Asymptotic methods in the Griffith problem. (English. Russian original) Zbl 0722.73049
J. Appl. Math. Mech. 53, No. 4, 520-525 (1989); translation from Prikl. Mat. Mekh. 53, No. 4, 665-671 (1989).
Summary: The problem of a crack in an elastic plane stretched at infinity is examined, taking sufficiently detailed account of the action of the interatomic cohesion forces between the crack edges. With respect to the functions characterizing the opening of the crack, the problem is reduced to a nonlinear singular integrodifferential equation containing two- dimensional parameters. It is shown by using asymptotic methods of regular and matched asymptotic expansions that this equation has two solutions besides the trivial one. They correspond to cracks with respect to small and large openings. Fracture criteria are obtained by using the condition of smooth closing of the crack edges.
MSC:
| 74R99 | Fracture and damage |
| 74B99 | Elastic materials |
| 74H99 | Dynamical problems in solid mechanics |
| 45M05 | Asymptotics of solutions to integral equations |
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 45G05 | Singular nonlinear integral equations |
Keywords:
elastic plane stretched at infinity; interatomic cohesion forces; nonlinear singular integrodifferential equation; two-dimensional parameters; regular; matched asymptotic expansions; small and large openings; smooth closing of the crack edgesReferences:
| [1] | Panasyuk, V. V., Ultimate Equilibrium of Brittle Bodies with Cracks (1968), Naukova Dumka: Naukova Dumka Kiev |
| [2] | Morozov, N. F., Mathematical Problems of the Theory of Cracks (1984), Nauka: Nauka Moscow · Zbl 0566.73079 |
| [3] | Zheltov, Yu. P.; Khristianovich, S. A., On the hydraulic discontinuity of an oil-bearing stratum, (Otd. Tekh. Nauk, 5 (1955), Izv. Akad. Nauk SSSR) |
| [4] | Bilby, B.; Eshelby, J., (Dislocations and Theory of Fracture. Fracture, 1 (1973), Mir: Mir Moscow) |
| [5] | Belotserkovskii, S. M.; Lifanov, I. K., Numerical Methods in Singular Integral Euations (1985), Nauka: Nauka Moscow · Zbl 0578.65140 |
| [6] | Bellman, R.; Kalaba, R., Quasilinearization and Non-linear Bounary Value Problems (1968), Mir: Mir Moscow · Zbl 0165.18103 |
| [7] | Aleksandrov, V. M.; Mkhitaryan, S. M., Contact problems for Bodies with Thin Coatings and Interlayers (1983), Nauka: Nauka Moscow |
| [8] | Naifeh, A. H., Perturbation Methods (1976), Mir: Mir Moscow |
| [9] | Kudish, I. I., Interaction of facings found under conditions of steady-state non-linear creep with an elastic half-plane, (Mekhanika, 32 (1979), Izv. Akad. Nauk Arm SSR), 2 · Zbl 0414.73034 |
| [10] | Kudish, I. I., Numerical methods of solving a class of non-linear integral and integrodifferential equations, Zh. vychisl. Mat. mat. Fiz., 26, 10 (1986) · Zbl 0634.65139 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
