An analytical model for interfacial stresses in double-lap bonded joints

Due to their many advantages, adhesively bonded joints are widely used to join components in composite structures. However, premature failure due to debonding and peeling of the joint is the major concern for this technique. Existing analytical models suffer from two major drawbacks: 1) not satisfying zero-shear stress boundary conditions at the adhesive layer’s free edges^[1] and 2) failure to distinguish the peel stress along two adherend/adhesive interfaces^[2]. In this study, we develop a novel three parameter elastic foundation (3PEF) model to analyze a representative adhesively bonded joint, the symmetric double-lap joint, which is believed to have relatively low peel stresses. Explicit closed-form expressions of shear and peel stresses along two adhesive/adherend interfaces are yielded. This new model overcomes the existing model’s major drawbacks by satisfying all boundary conditions and predicting various peeling stresses along two adherend/adhesive interfaces. It not only reaches excellent agreement with existing solutions and numerical results based on finite element analysis but also correctly predicts the failure mode of an experimentally tested double-lap joint. This new model therefore reveals the peel stresses’ significant role in the failure of the double-lap joint, but the classical 2PEF model cannot create it.     

Investigating the performances of stepwise patched double lap joint

An adhesive-bonded double-lap composites joint with stepwise attachments was proposed and investigated experimentally and numerically in this study. For the conventional double lap joint (DLJ), the high shear stress and peel stress taking place in the adhesive layer near the patch termination significantly influenced the joint strength. In order to diminish the amount of the stresses, a new design of stepwise patch was introduced in the fabrication of the double-lap joint. Based on the finite element stress analysis (FEA), it was found that both shear stress and peel stress within the adhesive layer were reduced appreciably by the employment of the stepwise attachment. In addition, experimental results illustrated that the double lap joint with stepwise patch exhibited not only higher joint strength, but also it showed longer fatigue life than the conventional double lap joints.     

Design of adhesive joints based on peak elastic stresses

The paper is focused on the static strength of adhesively bonded structural joints and seeks a simple calculation rule that can assist the designer in everyday engineering practice. The work encompasses three steps. In the first step, an experimental campaign is carried out on an assortment of customized bonded joints (single lap and T-peel) made of steel strips bonded by an acrylic structural adhesive. The dimensions of the joints are chosen so as to produce a wide range of combinations of shear and peel stresses in the adhesive layer. In the second step, the stress analysis of the joints is performed by means of a sandwich model that describes the variability of shear and peel stresses over the overlap length but disregards the stress singularities at the corners. In the third step, a design rule is inferred by noting that, in a chart having as axes the peak values of the peel and shear components in the adhesive at failure, the points—calculated for each joint at the 2% (deviation from linearity) proof load—define a limit zone. The inferred design rule is that the adhesive withstands the load if the representative point of the stress state lies inside this zone. For the tested case, the envelope of the limit zone has an approximately rectangular shape. This criterion predicts the failure load of the joints far better than the simplistic approach based on the nominal stress calculated as the ratio of the load to the bonded area.The paper also discusses the response which is obtained by applying, to the same experimental data, the traditional calculation based on the mean stress (force to area ratio), and the more sophisticated approach based on the stress intensity factor, which accounts for the singularity of the stress field. Applied to our experimental data, the performance of both has been unsatisfactory.     

Stress Distribution in Single Lap Joints with a Cracked Composite Adherend—Part I: Single Lamina Adherends

The effect of a crack in the overlap region of a single step lap joint is studied on the shear distribution in the adhesive layer. Each adherend is considered to be a lamina with unidirectional fibers aligned in the direction of the applied load. Crack location is selected to be in the top adherend, in the form of cut fibers and matrix bays. The shear-lag model is used to derive the equilibrium equations which are then solved using eigenvector expansion. Additionally, a finite element model of the lap joint was prepared and solved using ANSYS. The results of the two methods perfectly match each other. The effects of crack location along the length of the overlap, crack size, edge cracks, adhesive thickness, and type of fibers were investigated on the shear distribution in the adhesive layer and its corresponding peak values. The effect of dissimilar adherends was also investigated on the adhesive shear stress distribution. According to the results, in the presence of a crack, the peak adhesive shear stress is very susceptible to adhesive thickness and type of fibers used in each adherend. Other factors also influence the peak shear stress to some degree.     

