Modelling adhesive joints with cohesive zone models: effect of the cohesive law shape of the adhesive layer

Modelling adhesive joints with ohesive zone models: effet of the ohesive law shape of the adhesive layer R.D.S.G. Campilho, M.D. Banea, J.A.B.P. Neto, L.F.M. da Silva ABSTRACT Adhesively-bonded joints are extensively used in several fields of engineering. Cohesive Zone Models (CZM) have been used for the strength predition of adhesive joints, as an add-in to Finite Element (FE) analyses that allows simulation of damage growth, by onsideration of energeti priniples. A useful feature of CZM is that different shapes an be developed for the ohesive laws, depending on the nature of the material or interfae to be simulated, allowing an aurate strength predition. This work studies the influene of the CZM shape (triangular, exponential or trapezoidal) used to model a thin adhesive layer in single-lap adhesive joints, for an estimation of its influene on the strength predition under different material onditions. By performing this study, guidelines are provided on the possibility to use a CZM shape that may not be the most suited for a partiular adhesive, but that may be more straightforward to use/implement and have less onvergene problems (e.g. triangular shaped CZM), thus attaining the solution faster. The overall results showed that joints bonded with dutile adhesives are highly influened by the CZM shape, and that the trapezoidal shape fits best the experimental data. Moreover, the smaller is the overlap length (LO), the greater is the influene of the CZM shape. On the other hand, the influene of the CZM shape an be negleted when using brittle adhesives, without ompromising too muh the auray of the strength preditions. Keywords: Epoxy/epoxides, Composites, Finite element stress analysis, Frature 1. Introdution Adhesively-bonded joints are extensively used in several fields of engineering, suh as automotive, aeronautial and spae strutures, as an easy method to join omponents, assuring at the same time the design requirements for the struture [1]. General apabilities of this joining method involve more uniform stress fields than fastening or riveting, fluid sealing, high fatigue resistane, and possibility to join different materials on aount of orrosion prevention and aommodation of different thermal expansion of the adherends [2,3]. The tehniques for strength predition of bonded joints also improved. Initially, theoretial methods (mainly losed-form) were proposed for stress distributions in the adhesive for simple geometries suh as the single or double-lap joint, and failure estimation was arried out by omparison of the maximum stresses with the material strengths [4]. Some deades later, the FE initiated its inursion in the analysis of adhesively-bonded joints (e.g. the work of Wooley and Carver [5]), by onsideration of stress/strain or frature mehanis riteria for failure predition [6]. Even though these analyses were promising, they had few limitations: stress/ strain preditions depend on the mesh size at the ritial regions, while frature riteria suh as the Virtual Crak Closure Tehnique (VCCT) are restrited to Linear Elasti Frature Mehanis (LEFM) and need an initial rak. CZM have been used in the last deade for the strength predition of adhesive joints, as an add-in to FE analyses that allows simulation of damage growth within bulk regions of ontinuous materials or interfaes between different materials [7,8]. Compared to onventional FE, a muh more aurate predition is ahieved, sine different shapes an be developed for the ohesive laws, depending on the nature of the material or interfae to be simulated. The triangular and trapezoidal CZM shapes are most ommonly used for strength predition of typial strutural materials. For the appliation of this tehnique, tration-separation laws with a pre-defined shape are established at the failure paths, and the values of energy release rate in tension and shear (G n and G s, respetively) along the frature paths and respetive ritial values or toughness (G and n

G s) are required. The ohesive strengths in tension and shear repairs strength of nearly 2%, on aount of exessive plasti (t n and t s, respetively) are equally needed and they relate to degradation at the bond edges that was not observed in the real damage initiation, i.e. end of the elasti behaviour and beginning joints. Regarding the appliation of CZM for strength predition of of damage. Different tehniques are nowadays available for the adhesive bonds, trapezoidal laws are reommended for dutile definition of the ohesive parameters (G n, G s, t n and t s ), suh as adhesives [8,25], and this is partiularly ritial when onsidering the property identifiation tehnique, the diret method and the inverse method. These methods usually rely on the Double- Cantilever Beam (DCB), End-Nothed Flexure (ENF) or single-lap speimens, generally with good results [9 13]. The property identifiation tehnique onsists on the separated alulation of eah one of the ohesive law parameters by suitable tests, while in the inverse method the CZM parameters are estimated by stiff adherends, due to the pratially absene of differential deformation effets in these omponents along the overlap [23,26]. In ontrast, triangular CZM are effiient for brittle materials that do not plastiize by a signifiant amount after yielding [27], and also for the intralaminar frature of omposite adherends in bonded strutures, due to their intrinsi brittleness [28]. For adhesives that exhibit a relatively brittle behaviour in iterative fitting the FE predition with experimentally measured tension while showing large plasti