Published in NAFEMS advanced workbook of examples (Volume 1)
(NAFEMS is the National Association for Finite Elements Methods and Standards - please see www.nafems.org for more information).
1. Modelling adhesively bonded joints
Adhesives are defined in a British Standard 
as 'non-metallic substance capable of joining materials by surface bonding, the bond possessing adequate internal strength'. This definition covers a wide range of substances, but in this article the emphasis will be on polymeric adhesives. Adhesives are used in microelectronics through to large structures; they are used for many reasons, some of which are:
- Ease of assembly (fills gaps)
- Load spreading (compared to mechanical fastening)
- Because of an inability to join the materials in other ways
- Suitable for making joints with dissimilar materials.
- Usually a low temperature joining process (<200°C).
Structural bonds are used to illustrate some of the complexity in modelling adhesive joints. The examples are not exhaustive so the analyst should always be looking for other problems. An introduction to bonding can be found in reference  . More detailed guides to modelling can be found in references  and  .
Sections of bonded joints are illustrated in Fig 1.1; the bonds may also curve to form non-flat, non-straight bond surfaces. In the figure the lighter parts are the parts being bonded (adherends), and the darker parts the adhesive.
Fig. 1.1. Examples of adhesively bonded joints
(a) simple lap,
(c) tubular sleeve joint and
(d) a more complex joint.
1.2 Planning the modelling
It is essential to plan any modelling to produce the required information without an overly complicated and expensive model. Planning an analysis of an adhesive bonded component is not significantly different to that for other components; the reader should consult another NAFEMS booklet 
Including all of the detail of adhesively bonded joints into a model of a component will complicate an analysis because of several factors:
- The joints have features that are much smaller than the dimensions of the structure. In '3D' solid models, thin bond-lines will require large numbers of elements model the adhesive.
- Misaligned load paths across the joints produce local bending stresses.
- There is often a large mismatch in material properties in the joint.
In addition 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. For instance bond-line thickness can vary considerably and the adhesive fillet or spew may not be controlled.
- The properties of adhesives are not readily available and may be sensitive to cure conditions, temperature, strain rate, moisture up-take etc.
- The joint may be incompletely filled and the adhesive may contain voids or regions of poor bonding.
Practicalities such as how well the thickness of the bond-line can be controlled will need to be considered. An example of how a relatively small change in the detail of a bonded joint can affect its behaviour is shown in section 1.4. The following examples will concentrate on modelling issues rather than how well the bonding process can be controlled, but the analyst should be aware of these issues.
1.3 Geometry 1: 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 the joints are usually simplified. At the lowest level the joint can be left out of the model; however, where an over-lap joint causes local thickening it is prudent to locally thicken the elements to represent the extra stiffening. This is illustrated in Fig 1.2
, which shows models of a pressure-loaded panel. The model on which the example is based was much larger and a solid (brick) element model was impractical. Thickening the bond region in the shell model had a minimal effect on computation time but introduced stiffening to make the model more realistic.
Fig. 1.2. A simplified model of the floor of a bonded pressurised vessel. The floor has a lap bond along its centre and is simply supported around the periphery.
(a) shell model where the bond features are ignored,
(b) shellmodel with local thickening and offset and
(c) a solid element model.
1.4 Geometry 2: detailed models of bonded joints
Some components can be modelled with 2D meshes (plane strain, axisymmetric etc); this makes model size considerations less onerous and any bonds can be explicitly modelled. Also detailed models of sections of the joint may be produced to investigate the influence of joint parameters. These models may also be produced to determine stiffness characteristics of the joint so they can be fed into a larger component model (super-elements or sub-structuring). Hence there are a number of reasons for making detailed models of the joints. This example illustrates an extreme case of how joint detail can influence the joint behaviour.
Figure 1.3 shows two half-symmetric models of a T-joint, the adhesive is on the centre-line. In (a) a small amount of adhesive spew is present between the radii (as in Fig 1.1(b)), and in (b) it is not present. This is the sort of detail that can easily change in 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 and the boundary conditions.
Fig. 1.3. Two models of T-peel joints: both represent 2mm thick steel bonded with epoxy resin.
Model (a) has an adhesive fillet extending 1/ 3 of the way up the radius whereas model (b) lacks the fillet. The model (a) exhibits over double the stiffness of (b). Both meshes shown at x20 displacements.
1.5 Material Properties 1: sensitivity
Even for some metals, reliable materials data to use in FEA models can be difficult to find. With adhesives the problems can be magnified ten fold! Often the properties are dependent on hardener-resin ratio, cure conditions, strain rate, water up-take etc. Obviously different materials are affected by different amounts and over different time scales. For instance most epoxies will be fully cured in a few days but some acrylic adhesives may carry-on hardening for many weeks or months. The effects of cure schedule and temperature on the stress-strain properties are illustrated in Fig 1.4. The dramatic change in strength and ductility with temperature of some adhesives is illustrated in Fig 1.4(b); large changes are often attributed to crossing the adhesive glass transition temperature.
For many materials only rudimentary property data are available, say from a manufacturer's data sheet. The analyst is further warned that such data can be misleading because it may be based on the bulk properties and not on joint properties. For instance the thermal resistance of an adhesive in a joint can be many times higher than the equivalent bulk material property would suggest because of porosity (entrapped gas) and/or contact resistance at the bonds. Another reason is that large samples of adhesive will cure differently to small amounts in a joint (eg exothermic reaction raising the temperature of adhesive castings during cure).
