Method of Moments for Analysis of Multiple Crack Interaction in an Isotropic Elastic Solid

The problem of N cracks interaction in an isotropic elastic solid is decomposed into a subproblem of a homogeneous solid without crack and N subproblems with each having a single crack subjected to unknown tractions on the two crack faces. The unknown tractions, namely pseudo tractions on each crack are expanded into polynomials with unknown coefficients, which have to be determined by the consistency condition, i.e. by the equivalence of the original multiple cracks interaction problem and the superposition of the N+1 subproblems. In this paper, Kachanov-s approach of average tractions is extended into the method of moments to approximately impose the consistence condition. Hence Kachanov-s method can be viewed as the zero-order method of moments. Numerical results of the stress intensity factors are presented for interactions of two collinear cracks, three collinear cracks, two parallel cracks, and three parallel cracks. As the order of moment increases, the accuracy of the method of moments improves.

Authors:

• Weifeng Wang
• Xianwei Zeng
• Jianping Ding

Keywords:

• Crack interaction
• stress intensity factor
• multiplecracks
• method of moments.

References:

[1] M. Kachanov, "Effective elastic properties of cracked solids: critical
review of some basic concepts," Appl. Mech. Review, vol. 45, pp.
304-335, 1992.
[2] N. Horri, and S. Nemat-Nasser, "Interacting microcracks near the tip in
the process zone of a macro-crack", J. Mech. Phys. Solids, vol. 35, pp.
601-629, 1987.
[3] J W. Hutchinson, "Crack tip shielding by microcracking in brittle soilds",
Acta. Metall., vol. 35, pp. 1606-1619.
[4] Y. Z. Chen, "Integral equation methods for multiple crack problems and
related topics," Appl. Mech. Review, vol. 60, pp. 172-194, 2007.
[5] Y. Z. Chen, "Complex potentials in plane elasticity by distribution of
dislocation or force doublet along a curve," Int. J. Eng. Sci., vol. 36, pp.
23-31, 1998.
[6] H. Horri, and S. Nemat-Nasser, "Elastic fields of interacting
imhomogeneities," Int. J. Solids Struct., vol. 21, pp. 731-745, 1985.
[7] D. Gross, "Stress intensity factors of system of cracks," Ing. Archs., vol.
51, pp. 301-310, 1982.
[8] Y. Benveniste, G. J. Dvorak, J. Zarzour , and C. J. Wung, "On interacting
cracks and complex crack configurations in linear elastic media," Int. J.
Solids Struct., vol. 25, pp. 1279-1293, 1989.
[9] M. Kachanov, "Elastic soilds with many cracks: a simple method of
analysis," Int. J. Solids and Struct., vol.23, pp. 23-43, 1987.
[10] Y. P. Li, L. G. Tham, Y. H. Hwang, and Y. Tsui, "A modified Kachanov
method for analysis of solids with multiple cracks," Eng. Frac. Mech., vol.
70, pp. 1115-1129, 2003.
[11] N. I. Muskhelishvili, Some Problems in the Mathematical Theory of
Elasticity. Noordhoff, Groningen, 1953.
[12] C. Hwu, "Collinear cracks in anisotropic bodies," Int. J. Frac., vol. 52, pp.
239-251, 1991.
[13] G. C. Sih, Boundary problems for longitudinal shear cracks. Proceedings,
Second Conference on Theoretical and Applied Mechanics, New York:
Pergamon, 1964.