flow in shear, the data (typially the load displaement, P d, urve) up to an aurate representation. Both of these approahes begin with the assumption of a CZM shape to simulate a speifi material, whih approximately repliates it in terms of post-elasti behaviour [14]. On the other hand, the diret method gives the preise shape of the CZM laws of a speifi material or interfae, sine these are estimated from the experimental data of frature tests suh as the DCB or ENF [15]. This is done by differentiation of G n (tension) or G s (shear) with respet to the relative opening of the rak (d n for tension or d s for shear). Nonetheless, it is usual to onvert the obtained shape in an approximated parameterized shape for introdution in the FE software. Carlberger and Stigh proper seletion of the CZM parameters and also the minimisation of the onstant stress (plasti flow) region in the tensile law result on a good representation of the adhesive behaviour. The material/interfaial behaviour that the CZM law is simulating should always be the leading deision fator to selet the most appropriate shape. Despite this fat, other issues should be taken into aount [7]. In fat, the CZM law shape also influenes the iterative solving proedure and the time required to attain the solution of a given engineering problem: larger onvergene diffiulties in the iterative solving proedure usually take plae for trapezoidal rather than triangular CZM laws, due to the more abrupt hange of stiffness in the ohesive elements during stress [16] estimated the CZM law shapes of a thin adhesive layer in softening. Atually, for a fixed value of the material properties G n tension and shear with DCB and ENF tests, respetively, for an and Gs, the larger the onstant stress length of the trapezoidal adhesive thikness, t A, between.1 and 1.6 mm. The ohesive law, the bigger is the desending slope. Additionally, exponential laws were found by a diret method based on the differentiation and trapezoidal CZM are more diffiult to formulate and impleof the G n/g s vs. d n/d s data. The CZM shapes and respetive ment in FE software. parameters signifiantly varied with t A, ranging from a rough triangular shape for the smaller values of t A to a trapezoidal shape for bigger values of t A. It is thus reommend adjusting the shape of the CZM laws to This work studies the influene of the CZM shape (triangular, exponential or trapezoidal) used to model a thin adhesive layer in single-lap adhesive joints, for an estimation of its influene on the strength predition under different material onditions. The FE onform to the behaviour of the thin material strip or interfae software Abaqus s (Providene, RI, USA) and its embedded CZM they are simulating. Developed CZM inlude triangular [17], formulation were used in this work. As a result of this study, some linear-paraboli [18], polynomial [19], exponential [2] and onlusions were established to assess the importane of using a trapezoidal laws [21]. Thus, CZM an also be adapted to simulate CZM shape for a given adhesive that aurately predits the joint dutile adhesive layers, whose behaviour an be approximated strength, under different material onditions (i.e., onsidering a with trapezoidal laws [14]. Although it is always advised the use brittle and a dutile adhesive). of the most suitable CZM shape and to perform aurate parameter estimations, few works showed aeptable preditions for small variations to the optimal CZM parameters and shapes [11,22]. On the other hand, the effet of the CZM law shape on the strength preditions signifiantly varies depending on the struture geometry and post-elasti behaviour of the materials. 2. Experimental work 2.1. Materials haraterisation These issues beame evident in the experimental and FE study of Pinto et al. [23], whose objetive was the strength omparison of Unidiretional arbon epoxy pre-preg (SEAL s Texipreg HS single-lap joints with similar and dissimilar adherends and values 16 RM; Legnano, Italy) with.15 mm thikness was onsidered of adherend thikness, t P, bonded with the adhesive 3M DPfor the omposite adherends of the single-lap joints, with the 85 s. The aurate shape of the CZM law was onsidered [] 16 lay-up. Table 1 presents the elasti properties of a unidirefundamental for the strength predition and P d response of the tional lamina, modelled as elasti orthotropi in the FE analysis struture when using stiff adherends. Under these onditions, [29]. Two epoxy adhesives were onsidered. The adhesive peel stresses are minimal and, due to the large longitudinal Araldite s AV138 is a two-part (resin þ hardener) brittle and stiffness, shear stresses distribute more evenly along the bond high strength adhesive suited to bond a large variety of materials length. Thus, the P d urve is very similar in shape to the hosen suh as metals or polymers/polymer omposites. The adhesive shear CZM law. On the other hand, ompliant adherends led to Araldite s 215 is equally a two-part strutural adhesive, showing large shear and peel stress gradients. Sine this implies different damage states along the adhesive layer, using an inaurate CZM law gives adhesive stresses that are over predited at some elements and under predited at others. Thus, by using ompliant adherends the overall behaviour gave smaller errors. Ridha et al. [24] onsidered sarf repairs on omposite panels bonded with the high elongation epoxy adhesive FM s 3M (Cyte). CZM laws with linear, exponential and trapezoidal softening were ompared, and linear degradation resulted in under preditions of the Table 1 Elasti orthotropi properties of a unidiretional arbon epoxy ply aligned in the fibres diretion (x-diretion; y and z are the transverse and through-thikness diretions, respetively) [29]. Ex ¼ 1.9E þ 5 MPa nxy ¼.342 Gxy ¼ 4315 MPa Ey ¼ 8819 MPa nxz ¼.342 Gxz ¼ 4315 MPa Ez ¼ 8819 MPa nyz ¼.38 Gyz ¼ 32 MPa

