Modelling bonded joints - don't become unstuck

TWI Bulletin, November/December 2001

Terry Dickerson joined TWI in 1990, previously studing for a BSc in Mechanical Engineering at Coventry Polytechnic, followed by a PhD at Nottingham University. While at TWI, Terry has worked in many technical areas; however his core proficiency is finite element modelling.

Stephanie Feih is a PhD student at Cambridge University. Her research focuses on the design and numerical optimisation of composite adhesive joints. The project is funded by EPSRC in collaboration with TWI as part of the postgraduate training programme (PTP) established in the UK. Stephanie has a diploma degree in Mechanical Engineering from the University of Darmstadt, Germany, and a Master's degree from Cornell University, USA. Her previous research projects focused on composites and numerical modelling.

Adhesives are defined in a British Standard ^[1] as 'a non-metallic substance capable of joining materials by surface bonding, the bond possessing adequate internal strength'. As Terry Dickerson and Stefanie Feih report this covers a wide range of substances used in diverse applications from microelectronics through to large structures such as aircraft.

A typical industrial application - bonding of a car roof on to the body shell - is shown in Fig.1. Some of the advantages of adhesive bonding are:

• Ease of assembly (fills gaps)
• Load spreading (compared to mechanical fastening) and therefore improved fatigue life
• Suitable for joining dissimilar materials
• Low temperature joining process (<200°C) compared to alternative welding techniques

Introductions to adhesive bonding are given in refs ^[2-6] . Typical sections of bonded joints are shown in Fig.2. The lighter coloured parts are the parts being bonded (adherends), and the darker parts are the adhesive. Also, as shown in Fig.1, adhesive bonds can form non-flat, non-straight bond surfaces.

Adhesives can be relatively easy to use and in some cases minimal joint design is needed. However, in most cases a low-cost, efficient and reliable design is required and joints should be specifically designed for bonding. Non-critical or straightforward joints can be designed using simple formulæ relating the load to the joint area and the bond strength. Furthermore, there are analytical solutions for relatively complex loading conditions on idealised joints. However, such formulæ contain simplifications and many joints and structures containing joints are not suitable for detailed analyses using analytical solutions. In such cases it is routine to use numerical modelling methods to carry out any analysis of the joint or to incorporate the analysis of the bonds into component or assembly models.

The numerical models will typically use finite element modelling (FEM) techniques that are now commonplace and use commercial programs. However, modelling adhesive bonds takes specialist knowledge. This article passes on some of the tricks-of-the-trade and highlights potential pitfalls when modelling adhesively bonded joints. The advice is mainly given in the form of examples with problems and solutions. However, this article does not cover all aspects of modelling bonded joints and more detailed guides can be found in refs seven and eight.

Planning the analysis

Two major questions need to be asked before embarking on any analysis:

• What is the reason for any analysis and what do you want the analysis to tell you?
• How will the output of the analysis ( eg displacements, stresses...) be used to help answer the first point?

There will be more detailed modelling points that will need to be answered, but now let us get more specific and consider bonded joints. Planning an analysis of an adhesive bonded component is not significantly different to that for other components; if needed the reader can consult a NAFEMS booklet [9] for guidance. Including all details of adhesively bonded joints into a model of a component will complicate an analysis because:

• Joints have features that may be much smaller than the dimensions of the structure. For instance, in 3D solid models, thin bond lines will require large numbers of elements to model the adhesive without having severely distorted mesh.
• In-plane loading of joints with misaligned load paths ( ie the single lap joint shown in Fig.2a) produces local bending stresses. A non-linear analysis is normally required to describe this behaviour accurately if these deformations are sufficiently large.
• There is often a large mismatch in material properties in the joint, which can lead to numerical complications as some elements deform excessively during the analysis.

There can be a number of unknowns related to the component for which the model is being produced:

• Joint geometry may vary due to manufacturing tolerances - bond line thickness can vary considerably and the adhesive fillet or spew may not be controlled or of complex shape.
• The properties of adhesives are not readily available and may be sensitive to cure conditions, temperature, strain rate, or moisture up-take.
• The joint may be incompletely filled, and the adhesive may contain voids or regions of poor bonding.

The modelling objective is of primary importance when deciding on the required accuracy of the numerical model. If the overall deflection or stiffness of a structure containing joints is required, most joint details can normally be omitted. If the analysis has the aim of performing an investigation of adhesive stresses and joint failure, detailed modelling is of importance.

