Weight Functions for Three-Dimensional Symmetrical Crack Problems

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Abstract
The knowledge of weight functions would be of great help for the solution of three-dimensional crack problems. A numerical method of computation of these functions by finite elements is developed, starting from the simple consideration of concentrated forces applied to the crack and the energy released by local extensions of the crack. Particular attention is paid to the consideration of the asymptotic value of the weight functions at the crack tip, which allows the definition of dimensionless weight functions and makes the numerical calculation easier. Special weight functions are considered for the case when the applied stress depends on one coordinate only. The method is checked by comparing the computed weight function for a penny-shaped crack in an infinite solid with the known closed form solution. It seems that the accuracy obtained could allow for solution of engineering problems; however this should be checked by other tests especially in the region of the singularity. The computing time, however, is long because of the large number of nodes needed in three-dimensional bodies, and the calculation is costly. It seems advisable to investigate the possibilities of other methods of solution of elasticity problems, such as the method of boundary integral equations, for changing the order of magnitude of the computing time and the cost of the calculation.