Skip to main content
SpringerLink
Log in

Optimum Shapes of Elastic Bodies: Equistrong Wings of Aircrafts and Equistrong Underground Tunnels

  • Published:
Physical Mesomechanics Aims and scope Submit manuscript

Abstract

Since the times of Plato (424?-347 BC) and Aristotle (384-322 BC), the form has been considered as a fundamental notion of not only the physical universe, but also the spiritual world. The forms and perfect shapes are like jewels in the rock—their search and discovery make up the highest delight to human beings. This is what constituted the motive and driving force of science and scientists beginning from Pythagoras (ca. 570-ca. 500 BC), Archimedes (ca. 287-ca. 212 BC), Euclid (ca. 330-ca. 260 BC) and all that came after. In the introduction to the present article, a brief account of Plato’s theory of forms and Aristotle’s addition to this theory are given; the theory of optimum shapes of elastic bodies can be considered as a footnote to the Plato’s theory. In the framework of the theory of elasticity, the optimal shape of a body is the shape that meets the principle of equal strength or equistrength advanced by this author in 1963. According to this principle, the safety criterion like ultimate or failure stress is simultaneously satisfied in the utmost part of the body—this body or structure is called equistrong in this case. The equistrong structure has a minimum weight for a given material and safety factor, or a maximum safety factor for a given material and weight. As distinct from traditional problems, there are no existence theorems for equistrong shapes—a success in their search depends on skills of a researcher. In the present paper, a summary of common equistrong shapes and structural elements is brought out, namely: equistrong cable of bathysphere, equistrong tower or skyscraper, equistrong beam by Galileo Galilei, equistrong rotating disk, equistrong heavy chain, equistrong pressure vessels, equistrong arcs, plates and shells, equistrong underground tunnels, equistrong perforated plates, and others. A variety of the swept wings of aircrafts is found out to be equistrong; the front and rear edges of such wings are rectilinear in the plan view, and their chord in the flight direction depends on the task of an aircraft; the equistrong design exists for any task, from transport aviation to hypersonic jet fighters. Some new equistrong shapes of elastic solids with any number of infinite branches being pulled out of a body are also discovered for plane strain and plane stress.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bienzeno, C.B. and Grammel, R., Engineering Dynamics. Vol. II. Elastic Problems of Single Machine Elements, Glasgow: Blackie&Sons Ptd., 1956.

    Google Scholar 

  2. Cherepanov, G.P., An Inverse Elastic-Plastic Problem under Plane Strain, Mekh. Mashinostroenie, 1963, no. 1, pp. 63–71.

    Google Scholar 

  3. Cherepanov, G.P., To the Solution of Some Problems of the Theory of Elasticity and Plasticity with an Unknown Boundary, J. Appl. Math. Mech., 1964, vol. 28, no. 1, pp. 141–144.

    Article  MathSciNet  Google Scholar 

  4. Cherepanov, G.P., Some Problems with an Unknown Boundary in the Theory of Elasticity and Plasticity, Applications of the Theory of Functions of a Complex Variable in Continuum Mechanics. vol. 1, Moscow: Nauka, 1965.

    Google Scholar 

  5. Cherepanov, G.P., Equistrong Mine in a Rock, in Problems of Rock Mechanics, Erzhanov, A., Ed., Alma-Ata: Nauka, 1966, pp. 166–178.

    Google Scholar 

  6. Cherepanov, G.P., One Inverse Problem of the Theory of Elasticity, Eng. J. Solid Mech., 1966, no. 3, pp. 118–130.

    Google Scholar 

  7. Cherepanov, G.P. and Liberman, L.K., Mathematical Method of Optimal Design in Mining, Mining J., 1971, no. 9, pp. 987–992.

    Google Scholar 

  8. Cherepanov, G.P., Inverse Problems of the Plane Theory of Elasticity, J. Appl. Math. Mech., 1974, vol. 38, pp. 963–979.

    Article  MathSciNet  Google Scholar 

  9. Cherepanov, G.P. and Tagizade, A.G., Extension of Plates and Shells of Uniform Strength, Lett. Appl. Eng. Sci.,1975, no. 3, pp. 64–69.

    Google Scholar 

  10. Cherepanov, G.P. and Tagizade, A.G., Bending of Plates and Shells of Uniform Strength, Lett. Appl. Eng. Sci., 1975, no. 2, pp. 38–43.

    Google Scholar 

  11. Vigdergauz, S., Integral Equations of the Inverse Problem of the Theory of Elasticity, J. Appl. Math. Mech., 1976, vol. 40, no. 3.

    Google Scholar 

  12. Cherepanov, G.P., Smolsky, V.M., and Tagizade, A.G., Optimum Design of Some Engineering Materials, Izv. Acad. Sci. Armenian SSR, 1976, vol. 29, pp. 248–265.

    Google Scholar 

  13. Wheeler, L.T., On the Role of Constant-Stress Surfaces in the Problem of Minimizing Elastic Stress Concentration, J. Solids Struct., 1976, vol. 12, pp. 779–789.

