An investigation into using finite element analysis for epoxy resin joints

An investigation into using finite element analysis for epoxy resin joints C. W i l l i a m s o n a n d A . D a a d b i n Department of Mechanical Engineering and Manufacturing Systems, Northumbria, Newcastle-upon-Tyne, NE1 8ST, UK Received 18 October 1992; accepted 13 November 1992

University of

Finite element analysis (FEA) is a well-established tool for the study of engineering components, but designs often involve the use of materials that are bonded together using adhesives. The accuracy of using linear elastic FEA to model epoxy/aluminium bonds is investigated by the comparison of actual results with computer-predicted data. From the results obtained it is shown that it is possible to model epoxy resin joints, provided that careful mesh refinement is used in the interface region so that the effect of nodal stress averaging is overcome. The FE models were then used to study failure modes and it was established that both elastic or plastic failure of the joints can occur depending on the magnitude of any interface stress concentrations present.

Keywords: epoxy resin; f in i te e lement model; f rac ture cr i ter ia

NEI Victor of Wallsend manufacture flame- and explosion-proof industrial light fittings for the mining and petrochemical industries. The basic construction of the fittings is a glass dome or tube (depending on style) bonded to an aluminium body using a 5 mm thick resin interface. A requirement of the British Standard is that all fittings be pressure tested to simulate an in-service explosion ~.

NEI Victor use finite element analysis in the design of these fittings and are interested in accurate modelling of the resin joints, which are critical to the safety of the fitting. The behaviour of the joints under load is the main area of interest and needs to be investigated to determine how and why failure occurs.

The effect of nodal stress averaging caused by the modelling of dissimilar material interfaces has been stu- died fully. This may not cause a problem when the moduli of elasticity are similar (e.g. aluminium and glass) but distorts the results when using epoxy resin where the moduli may differ by a factor of 15.

Determination of mechanical properties of epoxy resin The need to test the mechanical properties of the resin was the fact that Ciba-Geigy, the resin manufacturer, only supply data on flexural modulus and not tensile modulus E and the Poisson ratio, v, as required by the finite element program. BS 2782:1976 method 335A (ISO R 527): describes a method for determining the tensile properties of plastics in the form of standard test specimens. A stress strain curve was also required to determine whether a non-linear analysis was necessary.

Description of epoxy resin From Ciba-Geigy ~ the resin is a general-purpose easy-to- pour flame-retardant encapsulant. It provides an alterna- tive to the established Araldite CW 1302 + Hardener HY 1300 system for applications requiring lower visco- sity and/or faster curing.

Correspondence to (7. Williamson

0261-3069/93/020091-05 O 1 993 Butterworth-Heinemann

Apparatus From BS 2782 the required mechanical properties were obtained by the use of standard dumb-bell specimens. The testing machine used was the Instron tensile/com- pressive with a 5 kN load cell. The Araldite CW 1302 resin with HY 2992 Hardener was mixed 14:1 with a mass ratio of 5 g of hardener to 70 g of resin to avoid excessive waste as recommended by Ciba-Giegy.

Method to determine Young's modulus E The resin was mixed as above for 3 min to ensure good circulation of hardener. A 2 kW hot-air blower was then used to heat the mould to 80°C for 75 min. The heater was then switched off and the mould allowed to cool for 45 min. The specimens were then removed from the mould and left in a warm room at 20-30°C for 24 h. The excess flash was then removed with 100 grade wet and dry paper. Five specimens were then tested in the Instron tensile testing machine with the following parameters:

1. 5 kN load cell 2. Feed rate 0.5 mm/min 3. Effective gauge length 125 mm

The specimens were tested to destruction and the load extension curve is shown in Figure 1. Linear regression was used on the data and Young's modulus E was found to be 5200 N/ram 2. A general value for epoxy resin was quoted by Ciba-Geigy 4 to be between 3000 and 8000 N/ mm 2. The range of tensile strengths was 19-24.2 N/mm 2 with an average value of or1- = 21.6 N/ram 2 and the range of yield strengths was 15.4--18.8 N/ram 2 with an average value of ~ = 17.6N/mm-'.