Fig. 1.4. Examples of stress-strain behaviour of adhesives.
(a) Influence of cure schedule for a hybrid acrylic-epoxy adhesive and (b) temperature dependence of an anaerobic adhesive.
1.6 Material properties 2: insensitivity
The information in this section contradicts what was said in section 1.5 as it shows that in some circumstances the joint performance can be insensitive to the properties of the adhesive. Figure 1.5 shows two similar models of a microelectronic component; on a large scale there is little difference between deformations and stresses predicted, even though a very low adhesive yield stress was used in one model. This demonstrates that joints can be insensitive to adhesive behaviour; in this case the reason is constraint offered by the adherends which prevents 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.
Fig. 1.5. Two models of a silicon die bonded on to an alumina ceramic substrate and isothermally cooled from the cure temperature of 150°C to room temperature.
(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.
1.7 Meshing: element density and type
A major consideration for any model should be the adequacy of the mesh used, both in terms of the mesh density and the type of elements.
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 detail of the joint cannot be represented making very detailed mesh refinement unnecessary. However it is useful to be able to define the location of the joints so that, for instance, local thickening can be applied, see section 1.3.
A potential problem associated with meshing of component models is when solid and shell elements are combined, where the solid elements represent the adhesive. The element types have different active DOF and are not strictly compatible. Some FEA programs have multi-point constraints (MPCs) to enforce displacement compatibility, but these do complicate model building. However, given the crudeness of this way of representing the joint, it is often possible to accept the incompatibility - providing the joint is not 'hinged'. The analyst should 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 in error!
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. Mesh density is a first 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 surprisingly long (high aspect ratio) because the stress gradients are usually quite low, this is illustrated at A in Fig 1.6. Hence mesh refinement should be concentrated at the joint edges where the stress concentrations occur (but also see section 1.8). The level of refinement will have to be investigated and will be constrained by the objectives of the modelling and the available computer power.
Fig. 1.6. The predicted peel stress in a single lap-shear joint. Results from two solid models with different mesh densities are shown. Symbols indicate nodal positions and hence element density.
Note: the peel stress is that prising the adherends apart, normal to the bond-line
The second 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, these elements have poor shear behaviour. Anyway the bond will usually be amenable to meshing with quadrangle based elements. For 3-D meshes, reduced integration elements can significantly reduce computer run time and still maintain accuracy; however, they do not work well when distorted so use them sparingly if large element distortions are expected (e.g. with rubbery adhesives). With rubbery adhesives that are almost incompressible ( ν ≥0.48), the analyst should consider using incompressible or hybrid elements.
1.8 Meshing: singularities
By modelling the joint ends are geometrically sharp, as indicated in Fig.1.7
, material and geometric discontinuities can give rise to strain singularities. In elastic analyses the strain singularity have a related stress singularity. These singularities are 'theoretical' because the strains and stresses in real components cannot be infinite - but they can introduce problems that are real enough in the models. The phenomenon is covered in more detail elsewhere [3,6]
The strains and stresses are calculated at Gauss points, which are located 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 change the reported strain at the singular point. Figure 1.6 illustrates this effect, further refinement would only increase the stress at B (as the mesh gets finer
Fig. 1.7. Representations of the adhesive fillet. (a) no fillet, (b) square fillet, (c) angled and (d) concave curved. The 's' indicates potential singularity points.
In elastic perfectly-plastic analyses, even if a strain singularity exists, the stress will be limited to the yield stress; but the yielding and its extent 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 not be produced. Furthermore some joints have corners rounded to lower the local stresses; the singular behaviour can therefore be eliminated from models, as shown in Fig 1.7(d). However the model geometry should have a physical basis or else the results will not reflect the real stresses.
1.9 Interpreting results
Interpreting output from models of bonded components and structures can be difficult. In addition to the problems mentioned above, this can be attributed to:
- Failure can initiate from one of three places: (a) in the adherends, (b) at the adherend-adhesive interface and (c) 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.
- 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 taken regarding results averaging at these nodes. The analyst should be clear on whether the particular results are for the adherend or the adhesive. Table 1.1 describes the results that can be averaged and those that 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.
Table 1.1 Descriptions of strain and stress quantities that can or should not be averaged at common nodes between the adhered and adhesive.
Analysis results can be interpreted in a number of ways depending on the loading and environment of the components modelled. However, the results may be considered to be either 'global' or 'local' parameters. Global parameters are not dependent on the detail of the joint, the total bending deflection or the stress acting on but remote from the joint are typical examples. 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.
| ||Averaging acceptable for:|
|strain and stress acting in|
the plane of the bond-line
|strain and stress acting|
normal to the bond-line
||Yes - displacement compatibility
||Yes - force equilibrium
Assessment by global methods do not cause too much problem, usually the model output will be compared to established failure criteria. An example of an established failure criterion is the fatigue S-N curve for adhesive joints defined in  the draft Eurocode 9. However, Standards based failure criteria are usually very conservative as they are catchall rules. More specific failure data may be taken from manufactures data sheets, but this is usually limited to basic test data. More specific data must be generated by testing.
Local assessment methods are wide and varied, and most are still at the research stage. There is no one accepted method of assessing and predicting strength or life of adhesive bonded joints. Some techniques use local stresses or strains  where others use methods based on fracture mechanics principles [4,8] .
All of the other problems of modelling adhesive joints almost pale into insignificance when compared to the problems of interpreting the results. This is particularly true where failure prediction is required.
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