Practicalities such as how well the thickness of the bond line and fillet details can be controlled have to be considered.

Examples

Incorporating joints into component models

To model components using 3D solid meshes, it is usually impractical to include full geometric detail of the joints because the model size would increase to unmanageable proportions. Also many bonded components are made of sheet and hence are amenable to modelling with plates or shells. However, the adhesive is better represented with solid elements. It is not always easy or possible to combine the two types of element. Therefore joints are usually simplified. At the simplest level the joint can be left out of the model; however, where an overlap joint causes local thickening it is prudent locally to thicken the elements to represent the extra stiffening. Figure 3 shows models of a pressure-loaded panel. The model, on which the example is based, was much larger and the solid element model as shown in Fig.3c was impractical. Thickening the bond region in the shell model had minimal effect on computation time but introduced some stiffening to simulate the model deformation more accurately. This example shows that even though the joint is not explicitly modelled some features can still be included without significantly increasing the model size.

Detailed models of bonded joints

If the bond line in a structure is reasonably wide (more than ten times joint thickness), edge effects can normally be neglected. In addition, axisymmetric components have no edge. In such cases joints can usually be modelled with 2D meshes, which reduces computation time. Consequently any bonds can usually be explicitly modelled while the model size is kept manageable. The component may be modelled to investigate the influence of joint parameters and hence subsequently optimise the joint design. Detailed models may also be produced to determine the stiffness characteristics of a joint. These stiffnesses can be used in larger component models as, for instance, linear or non-linear springs to represent the joint flexibility or by using a sub-structuring technique. Hence there are a number of reasons for producing detailed models of joint regions.

In some cases, joint behaviour can be extremely sensitive to model details. Figure 4 shows two half-symmetric models of a T-joint, where the adhesive layer is situated along the centre-line. In (a) a small amount of adhesive spew is present between the radii, which in (b) is not present. This is the sort of detail that can easily change in a batch of production joints. The spew changes the load flow path across the joint influences the joint stiffness and the stresses. It should be emphasised that these kinds of influences will be dependent on the relative stiffness of the adhesive compared to the rest of the joint, the innate stiffness of the component and the boundary conditions.

Traditionally, fillet optimisation has only been performed theoretically, as manufacturing problems prevent full use of this information. However, TWI recently filed a patent for its latest joining system - AdhFAST TM. This novel, three-in-one fastener allows adhesive to be injected through the middle of the device whilst retaining the joint and controlling bond line thickness. Additionally, the bond line is kept clean. Sealing the area prior to the injection with appropriate shaping materials can produce requested fillets. Figure 5 shows this technique with a single L-joint, a variation of the T-joint shown earlier. The radius is specified during production by adding a fillet shaped aluminium plate on the bottom, which is removed after the adhesive has cured. Fillets with about 3mm radius show excellent resistance against different loading conditions. The bond line thickness is established by the AdhFAST system. Injection is achieved from the top corner of the curved adherend. With such techniques any fillet optimisation carried out by numerical modelling can be defined and controlled in production.

Adhesive material properties

Even for some metals, reliable materials data to use in FEA models can be difficult to find. With adhesives the problems can be magnified many fold! The material properties are often dependent on hardener-resin ratio, cure conditions, strain rate, or water up-take. Different materials are affected by different conditions and over different periods. For instance, most epoxies will be fully cured within a few days, but some acrylic adhesives may carry on hardening for weeks or months. The effects of cure schedule and temperature on the stress-strain properties are shown in Fig.6a. The dramatic change in strength and ductility with temperature of some adhesives is shown in Fig.6b. Large changes are often attributed to crossing the adhesive glass transition temperature. TWI has a database of properties for some common adhesives from which Fig.6 was constructed, but these are always used with caution.

