Numerical analysis of the structural performance of adhesive T-joints under bending loads

Numerical analysis of the structural performance of adhesive T-joints under bending loads L.M. Ferreira a,b,* , R.D.S.G. Campilho c,d , M. Mu˜ noz-Reja b , P.N.B. Reis e a Grupo de Elasticidad y Resistencia de Materiales, Escuela T´ ecnica Superior de Ingeniería, Universidad de Sevilla, Camino Descubrimientos, S/N, Sevilla, 41092, Spain b Escuela Polit´ ecnica Superior, Universidad de Sevilla, C/ Virgen de ´ Africa, 7, Sevilla, 41011, Spain c School of Engineering, Polytechnic of Porto, CIDEM, ISEP, R. Dr. Ant´ onio Bernardino de Almeida, 431, 4200-072, Porto, Portugal d INEGI—P´ olo FEUP, Rua Dr. Roberto Frias, 400, 4200-465, Porto, Portugal e Department of Mechanical Engineering, CEMMPRE, ARISE, University of Coimbra, 3030-788, Coimbra, Portugal ARTICLE INFO Keywords: Adhesive joint Structural adhesive T-joint Finite element method Cohesive zone models ABSTRACT T-joints are a good alternative to conventional adhesive joints, such as single-lap joints, due to their ability to efficiently transfer bending, compressive, shear, and tensile loads between the T-profile stiffener and the base plate, all while maintaining cost-effectiveness. However, for widespread adoption, a comprehensive understanding of the joint strength and damage mechanisms is essential. Therefore, this study aims to numerically analyse the structural performance of adhesive T-joints subjected to 3-point bending loads using two-dimensional (2D) finite element models, and cohesive zone modelling (CZM) to simulate the behaviour of the adhesive. CZM validation was also accomplished by comparing the joints’ behaviour with experimental data. Numerically, peaks of peel stresses at the overlap edges were observed, with sharp gradients toward the inner bond region, as well as shear stresses due to the geometric discontinuity. Adhesives with higher stiffness provide higher peak values of peel and shear stresses, while more ductile ones lead to higher peak loads. The CZM was successfully validated with experiments, and a correlation was observed between energy dissipation and damage propagation, according to a non-symmetrical pattern, and for conditions that minimize the system’s functional energy. 1. Introduction The industrial sector is increasingly replacing traditional joining methods with adhesive joints due to their enormous advantages. In this context, single lap joints are the most used due to their simplicity and low cost [1,2], but they are limited to their preferred loading mode, i.e., tensile loading in the longitudinal direction. Therefore, T-joints are a good alternative because they can transfer bending, compressive, shear, and tensile loads between the T-stiffener profile and the base panel at very economical prices [3]. In this context, it is not surprising that its use has already spread to the aeronautical, automotive, and maritime sectors [4–6]. Nevertheless, for widespread use to be possible, a comprehensive understanding of joint strength and damage mechanisms is needed. With this purpose, experimental analysis or analytical/numerical modelling can be undertaken. Numerical modelling is particularly helpful for the design process of adhesive joints by quickly analysing the effect of geometrical and material variations on different responses, such as strength or stiffness [7]. A spectrum of methodologies has emerged to predict static strength, each with distinct principles, advantages, and limitations, aimed at assessing stresses, strains, and predicting maximum loads and failure paths. Continuum mechanics approaches are common to estimate stresses within adhesive layers and predict joint strength. However, challenges arise due to stress singularities at sharp corners, impeding stress convergence with reducing element sizes [8]. Recent innovations propose novel failure criteria, such as the Critical Longitudinal Strain (CLS) and Critical Normal Strain (CNS) criteria, offering precise results for specific joints, despite limitations [9]. Conventional fracture mechanics, distinct from continuum mechanics, deals with discontinuities in materials. Stress Intensity Factors (SIF) and Strain Energy Release Rates (SERR) form the primary methodologies, with studies like the Virtual Crack Closure Technique (VCCT) yielding promising results for brittle adhesives [10]. Damage mechanics simulate adhesive degradation by progressively reducing stiffness in finite * Corresponding author. Grupo de Elasticidad y Resistencia de Materiales, Escuela T´ ecnica Superior de Ingeniería, Universidad de Sevilla, Camino Descubrimientos, S/N, Sevilla, 41092, Spain. E-mail address: [email protected] (L.M. Ferreira). Contents lists available at ScienceDirect International Journal of Adhesion and Adhesives journal homepage: