Keywords: fracture mechanics, finite element, boundary element, stress intensity factor, singular and non-singular elements.

The determination of the stress intensity factor (K) in Linear Elastic Fracture Mechanics (LEFM), is very useful [1] for the investigation of crack propagation and structural integrity in pressure vessels, pipelines, aircraft fuselages, concrete dams and other structures. Crack propagation can be evaluated using the Fracture Mechanics parameters such as the Stress Intensity Factor (K) and the J-Integral (J) [2]. Obtaining K experimentally is expensive and time consuming. A fast, low cost, and economical alternative is the use of numerical methods for the calculation of K. In this paper, the numerical methods used to this end are the FEM and BEM [3].The BEM promises to bring improved accuracy as the fundamental solutions employed in the BEM enable the stresses in areas of high stress gradients to be calculated accurately. In spite of this, the use of special elements called quarter-point elements is common, both in the FEM [4] and the BEM [5] fracture mechanics applications to model the singularities at the crack tip [6]. Other advantages [3,7] of the BEM are the reduction of the dimensionality of problems; ease of modelling regions where parameters such as stresses or strains vary rapidly; ease of re-meshing where necessary, and others. These advantages of the BEM can be fully exploited in fracture mechanics applications. The ease of re-meshing is particularly important for following the propagation of cracks, as is the fact that it is only necessary to define elements on the boundary, avoiding the discretization of the complete domain.

In this paper, several techniques are used for the calculation of the Stress Intensity Factor KI (Deformation Mode I) [2] using both the FEM and the BEM. The results obtained lead to some practical conclusions which may be easily employed by engineers in the day to day evaluation of structural integrity. The use of special quarter-point elements is also considered. The FEM results are obtained using the ANSYS commercial software, which calculates automatically results for KI using the J-Integral [8,9] technique employing both common and quarter-point quadratic elements. The BEM results are obtained using a classic plane stress FORTRAN code adapted for fracture mechanics problems to employ the COD (Crack Opening Displacement) method, Nodal Stresses and the J-Integral technique to calculate KI. The code employs common quadratic elements, but the calculation of the derivatives of the displacements ( ) and ( ) which appear in the J-Integral is accomplished in an accurate way by differentiating the BEM fundamental solutions [10].

The results obtained here show that accuracy of the results improves when quarter point elements are employed and also if finer discretizations are used. It is noted that the simple methods such as COD and the Nodal Stress method can give reasonable values of KI when refined meshes and quarter-point elements are employed with the FEM. The technique which relates the value of KI to the J-Integral requires a larger computational effort, but produces much more accurate results. Using the FEM, the results obtained using the J-Integral and employing quarter-point elements contained only small errors in comparison with solutions available in the literature. Even considering only common elements, the formulation presented for BEM give results compatible with the best results obtained using the FEM. The BEM produced good results for discretizations employing only a few elements using implicit differentiation of the fundamental solutions for the calculation of and in the calculation of the J-Integral. Considering the results obtained one can ask if the search for the singularity at the crack tip in numerical methods is essential for the calculation of K via the J-Integral.

1

Saouma, V.E. (2000) Fracture Mechanics Lecture Notes, Dept of Civil, Environmental and Architectural Eng., University of Colorado, Boulder, USA

2

Broek, D, (1989) The Practical use of Fracture Mechanics. Kluwer Academic Publishers, USA

3

Brebbia, C.A. & Domingues J, (1989) Boundary Elements: An Introductory Course - Comp. Mech. Publications and McGraw-Hill, Southampton, UK

4

Barsouum, R.S. (1976) On the use of isoparametric finite elements in linear fracture mechanics, Inter. J. for Num. Methods in Engineering, 10, pp 25-37. doi:10.1002/nme.1620100103

5

Blandford, G.E. (1981) Two-Dimensional stress intensity factor computations using the boundary element method. International Journal for Numerical Methods in Engineering, 11 pp 387-404. doi:10.1002/nme.1620170308

6

Martinez J. & Domingues, J. (1984) On the use of quarter-point boundary elements for stress intensity factor computation. International Journal for Numerical Methods in Engineering, 20, pp 1941-1950. doi:10.1002/nme.1620201013

7

Banerjee, P.K (1993) Boundary Element Methods in Engineering, McGraw-Hill, London, UK

8

Rice, J.R. (1968) A path independent integral and the approximate analysis of strain concentrations by notches and cracks, J. App. Mech., 35, pp379- 386.

9

Ewalds, H.L. & Wanhill, R.J.H. (1984) Fracture Mechanics Arnold Edition, London, UK

10

Saigal, S., Aithal, R & Kane J.H. (1989) Conforming boundary elements in plane elasticity for shape design sensitivity, Inter. J. for Num. Meth. in Eng., 28, pp 2795-2811. doi:10.1002/nme.1620281206

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