Structural adhesives is a term that is used for any type of adhesive used to bond a load-bearing joint necessary for the structural integrity of a product (Hernon.com, 2018). In a wide range of applications in the automotive, aerospace and marine industries, structural adhesives have slowly begun replacing the conventional fastening method such as welding and riveting onto their structures. The implementation of welding or riveting method in a structure causes stress concentrations at the joint, which may affect the joint strength. Due to the stress concentration, welded joints are prone to fatigue cracking if subjected to high cyclic loads. Hence, as an alternative to reduce the stress concentration, adhesively bonded joints are now widely used in the industries. Besides reducing the stress concentration, the overall cost of a structure can be reduced as mechanical fasteners, such as rivets and bolts, are expensive on a large-scale which in turn reduces the overall weight of the structure. With the elimination of mechanical fasteners, the aesthetics of a structure is enhanced as adhesives are hidden in between adherends.

In industry, with respect to the applied load, joint geometry and component properties, the adhesively bonded joints will experience three different failure types: adherend failure, adhesive failure and cohesive failure. The different failures are illustrated below in Figure 1 (Ghorbani, 2018). Hence, to determine the failure mechanism of a typical bonded joint such as single-lap, double-lap and scarf joints shown in Figure 2, a stress-strain distribution obtained from a finite element analysis based on the cohesive zone model will be carried out.

Figure 2: (a) Single-lap joint; (b) Double-lap joint; (c) Scarf joint (Gunnion and Herszberg, 2006)

2 Literature review

The subject matter of this literature review is based on the existing approaches for a failure analysis of adhesively bonded joints. The failure analysis is conducted by the use of a finite element model such as the cohesive zone model. A parametric study on the bonded joints is also illustrated below.

2.1 Analytical approach for adhesively bonded joints

2.1.1 Elastic analysis

2.1.1.1 Volkersen model

Figure 3: Volkersen model (Quispe Rodriguez et al., 2012)

Volkersen’s method shown in Figure 3 is the first analytical method to determine the stress analysis of adhesively bonded joints. It is also known as the shear-lag model, which introduced the concept of a differential shear where it was assumed that the adhesive could only deform in shear and the adherend deforms in tension due to its elasticity (da Silva et al., 2009).

Figure 4: Deformations in loaded single-lap joints with elastic adherends (da Silva et al., 2009)

Figure 4 shows that the maximum tensile stress is in the upper adherend at point A which decreases to zero at B. This shows a progressive reduction in strain from A to B. The continuity of the adhesive-adherend interface causes a non-uniform shear strain distribution in the adhesive with maximum shear stresses at the ends of the overlap and a much lower shear strain in the middle. However, the eccentric load path governed by the bending effect is not considered in a single-lap joint (da Silva et al., 2009).

2.1.1.2 Goland and Reissner model

Figure 5: Goland and Reissner’s model (da Silva et al., 2009)

Figure 5 shows the Goland and Reissner’s model where the effects due to rotation of the adherends were considered. The model was divided into two parts: the determination of loads at edges of the joints with the use of a finite deflection theory of cylindrically bent plates and the determination of joint stresses due to applied loads. A bending moment factor, k, and a transverse force factor, k’, were used to relate to the applied tensile load per unit width, $\overset{̅}{P}$

to bending moment, M, and transverse force, V, at the overlap ends (Quispe Rodriguez et al., 2012).

2.1.2 Non-linear adhesive analysis

2.1.2.1 Hart-Smith model

In the Hart-Smith model, the adhesive plasticity was accounted for where the shear stress in the joints is modelled by using a bi-linear elastic-plastic shear stress model. The overlap between the adherend is divided into three regions, a central elastic region with a length d and two outer plastic regions as illustrated in Figure 6 (da Silva et al., 2009). In this stress model, the maximum lap-joint strength was determined by using the maximum shear strain as the failure criterion.

Figure 6: Plasticity in the adhesive as introduced by Hart-Smith (da Silva et al., 2009)

In the case of a double-lap joint, thermal mismatch between adherends decreases the overall joint strength and is significant with the increase of adherend thickness and stiffness. For a single-lap joint, the effects of the peel stress are more significant due to the eccentric load path which causes a problem as adherend thickness increases; the total in-plane load carried is able to increase with thickness. However, the through-thickness tensile stresses due to load transfer mechanisms are limited by the transverse tensile strength of the composite (da Silva et al., 2009).

