Skip to main content
SpringerLink
Log in

Large Strains of a Spherical Shell with Distributed Dislocations and Disclinations

  • Chapter
  • First Online:
Sixty Shades of Generalized Continua

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 170))

  • 418 Accesses

Abstract

A theory of nonlinear deformation of the elastic Cosserat shells with continuously distributed dislocations and disclinations is formulated. Displacements, rotations, and strains are considered to be arbitrarily large, and the rotation field is kinematically independent of the displacement field. A system of nonlinear differential equations is derived that describes the stress state of an elastic shell with given external loads and given dislocation and disclination densities. This system consists of equilibrium equations and incompatibility equations and contains, as unknown functions, the tensor fields of metric and flexural strains of the elastic shell. The general theory is illustrated by solving a nonlinear problem of the equilibrium of a spherical shell with a spherically symmetric distribution of dislocations and disclinations.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Shilpi S, Jain A, GuptaY, Jain SK (2007) Colloidosomes: An emerging vesicular system in drug delivery, Critical Reviews in Therapeutic Drug Carrier Systems 24(4):361-391. DOI https://doi.org/10.1615/critrevtherdrugcarriersyst.v24.i4.20

  2. Leiderer P (1995) Ions at helium interfaces, Zeitschrift für Physik B Condensed Matter 98(3):303-308. DOI https://doi.org/10.1007/BF01338394

  3. Rozynek Z, Mikkelsen A, Dommersnes P, Fossum JO (2014) Electroformation of Janus and patchy capsules, Nature Communications 5(1):3945. DOI https://doi.org/10.1038/ncomms4945

  4. Keber FC, Loiseau E, Sanchez T, DeCamp SJ, Giomi L, Bowick MJ, Marchetti MC, Dogic Z, Bausch AR (2014) Topology and dynamics of active nematic vesicles, Science 345(6201):1135-1139. DOI https://doi.org/10.1126/science.1254784

  5. Zandi R, Dragnea B, Travesset A, Podgornik R (2020) On virus growth and form, Physics Reports 847:1-102. DOI https://doi.org/10.1016/j.physrep.2019.12.005

  6. Azizi A, Zou X, Ercius P, Zhang Z, Elías AL, Perea-López N, Stone G, Terrones M, Yakobson BI, Alem N (2014) Dislocation motion and grain boundary migration in two-dimensional tungsten disulphide, Nature Communications 5(1):4867. DOI https://doi.org/10.1038/ncomms5867

  7. Butz B, Dolle C, Niekiel F, Weber K, Waldmann D, Weber HB, Meyer B, Spiecker E (2014) Dislocations in bilayer graphene, Nature 505(7484):533-537. DOI https://doi.org/10.1038/nature12780

  8. Kim T,ParkJY,Hwang J,Seo G,KimY(2020) Supramolecular two-dimensional systems and their biological applications, Advanced Materials 32(51):2002405. DOI https://doi.org/10.1002/adma.202002405

  9. Negri C, Sellerio AL, Zapperi S, Miguel MC (2015) Deformation and failure of curved colloidal crystal shells, Proceedings of the National Academy of Sciences of the United States of America 112(47):14545–14550. DOI https://doi.org/10.1073/pnas.151825811

  10. Xiong Z, Zhong L, Wang H, Li X (2021) Structural defects, mechanical behaviors, and properties of two-dimensional materials, Materials 14(5):1192. DOI https://doi.org/10.3390/ma14051192

  11. Gutkin MY, Ovid’ko IA (2004) Plastic Deformation in Nanocrystalline Materials. Springer, Berlin, Heidelberg. DOI https://doi.org/10.1007/978-3-662-09374-0

  12. Qi W-K, Zhu T, Chen Y, Ren J-R (2009) Topological aspect of disclinations in two-dimensional crystals, Chinese Physics B 18(3):1002-1008. DOI https://doi.org/10.1088/1674-1056/18/3/026

  13. Petrov CY, Shevchenko IA (2013) Topological defects in the two-dimensional spherical crystals, Engineering Journal of Don 26(3):103.