www.elsevier.com/locate/ijadhadh https://doi.org/10.1016/j.ijadhadh.2025.104064 Received 26 December 2024; Received in revised form 1 May 2025; Accepted 12 May 2025 International Journal of Adhesion & Adhesives 141 (2025) 104064 Available online 13 May 2025 0143-7496/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). elements until complete failure. The use of the Finite Element (FE) method combined with damage mechanics offer efficient and accurate results by replicating arbitrary crack paths in materials [11]. Cohesive Zone Modelling (CZM) emerges as a robust technique for static fracture modelling of adhesive joints. Provided that a minimum mesh refinement is provided at the crack propagation regions, this technique excels in mesh-independent strength predictions by utilizing cohesive laws based on mixed-mode stress criteria for damage initiation and crack propagation [12–14]. In enriched CZM, instead of a simple linear relationship, the models can incorporate more complex relationships, such as non-linear material behaviour and the influence of external factors (e.g., fluid pressure or confining stresses). The enriched formulation is often more accurate in simulating complex scenarios involving fracture, delamination, and other failure mechanisms, especially when the standard model’s assumptions are not valid [15]. The Extended FE Method (XFEM) utilizes an enriched displacement field near the crack, demonstrating advantages over CZM in mesh-independent crack propagation. Studies on stepped-lap joints and single-lap joints showed XFEM’s potential in accurate load predictions and crack path determination [16–19]. Meshless methods, despite not being extensively explored in the context of bonded joints, show promise in fracture modelling. These methods connect nodes using influence domains instead of traditional FE elements, as seen in recent applications [20]. Furthermore, to complete the landscape of available modelling strategies, approaches based on Finite Fracture Mechanics (FFM) and Phase-Field Modelling (PFM), as well as hybrid PF–CZM formulations, have emerged as powerful alternatives. These methods provide robust tools to simulate both the nucleation and propagation of damage in the bulk and at interfaces, with recent studies demonstrating excellent agreement with experimental results in delamination, interfacial and cohesive failure scenarios [21–23]. When comparing the strength between adhesive and mechanically fastened T-joints, for example, Chaves et al. [24] found that the first ones have equal or better mechanical response than the traditional mechanically fastened T-joints. Moreover, by adding spew fillets to the ends of the overlap, adhesive T-joint strength can be significantly improved. According to Shenoi and Hawkins [25], the fillet radius and the thickness of the overlaminate are the most important parameters to increase the load-bearing capacity of the joint, while the gap between panels and the edge preparation of the T-component are the least decisive. Studies developed by Dodkins et al. [26] showed that the overlamination and the fillet radius are critical variables because, in the first case, they determine the joint performance of the joint, while the fillet radius determines the load-bearing capacity of the joint. Nevertheless, the literature is consensual that the loading mode determines the most important variables, as well as the stiffness, strength, and failure region of the T-joint [27–30]. Serra et al. [31] studied adhesive T-joints involving different adherends and adhesives and observed that, in all cases, the cracks appeared at the end of the T-element and propagated toward the interior of the T-joint. Since this study was carried out in bending mode, the simultaneous action of tensile peel and shear stresses led to cracking at the end of the T-element [32–34], due to the higher peel stress concentrations in this region [35]. Zhan et al. [36] found that the stress field, stress concentrations, and load capacity are strongly influenced by the geometry of the bondline and, for example, the average failure load increases as the average bond area increases. Ferreira et al. [37], using the CZM approach, observed that the maximum load increases by 94.1% when the flat adherend thickness increases from 1 to 4 mm, decreases by 27.3% when the stiffener thickness increases from 0.5 mm to 2.5 mm, increases by 94.1% when the overlap length increases from 10 to 20 mm, and increases by 135.4% when the curved deltoid radius increases from 1 to 3 mm. These benefits were explained by the authors due to the favourable rearrangement of the stress distributions in the joint. By combining CZM with the XFEM approach, Ma et al. [38] were able to show that the crack propagates vertically toward the skin, creating an L-shaped separation between the two