Construction and Building Materials 25 (2011) 432–441

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Construction and Building Materials

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Experimental and numerical investigation of shear strain along anelasto-plastic bonded lap joint

Sylvain Chataigner a,b,*, Jean-Francois Caron a, Van Anh Duong a, Alberto Diaz Diaz c

a Université Paris-Est, UR Navier, UMR CNRS-LCPC, Ecole des Ponts Paristech, 6-8 Avenue Blaise Pascal, 77455 Marne La Vallée Cedex 2, Franceb LRPC d’Autun, Boulevard de l’Industrie, BP 141, 71 405 Autun, Francec CIMAV, Miguel de Cervantes 120, Complejo Industrial Chihuahua, 31109 Chihuahua, Chihuahua, Mexico

a r t i c l e i n f o a b s t r a c t

Article history:Available online 20 June 2010

Keywords:Structural bondingShear strainAdhesive plasticityModeling

0950-0618/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.conbuildmat.2010.05.021

* Corresponding author at: LRPC d’Autun, BoulevardAutun, France.

E-mail address: sylvain.chataigner@developpemtaigner).

This study investigates the use of an elasto-plastic adhesive in a bonded double lap joint to increase itsultimate capacity. The first section describes the experimental work we performed to characterize thematerials and measure the shear strain along the joint. A comparison between an elastic adhesive anda highly plastic adhesive is made and the effect of the type of behaviour on both the capacity and thecharacteristic anchorage length is studied. The second section then compares the experimental resultswith a classical 3D finite element model. Good correlation was found between numerical expectationsand experimental measurements, and the modelling allowed us to highlight typical phenomena linkedto the use of an elasto-plastic adhesive. In addition, a parameter based on a microscopic measurementhas been defined which allows us to describe the onset of plasticity and predict maximum capacity.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The technology of adhesive bonding is currently under closeinvestigation in order to obtain a good understanding of the mech-anisms at work in this kind of assembly and identify the keyparameters for proper design, high quality manufacture and longservice life.

In view of the fact that different kinds of adhesion forces exist[1,2], and that these are highly influenced by the surface prepara-tion, it is easy to see the importance of the manufacturing stagewith regard to optimizing joint capacity. The quality of manufac-ture is mainly determined by the surface preparation and the cur-ing conditions of the adhesive. Similarly, the service life will bedetermined by the exposure of the adhesive, which is an organicmaterial, to extremes of humidity, solar radiation, and tempera-ture. In the case of structural adhesive bonding, the quality ofmanufacture and service life are thus closely linked to the charac-teristics of the material and the adhesion forces.

Proper design is achieved more by reducing stress concentra-tions and studying the stress distribution within the joint. This as-pect has been investigated by a large number of researchers whohave highlighted the importance of detailing geometries andchoosing materials with suitable characteristics when designing

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de l’Industrie, BP 141, 71 405

ent-durable.gouv.fr (S. Cha-

the bonding technology so as to reduce the concentration of bothshear and peel stresses. As adhesive joints work best in shear, mostof the studies have focused on determining the shear stress, whichis nonlinear along the lap length [3–5]. The following approach wasthus adopted to reduce the observed shear concentrations on theedges of the lap length. First, some researchers worked on thegeometry of the adhesive fillets [6,7] or the adherends at the endsof the joint [8]. Others then studied the use of adherends with var-iable thickness [9] or variable modulus [10]. The geometry and theflexibility of the adherends has therefore been thoroughly investi-gated. As far as the adhesive is concerned, some studies have ana-lyzed the use of two adhesives with different moduli [11] or gradedmodulus adhesive [12].

The considerable number of difficulties associated with adhe-sively bonded joints mean their design is a difficult process. For ci-vil engineering applications, the design process has to be bothsimple and reliable. The authors of this study have conducted alarge number of investigations to obtain a better understandingof the way adhesively bonded joints work either in the case of con-crete reinforcement using a composite material [13] or the assem-bly of composite structures [14]. This understanding means thatsimplified modelling procedures often based on analytical meth-ods can be applied for adhesively bonded joints. One such ap-proach, which has recently been patented, has been used todesign an optimized adhesively bonded anchorage [15].

