An alternative solution for the edge moment factors of the unbalanced adhesive single-lap joint in tension

An evaluation of the existing theoretical solutions and a proposal of an improved one for edge moment factors of the unbalanced adhesive single-lap joint are performed. Firstly, the existing classical solutions are reviewed and studied in detail. The scope of applications and limitations related to the classical solutions are identified. Meanwhile, the determination for the long and short unbalanced single-lap joint (SLJ) is performed. Then, through removing disadvantages of the existing theoretical solutions, an improved theoretical solution considering the effect of large deflection for overlap is proposed. Meanwhile, the adherends of the overlap regions are treated as individual beams in the improved theoretical solution. The fully-coupled nonlinear formulations for determining adherend displacements (includes axial deformation and transverse deformation) in the overlap and edge moment factors are constructed using the improved theoretical solution. Finally, the results of the existing solutions and the improved theoretical solution are compared with the results of the finite element analysis.     

Closed-form solutions for elastic stress-strain analysis in unbalanced adhesive single-lap joints considering adherend deformations and bond thickness

A theoretical model is developed for the stress analysis in adhesive-bonded single-lap joints under tension, for which the two adherends could have different thicknesses and consist of different materials. A two-dimensional (2D) elasticity theory is adopted in the analysis, which simultaneously incorporates the complete strain-displacement and the complete stress-strain relationships for the adherends and adhesive. The approach provides a unified treatment for any possible adhesive layer flexibility and capable of satisfying the stress-free condition at the ends of the bondline. An explicit closed-form analytical solution is formulated for upper and lower adherends/adhesive stresses (strains) and tensile, shear and bending loads acting on the adherends along the overlap and then simplified for practical applications, and simple design formulae for adhesive stresses are produced. The results predicted by the present full and simplified solutions were compared with the previously theoretical solution by Bigwood and Crocombe (1989) [35], and the 2D geometrically nonlinear finite element model using MSC/NASTRAN. The agreement validates the present formulation and solutions for unbalanced bonded joints. The effects of the stiffness unbalanced parameters on the adhesive stress distributions were also discussed.     

Improved one-dimensional beam model for unbalanced composite single-lap joint

An analytical nonlinear solution was provided for unbalanced composite single-lap joint (CSLJ) using an improved one-dimensional beam model, which incorporated the effect of bending–tension coupling. The bending–tension coupling stiffness was introduced to characterize the coupling bending and tension behavior induced by the un-symmetric stacking sequence of composite laminates. The governing differential equations captured the bending–tension coupling behavior and the geometrically nonlinear features were constructed based on the displacement compatibility conditions of flexible interface. The transverse deformation in overlap region, edge moment factors and adhesive stress distributions for the unbalanced CSLJ with inflexible, intermediate flexibility and flexible adhesive can be determined by the present one-dimensional beam model. The accuracy of the present model was validated by the comparison with nonlinear finite element model. The effect of bending–tension coupling on edge moment factors and peak values of adhesive stresses was shed light on with the present model.     

The use of flexible interface theory for fully coupled nonlinear analysis of balanced adhesive single-lap joint

A theoretical model is presented for determining the edge moment factors, the transverse deflections and the interfacial stresses of the balanced adhesive single-lap joint (SLJ). Based on the flexible interface theory, the improved one-dimensional beam model incorporates simultaneously the effects of interfacial compliances, the overlap geometric nonlinearity and the transverse shear deformations for the adherends. On the basis of normal and tangential displacement compatibility condition for the flexible interface, two sets of fully coupled governing equations concerning rotation of transverse normal and longitudinal displacement of adherends are constructed, from which the improved solutions for the edge moment factors, the transverse deflections, and the interfacial stresses can be obtained. The applicability and accuracy of the improved one-dimensional beam model are validated by comparing the present solutions with the results of the classical model, non-linear finite element analysis, and experimental results. Finally, the effects of the interface compliances on the adhesive stresses distributions of the balanced SLJ are studied.     