a smaller ultimate strength than the previous, but allowing large plasti flow prior to failure. This is an important feature for bonded joints as it allows redistribution of stresses at stress onentration regions, whih usually takes plae beause of the sharp edges at the overlap ends and also joint asymmetry/distint deformation of the adherends along the overlap. The adhesives were previously haraterised regarding the elasti moduli in tension and shear (E and G, respetively), the failure strengths in Fig. 1. Geometry and harateristi dimensions of the single-lap joints. tension and shear (orresponding to t n and t s ) and the values of G n and G. For the adhesive haraterisation, bulk tests were s performed to haraterise the adhesives in tension and Thik Adherend Shear Tests (TAST) were hosen for shear haraterisation. It should be pointed out that the ohesive strengths of thin adhesive layers and the bulk strengths of adhesives are different quantities [3]. This is beause bulk adhesives are homogeneous materials raking perpendiularly to the maximum prinipal stress diretion, while adhesives as thin layers are highly onstrained between stiff adherends and damage growth under these onditions ours under mixed-mode (tension plus shear) and along the predefined path of the bonding diretion. In this work, the ohesive strengths of the adhesives were assumed as equal to their bulk quantities as an approximation. The good orrespondene that was observed by the omparisons to the experimental data allowed to assume that a fair approximation was attained and to orroborate the use of these properties. The bulk speimens were manufatured following the NF T 76-142 Frenh standard, to prevent the reation of voids. Thus, 2 mm thik plates were fabriated in a sealed mould, followed by preision mahining to produe the dogbone shape desribed in the standard. The TAST haraterisation of the adhesive was arried out aording to the 113-2:1999 ISO standard, onsidering DIN Ck 45 steel adherends. More details about the fabriation and testing proedures an be found in referene [31]. Charaterisation of the adhesives regarding the elasti onstants, strengths and strains in tension and shear, was previously onduted in the work of da Silva et al. [32] (Araldite s AV138) and by the authors in a previous work [33] (Araldite s 215). The values of G n and G s for the AV138 were determined by the authors in [27] by numerial fitting proedures. The authors also esti 2.2. Joint fabriation and testing mated in a previous work the values of G n and G s for the 215 3. Numerial study [33], by DCB (G n) and ENF tests (G s) using different Frature Mehanis data redution methods. The relevant mehanial properties of these adhesives, whih were used to onstrut the ohesive laws, are summarised in Table 2 (the initial yield stress was alulated for a plasti strain of.2%). The large differene The single-lap joint geometry and harateristi dimensions are represented in Fig. 1. The following dimensions were onsidered (mm): L O ¼ 1 8, width b ¼ 15, total length between gripping points L T ¼ 24, t P ¼ 2.4 and t A ¼.2. Eight different values of L O were evaluated (1, 2, 3, 4, 5, 6, 7 and 8 mm). The joints were fabriated by the following steps: (1) the surfaes to be bonded were roughened by manual abrasion with 22 grit sandpaper and leaned with aetone, (2) the joints were bonded in an apparatus for the orret alignment, and the desired value of t A was ahieved during assembly with a dummy adherend and a.2 mm alibrated spaer under the upper adherend, jointly with the appliation of pressure with grips and (3) tabs were glued at the speimen edges for a orret alignment in the testing mahine. The reported method for the joints assembly assured the preision of the obtained t A values, to redue test data satter to a minimum. The joints were left to ure at room temperature for 1 week to assure omplete uring, and the exess adhesive at the bonding region was then removed by preision milling to provide square-edges at the overlap edges. Tensile testing to the joints was arried out in an Instron s 428 (Norwood, MA, USA) eletro-mehanial testing mahine with a 1 kn load ell, at room temperature and under displaement ontrol (.5 mm/min). The testing mahine grips displaement was onsidered to build the P d urves. For eah value of L O, six speimens were tested, with at least four valid results. 3.1. FE simulation The FE software Abaqus s was onsidered for this study, to evaluate the modelling auray of its CZM embedded formula between G n and G s observed in Table 2 is typial of dutile tion when stipulating different CZM shapes to model the adhesive strutural adhesives, whih show a signifiantly larger plasti layer in single-lap joints. A geometrially non-linear stati analyflow in shear than in tension [34]. sis was performed [14,35], modelling the adherends with the elasti orthotropi properties of Table 1. Fig. 2 depits an example of FE mesh for the L O ¼ 1 mm joint. The meshes for all FE models Table 2 were automatially reated by the software onsidering bias Properties of the adhesives Araldite s AV138 and 215 [27,32,33]. effets, with smaller sized elements near the overlap edges and in the thikness diretion near the adhesive. Atually, it is known Property AV138 215 that the overlap edges are theoretially singularity spots with large stress variations [36]. To provide idential modelling on- Young s modulus, E [GPa] 4.897.81 1.857.21 Poisson s ratio, n b.35.33 