Comparative Analysis of Different Adhesive Model Representations in Single Lap Joints Using Finite Element Analysis

Abstract

This study addresses an existing gap in the literature by providing a comparative analysis of various adhesive model representation approaches, using cohesive zone models—both local and continuum models. Through a systematic investigation of stress distribution and force–displacement characteristics across different modeling techniques, we reveal the advantages and limitations of each method. This study provides a comparison of various adhesive modeling approaches, including single-row cohesive elements, interfacial elements, middle cohesive elements, and single-row continuum solid elements, highlighting their effects on stress distribution and failure modes in single lap joints across a range of adherend thicknesses and overlap lengths. The findings demonstrate that the choice of modeling techniques yields a similar prediction of failure modes in single lap joints under tensile loading. Consequently, choosing among these methods can be guided by the level of detail in capturing localized damage mechanisms. The results offer a foundation for informed decision making in adhesive modeling, with implications for improving joint design and reliability in real-world applications.

1. Introduction

Adhesive bonding is an important technology utilized in diverse engineering disciplines such as aerospace, automotive, marine, and construction. It has many benefits over various conventional mechanical fastening techniques such as riveting, welding, screws, bolts, and nuts due to the flexibility to join different materials, homogeneous and minimal stress concentration, enhanced fatigue resistance, and lighter weight [1,2,3]. Adhesive joints, when employed in dynamic applications, possess higher fatigue resistance and extended fatigue life compared to traditional joining methods. They also offer light weight, effective bonding of thin and dissimilar materials, reliable sealing, cost-efficient manufacturing, and notable vibration-damping properties [4,5]. The integrity and longevity of adhesive joints are crucial in diverse engineering applications. Single lap joints (SLJs) are the most often utilized type of adhesive joint because of their ease of usage and efficiency in transferring load. However, the ability to accurately forecast how the adhesive will behave under various loading scenarios is crucial to the functionality of these joints [6,7]. Adhesive joints present an attractive substitute; nevertheless, accurately modeling their strength and failure is challenging due to several factors, including the difficulty in testing the adhesive as a standalone material, uncertainties in translating standalone adhesive properties to the bonded configuration, the thinness and varying thickness of the adhesive layer, and the complexity of ensuring proper bonding chemistry and surface preparation. Furthermore, understanding how the adhesive behaves under real-world loading and environmental conditions remains an active area of research.

Finite element (FE) methods utilizing continuum mechanics approaches are commonly used to analyze adhesive joints. The assessment of joint failure usually relies on various criteria for the constituent materials, such as maximum strain, maximum principal stress, and ductile damage. These criteria are not exhaustive and may include additional factors. Despite their widespread use in the past [8], these element-based failure criteria have several drawbacks, including their dependence on mesh size [9,10], perfect adhesive-surface bonding assumptions, and typical usage of bulk properties of the constituent materials as input for modeling.

Cohesive zone (CZ) modeling has become a viable strategy for dealing with these issues. A traction–separation law, which describes the connection between stress (traction) and separation (displacement) across the adhesive contact, is used in CZ modeling to represent the adhesive layer. With a more realistic representation of the initiation and propagation of cracks inside the adhesive layer, this technique offers a greater understanding of the failure mechanisms. The application of CZ modeling to adhesive joint analysis has demonstrated a great deal of promise for improving the joint performance prediction accuracy under different loading scenarios [11,12]. For example, Kafkalidis and Thouless [13] provide a thorough discussion of the mechanics of lap-shear joints using a CZ method.

In addition to taking geometry into account, FE models containing CZ elements also enable the treatment of the interface’s cohesive characteristics and the adherends’ plastic deformation. The localized plastic and damage behaviors along the interface were described by Sun et al. [14] using CZ modeling, and the adhesive thickness is dependent on CZ hardening, which is incorporated into the suggested numerical technique to forecast the adhesive joint failure. Typically, when predicting the failure load in the composite SLJ using CZ modeling, the CZ elements are utilized to simulate the adhesive layer while the composite adherend is modeled using linear elastic orthotropic elements [15,16]. Moya-Sanz EM et al. [16] examined how the shape of the adherend affects the peel and shear stress and the overall failure load of the composite SLJ when subjected to tension.

Campilho et al. [17] conducted a comprehensive study that included both experimental and computational studies to investigate the tensile properties of CFRP single-strap repairs. The study examined various patch thicknesses and overlap lengths. A mixed-mode cohesive damage model was created to simulate the initiation and propagation of cracks accurately in structural repairs. The experimental and numerical findings showed a strong agreement in terms of failure modes, failure load, and equivalent stiffness of the repair. Using experimentation and numerical analysis, Neto et al. [18] investigated the adherend failure and adhesive failure on composite SLJ material. They tested SLJs experimentally using various adhesives and overlap lengths, and they contrasted the experimental findings with basic analytical predictions and numerical predictions.

Kemiklioglu et al. [19] conducted an experimental study to assess the stiffness of adhesives on the impact performance of bonded joints. They examined two types of adhesives: one that was brittle and another that was ductile. The joints were thereafter subjected to multiple impacts at varying energy levels. Although the more compliant and ductile adhesive performed better in the initial impact, the stiffer and brittle adhesive outperformed it in subsequent impacts. Carlberger and Stigh [20] utilized a closed form analytical solution to apply three distinct constitutive models to an epoxy adhesive: a basic model, a model with linear elastic characteristics for the adhesive, and a model with linear softening. Validation of all the models was conducted by comparison with analytical solutions. The researchers discovered that when stiffness values are high, the precision of the Young’s modulus value does not affect the results. Araújo et al. [21] conducted a study on the precise application of adhesive joints in the automotive industry. A flexible epoxy glue was utilized in a CFRP single lap joint with multiple overlap lengths. Numerical and experimental experiments were conducted under both quasi-static and dynamic circumstances, including impact scenarios. The CZ numerical models were verified for both situations. An augmentation in the maximum load was seen when the overlap was increased for both static and impact loadings. Furthermore, it was determined that carbon fiber-reinforced polymer adherends exhibit superior vibration absorption capabilities in comparison to steel adherends.

Various traction–separation law (TSL) shapes have been proposed to represent the interfacial behavior of adhesive layers, each capturing the details of damage initiation and propagation. For instance, bilinear, trapezoidal, and exponential TSLs [22] offer distinct ways to model the adhesive’s brittle or ductile response, influencing the predicted stress distributions and the onset and growth of damage. Although the present study does not explore the geometric placement of cohesive elements or their local versus continuum representations, the chosen TSL shape still significantly impacts the predicted joint performance. In particular, accurately anticipating adhesive debonding, cohesive fracture, or mixed-mode failures depends on selecting a TSL that best reflects the material’s fracture process.

