Static strength prediction of curved composite joints under internal pressure

ABSTRACT

Nowadays, adhesive joints are largely applied in the automotive, aircraft and aerospace, civil, and naval industries. Although several applications involve flat (plane) adherends, curved joints play an important role in many engineering applications like civil and aircraft construction. This work aims to investigate three structural adhesives, ranging from brittle to ductile, applied in curved joggle-lap joints between carbon fibre reinforced polymer adherends and subjected to internal pressure, to validate a robust design procedure and provide project guidelines for this complex structural, geometrical, and loading system, which constitute the main novelty of the proposed work. A numerical cohesive zone model parametric analysis was undertaken by varying the overlap length, thickness of the adherends, and adherends’ curvature, including the evaluation of peel and shear stresses in the adhesive, failure mode comparison, maximum load, and energy dissipated after failure. Validation of the cohesive zone model technique was initially performed using flat single-lap joint under tensile loads. A significant effect of the overlap length and thickness of the adherends on the maximum load was found, while the adherends’ curvature effect on the maximum load was negligible. Ductile adhesives, although less strong, generally performed better in terms of maximum load and dissipated energy.

KEYWORDS:

• Adhesive bonding
• aerospace materials
• composite structures
• finite element analysis (FEA)
• material modelling
• numerical analysis

1. Introduction

Adhesive bonding is a versatile joining technique and shows interesting characteristics and performance with respect to mechanical fastening or even welding. Nowadays, adhesive joints are largely applied in the automotive,^[1] aircraft and aerospace,^[2] civil,^[3] and naval industries.^[4] Like in other joining processes, adhesive bonding must be tailored according to the load solicitations, being the adhesive joints stronger in shear. The single-lap joint (SLJ) is one of the most employed because of its simplicity.^[5] In this type of joint, the adhesive is mainly loaded in shear.^[6] However, its geometry induces peel (σ_y) stresses, reducing the joint’s performance.^[7] Other joint configurations exist, whose application depends on the load solicitation. The most common joint configurations were described by Adams and Wake.^[8] Even though tensile loadings are most common, other loadings for specific applications are addressed in the literature, such as compression.^[9]

Although several applications involve flat (plane) adherends, curved joints play an important role in many engineering applications like civil and aircraft construction.^[10] In civil construction, curved joints can be used for reinforcement of existing masonry and concrete structures and to fabricate new structures of composite materials like fibreglass.^[11] In aircraft construction, curved joints are employed in the manufacture of fuselages,^[12] often between carbon fibre reinforced polymer (CFRP) structures. Furthermore, repairs on pipelines sometimes are performed using adhesive joints, providing advantages over traditional methods.^[13] Studies are available on the behaviour of these repairs on pressurised pipelines, indicating that the repairs can sustain higher pressures than the pipeline’s working pressure.^[13,14]

The joggle-lap joint (JLJ) is a variation of the SLJ that, although geometrically similar, has the advantage of aligned adherends, reducing transverse deflection. JLJ is used in aircraft construction because it allows joining curved segments of the fuselage whilst keeping the aerodynamic profile,^[15] and joining the fuselage to the half-wing.^[16] Taib et al.^[15] performed a study comparing JLJ to SLJ, both in their flat configurations, aiming to study the addition of a spew fillet on the JLJ performance. The study was numerical and performed using commercial software. The models were load-driven, enabling to identify differences in the bond line stresses, but not to calculate the maximum load (P_m). The stress distributions in JLJ were more uniform than those of SLJ, but the peak stresses at the knee side were higher, whilst those at the other free end were slightly lower, consequently reducing P_m. Nevertheless, leaving the spew between the knee and the upper adherend significantly reduced the peak stresses in the bond line, increasing them in the opposite free-end.^[15] On the other hand, the geometry of JLJ caused σ_y stresses to increase at the knee vicinities, increasing the chances of adherend damage when composite adherends are involved.^[16] Therefore, stress analyses should be performed while designing these joints. Although joints with curved adherends show similarities to standard or flat joints, there are a few challenges due to the curvature, being mostly related to fibre orientations on composite adherends, and loading conditions.

Stresses in adhesive joints, mainly in the bond line, can be obtained using analytical^[17] or numerical (finite element method or FEM) approaches,^[18] whilst joint strength can be determined through continuum mechanics, fracture mechanics, or cohesive zone model (CZM) techniques. Amongst these techniques, the latter delivers strength predictions closer to experimental values provided that the cohesive law is properly selected. The triangular cohesive law is one of the most used since it requires fewer parameters than the other cohesive laws and it gives accurate results for brittle and slightly ductile adhesives.^[19] However, it underpredicts P_m in joints with highly ductile adhesives. The different cohesive laws were described in detail by Alfano.^[20] A detailed description of the method was reported by Campilho et al.^[21] Although CZM is widely adopted in the literature, inconveniently, the failure path should be predefined beforehand, which in some cases is not adequate. Moreover, the required parameters for the cohesive laws must be determined from experimental tests. A comprehensive description of the process to characterise an adhesive is provided by Faneco et al.^[22] The double cantilever beam (DCB) for mode-I and the end-notched flexure test (ENF) for mode-II are frequently used for fracture characterization. More recently, Monsef et al.^[23] studied mode-II experimental tests in adhesive joints. The ENF and the end-loaded split (ELS) were the chosen tests. In addition, an analytical methodology to obtain the cohesive law was developed, which took advantage of the experimental load-displacement (P-δ) data. The results indicated that the cohesive law is independent of the test type, and the results also agreed with the J-integral method, validating the proposed methodology.

