Standard finite element techniques for efficient stress analysis of adhesive joints

The paper documents ongoing research in the field of stress analysis of adhesive bonded joints and aims at developing efficient and accurate finite element techniques for the simplified calculation of adhesive stresses. Goal of the research is to avoid the major limitations of existing methods, in particular their dependency on special elements or procedures not supported by general purpose analysis packages. Two simplified computational methods, relying on standard modelling tools and regular finite elements are explored and compared with the outcome of theoretical solutions retrieved from the literature and with the results of full, computationally intensive, finite element analyses. Both methods reproduce the adherends by means of structural elements (beams or plates) and the adhesive by a single layer of solid elements (plane-stress or bricks). The difference between the two methods resides in the thickness and in the elastic properties given to the adhesive layer. In one case, the adhesive thickness is extended up to the midplane of the adherends and its elastic modulus is proportionally increased. In the other case, the adhesive layer is maintained at its true properties and the connection to the adherends is enforced by standard kinematic constraints. The benchmark analyses start from 2D single lap joints and are then extended to 3D configurations, including a wall-bonded square bracket undergoing cantilever loading. One of the two simplified methods investigated provides accurate results with minimal computational effort for both 2D and 3D configurations.     

Analytical solutions for nonlinear analysis of composite single-lap adhesive joints

This paper presents analytical nonlinear solutions for composite single-lap adhesive joints. The ply layups of each composite adherend can be arbitrary, but in the overlap region the ply layups of the upper and lower adherends are assumed to be symmetrical about the adhesive layer. In the present formulation, equilibrium equations of the overlap are derived on the basis of geometrical nonlinear analysis. The governing equations are presented in terms of adherend displacements by taking into account large deflections of the overlap adherends and adhesive shear and peel stresses simultaneously. Closed-form nonlinear solutions for adherend displacements, an edge moment factor and adhesive stresses are formulated and then simplified for practical applications. To verify the present analytical solutions for nonlinear analysis of composite single-lap joints, the geometrically nonlinear 2D finite element analysis is conducted using commercial package MSC/NASTRAN. The numerical results of the edge moment factor, deflections and adhesive stresses predicted by the present solutions correlate well with those of the geometrically nonlinear finite element analysis. This indicates that the present analytical solutions capture key features of geometrical nonlinearity of composite single-lap adhesive joints.     

Elastic analysis and engineering design formulae for bonded joints

The general elastic plane strain problem of adhesively bonded structures which consist of two different adherends is considered. To facilitate a truly general approach the adhesive joint is modelled as an adherend-adhesive sandwich with any combination of tensile, shear and moment loading being applied at the ends of both adherends. A full elastic analysis is presented which calculates the adhesive shear and tensile stresses in the overlap region, this analysis has been validated for a range of load cases using a finite element program. Basic design approaches are outlined and explicit expressions are developed which enable the simple evaluation of the stress distributions in the adhesive overlap. Simplified two parameter design formulae are also produced which accurately describe the peak stresses at the ends of the adhesive overlap in both the transverse and longitudinal shear directions. In all of the analyses the adherends are assumed to behave as linear elastic cylindrically bent plates with the adhesive forming an elastic interlayer between them. In the simplified analyses only one component of adhesive stress is considered, while in the full elastic analysis two components of stress are considered with a consequent increase in the complexity of the required solution method, but also an increase in accuracy over the simplified analyses for a wider range of joint configurations.     

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.     

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.     

A Comparison of Numerous Lap Joint Theories for Adhesively Bonded Joints

Numerous authors have investigated the state of stress in the adhesive of adhesively bonded joints. They have made various assumptions concerning the behavior of the adhesive and adherends to yield tractable differential equations which remove the stress singularities which occur at the edges of the bi-material interfaces. By examining several test problems, this paper investigates the effect of these assumptions on predicted adhesive stress. It was found that predicted maximum adhesive shear stress is insensitive to underlying assumptions and that maximum adhesive peel stress is relatively unaffected by most assumptions except that neglecting shear deformation of the adherends can affect results by as much as 30%. Peel stresses from the well known theory of Goland and Reissner which neglects shear deformation of the adherends and makes several inconsistent assumptions vary as much as 30% from stresses from a consistent lap joint theory which considers shear deformation of the adherends. However, in most cases the effects of the inconsistencies cancel the effects of neglecting the shear deformation of the adherends and the variation is less than 15%. This paper points out that finite element analyses of bonded joints where one layer of 4 node isoparametric elements are used to model the adhesive give results very close to those from consistent lap joint theories.     