This study focuses on the identification of conveniently andaccurately measurable macroscopic values to characterize abonded joint for the design of an anchorage for a pultruded carbon

S. Chataigner et al. / Construction and Building Materials 25 (2011) 432–441 433

plate on a pedestrian footbridge [16]. More particularly, we haveinvestigated the role of the plasticity of the adhesive in bondedlap joints. This topic was first considered by Hart-Smith [17]. An-other analytical study which has examined the influence of plastic-ity on the anchorage length considers multilinear behaviour of theadhesive and is described in [14]. Both these studies were con-cerned only with determining stress profiles. Shear strain alongthe lap length is dictated by the plastic flow rule which these ana-lytical procedures do not consider.

In order to compare the experimentally measured strain to thetheoretically expected values it is necessary to use numerical mod-elling, for example finite element techniques. Among the first toconduct such modelling were Bigwood and Crocombe in [18].The importance of the use of nonlinear models for some of theadhesives used in the industry has been highlighted in [19]. Theseauthors described the inadequacy of the simple Von Mises yieldcriterion. However, in the modelling presented below, this crite-rion has been adopted to make an approximate comparison be-tween different models. Further work will address the issue ofthe most suitable criterion, taking account of the sensitivity ofthe adhesive to hydrostatic stress and using a Drucker–Prager yieldcriterion as described in [15,20].

In the first part of this paper, we have shown that plasticitychanges in the adhesive drastically modify the behaviour of thebonded joints. In particular, they affect the characteristic anchor-age length, and hence the maximum capacity. With regard to thedesign procedure, our work concentrated on understanding theadhesive behaviour of the adhesive. Combining experimental re-sults obtained from characterization tests and the observation ofstrain along bonded double lap joints with theoretical modellingenabled us to represent the effect of nonlinear behaviour of thistype. The modelling was shown to closely match the strain mea-surement based on microscopic study of the adhesive in shear.

We shall begin by presenting the experimental investigations.These involved several steps: the characterization of the differentmaterials, bonded joint failure tests, and measurement of shearstrain along the lap lengths for both elastic and elasto-plastic adhe-sives. A global quantity related to the 3D shear strain, the slip, wasidentified, rigorously defined and chosen as a potential global mac-roscopic parameter for characterizing a joint. The second sectionthen describes the use of finite element modelling to analyze theseexperimental results and investigates the usefulness of slip as a de-sign parameter. This finite element modelling took account of thenonlinear behaviour of the adhesive in order to determine theshear strain along the lap length. Finally, a comparison has beenmade between the experimental results and those obtained fromthis numerical modelling. Some of our observations have also beentied in with other theoretical results.

2. Experimental investigations

The first stage of the study was experimental and consisted ofusing an elastic and an elasto-plastic adhesive with unidirectionalpultruded carbon plates as adherends in a bonded lap joint. Tensiletests were conducted to observe the behaviour of the adhesive.Obviously, its shear behaviour may differ from its tensile behav-iour, particularly in the case of thin adhesive layers. However,the adhesive joints used in civil engineering are relatively thickand the anisotropy of the adhesive material observed in some stud-ies [21] is certainly less likely to occur. For this reason it wasdecided that it would be acceptable to adopt the hypothesis of isot-ropy in this study to simplify calculations. After the characteriza-tion of the materials alone, several series of bonded joints weretested to find the differences in terms of maximum capacitiesand to study the strain profiles along the lap lengths. The proce-

dure used to manufacture the bonded joints was very similar tothat applied in [14] and will be described more precisely at theend of this section.

2.1. The used materials

The adherends and the two adhesives were subjected to tensiletests. The tensile tests were conducted at low speed to limit theimpact of viscosity effects on the results. Although viscosity effectsmay be relatively large for this kind of polymer, it was decided notto include them in the initial modelling. More details about viscos-ity may be found in [22]. The strain was measured using conven-tional strain gauges.