    Google Scholar 

  14. Romanov A, Rozhkov M,Kolesnikova A (2018) Disclinations in polycrystalline graphene and pseudo-graphenes. Review, Letters on Materials 8(4):384-400. DOI https://doi.org/10.22226/2410-3535-2018-4-384-400

  15. Roychowdhury A, Gupta A (2018) On structured surfaces with defects: Geometry, strain incompatibility, stress field, and natural shapes, Journal of Elasticity 131(2):239-276. DOI https://doi.org/10.1007/s10659-017-9654-1

  16. Roshal DS, Konevtsova OV, Myasnikova AE, Rochal SB (2016) Assembly of the most topologically regular two-dimensional micro and nanocrystals with spherical, conical, and tubular shapes, Phys. Rev. E 94:052605. DOI https://doi.org/10.1103/PhysRevE.94.052605

  17. Lidmar J, Mirny L, Nelson DR (2003) Virus shapes and buckling transitions in spherical shells. Phys. Rev. E 68:051910. DOI https://doi.org/10.1103/PhysRevE.68.051910

  18. Å iber A (2006) Buckling transition in icosahedral shells subjected to volume conservation constraint and pressure: Relations to virus maturation, Phys. Rev. E 73:061915. DOI https://doi.org/10.1103/PhysRevE.73.061915

  19. Perotti LE, Aggarwal A, Rudnick J, Bruinsma R, Klug WS (2015) Elasticity theory of the maturation of viral capsids, Journal of the Mechanics and Physics of Solids 77:86-108. DOI https://doi.org/10.1016/j.jmps.2015.01.006

  20. Zubov LM (2007) Von Kármán equations for an elastic plate with dislocations and disclinations, Doklady Physics 52(1):67-70. DOI https://doi.org/10.1134/S102833580701017X

  21. Zubov L, Stolpovskii A (2008) A theory of dislocations and disclinations in elastic plates, Journal of Applied Mathematics and Mechanics 72(6):724-737. DOI https://doi.org/10.1016/j.jappmathmech.2009.01.005

  22. Zubov L (2010) The linear theory of dislocations and disclinations in elastic shells, Journal of Applied Mathematics and Mechanics 74(6):663-672. DOI https://doi.org/10.1016/j.jappmathmech.2011.01.006

  23. Zubov LM (2012) Large deformations of elastic shells with distributed dislocations, Doklady Physics 57(6):254-257. DOI https://doi.org/10.1134/S1028335812060092

  24. ZubovLM(1997) Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies, Springer, Berlin. DOI https://doi.org/10.1007/978-3-540-68430-5

  25. Zubov LM (2011) Continuum theory of dislocations and disclinations in nonlinearly elastic micropolar media, Mechanics of Solids 46(3):348-356. DOI https://doi.org/10.3103/S0025654411030022

  26. Karyakin MI, ZubovLM (2011) Theory of isolated and continuously distributed disclinations and dislocations in micropolar media, In:HAltenbach,GAMaugin, V Erofeev (eds.) Mechanics of Generalized Continua, pp 275-290. Springer, Berlin, Heidelberg. DOI https://doi.org/10.1007/978-3-642-19219-7_14

  27. Zelenina AA, Zubov LM (2018) Spherically symmetric deformations of micropolar elastic medium with distributed dislocations and disclinations, In: F dell’Isola, VA Eremeyev, A Porubov (eds.) Advances in Mechanics of Microstructured Media and Structures, pp 357-369. Springer International Publishing, Cham. DOI https://doi.org/10.1007/978-3-319-73694-5_19

  28. Zhilin PA (1976) Mechanics of deformable directed surfaces, International Journal of Solids and Structures 12(9):635-648. DOI https://doi.org/10.1016/0020-7683(76)90010-X

  29. Altenbach J, Altenbach H, Eremeyev VA (2010) On generalized Cosserat-type theories of plates and shells: a short review and bibliography, Archive of Applied Mechanics 80(1):73-92. DOI https://doi.org/10.1007/s00419-009-0365-3

  30. Chróscielewski J, Makowski J, Pietraszkiewicz W (2004) Statics and dynamics of multyfolded shells. Nonlinear theory and finite element method (in Polish), Wydawnictwo IPPT PAN, Warszawa.