These distinctions are particularly important in experimental tests where these failure modes are frequently observed. Moreover, each adhesive model representation inside the CZ model framework provides distinct perspectives on the mechanical behavior of the joint under different loading circumstances. For example, a bilinear model could effectively capture the initial stiffness and linear softening behavior, whereas a trapezoidal model may more accurately depict the plateau in the stress–strain curve, which indicates plastic deformation before total separation. An exponential model would more accurately represent the progressive decline in stress as separation increases [23]. Through the application of finite element analysis, researchers can gain a deeper understanding of the impact of adhesive characteristics and failure criteria on the overall structural integrity of the joint. More information on the behavior of single lap joints and their numerical simulation can be found in the review articles by He [24], Ramalho et al. [25], Budhe et al. [26], and Karthikeyan et al. [27].

Based on the detailed overview of the above-mentioned studies on how CZ modeling approaches were employed to model various adhesive failures in single lap joints, it can be seen that numerous adhesive models were proposed by the researchers [24], with each compared with relevant experimental results. However, none of them compare the suitability of various adhesive model representations to a lap joint problem because some of the models have limitations. This gap makes it difficult to quantitatively assess the relative suitability of each modeling representation. In this study, a cohesive zone interface model in Abaqus was calibrated and validated using prior experimental and modeling work from [28], ensuring that the constitutive parameters accurately reflect adhesive behavior. Then, this validated model is extended to investigate a broader range of joint geometries by varying adherend thicknesses and overlap lengths to assess how these parameters influence stress distribution, failure modes, and overall joint performance. By systematically analyzing both continuum and cohesive element strategies in this context, this study provides practical guidance on selecting an appropriate modeling approach for specific adhesive systems. This framework also helps bridge the gap between purely numerical analyses and real-world applications, offering a roadmap for optimizing adhesive joints in terms of durability, manufacturing feasibility, and cost effectiveness.

Detailed numerical and experimental validations are conducted to provide a comprehensive assessment of different CZ modeling approaches and identify the most suitable CZ model for specific adhesive types. In addition to numerical simulations, incorporating an analytical approach provides a more comprehensive understanding of the stress distribution in adhesive joints. Analytical models, particularly those that describe peel and shear stresses, offer a simpler yet effective means to validate numerical results, ensuring that the finite element model accurately represents the physical behavior of the system. The classical formulations by Goland and Reissner [29], along with subsequent developments in the literature [30,31,32,33], have laid a strong foundation for analyzing cemented joints. By leveraging these analytical solutions, we can enhance the robustness of the numerical analysis, particularly in predicting failure modes and optimizing joint design. This comparison research not only improves the capacity to forecast numerical simulations but also provides guidance for designing and optimizing adhesive joints to boost durability and performance.

The remainder of the manuscript is organized as follows. Firstly, the geometry and boundary conditions of the single lap joint model are discussed along with the various cases of cohesive zone representation within the adhesive layer that are explored. Secondly, these CZ representations are first validated by comparison to published experiments and then are studied numerically by finite element models in comparison with a simplified analytical model of SLJ. An array of lap joint dimensions is explored to assess the generalizability of the behavior of the CZ representations. Lastly, conclusions are drawn about the comparisons of the CZ approaches.

2. Numerical and Analytical Methods Description

Among various joints configurations, the single lap joint (SLJ) is frequently employed due to its simplicity and effectiveness in load transfer, making it a standard specimen in both experimental and numerical studies, according to the ASTM D1002 [34]. The single lap joint considered in this study represents a common configuration used to evaluate adhesive performance under different loading conditions. Aluminum alloys, such as the one used in this study, are often selected for lap joints due to their favorable strength–weight ratio, corrosion resistance, and widespread application in industries where joint reliability is critical. The chosen aluminum alloy adherend, simulated as an elastic-plastic isotropic material, reflects the material’s typical mechanical properties, ensuring that the findings of this study are relevant to real-world applications. The adhesive used in this study is FM 73M OST, a commercially available film adhesive frequently employed in aerospace and automotive applications for its robust mechanical properties and strong bonding performance with aluminum alloys. Its widespread industrial use reflects a balance of cost, manufacturability, and mechanical performance factors which are crucial for many large-scale engineering applications [28].

In a typical SLJ experiment, adherends (often metallic, composite, or a combination) are bonded using a structural adhesive over a prescribed overlap length, then cured according to the adhesive manufacturer’s specifications—commonly under elevated temperature and pressure, although some adhesives are ambient-cured. After curing, the joints are cut or machined to ensure consistent dimensions and then tested on a universal testing machine such as an MTS or Instron system. The ends of the adherends are clamped in grips with sufficient friction to prevent slippage, and a uniaxial tensile load is applied at a controlled displacement rate until failure [35].

In this study, the commercial software Abaqus (v2019) was used to perform this numerical analysis due to the embedded sophisticated cohesive zone models. The purpose of these analyses is to offer a thorough examination of the behavior of joints and to compare different adhesive model representations and values of overlap length based on the provided criteria. This study used a simulation that was three-dimensional and geometrically non-linear. The elastic properties of the aluminum and the FM 73M OST are provided in

Table 1. Aluminum alloy and adhesive elastic properties.

	Aluminum Alloy (MPa)	FM 73M OST
Elastic modulus	70,000	2000
Poisson’s ratio	0.3	0.38

. The aluminum alloy substrate used was simulated as an elastic-plastic isotropic material, and the longitudinal plasticity parameters are shown in

Table 2. Plasticity data for Al 2024-T3 [36].

Longitudinal		Transverse
Flow Stress (MPa)	Plastic Strain	Flow Stress (MPa)	Plastic Strain
300	0.000	290	0.000
330	0.003	300	0.003
370	0.015	340	0.011
420	0.043	390	0.035
440	0.1	430	0.100

. Longitudinal and transverse labels for the plastic strain versus flow stress-tabulated values refer to the rolling direction of the aluminum plates under consideration [36].

In this study, the SLJ follows the guidelines of ASTM D1002 [34] as depicted in Figure 1, while specific geometrical parameters are varied to assess the consistency of different modeling methods. The adherend thicknesses (t_p) are varied among four values (0.5 mm, 1 mm, 2 mm, and 3 mm) and the overlap (L_o) is investigated at four increments (10 mm, 15 mm, 20 mm, and 25 mm). These variations enable the study of how each adhesive modeling approach responds to a broad range of material and geometric dimensions. Meanwhile, the adhesive thickness, t_A = 0.2 mm, the grip length, L_G = 25 mm, and the adherend length, L_A = 100 mm, are held at constant values according to the standard ASTM D1002 [34]. An adhesive thickness of 0.2 mm was chosen to balance the competing demands of mechanical performance, manufacturability, and practical application. Banea et al. [37] noted that although a 2 mm bond line might not yield the highest fracture toughness, scaling the adhesive layer beyond that thickness often introduces defects and proves impractical for most engineering purposes.