Method to determine the Poisson ratio v The previous moulding procedure was repeated to produce the specimens. When the specimens were cured, shear strain gauges were bonded to their flat surface (KFC-2-CI-11 gauge factor = 2.04). Wires were then attached to the strain gauges and connected to the data logger. The data logger was set to a one-quarter bridge arrangement of measure the X and Y strains. Linear regression was again used on the data and the Poisson

Materials & Design Volume 14 Number 2 1993 91

Using FEA for epoxy resin joints: C. Williamson and A. Daadbin

1200

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600

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Figure 1 Tensile testing on epoxy resin

ratio was found to be v = 0.379. A general value was quoted by Ciba-Geigy 4 to be between 0.35 and 0.38.

From the above results it is clear that the epoxy resin has a good linear behaviour and does not exhibit much yielding before fracture. Therefore a linear elastic analysis can be used on the finite element models. The values for Young's modulus and the Poisson ratio lie between the general values quoted by Ciba-Geigy.

Model of tensile test specimen A finite element model was constructed of the tensile test specimen for the purpose of determining the following:

1. Verification of Young's modulus E and Poisson's ratio v.

2. How to model the force applied by the tensile testing machine.

The model uses six-node 2-D triangular solid elements 5 assuming a plane stress with a thickness of 3 mm. The test piece is symmetric about the X and Y axes, therefore only a one-quarter model is required. A 500 N load is applied at the nodes in the 20 mm width section of the specimen. The nodes have their vertical displacement (Uy) coupled together to simulate the action of gripping the specimen in the jaws of the machine.

A solution was run for the model and a load/extension graph plotted against the results obtained from the tensile test specimens. From Figure 1 the model shows a good approximation to the actual specimen. The Ansys program 6 has a function for determining the error of the mesh used. This function is called the normal error and is usually expressed as a percentage. It is obtained from the following formula:

Error Ratio (ERAT) = (TSE/(TSE + ENR)) ':2100% where TSE = sum of total strain energy within the model and ENR = sum of total error within all elements.

All the models used converged to < 10% on strain energy and displacement levels.

The model of the tensile test specimen had an error ratio of 6.67%, and from the FE model it was shown that the maximum error occurred at the loading points. As

a

b

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Tensile butt joint specimen

Tensile notch specimen

T h r e e - p o i n t bend bu t t joint

T h r e e - p o i n t bend notch joint

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Three-point bend reverse notch

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Figure 2 Different types of specimen

lines

the error was below the acceptable level of 10% no mesh refinement was required. The results confirm Young's modulus and the Poisson ratio and also show that the gripping of the machine jaws can be modelled as des- cribed previously.

Modelling of epoxy resin joints The main objective of this section is to determine how accurately it is possible to model an epoxy resin joint using FEA. The failure of the joints is discussed briefly but a more detailed explanation is given in the next section. Various specimens are used, starting with tensile test specimens and then using three-point bend specimens.

Tensile specimens Model of circular butt joint specimen. Five specimens were prepared as shown in Figure 2(a). The first problem encountered when the specimens were tightened into place was that as the jaws closed they also moved up in a vertical direction. This motion and its associated force was too much for the specimens to hold and they all

92 Materials & Design Volume 14 Number 2 1993

Using FEA for epoxy resin joints." C. Williamson and A. Daadbin

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Three-poin t bend test: 10 mm but t jo in t

broke before being tested. Therefore no element model was produced. Model of circular notch .joint specimen. Five specimens were prepared as shown in Figure 2(b). This time the specimens were able to withstand the axial precompression of the jaws and the tests were completed to destruction. A finite element model was constructed for the specimen. The load-extension line obtained was plotted against actual results as shown in Figure 3. The results showed a large scatter of up to 50% in both maximum carried loads and fracture extensions of the specimens of the joints. It was thought that due to the action of the testing machine jaws, the specimens were suffering from axial precompression of various magnitudes due to different clamping forces. The scatter can also be caused by resin porosity: but upon investigation of the fractured joints this was found not to be the case. The reasons for this conclusion are that:

1. When the FEA solution was compared with the experimental results it was found to be the approxi- mate mean of the actual results.