These comments regarding material behaviour suggest that it is nearly impossible to model adhesive joints accurately. However, it can be shown that adherends in many joints constrain the adhesive's deformation in such a way that the adhesive becomes insensitive to its specified properties. Figure 7 shows two similar models of a microelectronic component. On a large scale there is little difference between deformations and stresses predicted, even though elastic properties were used in one model and a very low adhesive yield stress was used in the other. This demonstrates that joints can be insensitive to adhesive characteristics. In this case constraint is offered by the adherends that prevent the adhesive yielding over most of the bond area. However, if the models were examined more closely at the free edge, local adhesive yielding would be seen and this limits the local stress at the joint edge, which is usually where failure initiates in these components.

a) Linear-elastic model
b) Elastic-plastic adhesive with yield = 15 N/mm ².
Note: In this model, on the die top-surface a combination of tension caused by bending and membrane compression interact to give a stress that is almost zero

Meshing: Element type and density

• Type of elements used for the analysis
• Mesh density

For models of components built with beam or shell elements, meshing is generally no more problematic than for any other model. This is because the geometric details of the joint normally cannot be represented with a beam or shell model, detailed mesh refinement becomes unnecessary. However, it is useful to be able to define the location of the joints so that, for instance, local thickening can be applied. This was seen earlier.

To save computational time, shell and solid elements can be combined. The latter are required to resolve the three-dimensional stress field in the adhesive if these stresses are of interest. A potential problem is caused as the element types have different active degrees of freedom and so are not strictly compatible. Some FEA programs have multi-point constraints (MPCs) to enforce displacement compatibility, but these complicate the building of the model. However, given the crudeness of this way of representing the joint, it is often possible to accept the incompatibility provided the joint is not 'hinged'. The analyst should in this case only rely on this method to represent global stiffness. The local strain and stresses at the nodes where the two types of elements join will be inaccurate.

The implications of meshing decisions for solid models are more profound - more geometric details are being represented so the models are more sensitive to mesh detail. The first major consideration is the element type, of which there are many variations. At this stage the analyst is warned not to use triangular based elements in the bond line, as they have poor shear behaviour. Load transfer in an adhesive bond line primarily occurs under shear, so an accurate prediction of shear stress components is important. The bond line will usually be amenable to meshing with quadrangle based elements. For 3D meshes, reduced integration elements can significantly decrease computer run time and still maintain accuracy. However, they do not work well when distorted so they should be used sparingly if large element distortions are expected ( eg with rubbery adhesives). With rubbery adhesives that are almost incompressible (Poissons ratio ν ≥ 0.48), the analyst should consider using incompressible or hybrid elements instead.

Mesh density is the second consideration. In general, for elastic analyses with small displacements, three nodes across the adhesive thickness will be adequate ( ie two linear or one quadratic element). In the centre of the joint the elements can be generated quite long (high aspect ratio), because the stress gradients are usually quite low (A in Fig.8). Hence mesh refinement can be concentrated at the joint edges where the stress concentrations occur.

Mesh refinement causes an undesirable numerical effect of stress or strain singularities to become apparent. This can be seen in Fig.8 where the maximum stress at 'B' increases with a finer mesh. Unlike other values in the model, this value will not converge, as the mesh is refined. The level of necessary mesh refinement has to be investigated and will be constrained by the objectives of the modelling and the available computer power.

Meshing: Singularities

By modelling the joint ends as geometrically sharp ( Fig.8 and 9) material and geometric discontinuities can give rise to strain singularities, which means that the values do not converge with increasing mesh refinement. In elastic analyses the strain singularity has a related stress singularity. These singularities are a numerical artefact as the strains and stresses in real components cannot be infinite.

The strains and stresses are calculated at Gauss points, which are inside the elements, where they must be finite. The strains and stresses are then extrapolated to the nodes using the element's shape function. The shape functions are normally polynomials and infinite strains at the singular points cannot be attained. Therefore, refining the model's element density will only serve to increase the reported strain at the singular point. Figure 8 shows this; further refinement would only increase the stress at B (as the mesh size ⇒ 0, σ _B ⇒ ∞). If the maximum stress is used as a failure criterion, this causes two problems when trying to predict adhesive failure:

• Predicted failure load becomes dependent on the mesh refinement
• Failure initiation site can be predicted incorrectly at the point of singularity, which is not always the case in the physical joint

In elastic perfectly-plastic analyses, even if a strain singularity exists, stress will be limited to the yield stress. The yielding and its extent however will depend on the reported strain, which is mesh dependent. Hence there are good reasons for avoiding these singular points in models.