ditions, the FE elements size in all models was made equal at the Tensile yield strength, sy [MPa] 36.49 72.47 12.637.61 overlap edges (approximately.2 x.2 mm 2 elements), thus Tensile failure strength, sf [MPa] 39.45 73.18 21.6371.61 allowing to aurately apture stress variations [29]. The joints Tensile failure strain, ef [%] 1.217.1 4.777.15 Shear modulus, G [GPa] 1.567.1.567.21 were simulated with two-dimensional FE models, using 4-node Shear yield strength, ty [MPa] 25.17.33 14.67 1.3 plane-strain elements (CPE4 from Abaqus s ) and COH2D4 4-node Shear failure strength, tf [MPa] 3.27.4 17.971.8 ohesive elements, ompatible with the CPE4 elements [33]. Shear failure strain, gf [%] 7.87.7 43.973.4 Boundary onditions inluded lamping the joints at one of the Toughness in tension, G n [N/mm].2 a.437.2 edges, to reprodue the testing mahine gripping, while the a Toughness in shear, Gs [N/mm].38 4.77.34 a Estimated in referene [27]. b Manufaturer s data. opposite edge was pulled in tension together with lateral restraining (Fig. 1). The adhesive layer was modelled with a single row of ohesive elements [27] and a damage model

between eah set of paired nodes with varying CZM shape, as hi are the Maaulay brakets, emphasising that a purely defined in Setion 3.2. This tehnique is implemented in Abaqus s ompressive stress state does not initiate damage. Thus, initiation CAE and will be briefly desribed for the different types of of damage is oupled between tension and shear [41]. After the ohesive laws evaluated. riterion of Eq. (2) is met, the material stiffness initiates a degradation proess. However, from this point on, an unoupled tensile/shear behaviour was used, in whih the tensile and shear 3.2. CZM implementation in the FE analysis behaviours of the CZM elements are independent up to failure. This hoie was made beause of the Abaqus s unavail- CZM reprodue the elasti loading up to a peak load, damage ability of mixed-mode oupling riteria for the trapezoidal CZM onset and rak growth due to loal failure. CZM are typially formulation. founded on a relationship between stresses/ohesive trations The softening regions of the CZM laws are defined in Abaqus s and relative displaements (in tension or shear) that onnet homologous nodes of the ohesive elements, to simulate the by speifiation of the damage variable (d n for tension or d s for shear), as a funtion of d n d n d s d s (tension) or (shear), i.e., as a elasti behaviour up to t n (tension) or t s (shear) and subsequent funtion of the effetive displaement beyond damage initiation stiffness redution, related to the progressive material degrada- (d n and d s represent the damage onset relative displaements in tion up to final failure [37,38]. In this work the triangular, linear- tension and shear, respetively). This is desribed by the following exponential and trapezoidal shapes were evaluated (Fig. 3 she- formulae. Fig. 4 pitures the definition of d n for the triangular law, matially represents these three CZM shapes with the assoiated although it an be extrapolated to d s [39] nomenlature). As shown in Fig. 3, the linear-exponential law is und linear up to t n or t s, and afterwards undergoes an exponential softening up to failure. This shape is an approximation of the fullexponential law [2], providing in this ase a more abrupt stress s drop than the triangular law, after the peak loads are ahieved. G n and G s are the areas under the CZM laws in tension or shear, where t und n and t und s are the urrent ohesive trations in tension respetively. The definition of the normal or shear maximum and shear, respetively, without stiffness degradation. In this f f expression, d n,s ¼ for an undamaged material (in the elasti relative displaements (d n and d s, respetively) is arried out by region) and d n,s ¼ 1 for a fully damaged material. By this priniple, making G n ¼ G n for tension or G s ¼ G s for shear. The initial linear the generi expression (in tension or shear) of d n,s for the elasti behaviour in the CZM laws (notwithstanding their shape) triangular law takes the form [39] is defined by an elasti onstitutive matrix relating the urrent stresses and strains in tension and shear aross the interfae (subsripts n and s, respetively) [39] For the exponential law, the expression of d n,s gives [39] t n and t s represent the ohesive trations in tension and shear, respetively, whilst e n and e s are the tensile and shear strain, in the same order. The stiffness matrix, K, ontains the adhesive stiffness parameters. A suitable approximation for thin adhesive layers is provided with K nn ¼ E, K ss ¼ G, K ns ¼ [21,4]. For all of the three CZM shapes, initiation of damage was evaluated by the following quadrati nominal stress riterion, previously tested for auray [14], and expressed as [39] where a is a non-dimensional parameter, related to a speifi material, that establishes the rate of damage evolution with d n,s (for a¼ a triangular law is attained). In this work, a ¼ 7 was hosen to provide a signifiant differene to the triangular shape, by a signifiantly faster degradation after t n,s is reahed. For the trapezoidal law, the stress softening displaements in tension and s s shear, d n and d s, respetively, are introdued. The value of d n,s is s divided into the onstant stress region (d n,s odrd n,s; Fig. 3) and s f softening region (d n,s odrd n,s; Fig. 3) as follows [39]: : Fig. 2. Detail of the mesh for the LO ¼ 1 mm model. The values of m and b relate to the straight line equation of the deaying portion of the CZM law with respet to the t d plot Fig. 3. CZM laws with triangular, exponential and trapezoidal shapes available in Abaqus s.

