Non-linear Modeling of Tubular Adhesive Scarf Joints Loaded in Tension

In this paper, a simple analytical model is developed to determine the adhesive shear strain distribution of a tubular adhesive scarf joint loaded in tension. The approach is an extension of the original well-recognized Volkersen's shear lag analysis for a shear loaded joint, which is frequently applied to adhesively-bonded joints. A mathematical representation consisting of linear and exponential functions is employed to model the elastic-plastic behavior commonly observed in structural adhesives. The governing equation is found to be in the form of a non-linear second-degree ordinary differential equation with variable coefficients. A numerical method required for solving this equation is also introduced. Numerical predictions of shear strain distributions are compared with results from non-linear Finite Element Analysis (FEA), utilizing the commercially available software, ANSYS 5.6, a general-purpose software system. It is shown that both the linear and non-linear approximate solutions are closely comparable with the FEA results for a 10°-scarf angle and elastic isotropic adherends. In concurrence with previous work on flat adherends, the present work demonstrates that the scarf joint develops more uniform shear stress and strain distributions with a consequent reduction in peak values than those for the conventional lap joint. In contrast, the conventional lap joint with the equivalent bonded surface area experiences a more substantial elastic trough, which can provide a more stable configuration for, sustained long term loading applications.     

Shear and peel stress analysis of an adhesively bonded scarf joint

The shear and peel stress distributions in a scarf joint made of two isotropic adherends with blunt adherend tips are analysed using a linear elastic analysis. The limits of the analysis with respect to adherend tip thickness have been investigated. A finite difference method is used to solve the differential equations for the shear and peel stress distributions over the joint. The boundary conditions used limit the analysis to the two adherends having the same thicknesses, lengths, and material properties. The adherends are modelled as plates with extensional and bending stiffnesses bonded together with an elastic interlayer. The stresses across the adhesive layer are assumed to be constant. The current analysis applied to cases known from the literature shows good agreement with the shear stresses but the peel stresses are overestimated.     

The determination of dynamic strength of single lap joints using the split Hopkinson pressure bar

The current investigation focuses on the determination of the strength of adhesive-bonded single lap joints under impact with the use of a split Hopkinson pressure bar (Kolsky bar). For this, experiments were conducted at different loading rates, for identical metallic adherends bonded by a two-part epoxy adhesive. Four different types of specimens were adopted, all with a given adhesive thickness. The length of overlap and the width of the adherends were varied resulting in four different areas of overlap. It was found that the average strength, as calculated from the readings obtained from a Kolsky bar, increases with decrease of overlap area. An elastodynamic model for the shear strain of the adhesive-bonded single lap joint was developed to investigate this drastic effect of overlap area on the average strength of the joint. The mathematical model was found to be dependent on both the material properties of the adherend and adhesive, as well as the structural properties of the joint, viz. the width and the thickness of the adhesive layer. A combined experimental-numerical technique was used to predict the strain distribution over the length of the bond in the adhesive. It was found that the edges of the adhesive were subjected to maximum strain, while a large part of the adhesive was found to exhibit zero shear strain. The effect of the lap length and the width was studied individually. The cumulative effect of averaging the strain over the entire overlap area, was decreased shear strain for an increased overlap area. The Kolsky bar was identified to give conservative values of the shear strength of an adhesive bonded lap joint under high rates of loading.     

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.     

Stress Analysis of Generally Asymmetric Single Lap Adhesively Bonded Joints

An analysis is presented that predicts shear and peel stresses in an adhesively bonded single lap joint having general asymmetric configuration. The single lap joint is under tension loading together with moments induced by geometric eccentricity. Because these eccentricity moments are the key elements of this analysis, a general relationship between the eccentricity moments and simple geometric moments has been determined with the aid of finite element analysis (FEA). Example calculations show that the shear- and peel-stress profiles from the closed-form model are well matched to FEA results except in the small regions near the free ends of the joints, because of the shear lag basis of the model. For asymmetric joints, the model predictions are more accurate for the case of modulus eccentricity than thickness eccentricity. Elastic-limit load predictions accounting for both shear and peel stress in the adhesive have been used to find optimal joint configurations between asymmetric adherends.     