In real joints the corners will not be perfectly sharp and singularities will therefore not be produced. Some joints have corners rounded to lower the local stresses; and the singular behaviour can therefore be eliminated from models as shown in Fig.9d. The maximum stress value in this case is a function of the radius.

a) No fillet
b) Square fillet
c) Angled
d) Concave curved. 's' indicates potential singularity points

Interpreting results and failure prediction

• Failure can initiate from any one of three places:
  - in the adherends
  - at the adherend-adhesive interface
  - in the adhesive itself

• There is usually a lack of accurate and reliable material property data and a lack of information on yield envelopes and failure criteria. Failure criteria might furthermore become mesh-dependent at points of singularity
• Different ways of loading may initiate different failure modes
• The long-term durability of adhesive bonds is not well understood

It is important that the correct results are used for the result interpretation. A critical region is at the boundary between the adhesive and adherends. In a model the elements in these two regions will share nodes and so care should be taken regarding results, which are averaged at these nodes. The analyst should be clear on whether the particular results are for the adherend or the adhesive. The Table describes the results that can and should not be averaged. For instance displacement compatibility means the strains at and in the plane of the bond line are identical in both the adherend and the adhesive and therefore may be averaged. Similarly, normal to the bond line there must exist local force equilibrium and so the stresses in the adherend and adhesive must be identical at the bond interface and may be averaged.

Analysis results can be interpreted in a number of ways depending on the loading and environment of the components modelled. The results may be considered either 'global' or 'local' parameters. Global parameters are not dependent on the detail of the joint ( ie the total bending deflection or the far-field strain). Local parameters require the joint to be modelled in detail. Examples are the peeling stress at a particular place in the joint or the maximum adhesive temperature in a heat flow analysis.

Table: Description of strain and stress quantities that can and should not be averaged across the bond line

Global methods

Local methods

Fracture mechanics based approaches assume a pre-existing crack and determine whether the conditions in the structure are suitable for crack growth and failure, generally using critical strain energy release rate G _c. A few limitations arise due to the constraining effect of the substrates:

• Firstly, G _c is generally regarded as a material property, which depends on the plastic zone developing in the material. Clearly, in a thin bond line, this zone will be restricted, which introduces geometry dependence for the critical energy release rate.
• Secondly, the crack mode might switch from mode I to a mixed mode during crack development due to the confined space. This changes the value of the required critical energy release rate during the analysis. A pre-existing crack has to be specified, which in practice alters with changing loading conditions and fillet size and shape. For most practical joints, no crack of sufficient size exists. Fracture mechanics then rely on the concept of 'inherent flaws', where a flaw size is calibrated with experimental results. This method is rarely applied in practice as it is very tedious and the precise location at which to insert the crack might not be obvious.

For local methods using continuum variables, adhesive failure is assumed to occur when a critical value of stress or strain or a combination of the components (effective value) with appropriate weighting factors attains its limiting value. One of the best known failure criteria is the von Mises criterion. This type of failure criterion has been used in finite element analyses and is one of the standard failure criteria in commercial software programs. The criterion can easily be applied in conjunction with geometric and material non-linear finite element analysis to perform failure analysis of adhesively bonded joints. Other proposed criteria include a maximum shear, peel or principal stress criterion. These critical values can be determined during the adhesive testing stage. Throughout the literature, failure criteria have also been established in strain-based form. Criteria like failure by a maximum principal strain are more unreliable because the experimental data regarding failure strain values show large standard variations.

Special attention has to be paid to the fact that these criteria can only be used to determine adhesive failure if the maximum fillet stress or strain is not present at a point of singularity. For most joints this will not be the case. Alternative approaches with local criteria determine the maximum value for a given mesh size, which is a common way to scale mesh dependent results. This method can work for one type of joint, but has to be calibrated repeatedly when changing the joint geometry or material.

To avoid the problem of singularities, many researchers suggest the use of zone criteria instead of point-wise criteria. In this case the critical value over an area ^[12] or the average value by integration over an area ^[13] of a stress component is calculated, eliminating mesh dependency. These methods introduce another parameter, eg critical zone, into the problem. These types of criteria have shown success for certain joint geometries, and the required zone parameter can be determined experimentally for some of these methods. ^[12] A continuum-based approach has the advantage that it can be easily incorporated into a finite element code during the analysis to predict joint failure. This makes the approach generally more applicable for industry compared to fracture mechanics.

Adhesives should be considered as a whole family of materials with a vast range of properties. Hence it is not surprising that there is no single accepted method of assessing and predicting strength or life of adhesive bonded joints. Often the best approach to failure prediction is to calibrate a test geometry using experimental data; and to extrapolate this to the component geometry. It is unlikely that a single generally applicable failure criterion usable for a wide range of joints and materials will ever be established.

Summary