elements. Subsequently, when the fracture reaches the skin, debonding occurs at the stringer-skin contact and progresses in both directions until the final rupture. This damage process began at the site of maximum principal strain, whose region encompassed both the filler and the skin region. According to the described literature review, to guarantee the structural integrity of the T-joint, it is crucial to estimate the stress and strain fields so that designers can accurately predict the strength of a Tjoint. Therefore, the aim of this work is to numerically study, using FE models with a CZM approach, the mechanical behaviour of adhesive Tjoints involving different adhesives and subjected to 3-point bending (3PB) test, which mainly induces peel stresses in the adhesive layer bonding the parent base plate to the T-stiffener profile. The numerical data is initially validated with experimental results for aluminium-alloy adherends bonded with two structural adhesives. The numerical twodimensional (2D) static analysis employs CZM elements with a triangular law to simulate base plate/T-stiffener profile decohesion. Initially, peel and shear stresses are taken at different stages of loading, for understanding of the failure process, followed by static strength prediction, and energy dissipated at failure. 2. Experimental procedure Experimental evaluation of adhesive T-joint mechanical performance was carried out through static 3PB tests. The T-joints were produced with a base plate and a T-profile stiffener, both manufactured from an aluminium alloy of grade 6063-T5, sourced from Alu-Stock in Madrid, Spain. Fig. 1 provides a representative photo of the specimen and testing apparatus (Fig. 1a), and of the respective schematic representation of the adhesive T-joint assembly (Fig. 1b), showcasing the dimensions of its constituent parts and of the testing apparatus. It should be noted that the consistency of the results was ensured by the quality of the specimens, which in dimensional terms was ensured by the standardized dimensions of the profiles (assured by the supplier) and, when these had to be cut to specific dimensions (length of the base plate and width of the T-profile stiffener), the values never exceeded a maximum standard deviation of 1%. Prior to the bonding process, the surfaces of the aluminium parts that contact with the adhesive were treated by subjecting them to abrasive preparation using P220 silicon carbide paper. Following the abrasive preparation, the surfaces underwent a cleansing process involving dry air and alcohol. This process intends to remove any potential contaminants and weak boundary layers. It should be emphasized that this methodology maintains the chemical composition of the treated surfaces unaltered [1,39–42]. Two distinct adhesive materials were employed for the bonding process, namely, the "Araldite® AV 4076-1 resin/HY 4076 hardener" and the "Araldite® AW 106 resin/HV 953 U hardener". These adhesives exhibit markedly different mechanical behaviours. The former displays a brittle behaviour, whereas the latter exhibits a ductile behaviour. More information about the mechanical and chemical properties of these structural adhesives can be found in the materials technical data sheets [42,43]. To maintain a consistent bondline thickness, a uniform pressure of 0.11 MPa was applied for 16 hours and at 40◦C by black metal dovetail clips to all the T-joint specimens during the adhesive curing process, leading to an average bondline thickness of 85 μ m, with a standard deviation of about 4,28 μ m, which corresponds to about 5% of the average thickness and is within desirable values for many industrial applications. Importantly, this value was obtained prior to testing and using post-testing, employing the measurement capabilities of a Mitutoyo Surftest® SJ-500 surface measuring instrument, sourced by Kawasaki, Japan. This approach is consistent with established practices, as corroborated by the literature [44–46]. The static 3PB tests were carried out at room temperature, employing a universal testing machine, Shimadzu AGS-X procured by Shimadzu, Japan. These tests utilized a 100 kN load cell and a consistent L.M. Ferreira et al. International Journal of Adhesion and Adhesives 141 (2025) 104064 2 displacement rate of 5 mm/min was maintained for the crosshead. A total of 6 specimens were subjected to testing, with 3 specimens dedicated to each adhesive type. 3. Numerical models Based on the static 3PB tests conducted on the adhesive T-joints, as detailed in Section 2, 2D models were developed using the ABAQUS® FE software [47]. A geometrically non-linear static analysis was subsequently performed. The aluminium adherend parts, comprising the base plate and a T-profile stiffener, were modelled using 4-node plane strain elements (CPE4 ABAQUS® element). The