deaying slope up to dn,s origin, given by [39] f (e.g. idential slope between the tensile t and shear CZM laws, if only tensile data is available) [43]. In this s f - n,s work, the first approah was adopted, onsidering d n,s/d n,s ¼.8. f s d Table 3 shows the onsidered values for the adhesive layer CZM The values of d f n,s are found by onsideration of the area under the t d plot to be equal to G. n,son the other hand, several tehniques are available for the definition of d s n,s (trapezoidal s f law), suh as pre-established ratios between d n,s and d n,s [42], use of experimental failure strain data [21], or pre-established laws, estimated from the experimental data of Table 2 and onsidering the average values of the experiments. Fig. 5 details the CZM laws with different shapes for the adhesives 215 and AV138 in tension (a) and shear (b). 3.3. Feasibility of unoupling the damage evolution As it was mentioned in the previous setion, the CZM formulation adopted in this work is unoupled in tension and shear after damage initiation. To assess the influene of this simplifying assumption and to guarantee the validity of the FE preditions throughout this work, a numerial analysis was performed by omparing, for the Araldite s 215 and the entire range of L O values onsidered in the analysis, the predited values of maximum load (P m) between the unoupled and full mixedmode formulations. This omparison is performed onsidering the triangular law as an example, sine it is readily available with a mixed-mode formulation. Moreover, only the Araldite s 215 2 12 16 8 Fig. 4. Definition of the damage variable in tension, dn, in Abaqus s (extrapolation is possible for ds). Table 3 Cohesive parameters of the adhesives Araldite s AV138 and 215 for CZM modelling. Property AV138 215 E [GPa] 4.89 1.85 G [GPa] 1.56.56 t [MPa] 39.45 21.63 n ts [MPa] 3.2 17.9 G n [N/mm].2.43 Gs [N/mm].38 4.7 P m [kn] 12 4 8 4-4 -8 1 2 3 4 5 6 7 8 L O [mm] Unoupled CZM Mixed-mode CZM Deviation [%] Fig. 6. Numerial omparison between the unoupled and full mixed-mode formulations for the Araldite s 215 and respetive deviation. Deviation [%] t n [MPa] t n [MPa] 5 4 AV138 3 2 1.1.2.3.4.5 n [mm] 5 4 215 3 2 1.1.2.3.4.5 n [mm] t s [MPa] t s [MPa] 5 4 3 2 1 5 4 3 2 1 AV138.5.1.15.2 s [mm] 215.2.4.6.8 s [mm] Fig. 5. CZM laws in tension (a) and shear (b) for both the adhesives tested.