The adhesive, on the other hand, was modeled in four different ways to compare the similarities and differences in capturing various failure modes, in accordance with ASTM D5573-99 [38]. The considered adhesive representations were modeled as single-row cohesive elements (SRCE), continuum single-row solid elements (Continuum), interface cohesive elements between substrate and adhesive (Inter-CZ), and cohesive elements in the middle of the adhesive solid elements layer (Middle-CZ). The models were constructed using eight-node brick elements (C3D8) for the adherends. For the initial elastic stress analysis, a three-dimensional finite element mesh with standard continuum elements was employed to capture the baseline stress distribution in both the adherends and the adhesive layer. Subsequently, for the damage and strength analysis, 8-node cohesive interface elements were introduced at a prescribed distance from the adhesive/adherend interface to simulate the onset and propagation of failure. These cohesive elements, commonly used in fracture mechanics and detailed in the works of Campilho et al. [22,39] and others, incorporate traction–separation laws that govern damage initiation and progression.

To ensure reliable stress distribution and failure predictions in the single lap joint models, a mesh sensitivity study was conducted. Initially, a coarse mesh at the adhesive region (0.5 mm × 0.5 mm × 0.2 mm elements) was used to establish baseline computational cost and to pinpoint regions of highest stress gradients, primarily at the overlap edges. Subsequently, the mesh was incrementally refined in these critical zones and monitored changes in peak stress, strain distribution, and predicted failure loads. Once the refinement produced less than a 3–5% variation in the failure load using 0.2 mm × 0.2 mm × 0.2 mm elements, it was deemed sufficiently converged. Simultaneously, the computational run times were tracked, noting that excessive refinement in low-gradient regions (0.1 × 0.1 × 0.2 mm elements) unnecessarily inflated computational costs without improving accuracy. As a result, the final mesh struck a balance between accuracy and efficiency by using finer elements only in the overlap region and coarser elements elsewhere. This approach follows established guidelines for performing convergence studies and optimizing mesh design in finite element models.

The finite element models were constructed using eight-node brick elements for the adherends and were also considered for the adhesive in places where the adhesive layer is treated as a continuum. Fully integrated elements give better results and capture bending but at a high computational cost. SLJ generally has high stress concentrations at the edges due to the sharp gradients, and 0.2 × 0.2 × 0.2 mm elements were used for the adhesive layer to account for this. In addition, the same mesh size was used across the different overlaps and the different thicknesses of the adherends, resulting in meshes containing on the order of 200,000 elements.

For the adherends, a biased seeding was used towards the overlap region to capture stress concentrations and variations due to the bending because of the applied displacement (Figure 2). The load eccentricity makes the single lap joint bend at the edges of the overlap, which can be attributed to the asymmetry of the joint.

2.1. Description of the Modeling Approaches

The complexity of adhesive behavior under different loading conditions necessitates the use of advanced modeling techniques. This section explores four major adhesive model representations combining both local and continuum approaches as shown in Figure 3. The first approach represents the adhesive layer using a finite-thickness single-row cohesive element configuration, wherein a single layer of cohesive elements (SRCE) is placed between the adherends [40,41]. These elements are assigned a small finite thickness to approximate the real bond line. A bilinear traction–separation law governs the onset and progression of damage, allowing this method to capture post-failure fracture when the local stresses exceed critical values [40]. However, while computationally efficient and straightforward to implement, this model may not fully capture complex through-thickness effects in a thicker adhesive layer.

In the SRCE approach, Abaqus allows the user to specify a finite thickness, enabling the element itself to represent both the bulk adhesive layer and its overall failure mode through a single feature in the model. This finite element configuration incorporates the adhesive’s elastic modulus to reflect the mechanical behavior across the thickness in an average sense. Thus, the effective elastic stiffness within the cohesive zone model is defined directly from the adhesive modulus divided by the actual thickness dimension. By contrast, the continuum single-row element approach models the adhesive as a single layer of 3D elements, to which elastic properties are assigned to reflect the stiffness of the adhesive layer. Without a means to represent plasticity or damage, the continuum single-row element approach serves simply as a reference for the response of the SLJ at low applied loads; this approach is unsuitable for tracking post-failure crack propagation. Where a more refined depiction of fracture is needed, the middle cohesive element strategy (Middle-CZ) inserts a zero-thickness cohesive interface within an otherwise continuum adhesive mesh, facilitating progressive fracture through the adhesive’s interior [39]. This hybrid approach is particularly useful for capturing damage that initiates away from the adherend–adhesive interface or when the adhesive thickness is substantial enough to warrant more detailed modeling.

Finally, the interfacial cohesive element technique (Inter-CZ) locates zero-thickness cohesive layers at the boundaries between the adherends and the adhesive itself, an approach well suited for conditions in which debonding consistently begins at or near the interface [40,41]. While interfacial cohesive methods precisely track fracture at the interface, they can neglect any potential bulk adhesive damage unless cohesive elements are also introduced elsewhere in the bond line. Although each modeling method presents distinct advantages and limitations, their collective application provides a view of adhesive joint performance under various loading conditions, enabling tailored modeling choices that align with specific design objectives and experimental observations.

2.2. Implementation of the CZ Model in Finite Element Analysis

The CZ model represents the initial elastic response of an interface layer until the local stress reaches a critical value, at which point damage initiates and cracks can propagate. In this framework, zero-thickness cohesive elements are inserted along potential fracture paths, often at the interface between the adherends [39]. These elements are governed by traction–separation laws, which correlate cohesive traction (stresses) with relative displacements across the interface. During the early stages of loading, the stiffness of these elements is high to approximate a near-elastic response. As the stress approaches the critical value, the element stiffness gradually degrades, simulating the progression of damage until ultimate failure. This sequence—from initial elastic behavior through damage initiation and propagation—captures the essential fracture mechanics of bonded interfaces in single lap joints and other adhesive structures.

The traction–separation law (Figure 4) is defined by three key parameters: the maximum traction (T_max), the corresponding separation at maximum traction (δ_max), and the critical separation (δ_c) at which complete decohesion takes place. Although several traction–separation curve shapes exist such as bilinear, trapezoidal, or exponential, this study specifically adopts a bilinear law. During the analysis, as the applied load grows, the cohesive parts undergo a progressive increase in tractions and separations until they reach the maximum traction, which signifies the beginning of damage. After reaching the critical separation, the cohesive elements deteriorate in accordance with the stated softening law, resulting in full separation. This methodology enables the precise replication of both the commencement and expansion of fractures, offering in-depth understanding of the causes of failure in adhesive connections.

A quadratic stress criterion is optimal for simulating a single lap joint because it effectively captures the mixed-mode nature of the stresses experienced in such joints [22]. SLJ often encounter a combination of tensile (mode I) and shear (mode II) stresses due to the eccentric load path and the resulting peel and shear forces. The quadratic stress criterion integrates the contributions of both modes by employing a combined stress approach, where the failure criterion is defined as the square root of the sum of the squares of the normalized mode I and mode II stresses. This allows for a more accurate representation of the interaction between different stress components and their cumulative effect on the joint’s failure. Moreover, the quadratic stress criterion ensures a more realistic simulation of the failure envelope for adhesive joints, accounting for the non-linear interaction between tensile and shear stresses [22]. By considering the normal and shear stress components simultaneously through a squared, normalized summation that reaches a critical threshold, the criterion captures how multiple stress modes collectively contribute to damage onset.

