Finite element analysis (FEA) is used when analytical solutions are not available or more detailed information is required for a joint in a structure. Finite element analysis can be used to obtain more detailed stress and strain information in an adhesive joint and to derive fracture mechanics parameters.
The main inputs to the FEA are the geometry, loading and material model parameters. A range of material models are available that may be used to represent various aspects of the behaviour of the adhesive.
Elastic - Models adhesive behaviour in linear region
Hyperelastic - May be required to model the nonlinear behavior of flexible adhesives
Elastic-plastic - These models describe non-linear stress-strain behaviour in glassy adhesives arising from yielding and flow. Different elastic-plastic models employ different yield criteria and flow laws. The simplest version uses the von Mises criterion, but the accuracy of predictions using this model is restricted because the yield behaviour of adhesives is known to be sensitive to the hydrostatic component of stress. Alternative models with hydrostatic stress sensitivity are the linear and exponent Drucker-Prager models. The exponent Drucker-Prager model has been shown to be more accurate than other models for predicting deformation behaviour under stress states with a significant hydrostatic component as occur in regions of high peel stress (see NPL Good Practice Guide).
Examples of FE stress analysis
Illustrations are given below of stress distributions in selected single lap, scarf and T-peel joint geometries calculated using FE analyses.
Single lap shear joint. The geometry and dimensions of the specimen used for the FE analysis of a single lap shear specimen are shown below. Small radii were imposed on the end faces of the adhesive and the adherends to avoid strain singularities in the analysis. The FE mesh used to describe the adhesive layer is shown in the stress map below.
The contour map shows the distribution of maximum principal stress in the adhesive in the region of the fillet. Units are GPa. The adhesive is a rubber-toughened adhesive and the adherends are steel. The predictions were obtained at a calculated load of 6kN and an extension of 0.085 mm in a gauge length of 25 mm centred around the adhesive layer. The calculations have been made using the exponent Drucker-Prager model. The site of failure initiation is revealed by the crack in the photograph and is seen to occur in one of the regions of peak stress. Studies such as this in a variety of joint geometries should aid the identification of criteria for crack initiation in the adhesive.
Scarf joint. The geometry and dimensions of the specimen used for the FE analysis of a scarf joint specimen are shown below. Small radii were again imposed on the end edges of the adherends to avoid strain singularities in the analysis. The FE mesh used to describe the adhesive layer is shown in the stress map below.
The contour map shows the distribution of maximum principal stress at one end of the adhesive layer. The units of stress are GPa and the adherends are aluminium. The calculations were made using the exponent Drucker-Prager model and correspond to a load of 8kN and an extension of 0.045 mm in a 25 mm gauge length centred on the adhesive layer.
T-peel joint. The geometry and dimensions of the specimen used for the FE analysis of a T-peel joint specimen are shown below. The FE mesh used to describe the adhesive layer is shown in the stress map.
The contour maps show distributions of maximum principal stress and hydrostatic stress in the region of the fillet. The units of stress are GPa and the adherends are aluminium. The calculations were made using an elastic-plastic model that takes account of the influence of cavitation in rubber particles on yielding and flow in the adhesive. Predictions correspond to a load of about 1.5 kN and an extension of 0.23 mm in a 25 mm gauge length centred on the adhesive layer. The photograph shows the location of crack initiation in the joint. Although the crack formed in a region of maximum stress, it propagated so rapidly that it was not possible to ascertain whether it was initiated by a critical level of hydrostatic stress or maximum principal stress.
Fracture mechanics parameters from FEA
Figure 2 shows how FEA is used to determine the relationship between crack length (a) and strain energy release rate (G), in this case for the 2-dimensional analysis of a T-peel joint. Cracks of various lengths are introduced into the model, within the adhesive layer, and G is calculated using the virtual crack closure technique (VCCT). This technique basically calculates the energy required to close the crack, based on the FEA cleavage forces and the displacements around the crack tip. Shear forces and displacements can used in a similar way to calculate the mode II contributions to G.
Figure 3 Plot G vs a for a T-Peel sample with a crack along the centre of the bondline.
Fatigue crack growth testing is performed to determine the relationship between G and crack growth rate (da/dN). This data is known as the materials model for the adhesive system.
Analyses is typically performed on component joints to determine the regions of maximum mode I cleavage/peel strain. Cracks are entered at these locations and the G levels (all modes) determined. Joint design can be modified to ensure the maximum predicted service G value is well below any threshold value, measured from fatigue testing. Deeper cracks can be introduced to generate a life prediction curve.