P m [kn] 2 15 1 5 1 2 3 4 5 6 7 8 L O [mm] Fig. 7. Experimental plot of the Pm LO values for the adhesives AV138 and 215. results are presented here sine, owing to its dutility, it yields bigger differenes than the Araldite s AV138, whose variation between the two approahes was negligible (fewer than.5%). For the mixed-mode formulation, failure was predited by a linear power law form of the required energies for failure in the pure modes [39] Fig. 6 shows the numerial omparison of P m values between the unoupled and full mixed-mode formulations for the Araldite s 215 and respetive deviation. The deviation plot shows that for small values of L O the unoupled formulation tends to slightly underestimate P m (up to 2.76%, for L O ¼ 2 mm). On the other, for bigger values of L O, the unoupled response gradually overshoots by an inreasing amount the P m predition, up to 3.1% for L O ¼ 8 mm. On aount of these results, the authors thus onlude that, for the range of L O values addressed in this work, the unoupled formulation is a suitable approximation. 4. Results and disussion 4.1. Joint strength 215 AV138 All the joints experiened a ohesive failure of the adhesive layer. Previously to the analysis, a mesh dependeny study was arried out to asertain if the seleted mesh refinement is enough to ensure onvergene to the right solution. This analysis onsidered the joints with both adhesives and with L O ¼ 1 and 8 mm, whih give the most signifiant differene in the adhesive layer stress state. Inreasing refinements were onsidered, with element lengths at the overlap edges of.5,.1,.2 and.4 mm. Maximum deviations of.3% were found relatively to the average value of P m between the four mesh sizes (independent analysis for eah adhesive). This behaviour was expeted, sine in CZM modelling an energeti riterion is used for damage propagation, based on the input values of G n and G s. Sine the energy required for rak growth is averaged over the damaged area, results are mesh independent provided that a minimum refinement is used, more speifially if a minimum of three to four elements are undergoing the softening proess at the damage front [21,44]. Fig. 7 reports P m as a funtion of L O for both adhesives tested, showing a nearly linear inrease of P m with L O. The non-existene of a limiting P m value in the P m L O urves is justified by the high strength of the CFRP (i.e., the tensile strength of the laminates was not attained for the tested L O values up to failure in the adhesive layer). Atually, the maximum value of longitudinal axial stresses (s x) was found for the joints with L O ¼ 8 mm (61.7 MPa for the joints with Araldite s 215 and 393.9 MPa for the joints with Araldite s AV138), being muh smaller in magnitude to the tensile strength of the employed omposite, of E 2 MPa [14]. Adding to this, a detailed stress analysis to the joints for the range of L O values used in this work was undertaken. It was possible to onlude that, owing to the adherends stiffness and respetive lak of plasti flow, the average shear stress along the bondline for both adhesives tends to diminish at the time of joint failure, but this redution is exponential, with smaller redutions for the bigger L O values (e.g. for the AV 138 between L O ¼ 7 and 8 mm, the redution of the average shear stress was only 6.9%). By the ombined effet of this redution and of the inrease of L O, an improvement of P m was found for the entire range of L O values tested. However, for different overall onditions, e.g. adherends with a smaller value of E, a onstant value of P m ould be attained in the P m L O plots at a relatively small value of L O on aount of a steeper redution of the average shear stress along the bondline at failure with the inrease of L O [45]. Analysing Fig. 7 in more detail, for L O ¼ 1 mm the AV138 shows a larger P m value than the 215, whih is aredited to the bigger adhesive strength (Table 3), and to the fat that shear stresses, whih rule the failure proess, are nearly onstant over the overlap for very short overlaps [3]. As a result, failure depends almost exlusively on the adhesive strengths, whilst the frature toughness (muh bigger for the 215) beomes irrelevant. For inreasing values of L O, the 215 shows a steeper inrease of P m than the AV138, beause it is extremely dutile (Table 2) and the joints fail with a signifiant degree of plasti flow in the adhesive layer [45]. In fat, adhesive plastiisation takes plae at the overlap edges where stresses peak, together with redistribution of stresses in the adhesive layer towards the inner overlap regions [46,47]. Beause of this issue, a nearly proportional relationship exists between P m and L O in Fig. 7. To further orroborate this fat, frature was always abrupt in the test speimens, only with a negligible rak growth before P m for the bigger values of L O. This shows that the adhesive plastiity always held up rak initiation at the overlap edges up to P m, keeping these regions at the peak strength, while stresses inreased at the inner regions [48]. On the other hand, the AV138 shows a steady but signifiantly smaller improvement of P m with L O, due to its brittleness, testified by the orresponding values of Gn,s (Table 2). Adding to this, s ypeak stresses at the overlap ends progressively inrease in magnitude with L O [3], as they gradually onentrate in a smaller region beause of more loalised bending of the adherends at the overlap edges. This, added to the redued allowane of adhesive plasti flow (small value of Gn ), gives a lesser advantage in the single-lap joints strength with the inrease of L O. 4.2. Cohesive law shape effets The CZM law shape influene on the strength preditions was arried out onsidering triangular, exponential and trapezoidal CZM, for a pereption of the influene of this hoie on the auray of the FE simulations under different material/geometrial onditions. Initial emphasis is given to the size of the predited length of the proess zone at P m (immediately before frature), for a better understanding of the failure proesses and differenes between the FE models with distint CZM shapes for the adhesive layer. To this end, the shear proess zone was onsidered, as it is the most signifiant to the failure proess, and it was measured for the joints with L O ¼ 1 and 8 mm, whih represent the limiting senarios of L O. Only the results for the Araldite s 215 are presented, beause any differene between models is more easily deteted on aount of the adhesive dutility, although the onlusions are idential to the Araldite s