    Article  MATH  Google Scholar 

  14. Cherepanov, G.P. and Ershov, L.V., Fracture Mechanics, Moscow: Mashinostroenie, 1977.

    Google Scholar 

  15. Vigdergauz, S., On a Case of the Inverse Problem of Two-Dimensional Theory of Elasticity, J. Appl. Math. Mech., 1977, vol. 41, no. 5, pp. 902–908.

    Article  MathSciNet  Google Scholar 

  16. Wheeler, L.T., On Optimum Profiles for the Minimization of Elastic Stress Concentration, ZAMM, 1978, pp. 235–236.

    Google Scholar 

  17. Cherepanov, G.P., Mechanics of Brittle Fracture, New York: McGraw-Hill, 1978.

    Google Scholar 

  18. Mirsalimov, V.M., Equistrong Excavation in Rock, FTPRPI, 1979, no. 4, pp. 24–29.

    Google Scholar 

  19. Banichuk, N.V., Shape Optimization of Elastic Solids, Moscow: Nauka, 1980.

    Google Scholar 

  20. Cherepanov, G.P., Stress Corrosion Cracking, in Comprehensive Treatise of Electrochemistry. V. 4. Electrochemical Materials Science, Bockris, O.M., Ed., New York: Plenum Press, 1981, pp. 333–406.

    Google Scholar 

  21. Ostrosablin, N.I., An Equistrong Hole in a Plate Under Non-Uniform Stress State, J. Appl. Mech. Tech. Phys., 1981, no. 2, pp. 155–163.

    MathSciNet  Google Scholar 

  22. Vigdergauz, S., Equistrong Hole in a Half-Plane, Mech. Solids, 1982, vol. 17, no. 1, pp. 87–91.

    MathSciNet  Google Scholar 

  23. Wheeler, L.T. and Kunin, I.A., On Voids of Minimum Stress Concentration, J. Solids Struct., 1982, vol. 18, pp. 85–89.

    Article  MATH  MathSciNet  Google Scholar 

  24. Vigdergauz, S., An Inverse Problem of the Three-Dimensional Theory of Elasticity, Mech. Solids, 1983, vol. 18, no. 2, pp. 83–86.

    Google Scholar 

  25. Vigdergauz, S., Effective Elastic Parameters of a Plate with a Regular System of Equistrong Holes, Mech. Solids, 1986, vol. 21, no. 2, pp. 165–169.

    Google Scholar 

  26. Cherepanov, G.P. and Annin, B.D., Elastic-Plastic Problems, New York: ASME Press, 1988.

    Google Scholar 

  27. Vigdergauz, S., The Geometrical Characteristics of Equistrong Boundaries of Elastic Bodies, J. Appl. Math. Mech., 1988, vol. 52, no. 3, pp. 371–376.

    Article  MATH  MathSciNet  Google Scholar 

  28. Wheeler, L.T., Stress Minimum Forms for Elastic Solids, ASME Appl. Mech. Rev., 1992, vol. 45, pp. 110–125.

    Article  Google Scholar 

  29. Vigdergauz, S., Optimal Stiffening of Holes under Equibiaxial Tension, Int. J. Solids Struct., 1993, vol. 30, no. 4, pp. 569–577.

    Article  MATH  Google Scholar 

  30. Cherepanov, G.P., Optimum Shapes of Elastic Solids with Infinite Branches, J. Appl. Mech. Transactions ASME, 1995, vol. 62, pp. 419–422.

    Article  ADS  MATH  Google Scholar 

  31. Savruk, M.P. and Kravets, V.S., Application of the Method of Singular Integral Equations to the Determination of the Contours of Equistrong Holes in Plates, Mater. Sci., 2002, vol. 38, no. 1, pp. 34–52.

    Article  Google Scholar 

  32. Cherepanov, G.P. and Esparragoza, I.E., Equistrong Tower Design, Theor. Appl. Mech., 2007, pp. 3–9.

    Google Scholar 

  33. Cherepanov, G.P., Fracture Mechanics, Izhevsk-Moscow: IKI, 2012.

    Google Scholar 

  34. Odishelidze, N., Criado-Aldeanueva, F., and Sanchez, J.M., A Mixed Problem of Plate Bending for a Regular Octagon Weakened with a Required Full-Strength Hole, Acta Mech., 2013, vol. 224, pp. 183–192.

    Article  MATH  MathSciNet  Google Scholar 

  35. Cherepanov, G.P., Methods of Fracture Mechanics: Solid Matter Physics, Dordrecht: Kluwer Publ., 1997.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The New York Academy of Sciences, New York, 10007-2157, USA

    G. P. Cherepanov

Authors
  1. G. P. Cherepanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. P. Cherepanov.

Additional information

Original Text © G.P. Cherepanov, 2015, published in Fizicheskaya Mezomekhanika, 2015, Vol. 18, No. 5, pp. 114-123.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cherepanov, G.P. Optimum Shapes of Elastic Bodies: Equistrong Wings of Aircrafts and Equistrong Underground Tunnels. Phys Mesomech 18, 391–401 (2015). https://doi.org/10.1134/S1029959915040116

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1029959915040116

Keywords

Search

Navigation