adhesive, on the other hand, was modelled using 4-node two-dimensional cohesive elements (COH2D4 ABAQUS® element) with a constant element thickness of 85 μ m, and a length of 0.1 mm. The adhesive layer thickness of 85 μ m corresponds to the actual geometrical separation between adherends and was also used as the effective thickness in the definition of the initial stiffness of the traction–separation law within the cohesive elements. It should be noted that a mesh sensitivity analysis was conducted in [14], where average element lengths ranging from 0.05 mm to 0.4 mm were evaluated at the adhesive bond of a single lap joint. The results demonstrated that the influence of mesh size on the numerical strength predictions was negligible, with a maximum variation of 0.4%. Additionally, the crosshead and the supports were simulated using discrete 2-node linear rigid elements (R2D2 ABAQUS® element). It should be noted that all the considered elements feature two degrees of freedom per node (translational displacements in the X and Y directions), linear interpolation and full integration. Fig. 2 presents the complete FE model, including detailed views of the mesh discretization. As shown, quadrilateral elements were employed throughout the model. Particular attention was given to the adhesive layer at the overlap edges (contact between the base plate and the T-profile stiffener), as well as the regions in contact with the supports and the crosshead. In these areas, a finer mesh (elements with approximately 0.1×0.1 mm 2 ) was utilized to accurately capture the significant stress variations that occur due to their nature as theoretical singularity Fig. 1. a) Representative photo of the specimen and testing apparatus; b) Schematic representation of the 3PB test with the geometric parameters of the specimens. L.M. Ferreira et al. International Journal of Adhesion and Adhesives 141 (2025) 104064 3 points [48,49]. A coincident mesh was used at the interfaces between the cohesive layer and the adjacent adherends to ensure proper stress transfer and numerical stability. The imposed boundary conditions served to reproduce the experimental testing setup presented in Fig. 1. In this way, both reference points of the supports were constrained on all displacements and rotations (ux=uy=rz=0). Additionally, a displacement uy with a ramp amplitude was applied to the crosshead. A CZM approach was employed to simulate the behaviour of the adhesives, utilizing a single row of cohesive elements as proposed by Campilho et al. [50]. This implementation is shown in “detailed view A ″ of Fig. 2. Consequently, it incorporates a bilinear (triangular) traction-separation constitutive model between each set of paired nodes, which delineates how the tractions ( τ )along the interface evolve with separation (δ), as illustrated in Fig. 3. The shape of the cohesive law can be customised to replicate the behaviour of the interface it represents [51]. Actually, it is known that highly ductile adhesives, for instance, are best modelled using a trapezoidal CZM law. Nonetheless, in this work, a bilinear (triangular) law was applied due to two distinct reasons: (1) trapezoidal laws with mixed-mode coupling are not widespread in commercial software, which hinders their direct use by practitioners and industrialists, and (2) previous studies showed that using triangular laws with ductile adhesives actually is not the best choice, but the maximum load prediction errors are typically below or near to 10%, since failure is ruled by an area/energetic concept [52]. The initial elastic linear response (k)can be defined by the stiffness parameters of the thin adhesive layer, given by the axial and shear elastic moduli (E and G, respectively) [50,53]. The initial linear elastic response (k) was defined based on the mechanical properties of the adhesive layer, considering an effective adhesive thickness t=85 μ m. The normal and shear stiffness values were computed as kn=E/t and ks=G/t where E and G are the Young’s and shear moduli of the adhesive, respectively, and t is the physical thickness of the adhesive layer. Although this formulation neglects potential finite-thickness effects that may influence the effective peel stiffness, it was adopted for consistency with previous studies and yielded satisfactory agreement with the experimental results. A nominal stress-based quadratic failure criterion was used to predict damage initiation, expressed as [47], {〈 τ n〉 τ 0 n}2 +{ τ s τ 0 s}2 =1 (1) where τ n and τ s denote the normal and shear contact stresses of the cohesive layer, respectively, while τ 0 n and τ 0 s represent their corresponding strengths. The Macaulay brackets <> indicate that compressive stresses do not initiate damage [55]. Once the cohesive strength is attained, the damage evolution is controlled