The damage models used for CZM are also applicable to the eXtended Finite Element Method (XFEM).^[24] The XFEM is based on the concept of partition-of-unity proposed by Belytschko.^[24] Although the initial proposal was not suitable for long cracks nor 3D cases, the method was expanded to such cases by Moes et al.^[25] Unlike the CZM, the XFEM allows to simulate crack initiation and growth without the need of a predefined path, resulting on successful implementations for crack propagation, as described by Rabczuk et al.^[26] Whereas the theory and formulations needed to implement the XFEM are out of the scope of this work, they are available in references.^[25,26] Furthermore, the ability to predict crack propagation through the XFEM has been employed to study adhesive joints. For example, Machado et al.^[27] recently validated the XFEM as a means to estimate P_m in SLJ. Three adhesive systems were used, and several damage initiation criteria and damage propagation laws were tested. The results indicated that this numerical methodology provides acceptable strength predictions. However, the accuracy of the prediction depends on the damage initiation criterion, being those stress-based the ones yielding the best results. Later, Sadeghi et al.^[24] performed a parametric study to SLJ using FEM-based techniques like FEM, CZM, and XFEM, among others. The results were compared against experimental data acquired during the study. Although all the cases tested presented differences with the experimental values, the cases corresponding to XFEM for an adhesive layer thickness of 0.2 mm were similar. The XFEM estimated joint strength close to experimental values, indicating its suitability to study adhesive joints.

Analytical and numerical modelling of curved bonded joints is scarce in the literature. Liu et al.^[28] addressed the effect of adherends’ curvature (R), size, and free ends of composite curved lap joints such as those employed in aircraft fuselages under circumferential tension. The proposed methodology used numerical models of flat joints (validated with experimental data) as a base for further analyses of other joint configurations, as curved joints, thus reducing the number of required experiments and allowing to analyse larger and more complex components. The numerical models were based on CZM. Three R values were used: 1000 mm, 2000 mm, and 3000 mm. The analysed R are relevant to study aircraft cylindrical surfaces, but no relevant effect on P_m was observed. Conversely, the joints’ compliance increased with R. Later, the performance of three adhesive systems (Araldite® AV138, Araldite® 2015, and Sikaforce® 7888) on curved SLJ with CFRP adherends was evaluated by Correia et al.^[29] for identical loading conditions. overlap length (L_O), thickness of the adherends (t_P), and R effects were evaluated by CZM. The numerical technique was validated with experimental data from flat SLJ. The results indicate that the energy dissipated after failure (U) is proportional to L_O when using ductile adhesives, whilst t_P has the opposite effect. Conversely, in the presence of a brittle adhesive, these parameters had no significant effect on P_m. On the other hand, R higher than 2000 mm had no effect on P_m as the geometry further approaches a flat SLJ. Curved bonded joints, either in the overlap or joggle configurations, find application in aircraft fuselages, where they are subjected to internal pressure while in service. The internal pressure inside a commercial aircraft (airliner) is around 0.75 bar whilst the pressure at cruising altitude (40,000 ft) is 0.2 bar. Thus, the associated differential is 0.55 bar.^[30] Although internal pressure is important, it is not considered in the available studies. It is thus considered that the described numerical methodologies, mainly CZM, are good candidates to evaluate the added effect that the internal pressure has on material and geometrical variables related to curved joints.

This work aims to investigate three structural adhesives, ranging from brittle to ductile, applied in curved JLJ between CFRP adherends and subjected to internal pressure, to validate a robust design procedure and provide project guidelines for this complex structural, geometrical, and loading system, which constitute the main novelty of the proposed work. A numerical CZM parametric analysis was undertaken by varying L_O, t_P, and R, including the evaluation of σ_y and shear (τ_xy) stresses in the adhesive, failure mode comparison, P_m, and U. Validation of the CZM technique was initially performed using flat SLJ under tensile loads.

2. Experimental work

The curved joints to analyse in this work are subjected to mixed-mode loading, making it possible to experimentally validate the numerical technique using a slightly simpler joint geometry also loaded in mixed-mode, in this case the SLJ. The latter joint possesses similar geometry and loading conditions to the curved joints, but it is simpler to manufacture and test. In addition, this joint configuration is more readily available in the literature, and so more data exist for comparison.

2.1. Joint materials

In this work, curved joints with CFRP adherends bonded with three different adhesives are analysed. Consequently, the tested SLJ were made with the same type of materials. The adherends are composed by CFRP sheets (SEAL Texipreg HS 160 RM, Legnano, Italy) having a unit thickness of 0.15 mm, to produce a [0]₁₆ configuration. The adherends’ manufacture consisted of stacking the composite sheets using the desired configuration and pressing the laminate in a press with 130°C and 4 bar for 1 hour, following the procedure described by Campilho et al.^[31] The elastic properties of the composite were also reported by Campilho et al.,^[31] while t_P for the SLJs was 2.4 mm.

The Araldite® AV138 and Araldite® 2015 adhesives were used in the CZM validation process, while the adhesive Sikaforce® 7752 was added for comparison in the numerical curved joint analysis. All relevant properties were experimentally acquired in former works,^[21,22,32] and are listed in Table 1. A comprehensive description of the testing processes for all the parameters, including the respective standards, is provided by Faneco et al.^[22]

Table 1. Mechanical and fracture properties of the adhesives Araldite® AV138,^[32] Araldite® 2015,^[21] and Sikaforce® 7752.^[22].

Property	AV138	2015	7752
Young’s modulus, E (GPa)	4.89 ± 0.81	1.85 ± 0.21	0.493 ± 0.089
Poisson’s ratio, ν	0.35 ^a	0.33 ^a	0.33 ^a
Tensile yield strength, σe (MPa)	36.49 ± 2.47	12.63 ± 0.61	3.24 ± 0.5
Tensile failure strength, σf (MPa)	39.45 ± 3.18	21.63 ± 1.61	11.49 ± 0.3
Tensile failure strain, εf (%)	1.21 ± 0.10	4.77 ± 0.15	19.18 ± 1.4
Shear modulus, G (GPa)	1.56 ± 0.01	0.56 ± 0.21	0.188 ^c
Shear yield strength, τe (MPa)	25.1 ± 0.33	14.6 ± 1.3	5.16 ± 1.1
Shear failure strength, τf (MPa)	30.2 ± 0.40	17.9 ± 1.8	10.17 ± 0.6
Shear failure strain, γf (%)	7.8 ± 0.7	43.9 ± 3.4	54.82 ± 6.4
Toughness in tension, GIC (N/mm)	0.20 ^b	0.43 ± 0.02	2.36 ± 0.2
Toughness in shear, GIIC (N/mm)	0.38 ^b	4.70 ± 0.34	5.41 ± 0.5