[%] 3 2 1-1 -2 1 2 3 4 5 6 7 8 L O [mm] implementation, time of alulation, CZM parameter definition and availability in ommerial FE odes. The use of more ompliant adherends would redue this deviation even further beause of bigger stress gradients along the overlap [23]. On the other hand, adherends suh as steel would inrease this deviation. The exponential CZM gave opposite results for the range of L O values evaluated. For small values of L O, P m was numerially overestimated (maximum D of 27.9% for L O ¼ 1 mm). The D values onsistently redued and approahed the experimental results for L O ¼ 5 mm. From this point, under preditions of P m were obtained with exponential softening (reahing D¼- 6.8% for L O ¼ 8 mm). Analysis of the FE results showed that the over estimation of P m for the smaller L O values is due to the following motives: Fig. 8. Perentile deviation between the experimental and FE Pm values for the adhesive 215. AV138. For L O ¼ 1 mm, the proess zone extents at P m were as follows (averaged over L O): trapezoidal law 79.1%, triangular law 88.8% and exponential law 1%. For L O ¼ 8 mm, the following data was obtained: trapezoidal law 9.2%, triangular law 94.5% and exponential law 1%. These results are in agreement with the plotted laws used to model the adhesive (Fig. 5), showing that the extent of damage is largest for the exponential law, followed by the triangular and trapezoidal laws, by the respetive order. However, as it will be disussed further in (1) With the redution of L O, peel peak stresses develop at a larger normalised region of L O [3]. With the inrease of L O, peel peak stresses onentrate at smaller normalised regions of L O. This differene makes the preponderane of peel stresses not negligible for small L O values. The over estimation of P m for small L O values is thus linked to the bigger value of d f n for the exponential law (Fig. 5a), whih leads to failure at the overlap edges at higher values of P m. The peel stresses extension (normalised over the overlap) rapidly diminishes with the inrease of L O, reduing the error of the CZM preditions with the exponential law. (2) With the redution of L O, owing to the bigger value of d f for s the exponential law indued by the steeper redution of t s this Setion, this does not neessarily implies bigger values of after t s is attained, and also to a state of approximately predited P m by the exponential law, as the triangular and more speifially the trapezoidal laws allow bigger transmission of loads at the initial stages of damage (Fig. 5). onstant shear stresses [3], the CZM elements of the inner overlap region at the time of failure show smaller degradation (i.e., higher transmitted loads), and thus the predited P m values artifiially inrease. 4.2.1. Single-lap joints with the dutile adhesive Fig. 8 reports the perentile deviation (D) between the experi- Figs. 9 and 1 show the P d urves for the joints with L O ¼ 1 mental and FE P m values for the adhesive 215 (averaged by the and 8 mm, respetively, bonded with the Araldite s 215. The experimental and FE urves are in reasonable agreement, in respetive experimental P m values). The slight inonsistent trend whih regards to the joint stiffness and values of P m (this issue is of the P m L O plots is related to the alulation proess to average disussed in detail further in this work). On the other hand, the experimental data, giving natural osillations. Results show that experimental failure displaements are slightly bigger than the FE the trapezoidal law approximates the best the experimental data. ones, owing to minor slippage of the speimens in the mahine The perentile errors between the experimental and FE data are grips. It is also observed that the FE stiffness between CZM shapes negligible, with a maximum of 1.9% for L O ¼ 8. These results are onsistent with previous observations for these types of adhesives n,s is idential up to P m, whilst the preditions mainly differ in the value of P m. These figures were hosen beause they represent the [21,24]. Thus, the large plasti flow of the adhesive at a onstant level of stresses after attaining the peak strength is aptured by minimum and maximum values of L O for the Araldite s 215, the FE simulations, by using damage definitions that orrespond to a onstant level of stresses at the end of the elasti region. The use of a triangular law showed to onsistently underestimate P m, allowing the analysis of the predited behaviour for the limit 4 with a lear tendeny for bigger disrepanies with larger values of L O (D¼- 2.2% for L O ¼ 1 mm, growing steadily for bigger L O 3 values; D¼- 5.5% for L O ¼ 8 mm). The desribed tendeny is justified in light of the typial stress distributions (namely shear stresses) for single-lap joints. As a fat, for small values of L O, the 2 nearly onstant level of shear stresses between overlap ends [3] makes the CZM law shape pratially irrelevant beause at the time P m is attained, the adhesive is evenly loaded in all its length. 1 In the FE analyses, this orresponds to a senario in whih the stress levels are lose to t along the entire bond, whih renders the softening shape of the CZM law not so important. With bigger values of L O the stress gradients inrease [3] and the deviation to the experimental data enlarges as well. Despite the variations to the experimental results, the triangular law still manages to predit P m with an aeptable auray, whih is an important feature to mention, as it is the easiest CZM law to use in terms of P [kn].1.2.3.4.5 [mm] Experimental FE -triangular FE -Exponential FE -trapezoidal Fig. 9. Experimental and numerial P d urves for the joints with LO ¼ 1 mm bonded with the Araldite s 215.