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2.1.2.2 Bigwood and Crocombe model

Bigwood and Crocombe improved the closed-form elastic analysis to account for the non-linear behaviour present in the bonded joints. In the stress model, the non-linear stress response of the adhesive was accounted for and also subjected to different forms of loading (da Silva et al., 2009). To determine the non-linear behaviour of the adhesive, a hyperbolic tangent approximation is used. The yielding of the adhesive was modelled using the maximum von Mises stress criterion. However, the von Mises criterion does not consider the hydrostatic stress that the adhesives are dependent on (Bigwood and Crocombe, 1990).

2.1.2.3 Adams and Mallick

The adhesive joint stresses such as shear, peel and longitudinal are defined as four independent functions. The influence of the adhesive plasticity was taken into account by means of an iterative procedure. Hence, successive load increments were applied onto the model until the maximum stress or strain reaches a failure condition or until the full load is applied. From the stress model, it was found that the adhesive shear strain from the model followed the shear strain distribution predicted by Volkersen as explained in 2.1.1.1 (da Silva et al., 2009).

2.1.3 Non-linear adhesive and adherend analysis

2.1.3.1 Wang et al. model

From the theory proposed by Bigwood and Crocombe, Wang et al. further extended their work to take into account of the adherend shear deformation to predict the adhesive failure in arbitrary joints subjected to a large-scale adherend yielding. Both the adhesive and adherend yielding were modelled based on the von Mises criterion with the hyperbolic tangent approximation for adhesive yielding and a bi-linear model for adherend yielding (da Silva et al., 2009).

2.1.3.2 Adams et al. model

Adams et al proposed a simple design methodology based on adherend yielding. The methodology proposed is applicable to non-yielding adherends with ductile adhesives with 10% or more shear strain to failure. It is also applicable to adherends that are able to yield with any type of adhesive, either brittle or ductile. This can be done with simple calculation. However, with intermediate or brittle non-yielding adherends and adhesives, a finite-element method was found to be more efficient and accurate (da Silva et al., 2009).

2.2 Failure criteria for adhesively bonded joints

2.2.1 Maximum stress or strain criterion

The maximum stress or strain criterion is known to be the starting point for joint strength predictions as it assumes that the adhesively bonded joint will fail when a critical value of stress or strain is reached at any point within the joint. Within this criterion, there are different quantities that were used to predict the bonded joint strength:

Maximum shear stress criterion: Based on the approach used by Greenwood et al., the maximum shear stress was assumed to be the main failure criteria in the bonded joints. Through the formulation developed by Goland and Reissner, it was determined that the maximum shear stress occurs at about 45° across the adhesive layer (Quispe Rodríguez et al., 2012).
Maximum peel stress criterion: Peel stress is a stress that is concentrated along a thin line at the edge of the bond where one substrate is flexible. Hence, this stress acts at a smaller area of the bond, which cause failure at a lower force level (Adhesives.org, 2018). Hart-Smith considered this criterion where he distinguished between the failure occurring in the adhesive or in the adherend.
Maximum von Mises stress criterion: The von Mises stress or equivalent stress is used as a failure criterion for bonded scarf joints. This criterion neglects the hydrostatic stress, which affects the yield and deformation behaviour of polymers. Hence, it is ineffective on double lap joints (Crocombe and Kinloch, 1994).
Maximum effective uniaxial plastic strain criterion: Crocombe and Adams used the maximum effective uniaxial plastic strain to predict the failure in peel joints by using a large displacement elastoplastic finite element analysis (Crocombe and Kinloch, 1994).

2.2.2 Critical stress or strain at a distance over a zone criterion

Due to the presence of singularities, this type of criterion is used because of the high dependency on the mesh with the maximum value criteria shown in 2.2.1. Three approaches were used to determine this criterion:

Zhao, Adams and da Silva proposed using a weighted average maximum stress criterion where the adhesive thickness is used as the distance over which the stresses are averaged and is compared to the adhesive yield strength. However, it was determined that the averaged stress failure criterion only applied to sharp corners and to a small radius configuration. For a larger radius configuration, the maximum stress criterion must be used instead (Zhao et al., 2011).
Clarke and McGregor proposed a different approach where the maximum principal stress from an elastoplastic analysis must exceed the ultimate tensile stress of the adhesive over a finite zone where the zone size is independent of the joint geometry. In this approach, detailed finite element models of the joints were constructed and the stress patterns were compared in the adhesive by means of measured failure loads (Clark and McGregor, 1993).
Towse et al. proposed to use the critical strain at a distance for a double-lap joint. With the use of a non-linear analysis along with the effect of residual thermal stresses, the joint was shown to fail when the strain near the singularity reached the adhesive ultimate strain (Quispe Rodríguez et al., 2012).