Once damage initiates, the model uses a power-law formulation for the evolution of damage, relating the energy dissipated in each stress mode to its corresponding critical fracture energy. In this study, the key parameters for the cohesive zone model (CZM) including critical fracture energies and maximum stresses were selected based on established literature for aluminum adherend systems [28,42]. During loading, partial fracture energy is consumed in the normal and shear directions. When the sum of these mode-based energy terms, weighted by an exponent (α) that dictates their mixed-mode interaction, meets or exceeds unity, the adhesive fully debonds [43]. This approach aligns well with experimental observations of progressive failure, in which local cracking initiates at high-stress regions and then spreads as more fracture energy becomes available. From tensile tests of bulk FM 73M OST adhesive, it was found that the tensile strength ranged from 49 to 53 MPa, while the Young’s modulus ranged from 2000 to 2300 MPa and the Poisson’s ratio was 0.4 [28]. The relevant cohesive parameters, such as the allowable stresses and fracture energies for each mode, are summarized in

Table 3. CZM baseline properties of adhesive FM 73M OST [28].

Kn (N/mm3)	Ks (N/mm3)	Kt (N/mm3)	Tn,max (N/mm2)	Ts,max (N/mm2)	Tt,max (N/mm2)	GIC (kJ/m2)	GIIC (kJ/m2)	GIIIC (kJ/m2)	α
500,000	178,750	178,750	49	28	28	2.5	5	5	1.5

.

The SRCE employs the adhesive’s elastic modulus as its stiffness and takes advantage of Abaqus’s capability to specify the finite thickness of the adhesive layer directly in the cohesive section. By contrast, in the interfacial and middle cohesive element configurations, the adhesive layer is treated as a continuum with the same elastic properties, but a zero-thickness cohesive element defined by the parameters in Table 3 is introduced at the interface to capture failure. This setup allows the continuum portion to represent the bulk elastic behavior of the adhesive, while the cohesive interface isolates and simulates the damage and fracture processes. Choosing the appropriate penalty stiffness for zero-thickness cohesive elements is crucial to avoid under- or over-conditioning the system.

In this study, we define the normal (K_n), shear (K_s), and tangential (K_t) penalty stiffnesses in Table 3, listing their respective units [N/mm³] and explaining that they are typically several orders of magnitude larger than the stiffness of the adherends. These high stiffness values ensure that no significant penetration or gap forms at the interface under small deformations. However, the chosen penalty values must not be so large as to introduce numerical instability or convergence issues. Often, an initial estimate of penalty stiffness is made based on material properties and then adjusted through empirical testing and calibration, and this is done in this study by employing the DEIP (v4.4) interface element software [44]. Convergence studies can help identify the optimal range of penalty stiffness that balances accuracy and stability [45]. Also, the DEIP (v4.4) software was combined with Abaqus to insert the zero-thickness cohesive elements into the single lap joint finite element mesh.

2.3. Analytical Method Description

The analytical model used in this study is based on the well-established formulation by Goland and Reissner, which describes the distribution of peel and shear stresses within the adhesive layer of a single lap joint. The peel stress and shear stress can be derived by considering the equilibrium of forces and moments in the adherends and the adhesive layer along with some simplifying kinematic assumptions. These stresses are typically expressed as functions of the applied load, adhesive thickness, and adherend stiffness, and are defined based on the bending moments induced by the eccentric loading of the joint, and the load transfer through the adhesive layer. The final expressions for the shear stress τ and peel stress σ are given by the following:

$$\tau ={\displaystyle \frac{{\beta }_{\tau }\left(F{t}_{\mathrm{p}}+6M_k\right)\cosh {\beta }_{\tau }x}{8t_{\mathrm{p}}\sinh {\beta }_{\tau }c}}+{\displaystyle \frac{3\left(F{t}_{\mathrm{p}}-2M_k\right)}{8t_{\mathrm{p}}c}}$$

(1)

$$\sigma ={\displaystyle \frac{2E_{\mathrm{A}}}{t_{\mathrm{A}}}}\left(B_{\sigma 1}\sinh {\beta }_{\sigma }x+B_{\sigma 4}\cosh {\beta }_{\sigma }x\cos {\beta }_{\sigma }x\right)$$

(2)

where $B_{\sigma 1}\mathrm{and}{B}_{\sigma 4}$ are integration constants [31].

${\beta }_{\tau }\mathrm{and}{\beta }_{\sigma }$ are eigenvalues and are defined as ${\beta }_{\tau }=\sqrt{{\displaystyle \frac{8G_A}{{E_{\mathrm{p}}t}_{\mathrm{A}}{t}_{\mathrm{p}}}}}$; and ${\beta }_{\sigma }={\displaystyle \frac{\sqrt{2}}{2}}\sqrt[4]{{\displaystyle \frac{24E_{\mathrm{A}}}{E_{\mathrm{p}}{t}_{\mathrm{p}}^3{t}_{\mathrm{A}}}}}$

$$k={\displaystyle \frac{1}{1+2\sqrt{2}\tanh \left({\beta }_{k}c/2\sqrt{2}\right)}}$$

(3)

where F = axial force, t_p = thickness of the adherend, t_A = thickness of the adhesive, E_A = adhesive Young’s modulus, E_p = adherend Young’s modulus, G_A = adhesive shear modulus, c = half of the adhesive overlap, x = position along the overlap from the midpoint, and k = edge moment factor formula.

Several key assumptions are made to simplify the analytical formulation of the single lap joint. First, the adherends are assumed to be linearly elastic, isotropic, and of uniform thickness, which allows for the use of classical beam theory in their analysis. The adhesive layer is also assumed to be linearly elastic and of uniform thickness, with no initial defects or voids. Furthermore, the model assumes that the adhesive deforms primarily in shear, with negligible normal strains. The overlap length of the joint is considered sufficient to develop the full strength of the adhesive, and the effects of edge stresses are neglected. While the assumption of elasticity will be violated in the FE simulations of this study at higher applied displacement levels, the FE simulations will be assessed for their consistency with this analytical model at lower applied displacements for which all stresses are below the elastic limit of the adhesive and adherends.

3. Results and Discussion

The study of adhesive model representations within single lap joints (SLJ) begins by reproducing simulation results corresponding to tensile experiments conducted in a published article, followed by an assessment of boundary conditions representing the one-inch gripped zone of the SLJ. Then, one scenario of overlap length and adherend thickness is investigated closely to understand the general mechanical response of the adhesively bonded aluminum SLJ during its loading stages. Next, the four adhesive model representations are compared with respect to their force–displacement curves and stress distribution within the bond line to identify distinctions in their behavior. These comparisons are made across a range of adherend thickness and overlap lengths to assess the generality of the observed trends. Stress distributions obtained within the elastic region of the aluminum adherends are also validated against the referenced analytical model. Lastly, strain contours from the external surfaces of the SLJ are analyzed for select geometrical configurations as preliminary results for future comparison with experiments conducted using Digital Image Correlation (DIC). To aid in this study serving as a benchmark for the modeling of adhesively bonded SLJ, the major relevant data are provided at https://zenodo.org/records/14927988 (accessed on 25 February 2025), including all input files and a selection of the output files from the analyses herein.