Extended analytical model for interfacial stresses of double-lap joints under harmonic loads

In this work, a modified analytical model with closed-form solution is proposed by following the existing framework in the prior works, to analyze the dynamic responses of interfacial shear and peel stresses in adhesively bonded double-lap joints subjected to harmonic axial load. By applying the dynamic equilibrium along the through-thickness direction, the differential equation governing adhesive peel stress is first contained in this model. The dynamic responses of interfacial stresses obtained by finite element simulations under four different loading conditions are used to validate this analytical model. The parametric study based on the proposed model is also implemented to assess the effect of some geometrical parameters on the dynamic response of adhesive interfacial stresses.     

Analysis of the Effect of Piezoelectric Patches on the Behavior of Adhesively Bonded Scarf Joints — An Analytical Study

In order to reduce the maximum peel and shear stress concentrations in the adhesive layer, a smart adhesively bonded scarf joint system was developed by surface bonding of piezoelectric patches onto a typical scarf joint. The forces and bending moments at the edges of the developed smart joint system can be adaptively controlled by adjusting the applied electric field on the piezoelectric patches, thus reducing the peel and shear stresses concentration in the adhesive layer. In order to verify the effect of surface bonding of piezoelectric patches in smart scarf adhesive joints, an analytical model was developed to evaluate the shear stress distribution and to predict the peel stress. It was established that the piezoelectric patched joint could reduce the stress concentrations at the scarf joint edges. The influence of the electric field and the effects of the scarf angle and the adherend Young's modulus on the peel and shear stresses were investigated. It was found that the effect of scarf angle is more significant at higher angles to raise the stresses. The effect of the electric field on the shear stress is more significant than on the peel stress.     

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.     

An efficient analysis model for functionally graded adhesive single lap joints

Employing a functionally graded adhesive the efficiency of adhesively bonded lap joints can be improved significantly. However, up to now, analysis approaches for planar functionally graded adhesive joints are still not addressed well. With this work, an efficient model for the stress analysis of functionally graded adhesive single lap joints which considers peel as well as shear stresses in the adhesive is proposed. Two differential equations of the displacements are derived for the case of an axially loaded adhesive single lap joint. The differential equations are solved using a power series approach. The model incorporates the nonlinear geometric characteristics of a single lap joint under tensile loading and allows for the analysis of various adhesive Young׳s modulus variations. The obtained stress distributions are compared to results of detailed Finite Element analyses and show a good agreement for several single lap joint configurations. In addition, different adhesive Young׳s modulus distributions and their impact on the peel and shear stresses as well as the influence of the adhesive thickness are studied and discussed in detail.     

Three-parameter,elastic foundation model for analysis of adhesively bonded joints

A novel three-parameter, elastic foundation model is proposed in this study to analyze interface stresses of adhesively bonded joints. The classical two-parameter, elastic foundation model of adhesive joints models the adhesive layer as a layer of normal and a layer of shear springs. This model does not satisfy the zero-shear-stress boundary conditions at the free edges of the adhesive layer due to the inherent flaw of the two-parameter, elastic foundation model, which violates the equilibrium condition of the adhesive layer. To eliminate this flaw, this study models the adhesive layer as two normal spring layers interconnected by a shear layer. This new three-parameter, elastic foundation model allows the peel stresses along the two adherend/adhesive interfaces of the joint to be different, and therefore, satisfies the equilibrium condition of the adhesive layer. This model regains the missing “degree of freedom” in the two-parameter, elastic foundation model of the adhesive layer by introducing the transverse displacement of the adhesive layer as a new independent parameter. Explicit closed-form expressions of interface stresses and beam forces are obtained. The new model not only satisfies all boundary conditions, but also predicts correctly which interface has the strongest stress concentration. The new model is verified by continuum models existing in the literature and finite element analysis. The new three-parameter, elastic foundation model provides an effective and efficient tool for analysis and design of general adhesive joints.     

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.     

The Adhesive Bonding of Thermoplastic Cornposites

The present paper reports some initial results on the adhesive bonding of thermoplastic composites, based upon carbon-fibre in a matrix of poly(aryl ether-ether ketone). Both single-and double-overlap joints have been employed and the mechanisms of failure studied using scanning electron microscopy. Further, a theoretical model, based upon a shear-lag analysis, has been used to predict the strength of the double-lap joints as a function of the overlap length and the theoretical results are compared to the experimental data.     