values of L O, and learly showing the differenes in P m reported in Fig. 8. The orrelations obtained for these two onditions are also valid for the other values of L O and for the Araldite s AV138, although the urves are not presented here. 4.2.2. Single-lap joints with the brittle adhesive Fig. 11 provides an idential omparison for the adhesive AV138, in whih the osillations are due to the aforementioned experimental variations. A large disrepany an be readily observed in whih regards the order of magnitude of D, sine for the AV138 the maximum deviation is near 3%, ompared to the approximate 3% for the 215. On the other hand, the results of all the three CZM onfigurations follow the same tendeny for [%] P [kn] 25 2 15 1 5.4.8 1.2 1.6 2 [mm] Fig. 1. Experimental and numerial P d urves for the joints with LO ¼ 8 mm bonded with the Araldite s 215. 1 5-5 Experimental FE - triangular FE - exponential FE - trapezoidal -1 1 2 3 4 5 6 7 8 L O [mm] Fig. 11. Perentile deviation between the experimental and FE Pm values for the adhesive AV138. the entire range of L O values. This is related to the brittleness of the AV138, espeially when ompared to the large dutility of the 215, whih an be testified in Fig. 5 by disparity in the f f d n,s values. Atually, for the shear behaviour (Fig. 5b), d s for the 215 is more than one order of magnitude higher than for the AV138. As a result of this differene, the CZM shape of the AV138 is muh less influent beause the region under softening is negligible when ompared to that of the 215. This an be observed in Fig. 12, whih ompares joints with L O ¼ 8 mm bonded with the 215 (a) and AV138 (b) when P m is attained (trapezoidal CZM). The parameter SDEG orresponds to d s, i.e. the stiffness degradation in shear, with SDEG ¼ relating to the undamaged material and SDEG ¼ 1 to omplete failure. Sine the region of influene of the CZM laws for the AV138 is restrited to a small portion of the overlap, any differenes in shape have a redued effet. The same tendeny between all three CZM shapes is also a result of this, although a slight redution of D between the three shapes ours with the inrease of L O, with negligible variations for the bigger values of L O. This variation an be aredited to the inreasing degree of stress gradients in the adhesive bond, both peel and shear [3], whih further redues the bond length under softening, where the differenes between the three CZM shapes appear. Under brittle onditions, all the CZM shapes revealed to be aurate in prediting the measured response of the joints, although the best results (espeially for small values of L O) were found with the triangular law (maximum value of D of - 1.9% for L O ¼ 1 mm). Compared to these and the experiments, the trapezoidal results showed a slight under predition (maximum D¼- 2.9% for L O ¼ 1 mm). The exponential CZM further under predits P m (maximum D¼- 3.2% for L O ¼ 1 mm), although following the very same trend of the previously reported data. 5. Conluding remarks The main purpose of this work was to evaluate the influene of the CZM shape used to model a thin adhesive layer in single-lap joints on the strength preditions, for different geometry/adhesive ombinations. With this purpose, single-lap joints were bonded with a brittle and dutile adhesive and tested under tension, onsidering a large range of L O values, whih allowed to test different bond solutions in whih regards to stress distributions (short overlaps are usually related to small shear stress gradients, while large overlaps give rise to large stress onentrations). The experimental results initially showed a markedly different trend for both adhesives as a funtion of L O, sine the brittle adhesive resulted in a smaller improvement of P m with L O, as the joints failed soon after the attainment of the adhesive strengths at the overlap ends. Oppositely, the joints bonded with the dutile adhesive showed a major strength improvement with Fig. 12. Stiffness degradation for the joints with LO ¼ 8 mm bonded with the 215 (a) and AV138 (b) when Pm is attained (trapezoidal CZM).

L O on aount of failure ruled by allowane of large plasti flow in the adhesive layer. Regarding the different CZM shapes, these showed a signifiant influene on the results for the joints bonded with the adhesive 215. These were more preisely modelled by the trapezoidal CZM that aptured the adhesive plasti flow at the end of the elasti region, whilst the triangular CZM under predited P m up to D¼- 5.5% for L O ¼ 8 mm. The exponential CZM showed over preditions of P m for short overlaps (up to 27.9%) and under preditions for long overlaps (up to - 6.8%). For the AV138, the triangular CZM showed to be the most suited, although the results were very lose between all CZM shapes tested (maximum deviations of - 1.9%, - 2.9% and - 3.2% for the triangular, trapezoidal and exponential CZM, respetively). As a result of this study, some onlusions were established to properly selet the CZM shape for a given adhesive, depending on its harateristis, but the importane of using the most suited CZM shape will invariably depend on the required preision and on CZM availability/easiness to use. Atually, triangular CZM are more widespread in ommerial software, they are more straightforward to formulate, and give results faster on aount of easier onvergene. Overall, it was found that the influene of the CZM shape an be negleted when using brittle adhesives without ompromising too muh the auray, whilst for dutile adhesives this does not our. Additionally, the smaller the value of L O and the adhesive dutility, the greater is the influene of the CZM shape. In the end, any use of a CZM shape not suited to the material/interfae to be simulated has to be balaned in these issues and expeted variations in auray. Referenes [1] Lee MJ, Cho TM, Kim WS, Lee BC, Lee JJ. 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