the adhesive layer behaviour in the elas-tic region. These are obtained from the ratio between the elastic modulus of thematerial in tension or shear (E or G, respectively) and tA. To fully characterizethe cohesive laws in modes I and II, the local strengths (σu,i), the second inflexionpoints (δ2,i) and the fracture energies (Jic) must also be determined. While Jic canbe obtained by standardized tests, different approaches can be followed to obtainexperimentally σu,i and δ2,i . These quantities can be equalled to the correspond-ing bulk properties, or inverse methods can be used. The inverse techniques aremore advisable, since it is known that adhesives as a bulk behave differently thanas thin layers [30, 31, 33]. To simulate numerically the mechanical behaviour ofthese repairs, the cohesive laws in pure modes I and II for the adhesive layer, com-posite interlaminar and composite intralaminar (in the transverse direction) weredetermined with DCB and ENF tests, respectively, using an inverse method. For theadhesive layer laws, it is emphasized that cohesive failures were always obtained,which is essential to characterize the adhesive layer accurately.

This technique can be applied in two steps (for pure mode I or II cohesive laws).The fracture energies in pure mode I or II (JIc and JIIc, respectively) are initiallyobtained from standardized DCB or ENF tests, respectively. The geometry anddimensions of the DCB and ENF specimens used in the adhesive layer fracturecharacterization procedure are presented in Fig. 2. For the interlaminar and trans-verse intralaminar failures, the same geometries were used, except for the adhesivelayer and values of arm thickness (h = 1.8 mm for the interlaminar and h = 3.6 mmfor the intralaminar fracture characterization). In these cases, the initial crack (a0)

was introduced during the plates manufacturing procedure, using a 25 µm thicknessteflon strip. Known data reduction schemes, such as the Compliance CalibrationMethod (CCM) and Corrected Beam Theory (CBT), along with the Compliance-Based Beam Method (CBBM), previously developed [34], were used to extractthe average JIc and JIIc values for the adhesive layer, and also composite inter-laminar and composite intralaminar (in the transverse direction) failures. For the


(a)

(b)

Figure 2. Geometry of the DCB (a) and ENF (b) tests for the adhesive layer characterization (dimen-sions in mm).

adhesive layer fracture characterization, the same value of tA for the specimens tobe simulated must be used in these tests, due to the known dependency of the ad-hesive layer behaviour on its thickness [35]. The fracture energy is then used asan input parameter in numerical DCB or ENF models. These models, having themeasured dimensions of each tested specimen, include the respective pure modecohesive law with the fracture energy previously determined and typical values forσu,i and δ2,i . These properties are determined by performing a few numerical iter-ations until a good accuracy between the numerical and experimental P –δ curvesis obtained. Figures 3 and 4 show, for a single tested specimen, the experimentaland numerical P –δ curves for the adhesive layer characterization procedure in puremodes I (DCB test) [36] and II (ENF test) [37], respectively, after the fitting pro-cedure. In all cases, at least five specimens were tested and the average values ofeach cohesive parameter were used to build the respective cohesive law. A detaileddescription of this methodology is presented in [34]. Table 1 presents the cohesiveparameters of the traction–separation laws in pure mode I and II used to simulatethe adhesive layer. The elastic moduli in tension and shear were determined ex-perimentally [38] with bulk tensile and Thick Adherend Shear Test (TAST) tests,respectively (E = 1850 MPa; G = 650 MPa). The cohesive laws for the interlam-inar and transverse intralaminar failures are also presented in Table 1. However,in these situations, a penalty function method was used for the initial ascendingpart of the cohesive laws (considering ei = 106 N/mm3), since the interface fi-nite elements simulate a zero thickness interface instead of a finite thickness layer.



Figure 3. Experimental and numerical P –δ curves comparison for a single specimen tested (DCBtest) [36].

Figure 4. Experimental and numerical P –δ curves comparison for a single specimen tested (ENF test)[37].

Table 1.Cohesive parameters in pure modes I and II used to simulate different failures

Cohesive laws i Jic (N/mm) σu,i (MPa) δ2,i (mm)

Adhesive layer (Araldite® 2015) I 0.43 23.0 0.01870II 4.70 22.8 0.1710

Interlaminar (∗) I 0.33 25.0 0.00008II 0.79 13.5 0.00038

Transverse intralaminar (∗) I 0.54 42.6 0.0012II 0.93 39.3 0.0022

Fibre (∗) I 20.0 2000.0 0.004II 4.0 100.0 0.0005

∗ SEAL® Texipreg HS 160 RM.


Moreover, triangular traction–separation laws were used, due to the brittle natureof these interfaces. Table 1 also includes the cohesive parameters used to simulatefibre failure. This failure option will be introduced later on in the numerical mod-els in the 0◦ plies of the laminates and patch, even though it was not observed inthe experiments. The fibre properties were not obtained experimentally; they wereestimated from typical values in the literature [32].