by the fracture energy (Gc), representing the area under the bilinear traction separation law. Predictions of complete separation under mixed-mode loading are based on a power-law formulation of the energies required for failure in pure modes, expressed as [47], Gn Gc n +Gs Gc s =1 (2) where Gn and Gs denote the strain energy release rates under tension and shear, respectively, while Gc n and Gc s signify the corresponding critical strain energy release rates. The material properties required to characterize the CZM for the two adhesives, AV 4076-1 and AW 106 in ABAQUS®, in ABAQUS®, are listed in Table 1. The Young’s modulus E, shear modulus G, and adhesive layer thickness t =0.085 mm (equivalent to 85 μ m) were used to compute the initial normal and shear stiffnesses as kn=E/t and ks= G/t. The peak tractions τ 0 n , τ 0 s , and critical fracture energies Gc n, Gc s were directly assigned based on literature and experimental data. The complete set of six parameters (kn, ks, τ 0 n, τ 0 s, Gc n, Gc s) was introduced directly into ABAQUS®. The final opening displacements δf were not defined Fig. 2. Finite Element model of the T-joint with detailed views of the mesh discretization. Fig. 3. Bilinear traction-separation law of the cohesive elements [44,51]. L.M. Ferreira et al. International Journal of Adhesion and Adhesives 141 (2025) 104064 4 explicitly but rather computed internally by the software to ensure consistency with the specified fracture energies in each mode. As such, the parameter set is energetically consistent within the standard bilinear cohesive formulation implemented in ABAQUS®. The aluminium 6063-T5 adherends were modelled as a linear elastic isotropic material with the following elastic constants: an elastic modulus Eal =70 GPa and a Poisson’s ratio ν al =0.3. It is important to note that the numerical models did not account for adherend plasticization, as experimental evidence indicated linear behaviour of the Tjoints up to failure. 4. Results The behaviour of the T-joints bonded with the AV 4076-1 and AW 106 under 3PB loading conditions is analysed in this section. It includes the numerical-experimental correlation, stress distribution within the adhesive layer, and the overall energy balance. 4.1. Numerical-experimental correlation Numerical predictions of the static properties, including maximum flexural force, stiffness, and displacement, are validated with the experimental results reported in [31]. Fig. 4 presents a comparison between representative experimental flexural force-displacement curves (whose force was obtained directly by the machine’s load cell) and their corresponding numerical predictions. For both adhesives, the flexural force exhibits a linear increase up to a peak value, which corresponds to the onset of debonding. Notably, the ductile nature of the AW 106 results in higher elongation compared to the brittle AV 4076-1. The discrepancy in peak loads between the two adhesives can be attributed to the ductility of the AW 106, which allows a more effective stress redistribution as the applied load increases [56,57]. This behaviour is further analysed in Section 4.2. Table 2 summarises the static properties, including both experimental and numerically predicted values. For the AW 106, the percentile differences for the maximum flexural force and stiffness are below 1%, indicating a strong numerical-experimental correlation. Although the percentile difference for the maximum displacement is approximately 9.2%, the numerical prediction remains within the range of experimental uncertainties. In the case of the AV 4076-1, despite larger percentile differences, particularly in the maximum flexural force and displacement (4.5% and 8.4%, respectively), the numerical predictions still fall within the range of the experimentally observed data. In Fig. 5, the predicted damage evolution of the AV 4076-1 is compared with experimental observations at various representative load stages, denoted as points A, B, and C in Fig. 4. The numerical predictions indicate no damage in the adhesive up to point A, as evidenced by the CZM stiffness degradation given by the SDEG - Scalar Damage Variable for Energy-based Damage Models ABAQUS® output variable being zero for all elements. Up to this loading stage, the cohesive elements exhibit a linear behaviour, resulting in a linear elastic response of the T-joint, as shown in Fig. 3. Just after point A, damage onset occurs, indicating that the nominal stress-based quadratic failure criterion, as shown in Eq. (1), has reached a value of 1. Between points A and B, damage evolves, leading to a progressive reduction in stiffness and strength based on the specified fracture energies. At point B, corresponding to the peak load P m , the SDEG reaches a value of 1 at the ends of the bond region, indicating