    Google Scholar 

  31. Eremeyev VA, Zubov LM (2008) Mechanics of Elastic Shells (in Russ), Nauka, Moscow.

    Google Scholar 

  32. Libai A, Simmonds JG (1998) The Nonlinear Theory of Elastic Shells (2nd edn.), Cambridge University Press.

    Google Scholar 

  33. Zubov LM, Eremeyev VA (2003) Mechanics of elastic micropolar shells, Far Eastern Math. J. 4(1):182-225.

    Google Scholar 

  34. Altenbach H, Eremeyev VA (2014) Vibration analysis of non-linear 6-parameter prestressed shells, Meccanica 49(8):1751-1761. DOI https://doi.org/10.1007/s11012-013-9845-1

  35. Eremeyev VA, Lebedev LP, Cloud MJ (2015) The Rayleigh and Courant variational principles in the six-parameter shell theory, Mathematics and Mechanics of Solids 20(7):806-822. DOI https://doi.org/10.1177/1081286514553

  36. Zubov LM, Karyakin MI (2006) Tensor Calculus (in Russ), Vuzovskaya Kniga, Moscow.

    Google Scholar 

  37. Zubov LM (1982) Methods of Nonlinear Theory of Elasticity in the Theory of Shells (in Russ), Rostov Univ. Press., Rostov-on-Don.

    Google Scholar 

  38. deWit R (1973) Theory of disclinations: II. Continuous and discrete disclinations in anisotropic elasticity, J ResNatl Bur StandAPhys Chem 77A(1):49-100. DOI https://doi.org/10.6028/jres.077A.003

  39. deWit R (1973) Theory of disclinations: III. Continuous and discrete disclinations in isotropic elasticity, J Res Natl Bur Stand A Phys Chem 77A(3):359-368. DOI https://doi.org/10.6028/jres.077A.024

  40. Zubov LM (1989) Nonlinear theory of isolated dislocations and disclinations in elastic shells, Izv. AN SSSR. MTT pp 139–145.

    Google Scholar 

  41. Zubov LM, Karyakin MI (2023) Theory of Cosserat-type elastic shells with distributed dislocations and disclinations, In: H Altenbach, SM Bauer, AK Belyaev, DA Indeitsev, VP Matveenko, YV Petrov (eds.) Advances in Solid and Fracture Mechanics – A Liber Amicorum to Celebrate the Birthday of Nikita Morozov,Advanced Structured Materials. vol. 180, pp 259-278. Springer, Cham. DOI https://doi.org/10.1007/978-3-031-18393-5_17

  42. Goloveshkina EV, Zubov LM (2021) Spherically symmetric tensor fields and their application in nonlinear theory of dislocations, Symmetry 2021(13):830. DOI https://doi.org/10.3390/sym13050830

Download references

Author information

Authors and Affiliations

  1. Southern Federal University, Rostov-on-Don, Russia

    Leonid M. Zubov

  2. South Mathematical Institute, Vlkadikavkaz & Southern FederalUniversity, Rostov-on-Don, Russia

    Mikhail I. Karyakin

Authors
  1. Leonid M. Zubov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikhail I. Karyakin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leonid M. Zubov .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

    Holm Altenbach

  2. Tallinn University of Technology, Tallinn, Estonia

    Arkadi Berezovski

  3. Università degli Studi dell’Aquila, L'Aquila, Italy

    Francesco dell'Isola

  4. Institute of Problems of Mechanical Engineering, Saint Petersburg, Russia

    Alexey Porubov

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Zubov, L.M., Karyakin, M.I. (2023). Large Strains of a Spherical Shell with Distributed Dislocations and Disclinations. In: Altenbach, H., Berezovski, A., dell'Isola, F., Porubov, A. (eds) Sixty Shades of Generalized Continua. Advanced Structured Materials, vol 170. Springer, Cham. https://doi.org/10.1007/978-3-031-26186-2_45

Download citation

Publish with us

Policies and ethics

Search

Navigation