Three-dimensional finite element analysis of single-lap adhesive joints under impact loads

This paper deals with the stress wave propagation and stress distribution in single-lap adhesive joints subjected to impact tensile loads with small strain rate. The stress wave propagations and stress distributions in single-lap joints have been analyzed using an elastic three-dimensional finite-element method (DYNA3D). An impact load was applied to the single-lap adhesive joint by dropping a weight. One end of one of the adherends in the single-lap adhesive joint was fixed and the other adherend to which a bar was connected was impacted by the weight. The effects of Young's modulus of the adherends, the overlap length, the adhesive thickness and the adherend thickness on the stress wave propagations and stress distributions at the interfaces have been examined. It was found that the maximum stress occurred near the edge of the interface and that it increased with an increase of Young's modulus of the adherends. It was also seen that the maximum stress increased as the overlap length, the adhesive thickness and the adherend thickness decreased. In addition, strain response of single-lap adhesive joints subjected to impact tensile loads was measured using strain gauges. Fairly good agreements were observed between the numerical and experimental results.     

On the non-linear elastic stresses in an adhesively bonded T-joint with double support

Structures consisting of single or more materials, such as adhesive joints, may undergo large displacements and rotations under reasonably high loads, although all materials are still elastic. The linear elasticity theory cannot predict correctly the deformation and stress states of these structures, since it ignores the squares and products of partial derivatives of the displacement components with respect to the material coordinates. When these derivatives are not small, these terms result in a non-linear effect called geometrical non-linearity. In this study, the geometrically non-linear stress analysis of an adhesively bonded T-joint with double support was carried out using the incremental finite element method. Different T-joint configurations bonded to a rigid base and to a flexible base were considered. For each configuration, linear and geometrically non-linear stress analyses of the T-joint were carried out and their results were compared for different horizontal and vertical plate end conditions. The geometrically non-linear analysis showed that the large displacements had a considerable effect on the deformation and stress states of both adherends and the adhesive layer. High stress concentrations were observed around the adhesive free ends and the peak adhesive stresses occurred inside the adhesive fillets. The adherend regions corresponding to the free ends of the adhesive–plate interfaces also experienced stress concentrations. In addition, the effects of the support length on the peak adhesive and adherend stresses were investigated; increasing the support length had a considerable effect in reducing the peak adhesive and adherend stresses.     

Through-the-width delamination damage propagation characteristics in single-lap laminated FRP composite joints

Three-dimensional non-linear finite element analyses have been carried out to study the effects of through-the-width delaminations on delamination damage propagation characteristics in adhesively bonded single-lap laminated FRP composite joints. The delaminations have been presumed either to pre-exist or to get evolved due to coupled stress failure criteria in the laminated FRP composite adherends near the overlap ends beneath the ply adjacent to the overlap region. The out-of-plane stresses in the adhesive layer, the interlaminar stress distributions along the delamination fronts and the strain energy release rates (SERRs) corresponding to the three individual modes have been evaluated for varying positions of the delaminations pre-embedded in either of the adherends. A good matching between the present 3D results and experimental and analytical solution of the literature has been established for the undamaged and a damaged model. A significant difference in the interlaminar stresses and the SERR values has been observed and is largely dependent on the adherends (bottom or top) possessing the through-the-width delamination damages. Also, the interlaminar stresses and SERR values along the two corresponding delamination fronts are different. Accordingly, it can be concluded that the positions of the through-the-width delaminations significantly influence the delamination damage propagation behaviour vis-a-vis the performance of the composite joint.     

Effect of adherend's rigidity on the shear strength of single lap adhesive joints

The present paper compares the tensile shear strength of single-lap joints with different adherends. Three materials were combined in the single lap joints: a carbon/epoxy laminated composite, a high elastic limit steel and the 6082-T6 aluminium alloy. The shear strength of joints was influenced by the adherend stiffness and the highest shear strengths were obtained using high stiffness adherend materials. The overlap length influenced the shear strength in different ways depending on the adherend materials. Numerical analysis concluded that the increase in the rigidity of the adherends decreases the rotation of the specimen and promotes a more uniform distribution of stresses in the glue. In joints with distinct materials, the less stiff material was found to determine the strength of the appropriate joint.     

