Computation of Stress Intensity Factors for Three-Dimensional Bodies Using Crack-Displacements

In welded structures, surface or embedded cracks usually initiate from defects near the heat-affected zone. To estimate the life of crack propagation and fracture strength, the variation of the stress intensity factor along the crack front needs to be computed, which in most cases requires a numerical solution.

The finite element method (FEM) is one of the most widely used numerical method. It is commonly accepted that an adequate modeling of the singularity is essential [1] when evaluating stress intensity factors using the FEM. With exception of the J-integral which does not require the knowledge of the local deformation near the crack tip, this usually requires the use of crack elements and a fine mesh near the crack tip region. In the past decade two approaches [2, 3] were developed using crack elements to compute stress intensity factors for an embedded or surface elliptical crack. The approach of Raju and Newman [2] uses the nodal forces acting on several elements lying on a trajectory normal to the crack front. For a crack with an elliptical front the trajectory can be obtained directly using a transformation from a circular crack. However, for an arbitrary crack shape such a transformation may not readily be available. The other approach from Abdel, et al. [3] uses the displacements behind the crack tip and the nodal forces ahead of the crack tip. When the area of the elements behind the crack tip is different from that ahead of the crack tip, a correction of the resultant nodal forces on the element ahead of the crack, as described in [3], becomes necessary. For embedded and surface cracks in infinite bodies the two approaches lead to results that are in close agreement, with a difference typically less than 4%. For a deep semi-elliptical crack in a plate of finite thickness the stress intensity factors were obtained by Raju and Newman in [2]. Their results, though in disagreement with other researchers’ work [4-6] by 50% to 100%, were found to agree with experimental data within 10% [7].

Recently an exact expression was derived for 2-D problems in [8] to relate the stress intensity factors to the change of the displacements due to an increment of the crack. The procedure involves computations of displacements for both the actual loading and a virtual force at a location away from the crack plane. Since the solution is independent of the location of the displacements, stress intensity factors can be obtained accurately from the displacements remote from the crack tip region without using crack tip elements or a fine mesh near the crack.
In this paper we will extend significantly the approach developed in [8] to include the computation of the stress intensity factor for a 3-D body containing a crack of arbitrary shape. First, an expression relating the change of the displacements to the stress intensity factor at a location along the crack front is obtained. Computations are then carried out for several 3-D crack configurations. These include an embedded circular or elliptical crack in an infinite body, a semi-circular or semi-elliptical surface crack in a half space and a deep semi-elliptical crack in a plate of finite thickness. In all cases the variation of the stress intensity factor along the crack front is obtained and compared with the exact solution [9] and numerical results available in literature [2, 4-6, 10].

1. O. C. Zienkiewicz, The Finite Element Method, 3rd edition, McGraw-Hill Book Company Limited. p.669 (1977).
2. I. S. Raju and J. C. Newman, Jr., “Stress-Intensity Factors for a Wide Range of Semi-Elliptical Surface Cracks in Finite-Thickness Plates,” Eng. Fracture Mechanics, 11, pp. 817-829 (1979).
3. M. M. Abdel Wahab and G. De Roeck, “A Finite Element Solution for Elliptical Cracks Using the ICCI Method,” Engineering Fracture Mechanics, 53, pp. 519-526 (1996).
4. F. W. Smith and D. R. Sorensen, “Mixed Mode Stress Intensity Factors for Semi-Elliptical Surface Cracks,” NASA CR-134684 (1974).
5. A. S. Kobayashi, “Surface Flaws in Plates in Bending,” Proc. 12th Annual Meeting of the Soc. Of Engng Sci., Austin, Texas (1975).
6. K. Kathiresan, “Three-dimensional Linear Elastic Fracture Mechanics Analysis by a Displacement Hybrid Finite Element Model,” Ph. D. Thesis, Georgia Institute of Technology (1976).
7. J. C. Newman and I. S. Raju, “An Empirical Stress-Intensity Factor Equation for the Surface Flaw,” Engineering Fracture Mechanics, 15, pp. 185-192 (1981).
8. W. Cheng and I. Finnie, “Computation of Stress Intensity Factors from Displacements at an Arbitrary Location,” submitted to Int. J. of Fracture.
9. A. E. Green and I. N. Sneddon, “The Distribution of Stress in the Neighborhood of a Flat Elliptical Crack in an Elastic Solid,” Proc. Cambridge Phil. Soc. 46 (1959).
10. C. Newman, Jr. and I. S. Raju, “Stress-Intensity Factor Equations for Cracks in Three-Dimensional Finite Bodies,” Fracture Mechanics: Fourteenth Symposium I, ASTM STP 791, I238-I265 (1983).

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