^aproducer information

^bdefined in reference,^{[32] c} calculated using Hooke’s law

2.2. CZM validation and curved geometries

Figure 1 reports the flat SLJ (a) and curved JLJ (b) used for CZM validation and numerical evaluation, respectively. The base dimensions of both joints are as follows: joint length L_T = 200 mm, joint width B = 15 mm (flat SLJ) or B = 25 mm (curved JLJ), adhesive thickness t_A = 0.2 mm, t_P = 2.4 mm, and L_O such that 10≤ L_O≤80 mm with 10 mm increments. The numerical analysis to the curved joints involved the same L_O range, t_P = 1.2, 2.4 and 3.6 mm, and R = 1000, 2000 and 3000 mm. The R parametric analysis aims to check the influence of the joint curvature (arising from different structure geometries to bond) on the joint response. While the CZM approach was validated with flat SLJ subjected to a tensile load, the curved JLJ were subjected to an internal pressure of 1 MPa.

Figure 1. Geometries of the flat SLJ for the CZM validation process (a) and curved JLJ for the CZM parametric analysis (b).

2.3. Specimen fabrication and tests

The adherends for the flat SLJ used in the CZM validation process were cut to size to allow L_T = 200 mm (Figure 1). Therefore, the adherend length is specific for each L_O. The adherends were cut into strips with B = 15 mm. Subsequently, the areas where the adhesive was to be applied were sanded, and then the whole adherend was thoroughly cleaned and degreased with acetone. The adherends were aligned on a flat surface, and calibrated shims were used to ensure a uniform t_A of 0.2 mm. The adhesive then was applied and the adherends bonded together. The joint was aligned using slight pressure. Then, the alignment was kept using spring clamps during the whole curing process. The joints were cured for a week at room temperature. After this procedure, the excess of adhesive was mechanically removed from the joint by milling with mounted points.

Five specimens were manufactured per joint configuration, thus leading to 80 fabricated and tested specimens and 16 different configurations, due to using two adhesives (Araldite® AV138 and Araldite® 2015) and eight L_O (between 10 and 80 mm). The SLJ were tested to failure using a Universal Testing Machine (UTM) Shimadzu AG-X 100 equipped with a load cell of 100 kN. The testing speed was 1 mm/min. P and δ were recorded during the tests, providing the P-δ curves for all the specimens. P_m was considered as the maximum load sustained by each joint. Finally, the P_m per configuration was calculated as the average of the corresponding samples.

3. Numerical work

3.1. Pre-processing

The flat SLJ for CZM validation and the curved JLJ for the CZM parametric analysis were developed using the same numerical principles in Abaqus® using two-dimensional (2D) models and plane-strain assumptions for all joint components. In the first case, the models were used to assess the failure mode and P_m, while in the second case these aimed to obtain the stress distributions in the adhesive layer (during elastic loading), followed by the failure modes, P_m, and U.

For the stress distributions required for the CZM parametric analysis, quadrilateral elements (CPE4) were considered in the adherends and adhesive layer, while σ_y and τ_xy stresses were evaluated at the middle of t_A. For all the other SLJ and JLJ analyses, the CPE4 elements were only equated for the adherends, while for the adhesive layer, a single row of CZM elements (COH2D4) with t_A = 0.2 mm was inserted between the adherends. Triangular, i.e., bilinear traction-separation laws were considered for the CZM elements. Moreover, thin (0.02 mm) CZM layers with CFRP interlaminar properties were placed in the adherends, between the two plies closest to the adhesive, to make possible this type of failure in the numerical modelling, following experimental occurrence in some joint configurations.

The mesh refinement was also different between models: stresses were evaluated with a highly refined mesh in the adhesive layer and neighbouring regions, to assure a high accuracy, while all other analyses were run with a coarser mesh, due to the insensitivity of this technique to peak stresses, if the models assure a minimum refinement.^[33] In the first scenario, the adhesive was modelled with ten solid elements through-thickness (element side dimensions of 0.02 mm), compared to the use of one element for the other analyses (thus, side dimensions of 0.2 mm). Element size grading was also applied to provide a higher refinement at the overlap edges (along the adhesive’s length), and close to the adhesive layer (in the transverse direction). In the unbonded regions, the element size was bigger to save computational resources.

To correctly describe the internal pressure event and respective restraints in a circular component, firstly a polar coordinate system was created, as shown in Figure 1. Then, both joint ends were constrained in the tangential direction whilst left free in the radial direction; thus, allowing the expansion of the joint due to the internal pressure (Figure 2a). Subsequently, a pressure of 1 MPa was assigned, capable of causing failure in all models. This pressure was applied perpendicularly to all internal faces of the joint, including the butt portion of the inside adherend (Figure 2b). In all cases, the models were solved considering geometrical non-linearities, i.e., large deformations solution, which is necessary to consider the expansion of the joint due to the internal pressure, otherwise, the result would be underpredicted. Further description of geometrical non-linearities is provided by de Sousa Neto et al.^[34]

Figure 2. Boundary conditions and loads applied to the joints, with details at the model supports (a) and overlap (b).

3.2. CZM formulation

Although cohesive formulations can represent pure or mixed modes, the latter is typically required for most adhesive joint applications. Between the several available cohesive laws, the triangular law provides a good representation of the actual behaviour of adhesives in bonded joints.^[35] This law requires fewer parameters than other cohesive laws, reducing the complexity of their experimental estimation, and consequently simplifying the CZM implementation in the computational software.