Three-dimensional finite element stress analysis of single-lap adhesive joints of dissimilar adherends subjected to impact tensile loads

The stress-wave propagations and stress distributions in single-lap joints of dissimilar adherends were analyzed using an elastic three-dimensional finite-element method (DYNA3D). An impact tensile load was applied to the single-lap adhesive joint by dropping a weight. One end of the upper adherend in the single-lap adhesive joint was fixed and the other adherend (lower adherend) which was connected to a bar was impacted by the weight. The effects of Young's modulus and the thickness of each adherend on the stress wave propagations and stress distributions at the interfaces were examined. It was found that the maximum value of the maximum principal stress occurred near the edge of the interface of the fixed adherend. The maximum principal stress increased as Young's modulus of the fixed adherend increased. It was also observed that the maximum principal stress increased as the fixed adherend thickness decreased. In addition, strain responses in the single-lap adhesive joints of dissimilar adherends subjected to impact tensile loads were measured using strain gauges. Fairly good agreements were found between the FEM calculations and the experimental measurements.     

Thermal non-linear elastic stress analysis of an adhesively bonded T-joint

Adhesive joints consist of adherends and an adhesive layer having different thermal and mechanical properties. When they are exposed to uniform thermal loads the mechanical-thermal mismatches of the adherends and adhesive layer result in uniform but different thermal strain distributions in the adhesive and adherends. The thermal stresses arise near and along the adherend-adhesive interfaces. The present thermal stress analyses of adhesively bonded joints assume a uniform temperature distribution or a constant temperature imposed along the outer boundaries of adhesive lap joints. This paper outlines the thermal analysis and geometrically non-linear stress analysis of adhesive joints subjected to different plate edge conditions and varying thermal boundary conditions causing large displacements and rotations. In addition, the geometrically non-linear thermal stress analysis of an adhesively bonded T-joint with single support plus angled reinforcement was carried out using the incremental finite element method, which was subjected to variable thermal boundary conditions, i.e. air streams with different temperatures and velocities parallel and perpendicular to its outer surfaces. The steady state heat transfer analysis showed that the temperature distribution through the joint members was non-uniform and high heat fluxes occurred inside the adhesive fillets at the adhesive free ends. Based on the geometrically non-linear stress analysis of the T-joint bonded to both rigid and flexible bases for different plate edge conditions, stress concentrations were observed at the free ends of adhesive-adherend interfaces and inside the adhesive fillets around the adhesive free ends, and the horizontal and vertical plates also experienced considerable stress distributions along outer surfaces. In addition, the effect of support length on the peak thermal adhesive stresses was found to be dependent on the plate edge conditions, when a support length allowing moderate adhesive stresses was present.     

Experimental and numerical investigations of adhesively bonded CFRP single-lap joints subjected to tensile loads

In this study, both experimental tests and numerical simulation are implemented to investigate the tensile performance of adhesively bonded CFRP single-lap joints (SLJs). The study considers 7 different overlap lengths, 5 adherend widths and 3 stacking sequences of the joints. Three-dimensional (3D) finite element (FE) models are established to simulate the tensile behavior of SLJs. The failure loads and failure modes of SLJs are investigated systematically by means of FE models and they are in good agreement with those of experiments, proving the accuracy of finite element method (FEM). It is found that increasing the adherend width can improve the load-carrying capacity of the joint better than increasing the overlap length does. Moreover, choosing 0° ply as the first ply is also beneficial for upgrading joint's strength. With respect to failure modes, cohesive failure in adhesive and delamination in adherend take dominant, while matrix cracking and fiber fracture only play a small part. With overlap length increasing or adherend width decreasing, cohesive failure takes up a smaller and smaller proportion of whole failure area, but the opposite is true for delamination. SLJs bonded with [0/45/-45/90]_3S adherends are prone to cohesive failure, and [90/-45/45/0]_3S adherends are easy to appear delamination. Both shear and peel stress along the bondline indicate symmetrical and non-uniform distributions with great stress gradient near the overlap ends. As the load increases, the high stress zone shifts from the end to the middle of the bondline, corresponding to the damage initiation and propagation in the adhesive layer.     

Stress analysis and strength evaluation of single-lap band adhesive joints of dissimilar adherends subjected to external bending moments

Single-lap band adhesive joints of dissimilar adherends subjected to external bending moments are analyzed as a four-body contact problem using a two-dimensional theory of elasticity (plane strain state). In the analysis, the upper and lower adherends and the adhesive which are bonded in two regions are replaced by finite strips. In the numerical calculations, the effects of the ratio of Young's moduli of the adherends, the ratio of the adherend thicknesses, and the ratio of the band length to the half lap length on the stress distributions at the interfaces are examined. A method for estimating the joint strength is proposed using the interface stress and strain obtained by the analysis. An elasto-plastic finite element analysis (EP-FEA) was conducted for predicting the joint strength more exactly. Experiments to measure strains and the joint strength were also carried out. The results show that the strength of a single-lap band adhesive joint is almost the same as that of a single-lap adhesive joint in which the two adherends are completely bonded at the interfaces. Thus, the single-lap band adhesive joints are useful in the design of single-lap joints.     

