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Thin Three-Dimensional Plate with a Crack along the Clamped Zone on Its Lateral Side

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Abstract

Asymptotic terms are constructed for a solution to the three-dimensional elasticity problem for a thin cylindrical isotropic plate clamped at its lateral side except for a part (crack) that is in contact with a rigid profile without friction. The existence and uniqueness theorems are established. The behavior of solutions at infinity is studied. Bibliography: 38 titles. Illustrations: 1 figure.

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References

  1. A. L. Gol'denvejzer, “Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity,” Prikl. Mat. Mekh. [in Russian], No. 4, 668–686 (1962) [English Translation: J. Appl. Math. Mech., 1000–1025 (1963)].

  2. A. L. Gol'denvejzer, Theory of Elastic Thin Shells, Nauka, Moscow (1976).

    Google Scholar 

  3. B. A. Shoikhet, “On asymptotically exact equations of thin plates of complex structure,” J. Appl. Math. Mech., No. 5, 867–877 (1973).

  4. P. Destuynder, “Comparision entre les modeleles tridimensionnels et bidimensionnels de plaques en elasticite,” RAIRO Analyse Numerique, 331–396 (1981).

  5. S. A. Nazarov, Introduction to Asymptotic Methods of the Elasticity Theory [in Russian], Leningrad, 1983.

  6. S. N. Leora, S. A. Nazarov, and A. V. Proskura, “Derivation of limiting equations for elliptic problems in thin domains using computers,” Zh. Vych. Mat. i Mat. Fiz., No. 7, 1032–1048 (1986) [English Translation: USSR Comput. Math. Math. Phys., No. 4, 47–58 (1986)].

  7. I. S. Zorin and S. A. Nazarov, “Edge effect in the bending of a thin three-dimensional plate,” Prikl.Mat.Mekh. [in Russian], No. 4, 642–650 (1989) [English Translation: J. Appl. Math. Mech., No. 4, 500–507 (1989)].

  8. P. G. Ciarlet, Plates and Junction in Elastic Multi Structures, Masson, Paris (1990).

    Google Scholar 

  9. W. G. Mazja, S. A. Nazarov, and B. A. Plamenevskii, Asymptotische theorie elliptischer randwertaufgaben in singular gestrorten Gebienten. Bd. 2. Berlin: Akad.-Verlag, Berlin (1991) [English Translation: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domain. Vol. 2. Birkhauser Verlag, Basel-Boston-Berlin (2000)].

    Google Scholar 

  10. M. Dauge and I. Gruais, “Developpment asymptotique d'order arbitrare pour une plaque elastique mince encastree,” C. R. Acad. Sci. Paris. Ser. 1, 375–380 (1995).

  11. S. A. Nazarov, “Elastic capacity and polarization of a defect in an elastic layer,” Mekhanika Tverd. Tela [in Russian], No. 5, 57–65 (1990).

  12. S. A. Nazarov, “The spatial structure of the stress field in a neighborhood of the corner point of a thin plate,” Prikl. Mat. Mekh. [in Russian], No. 4, 653–661 (1991) [English Translation: J. Appl. Math. Mech., No. 4, 523–530 (1991)].

  13. S. A. Nazarov, “On three-dimensional effects near the vertex of a crack in a thin plate,” Prikl. Mat. Mekh. [in Russian], No. 3, 500–510 (1991) [English Translation: J. Appl. Math. Mech., No. 3, 407–415 (1991)].

  14. S. A. Nazarov, “Asymptotics of the solution to a boundary value problem in a thin cylinder with nonsmooth lateral surface,” Izv. Ross. Akad. Nauk. Ser. Mat. [in Russian], No. 1, 202–239 (1993) [English Translation: Math. Izvestiya, No. 1, 183–217 (1994)].

  15. S. A. Nazarov, “Korn inequalities that are asymptotically exact for thin domains,” Vestn. St.Petersburg Univ. [in Russian], No. 8, 19–24 (1992) [English Translation: Vestn. St. Petersburg Univ. Math., No. 2, 18–22 (1992)].

  16. V. A. Kondrat'ev, “Boundary value problems for elliptic equations in domains with conical or angular points,” Tr. Mosk. Mat. O.va [in Russian], 219–292 (1967).

  17. V. G. Maz'ja and B. A. Plamenevskii, “Estimates in L p and in Hölder classes, and the Miranda-Agmon maximum principle for the solutions of elliptic boundary value problems in domains with singular points on the boundary,” Math. Nachr., 25–82 (1978).

  18. S. A. Nazarov and B. A. Plamenevskii, Elliptic Problems in Domains with Piecewise Smooth Boundary [in Russian], Nauka, Moscow (1991) [English Translation: EllipticProblems in Domains with PiecewiseSmooth Boundaries, De Gruyter, Berlin (1994)].