Stress Distribution in Single Lap Joints with a Cracked Composite Adherend—Part II: Laminated Adherends

The effect of a crack in the overlap region of an adhesive single lap joint is studied on the shear stress distribution in adhesive layer. Each adherend is considered to be a laminated composite material with unidirectional fibers aligned in the direction of the applied load. Crack location is selected to be in the top adherend laminate, in the form of cut fibers and matrix bays. The crack can occur in any layer. The shear-lag model is used to derive the equilibrium equations which are then solved by means of eigenvector expansion. The effects of adhesive thickness, crack size, and location in the adherend, total number of layers in each adherends, volume fraction of fibers, and type of fibers are investigated on the shear distribution in the adhesive as well as load distribution in the intact fiber at the crack tip located in the top adherend. The effect of dissimilar laminated adherends is also investigated on the adhesive shear stress distribution. According to the results, in the presence of a crack, the peak shear stress in the adhesive layer and load concentration in the fibers are very susceptible to the adhesive thickness and number of layers in laminated adherends.     

The Adhesive Bonding of Thermoplastic Cornposites

The present paper reports some initial results on the adhesive bonding of thermoplastic composites, based upon carbon-fibre in a matrix of poly(aryl ether-ether ketone). Both single-and double-overlap joints have been employed and the mechanisms of failure studied using scanning electron microscopy. Further, a theoretical model, based upon a shear-lag analysis, has been used to predict the strength of the double-lap joints as a function of the overlap length and the theoretical results are compared to the experimental data.     

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.     

Shear stress distribution in adhesive layers of a double-lap joint with void or bond separation

In this work, stress distribution in adhesive layers of a double-lap joint subjected to tension and suffering from a void or a partial debond at the adhesive–adherend interface is examined. For symmetric voids, the deduced equilibrium equations are decoupled for better application of boundary conditions at the extreme ends of each adhesive layer. The proposed method of solution has resulted in better estimates on peak shear stress developed in the adhesive layers. The results based on analytical solution are compared with those of finite element findings. Very good agreement is observed between the two. The major difference between stresses stemming from debonds and voids occurs at the edge of the large size defects. For small central defects, it is hardly discernible by the stresses to distinguish the type of defect. Moreover, there appears to be an optimum length to thickness ratio for each adhesive layer which produces minimum peak interfacial shear stress. This value seems to be a function of defect size and location. A double-lap joint shows to experience smaller interfacial shear stresses due to a single void or debond in comparison with a single-lap joint with a similar defect. The peak interfacial shear stress in a double-lap joint suffering from symmetric voids or debonds is still lower than that of a single-lap joint with a single defect of the same size and location.     

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.     

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.     

The effects of partial bonding in load carrying capacity of single lap joints

The effects of the presence and size of gaps in the band single lap joint geometry were studied. Two types of adhesives: a deformable, acrylic tape and epoxy putty were used as model adhesives. When using the epoxy putty, the substrate overlap end conditions were also varied by machining 10° end tapers in some joints. For both adhesive types, the introduction of the gap had a moderate negative effect on the load carrying characteristics of the joint, but joints utilizing the epoxy putty maintained joint strength as the gap size was increased to 9.53 mm (38% gap), while the highly deformable acrylic tape case displayed a constant decline and maintaining constant ultimate shear stress values. We suspect that this variation is due to a combination of the different failure modes of each adhesive and their differing moduli, as well as how these relate to the peeling stresses at the ends of the bond length. In the epoxy putty series, the samples with tapered substrates consistently carried higher loads than those with unmodified substrates. This improvement is a manifestation of the ability of the tapered joint geometry to reduce peeling stresses experienced within the adhesive layer.     

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.     

Thermal peel,warpage and interfacial shear stresses in adhesive joints

Thermal stresses are determined in a single lap joint with identical adherends, which are due solely to temperature changes. The simple bending model used here includes bending and extension of the adherends and extensional and shear strains in the adhesive. The analytical solution shows 'sinusoidal' deformation consistent with warpage (bending) of the adherends due to thermal mismatch. While a modified shear lag model (MSLM) with no adherend bending leads to peak bondline shear stresses which occur only at the ends of the overlap, the bending model shows that such stresses occur not only near the ends, but also at interior points of the overlap region. Results for aluminum adherends and an epoxy adhesive show how the peel, warpage and interfacial shear stresses are distributed over the overlap region.