Progressive failure and experimental study of adhesively bonded composite single-lap joints subjected to axial tensile loads

Experimental tests and finite element method (FEM) simulation were implemented to investigate T700/TDE86 composite laminate single-lap joints with different adhesive overlap areas and adherend laminate thickness. Three-dimensional finite element models of the joints having various overlap experimental parameters have been established. The damage initiation and progressive evolution of the laminates were predicted based on Hashin criterion and continuum damage mechanics. The delamination of the laminates and the failure of the adhesive were simulated by cohesive zone model. The simulation results agree well with the experimental results, proving the applicability of FEM. Damage contours and stress distribution analysis of the joints show that the failure modes of single-lap joints are related to various adhesive areas and adherend thickness. The minimum strength of the lap with defective adhesive layer was obtained, but the influence of the adhesive with defect zone on lap strength was not decisive. Moreover, the adhesive with spew-fillets can enhance the lap strength of joint. The shear and normal stress concentrations are severe at the ends of single-lap joints, and are the initiation of the failure. Analysis of the stress distribution of SL-2-0.2-P/D/S joints indicates that the maximum normal and shear stresses of the adhesive layer emerge on the overlap ends along the adhesive length. However, for the SL-2-0.2-D joint, the maximum normal stress emerges at the adjacent middle position of the defect zone along the adhesive width; for the SL-2-0.2-S joint, the maximum normal stress and shear stress emerge on both edges along the adhesive width.     

A two-dimensional stress analysis of single-lap adhesive joints of dissimilar adherends subjected to tensile loads

Single-lap adhesive joints of dissimilar adherends subjected to tensile loads are analyzed as a three-body contact problem using the two-dimensional theory of elasticity. In the numerical calculations, the effects of Young's modulus ratio between different adherends, the ratio of the adherend thicknesses, the ratio of the adherend lengths, and the adhesive thickness on the contact stress distributions at the interfaces are examined. As a result, it is found that (1) the stress singularity occurs near the edges of the interfaces and it increases at the edge of the interface of an adherend with smaller Young's modulus; (2) the stress singularity increases at the edge of the interface of an adherend with thinner thickness; (3) the singular stresses increase at the edges of the two interfaces as the ratio of the upper adherend length to the lower one decreases; and (4) the singular stresses increase at the edges of the two interfaces as the adhesive thickness decreases when the adhesive is thin enough, and they also increase as the adhesive thickness increases when the adhesive is thick enough. In addition, the singular stresses obtained from the present analysis are compared with those obtained by Bogy. Fairly good agreement is seen between the present analysis and the results from Bogy. Strain measurement and finite element analysis (FEA) were carried out. The analytical results are in fairly good agreement with the measured and the FEA results.     

Low-speed bending impact behavior of adhesively bonded single-lap joints

This study addresses the low-speed impact behavior of adhesively bonded single-lap joints. An explicit dynamic finite element analysis was conducted in order to determine the damage initiation and propagation in the adhesive layers of adhesive single-lap joints under a bending impact load. A cohesive zone model was implemented to predict probable failure initiation and propagation along adhesive–adherend interfaces whereas an elasto-plastic material model was used for the adhesive zone between upper and lower adhesive interfaces as well as the adherends. The effect of the plastic deformation ability of adherend material on the damage mechanism of the adhesive layer was also studied for two aluminum materials Al 2024-T3 and Al 5754-0 having different strength and plastic deformation ability. The effects of impact energy (3 and 11 J) and the overlap length (25 and 40 mm) were also investigated. The predicted contact force-time, contact force-central displacement variations, the damage initiation and propagation mechanism were verified with experimental ones. The SEM and macroscope photographs of the adhesive fracture surfaces were similar to those of the explicit dynamic finite element analysis.     