2.2.3 Limit state criterion

Crocombe introduced the limit state analysis that was known as the global yielding failure criterion. Global yielding is applied when a path of adhesive along the overlap region reaches the state in which the adhesive can no longer sustain any significant increase in applied loads. Three different joint geometries, single-lap joint, double-lap joint and compressive shear were analysed using non-linear finite element techniques (Crocombe, 1989).

The single-lap joint study was to understand the effect of adhesive layer thickness on the joint strength. From the study, it was shown that the thinner adhesive layer yielded at an earlier load. However, the thicker adhesive layer was the first to reach a state of complete yield (Crocombe and Kinloch, 1994).
The double-lap joint study was to investigate the effect of adherend cladding on the joint strength. The double-lap joint model was modelled similarly to a single-lap joint study but with different constraints applied to the nodes and along the line of symmetry. In this mode of study, the adhesive plasticity was modelled while the adherend remained elastic. The limit state criterion is only applicable to a limited range of adhesive joints.
The compressive shear test was carried out to investigate an alternative method of loading a joint in shear whilst avoiding the adhesive tensile transverse direct stresses that are present in single and double lap joints. In this study, it was shown that the yielding of the adhesive is that the initial yield does not occur at the loaded adherend corner but is within the adhesive itself (Crocombe, 1989).

2.2.4 Fracture mechanics criterion

Fracture mechanics is the study of the strength of structures that contains flaws such as cracks. Hence, it is assumed the flaws present in the adhesive joint are a result of imperfect bonding or manufacturing defect. The flaws can grow slowly at first during service, then rapidly upon reaching a critical size which is dependent on the material properties and the nature of loading. The critical stress intensity factor, K_c is used to predict failure in fracture mechanics that determines the onset of rapid fracture (Sih, 1980). In a principle set out by Griffith, a brittle system that contains flaws will fail when the energy of the structure supplied to the crack tip under a loading (energy release rate G) is equal to the energy required for the crack to propagate (critical energy release rate G_c) (Crocombe and Kinloch, 1994).

2.2.5 Damage mechanics criterion

The damage mechanics criterion can be determined at different levels of complexity. The first step would be to remove the material when a critical condition is exceeded. As a continuation to that step would be to allow the material to soften gradually up to a state when it can no longer carry any load. The last step, also known as the continuum damage mechanics, would be to define the damage criterion and combine with the yield criterion in a constitutive model (Crocombe and Kinloch, 1994). The continuum damage criterion has proved to be efficient in damage modelling and is based on the damage parameter that is introduced as a new state variable, which is defined as the effective surface of micro-cracks and cavities that intersect with a plane. The damage parameter that is introduced is to describe the evolution of stress as damage progresses, with the concept of an effective stress. The initiation of a macro-crack takes place when an accumulated damage reaches a critical value where the accumulation of damage is expressed in terms of number of cycles to failure in fatigue cases or time to rupture in creep cases (Khoramishad et al., 2011).

2.3 Finite element analysis

2.3.1 General

Finite element analysis is a computational method that subdivides a CAD model into very small but finite-sized elements of geometrically simple shapes, which constitutes into a finite-element mesh (Sjodin, 2016). To determine the failure mechanism in a typical bonded joint, a numerical analysis can be conducted to obtain the stress distribution and the failure criterion. In ABAQUS, a finite element modelling software, a damage-modelling database known as the cohesive zone model can be used to analyse an adhesively bonded joints.

2.3.2 Cohesive zone model

The cohesive zone model is a standard model in ABAQUS used to describe the crack tip process zone that assumes the bonds stretch orthogonally to the crack surfaces until they break at a characteristic stress level. A pre-defined crack path has to be defined to model the progressive damage and failure in a CZM approach. The CZM simulates the macroscopic damage along the path by specification of a traction-separation response between initially coincident nodes on either side of the pre-defined crack path (Liljedahl et al., 2006). Linear elastic fracture mechanics (LEFM) is an efficient tool for solving fracture problems provided a crack-like notch or flaw exists in the body and the non-linear zone ahead of the crack tip is negligible (Elices et al., 2002). The singular region introduced from linear elastic fracture mechanics (LEFM) can be replaced by a lateral region over which non-linear phenomena occurs. In the simple formulation of a CZM, the fracture process is merged into the crack line and is characterised by a cohesive law that relates tractions (T) and displacements ( $∆$

) jump across cohesive surfaces. The whole body volume remains elastic while the non-linearity that is embedded in the cohesive law is the cohesive strength of the material $σ_{c}$

while the area under the curve is the cohesive fracture energy, $Γ$

_c (Alfano et al., 2007).