3. Experimental Work

Figure 5 shows the geometry of the scarf repairs (L = 170 mm, b = 15 mm,tP = 2.4 mm, tA = 0.2 mm and e = 10 mm). L corresponds to the test lengthbetween the grips edges. The total length of the specimens, including the edgesclamped in the grips, is 270 mm. Scarf angles (α) of 2, 3, 6, 9, 15, 25 and 45◦ wereevaluated. Smaller scarf angles were not tested, since the repair lengths neededwere not compatible with the chosen value of L. Additionally, using higher valuesof L, the increased buckling of the specimens would significantly diminish the re-pairs strength [24]. The laminates and patches were fabricated using carbon/epoxypre-preg (Texipreg HS 160 RM from SEAL®, Legnano, Italy) with 0.15 mm plythickness and a [02,902,02,902]S lay-up. The respective mechanical properties arepresented in Table 2 [29]. The laminates and patches were cut with a diamond discsaw from bulk plates fabricated by hand lay-up and cured in a hot-plates pressfor one hour at 130◦C. Following this, a stone grinding wheel was used to grindthe laminates and patches to the chosen scarf angles. Since this procedure led to a

Figure 5. Scarf repair geometry.

Table 2.Mechanical properties of a 0◦ CFRP ply [28]

E1 = 1.09E+05 MPa ν12 = 0.342 G12 = 4315 MPaE2 = 8819 MPa ν13 = 0.342 G13 = 4315 MPaE3 = 8819 MPa ν23 = 0.380 G23 = 3200 MPa



rough bonding surface, enough to avoid adhesion failure between the adhesive andcomposite, the surface preparation involved only cleaning with acetone. The spec-imens were bonded with the ductile adhesive Araldite® 2015 (from Huntsmann,Basel, Switzerland) using a device that guaranteed the laminates and patches align-ment in both the thickness and width directions. The 0.2 mm adhesive thicknesswas achieved using a differential length approach, measuring the total length of thespecimens with a digital micrometer in the device before and after placing the adhe-sive. The specimens were cured at room temperature. The compression tests wereperformed with an Instron® (Norwood, MA, USA) 8801 hydraulic testing machineequipped with a 100 kN load cell. The testing setup guaranteed that the grips wereperfectly aligned during the test and that no rotation of the grips occurred. The spec-imens were tested at room temperature under displacement control (0.5 mm/min).The P –δ data were extracted from the load cell measurements and the grips dis-placement, with a sample rate of 5 points per second. Six specimens were tested foreach geometry and at least four valid results were always obtained.

4. Numerical Analysis

A numerical analysis was carried out in ABAQUS® (from Dassault Systèmes,Suresnes, France) to simulate the mechanical behaviour of bonded repairs usingthe methodology presented in Section 2. The laminates and patches were modelledwith plane-stress 8-node rectangular and 6-node triangular solid finite elements.Geometrical and material non-linearities were included in the numerical analyses.Figure 6 shows a detail of the mesh at the lower scarf tip (region B in Fig. 7)for the 9◦ scarf angle repair. Eighty solid finite elements were used along thebond length near the adhesive layer, due to the high stress gradient in this region[14, 15, 19]. However, since stresses are approximately constant far from this re-gion, a coarsening technique was used to reduce the number of elements in themodels (Fig. 6). The laminates and patches were modelled as orthotropic materi-als (considering the properties in Table 2), using one element through-thicknessfor each set of two equally-oriented plies. Figure 7 represents the symmetry andboundary conditions used to simulate the scarf repairs compressive tests. Due tothe symmetry of this geometry, only half the specimen was considered, applyingsymmetry conditions at the middle of the repair (line A–A in Fig. 7). Moreover,at the edge of the specimens, the displacements were restrained in direction y

and a compressive displacement was applied. The placement of the interface fi-nite elements is presented in Fig. 8. The adhesive layer elements were placed alongthe bond length replacing the adhesive layer, the interlaminar elements were posi-tioned between differently oriented plies, the transverse intralaminar elements wereused vertically in the 90◦ plies to simulate the intralaminar matrix cracking andthe fibre elements were placed vertically in the 0◦ plies to simulate fibre crack-ing.


Figure 6. Detail of the mesh at the overlap edge for the 9◦ angle repair.

Figure 7. Symmetry and boundary conditions for the simulation of the scarf repair compression test.

Figure 8. Placement of the interface elements with different cohesive laws in the numerical models.