3.1. Validation of the Cohesive Parameters

To validate the finite element analysis, the simulation data were meticulously reproduced using parameters from a published article [28]. The single lap joint specimens were modeled with precise geometric details to match the reference study, with mechanical properties from Table 2 and Table 3, and the geometrical properties are listed as follows based on Figure 1: the adherend thicknesses, t_p = 4.7 mm, the adhesive thickness, t_A = 0.2 mm, the grip length with shims, L_G = 40 mm, the adherend length, L_A = 100 mm, and the overlap distance, L_o = 30 mm. In the approach of this reference study, the interfacial and middle cohesive elements were used for the analysis to simulate the damage within the adhesive layer. These dimensions were chosen to ensure consistency with the reference study, allowing for a direct comparison of results with our model.

While the reference study utilized a 2D plane stress model, our approach involved a 3D finite element model. This transition from 2D to 3D is significant as 3D models capture out-of-plane stresses and deformations that 2D models, constrained by plane stress or plane strain assumptions, may not fully represent. These out-of-plane effects in the 3D often result in a stiffer response in the force–displacement behavior of the joint, especially in regions where stress concentrations occur due to geometry transitions. The finite element mesh in our study was carefully constructed, adhering to standard criteria such as aspect ratio and element distortion, to ensure numerical accuracy and stability across the simulations.

Upon reproducing the simulation results, as seen in Figure 5, a close examination of the force–displacement curves revealed slight variations in the slopes compared to the published data. These variations can be attributed to the differences between the 2D and 3D analyses. While the 2D plane stress model from the reference study inherently provides a softer response due to its inability to capture the full three-dimensional stress state, our 3D model incorporates these effects, providing a more realistic representation of the joint behavior under load. The boundary conditions, including the application of load and constraints, were carefully replicated to match the reference study, ensuring that any differences observed are due to the dimensional transition rather than discrepancies in setup. Thus, our model is validated against the real scenario presented in the reference study.

3.2. Effects of Boundary Condition on SLJ

An investigation was conducted to assess the impact of one-inch grips on the force–displacement response of single lap joints. By systematically adding and removing the grips in the finite element model, it was determined that both scenarios produced similar force–displacement curves and identical stress distributions in the overlap region, indicating that the grips do not affect the critical behavior being studied. The model with grips used 162,099 elements, while the model without grips used 122,409 elements, resulting in a significant reduction in computational cost. Parallel CPU usage and runtime for the two models further highlighted the efficiency gained by excluding the grips, with the model without grips running 20% faster. These findings support the decision to exclude grips for enhanced computational efficiency without compromising the accuracy of the results. The single-row cohesive element layer was employed to simulate the adhesive, integrating the traction–separation law to represent the adhesive behavior precisely.

The simulations were carried out under displacement-controlled loading, with both the grip-inserted and grip-removed scenarios being subjected to identical loadings (Figure 6). Key parameters included the material properties of the adhesive and adherends and the geometry condition listed as follows: L_G = 25 mm, L_o = 25 mm, L_A = 100 mm, t_p = 2 mm, and t_A = 0.2 mm. Boundary conditions were applied to simulate the set-up accurately, with the displacement-controlled BC being incrementally applied until failure occurred. The force–displacement response was recorded for each scenario to facilitate a comparative analysis.

The parametric study results revealed that the force–displacement curves for the grip-inserted and grip-removed scenarios were virtually identical, indicating that the presence of the 25 mm grips does not significantly affect the mechanical response of the single lap joint, as seen in Figure 7. This study suggests that the grips can be removed from the model without compromising the accuracy of the simulation results. By excluding the grips, the computational time and resources required for the simulations can be reduced, thereby enhancing the efficiency of the finite element analysis.

3.3. Mechanical Behavior of SLJ Reference Configuration

In analyzing the force–displacement response of lap joints, the transition of stresses, particularly von Mises and shear stresses, can be observed distinctly across the elastic (A), hardening (B), and maximum failure regions (C). Initially, in the elastic region, the applied load causes a proportional increase in both von Mises and shear stresses within the adhesive layer. The SLJ configuration considered in Figure 7 will be explored in this section to illustrate these three regions.

The different stress variations are shown in Figure 8. The shear stress distribution exhibits a pattern which shows that the shear stress is dominant in the mechanics of the lap joint. In the elastic range, there are high stresses at the edges, primarily due to the non-uniform distribution of stress due to the eccentric load path and the geometric discontinuity. At the start of the analysis, the overlap region experiences a combination of normal and shear stresses, and the abrupt changes at the edges create a stress riser, leading to an increase in the shear stress values compared to the central region of the joint. Similarly, shear stress concentration occurs at the edges because the adhesive layer must accommodate differential deformations between the adherends. The effect of the shear stresses at the edges makes these regions critical for initiating damage and failure in the adhesive layer, which is visible in the peel stress plot and the adhesive layer representation. The graphics in Figure 9 correlate with the plots in Figure 8, illustrating the variations in adhesive stress distributions.

As the load increases in the single-row cohesive zone model, stress concentrations develop at the overlap edges, but the plastic behavior is localized in the adherends rather than in the CZ adhesive layer. Because a single-row cohesive element uses a traction–separation law to represent the adhesive, there is no provision for plasticity or strain hardening within that interface. Instead, the von Mises stress exceeding the yield strength applies primarily to the adherends just outside the overlap. This localized plasticity in the adherends can exhibit strain hardening, further raising the overall load-bearing capacity. Meanwhile, the shear stress in the cohesive elements continues to increase, since the critical stress is a higher value than the adherend yield stress. Hence, the adherends not the cohesive adhesive layer undergo plastic deformation and hardening, even though their deformation may appear closely coupled with the failure process at the joint edges.

Upon reaching the maximum failure region, the effects of peel and shear stresses cause the cohesive elements at the edges to degrade fully. This leads to crack initiation and propagation from the edges towards the center of the lap joint. The adhesive layer starts to experience significant damage, with micro-cracks coalescing into larger fractures. As the load continues to increase, these cracks propagate, reducing the effective load-carrying area of the adhesive layer and further increasing the local stress concentration, as shown in Figure 9. Eventually, the lap joint reaches its ultimate load-bearing capacity, and complete separation occurs, characterized by a rapid decline in both von Mises and shear stresses as the adhesive layer fails. This failure process highlights the critical role of edge stress concentrations in determining the overall structural integrity and failure behavior of single lap joints.

3.4. Strength Prediction

A finite element analysis was conducted to investigate the force–displacement behavior of single lap joints with various adherend thicknesses and overlap lengths. The analysis incorporated the four different modeling approaches for the adhesive layer, and the objective was to evaluate the joints’ performance and failure mechanisms under various configurations. The results presented here highlight how changes in adherend thickness and overlap affect the load capacity, stiffness, and predominant failure mode in SLJ.

