Skip to main content
SpringerLink
Log in

Optimal exponentials of thickness in Korn’s inequalities for parabolic and elliptic shells

  • Published:
Annali di Matematica Pura ed Applicata (1923 -) Aims and scope Submit manuscript

Abstract

We consider the scaling of the optimal constant in Korn’s first inequality for elliptic and parabolic shells which was first given by Grabovsky and Harutyunyan with hints coming from the test functions constructed by Tovstik and Smirnov on the level of formal asymptotic expansions. Here, we employ the Bochner technique in Remannian geometry to remove the assumption that the middle surface of the shell is given by one single principal coordinate, in particularly, including closed elliptic shells.

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. Korn, A.: Solution gnrale du problme dquilibre dans la thorie de llasticit dans le cas o les erts sont donns la surface. Ann. Fac. Sci. Toulouse 10, 165–269 (1908)

    Article  Google Scholar 

  2. Korn, A.: Uber einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen. Bull. Int. Cracovie Akademie Umiejet, Classe des Sci. Math. Nat., 705-724 (1909)

  3. Friedrichs, K.O.: On the boundary-value problems of the theory of elasticity and Korns inequality. Ann. Math. 48(2), 441–471 (1947)

    Article  MathSciNet  Google Scholar 

  4. Kondratiev, V.A., Oleinik, O.A.: Boundary value problems for a system in elasticity theory in unbounded domains. Korn inequalities. Usp. Mat. Nauk 43(5), 55–98 (1988)

    MathSciNet  MATH  Google Scholar 

  5. Kondratiev, V.A., Oleinik, O.A.: On Korns inequalities. C. R. Math. Acad. Sci. Paris Ser. I 308, 483–487 (1989)

    MathSciNet  MATH  Google Scholar 

  6. Kohn, R.V.: New integral estimates for deformations in terms of their nonlinear strains. Arch. Ration. Mech. Anal. 78(2), 131–172 (1982)

    Article  MathSciNet  Google Scholar 

  7. Payne, L.E., Weinberger, H.F.: On Korns inequality. Arch. Ration. Mech. Anal. 8, 89–98 (1961)

    Article  MathSciNet  Google Scholar 

  8. Bauer, S., Neff, P., Pauly, D., Starke, G.: Some Poincar type inequalities for quadratic matrix fields. Proc. Appl. Math. Mech. 13, 359–360 (2013)

    Article  Google Scholar 

  9. Conti, S., Dolzmann, G., Mller, S.: Korns second inequality and geometric rigidity with mixed growth conditions. Calc. Var. Part. Differ. Equ. 50, 437–454 (2014)

    Article  MathSciNet  Google Scholar 

  10. Neff, P., Pauly, D., Witsch, K.-J.: A canonical extension of Korns first inequality to H(Curl) motivated by gradient plasticity with plastic spin. C. R. Math. Acad. Sci. Paris Ser. I 349, 1251–1254 (2011)

    Article  MathSciNet  Google Scholar 

  11. Neff, P., Pauly, D., Witsch, K.-J.: A Korns inequality for incompatible tensor fields. In: Proceedings in Applied Mathematics and Mechanics, 6 June (2011)

  12. Friesecke, G., James, R., Muller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Commun. Pure Appl. Math. 55, 1461–1506 (2002)

    Article  MathSciNet  Google Scholar 

  13. Cioranescu, D., Oleinik, O., Tronel, G.: On Korns inequalities for frame type structures and junctions. C. R. Acad. Sci. Paris Ser. I Math. 309(9), 591–596 (1989)

    MathSciNet  MATH  Google Scholar 

  14. Lewicka, M., Muller, S.: On the optimal constants in korns and geometric rigidity estimates, in bounded and unbounded domains, under neumann boundary conditions. Indiana Univ. Math. J. 65(2), 377–397 (2016)

    Article  MathSciNet  Google Scholar 

  15. Nazarov, S.A.: Weighted anisotropic Korns inequality for a junction of a plate and a rod. Sbornik: Math. 195(4), 553–583 (2004)

    Article  MathSciNet  Google Scholar 

  16. Nazarov, S.A.: Korn inequalities for elastic junctions of massive bodies, thin plates, and rods. Russ. Math. Surv. 63(1), 35 (2008)