complete damage of these cohesive elements. The results further demonstrate that this complete damage initiates symmetrically at both edges. From point B to point C, which corresponds to the failure of the Tjoint, the complete debonding propagates inward, displaying a nonsymmetric behaviour, consistent with the experimental evidence shown in Fig. 6. These damages are representative of all T-joints tested and can be characterized by a mixed adhesive/cohesive failure mode. Cohesive failure occurs when the bond between molecules within the adhesive fails due to a higher external force, whereas adhesive failure happens when the forces acting on the joint are higher than those between the adherend and adhesive, i.e., when the maximum adhesive strain exceeds its limit [58]. More details about the damage mechanism can be found in [31]. Overall, the results underscore a good correlation between the experimental findings and the numerical predictions, demonstrating the accuracy and reliability of the developed FE models in capturing the behaviour of the T-joint (base plate and T-profile stiffener) and of both adhesives under the loading conditions imposed by the 3PB test. 4.2. Stress analysis Peel ( σ y ) and shear stresses ( τ xy ) were analysed at the adhesive layer mid-thickness for the T-joints bonded with both adhesives, with the objective of gaining a detailed understanding of the failure process. Stresses were evaluated at the peak load P m and across various loading stages. In all stress plots, the adhesive layer length is defined as x/L, where x is the length coordinate and L is the bonding length between the base and T adherends, hence 0 ≤x /L≤1. σ y and τ xy stresses at P m for the T-joints bonded with the AV 4076-1 and AW 106 are represented in Fig. 7. To improve the understanding of the results, Fig. 8 illustrates both σ y and τ xy stresses along the T-joint bonded with the AW 106 as an example. Despite the marked peel loading applied to the T-joints, significant shear effects are observed in the adhesive layer due to the shear sliding between the base plate and T-stiffener profile adherends, driven by the bending effect. In this context, the mostly symmetric nature of τ xy stresses at both sides of the applied load is due to the opposite directions of sliding. These τ xy stresses are concentrated at the overlap edges of the bond due to geometric discontinuity at the ends of the bonded reinforcement, with the lowest stress values occurring at the inner bond Table 1 Properties of the adhesives AV 4076-1 and AW 106 used for the CZM [39,40]. Property Units AV 4076-1 AW 106 EGPa 1.40 1.90 GGPa 0.54 0.7 τ 0 nMPa 20 25 τ 0 sMPa 15 20 Gc nN/mm 0.26 0.88 Gc sN/mm 0.52 1.76 knN/mm 3 0.016.1060.22⋅106 ksN/mm 3 0.006.1060.008.106 Fig. 4. Experimental and numerically predicted flexural force-displacement curves for the T-joints with the AV 4076-1 and AW 106. L.M. Ferreira et al. International Journal of Adhesion and Adhesives 141 (2025) 104064 5 region. At this region, the presence of the T-stiffener slightly disrupts τ xy stresses. σ y stresses are slightly lower in magnitude than τ xy stresses, but similarly exhibit peaks at the overlap edges, where the separation between the base plate and the T-stiffener profile is more pronounced. However, unlike τ xy , the gradients of σ y toward the inner bond region are not as smooth, showing distinct peaks. σ y stresses are nearly identical in magnitude across both halves of the bond, maintaining a peel-dominant nature. A slight discontinuity in the inner bond region is also noted, due to the presence of the T-stiffener reinforcement. When comparing adhesives, the higher stiffness of the AW106 relatively to the AV 4076, as shown in Table 1, results in higher σ y and τ xy peak stresses. However, its higher ductility plays a significant role in influencing P m . Figs. 9 and 10 show τ xy (a) and σ y (b) stress distributions during the loading and failure process for T-joints bonded with the adhesives AW 106 and AV 4076-1, respectively. These figures evaluate different P/P m ratios, expressed as percentages, where 100% corresponds to P m . Notably, despite symmetrical loading and geometry of the developed FE models, numerical failure is predicted on one halve of the joint, consistent with the experimental evidence. For both adhesives, the highest percentile value relates to complete debonding (109.1% for the AW 106 and 114.4% for the AV 4076-1). The results show that the P/P m ratio at failure is higher for the brittle adhesive AV 4076-1, which is attributed to the adhesive’s lower stiffness, as detailed in Table 1. The stress evolution for both τ xy and σ y stresses follow a similar quantitative pattern for both adhesives. Peak stresses initially appear at the edges of the adhesive bondline and propagate inward as the load increases, reaching