Application a direct/cohesive zone method for the evaluation of scarf adhesive joints
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With the increasing use of structures with adhesive bonds at the industrial level, several authors in the last decades have been conducting studies concerning the behaviour and strength of adhesive joints. Between the available strength prediction methods, cohesive zone models, which have shown good results, are particularly relevant. This work consists of a validation of cohesive laws in traction and shear, estimated by the application of the direct method, in the strength prediction of joints under a mixed-mode loading. In this context, scarf joints with different scarf angles (α) and adhesives of different ductility were tested. Pure-mode cohesive laws served as the basis for the creation of simplified triangular, trapezoidal and exponential laws for all adhesives. Their validation was accomplished by comparing the numerical maximum load (Pm) predictions with the experimental results. An analysis of peel (σ) and shear (τ) stresses in the adhesive layer was also performed to understand the influence of stresses on Pm. The use of the direct method allowed obtaining very precise Pm predictions. For the geometric and material conditions considered, this study has led to the conclusion that no significant Pm errors are incurred by the choice of a less appropriate law or by uncoupling the loading modes.
KeywordsAdhesive joint Structural adhesive Finite elements Cohesive zone model Scarf joint
initial crack length
cohesive zone models
damage variable in tension
damage variable in shear
Young’s modulus of the adherends
finite element method
tensile strain energy release rate
critical tensile strain energy release rate
shear strain energy release rate
critical shear strain energy release rate
length (for the DCB specimen) or mid-span (for the ENF test)
load per unit width applied at the specimen ends
cohesive stress in tension
cohesive strength in tension
current value of tn without stiffness loss
cohesive stress in shear
cohesive strength in shear
current value of ts without stiffness loss
coordinate tangential to the adhesive layer
extended finite element method
percentile difference between Pm for each independent model and the mixed-mode triangular model
parameter to specify the softening evolution
relative displacement in tension
relative displacement at t n 0
tensile relative displacement at failure
tensile stress softening onset displacement of the trapezoidal CZM law
relative displacement in shear
relative displacement at t s 0
shear relative displacement at failure
shear stress softening onset displacement of the trapezoidal CZM law
relative rotation of the adherends at the crack tip
relative rotation of the adherends measured at the loading line
average shear stress in the adhesive layer
The use of adhesive bonds in the aerospace, aeronautical and automotive industries, among others, has assumed a high preponderance to the detriment of conventional joining methods such as riveting, bolting, brazing and welding. In fact, adhesive bonds offer several advantages such as the reduction of stress concentrations, good response to fatigue stresses, ability to bond dissimilar materials and lightness of structures. However, they also present some limitations, namely the difficulty of disassembly, high cure times (in some cases) and limited temperature and humidity conditions . The strength and behaviour of adhesive bonds depends on several factors, namely the type of adhesive used, the material of the adherends, the joint configuration and dimensional factors, such as the overlap length (LO) and the adhesive and adherends’ thickness (tA and tP, respectively). There are several types of joint geometries, although the most common are single-lap, double-lap and scarf [2, 3, 4, 5]. Scarf joints are modified butt joints that are more time-consuming to fabricate (i.e., they require milling operations). This type of joint, when loaded, has the ability to keep the axis of loading aligned with the joint . Besides that, it allows a practically constant stress distribution when manufactured with an optimized value of α. Because of this, scarf joints are popular to join composite parts . However, this joint type fails to work under bending, because this loading induces cleavage stresses in the adhesive .
Over the years, reliable and accurate strength prediction techniques were developed. Two distinct approaches can be followed: analytical and numerical. The analytical techniques are able to easily obtain the stress state in bonded structures due to assuming simplifying assumptions in terms of joint geometry, loading and boundary conditions, allowing to obtain an explicit analytical solution for the behaviour in the elastic domain [8, 9]. Nowadays, numerical methods are often preferred over analytical ones, since they overcome some limitations of the analytical methods, such as the possibility to include some effects such as non-linearity of the adhesive and adherends, or to consider geometrical non-linearities . The most common numerical method applied to adhesive bonds is CZM, as a modification of the conventional continuum-based finite element method (FEM) formulation [11, 12]. The extended finite element method (XFEM) is a more recent technique that has showed promising results in the simulation of bonded joints [13, 14]. CZM has been extensively used in the simulation of the structures’ behaviour until failure, since it allows to include in the numerical models the possibility of multiple failure paths in different regions of the materials or between interfaces, which is very helpful for adhesive joints . CZM laws are based on a relationship between the cohesive stresses in tension (tn) and shear (ts) with the relative displacements in tension (δn) and shear (δs) that link homologous nodes of the cohesive elements . The work of Rocha and Campilho  analysed the effect of using different CZM conditions in modelling single-lap joints under a tensile loading. The following analyses were made: variation of the CZM laws’ elastic stiffness, different mesh refinements in the crack paths, study of the adherends’ element type, and evaluation of several damage initiation and growth criteria. It was shown that CZM is suitable for static strength prediction of bonded joints, and the best set of numerical conditions for this purpose was pointed out.