Comparison of fracture behavior between acrylic and epoxy adhesives

An experimental study was conducted on the strength of adhesively bonded steel joints, prepared epoxy and acrylic adhesives. At first, to obtain strength characteristics of these adhesives under uniform stress distributions in the adhesive layer, tensile tests for butt, scarf and torsional test for butt joints with thin-wall tube were conducted. Based on the above strength data, the fracture envelope in the normal stress-shear stress plane for the acrylic adhesive was compared with that for the epoxy adhesive. Furthermore, for the epoxy and acrylic adhesives, the effect of stress triaxiality parameter on the failure stress was also investigated. From those comparison, it was found that the effect of stress tri-axiality in the adhesive layer on the joint strength with the epoxy adhesive differed from that with the acrylic adhesive. Fracture toughness tests were then conducted under mode l loading using double cantilever beam (DCB) specimens with the epoxy and acrylic adhesives. The results of the fracture toughness tests revealed continuous crack propagation for the acrylic adhesive, whereas stick-slip type propagation for the epoxy one. Finally, lap shear tests were conducted using lap joints bonded by the epoxy and acrylic adhesives with several lap lengths. The results of the lap shear tests indicated that the shear strength with the epoxy adhesive rapidly decreases with increasing lap length, whereas the shear strength with the acrylic adhesive decreases gently with increasing the lap length.     

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.     

Scarf repairs to highly strained graphite/epoxy structure

Scarf joints representative of repairs to graphite/epoxy (gr/ep) honeycomb structure, typical of that used in the F/A-18 horizontal stabilator, have been investigated using mechanical tests and finite-element (FE) modelling. The load capacity of such scarf repairs is marginal when tested under hot/wet conditions compared to the required design ultimate strain of 5200 με. An analysis of the scarf joint using FE methods predicts that high stresses will occur at ply drop-offs and at the top of the scarf and good correlation is observed between the FE models and experimental results. Detailed predictions of the shear stress distribution within the joint have also been obtained from the FE models. These show that the shear stress in the adhesive is not uniform over the length of the scarf, as would be the case for isotropic adherends, but rather varies with position by up to 250%, due to the varying longitudinal compliances of the plies within the gr/ep adherends. Experimental evidence indicating the occurrence of stress relaxation or creep within the scarf joints is also reported. An F/A-18 stabilator containing a scarf repair has been loaded satisfactorily to design limit load without failure. External patches have also been shown to be effective as temporary repairs.     

Progressive damage modeling of adhesively bonded lap joints

A continuum damage model for simulating damage propagation of bonded joints is presented, introducing a linear softening damage process for the adhesive agent. Material models simulating anisotropic non-linear elastic behavior and distributed damage accumulation were used for the composite adherends as well. The proposed modeling procedure was applied to a series of lap joints accounting for adhesion either by means of secondary bonding or co-bonding. Stress analysis was performed using plane strain elements of a commercial finite element code allowing implementation of user defined constitutive equations. Numerical results for the different overlap lengths under investigation were in good agreement with experimental data in terms of joint strength and overall structural behavior.     

Elasto-Plastic Analysis of Adhesively Bonded Symmetric Single Lap Joints Under In-Plane Tension and Edge Moments

An analysis is presented that predicts adhesive shear and peel stresses and strains in an adhesively bonded single lap joint having symmetric configuration with adhesive behavior. The single lap joint is under tension loading together with moments induced by the interactions of the geometric eccentricity and the boundary conditions of the joint. The von Mises yielding criterion is used to relate the adhesive stress components within the yielded region. The adhesive strains are computed from the relative displacements of the adherends and can be considered as an average of the strain variation through the adhesive thickness direction. Example calculations show that the predicted adhesive shear and peel stress and strain profiles are well matched to detailed finite element analysis results. Generally, the analytical model predictions are found to be more accurate when the adhesive thickness is small.     