    Article  Google Scholar 

  17. Paroni, R., Tomassetti, G.: Asymptotically exact Korns constant for thin cylindrical domains. Comptes Rendus Mathematique 350(15), 749–752 (2012)

    Article  MathSciNet  Google Scholar 

  18. Paroni, R., Tomassetti, G.: On Korns constant for thin cylindrical domains. Math. Mech. Solids 19(3), 318–333 (2014)

    Article  MathSciNet  Google Scholar 

  19. Grabovsky, Y., Harutyunyan, D.: Korn inequalities for shells with zero Gaussian curvature. Ann. Inst. H. Poincar Anal. Non Linaire 35(1), 267–282 (2018)

    Article  MathSciNet  Google Scholar 

  20. Grabovsky, Y., Harutyunyan, D.: Exact scaling exponents in Korn and Korn-type inequalities for cylindrical shells. SIAM J. Math. Anal. 46(5), 3277–3295 (2014)

    Article  MathSciNet  Google Scholar 

  21. Tovstik, P.E., Smirnov, A.L., Asymptotic methods in the buckling theory of elastic shells, volume 4 of Series on stability, vibration and control of systems. World Scientific, Singapore 2001

  22. Harutyunyan, D.: Gaussian curvature as an identifier of shell rigidity. Arch. Ration. Mech. Anal. 226(2), 743–766 (2017)

    Article  MathSciNet  Google Scholar 

  23. Wu, H.: The Bochner Technique in Differential Geometry, Mathematical Reports, Vol. 3, part 2, Harwood Academic Publishers, London-Paris, (1988)

  24. Yao, P.F.: Modeling and control in vibrational and structural dynamics. A differential geometric approach. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. CRC Press, Boca Raton, FL, (2011)

  25. Yao, P.F.: Linear Strain Tensors and Optimal Exponential of thickness in Korn’s Inequalities for Hyperbolic Shells, arXiv:1807.11115 [math-ph]

  26. Harutyunyan, D.: On the Korn interpolation and second inequalities for shells with non-constant thickness, arXiv:1709.04572 [math.AP]

  27. Koiter, W.T.: On the nonlinear theory of thin elastic shells I. Proc. Kon. Ned. Akad. Wetensch. B 69, 1–17 (1966)

    Article  MathSciNet  Google Scholar 

  28. Harutyunyan, D.: New asyptotically sharp Korn and Korn-like inequalities in thin domains. J. Elast. 117(1), 95–109 (2014)

    Article  Google Scholar 

  29. Chen, W., Jost, J.: A Riemannian version of Korn’s inequality. Calc. Var. Part. Differ. Equ. 14(4), 517–530 (2002)

    Article  MathSciNet  Google Scholar 

  30. Yao, P.F.: Linear Strain Tensors on Hyperbolic Surfaces and Asymptotic Theories for Thin Shells. SIAM J. Math. Anal. 51(2), 1387–1435 (2019)

    Article  MathSciNet  Google Scholar 

  31. Lee, J.M.: Introduction to smooth manifolds. Second edition. Graduate Texts in Mathematics, 218. Springer, New York, (2013)

  32. Ciarlet, P.G., Lods, V.: On the ellipticity of linear membrane shell equations. J. Math. Pures Appl. 75(2), 107–124 (1996)

    MathSciNet  MATH  Google Scholar 

  33. Lods, V., Mardare, C.: The space of inextensional displacements for a partially clamped linearly elastic shell with an elliptic middle surface. J. Elast. 51(2), 127–144 (1998)

    Article  MathSciNet  Google Scholar 

  34. Spivak, M.: A comprehensive introduction to differential geometry. Vols. III and IV. Second edition. Publish or Perish, Inc., Wilmington, Del., (1979). xii+466 pp. ISBN: 0-914098-83-7

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Systems and Control Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Peng-Fei Yao

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Peng-Fei Yao

Authors
  1. Peng-Fei Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peng-Fei Yao.

Ethics declarations

Conflict of interest

The author declares no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by the author.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the National Science Foundation of China, Grant Nos. 61473126 and 61573342, and Key Research Program of Frontier Sciences, CAS, No. QYZDJ-SSW-SYS011.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yao, PF. Optimal exponentials of thickness in Korn’s inequalities for parabolic and elliptic shells. Annali di Matematica 200, 379–401 (2021). https://doi.org/10.1007/s10231-020-01000-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10231-020-01000-6

Keywords

Mathematics Subject Classification

Search

Navigation