2. The scatter was too large to be due to experimental error, as some results were out by over 50%.

The residual stressing is not a constant value and seems to be mainly due to the vertical clamping forces which caused the breakage of the butt joint specimens. On the other hand, if it was due to these forces alone then the extensions would all have the same gradient with the intercepts offset by the amount of stressing. As this was not the case it was concluded that the residual stressing of the specimens is a function of the following parameters:

Residual stresses = •{Clamping force, thermal straining, mould shrinkage, porosity etcJ.

with the clamping force as the main contributor. This is also confirmed by the following:

1. It is later shown that using three-point bend specimens, which are not clamped into place, gives much better results.

2. Modelling of the thermal stresses using the FE program changed the gradient of the load extension line by 5%.

500~

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Three-poin t bend test: 20 mm but t .joint

. The three-point bend specimens are not as severely restrained during curing, thereby reducing curing stresses (mould shrinkagey.

Therefore it was decided to use three-point bend tests which would eliminate any clamping stresses, as the specimens are supported between rollers.

Three-point bend specimens Model Of lO and 20 mm butt, notch and inverse notch joint speeimens. Five specimens for each type of joint were prepared as shown in Figures 2(e) (e). They were tested to destruction using the lnstron tensile testing machine adapted to a three-point bend test and the load-extension curves produced are shown in Figures 4--7. A finite element model of the 20 mm butt joint specimen was constructed. As the model was the same basic construction as the 10 mm specimen (except for the obvious change in joint thickness) a powerful feature of the Ansys program was used. This is the Ansys Parametric Design Language (APDL), which allows the user to input sizes, forces or displacements (in fact, anything) as parameters. In this case the size of the joint was input as SIZE = 20. After a solution was found for the 20 mm model, the pre-pro- cessor database was updated with SIZE = 10 and a solution obtained for the 10 mm model. The results of



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Figure 7 Three-point bend test: inverse notch specimen

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Effect of nodal stress averaging While the above has shown that the load-extension curve of a joint can be predicted, estimation of the resin stress levels can be obscured by the problem of nodal stress averaging. In analysis of the lampglass, it was noted that stress levels at the resin interfaces were extremely high, predicting failure at a much lower test pressure than that actually observed. Upon investigation of the finite element results it became clear that the program averages the nodal stresses of elements along the resin/aluminium interface between the high stress value for the aluminium and the low stress value for the resin. This is because even though the elements have different materials they share common nodes at the interface. This results in displacement compatibility but causes a stress discontinuity, as a node cannot have two stress values. To eliminate this problem it is possible to select the nodes and elements of the resin and use the unaveraged results (Figure 8). This method was applied to an analytical solution for a composite cylinder 8 and the correlation between the FE model and the analytical results was very good.

Failure modes The established failure criterion for a homogeneous brittle material is the Rankine maximum principal stress criterion 8. This was used as the basis for the evaluation of joint failure modes. However, the above failures seemed to contradict each other, as some joint types failed in the elastic region whereas other failed in the plastic region. The 10 mm butt and the inverse notch joints all failed in the elastic region, whereas the 20 mm butt and the notch joints all failed in the plastic region. Ansys predicts yielding at a lower load than that observed but it is possible that, as the material is fairly brittle, small amounts of yielding remain undetected until a relatively large section has yielded. However, the ratio of the yield loads (Fy/Fy Ansys) gives a good linear relationship, as shown in Table 1.

the FE model were plotted against the actual results (Figures 4-7). It was found that much time was spent in re-meshing the models to obtain a suitable mesh accuracy (using the Ansys error norm). To overcome this problem a macro was written to automatically converge the solution to a user-defined accuracy.

Apart from the initial problem of clamping the specimens in the tensile machine, the above results show that it is possible to model the load-extension curves of the joints. There were failures in both the elastic and plastic ranges of the resin, which will be discussed in the next section.