Under pure mode, the triangular cohesive law relates the cohesive stresses in traction (t_n) or shear (t_s) with the relative displacements (δ_n or δ_s, respectively). The first segment corresponds to the elastic loading. Then, a second line connecting that point to the corresponding failure displacement (δ_n^f or δ_s^f for traction or shear, respectively) represents the material degradation until fracture. The area beneath the whole curve corresponds to the tensile fracture toughness (G_IC) or shear fracture toughness (G_IIC). The elastic loading is defined by the stiffness matrix K_COH established between the current stresses and strains in tension and shear (subscripts n and s, respectively) as

(1) $\mathrm{t}=\left\{\begin{array}{c}{t}_n\\ t_s\end{array}\right\}=\left[\begin{array}{c}{K}_{nn}{K}_{ns}\\ K_{ns}{K}_{ss}\end{array}\right].\left\{\begin{array}{c}{\epsilon }_n\\ {\epsilon }_s\end{array}\right\}=\mathrm{K}\epsilon .$(1)

ε_n and ε_s are the tensile and shear strain, respectively. For thin adhesive layers it can be stated that K_nn = E, K_ss = G, K_ns = 0 (E is the Young’s modulus and G is the shear modulus). Under mixed mode, both mode I (tension) and mode II (shear) contribute to material degradation until fracture. Finally, damage initiation is defined by a quadratic criterion combining the proportions of t_n and t_s with the respective cohesive strengths (t_n⁰ and t_s⁰) within an elliptic envelope, noting that compression does not contribute to damage initiation. Damage propagation is assessed by a linear power-law expression relating the tensile fracture energy (G_I) and shear fracture energy (G_II) with the respective limit values. This formulation is described with more detail in the literature.^[36] Whilst the adhesives’ CZM properties were taken from Table 1, considering E and G as the initial stiffness of the CZM laws, the CFRP interlaminar CZM properties were previously determined^[37] for the same material. In this case, the initial stiffness in both loading modes was assumed as 10³ GPa, using the penalty function method to avoid affecting the structural response of the specimens. Table 2 summarizes the CZM data used in the numerical models.

Table 2. CFRP interlaminar and adhesive layer CZM properties.

CZM property	CFRP	AV138	2015	7752
Knn (GPa)	10^3	4.89	1.85	0.493
Kss (GPa)	10^3	1.56	0.56	0.188
tn^0 (MPa)	42.6	39.45	21.63	11.49
ts^0 (MPa)	39.3	30.2	17.9	10.17
GIC (N/mm)	0.54	0.20	0.43	2.36
GIIC (N/mm)	0.93	0.38	4.70	5.41

4. Results

4.1. CZM validation

To enable the purely CZM analysis to the JLJ under internal pressure, initial CZM validation is carried out considering flat SLJ under tensile loads, bonded with the adhesives Araldite® AV138 and Araldite® 2015, representing examples of brittle and ductile adhesives, respectively.

4.1.1. Failure mode comparison

The first step in validating the CZM technique for the intended purpose consists of failure mode comparison between the simulations and experiments. The failure modes were identical and as follows:

• Adhesive Araldite® AV138 – experimental cohesive failure of the adhesive for L_O = 10 and 20 mm, and interlaminar failure in the entire overlap for 30≤ L_O≤80 mm, in some cases with small cohesive failure spots. The change in failure mode is linked to increasing gradients of both σ_y and τ_xy stresses taking place for bigger L_O. Figure 3(a) shows an example of cohesive failure in the adhesive layer for L_O = 10 mm and Figure 3(b) of interlaminar failure for L_O = 40 mm (emphasizing a cohesive failure spot of the adhesive at the leftmost overlap edge). In the simulations, it was possible to replicate this change in failure mode. Figure 3(c) shows the obtained numerical interlaminar failure for the conditions of Figure 3(b);

  Figure 3. Experimental cohesive failure in the adhesive layer for the adhesive Araldite® AV138 and L_O = 10 mm (a), experimental interlaminar failure in the CFRP for the adhesive Araldite® AV138 and L_O = 40 mm (b) and respective CZM prediction (c), experimental cohesive failure in the adhesive layer for the adhesive Araldite® 2015 and L_O = 20 mm (d) and respective CZM prediction (e).

  

• Adhesive Araldite® 2015 – experimental cohesive failure of the adhesive for all L_O, with a visible layer of adhesive layer in both failed adherends and absence of composite damage. Figure 3(d) relates to L_O = 20 mm. The CZM results confirmed the cohesive failure mode for all L_O (Figure 3(e) shows the CZM failure for the same L_O = 20 mm condition).

4.1.2. Maximum load comparison

CZM validation also involves evaluation of the CZM capacity to predict P_m. Initially, the numerical and experimental P-δ curves were compared. Figure 4 shows a sample comparison for the joints bonded with the adhesive Araldite® 2015 and L_O = 10 mm. The degree of correspondence of P_m between the experimental curves and numerical prediction is representative of all tested conditions. The initial stiffness of the joints shows a good match, although the experimental curves tend to lose stiffness near to P_m. After careful analysis to the failed specimens, namely at the grips region, it was found that the loss of stiffness was caused by minor slip in the testing grips, due to the high values of P attained in the tests. Due to the reduced hardness of the composite adherends, which is governed by the matrix, slipping marks could be seen in the adherends, induced by the serrated surface of the grips. A similar effect was reported by Kang et al.^[38] In the present work, it was found that the numerical P_m values are close to the experimental average. Figure 5 compares the experimental/numerical P_m data obtained from the P-δ curves for the two evaluated adhesives and all L_O, including the standard deviation associated to the test data.

Figure 4. Experimental/numerical P-δ curve comparison: example for the joints bonded with the adhesive Araldite® 2015 and L_O = 10 mm.

Figure 5. Experimental/numerical P_m vs. L_O evolution for the joints bonded with the adhesive Araldite® AV138 (a) and 2015 (b), including the standard deviation of the experimental data.

P_m has a monotonic evolution with L_O for both adhesives, although higher for the adhesive Araldite® 2015. The obtained absolute P_m increments were 13.1 and 16.6 kN for the adhesives Araldite® AV138 and Araldite® 2015, respectively. The adhesive Araldite® 2015 provided the worst result for L_O = 10 mm (P_m = 2.5 kN), while the adhesive Araldite® AV138 was higher by 18.3%. For an intermediate L_O (e.g., 40 mm), the adhesive Araldite® AV138 turns into the lowest performing adhesive (P_m = 8.8 kN), while the adhesive Araldite® 2015 exceeds this P_m by 5.9%. This difference increases by considering the highest L_O (80 mm), since the adhesive Araldite® AV138 gives P_m = 16.0 kN, while the adhesive Araldite® 2015 P_m is higher by 19.3%. The increasing offset with L_O is justified by the ductility of the adhesive Araldite® 2015, which promotes adhesive plasticization at the sites of stress concentrations instead of sudden failure, and thus P_m becomes higher because the average shear stress (τ_avg) at the time of failure is also higher.