For instance, in the case of the 3 mm thick adherend and the four different overlaps, cohesive failure was noticed in the adhesive layer, and no hardening was observed within the adherends (see Figure 10a). Because the adherend thickness is relatively large compared to the adhesive layer thickness, the adherends remain rigid throughout the loading process, causing the primary failure to be driven by the adhesive’s cohesive properties rather than by plastic deformation or hardening in the adherends.

The SRCE produced results nearly identical to the middle cohesive elements, as shown in Figure 10, demonstrating its reliability in capturing adhesive (cohesive) failure. The continuum solid elements adhesive model representation (continuum) only indicated the yielding of the adherend and the adhesive (the traction serving as the yield property of the adhesive); because it lacks a cohesive zone (CZ) framework, it cannot simulate failure initiation and propagation, and instead merely indicates yielding without progressing to a failure. Despite this limitation, the continuum model is included as a critical reference for verifying the pre-failure (elastic and early plastic) behavior of the other adhesive representations. By comparing the initial force–displacement response and the local stress/strain fields of the continuum model with those of the CZ-based methods, it is noticed that the cohesive element penalty stiffness is sufficiently high to approximate a nearly rigid interface during early loading. Consequently, the single-row cohesive, interfacial CZ, and middle cohesive models align closely with the continuum approach in the elastic region, typically within 5% discrepancy.

This validation step addresses an important philosophical point noted in both recent and foundational works on cohesive zone modeling [46]. Because a zero-thickness interface with a finite penalty stiffness can introduce artificial compliance if the stiffness is not chosen carefully, ensuring that the CZ layer’s initial slope parallels that of the full-continuum model bolsters confidence in the accuracy of the predicted joint stiffness prior to damage. Only once localized stress exceeds the adhesive’s cohesive strength do the CZ-based approaches diverge by simulating the failure initiation, something the continuum model cannot replicate. As a result, the continuum approach demonstrates its usefulness for gauging elastic performance, while the CZ methods become indispensable for analyzing post-failure mechanisms. By combining both perspectives, this study ensures simulation of both the early elastic phases and the discrete failure processes inherent to single lap joint configurations.

For larger overlaps (20–25 mm), the adherend consistently yielded without failing in the adhesive region, demonstrating the dominance of the adherend’s flexibility and its susceptibility to yielding, as shown in Figure 11. However, the adhesive failed for a 10 mm overlap, indicating that the shorter overlap could not provide sufficient load distribution. Also, the four methods produced the same force–displacement curve for larger overlaps since the adhesive did not fail due to the adherend’s thickness.

In summary, for the 0.5 mm thick adherends, none of the lap joints experienced adhesive failure across the tested overlaps. Instead, the adherends yielded plastically, preventing the stress from reaching levels sufficient to initiate cohesive damage in the adhesive layer. Consequently, the force–displacement responses were virtually identical among the different overlap lengths and adhesive modeling approaches (Figure 12). These observations underscore that thinner adherends can yield well before the adhesive’s ultimate strength is reached. By contrast, shorter overlaps in thicker adherend configurations (Figure 10) are more prone to early adhesive failure, as they concentrate stresses at the overlap edges. In the next section, we delve into the stress profiles in these joints to illustrate how overlap length and adherend thickness influence the failure mechanism. Then, we focus on the detailed analyses on thicker adherends, where the cohesive zone (CZ) representations display more pronounced differences in predicting the peak load and failure evolution.

3.5. Stress Distribution of the Adhesive

The stress distribution analysis for single lap joints using four different modeling methods revealed distinct behaviors across different adherend thicknesses and overlap lengths. Initially, all methods produced similar peel and shear stresses in the elastic region for a 3 mm thick substrate with a 25 mm overlap. However, significant differences emerged in the maximum force region. The SRCE method exhibited a notably high peel stress of 50 MPa, in contrast to the cohesive element methods, which produced peel stresses around 22 MPa as shown in Figure 13c. This discrepancy highlights the continuum solid element method’s inability to accurately capture failure or post-failure effects, underscoring the limitations of this approach in simulating adhesive joint behavior under high load conditions.

At the point of maximum force, the three cohesive element methods produced shear stresses at 28 MPa, which aligns with the adhesive’s shear stress threshold, indicating imminent failure. This consistency among cohesive methods demonstrates their efficacy in capturing the initiation and progression of failure within the adhesive layer. All these findings were further validated through an analytical formulation [29,30,31] showing the same shear stress distribution and validating the predictive capability of these cohesive models. In contrast, the continuum method’s peel stress at this stage remained significantly lower at the middle of the lap joints, reflecting its inadequacy in representing the true stress state of the joint. This finding underscores the importance of cohesive elements in accurately modeling the complex failure mechanisms of adhesive joints.

In Figure 13a,c, the observed trend where peel stresses started at approximately 30 MPa in the elastic region and decreased to 22 MPa at maximum force indicates that peel stresses are initially highest and reduce towards failure. This behavior is consistent across different overlap lengths and thicknesses, though the pattern varies slightly (Figure 14). This absence of a wavy pattern highlights the influence of overlap length on the initial stress distribution, with shorter overlaps uniformly distributing the stresses across the different regions of the overlap area.

A closer examination of Figure 13a reveals that the continuum approach consistently displays the same amplitude fluctuations with the other cohesive zone (CZ) models. Since SRCE is the most streamlined cohesive representation by assigning an equivalent traction–separation law to the entire adhesive layer, it naturally smooths out localized effects. Moreover, all the CZ-based methods show slightly higher peak stresses than the analytical model, which is common because the finite penalty stiffness of cohesive elements may introduce additional compliance, altering the stress distribution. Nonetheless, these differences remain small, indicating that the chosen cohesive stiffness is sufficiently high to mimic near-elastic behavior before failure initiation, a key objective in cohesive zone modeling, where the elastic branch of the traction–separation law is not meant to replicate the entire adhesive material’s elasticity in detail, but rather to enable subsequent failure prediction.

Figure 13c demonstrates a similar trend: the two middle and interfacial CZ approaches with comparable through-thickness representations exhibit closely matched amplitudes, while the SRCE model still yields a slightly smoother distribution. In this mid-load regime, the differences among the CZ approaches remain modest, reflecting the fact that none of the joints have fully transitioned into damage-dominated behavior. By contrast, the continuum single-row model shows higher amplitude stress emphasizing the purely elastic-plastic adherend response without cohesive failure in the adhesive.

Moving on to Figure 13d, the asymmetry in the continuum model’s stress distribution becomes more pronounced. In the CZ-based approaches, local adhesive damage triggers load redistribution, though the cohesive zone models allow for progressive edge softening rather than pure plastic rotation of the adherends. Consequently, each method’s stress contour evolves uniquely under large deformations, with the continuum model showing more pronounced asymmetry once plastic hinges begin to form in the adherend material near the overlap edges.