The experimental results for the three-point bend tests were repetitive. The FE results gave the following differ- ences when plotted against the actual ones:

l0 mm butt 20 mm butt Inverse notch Notch

Displacement 10% less than actual Displacement 20% less than actual Displacement 6% more than actual Displacement 0.25% less than actual

Therefore by selecting the three-point bend specimens the problem of clamping stresses seems to have been eliminated. However, this does not prove the presence of other residual stresses, which may be hidden by the scatter of the experimental values.

Plastic collapse of section An explanation for the failures in the plastic region is to be found in Benham et al. 8 who state that 'For failure to occur in a beam, a fully plastic bending moment must be developed across its section'. For an ideal rectangular beam the load to cause collapse is given by

Fp= 1.5Py

where the value 1.5 is termed the shape factor and varies for different cross-sections. Note that the equation Fp= 1.5Fy is not dependent on the material. By utilizing this basic formula, which is mainly used on ductile material, and allowing for experimental errors and different shape factors caused by the joint geometry the failure at the higher loads could be explained. However, it would not account for failure in the elastic region.

Proposed resin fracture criteria To explain the failures in the elastic region, re-investigation of the FE plots showed that the joint types that failed in this region (10 mm butt) had an interface stress concentration which was larger than those that failed in the plastic region. The size of the concentration also corresponds to that of exposed aluminium at the interface, observed after fracture. Crack initiation and

94 Materials & Design Volume 14 Number 2 1993


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Plot showing ..'"' = hoop s t ress ,.- - .

in composi te

b T h i c k c y l i n d e r at 400 psi ( 2 . S M P a )

Figure 8 Stress distribution in a composite cylinder

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growth were also observed during testing of both the 10 mm butt and inverse notch joint specimens. By using the assumption 'when the stress (or t = Rankine maximum principal stress) at the resin/aluminium interface exceeds the tensile strength of the resin crx', 'peeling' will occur along the interface. This is assumed to be a crack that

Table l

Joint type Actual Actual Ansys F~/F, failure yield yield Ansys (/7,) (N) (r,) (N) (F,) (N)

20 mm butt 345 248 19(1 1.30 10 mm butt 245 225 Notch 426 254 200 1.27 Inverse notch 320 300 240 1.25

E/E~ F,,Ik~ Ansys Crack size from FE

20 mm butt 1.39 1.8 1.3 10 mm butt 1.1 1.4 Notch 1.68 2.1 0.5 Inverse notch 1.07 1.3 1.4

will grow with the increasing load. The material will then fail when the crack length a reaches the critical value aft.

Ktc = Ycry (rtac) 12 where K~c = plane strain fracture toughness = 0.8 N/m -~/2, Y = geometric factor= between 0.7 and 1.5 and a~ = critical crack size.

The crack size is estimated from the FE model. This is done by plotting the distance along the interface against stress above cry, and it can be estimated whether fracture will occur:

1. Estimated crack length less than ac, material will fail in plastic region.

2. Estimated crack length greater than ac, material will fail in elastic region and an estimate must be found for the failure load, which is F = ac/a × Fy.

While the above fracture criteria must be refined by further work, the approximations made still give an idea of when and how failure will occur.

It is intended to model the interface of the joint using contact elements that are set to open when the interface stress reaches aT. It is hoped that this will then produce more accurate crack lengths and will verify the 'crack criteria'.

R e f e r e n c e s 1 BS 5501: Part I: 1977 2 BS 2782:1976 method 335A 3 Ciba-Giegy Product Manuals, Ciba-Geigy Plastics, Duxford, Cam-

bridge 4 Epoxy Resin Handbook, Ciba-Geigy Plastics, Duxford, Cambridge 5 Ansvs Primer/'or Stress Analysis, Swanson Analysis Systems 6 Ansys Reference Manual, Swanson Analysis Systems 7 Chan, Z. The Failure and Fracture Analysis of Adhesive Bonds,

Thesis, Bristol University, 1985 8 Benham, P.P. and Crawford, R.J. Mechanic's of Engineering

Materials, Longman, Harlow, 1987 9 Hertzburg, R.W. Deformation and Fracture ~fEngineering Mater-

ials, Wiley, Chichester, 1989


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An investigation into using finite element analysis for epoxy resin joints