The experimental and numerical data comparison for the adhesive Araldite® AV138 (Figure 5a) showed marginal deviations, particularly for the smaller L_O. The highest difference was −9.1% (L_O = 80 mm). The highest offset for the adhesive Araldite® 2015 (Figure 5b) was −7.0%, obtained for the joint with L_O = 70 mm. Thus, the main objective of this CZM validation process, consisting of assessing the technique’s capacity for predictive effects, was accomplished.

4.2. Numerical curved joint analysis

After the CZM validation concerning flat SLJ presented in section 4.1, a numerical study is performed to evaluate the effects of different geometrical parameters, such as L_O, t_P and R, on curved JLJ.

4.2.1. Elastic stresses

The relevant stress components (σ_y and τ_xy) are both normalized by τ_avg along the adhesive layer, for the respective L_O. The analysis graphically focuses on one of the three adhesives, since the stress distributions’ generic shape is identical between selected adhesives for the different parameters (L_O, t_P and R), peaking at the overlap edges. However, differences arise in the absolute values of peak stresses at the overlap ends due to the different stiffness, as shown in Table 1.^[39] For this section, only the adhesive Araldite® 2015 is considered, due to having an intermediate behaviour. Figure 6 reports σ_y (a) and τ_xy (b) stresses for the adhesive Araldite® 2015 considering fixed R = 2000 m and varying L_O (10 and 80 mm) and t_P (1.2 and 3.6 mm).

Figure 6. σ_y/τ_avg (a) and τ_xy/τ_avg (b) stresses for the 2015 considering fixed R = 1000 m and varying L_O (10 and 80 mm) and t_P (1.2 and 3.6 mm).

Normalized σ_y stresses (Figure 6a) are mostly negligible along x/L_O, except at the overlap edges. However, stresses peak at x/L_O = 0 (joggle curvature) and a smaller peak exists at x/L_O = 1. The locations of σ_y peak stresses were expected for lap joints.^[15] For L_O = 10 mm, σ_y/τ_avg peak stresses attained 19.1 and 10.3 for t_P = 1.2 mm and 3.6 mm, respectively (46% reduction between t_P). For L_O = 80 mm, peak stresses attain 147.0 for t_P = 1.2 mm and 70.1 for t_P = 3.6 mm (52% reduction). Higher L_O significantly increase σ_y stresses (up to ≈670% for t_P = 1.2 mm), which agrees with flat SLJ behaviour.^[40] Higher t_P also benefit σ_y stresses (reductions of ≈50%). Between adhesives, the adhesive Araldite® AV138 significantly increased peak σ_y/τ_avg stresses, up to 49.1% for the joint with L_O = 80 mm and t_P = 1.2 mm, while the adhesive Sikaforce® 7752 managed to highly reduce peak stresses up to 47.6% for the same joint configuration.

Similarly to σ_y stresses, τ_xy stresses (Figure 6b) have a smaller magnitude at the overlap centre, while peaking at the overlap edges.^[40] However, the variation is smoother and, especially for smaller L_O, the inner overlap is also loaded with less significant τ_xy stresses. The peak τ_xy/τ_avg stress asymmetry was also found. For L_O = 10 mm, peak τ_xy/τ_avg stresses are 12.7 and 7.2, respectively, for t_P = 1.2 mm and 3.6 mm, which corresponds to a drop of 43.3%. For L_O = 80 mm, these values increase to 92.5 for t_P = 1.2 mm and 48.9 for t_P = 3.6 mm (drop of 47.1%). Increasing L_O leads to major peak stresses (maximum difference to L_O = 10 mm of ≈630% for t_P = 1.2 mm), and increasing t_P reduces peak τ_xy/τ_avg stresses up to almost 50%. Peak stresses increased up to 96.5% for the adhesive Araldite® AV138 (L_O = 80 mm and t_P = 3.6 mm), and diminished up to 59.9% for the adhesive Sikaforce® 7752 (same geometry).

The R influence on σ_y and τ_xy stresses between joint conditions is compared in bar plots since the differences between curves are small and restricted to the peak values, which prevents the clear visualization in stress plots. Thus, the adhesive Araldite® 2015, L_O = 10 mm and t_P = 1.2 mm absolute peak data at x/L_O = 0 and x/L_O = 1 is summarized in Figure 7(a) for σ_y/τ_avg and (b) for τ_xy/τ_avg stresses. Note that the vertical axes are truncated to focus on the region of interest.

Figure 7. Peak σ_y/τ_avg (a) and τ_xy/τ_avg (b) stresses for the 2015 at x/L_O = 0 and x/L_O = 1 considering fixed L_O = 10 mm, t_P = 1.2 mm and R = 1000 m, and varying R (1000, 2000 and 3000 mm).

Figure 7(a) shows that peak σ_y/τ_avg stresses at x/L_O = 0 (tensile) are always higher than those at x/L_O = 1 (which are compressive), with a relative difference of ≈20 times. Increasing R leads to a reduction of peak σ_y/τ_avg stresses at x/L_O = 0 from 19.1 (R = 1000 mm) to 18.6 (R = 3000 mm), or 2.3% reduction. Higher R approach the flat SLJ condition without curve asymmetry. The same effect is found at x/L_O = 1. Consistency was found with the adhesives Araldite® AV138 and Sikaforce® 7752, equally with small peak σ_y/τ_avg stress reductions by increasing R. The adhesive Araldite® AV138 increased peak σ_y/τ_avg up to ≈49% and the adhesive Sikaforce® 7752 reduced up to 47%.