    Google Scholar 

  19. O. V. Isotova and S. A. Nazarov, “Energy release due to decreasing of the clamped zone of a bended plate,” Vestn. St. Petersburg Univ. [in Russian], No. 15, 64–67 (1999).

  20. H. F. Buecner, “A novel principle for computation of stress intensity factor,” Z. Angew.Math. Mech., 0, 529–546 (1970).

    Google Scholar 

  21. V. G. Maz'ja and B. A. Plamenevskii, “The coefficients in the asymptotics of solutions of elliptic boundary value problems with conical points,” Math. Nachr., 29–60 (1977).

  22. S. A. Nazarov, “Weight functions and invariant integrals,” Vychisl. Mekh. Deform. Tverd. Tela [in Russian], 17–31 (1990).

  23. K. O. Fridichs and T. F. Dressler, “A boundary-layer theory for elastic plates,” Comm. Appl. Math., No. 1, 1–33 (1961).

  24. A. L. Gol'denvejzer and A. V. Kolos, “On construction of two-dimensional equations of elastic thin shells,” Prikl. Mat. Mekh. [in Russian], No. 1, 141–161 (1965).

  25. R. D. Gregory and F. V. Wan, “Decaying states of plane strain in a semi-infinite strip and boundary condition for plate theory,” J. Elast., No. 1, 27–64 (1984).

    Google Scholar 

  26. I. S. Zorin and S. A. Nazarov, “Two-term asymptotics of the solution to the problem on longitudinal deformation of a plate clamped at the edge,” Vychisl. Mekh. Deform. Tverd. Tela [in Russian], 10–21 (1992).

  27. S. A. Nazarov, “Boundary layers and the hinge-support conditions for thin plates,” Zap. Nauchn. Semin. POMI [in Russian], 228–287 (1999).

  28. O. A. Ladyzhenskaya, BoundaryValue Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  29. J.-L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin (1972).

    Google Scholar 

  30. S. A. Nazarov, “Asymptotic expansions at infinity of solutions to the elasticity theory problem in a layer,” Tr. Mosk. Mat. O.va [in Russian], 0, 3–97 (1998) [English Translation: Trans. MoscowMath. Soc., 0, 1–85 (1999)].

    Google Scholar 

  31. S. A. Nazarov, “Asymptotics of the solution of a Dirichlet problem in an angular domain with a periodically changing boundary,” Mat. Zametki [in Russian], No. 5, 86–96 (1991) [English Translation: Math. Notes, No. 5, 502–509 (1991)].

  32. S. A. Nazarov, “Asymptotics at infinity of the solution to the Dirichlet problem for a system of equations with periodic coefficients in an angular domain,” Russian J. Math. Phys., No. 3, 297–326 (1995).

    Google Scholar 

  33. V. G. Mazja and S. A. Nazarov, “The asymptotic behavior of energy integrals under small perturbations of the boundary near corner points and conical points,” Tr. Mosk. Mat. O.va [in Russian], 0, 79–129 (1987) [English Translation: Trans. Moscow Math. Soc., 0, 77–127 (1988)].

    Google Scholar 

  34. G. C. Sih and H. Liebowitz, H. “Mathematical Theories of Brittle Fracture”, Fracture. 2, Academic Press, New York (1968). p. 67–190.

    Google Scholar 

  35. G. P. Cherepanov, “Crack propagation in continuousmedia,” Prikl. Mat. Mekh. [in Russian], No. 3, 476–488 (1967) [English Translation: J. Appl. Math. Mech., 503–512 (1967)].

  36. J. R. Rice, “A path independent integral and the approximate analysis of strain concentration by notches and cracks,” J. Appl. Mech., No. 2, 379–386 (1968).

  37. O. R. Polyakova and S. A. Nazarov, “On the equivalence of the fracture criteria for a mode-one crack in an elastic space,” Mekhanika Tverd. Tela [in Russian], No 2, 101–113 (1992).

  38. S. A. Nazarov, “Derivation of the variational inequality for small increase of mode-one crack,” Mekhanika Tverd. Tela [in Russian], No. 2, 152–160 (1989) [English Translation: Mech. Solids, 145–152 (1989)].

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Authors
  1. O. V. Izotova
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  2. S. A. Nazarov
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Izotova, O.V., Nazarov, S.A. Thin Three-Dimensional Plate with a Crack along the Clamped Zone on Its Lateral Side. Journal of Mathematical Sciences 105, 2398–2435 (2001). https://doi.org/10.1023/A:1011317330046

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