zero on one side of the bondline when the T-joint fails, at which point P rapidly decreases, leaving only the base plate adherend contributing to the T-joint stiffness. Meanwhile, stresses continue to develop at the opposite side of the bondline, as it remains intact. Figs. 9 and 10 clearly illustrate this behaviour, in which the left section of the bond, corresponding to x/L <0.4, shows zero τ xy and σ y stress values at the highest P/P m ratio, while the remaining bondline continues to develop stresses. These results align with the damage progression shown in Fig. 5, where complete debonding occurs at the same side of the specimen. Although the overall behaviour is comparable between adhesives, σ y and τ xy stresses for the AV 4076-1 tend to concentrate closer to the bond edge at P/P m =100% compared to the AW 106. Additionally, during the loading process, the load transfer area, represented by the x/L extension of σ y and τ xy stresses, is larger for the ductile AW 106 than for the brittle AV 4076-1, a characteristic that likely contributes to the higher strength of the T-joint bonded with the AW 106. Identical findings are available in the literature for different joint geometries, such as single-lap joints [59]. 4.3. Symmetric versus non-symmetric delamination In all the experimental tests carried out on the T-joints, a nonsymmetric delamination was observed, as shown in Fig. 5. However, it Table 2 Static properties of the T-joints with the AV 4076-1 and AW 106: Comparison of the experimental data and numerical predictions. Bending Properties AW 106 AV 4076-1 Exp [28]. Num. % Diff. Exp [31]. Num. % Diff. Maximum. Force (N) 1080.1±65.5 1071.4 0.8% 626.7±25.2 598.3 4.5% Maximum. Displacement (mm) 3.6±0.51 3.21 9.2% 1.9±0.13 1.74 8.4% Stiffness (N/mm) 386.3±13.2 390 0.9% 399.6±0.13 386 3.4% Fig. 5. Progressive damage for the T-joints bonded with the AV 4076-1: (a) Numerical predictions. (b) Experimental evidence [31]. L.M. Ferreira et al. International Journal of Adhesion and Adhesives 141 (2025) 104064 6 remains inconclusively whether this non-symmetry arises from the inherent randomness of the problem’s conditions or from the natural presence of minor defects and/or other sources of asymmetry. Previous studies have examined failure configurations in initially symmetric structures, where crack propagation occurs asymmetrically, thus leading to the loss of symmetry of the test [52–55]. These studies demonstrate that asymmetric crack growth tends to be the preferred mode, even when the initial configuration is perfectly symmetric. In this way, the present section aims to numerically analyse two interface delamination configurations, using the system’s energy balance: one symmetric and the other non-symmetric. Fig. 11 illustrates both the symmetric and non-symmetric numerical model configurations for the T-joints with the AV 4076-1, with the corresponding boundary conditions, at the instance where the interface failure is predicted to occur (on the left side). Notably, in the non-symmetric configuration, the length of delamination is defined as "a", corresponding to the total number of damaged interface elements, whereas in the symmetric configuration, the same delamination length is equivalent to half the total failure length, i.e., a/2. During the fracture process, the energy dissipated is supplied to the system by the external loads in the form of potential energy. The interface material undergoes elastic deformation, with a portion of this energy dissipated during the delamination process, assuming no other dissipative processes are present. Consequently, the system’s energy functional comprises of external work (E w ), recoverable strain energy (E s ), and energy dissipated during the delamination process (E d ). These Fig. 6. Typical failure morphologies observed for T-joints involving: a) Al 6063-T5 base plates bonded with AW 106; b) Al 6063-T5 base plates bonded with AV 4076-1. Fig. 7. τ xy and σ y stresses at peak load P m for the T-joints bonded with the AV 4076-1 and AW 106. Fig. 8. τ xy and σ y stresses distribution at peak load P m for the T-joints bonded with the AV 4076-1. Fig. 9. τ xy (a) and σ y stress (b) evolution during loading for the T-joints bonded with the AW 106. L.M. Ferreira et al. International Journal of Adhesion and Adhesives 141 (2025) 104064 7 energy terms correspond to the ABAQUS® outputs ALLWK, ALLSE, and ALLDMD, respectively. Fig. 12 shows the system’s energy functional as a function of total damage propagation along the interface for both the symmetric and nonsymmetric configurations. The energy functional is expressed as the sum of the available potential energy and the dissipated energy, formulated as Es−Ew+Ed. The delamination propagation is defined as the process that minimizes the energy functional through the creation of two new surfaces. As shown in Fig. 12, the system with the lower energy functional corresponds to the non-symmetric configuration. Additionally, Fig. 13 shows the relationship between the applied displacement and the total delamination size for both the symmetric and non-symmetric configurations. The results indicate that the preferential failure configuration is the one requiring a lower critical remote displacement for delamination initiation and propagation, which is associated with the non-symmetric configuration. 