5. Results

5.1. Stress Analysis

It is known that normal and shear stresses in the adhesive layer of scarf assembliespresent an almost constant profile for isotropic adherends [5, 6] or fibre reinforcedones, with unidirectional laminates [14]. This can be justified by a smaller loadeccentricity, compared to lap geometries [23, 29] and by the adherend tapering ef-fect in scarf assemblies [23]. Moreover, for small scarf angles, normal stresses arealmost nil and the adhesive layer is practically loaded in pure shear [12], whichleads to a high efficiency of these assemblies relatively to the bond area. However,considering layered materials with differently oriented plies, it is observed that thedifferences in compliance in the load direction of each ply lead to wavy stress dis-tributions [8, 9, 15, 16].



This section presents an elastic normal and shear stress analysis in the adhesivelayer along the bond length. The stresses were extracted from the adhesive layerinterface finite elements, using the coordinate system t–n, i.e., tangentially and nor-mally to the adhesive layer (Figs 5 and 7). Figures 9 and 10 present normal (σn) andshear (τtn) stresses, respectively, in the adhesive layer along the bond length as afunction of the scarf angle. Both stresses are normalized by the average shear stressof the respective scarf angle repair along the bond length (τavg). Compressive nor-mal stresses were generally observed in the adhesive layer, which is an advantagecompared to the tensile load. Indeed, under a tensile load, significant peel stressesreduce the repair strength [14]. Moreover, it can be observed that compressive nor-mal stresses are more significant near the 0◦ plies and are much less significant thanshear stresses for the smaller scarf angles [12, 14]. The compressive normal stressesincrease gradually with the scarf angle, until an approximate magnitude to the shearstresses is observed for a 45◦ scarf angle repair. Shear stresses exhibit similarly awavy distribution, peaking near the 0◦ plies. Also, the shear stress gradients be-tween the 0 and 90◦ plies increase gradually with the reduction of the scarf angle.This is caused by the higher scarf length for each set of two 90◦ adjacent plies, forthe smaller scarf angles. This reduces the deformation constraining effects of thestiffer 0◦ plies adjacent to the 90◦ plies, allowing their higher deformation, with thecorresponding shear stress reduction in these regions [9].

5.2. Mechanical Behaviour

Figures 11 and 12 show the numerical and experimental P –δ curves for the 45and 2◦ scarf angle repairs, respectively. Initially, an elastic region was observed,during which no global buckling of the assembly occurred. A plateau region fol-lowed, characterized by an approximate constant load. This region corresponded tothe beginning and development of the global buckling of the repairs between thetesting machine grips [24]. All the repairs studied presented this behaviour, eventhough significant differences were observed in the extent of buckling, dependingon the scarf angle. The two intermediate load reductions prior to failure, for the2◦ scarf angle repair, were caused by highly localized failures at the scarf edges,initially in region B and later in region A (Fig. 7). Figure 13 presents a comparisonbetween the numerical and experimental deformed configurations of the 2◦ scarfangle repair immediately before failure, allowing observing a good correspondencebetween the two. Whilst the premature failure in region A is easily identified inFig. 13, in region B the localized failure occurred mostly by shear in the adhesivelayer and is not clearly visible. Mainly two different behaviours were observed dur-ing the global buckling of the repairs, depending on the scarf angle. For the largerscarf angle repairs (15, 25 and 45o), cohesive failures in the adhesive layer nearthe patch/adhesive interface occurred, which were captured by the numerical sim-ulations. Figure 14(a and b) show experimental and numerical cohesive failures,respectively, for a 45◦ scarf angle repair. Moreover, only relatively small plateau re-


Figure 9. Normal stress (σn) distributions in the adhesive layer.

Figure 10. Shear stress (τtn) distributions in the adhesive layer.

gions were detected, corresponding to an equally minor buckling prior to the repairfailure. On the other hand, the 2, 3, 6 and 9◦ scarf angle repairs experienced a mixedcohesive and interlaminar/intralaminar failure of the patch, as obtained numerically.



Figure 11. Numerical and experimental P –δ curves for the 45◦ scarf angle repair.

Figure 12. Numerical and experimental P –δ curves for the 2◦ scarf angle repair.

Figure 15 shows a schematic representation of this failure, which was characterizedby a cohesive failure near the 0◦ plies and a mixed interlaminar/intralaminar failureof the 90◦ plies. Figure 16(a and b) represents the experimental and numerical fail-ures, respectively (2◦ scarf angle repair). The different failure modes observed forthe smaller scarf angle repairs are attributed to the pronounced global buckling ofthe repairs before failure, leading to premature intralaminar failures in the 90◦ plies


(a)

(b)

Figure 13. Experimental (a) and numerical (b) deformed configurations of the 2◦ scarf angle repairimmediately before catastrophic failure.

(a) (b)

Figure 14. Scarf angle repair (α = 45◦) after fracture due to a cohesive failure in the adhesive layer:experimental fracture (a) and numerical simulation (b).

in the repair region followed by interlaminar failures, prior to failure in the adhesivelayer.