The figure-to-figure consistency of the CZ models in the elastic regime—where they remain close to the continuum model and analytical predictions—confirms that their traction–separation “stiffness” is large enough to avoid artificial compliance before damage. The minor differences among CZ approaches arise largely from how each formulation handles local stress distributions and potential failure initiation. Hence, even though the SRCE model offers a more simplified representation of the adhesive layer, it still matches overall elastic behavior quite well, while the more refined cohesive models may capture localized stress peaks or failure paths in greater detail.

Analyzing the effect of different adherend thicknesses using the single-row cohesive element method revealed a progressive increase in peel and shear stress as the thickness decreased. For the same load step at the elastic region, the 3 mm thick adherend produced peel and shear stresses of 14.1 MPa and 7.6 MPa, respectively. In contrast, the 2 mm thick adherend showed peel and shear stresses of 14.3 MPa and 9.2 MPa, respectively. This trend continued with the 1 mm thick adherend, producing 14.6 MPa and 10.6 MPa, and the 0.5 mm thick adherend yielding the highest stresses of 14.8 MPa and 12.1 MPa, in Figure 15. These results indicate that thinner adherends experience higher stress, making them more susceptible to initiating degradation in the cohesive zone at the outer edges of the overlap under similar loading conditions compared to thicker adherends. The analytical formulation in Figure 15c,d also replicated this progressive increase in stress with decreasing thickness, further validating the numerical results and confirming the mechanical behavior predicted by the simulations.

In this thickness variation study, a 25 mm overlap was chosen in accordance with ASTM D1002 guidelines, which prescribe standardized dimensions for SLJ. Under these conditions, the distribution of peel stresses was found to vary with adherend thicknesses. For the thickest adherend (3 mm), peel stresses were more evenly distributed across the overlap length due to the larger size, whereas thinner adherends exhibited more pronounced peel stresses at the edges. Thicker adherends possess a large moment of inertia, thus exhibiting greater stiffness in bending. Consequently, they experience less out-of-plane displacement under tensile loading, effectively shifting the joint’s response toward shear-dominated behavior rather than bending-dominated behavior. As a result, the load transfer is more uniform, reducing stress concentrations at the overlap edges and improving the joint’s overall durability. By contrast, thinner adherends are more susceptible to edge-peel failures because their reduced bending stiffness localizes deformation at the edges, promoting higher stress concentrations that can trigger premature damage (Figure 15).

During the initial stages of loading, shear stresses are highest at the edges of the overlap, forming a characteristic “U-shaped” distribution across the width of the joint. As the load increases, these edge regions exceed the cohesive zone’s traction threshold, initiating local damage and triggering the softening branch of the traction–separation law. As a result, the degraded edges can no longer carry as much load, causing the shear stress to redistribute inward toward the middle of the overlap. Consequently, at higher load levels (Figure 13d), the shear stress profile appears inverted with the peak shifting to the central region.

This shift in stress concentration reflects a common cohesive damage progression: once the outer edges fail, the surviving material in the middle is forced to carry an increasing share of the load, thereby raising its stress level. Papers on cohesive zone modeling of single lap joints similarly report this pattern, attributing it to a combination of localized debonding at the edges and the adhesive’s progressive softening response [22]. Understanding this transition underscores why joint designs or reinforcements targeting edge-strength improvements can delay edge damage and maintain a more uniform stress distribution, potentially enhancing overall joint strength and durability.

The numerical results for different overlap lengths (Figure 16) for a 3 mm thick adherend reveal that shorter overlaps exhibit higher stress concentrations, primarily at the edges. When the overlap is small, a large bending effect develops because the eccentric load path has less adhesive area over which to distribute the forces, causing the peel stress at the edges to spike. Meanwhile, longer overlaps may display a “W-shaped” distribution early in the loading process, with pronounced tensile peaks near the ends of the overlap and a slightly compressed region at mid-span. As damage initiates at the overstressed edges, the load shifts toward the center of the joint, progressively flattening the outer peaks and producing a more “U-shaped” profile at higher loads. This transition mirrors the cohesive zone model’s softening behavior, wherein edge regions degrade first and compel the still-intact central region to carry a greater share of the load.

Furthermore, Goland and Reissner’s analysis [29] indicates that, in the absence of significant external peel forces, the net integral of peel stresses across the bond line must approximately balance to zero. Hence, an elevated tensile stress at the overlap edges is typically offset by a correspondingly compressive region, which is often observed in the central part of the overlap. Such behavior appears in our simulations, where shorter overlaps show large positive shear stresses and negative stresses in the middle, consistent with the need for equilibrium in bending moments. Although the analytical approach sometimes predicts slightly higher peak stresses than the numerical model (e.g., a 10% difference for the 10 mm overlap scenario in the shear stress contour), the overall distribution trends remain remarkably similar, underscoring the continued relevance of Goland and Reissner’s classical solution.

This stress distribution analysis, validated by the analytical formulation, underscores the critical impact of adherend thickness, overlap length, and modeling method on the mechanical performance and failure behavior of single lap joints. Cohesive element approaches consistently demonstrated superior accuracy in capturing adhesive failure mechanisms, while the single-row continuum method fell short in representing post-failure behavior. The analytical results not only confirmed the trends observed in the numerical simulations but also provided a theoretical foundation for understanding the mechanics of the joints, emphasizing the robustness of the cohesive models used in this study. This section highlights the importance of considering adherend thickness and overlap length in joint design to optimize stress distribution and enhance joint reliability.

3.6. Strain Contour Comparison

The strain contours obtained from the finite element analysis can be directly compared with those from Digital Image Correlation (DIC) to validate the numerical model against experimental observations. In Figure 17a,b, the dashed black outline indicates the overlap region, which is the critical zone where the adhesive is bonding the two adherends. Although the 0.5 mm and 3 mm adherend configurations exhibit different absolute strain levels due to variations in stiffness and load distribution, their strain distributions remain almost the same because both show similar longitudinal gradients and regions of elevated strain near the overlap edges. Additionally, the same color bar scale has been applied across these plots to facilitate visual comparison; thus, greens and blues correspond to comparable strain ranges for each thickness. As a result, while the thicker adherend case may show lower peak values than the thinner one, the overall pattern of increasing strain toward the overlap region is analogous in both figures, confirming that the single-row cohesive element model captures the fundamental deformation behavior in each scenario.

From Figure 17a,b, a simple analytical estimate for strain (Load × Length/Area × Modulus) yields approximately 1.12 × 10⁻³ mm/mm, which can be seen across the adherend surface of the two different thicknesses. The same applies to (c) and (d), where the same calculation yields 2.4 × 10⁻³ mm/mm. Although these single-lap joints have an overlap region where the adherends are effectively double their thickness, this analytical expression assumes a uniform cross-sectional area along the axial direction. The total length of the joint is significantly larger than the overlap zone, and the local stiffening effect of the double-thickness region is relatively small in comparison to the overall specimen length; using the single-thickness cross section still provides a reasonable first-order approximation of the average strain.