On the thermomechanical behavior of two-dimensional foam/metal joints with shear-deformable adherends: Model validation with FE analysis

The stress distributions in metal/adhesive/foam planar joints subjected to biaxial tensile load and thermal load was investigated through a semi-analytical model. The shear deformation of adherends was accounted for according to a linear law in order to obtain closed-form solutions. For the model validation, a comparative study with a finite element (FE) simulation was carried out. A 2D behavior of stress fields is observed due especially to the Poisson's ratio effects and the biaxial nature of loads. The through thickness shear stresses are comparable to normal stresses; therefore, the adherend shear deformation must be accounted for correct failure prediction. According to the comparison with FE results, the normal stress distributions at any location in the foam and the shear stresses in the foam regions close to the adhesive surface can be well predicted by the proposed model. The through thickness shear stresses, however, showed to vary according to a cubic law rather than a linear law.     

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.     

Development of a Computer Program for the Design of Adhesive Joints

Adhesive joints are increasingly being used due to their improved mechanical performance and a better understanding of the mechanics of failure. To predict the joint strength, one must have the stress distribution and a suitable failure criterion. The literature contains many closed-form solutions for the stress distribution. However, the models are sometimes difficult to implement and use. The objective of the present work was to compile existing models of increasing complexity into user friendly software. Three main situations were considered: elastic adherends and adhesive, elastic adherends with nonlinear adhesive and nonlinear analyses for both adherends and adhesive. The adherends were both isotropic (metals) and anisotropic (composites). The joints considered are the single and double lap joints for most of the models. However, a sandwich model initially proposed by Crocombe can be used for any type of joint provided the boundary conditions are known. For each model proposed the compatible failure criteria are included to enable the user not only to have the stress distribution but also the failure load for a given joint/load scenario. Experimental tests corresponding to the three cases described above were carried out to validate the models implemented.     

Study on the fatigue strength of AA 6082-T6 adhesive lap joints

A research study on the fatigue behaviour of aluminium alloy adhesive lap joints was carried out to understand the effect of surface pre-treatment and adherends thickness on the fatigue strength of adhesive joints. The adherend material used for the experimental tests was an aluminium alloy 6082-T6 in the form of thin sheets, and the adhesive used was a high strength epoxy (Araldite 420 A/B). The surface preparation included an abrasive preparation (AP joints) and sodium dichromate–sulphuric acid etch (CSA joints).A maximum fatigue strength was obtained for the CSA surface treatment with a 1.0 mm adherends’ thickness. The fastest fatigue damage was related with a high surface roughness and a high stress perpendicular to adhesive surface, which helps to promote the adhesive failure. A numerical analysis was also performed to understand the effect of the adherends thickness on the stress level. Results showed an increase of the out-of-plane peak stresses with the increase of adherends thickness.     

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.     

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.     

On the von Mises elastic stress evaluations in the bi-adhesive single-lap joint: a numerical and analytical study

Bi-adhesive joints are an alternative stress-reduction technique for adhesively bonded joints. The joints have two types of adhesives in the overlap region. The stiff adhesive should be located in the middle and the flexible adhesive at the ends. This study is the extension of our previous paper to the von Mises stress evaluation and discusses the values and importance of the von Mises stresses in the bi-adhesive single-lap joint. Both analytical and numerical analyses were performed using three different bi-adhesive bondline configurations. The Zhao’s closed form (analytic) solution used includes the bending moment effect. In the finite element models, overlap surfaces of the adherends and the adhesives were modeled using surface-to-surface contact elements. The contribution levels of the peel and shear stresses for producing a peak von Mises stress are also studied. It is concluded that the contribution level of the shear stress at where von Mises stress becomes peak is more than that of the peel stress. Joint strength analyses were performed based on the peak elastic von Mises stresses. It is seen that joint strength can be increased using bi-adhesive bondline. The analytical and numerical results show that the appropriate bond-length ratio must be used to obtain high joint strength.     