Figure 15. Schematic representation of a mixed failure.

(a)

(b)

Figure 16. Experimental (a) and numerical (b) mixed failures for a 2◦ scarf angle repair.

5.3. Summary of the Results

This section summarizes the results obtained in the experiments and the respectivestandard deviations, which are compared with the numerical predictions. The para-meters considered are the elastic stiffness of the repairs (K , Fig. 17), the maximumload (Pm, Fig. 18) and the corresponding displacement (δm, Fig. 19) and the plateaudisplacement (δp, Fig. 20). K represents the load/displacement ratio in the elasticregion of the P –δ curve. Pm is the maximum load sustained by the repairs andδm the respective displacement. δp corresponds to the failure displacement, duringwhich the repair is able to sustain loads. It was observed that K slightly increaseswith the reduction of the scarf angle, from the 45 to the 3◦ scarf angle repair. Below


Figure 17. Elastic stiffness as a function of the scarf angle. Experimental results, numerical predic-tions and respective tendency as a function of α.

Figure 18. Maximum load as a function of the scarf angle. Experimental results, numerical predictionsand respective tendency as a function of α.

this value, an abrupt drop of the elastic stiffness was observed [25]. This behaviouris attributed to the significant increase of the repair region, which is more compli-ant than the undamaged composite. On the other hand, it is known that the bucklingstrength of slender structural elements under pure compression is governed by theirstiffness, rather than the strength of the materials involved [25, 39]. Consequently,the maximum load of the repairs reflected the global trend of the elastic stiffness,i.e., a gradual increase from the 45 to the 3◦ scarf angle repair, and an abrupt drop



Figure 19. Maximum load displacement as a function of the scarf angle. Experimental results, numer-ical predictions and respective tendency as a function of α.

Figure 20. Plateau displacement as a function of the scarf angle. Experimental results, numericalpredictions and respective tendency as a function of α.

for the 2◦ scarf angle. For both these two parameters, only a small deviation in theexperimental data was observed, together with an excellent correlation with the nu-merical predictions. By analysing δm, a similar global tendency is identified bothexperimentally and numerically, characterized by a practically no influence of thescarf angle on δm from the 45 to the 15◦ scarf angle repair. Again a similar trendwas observed for δp, which revealed an increasing plateau displacement with thereduction of the scarf angle. This result can be explained by the reduction of both


the laminates and patch stiffness in the bond regions, allowing them to deform elas-tically of an increasing amount with the reduction of the scarf angle, prior to failurein the adhesive layer.

In summary, it is observed that the buckling strength of the scarf repairs slightlyincreases with the reduction of the scarf angle, up to a given value. Below this point,the abrupt drop of the repair compressive stiffness leads to a corresponding drop inthe buckling strength. Consequently, no significant strength advantage exists withthe reduction of the scarf angle, in contrast to the tensile case. In fact, under a ten-sile load, the repair strength increases exponentially with the reduction of the scarfangle [13–15], due to the corresponding increase in the bond area, which governsthe strength of these repairs under tension. However, reducing the scarf angle un-der compression significantly increases the displacement supporting capability ofthe repair before failure, which can be an advantage if the repaired region of thestructure is mainly forced to sustain a given displacement.

6. Concluding Remarks

In this work, a numerical methodology was presented to simulate the mechani-cal behaviour of the adhesive layer in bonded assemblies. A mixed-mode cohesivedamage model with a trapezoidal shape was employed to simulate the adhesivelayer behaviour. The methodology was extended to simulate the interlaminar, in-tralaminar and fibre fracture of the composite adherends, to fully reproduce theexperimental failure modes. The different traction–separation laws were determinedusing an inverse method, which consisted in obtaining the fracture toughness inpure modes I and II with Double-Cantilever Beam and End-Notched Flexure tests,respectively, and estimating the remaining parameters of the pure mode laws usinga fitting iterative procedure between the numerical and experimental P –δ curves.The mixed-mode behaviour of the adhesive layer or interfaces, typical in these as-semblies, was simulated with appropriate criteria. This numerical methodology wasvalidated by simulating the buckling behaviour of carbon–epoxy scarf repairs underpure compression, using different scarf angles, and comparing with experiments.The comparison was performed in terms of the repairs elastic stiffness, the maxi-mum load and the corresponding displacement, the plateau displacement, as wellas the failure path until complete failure. The results obtained allowed concludingthat the methodology presented in this work is adequate to simulate the mechanicalbehaviour of these assemblies.

Acknowledgements

The authors would like to thank the Portuguese Foundation for Science andTechnology for supporting the work presented here, through the individual grantSFRH/BD/30305/2006 and the research project PDTC/EME-PME/64839/2006.



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buckling behaviour of carbon–epoxy adhesively...

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