The surface variation is due to the eccentric load path, meaning lower strains can be noticed at the lower part of the overlap region. In contrast, high strain can be observed in the middle/upper region of the overlap area. Moreover, the color scale in these contour plots is normalized so that the green band corresponds approximately to the nominal strain predicted by the analytical estimate, making it easier to distinguish regions of higher strain than the expected elastic value. Collectively, these observations align with analytical considerations of single lap joint behavior, wherein the eccentricity amplifies local bending near the overlap edges, and any onset of yielding redistributes the load more evenly once plastic deformation sets in.

Although all the strain contours presented in Figure 18 are derived from the finite element analysis, these numerical results can still guide and enhance the interpretation of DIC data from physical experiments. Since DIC primarily captures surface strains, it can miss important information within the adhesive layer or at the interface between the adhesive and the adherends. By contrast, the FEA enables virtual sectioning of the joints at different through-thickness locations, illuminating sub-surface strain distributions that remain hidden from optical measurement methods. Consequently, combining DIC (for detailed surface strain measurements) with numerical strain contours (for internal deformations) offers a more comprehensive view of how single lap joints deform and fail. This synergy is particularly valuable for identifying the earliest stages of damage initiation, which may appear first in the subsurface regions before manifesting on the joint’s external surface. Additionally, the FEA can account for the effects of different adherend thicknesses on the overall strain distribution, demonstrating that the top surface strains remain similar despite variations in thickness, which should align with the experimental observations.

The identical strain pattern observed in the 25 mm overlap was likewise observed in the 10 mm overlap (Figure 18), regardless of the thickness. The anticipated strain value can be observed on the surface of the adherend. Initially, an examination of Figure 18 reveals that both the 25 mm and 10 mm overlaps display similar overall strain distributions at two different displacement levels (e.g., 0.12 mm and 0.3 mm). Nevertheless, there are key distinctions in how the strain gradients evolve once the load increases and potential plasticity arises. In particular, the longer overlap (25 mm) tends to distribute the load more uniformly, which delays the onset of localized high-strain regions. By contrast, the shorter overlap (10 mm) concentrates stresses more intensely near the edges, causing earlier local strain peaks.

Furthermore, the transition from the elastic regime to a possible plastic regime can be discerned by comparing the strain contours at 0.12 mm displacement where deformation likely remains primarily elastic with those at 0.3 mm displacement where either adherend yielding may begin. When the adherends yield, plastic hinges can form at or just beyond the overlap edges, thereby redistributing strain away from the damaged areas and causing the strain contours to deviate from a simple scaled version of the elastic pattern. Conversely, if cohesive damage initiates in the adhesive, local stresses at the edges drop, and the load shifts toward the middle of the overlap, also altering the contour shape.

Finally, comparing the 25 mm and 10 mm overlaps reveals that, despite both exhibiting the same general “pattern”, the shorter overlap reaches critical stress and strain thresholds sooner. This outcome aligns with simpler analytical expectations wherein a reduced bond area elevates local loads, increases stress gradients, and hastens potential damage. Consequently, pinpointing whether strain redistribution stems from adherend plasticity or cohesive failure requires analyzing the strain location (i.e., adherend or adhesive) and comparing the magnitude against known yield or damage criteria. These spatial strain distributions underscore how overlap length and loading magnitude interact to govern the onset and progression of deformation within single lap joints.

4. Conclusions

In this study, a finite element analysis of single lap joints was conducted to investigate how variations in adherend thickness and overlap length influence joint performance and failure modes. Four different adhesive modeling approaches, single-row cohesive element (SRCE), single-row continuum element, interfacial element, and middle cohesive element, were compared to highlight their relative strengths and limitations. The single-row continuum element method served as a baseline to verify the elastic response and assess the compliance of other cohesive element strategies. Although it cannot predict failure, it aligns closely with analytical models in the pre-failure region.

Comparing the four adhesive modeling approaches revealed some differences. The SRCE and middle cohesive element methods proved particularly effective at simulating the initiation and propagation of adhesive failure, offering a detailed portrayal of how failure evolved. Notably, although the middle cohesive element approach allows for a more refined through-thickness representation, this study showed that SRCE still captured the global response and failure mechanisms accurately for relatively thin adhesive layers. Meanwhile, the interfacial element method showed a tendency for larger displacements at failure due to the double cohesive layering, highlighting the need to carefully calibrate such models when interface-specific failures are the primary concern.

In addition, adherend thickness and overlap length play critical roles in determining the failure patterns. For instance, thicker adherends (2 mm and 3 mm) often exhibited cohesive failure within the adhesive layer when subjected to longer overlaps, underscoring the stronger load-bearing capacity of thicker materials. Conversely, thinner adherends (1 mm and especially 0.5 mm) tended to yield first, thereby preventing the adhesive from experiencing full cohesive damage in many cases. This difference highlights the importance of selecting optimal adherend dimensions in applications where joint reliability is paramount: thicker aluminum adherends tended toward cohesive failure within the adhesive layer, whereas thinner aluminum adherends often yield first, preventing full adhesive damage.

Moreover, shorter overlaps produced elevated peel and shear stresses near the overlap edges, which aligns closely with classical theories on bending-induced stress concentrations. The numerical results matched analytical predictions within approximately 10%, underscoring the variations in the modeling strategies. From a practical standpoint, these findings underscore the fact that smaller overlap lengths can lead to stress concentrations that may trigger premature failures unless designers incorporate mitigating features or select higher-strength adhesive systems.

Overall, these contributions enhance the existing body of knowledge on SLJ mechanics. While prior literature has addressed certain aspects of single lap joint behavior, this study provides a head-to-head comparison of multiple adhesive modeling techniques, thereby offering practitioners a clearer roadmap for choosing the most suitable numerical strategy. Looking ahead, there is potential to expand this framework by examining thicker bond lines, multi-material adherends, or varying thermal conditions to assess how each modeling approach scales under more complex scenarios. Ultimately, the work presented here reaffirms the importance of both geometric parameters and cohesive zone modeling in designing reliable adhesive joints and lays the groundwork for future explorations of bonded structures in advanced engineering applications.

To build on these insights, several avenues for future work are recommended. First, conducting controlled experimental studies to measure load-displacement behavior, strain fields, and failure progression would validate and refine the numerical models. Second, varying the adhesive thickness or using alternative adhesives would reveal how each modeling approach manages potential bulk adhesive deformation and through-thickness stress gradients. Third, evaluating different adherend materials (e.g., composites, steels) would help ascertain whether the modeled failure modes change substantially under altered stiffness contrasts or potential delamination mechanisms. Lastly, considering dynamic or shock loads would illuminate the need for rate-dependent cohesive laws or specialized continuum formulations to capture rapid crack propagation more accurately.

By addressing these topics, researchers can further enhance the accuracy, robustness, and real-world applicability of adhesive modeling strategies for single lap joints and other bonded assemblies.