For bonding, two-component epoxy adhesives were used inboth cases. The first, which exhibits elastic behaviour is more com-monly used for reinforcing concrete structures with carbon plates.Its tensile Young’s modulus was found to be 4940 MPa (Fig. 1). Thesecond, exhibits elasto-plastic behaviour, and is a commercialindustrial adhesive. Its tensile behaviour breaks down into twoparts: an elastic part and a plastic plateau. The elastic tensile mod-ulus was found to be 2500 MPa, and the plateau was reached at37 MPa (Fig. 1). The shear properties were obtained from the ten-sile properties, assuming the materials were isotropic. Althoughwe shall not consider it in depth in this study, readers should bearin mind that the question of whether adhesives have isotropicbehaviour or not in bonded joints is still unresolved, as mentionedin [21]. As the same study [21] has found the differences in shearmodulus between bulk material and joint material to be fairlysmall, the isotropic hypothesis seemed to be a reasonable startingpoint. Besides, in the special case of civil engineering applications,adhesive joint thicknesses are usually relatively large reducing thedanger of the alignment of internal loads which may be responsi-ble for anisotropic behaviour. Poisson’s ratio was considered tobe 0.3 for both adhesives. In the manufactured bonded joints, theadhesive thickness was 0.6 mm for the elastic adhesive, and0.25 mm for the elasto-plastic adhesive, fulfilling the recommen-dations in the technical data sheet.

The orthotropic behaviour of the pultruded carbon plate wasdetermined using two tensile tests, and the ‘‘rule of mixture” the-ory. The tensile Young’s modulus in the fibre direction was162,000 MPa, and in the transverse direction 10,600 MPa. ThePoisson’s ratio was found to be 0.325. The rule of mixture gavethen an out-of-plane shear modulus of 4077 MPa. The used pul-truded plate was 1.2 mm thick and 20-mm wide.

2.2. Bonded joints

Double lap bonded joints were chosen in order to minimize peelstresses and obtain shear stresses in the adhesive joint. A carefulmanufacturing procedure was applied to ensure good alignmentof the adherends. The bond thickness was controlled by embeddingtwo nylon wires within the joint, far from the edges to avoid dis-turbing these critical zones. In order to investigate the behaviourof the bonds thoroughly, a first series of tests was conducted tofind the optimum surface preparation for achieving cohesive ratherthan adhesive failure of the joint. The results were different for thetwo adhesives. While simple degreasing appeared to be sufficientfor the elasto-plastic adhesive, additional mechanical abrasionwas necessary in the case of the elastic adhesive to obtain cohesivefailure, which actually occurred as a result of interlaminar failureof the pultruded plate. The failure mode was checked using micro-scopic observation of the bonded joints after failure (Fig. 2).

Once the surface preparation was determined, several series oftests where the bonded length was varied were conducted. Theseseries provided a good idea of the typical anchorage length of theassembly for each of the two adhesives. For the elastic adhesive,

Fig. 1. The tensile behaviour of the two adhesives: the brittle elastic adhesive is on the left, and the elasto-plastic adhesive is on the right (for this adhesive, the measurementdevice was saturated and the ultimate strain was consequently not determined).

Fig. 2. Photos of specimens after failure showing its cohesive nature: intralaminar failure in the adherend.

Fig. 3. Theoretical failure capacity versus bonded length according to Volkersen’s elastic theory (—), and experimental results (�): The elastic adhesive is on the left and theelasto-plastic adhesive is on the right.

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Volkersen [4] has provided a satisfactory theoretical analysis ofshear stress along lap lengths. Using this theory, and the mechan-ical characteristics of the materials stated in the paragraph above,we plotted a theoretical curve of maximum capacity versus bondedlength for the elastic case. This theoretical elastic curve was thensuperimposed on the experimental results obtained for themaximum capacity of the specimens (Fig. 3). Here, the comparison

between theoretical expectations with experimental results essen-tially involved the comparison between theoretical and experi-mental anchorage lengths. In Fig. 3, the elastic theory matchedthe experimental results well in the first case (elastic adhesive).This is a very interesting result because it means that Volkersen’ssimplified approach, which ignores the shear stress singularity atthe edge of the joint and provides a finite value of this shear stress