Peak τ_xy/τ_avg stresses (Figure 7b) also reveal to be critical at x/L_O = 0, with a ≈ 20-fold relation to the peak values x/L_O = 1. At x/L_O = 0, a reduction is found from 12.7 (R = 1000 mm) to 12.5 (R = 3000 mm), i.e., reduction of 1.4%. However, a marked increase takes place at x/L_O = 1: from 0.67 (R = 1000 mm) to 0.75 (R = 3000 mm), i.e., difference of 11.9%. Nonetheless, the absolute peak values are not significant compared to x/L_O = 0. Between adhesives, peak τ_xy/τ_avg stresses increased up to ≈87% (adhesive Araldite® AV138) and reduced up to 57% (adhesive Sikaforce® 7752).

4.2.2. Failure modes

The numerical failure modes were analysed using the stiffness degradation (SDEG) variable associated to the cohesive elements representative of the adhesive layer and interlaminar fracture plane within the composite. This variable measures the stiffness loss of the cohesive elements, and it varies between 0 and 1, with the first representing an undamaged cohesive element, and the last representing a failed cohesive element. During the numerical analysis, two types of failure were registered, i.e., cohesive failure of the adhesive layer (Figure 8a) and interlaminar failure of the composite close to the adhesive layer (Figure 8b).

Figure 8. Numerical predictions of cohesive failure in the adhesive layer of a joint bonded with the adhesive Araldite® 2015, R = 1000 mm, L_O = 10 mm and t_p = 1.2 mm (a) and interlaminar failure in the CFRP of a joint bonded with the adhesive Araldite® 2015, R = 1000 mm, L_O = 10 mm and t_p = 2.4 mm (b).

The failure mode analysis involves four input variables: t_P, R, L_O, and adhesive, all with three levels except L_O containing eight. The discussion also includes the approximate length of the process zone at the overlap edges of the adhesive layer at the beginning of overlap edge crack propagation, taken from the models using the CZM elements satisfying 0< SDEG<1. The following conclusions were taken:

Upon analysing all the models, it was found that R has no effect on the failure mode, considering the two possibilities of failure: cohesive in the adhesive layer and interlaminar. All failures were cohesive in the adhesive layer;

t_P has some degree of influence on the failure mode. In brittle to moderately ductile adhesives (Araldite® AV138 and Araldite® 2015), for each t_P the failure mode kept unchanged throughout the entire L_O spectrum. For the adhesive Araldite® AV138, all joints with t_P = 1.2 mm and 2.4 mm presented cohesive failure in the adhesive layer, whilst those with t_P = 3.6 mm showed interlaminar failure. Due to the brittleness of this adhesive, the process zone length was approximately 0.2 mm for all models, i.e., close to t_A, for both cohesive and interlaminar failures (in which the adhesive also showed damage before interlaminar crack propagation). For the adhesive Araldite® 2015, all joints with t_P = 1.2 mm failed cohesively in the adhesive layer, while all joints with t_P = 2.4 mm and 3.6 mm showed interlaminar failure. In the joints with t_P = 1.2 mm, the process zone length reduced from a maximum of 4.5 mm (L_O = 10 mm) to a minimum of 1.5 mm (L_O = 80 mm), thus showing that the increase of peak stresses with L_O tends to further concentrate damage at the overlap edges. Nonetheless, the process zone is significantly bigger than t_A for all L_O, since this adhesive is moderately ductile. In the joints with higher t_P, showing interlaminar failure, adhesive damage between 0.2 and 0.4 mm was also found;

An interaction between L_O and t_P was found for the joints bonded with the adhesive Sikaforce® 7752, as shown in the failure plot of Figure 9. Although the shorter L_O present a failure in the adhesive layer whilst the longer L_O show interlaminar failure, the amount of L_O cases showing the former reduces as t_P increases. In consequence, for t_P = 3.6 mm, only the joint with the shortest L_O shows cohesive failure in the adhesive layer. In the cohesive failures of the adhesive, typically damage extended through the entire L_O, which is related to the high ductility of the adhesive. In the interlaminar failures, before composite failure the adhesive layer showed damage in a length between 2.5 and 5.5 mm.

Figure 9. Failure plot of the joints bonded with the adhesive Sikaforce® 7752 per t_P and L_O (dimensions in mm).

4.2.3. Maximum load

Figure 10 represents P_m as a function of L_O for all adhesives considering t_P = 1.2 mm (a), 2.4 mm (b) and 3.6 mm (c), and a fixed R = 1000 mm. The analysis of the presented data shows that P_m increases with L_O, although it tends to stabilize once a certain value is reached. This result agrees with common knowledge regarding flat SLJ, and is mostly related to the increase of shear-resistant area of the adhesive, although peak stresses tend to increase as well with L_O.^[40] The smaller P_m variation for t_P = 2.4 and 3.6 mm (Figure 10(b,c) relates to the predominantly interlaminar failure for higher t_P, leading to joint failure before the full adhesive strength can be developed, since peak stresses reduced, as reported in section 4.2.1. On the other hand, for the adhesives Araldite® 2015 and Sikaforce® 7752, t_P = 1.2 mm mostly leads to cohesive failures and improved P_m compared to higher t_P, since these adhesives can better absorb peak stresses due to their ductility. Between adhesives, the analysis is divided into t_P = 1.2 mm, and t_P = 2.4 and 3.6 mm, due to the similar results.

Figure 10. P_m vs. L_O comparison between the three adhesives for t_P = 1.2 mm (a), 2.4 mm (b), and 3.6 mm (c), considering fixed R = 1000 mm.