Elasto-Plastic Analysis of Adhesively Bonded Symmetric Single Lap Joints Under In-Plane Tension and Edge Moments

An analysis is presented that predicts adhesive shear and peel stresses and strains in an adhesively bonded single lap joint having symmetric configuration with adhesive behavior. The single lap joint is under tension loading together with moments induced by the interactions of the geometric eccentricity and the boundary conditions of the joint. The von Mises yielding criterion is used to relate the adhesive stress components within the yielded region. The adhesive strains are computed from the relative displacements of the adherends and can be considered as an average of the strain variation through the adhesive thickness direction. Example calculations show that the predicted adhesive shear and peel stress and strain profiles are well matched to detailed finite element analysis results. Generally, the analytical model predictions are found to be more accurate when the adhesive thickness is small.     

Application of the design of experiments procedure to the behaviour of adhesively bonded joints with plastically deformable adherends to enable further understanding of strain rate sensitivity

The study presented in this paper was carried out to investigate further the effects of strain rate on the strength of adhesively bonded single lap shear joints. Tests were carried out on two different configurations of adhesively bonded joints that were designed to exhibit different behaviours. In one configuration both adherends were made from a relatively low strength grade of aluminium such that both would exhibit significant plastic deformation prior to adhesive failure. The other configuration used one adherend that was significantly stronger such that only elastic deformation was exhibited prior to failure of the adhesive. The joint specimens were tested at several different strain rates using a servo-hydraulic test machine and the results analysed using statistical methods. To further understand the results Finite Element models of the joints were created using a Cohesive Zone Model to predict damage development and failure in the adhesive. The Design of Experiments procedure was used to study the effects of material parameters relating to both the adherends and the adhesive in the Finite Element models. The results of the testing suggested that the strength of joints formed from two adherends that exhibited plastic deformation prior to failure did not show statistically significant sensitivity to strain rate. Interpretation of the results of the Finite Element analyses suggested that the adherend yield was the main factor influencing failure load in the adhesive for joints of this type.     

The effect of adhesive thickness on tensile and shear strength of polyimide adhesive

The effect of adhesive thickness on tensile and shear strength of a polyimide adhesive has been investigated. Tensile and shear tests were carried out using butt and single lap joints. Commercially available polyimide (Skybond 703) was used as adhesive and aluminum alloy (5052-H34) was used as adherends. The tensile strength of the butt joints decreased with increasing adhesive thickness. In contrast, adhesive thickness did not seem to affect the shear strength of single lap joints. The fabricated joints using the polyimide adhesive failed in an interfacial manner regardless of adhesive thickness. The linear elastic stress analysis using a finite element method (FEM) indicates that the normal stress concentrated at the interface between the adherend and the adhesive. The FEM analysis considering the interfacial stress well explains the effect of adhesive thickness on the joint strength.     

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.     

Optimization of adhesively-bonded single lap joints by adherend notching

The effect of adherend notching on the strength and deformation behavior of single lap joints was investigated. First, a parametric study was conducted using finite element analysis (FEA). This initial part of the research into the effect of notches on joint behavior involved determination of the optimum notch location and notch dimensions. This was done by using FEA in a series of models with different notch positions and geometries. The results of this parametric study were used to select the most promising lap geometries for further study. Next, more detailed FEA were conducted on the selected lap geometries. These data were compared with the experimental single-lap shear test results to assess the applicability of different failure criteria. Three different model adhesives were used: a rubber toughened film epoxy with nylon carrier, a styrene-butadiene-styrene block copolymer based deformable 'gel' adhesive, and a two-part, metal filled brittle epoxy adhesive. The FEA for single lap joints containing 'top notches' on the unbonded, top side of the adherends, at locations corresponding to the overlap ends, and bonded with the two-part metal filled epoxy provided the best agreement with the experimental results. The experimental results showed a 29% increase in joint strength with the introduction of the notches, which matched very well with the 27% decrease in the peak peel stress observed by the FEA results. For this brittle adhesive, the peel stress is almost certainly the governing failure stress. This was confirmed by matching of the FEA peak peel stress ratios with the experimental load ratios, for both the notched and unnotched specimens.     