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even at the edge, is nevertheless valid and represents a convenientmodel for joint design. The maximum shear stress in Volkersen’sequation is used here as a coefficient to identify (here 39 MPa forthe elastic adhesive) which is able to predict the anchorage lengthsand thus the shape of the theoretical curves. The description of thejoint using elastic theory has been shown to be inadequate when anonlinear adhesive is used (Fig. 3). When this is the case, a maxi-mum shear stress parameter of this type is therefore inadequate,and a different approach must be applied to predict the ultimatecapacity. A second observation can be made regarding the maxi-mum capacity of the joint. For the second adhesive, the maximumcapacity is indeed four times higher than for the first adhesive. Thisincrease in maximum capacity was actually linked to the increasein the anchorage length and is described in greater detail in [14].

2.3. Shear strain measurements

In addition to failure tests on bonded joints, it was decided tomeasure the strains in the adhesive while applying a constant loadand after load/unload cycles. These measurements set out to checkqualitative expectations, to highlight plasticity in shear, to identifya global quantity that would be useful in design, and to study thedifferent modelling alternatives available. Some authors have ap-proached this problem using different approaches. In [23], theauthors measured the axial strain along the adherends in orderto deduce the shear strain in the adhesive. A similar method hasalso been used to characterize composite to concrete bonded jointsin the field of structural reinforcement [13]. Other researchers haveused Moiréinterferometry [6] or grid methods [24] to measure inplane surface deformation. From these full-field measurementsone can then extract characteristic parameters using, for instance,the virtual field methods described in [25].

In our study, measurement simply consisted of observing theadhesive joint with a microscopic magnification lens. A specificspecimen preparation procedure developed in [26] was adopted,as it proved to be necessary to carry out fine polishing of the jointsides. After this important polishing operation, lines were drawnon the joints using a fine-tipped pencil in order to measure themaximum strain (Fig. 4). This measurement method revealed themaximum strains on both sides of the joint. It was decided to mea-sure the relative displacement of the two interfaces. We shall referto this parameter as slip, and it includes an elastic and a nonlinearcomponent. The slip was measured both for the elastic and theelasto-plastic adhesives. However, in the case of the elastic adhe-sive, the variations in slip were too small for our measurementmethod to detect. Consequently, all the slip measures presentedbelow were made on joints bonded with the elasto-plasticadhesive.

Fig. 4. Slip measurement

The average shear strain in the thickness direction can be de-duced from the measured slip on the basis of the adhesivethickness.

c ¼ dec

ð1Þ

where c is the average shear strain, d is the slip, and ec the thicknessof the adhesive.

Several different tests were conducted measuring shear strainsat different points along the lap length. The first test aimed toascertain the nature of the strain (elastic, plastic, or damage-re-lated) in the adhesive by applying a load/unload cycle.

2.4. First results

At the outset, it was decided to make two measurements for anapplied force of about 5 kN. The first measurement of shear strainalong the lap length was made at first loading. The second mea-surement was then made after one loading/unloading cycle to26 kN. The aim of this approach was to detect residual strain andthus check the adequacy of plasticity models in the studied case.The results are given in Fig. 5.

The results clearly show the existence of a nonlinear phenome-non within the joint resulting in residual strain. The plasticityhypothesis may thus be valid in the studied case. Later, we willshow that damage does not seem to be responsible for this residualstrain because the rigidity of the joint was not affected. From Eq.(1) we know that the slip d and the average shear strain c are pro-portional to each other. The measured slip profiles d can thus be di-rectly related to average shear strain profiles c along the lap length.It is worth highlighting that the average shear strain profiles c aareconcentrated on the edges and have a similar form to Volkersen’saccount of expectations according to elastic theory [4] (Fig. 5). Inparticular, the lack of equilibrium in the joint leads to a higher con-centration of slip at the left end which is in our case the end wherethe load was applied and where failure seems to start, even if thisis difficult to verify experimentally.

3. Modelling the joint with elasto-plastic adhesive

To verify the validity of slip as a design parameter and betterunderstand its mechanical significance, it must be related to the3D nature of the joint. This was done by using a finite elementmodel. Obviously, the modelling in this case was still simplified,with relatively low mesh refinement and a simplified materialmodel for adhesive plasticity.