For t_P = 1.2 mm, the P_m improvements between L_O = 10 and 80 mm correspond to 219.1%, 170.6% and 295.6% for the adhesives Araldite® AV138, Araldite® 2015 and Sikaforce® 7752, respectively. Thus, despite the increase of peak stresses depicted in section 4.2.1 for higher L_O, the increase of bonding area superimposes to this effect and enables a higher absolute performance. The best results were found for the adhesive Araldite® 2015, followed by the adhesive Sikaforce® 7752, while the adhesive Araldite® AV138 results in the lowest P_m for most L_O. The failure modes were cohesive, although for the adhesive Araldite® 2015 and L_O≥50 mm interlaminar damage was also found, which limited P_m. In this particular adhesive, minor reductions in P_m were found for higher L_O, due to the observed effect of increasing peak σ_y and τ_xy stresses acting on the interlaminar layers, for which this adhesive revealed to be particularly sensitive to due to the interlaminar layer/adhesive vicinity. The same happened for the adhesive Sikaforce® 7752 for L_O≥40 mm, although always with a minor P_m improvement. Thus, for the adhesive Araldite® 2015 and t_P = 1.2 mm, the peak P_m at L_O = 40 mm is related to the modification of the failure mode. Since most failures are cohesive, the adhesive characteristics has the most effect on P_m: the adhesive Araldite® 2015 combines acceptable strength with moderate ductility, thus giving the best results. The adhesive Araldite® AV138 is strong but brittle, being highly affected by the peak stresses described in section 4.2.1. The highest P_m was thus found for the adhesive Araldite® 2015 and L_O = 40 mm, and for this L_O the difference between adhesives is highest. For these conditions, the other adhesives showed a P_m percentile difference of 22.3% (adhesive Sikaforce® 7752) and 49.7% (adhesive Araldite® AV138). However, these differences clearly diminish for smaller and higher L_O, reaching a minimum for L_O = 10 and 80 mm.

For t_P = 2.4 mm and 3.6 mm, the P_m tendencies with L_O and absolute magnitudes are similar. The P_m improvements with L_O using the adhesives Araldite® AV138 and Araldite® 2015 are reduced and identical, and the relative differences between limit L_O are only ≈35%. This similarity happens for two reasons. The first is related to the failure mode and explains the behaviour of the adhesive Araldite® 2015. Since the mode of failure in these joints is exclusively interlaminar, the full capabilities of this adhesive are not taken advantage of. The second reason relates to the inherent properties of the adhesive. Since the adhesive Araldite® AV138 is brittle, as soon as the elastic limit is reached, the adhesive fails. The adhesive Sikaforce® 7752 is less strong and, thus, due to the cohesive failure, for the smallest L_O it falls short of the other two adhesives (by ≈39%). However, despite the change to interlaminar failure for L_O = 30 mm (t_P = 2.4 mm) or L_O = 20 mm (t_P = 3.6 mm), it manages to attain higher P_m for 30≤ L_O≤80 mm (relative difference of ≈16%). This difference is related to the higher compliance of this adhesive, leading to a larger extent of interlaminar damage than for the other two adhesives. Between limit L_O, the relative improvement for this adhesive was nearly 210% for both t_P addressed. Thus, in the event of interlaminar failures, a compliant adhesive is recommended.

Figure 11 presents the P_m vs. L_O curves for the adhesive Araldite® 2015 and t_P = 1.2 mm as a function of R. Apart from the aforementioned behaviour related to interlaminar damage for L_O≥50 mm, added to the cohesive failure, the R effect is negligible for this adhesive. Between limit L_O, the percentile P_m improvement was between ≈160 and 170%. This ratio increased for the other two adhesives, whose failure mode was always the same for all L_O (≈210-225% for the adhesive Araldite® AV138 and 280–295% for the adhesive Sikaforce® 7752, disregarding R). Thus, it can be concluded that this geometric variable does not provide noticeable differences for the tested design values, despite small differences in the peak stresses (section 4.2.1).

Figure 11. P_m vs. L_O comparison for different R, considering fixed adhesive (2015) and t_P = 1.2 mm.

4.2.4. Dissipated energy

Figure 12 presents U as a function of L_O the three adhesives studied, with a fixed R = 1000 mm and t_P = 1.2 mm (a), 2.4 mm (b) and 3.6 mm (c). The magnitude of U depends on both P_m and failure displacement, since it is measured by the absorbed energy up to failure, thus being calculated by the area beneath the P-δ curve. Measurement of U was accomplished by the P-δ data, considering the division of the area beneath the curves into bars, defined by the product of P with difference of consecutive δ points, and respective summation to estimate U. Since all P-δ curves were built with at least 100 data points up to P_m, it is considered that this technique provides accurate U estimations. Similarly to the P_m analysis, U has a marked tendency to increase with L_O, but it tends to stabilize at a certain value. This behaviour suggests that increasing L_O has a decreasing influence on the gain of U, and the interlaminar failures failure or partial damage found for higher L_O and some adhesives also affected U. The shape of the U vs. L_O curve of the adhesive Araldite® 2015 for t_P = 1.2 mm, similarly to the P_m vs. L_O curve, is explained by interlaminar damage for L_O≥50 mm. The detailed analysis is divided into t_P = 1.2 mm, and combined discussion of t_P = 2.4 and 3.6 mm, whose results are very similar.

Figure 12. U vs. L_O comparison between the three adhesives for t_P = 1.2 mm (a), 2.4 mm (b), and 3.6 mm (c), considering fixed R = 1000 mm.

For t_P = 1.2 mm, there are considerable differences with L_O. The adhesive Araldite® AV138 steadily increases U between limit L_O, with a respective difference of 659.7%. The joints bonded with the adhesive Araldite® 2015 peak for L_O = 40 mm. The U reductions for bigger L_O are related to the onset of interlaminar damage accompanying the cohesive failure responsible for joint fracture, which limits the failure displacement of the joints. The U improvements between L_O = 10 and 40 mm, and between limit L_O, were 533.4% and 389.7%, respectively. The adhesive Sikaforce® 7752 failed cohesively in the adhesive layer up to L_O = 30 mm, while bigger L_O led to interlaminar failure. Thus, U was kept approximately constant for L_O≥40 mm. The U relative variation between L_O = 10 and 40 mm was ≈792.3%, and it was kept nearly equal up to the biggest L_O. Compared to the adhesive with the best overall performance (adhesive Araldite® 2015), the other adhesives presented an offset of ≈60% for L_O = 10 mm, 74.6% (adhesive Araldite® AV138) and 42.1% (adhesive Sikaforce® 7752) for L_O = 40 mm, corresponding to the peak U, and finally ≈25% for L_O = 80 mm.

