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The methods of complex potentials, singular integral equations and integral transformations in a series of problems for the reinforcement of cracked plates

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Abstract

We study the initiation and propagation of a vertical crack in an elastic semi-infinite plate, reinforced on its boundary by an infinite discontinuous stringer within the limits of the theory of brittle failure. The plate is subjected to uniform distributed tensile forces at infinity, as well as to contact stresses due to application of forces to the stringer. We find the appropriate loading of the coherent stringer, and consequently we consider a problem where the stringer is cracked and a vertical crack has developed within the plate. We deduce the exact analytical solution for the principal singular integral equation for this case; hence the stringer is perfectly rigid and we calculate characteristic parameters of the problem. The results show that the crack tip has a logarithmic singularity, and the tangential contact stresses under the stringer at that end point are finite and generally differ from zero.

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References

  1. Melan E.: Ein Beitrag zur Theorie geschweisster Verbindungen. Ing. Arch. 3(2), 123 (1932)

    Article  Google Scholar 

  2. Bufler H.: Zur Krafteinleitung in Scheiben über geschweisste oder geklebte Verbindunngen. Österreichisches Ing. Arch. 18(3–4), 284 (1964)

    MathSciNet  MATH  Google Scholar 

  3. Muki R., Steinberg E.: Transfer of load from an edge stiffener to a sheet. A reconsideration of Melan’s problem. J. Appl. Mech. Trans. ASME 34(3), 679 (1967)

    Google Scholar 

  4. Sternberg, E.: Load transfer and load diffusion in elastostatics. Proceedings of the Sixth U.S. National Congress of Applied Mechanics (1970)

  5. Erdogan, F.: Analysis of elastic over plates. Developments in Mechanics, vol. 6, pp. 817–830. In: Proceedings 12th Midwestern Mechanics Conference (1971)

  6. The development of theory of contact problems in USSR. Moscow, Nauka, 1983 (in Russian). Editor: Galine

  7. Grigolyuk, E.I., Tolkachev, V.M.: Contact problems of theory of plates and shells. Moscow, Mashinostroenie (in Russian) (1980)

  8. Cherepanov, G.P.: Fracture mechanics of composite materials. Moscow, Nauka (in Russian) (1983)

  9. Alexandrov, V.M., Mkhitaryan, S.M.: Contact problems for bodies with covers and layers. Moscow, Nauka (in Russian) (1983)

  10. Muki R., Sternberg E.: Load absorption by a discontinuous filament in a fiber reinforced composite. Z. Angew. Math. Phys. 22(809), 1971 (1971)

    Google Scholar 

  11. Greif, R., Sanders, J.L. Jr.: The effect of the stringer on the stress in a cracked sheet. J. Appl. Mech. Trans. ASME 32, 66, translated in Russian (1965)

  12. Delale F., Erdogan F.: The crack problem for a half plane stiffened by elastic cover plates. Int. J. Solids Struct. 18(5), 381 (1982)

    Article  MATH  Google Scholar 

  13. Theocaris P.S., Bardzokas D.: The influence of a finite stringer on the stress intensities around cracks in plates. Eng. Fract. Mech. 14, 493–508 (1981)

    Article  Google Scholar 

  14. Theocaris P.S., Bardzokas D.: The frictionless contact of cracked elastic bodies. ZAMM 63, 89–102 (1993)

    MathSciNet  Google Scholar 

  15. Bardzokas D., Parton V.Z., Theocaris P.S.: Integral equations of elasticity theory for the multiconnected domain with inclusions. P.M.M 53(3), 375–384, (1979) in Russian

    MathSciNet  Google Scholar 

  16. Bardzokas D., Exadaktylos G.E., Anastaselos G.: The effect of stringers and patches on the stress intensities around cracks in plates. Eng. Fract. Mech. 55(6), 935 (1996)

    Article  Google Scholar 

  17. Antipov Y.A., Bardzokas D., Exadaktylos G.E.: Partially stiffened elastic half-plane with an edge crack. Int. J. Fract. 85, 241–263 (1997)

    Article  Google Scholar 

  18. Lawn B.R., Wilshaw T.R.: Indentation fracture: principles and applications. J. Mater. Sci. 10(6), 1049 (1975)

    Article  Google Scholar 

  19. Kolesnikow, U.V., Morozov, E.M.: Mechanics of contact fracture. Moscow, Nauka (in Russian) (1989)

  20. Cherepanov G.P.: A problem of stamp indentation with formation of cracks. J. Appl. Math. Mech. (PMM) 27(1), 210–216 (1963)

    Article  MATH  Google Scholar 

  21. Erdogan F., Gupta G., Cook T.: The numerical solutions of singular integral equations. In: (eds) Methods of analysis and solutions of crack problems, pp. 368–425. Nordhoff Int. Publ., Leyden (1973)

    Google Scholar 

  22. Barenblatt, G.I.: Mathematical theory of equilibrium cracks, formed under the brittle fracture (in Russian). Appl. Mech. and Tech. Phys. (PMTP), No. 4, 3 (1961)

  23. Timoshenko S.P., Goodier J.N.: Theory of Elasticity. McGraw-Hill, New York (1970)

    MATH  Google Scholar 

  24. Baterman H., Erdelyi A.: Higher Transcendental Functions. McGraw-Hill, New York (1953)

    Google Scholar 

  25. Titchmarch, E.C.: Introduction to the theory of Fourier integrals. University of Oxford, 2nd edn

  26. Baterman, H., Erdelyi, A.: Tables of integral transforms. Vol. I, Moscow, Nauka (in Russian) (1969)

  27. Lighthill M.J.: Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press, Cambridge (1959)

    Google Scholar 

  28. Dundurs, J.: Elastic interaction of dislocations with inhomogeneities. In: Mura, T. (ed.) Mathematical Theory of Dislocations, pp. 70–115. ASME, USA (1969)

    Google Scholar 

  29. Gachov, F.D.: Boundary problems, Moscow, Nauka (in Russian) (1977)

  30. Bilby B.A., Eshelby J.D.: Dislocations and the theory of fracture. In: Liebowitz, H. (eds) Fracture: An Advanced Treatise, pp. 100. Academic Press, New York (1968)

    Google Scholar 

  31. Panasyuk, V.V.: Limiting equilibrium of brittle bodies with cracks (in Russian). Kiev, Naukova Dumka (1968)

  32. Gradshteyn I.S., Ryzhik I.M.: Tables of Integrals, Series and Products. Academic Press, London (1980)

    Google Scholar 

  33. Panasyuk, V.V., Savruk, M.P., Datsyshyn, A.P.: The distribution stresses nearby cracks in plates and shells (in Russian). Kiev, Naukova Dumka (1976)

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Authors and Affiliations

  1. National Technical University of Athens, Athens, Greece

    Dimosthenis I. Bardzokas & Georgios I. Sfyris

  2. Institute of Mechanics, National Academy of Sciences of Republic of Armenia, 24b Marshal Baghramianave., Erevan, 0019, Republic of Armenia

    S. M. Mkhitaryan

Authors
  1. Dimosthenis I. Bardzokas
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  2. S. M. Mkhitaryan
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  3. Georgios I. Sfyris
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Correspondence to Georgios I. Sfyris.

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D. I. Bardzokas: Deceased.

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Bardzokas, D.I., Mkhitaryan, S.M. & Sfyris, G.I. The methods of complex potentials, singular integral equations and integral transformations in a series of problems for the reinforcement of cracked plates. Acta Mech 221, 147–174 (2011). https://doi.org/10.1007/s00707-011-0473-3

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