Modelling was performed with the Abaqus finite element codeand prior to the experimental investigations (Class A). A 2D plane

along the lap length.

Fig. 5. Load history on the left, and experimentally measured slip at first loading and after one load/unload cycle to 26 kN for an applied load of 5 kN on the right.

Fig. 6. Finite element model showing the mesh density.

Fig. 7. Change in the shear strain profile throughout the adhesive thickness.

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strain model was used due to the fairly large width of the joint, andquadrilateral interpolation was used. The mesh density was chosenso that four reduced quadrilateral elements were obtained in thethickness of the adhesive as recommended in [27]. The classicalVon Mises plasticity model with an associated flow rule was usedin this study, although some authors have demonstrated that amodel that considers hydrostatic dependency and a non-associatedflow rule may more accurately describe adhesive plasticity in thecase of multi-axial loading [22]. Our approach was based on anassessment of the equivalent Von Mises stress and the experimen-tal tensile stress–strain curve data for the adhesive in question. Be-sides, in our study, only the shear stress was studied so thequestion of multi-axial loading did not arise. The adhesive filletwas not modelled here, in spite of its importance. As double lapjoints were tested, only half the experimental specimen was mod-elled due to symmetry considerations (Fig. 6).

In order to compare the finite element results dEF with the mea-sured slip data dexp, it is necessary to average the strain fields in theadhesive thickness with the classical finite element approach. Itwas observed that the 3D shear strains vary within the thicknessof the adhesive. To illustrate this, in Fig. 7 we have plotted four dif-ferent shear strain profiles obtained using the finite element mod-elling. The first three show the results along the lap length at threedifferent depths in the adhesive thickness. The fourth curve, whichis bolder, shows the average shear strain cEF as defined in theexperimental procedure: the total slip between the two interfacesdivided by the thickness of the adhesive layer which is then com-pared with the experimentally measured slip data cexp. The firstthree profiles highlight the differences that exist within the thick-ness of the adhesive and the difficulties due to the relationshipwhich exists between the mesh and the quantitative results inclassical finite element methods. Notice that cEF is close to the

average of the 3D shear strains, although greater mesh refinementwould be necessary to explore this assertion more fully. Althoughthe shear strain changes throughout the adhesive thickness, forshear stress it is important to note that there was little differencebetween the three positions chosen in the adhesive thickness forthe studied case. Indeed, when we conducted a similar study ofshear stress profiles for the three vertical positions using a differ-ent mesh refinement, no difference was found between the paths.Consequently, stress data was taken arbitrarily in the middle of theadhesive layer.

As already mentioned above, it can be seen in Fig. 7 that strain isnot uniform throughout the thickness of the adhesive. The plastic-

Fig. 8. Maximum plastic strain concentration in the adhesive: maximum values (over 0.2%) are shown in black.

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ity path can be followed using the first three strain profiles. Plastic-ity is first concentrated near the exterior adherend in the left cor-ner. It then moves nearer to the interior adherend. Averaging theplastic strain throughout the adhesive thickness conceals thisthrough the thickness localisation (Fig. 8). It is interesting to notethat experimentally it was also observed that in some cases theshear strain was effectively nonuniform throughout the thicknessof the adhesive (Fig. 9). Averaging strain (c) provides quantitativeand representative results that are independent of mesh refine-ment while reducing strain concentrations along the lap length.As the shear stress s remains constant throughout the thicknessof the adhesive, and as the measured slip is considered to be theaverage of the 3D strains in the adhesive thickness, the shear en-ergy, expressed as

Radhesivethickness s�dz can be simplified as sd. This

energetic consideration may thus justify a failure criterion basedon a critical slip value dc which thus appears to be a potentially rel-evant design parameter.

The following section, which consists of a comparison betweenmodelling results and experimental measurements will only focuson slip data, which removes the effects of plastic strain concentra-tion in the thickness of the adhesive.

