PSI - Issue 52

Akihide Saimoto et al. / Procedia Structural Integrity 52 (2024) 323–339 A. Saimoto et al. / Structural Integrity Procedia 00 (2023) 000–000

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The relationship between the SIF and h / b at the inner crack tip B is plotted in Fig.11 for α = 0 ◦ and Fig.12 fore α = 60 ◦ , respectively. The mode I SIF is considerably larger than that for tip A due to interference e ff ects. On the other hand, the mode II SIF is negative over a wide range of h / b , which may indicate a tendency for the two cracks to propagate in the direction of merging. Although these trends do not change much with changing the value of angle α , the SIFs are definitely di ff erent for each value of α , and it is recognized that it is necessary to accurately determine the SIF for each angle α .

4. Conclusion

Numerical analyses based on the body force method were performed for the case of two parallel cracks of an identi cal length in an elastic infinite plate with orthotropic anisotropy. The crack tip stress intensity factor was calculated for cracks located in a shielding or staircase positions. The material properties considered were taken from reference pub lications and the direction of the principal axis of the anisotropy and distance between cracks varied systematically. In the body force method, the relative displacement between the upper and lower crack faces is expressed by embedding three types of point force doublets, which were shown in Fig.3, along a boundary to be a crack. It is known that the accuracy of numerical analysis dramatically improved by introducing a fundamental density function that expresses the stress singularity at the crack tip exactly. In the present study, the boundary to be a crack is divided equally in length following a resultant force method proposed by Isida (1978), and the weight function of the body force doublets is supposed to vary linearly in each segment. Furthermore, all necessary boundary integrals were performed analytically so that numerical errors caused by numerical integrals could be completely eliminated. In many published solutions of crack problems for orthotropic materials, the accuracy of numerical solution is generally undefined, and therefore, there is no guarantee of reliability. In this study, an original computer program was developed to estimate a stress intensity factor at a crack tip, and the computation was continued until the obtained solution converged su ffi ciently by increasing the number of boundary divisions. For example, when two cracks of equal length are located in such a way that they shield each other, the accuracy of the numerical analysis is considerably low compared with a single-crack situation. This tendency becomes more apparent when the distance between cracks becomes smaller. However, as seen in Table 2-9, the stress intensity factor at the crack tip converges monotonically to a certain value with the increase of crack division number n . Inother words, the practically exact solution of the stress intensity factor can be obtained by simply increasing the number of divisions of the crack boundary. In discretization analysis procedures such as finite elements and boundary elements, the element size in the vicinity of a crack tip, and the element geometry may have a significant influence on the numerical accuracy. In stress analysis using such general-purpose programs, it is a major challenge for engineers to evaluate the reliability of numerical solutions. In such a situation, the existence of a practical exact solution plays an important role. For example, in the finite element analysis of the two-crack interference problem, the converged values in Tables 2-9 would suggest how detailed the element division near a crack tip should be prepared. As seen, the stress intensity factor solutions shown in the last lines of these tables can be regarded as practically exact. According to Fig.8, when two equal-length cracks are in a shielding location with each other, the mode I stress intensity factor monotonically increase while mode II decreases, with the increase of distance between two cracks, regardless of the material properties and direction of the material principle axis. That is, in the limit where the two cracks are su ffi ciently far apart, the stress intensity factor converges to that of a single crack. On the other hand, when the two cracks are placed in a staircase location, the values of stress intensity factor of both mode I and II are larger than those of an isolated crack.

References

Nisitani, H., 1967. The Two-Dimensional Stress Problem Solved Using an Electric Digital Computer. Journal of The Japan Society of Mechanical Engineers, 70 pp.627-632 (In Japanese). Isida, M., 1978. Analysis of Stress Intensity Factors of Plates Containing an Arbitrary Array of Line and Branched Cracks. Transactions of The Japan Society of Mechanical Engineers, 44(380) pp.1122-1132 (In Japanese). Sih, G.C., Paris, P.C.,Irwin, G.R., 1965. On Cracks in Rectilinearly Anisotropic Bodies. International Journal of Fracture Mechanics, 1,pp.189-203. Savin,G.N., 1961. Stress Concentration Around Holes. Pergamon Press.