The main parameters of the CZM laws to introduce in the numerical models are the cohesive strengths in tension (t n 0 ) and shear (t s 0 ), and the critical values of tensile and shear strain energy release rate (GIC and GIIC, respectively). The estimation of the referred cohesive parameters (GIC, GIIC, t n 0 and t s 0 ) can be performed by the property identification technique, the inverse method and the direct method. All these methods usually rely on double-Cantilever Beam (DCB) or End-Notched Flexure (ENF) tests. The property identification technique consists of the isolated definition of the cohesive laws parameters through appropriate tests, while the inverse method relies on estimating at least one parameter by an iterative adjustment procedure between experimental data of a given test and its FEM prediction . In a typical inverse method approach, GIC and GIIC are estimated by tests such as the DCB and ENF, respectively, using a suited method or theory. Then, the obtained values of GIC and GIIC are used to build CZM laws in pure modes (tension and shear), while typical values of the parameters t n 0 and t s 0 are initially assumed. In this process, the elastic stiffness of the tensile and shear CZM laws is usually directly inferred from the Young’s modulus (E) and shear modulus (G), respectively, both divided by tA. These CZM laws are next applied to numerical models replicating the fracture tests, created with the same dimensions of the real specimens. Following, for the definition of t n 0 and t s 0 , an adjustment is undertaken between the numerical and experimental load–displacement (P–δ) curves, producing a cohesive law that can accurately reproduce the adhesive layer in the respective loading mode. The assembly of a mixed-mode CZM from this information, including criteria for damage initiation and growth prediction, enables to design bonded structures under arbitrary loadings. This method was applied in the work of Moreira and Campilho , to assess the strength improvement of bonded scarf repairs in aluminium structures with distinct external reinforcements, using the adhesive Araldite® 2015 (Araldite® from Huntsman, Basel, Switzerland). Fairly accurate numerical approximations were attained, but generally with slightly smaller Pm values compared to the experiments. The maximum deviation attained was 20.2%, for a specific repair configuration. This allowed to conclude that the used numerical technique and respective inverse method for CZM laws establishment simulate with reasonable accuracy the behaviour of the bonded repairs. The direct method allows estimating the complete cohesive law for a given material or interface by differentiating the tensile strain energy release rate (GI) or the shear strain energy release rate (GII) with respect to δn or δs, after the polynomial fitting to the most precise degree. With this procedure, the tensile and shear cohesive laws can be obtained, for example, from DCB and ENF tests, respectively . The accuracy of the obtained cohesive laws can be verified by comparing the P–δ curves resulting from CZM laws simulation with the respective test results. A few works are available that estimate the CZM laws of adhesives by the direct approach from either pure or mixed-mode fracture tests. For instance, in the work of Ji et al. , the DCB specimen was used to estimate the CZM laws in tension of a brittle epoxy adhesive by a direct method. A clear tendency was achieved as a function of tA: (1) GIC increases up to tA = 1 mm and (2) t n 0 is highest for tA = 0.09 mm and slowly reduces by increasing tA up to equalling the bulk strength of the adhesive. Leffler et al.  used the ENF specimen to calculate GIIC and the CZM law in shear of an epoxy adhesive by the direct method. Constant displacement rate and constant shear deformation rate assumptions were compared, giving identical estimations of t s 0 . On the other hand, higher GIIC was found by testing at constant displacement rate. Jumel et al.  directly addressed the mixed-mode CZM laws of adhesive layers by using the mixed-mode bending (MMB) test. Gheibi et al.  proposed a new mode-dependent CZM for the simulation of adhesive joints, estimated by the direct method. This was accomplished by using DCB and ENF tests for the direct experimental extraction of the CZM laws (tensile and shear, respectively). The obtained laws were used to implement a simplified Park–Paulino–Roesler CZM. After deriving the mixed-mode parameters, the proposed model was implemented in Abaqus® (Dassault Systèmes, Vélizy-Villacoublay, France) and validated against experimental results of single-lap joints and scarf joints. The accuracy of the developed mixed-mode CZM model for was confirmed for different mode-mixity conditions. The direct method was also applied in the work of Carvalho and Campilho  to validate, with a mixed-mode geometry, tensile and shear cohesive laws obtained in pure mode. With this purpose, single lap-joints with different LO and adhesives were considered, ranging from brittle to ductile. To apply the direct procedure, the originally obtained pure-mode CZM laws were simplified to parameterized triangular, trapezoidal and linear–exponential CZM laws in order to evaluate which form fits better with the behaviour of each adhesive. The joints bonded with the brittle Araldite® AV138 were best modelled by a triangular CZM law shape, mainly due to the adhesive’s brittleness, while the Araldite® 2015 results were best fitted with a trapezoidal CZM, considering that this adhesive has some degree of ductility. The behaviour of the highly ductile Sikaforce® 7752 (Sikaforce® from Sika®, Baar, Switzerland) was more accurately reproduced by a trapezoidal CZM law. However, irrespectively of the adhesive, for the analysed joint geometry (single-lap joints), the errors incurred by applying a less suitable CZM law shape for a given adhesive were always under 10%.
