Geodesic
Asymptotics of the near crack-tip stress field of a~fatigue growing crack in damaged materials: numerical experiment and analytical solution
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 2, pp. 201-217.

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In this paper, the asymptotic analysis of the near fatigue growing crack-tip fields in a damaged material is done. The integrity parameter describing the damage accumulation process in the vicinity of a crack tip is incorporated into the constitutive law of the isotropic linear elastic material. The asymptotic solution based on the eigenfunction expansion method is obtained. It is shown that the problem is reduced to the nonlinear eigenvalue problem. The analytical solution of the nonlinear eigenvalue problem is found by the artificial small parameter method. The perturbation theory approach allows us to derive the analytical presentation of the stress and integrity distributions near the crack tip. The technique proposed permits us to find the higher-order terms of the asymptotic expansions of the stress components and the integrity parameter. 
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L. V. Stepanova; S. A. Igonin. Asymptotics of the near crack-tip stress field of a~fatigue growing crack in damaged materials: numerical experiment and analytical solution. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 2, pp. 201-217. http://geodesic.mathdoc.fr/item/SJVM_2015_18_2_a7/

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