Transverse cracking of a simple lap joint under axial load

The failure mode of axially loaded simple, single lap joints formed between thin adherends which are flexible in bending is conventionally described as one of axial peeling. We have observed – using high-speed photography – that it is also possible for failure to be preceded by the separation front, or crack, moving in a transverse direction, i.e. perpendicular to the direction of the axial load. A simple energy balance analysis suggests that the critical load for transverse failure is the same as that for axial separation for both flexible lap joints, where the bulk of the stored elastic energy lies in the adhesive, and structural lap joints in which the energy stored in the adherends dominates. The initiation of the failure is dependent on a local increases in either stress or strain energy to some critical values. In the case of a flexible joint, this will occur within the adhesive layer and the critical site will be close to one of the corners of the joint overlap from which the separation front can proceed either axially or transversely. These conclusions are supported by a finite element analysis of a joint formed between adherends of finite width by a low modulus adhesive.     

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.     

Failure load prediction of structural adhesive joints : Part 2: Experimental study

The objective of this study was to determine how the fracture of adhesive joints depends on elastic beam parameters describing the adherends and the applied loads. The basic specimen geometry was the cracked lap shear joint constructed of aluminium alloy with various adherend and bondline thicknesses. Loads were applied in different combinations of bending, tension and shear to generate a failure envelope for each adhesive and specimen geometry. It was found that crack propagation for precracked specimens occured at a critical strain energy release rate but was also a function of the G_I/G_II ratio and the bondline thickness. The experiments also showed that the loads required to propagate a crack in a precracked specimen were always lower than the loads required to break the fillet. Hence, by treating uncracked joints as being cracked, where the fictitious crack tip is assumed to coincide with the location of the fillet, a conservative estimate of the failure load is obtained.     

A method to predict fracture in an adhesively bonded joint

The application of a fracture criterion, formulated in terms of material-induced stress/strain singularities at the terminus of an adhesive joint, to cohesive fracture in a single lap joint is presented. The criterion can be interpreted physically in terms of the elastic strain energy density. The strength of the singularities depends on the elastic properties of the adhesive and adherends and the geometry of the bond terminus, but is independent of loading and global geometry. A finite element method is used to predict the limit load of an adhesively bonded single lap joint from a known value of Q_crit, the critical singular intensity factor. This method may be applicable to general joint geometries.     

Three-dimensional finite element analysis of single-lap adhesive joints subjected to impact bending moments

This paper deals with the stress wave propagations and stress distributions in single-lap adhesive joints subjected to impact bending moments with small strain rate. The elastic stress wave propagation and the stress distribution in single-lap adhesive joints of similar adherends subjected to impact bending moments are analyzed using three-dimensional finite-element method (FEM). A three-point impact bending moment is applied to the joint by dropping a weight. FEM code employed is DYNA3D. The effects of Young's modulus of the adherends, the lap length, the adherend thickness and the adhesive thickness on the stress wave propagation at the interfaces are examined. It is found that the maximum value of the maximum principal stress, σ₁, appears at the interface between the adhesive and the upper surface of upper adherend which is impacted. The maximum stress, σ₁, increases as Young's modulus of adherends, the lap length and the adhered thickness increase. It is also found that the maximum stress, σ₁ increases with decreasing adhesive thickness. In addition, experiments were carried out to measure the strain response of single-lap joints subjected to impact bending moments using strain gauges. A fairy good agreement was observed between the numerical and experimental results.