This work consists of a validation of cohesive laws in traction and shear, estimated by the application of the direct method, in the strength prediction of joints under a mixed-mode loading. In this context, scarf joints with different α values and adhesives of different ductility were tested. Pure-mode cohesive laws served as the basis for the creation of simplified triangular, trapezoidal and exponential laws that were tested for each of the adhesives. Their validation was accomplished by comparing the numerical Pm predictions with the experimental results. An analysis of σ and τ stresses was also performed in the adhesive layer in order to understand the influence of stresses on the joints’ strength.
Adherends and adhesives
Young’s modulus, E (GPa)
4.89 ± 0.81
1.85 ± 0.21
0.49 ± 0.09
Poisson’s ratio, ν
Tensile yield stress, σy (MPa)
36.49 ± 2.47
12.63 ± 0.61
3.24 ± 0.48
Tensile failure strength, σf (MPa)
39.45 ± 3.18
21.63 ± 1.61
11.48 ± 0.25
Tensile failure strain, εf (%)
1.21 ± 0.10
4.77 ± 0.15
19.18 ± 1.40
Shear modulus, G (GPa)
1.56 ± 0.01
0.56 ± 0.21
0.19 ± 0.01
Shear yield stress, τy (MPa)
25.1 ± 0.33
14.6 ± 1.3
5.16 ± 1.14
Shear failure strength, τf (MPa)
30.2 ± 0.40
17.9 ± 1.8
10.17 ± 0.64
Shear failure strain, γf (%)
7.8 ± 0.7
43.9 ± 3.4
54.82 ± 6.38
0.43 ± 0.02
2.36 ± 0.17
4.70 ± 0.34
5.41 ± 0.47
Joint geometry, fabrication and testing
All joints were tested in the Shimadzu AG–X test equipment (Shimadzu, Kyoto, Japan), with a 100 kN load cell. For the DCB and scarf tests, the machine was in the generic configuration to perform tensile tests. For the ENF tests, it was adapted to apply a three-point bending flexure loading. The equipment is connected to a data acquisition system that records the test time, applied load and grips’ displacement. These values will be used to plot the P–δ curves and perform all required data analysis. All the tests were carried out under conditions of room temperature and humidity. The test velocity was 1 mm/min. Five specimens were tested for each condition.
Direct method for the DCB and ENF tests
The ts(δs) curve or shear cohesive law of the adhesive layer is thus estimated by fitting the resulting GII–δs curve, and differentiation with respect to δs. Leitão et al.  presents the full details regarding the description of the direct method applied to the ENF specimen, as well as the algorithm to estimate δs.
Implementation of the model in Abaqus®
Results and discussion
CZM law estimation by the direct method
Values of GIC and GIIC (N/mm) for the three adhesives
0.245 ± 0.045
0.580 ± 0.090
0.533 ± 0.123
3.123 ± 0.203
3.770 ± 0.278
5.667 ± 0.459
0.249 ± 0.033
0.618 ± 0.069
0.539 ± 0.116
2.967 ± 0.273
3.683 ± 0.320
5.562 ± 0.356
A numerical stress analysis was initially carried out to study σ and τ stresses developing at the middle of the scarf adhesive layer, during elastic loading, as a function of the normalized scarf length x/LS (0 ≤ x ≤ LS), in which x is the coordinate tangential to the adhesive layer and LS the scarf length. Both σ and τ stresses were subjected to a normalization procedure, by dividing them by the average shear stress in the adhesive layer (τavg) for each α. In this work, only the stress distributions for the joints bonded with the Araldite® 2015 are shown. However, these stresses are also representative of the other two adhesives’ behaviour (although there are small differences in the peak stresses because of the differences in E; see Table 1). In general, it is known that the use of smaller values of α leads to higher LS and, thus, to higher joint strengths. However, it also provides longer machined lengths and required volume of adhesive .