4. Comparison between the experimental and finite elementstudies of slip along the adhesive layer during load/unloadcycles

This section compares the experimental slip values dexp given bythe theoretical predictions dEF using the measured geometry of the

Fig. 9. Photo of the measured defo

Fig. 10. Plot of load application on the left. On the right, the difference bet

tested specimen, and the mechanical properties obtained from ten-sile tests. Each material was considered to be isotropic, althoughthis is certainly not the case for the adherends and the adhesive.However, their anisotropic behaviour has little influence on theway stress is transferred in comparison with plasticity. It is impor-tant to note that these experimental investigations required highprecision and great care and that consequently only a few speci-mens could be tested for this study.

4.1. Horizontal slip profiles

Our experimental horizontal slip measurements were com-pared with the numerical expectations obtained from the finiteelement modelling. The different abscissas where slip was mea-sured were precisely determined and measurements were madeat intervals of approximately 5 mm. The total lap length was62 mm and the width 16.2 mm. It is important to note that per-fectly elasto-plastic behaviour was taken into account. Conse-quently, the plastic strain tended to be over-estimated forintermediate values. The simulation of an experiment includingone load/unload path to 26 kN was conducted (Fig. 10). The com-parison was made for six particular points: A, B, C, D, E and F.

From Fig. 11, it is clear that, although the numerical expecta-tions in terms of slip may be higher than the measured data forintermediate loading, correlation was good. The theoretical andexperimental results matched for low load levels (A), and high loadlevels (E and F). For intermediate levels (B, C and D), the yieldingprocess needs to be modelled more precisely if more accurate re-

rmation in the adhesive layer.

ween perfect elasto-plasticity and the real behaviour of the adhesive.

Fig. 11. Comparison between theoretical and experimental horizontal slip along the lap length (— : results from Abaqus; � : experimental measurements).

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sults are required (Fig. 10). In particular, a multilinear characteris-tic curve should be considered for the behaviour of the adhesive.Even if the chosen yield criteria and associated flow rule may notbe appropriate and further work should be carried out which con-siders other plasticity modelling theories, the slip can be proposedas a good candidate for describing the state of the joint, especiallyits ultimate capacity.

4.2. Other experimental investigations

Two additional tests were conducted using different loadingschemes to check the previous observations and the accuracy ofmodelling. For each test, the result of finite element modellingwas compared to the experimental results. The same materialsand specimen geometry were used. The first specimen was notsubjected to unloading. The second specimen was subjected totwo load/unload paths. For each, horizontal slips were measuredin different zones (different abscissas along the lap length:1.2 mm from the loaded end for the first and the second zones;

1.8 mm from the loaded end for the third zone). The results are gi-ven in Figs. 12 and 13. The numerical slip results are shown by thesolid line, and the experimental results by the dots (the verticallines represent an assessment of the measurement error range).For the sake of clarity, the experimental dots are joined by solidlines to show the path followed during the test.

On the basis of these two cases, we can state that the modelaccurately matches the experimental data for simple loading.Fig. 12 shows three graphs which plot the measured slip data ver-sus the applied force in the case of the same bonded joint for threedifferent zones and monotonic loading. It is important to note thatas the tests were conducted on double lap bonded joints, two jointswere tested at the same time. The first zone corresponded to1.2 mm on the abscissa for the first bonded joint, the second zonecorresponds to 1.2 mm on the abscissa for the second bonded joint,and the third zone corresponds to 1.8 mm on the abscissa for thefirst bonded joint. It is apparent that although the measurementmethod needs to be improved (using automatic procedures or im-age analysis techniques), the correlation was very impressive and

Fig. 12. Comparison between theoretical and experimental horizontal slip for different zones on specimen one (— : results from Abaqus results; � : experimentalmeasurements).

Fig. 13. Comparison between theoretical and experimental horizontal slip for different zones on specimen two (— : results from Abaqus results; � : experimentalmeasurements).

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no artificial parameter is required to make a comparison. For thedesigner, the link between the applied force and the slip seemsto be reliable and to offer interesting potential.