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.     

胶接面纤维铺层角度对复合材料胶接应力和强度影响分析

为研究复合材料胶接面铺层对接头强度的影响,以单搭接接头为研究对象,通过铺层设计使接头的被粘物具有相同的拉伸模量和弯曲模量,但胶接面的铺层角度不同,使用有限元法对不同铺层的接头进行建模,分析接头胶接面和胶层的应力,引入Tsai-Wu失效因子对胶接面铺层进行评估。结果表明:胶接面铺层角度对应力分布有一定影响,0°胶接面会造成较大的胶层应力,但胶接面的应力和失效因子较小;90°铺层下胶层应力最小,但胶接面的应力和失效因子水平较高;45°下胶接面的失效因子和胶层应力水平介于两者之间。通过与实验结果对比,得出了胶接面铺层角度影响接头强度及破坏模式的一般性规律。     

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.     

A two-dimensional stress analysis and strength evaluation of band adhesive butt joints subjected to tensile loads

The stresses in band adhesive butt joints, in which two adherends are bonded partially at the interfaces, are analyzed, using a two-dimensional theory of elasticity, in order to demonstrate the usefulness of the joints. In the analysis, similar adherends and adhesive bonds, which are bonded at two or three regions, are, respectively, replaced by finite strips. In the numerical calculations, the effects of the ratio of Young's moduli for adherends to that for adhesives, the adhesive thickness, the bonding area and position, and the load distribution are shown on the stress distributions at interfaces. It is seen that band adhesive joints are useful when the bonding area and positions are changed with external load distributions. Photoelastic experiments and the measurement of the adherend strains were carried out. The analytical results are in a fairly good agreement with the experimental results. In addition, a method for estimating the joint strength is proposed by using the interface stress distribution obtained by the analysis. Experiments concerning joint strength were performed and fairly good agreement is found between the estimated values and the experimental results.     

Analysis of tubular adhesive joints with a functionally modulus graded bondline subjected to axial loads

The strength and lifetime of adhesively bonded joints can be significantly improved by reducing the stress concentration at the ends of overlap and distributing the stresses uniformly over the entire bondline. The ideal way of achieving this is by employing a modulus graded bondline adhesive. This study presents a theoretical framework for the stress analysis of adhesively bonded tubular lap joint based on a variational principle which minimizes the complementary energy of the bonded system. The joint consists of similar or dissimilar adherends and a functionally modulus graded bondline (FMGB) adhesive. The varying modulus of the adhesive along the bondlength is expressed by suitable functions which are smooth and continuous. The axisymmetric elastic analysis reveals that the peel and shear stress peaks in the FMGB are much smaller and the stress distribution is more uniform along its length than those of mono-modulus bondline (MMB) adhesive joints under the same axial tensile load. A parametric evaluation has been conducted by varying the material and geometric properties of the joint in order to study their effect on stress distribution in the bondline. Furthermore, the results suggest that the peel and shear strengths can be optimized by spatially controlling the modulus of the adhesive.     

Two-dimensional stress analysis of adhesive butt joints subjected to cleavage loads

This paper deals with the two-dimensional stress analysis of adhesive butt joints subjected to cleavage loads. The purpose of the paper is to contribute to the establishment of fracture criteria of adhesive joints. Similar adherends and an adhesive bond are replaced with finite strips in the analysis. Stress distributions in adhesive joints are analysed using the two-dimensional theory of elasticity. The effects of the ratio of Young's modulus of adherends to that of an adhesive and the thickness of the adhesive bonds on the stress distributions are shown by numerical calculations. For verification, strains produced on adherends are experimentally measured and a photoelastic experiment is carried out. The analytical results are in fairly good agreement with experiment.