Damage variable analysis
Notwithstanding the adhesive type, the highest incidence of damage in this type of joint occurs at the adhesive layer’s ends, consistently to the stress distributions presented in “Introduction” section. Oppositely, at the intermediate zone of the bond, SDEG is typically zero. Between adhesives, it was found that the stiffness and brittleness of the Araldite® AV138 makes it very sensible to the peak stresses that take place near x/LS = 0 and 1 (Fig. 8 shows this effect, although for a different adhesive), such that the span of damage, i.e., length of x/LS between SDEG = 0 and 1, is very short. The total length under damage ranges between 11.1% (α = 3.43°) and 55.0% (α = 45°) of LS, reinforcing the lack of plasticization ability of this adhesive. The shift in the adhesives’ properties, particularly the increasing elastic compliance and ductility, enables damage to spread more evenly and to reduce the gradients along the bond length. Actually, on one hand, the higher compliance of the Araldite® 2015 and Sikaforce® 7752 reduces σ and τ peak stresses which, alone, helps in spreading damage more evenly. On the other hand, the higher ductility of these two adhesives, compared to the Araldite® AV138, enables the damaged regions to keep working while the inner regions of the bond are put under loads as well. As a result, the damage lengths for these adhesives range between 65.0% (α = 45°) and 87.2% (α = 3.43°) for the Araldite® 2015 and 15.0% (α = 45°) and 78.6% (α = 3.43°) for the Sikaforce® 7752. Between adhesives, the increased compliance of the Sikaforce® 7752 is responsible for the more uniform span of damage at Pm, which ranges between 0 and ≈ 12.5% compared to 0–100% for the Araldite® AV138 and 0–43.8% for the Araldite® 2015 (maximum values, obtained for specific α). Comparing the SDEG curves between the different α, the Araldite® AV138 shows a different behaviour to the other adhesives, in the sense that smaller α results in more concentrated damage and higher SDEG gradients. This is because of its brittleness, which clearly cannot accommodate the higher σ and τ peak stresses at the overlap ends (this effect can be identified in Fig. 8, despite this figure relates to the Araldite® 2015). Since σ and τ stresses slowly become more flat with increasing α, the damage also increases towards the central region of the bond. The ductile adhesives are not affected by these peak stresses, since the damage span across the bond does not vary much with α. Nonetheless, SDEG attains higher percentile values at Pm for smaller α also due to the aforementioned differences in σ and τ peak stresses.
Discussion on the joint strength
Evaluation of the different CZM law shapes
The main purpose of this work was the evaluation of different CZM formulations to predict the strength of adhesively-bonded scarf joints, considering the tensile and shear pure-mode laws were estimated by a direct method. Three adhesives were evaluated, whose behaviour ranged from strong and brittle to less strong and ductile. The results showed that the mechanical behaviour of the scarf joints is dependent on the type of adhesive and value of α. A more stiff and brittle adhesive leads to slightly higher peak stresses, while a more flexible and ductile adhesive gives more uniform stress distributions. Nonetheless, comparing with other joint configurations such as single-lap and double-lap, the stress distributions are much improved. Peak τ stresses along the bondline increase with the reduction of α, which negatively affects the joint strength, but the reduction of this parameter also increases exponentially LS, whose effect is much more preponderant and leads to a Pm improvement. The damage plots at Pm reflected the result of the elastic stress distributions, with maximum damage at the adhesive ends. The damage span was smallest for the Araldite® AV138, due to the combined effect of the highest peak stresses and reduced GIC and GIIC, while the other adhesives showed an improved behaviour by enabling a wide damage spread at Pm. Despite these facts, the experimental Pm results showed that the Araldite® AV138 achieves the highest Pm for all α between adhesives because, under the conditions of small stress gradients, the strengths of the adhesive are prevalent over the energy parameters. The numerical strength predictions were accurate in comparison with the experimental data, for all CZM law shapes and coupling modes. For the particular geometry tested in this work, the differences between CZM shapes were minimal. Thus, no significant errors are made in the choice of a less adequate law. However, it should be noted that this only occurs due to the particular load transfer characteristics of scarf joints. For other geometries, namely with higher stress gradients along the adhesive, non-negligible differences can be found between law shapes.
DFOS performed the experimental tests and respective data analysis, and he also performed the numerical analysis. RDSGC and UTFC developed the numerical technique and participated the paper writing. FJGdaS participated in the paper writing. All authors read and approved the final manuscript.
The authors would like to thank Sika® Portugal for supplying the adhesive.
The authors declare that they have no competing interests.
Availability of data and materials
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
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