5. Conclusions A numerical study was performed to evaluate the structural performance of T-joints composed of aluminium-alloy adherends bonded with two distinct adhesives, AV 4076-1 and AW 106, under 3 PB loading. For this analysis, 2D FE models were generated to replicate the experimental setup. The behaviour of the two adhesives, which exhibit markedly different mechanical properties, was simulated using a CZM approach. Fig. 10. τ xy (a) and σ y stress (b) evolution during loading for the T-joints bonded with the AV 4076-1. Fig. 11. Symmetric and non-symmetric delamination for the T-joints with the AV 4076-1. Fig. 12. Energy functional for the T-joints with AV 4076-1and for symmetric and non-symmetric configurations. Fig. 13. Applied displacement for the T-joints with AV 4076-1 and for symmetric and non-symmetric configurations. L.M. Ferreira et al. International Journal of Adhesion and Adhesives 141 (2025) 104064 8 The numerical predictions showed a good agreement with experimental data, particularly in terms of the flexural force-displacement curves and the progressive damage patterns observed throughout the loading process. σ y and τ xy stresses at the adhesive layer mid-thickness were analysed to better understand the failure process of the T-joints. Stresses were evaluated at P m and at various loading stages. Notable shear effects were observed due to sliding between the adherends, base plate and Tstiffener. Concentrating τ xy stresses were found at the overlap edges of the bond due to geometric discontinuity at the ends of the bonded reinforcement. σ y stresses also showed peaks at the overlap edges but with sharper gradients toward the inner bond region. Comparing adhesives, the AW 106’s higher stiffness led to greater σ y and τ xy peak stress values, while its ductility influenced higher P m . Stress evolution was found to be similar for both adhesives, with peak stresses initiating at the bond edges and progressing inward with increasing load. The numerical study of the energy of the system showed a correlation between energy dissipation and damage propagation: as damage propagates through the adhesive layer, a greater part of the energy is dissipated. Furthermore, it was shown that the preferential failure configuration is the one that minimizes the energy functional of the system, favouring a non-symmetric pattern of damage propagation. Overall, this study demonstrated the effectiveness of FE models using a well-identified CZM approach to predict the mechanical behaviour and damage evolution in adhesive T-joints subjected to bending loads. CRediT authorship contribution statement L.M. Ferreira: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Data curation. R.D.S.G. Campilho: Writing – review & editing, Writing – original draft, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Conceptualization. M. Mu˜ noz-Reja: Writing – review & editing, Writing – original draft, Software, Methodology, Investigation. P.N.B. Reis: Writing – review & editing, Writing – original draft, Supervision, Investigation, Formal analysis. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This research was sponsored by national funds through FCT—Fundaç˜ ao para a Ciˆ encia e a Tecnologia, under the project UIDB/ 00285/2020 and LA/P/0112/2020. This research was partially supported by the Spanish Ministry of Science, Innovation, and Universities, and European Regional development Fund (PID2021-123325OB-I00) Appendix The variability observed in the experimental force–displacement curves for the T-joints bonded with the adhesives AV 4076-1 and AW 106, is shown in Fig. 14. The representative curves of the tested specimens are represented along with the upper and lower limits of the measured data (shaded regions). Fig. 14. Experimental force–displacement curves of the T-joints bonded with AV 4076-1 and AW 106. Data availability Data will be made available on request. References [1] Reis PNB, Antunes FJV, Ferreira JAM. Influence of superposition length on mechanical resistance of single-lap adhesive joints. Compos Struct 2005;67(1): 125–33. [2] da Silva LFM, Critchlow GW, Figueiredo MAV. Parametric study of adhesively bonded single lap joints by the taguchi method. J Adhes Sci Technol 2008;22(13): 1477–94. 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