For t_P = 2.4 and 3.6 mm, the U behaviour is essentially similar as a function of L_O and between adhesives. The adhesives Araldite® AV138 and Araldite® 2015 show a similar and better behaviour for L_O = 10 mm (≈110% higher than the adhesive Sikaforce® 7752), which is related to the higher P_m (Figure 10). However, due to the ductility of the adhesive Sikaforce® 7752, P_m quickly surpasses the other two adhesives, which then reflects on U. For L_O = 20 mm, the three adhesives have a comparable performance, while for L_O≥30 mm the adhesive Sikaforce® 7752 clearly behaves better and surpasses the adhesives Araldite® AV138 and Araldite® 2015. Considering L_O = 30 mm, the adhesive Sikaforce® 7752 improves U of the other two adhesives by ≈54%, and this difference does not significantly vary up to L_O = 80 mm. Thus, for higher t_P, a compliant although less strong adhesive is recommended for high energy absorption.

Figure 13 shows U vs. L_O for the adhesive Araldite® 2015 and different R. Typically, U increases with L_O up to 40 mm and then slightly reduces, due to the harmful effect of interlaminar damage, and it increases with R. The U variation with R variation is marginally related to P_m (Figure 11), which is little affected by R. Actually, it is mostly affected by the failure displacement of the joints, showing a marked increase with the adhesive’s ductility. Between R, the best U results are always found with R = 3000 mm. The relative improvements for L_O = 10 mm are 199.9% (R = 1000 mm) and 52.3% (R = 2000 mm), for L_O = 40 mm are 164.9% (R = 1000 mm) and 40.8% (R = 2000 mm), and for L_O = 80 mm are 174.2% (R = 1000 mm) and 49.6% (R = 1000 mm). Thus, in relative terms, no significant differences were found. For the other adhesives, the same quantitative differences exist between different R. However, the adhesive Araldite® AV138 shows a steady increase of U with L_O for all R, while the adhesive Sikaforce® 7752 shows a stabilization for L_O≥40 mm.

Figure 13. U vs. L_O comparison for different R, considering fixed adhesive (2015) and t_P = 1.2 mm.

4.2.5. Discussion

The specific set of joint configurations/loading type analysed in this work finds no match in the literature, namely regarding the combination of composite material, including possibility of interlaminar failures, joggle geometry, and parametric study on the geometry and adhesive type, modelled by CZM. Nonetheless, few works can be found on the topic of curved bonded joints, in which few results can be compared. In the work of Taib et al,^[15] a FEM stress analysis was conducted on bonded joints between glass composites. The curved JLJ was one of the evaluated types of joint under tensile pulling, and the same stress asymmetry between the two overlap edges, with corresponding higher magnitude near the joggle. Additionally, the R effect was also tested, and it was found to have a negligible effect on P_m. Liu et al.^[28] addressed curved SLJ and JLJ CFRP joints under tensile loads, using the CZM technique to simulate cohesive failure of the adhesive. R showed a negligible effect on P_m, especially for R > 2000 mm, which agrees with the findings of the present work. The JLJ geometry led to smaller σ_y peak stresses and lower strain energy release rate at the joggle region. The more recent work of Correia et al.^[29] evaluated, by CZM, the tensile behaviour of curved overlap joints in CFRP structures, considering the adhesives Araldite® AV138 and Araldite® 2015 studied in this work, and also a high-strength and elongation polyurethane. Failure possibilities included cohesive failure of the adhesive layer and interlaminar failure close to the adhesive. The stress distributions were found to be asymmetric, with higher stresses at the adhesive edge corresponding to the end of the inner adherend. A strong L_O dependency was found, especially for ductile adhesives, in agreement with the findings of this work. Moreover, t_P reduced P_m and R improved strength, but for R > 2000 mm the different was not relevant. Thus, despite the different loading type, some qualitative tendencies obtained by varying the geometry and adhesive are consistent to the inner pressure loading case studied in this work.

5. Conclusions

This work presented a numerical parametric study of curved JLJ in CFRP structures and subjected to internal pressure. Adhesives with distinct ductility were evaluated and their performance assessed under different geometric configurations, including variation of L_O, t_P and R. A CZM validation process with experiments was initially undertaken for flat SLJ and varying L_O. The failure mode depended on both adhesive and L_O. Additionally, P_m was highly dependent on L_O. The triangular CZM approach revealed accurate for brittle and not too ductile adhesives. The parametric study was divided into four features, leading to geometrical and material guidelines for the design of curved joints:

• Elastic stresses – Major peak σ_y and τ_xy stresses were found, mostly at x/L_O = 0 (at the joggle curvature). Both σ_y and τ_xy stresses are negligible in magnitude at the inner overlap. The adhesives’ stiffness had a major effect on peak stresses, since stiffer adhesives such as the adhesive Araldite® AV138 led to higher peak values. In all cases, peak stresses highly increased with L_O (linearly), moderately diminished with t_P, and little diminished with R;

• Failure modes – Joint failures were either cohesive in the adhesive layer or interlaminar, depending on the selected adhesive, L_O and t_P. For the adhesives Araldite® AV138 and Araldite® 2015, the L_O effect was negligible, and failures were divided by t_P: small t_P led to cohesive failures and high t_P to interlaminar failures. For the adhesive Sikaforce® 7752, a joint L_O/t_P effect was found: higher t_P increase the preponderance of joints with interlaminar failure, but small L_O had topical cohesive failures in the adhesive;

• Joint strength – L_O showed a significant effect on P_m, especially for small t_P. For t_P = 1.2 mm, the adhesive Araldite® 2015 provided the highest P_m for all L_O. However, P_m stabilized for L_O≥50 mm due to concurrent interlaminar damage. Higher t_P reduced P_m due to the appearance of interlaminar failures. For these conditions, the adhesive Sikaforce® 7752 is mostly recommended due to its ability to spread stresses. The R effect is minimal for the range of tested geometries;

• Dissipated energy – The U output has large similarities with P_m, since it results from a combined effect of P_m and failure displacement. Thus, the general comments on the joint strength are applicable, although for high t_P, the U difference between the adhesive Sikaforce® 7752 and the less performing adhesives is larger on account of the higher failure displacements, added to the higher observed P_m.

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Additional information

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.