Three-dimensional fracture mechanics analysis of a cracked lap joint.

Pastrama, Stefan Dan ; Nutu, Emil ; Jiga, Gabriel 等

1. INTRODUCTION

When loaded in traction, lap joints take over substantial bending effect. In this case, two dimensional finite element models become irrelevant due to the non-uniformity of the stress distribution over the strip thickness. This non-uniformity leads to variations of the stress intensity factor through the thickness (Moreira et al., 2005).

Three dimensional fracture mechanics models require significant efforts for finite element analyses, due to the special modeling needed along the crack front. Moreover, in case of riveted joints, contact between joint components should be taken into account. Usually, fracture mechanics parameters, such as the stress intensity factor (SIF), are used in subsequent fatigue calculations in order to estimate the fatigue life. Calibration of SIF for different structures is presented in handbooks (Rooke and Cartwright, 1976, Murakami, 1987). For structures without known solutions for SIF, a numerical or experimental calibration is required if future lifetime predictions are needed. In this case, a parametric model is suitable for diminishing time costs and efforts.

This paper presents an application of parametric finite element modeling, in the case of a lap joint with one rivet and a symmetrical crack developed at the rivet hole in one strip.

2. THE MODEL DESCRIPTION

Three sets of parameters were defined to model the joint: Dimensional parameters (figure 1), including the total length L of the lap joint, the overlap length Ls, the width of the strips 2W, the thickness of the strips T1 and T2, the rivet hole radius R and the rivet head radius Rc.

Material properties parameters including the Young modulus and Poisson's ratios for each material (for strips and rivet).

Crack parameters including the effective crack length c and the half crack length a (c + R).

[FIGURE 1 OMITTED]

A linear-elastic behavior of the studied structure was taken into account. Due to the symmetry, only half of the structure was modeled.

The parameters values chosen for the analysis are: Dimensional parameters: L = 100 mm, Ls = 20 mm, W = 10 mm, T1 = T2 = 1.6 mm, R = 1.6 mm, Rc = 2.4 mm.

Material properties parameters: for the material of the strips (Aluminum 2024-T3), the following elastic properties were considered: E = 70.61 GPa and v = 0.33, while for the rivet material (Aluminum 2117-T4) the elastic constants E = 68.5 GPa and v = 0.3 were taken into account.

Crack parameters: Eight different analyses were performed, for different effective crack lengths, selected within the range [0.8 mm ... 2.2 mm] with a difference of 0.2 mm between two consecutive values.

3. MODEL MESH, LOADS AND BOUNDARY CONDITIONS

The 20 nodded structural brick element with midside nodes was chosen to mesh the structure. One should mention that, around the crack front, the midside nodes were moved to a distance from the crack front equal to a quarter of the element sides, in order to simulate the crack tip singularity (Owen and Fawkes, 1983).

A compromise between the special mesh around the crack front, as described above, and the total number of elements is needed in order to ensure a reliable evaluation of stress intensity factor and also to eliminate the unnecessary refined mesh over the sheets, where load transfer and material behavior is linear (the sheets region where no contact and no fracture mechanics phenomena is involved). As a consequence, the total number of solid elements is varying, being embedded within the range [2166 ... 2376], where the minimum number of elements corresponds to the maximum effective crack length model (2.2 mm) and the maximum number to the minimum crack length (0.8 mm) model.

Figure 2 shows the general mesh view and some suggestive mesh details corresponding to the 1.8 mm effective crack length.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

In order to obtain the stress intensity factor variation along the thickness of the strip, three layers of elements were created over the cracked strip thickness. The uncracked strip was also meshed with three layers of elements in order to efficiently equilibrate the numerical approximation of load transfer from the loaded strip through the rivet and foreword to the other strip.

Multiple contact pairs were created, resulting into a total of nine contact surfaces and a total number of contact elements varying within the range [2203 ... 2413] (depending on effective crack length), as follows: contact between sheets, contact between rivet body and each sheet, contact between rivet heads and each sheet.

4. RESULTS

The values of SIF were extracted corresponding to all three fracture modes at seven different levels of the crack strip thickness for each mode.

The effective values of the stress intensity factor Kejj were calculated through the strip thickness for all eight evaluated crack lengths using the equation (Broek, 1986):

[K.sub.aff] = [square root of [K.sup.2.sub.I] + [K.sup.2.sub.II] + [K.sup.2.sub.III]/1 - V (1)

In order to emphasize the evolution of the Mode I SIF [K.sub.I] and the effective SIF [K.sub.eff] against the effective crack length c, the plots of these fracture parameters are shown in figures 3 and 4. The [K.sub.I] and [K.sub.eff] variation against the thickness of the cracked strip is plotted in figures 5 and 6 respectively. Using these plots, one can determine the SIF values for any particular load.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Some future predictions can be done using the values of SIF's plotted above.

5. CONCLUSIONS

The paper presents an example of finite element parametrical modeling of a riveted single lap joint with symmetrical crack near the rivet hole, in one sheet. The results are expressed in terms of stress intensity factor values, as a fundamental parameter in linear elastic fracture mechanics.

As the model allows to quickly redefine several dimensions for the lap joint, materials, loads and boundary conditions, future analysis can be fulfilled in order to determine the joint fatigue life and its dependence of some parameters such as: the thickness of the sheets, the materials properties, the type of load cycle, the overlap length, the total length of the joint, etc. Whatever type of influence will be taken into consideration, preliminary experimental determinations should be accounted for in order to determine the conditions that lead to crack initiation and growth.

6. REFERENCES

Broek, D. (1986). Elementary Engineering Fracture Mechanics, Martinus Nijhoff, 4th rev. edition, ISBN 9789024726561, Dordrecht, The Netherlands

Moreira, P.M.G.P.; de Matos, P.F.P.; Camanho, P.P.; Pastrama, S.D.; de Castro, P.M.S.T. (2005). Fatigue analysis of an AL 2024-T3 Alclad cracked lap splice specimen, Proceedings of the International Conference on Structural Analysis of Advanced Materials ICSAM 2005, S.D. Pastrama (Ed), pp. 157-162, Bucharest, September 2005, Printech, Bucharest

Murakami, Y. (Ed.-in-chief) (1987). Stress Intensity Factor Handbook, Pergamon Books, ISBN 0080348092, London Owen, D.R.J. & Fawkes, A.J. (1983). Engineering Fracture Mechanics. Numerical Methods and Applications, Pineridge Press Ltd., ISBN 0906674263, Swansea, UK

Rooke, D.P. & Cartwright, D.J. (1976). Compendium of Stress Intensity Factors, Her Majesty's Stationery Office, ISBN 0117713368, London