Figure 7: Cohesive zone modeling of fracture (Alfano et al., 2007)

The fracture process can be seen in Figure 7:

Section 1: A linear elastic material response prevails
Section 2: As load increases, the crack initiates (T= $σ_{c}$
)
Section 3: Governed by a non-linear cohesive law, the crack evolves to complete failure
Section 4: Appearance of new traction free crack surfaces ( $∆$
= $∆_{c}$
)

2.4 Parametric study

There has been an increase in the use of adhesive bonding in different industries due to its suitability in different criteria such as a high strength to weight ratio, design flexibility, damage tolerance and fatigue resistance. Hence, the parameters that would affect the performance of the bonded joints are illustrated below.

2.4.1 Joint configuration

The strength of the joint is dependent on the applied load and its stress distribution within the joint, which is affected by the joint geometry. The adhesively bonded joints should be designed to reduce the stress concentration where peel and cleavage stress are minimised and shear and compressive stress are maximised. A single lap joint is the most common joint used in the industry due to its simplicity and efficiency. However, the stress distribution, shear and peel stress, in the joint is concentrated at the ends of the overlap. To improve the efficiency of the joint, different techniques have been proposed such as changing the adherend geometry, adhesive geometry and spew geometry (Banea and da Silva, 2009).

2.4.1.1 Adherend geometry

In a study carried out by Çalık, six models with different end geometries of an adhesively bonded single lap joint were generated as shown in Figure 8. Tapering, recessing and stepping are the ideal geometry alteration method, as fabrication of the joints would be made simpler. In the analysis, it was found that failure began at the ends of the overlap length as the maximum stress concentration occurs at the end of overlap length of the adhesively bonded joints. The peel stresses at the edges of the overlap are important as peel stresses may cause the initiation and propagation of failure in the region. It was concluded that altering the geometry of a bonded joints would have a significant effect on the stress concentration in the joints (Çalık, 2016).

Figure 8: Different geometries of adhesively single lap joints (Çalık, 2016)

2.4.1.2 Spew geometry

Spew is known as the portion of adhesive that is squeezed out from the lap area and forms a bead at the lap ends as the two adherends are assembled together. In a study carried out by Lang and Mallick, eight different spew geometries, as shown in Figure 9, were modelled and analysed.

Figure 9: Different spew geometries on adhesively bonded single lap joint (Lang and Mallick, 1998)

An initial model without the spew was generated to act as a constant for the experiment. The joint, both adherend and adhesive, were assumed to behave in linear elastic. From the experiment, it was determined that with the presence of spew in the bonded joints, there was a significant reduction on stress concentration within the joints. The most significant effect was the reduction in peel stress on the edges of the overlap of the adhesively bonded joints. This reduction is important during design, as the peel stresses are generally high in a single-lap joint, which causes premature failure in the joint. As the spew size increases, the larger the portion of load it can carry. Thus, the presence of spew reduces the stress concentration occurring on the bonded joints (Lang and Mallick, 1998).

3 Conclusion

This report has provided a brief literature review on the mechanics adhesively bonded joints. Three different analytical methods were carried out by different researchers: an elastic analysis by Volkersen and Goland and Reissner, a non-linear adhesive analysis by Hart-Smith and Bigwood and Crocombe along with Adams and Mallick and a full non-linear adhesive and adherend analysis by Wang et al. and Adams et al. The elastic analysis carried out became a starting point for the stress analysis of adhesively bonded joints but the limitation of an elastic analysis is that the variation of adhesive stresses through the thickness of the adhesive joints is not taken into account. To improve the stress analysis of the bonded joints, a non-linear adhesive and a full non-linear adherend and adhesive analysis were carried out which took into account the different adhesive stresses that are present within an adhesively bonded joints.

The failure criteria of the adhesively bonded joints form the basis of the analytical model of a stress analysis. It was determined that the maximum stress or strain criterions such as the von Mises and the peel stresses are commonly used as the failure criterion in the analytical mode. It was shown that, in a single-lap joint, the peel stress is the main cause of failure. This was due to a stress concentration that occurs at the edges of the overlap length. From this study, it can be determined that the stress concentration of bonded joint can be reduced with the alteration of adherend geometry and spew geometry as explained in 2.4.1.1 and 2.4.1.2 respectively. The stress analysis of adhesively bonded joint can be carried out by using a finite element analysis. In ABAQUS, the cohesive zone model is used to analyse the stress concentration in the adhesive joints.

To further improve the research, a stress analysis through the analytical method and finite element method will be carried out using different typical bonded joints such as the double-lap joints and scarf joints. Hence, the failure mechanism of the bonded joints can be determined from the analysis.

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