A Two-dimensional Stress Analysis and Strength of Single-lap Adhesive Joints of Dissimilar Adherends Subjected to External Bending Moments

The stress distributions of single-lap adhesive joints of dissimilar adherends subjected to external bending moments are analyzed as a three-body contact problem by using a two-dimensional theory of elasticity (plane strain). In the analysis, dissimilar adherends and an adhesive are replaced by finite strips. In the numerical calculations, the effects of the ratio of Young's moduli of adherends, the adherend thickness ratio and the adherend length ratio between dissimilar adherends on the stress distributions at the interfaces are examined. The results show that the stress singularity occurs at the ends of the interfaces, and its intensity is greater at the interface of the adherend with smaller Young's modulus. It is also noted that the singular stress is greater at the interface of the thinner adherend. It is found that the effect of the adherend length ratio on the stress singularity at the interfaces is very small. Joint strength is predicted by using the interface stress and it was measured by experiments. From the analysis and the experiments, it is found that the joint strength increases as Young's modulus of adherends and the adherend thickness increase while the effect of the adherend lengths on the joint strength is small. For verification of the analysis, a finite element analysis (FEA) is carried out. A fairly good agreement of the interface stress distribution is seen between the analytical and the FEA results.     

Single Lap Joints with Rounded Adherend Corners: Stress and Strain Analysis

One of the major difficulties in designing adhesive lap joints is the stress singularity present at the adherend corners at the ends of the overlap. One way to overcome this problem is to assume that the corners have a certain degree of rounding. The objective of the present study was to better understand the effect of the change in the geometry of the adherend corners on the stress distribution and, therefore, on the joint strength. Various degrees of rounding were studied and two different types of adhesives were used, one very brittle and another which could sustain a large plastic deformation. The study gives a detailed stress and strain distribution around the rounded adherends using the finite element method. The major finding is that the stresses or strains in the adhesive layer of a joint with rounded adherend corners are finite. In real joints, adherends generally have small rounded corners. Consequently, the model with small radius corners may be used to represent real adherends.     

The behaviour of single lap joints under bending loading

Four-point bend tests were performed on single lap joints with hard steel adherends and a structural epoxy adhesive. The effect of the overlap, the adherend thickness and the adhesive thickness was studied. It was found that the length of the overlap has no significant effect on the strength of the joints. This is because the load transfer is occurring in a very localised area around the edges of the overlap, being the failure governed by peel mechanisms. The thickness of the adherends strongly affects the strength of the joints. The thicker the adherend, the stronger is the joint. The experimental results are compared with a finite element model and reinforce the fact that the failure takes place due to local strains at the ends of the overlap in tension. An analytical model is also given to predict in a simple but effective way the joint strength and its dependence on the adherend thickness.     

A two-dimensional stress analysis of single-lap adhesive joints subjected to external bending moments

The stress distributions in single-lap adhesive joints of similar adherends subjected to external bending moments have been analyzed as a three-body contact problem using a two-dimensional theory of elasticity (plain strain state). In the analysis, both adherends and the adhesive were replaced by finite strips. In the numerical calculations, the effects of the ratio of Young's moduli of the adherends to that of the adhesive and the adhesive thickness on the stress distribution at the interfaces were examined. It was found that the stress singularity occurs at the edges of the interfaces and that the peel stress at the edges of the interfaces increases with decreasing Young's modulus of the adherends. It was noticed that the singular stress decreases at the edges of the interfaces as the adherend thickness increases. In addition, photoelastic experiments and FEM (finite element method) calculations were carried out and fairly good agreement was found between the analytical and the experimental results.     

Modeling of single-lap composite adhesive joints under mechanical and thermal loads

Two-dimensional (plane-stress and plane-strain) theoretical models are presented for stress analysis of adhesively bonded single-lap composite joints subjected to either thermal or mechanical loading or a combination thereof. The joints consist of similar/dissimilar orthotropic or isotropic adherends and an isotropic adhesive interlayer. The governing differential equation of the problem is obtained using a variational method which minimizes the complementary strain energy in the bonded assembly. In this formulation, through-thickness variation of shear and peel stresses in the interlayer is considered. Both shear and normal traction-free boundary conditions are exactly satisfied. Peel and shear stresses obtained from plane-strain analytical models considering a homogeneous adhesive interlayer are in close agreement with those of the finite element predictions. A systematic parametric study is also conducted to identify an ideal set of geometric and material parameters for the optimal design of single-lap composite joints.