In the case of unloading, the plastic strain seems to be slightlyover-estimated as we have seen above. In Fig. 13, two graphs are

given that plot the applied force against the measured slip fortwo identical bonded joints. The measurement was made for twopoints on the abscissa for the two bonded joints (the first zoneswere located 1.3 mm from the loaded end and the second zones2.1 mm from the loaded end), and two load/unload cycles were

440 S. Chataigner et al. / Construction and Building Materials 25 (2011) 432–441

conducted before the joint was loaded to failure. Even though theperfect elasto-plasticity hypothesis leads to some differences be-tween experimental and numerical results, the correlation wasagain good. The initial slope values clearly showed the effect ofshear concentration on the edges which gives rise to differencesbetween the two zones. In addition, this slope changed little duringthe unloading cycles, indicating that any damage was probablyminor. It is important to note that the finite element model didnot take account of damage or viscosity, but even so hysteresis oc-curred. This was linked to plasticity within the bonded joint anddue to structural factors rather than damage or the effect of viscos-ity, which, although it may exist, would be much smaller in ourcase than the plastic case.

5. Conclusion

Using two different adhesives, we have shown experimentallythat in the studied case the plastic behaviour of the adhesive in-creased the ultimate capacity of the bonded joints by a factor offour while maintaining the same failure mode (within the adher-ends). When the influence of the bonded length was observedmore closely, Volkersen’s theory based on elastic hypotheseswas, of course, shown to be inadequate in the case of an elasto-plastic bond material. However, in the elastic case, even if it iswell-known that a singularity, and hence infinite shear stress, ex-ists at the edge, this theory allows us to express a critical finiteshear stress value which seems to be a valid design criterion. Ourexperimental investigations using microscopic observation of thejoint also showed the presence of plasticity in the shear behaviourof the adhesive layer.

In the case of the highly ductile adhesive, these observations en-abled us to determine the slip along the lap length which is actu-ally linked to the shear strain. Although, as has been shown inthe section that deals with the numerical model, this parameterdoes reveal plastic strain concentration effects in the adhesivethickness, it provides an average strain value which does not de-pend on mesh density and which is closely linked to the shear en-ergy in the joint. This important point has been raised by manyauthors working on multilayer modelling approaches [28–30] orasymptotic analyses [31]. In some of these approaches, the shearstress at the interface is associated through interface behaviourwith a generalized strain that is equal, in the case of pure shearloading, to the slip. Two important studies, based on multilayermodelling, have recently been conducted to introduce nonlinearinterface behaviour. The first was based on an analytical develop-ment and is described in [32]. The second was based on a finite ele-ment development and is presented in [33]. A comparison with aclassical 3D finite element model was conducted and has shownsimilar results concerning the direct link between slip and shearenergy. An approach of this type reduces computing time thusfacilitating the design process. Based on the same modelling char-acteristics, a limit analysis has been conducted as described in [34]which could certainly offer good potential for the use of critical slipin order to study bonded joints.

Finally, the experimental measurements and theoretical resultswere compared in the case of several specimens. This revealed thatthere was a good match between them even for load/unload cyclesand even if only a few specimens were tested. Plastic strain wasslightly over-estimated for intermediate levels due to the choiceof a perfect elasto-plastic numerical model. Multilinear modellingof the adhesive would certainly be closer to the experimental re-sults and reduce the calculated plastic strains. However, for highstrains, the critical slip value seems to be a good candidate as a de-sign parameter. It is important to note that isotropic behaviour waschosen for the adhesive based on tensile testing results. In this

analysis, the tensile properties tend to reflect shear properties well,and the isotropic hypothesis seems to be valid. However, a consid-erable amount of work remains to be done with regard to the cho-sen yield criterion and flow rule in order to obtain results thatbetter match reality.

This analysis has shown that a simple characterization methodfor the adhesive and a simple design parameter (slip) can be a goodindicator of the load state within the joint. This has been validatedby a classic finite element approach and has been recently pro-posed using an analytical development based on a multilayer ap-proach [32]. The experimental and numerical residual strainvalues after one load/unload cycle demonstrate the presence ofplasticity and consequently the prestressing of the joint even whenit is unloaded. This last remark is extremely important if adhesivesof this type are to be used for structural applications as explainedin [35], and it must consequently be taken into account in the de-sign process.

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