[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Approximation of an Elastic Rod with Self-Contact

Impenetrable Elastic Rod

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The variational formulation of an elastic rod with an impenetrable surface surrounding the centerline corresponds to a nonsmooth optimization of cost functional \(\mathcal {J}\) with nonconvex inequality constraints and so presents many analytical and computational challenges in approximating the minima. We construct a sequence of approximate variational cost functionals \(\mathcal {J}_k\), corresponding to elastic rods with infinite energy barriers that enforce impenetrability constraints. Using this construction, we show strong convergence of minimizing configurations of \(\mathcal {J}_k\) to the minimizer of \(\mathcal {J}\) and weak* convergence in \([C(0,1)]^*\) of contact forces induced by the repulsive potential to the contact forces of the minimizing configurations of \(\mathcal {J}\).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Antman, S.S.: Nonlinear Problems in Elasticity, 2nd edn. Springer-Verlag, Berlin (2005)

    MATH  Google Scholar 

  2. Braides, A.: A handbook of \(\gamma \)-convergence. In: Chipot, M., Quittner, P. (eds.) Handbook of Differential Equations, Stationary Partial Differential Equations, vol. 3, pp. 101–214. Elsevier, Amsterdam (2006)

    Chapter  Google Scholar 

  3. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New Jersey (1983)

    MATH  Google Scholar 

  4. Cosserat, E., Cosserat, F.: Theories des Corps deformables. International Series in Pure and Applied Mathematics. A. Hermann et Fils, Paris (1909)

    MATH  Google Scholar 

  5. Cosserat, E.F.C.R., Cosserat, F.: Sur les equations de la theorie de l’elasticite. CR Acad. Sci. Paris 126, 1089–1091 (1974)

    MATH  Google Scholar 

  6. Goyal, S., Perkins, N.C., Lee, C.L.: Non-linear dynamic intertwining of rods with self-contact. Nonlinear Mech. 43, 65–73 (2008)

    Article  Google Scholar 

  7. Goyal, S., Perkins, N.C., Lee, C.L.: Nonlinear dyanmics and loop formation in Kirchhoff rods with implications to the mechanics of DNA cables. J. Comp. Phys. 209, 371–389 (2005)

    Article  Google Scholar 

  8. Hoffman, K.A., Seidman, T.I.: A variational rod model with a singular nonlocal potential. Arch. Ration. Mech. Anal. 200, 255–284 (2010)

    Article  MathSciNet  Google Scholar 

  9. Hoffman, K.A., Seidman, T.I.: A variational characterization of a hyperelastic rod with hard self-contact. Nonlinear Anal. A: Theory Methods Appl. 74(16), 5388–5401 (2011)

    Article  MathSciNet  Google Scholar 

  10. Mordukhovich, B.: Variational Analysis and Generalized Differentiation I. Springer Verlag, Berlin (2006)

    Book  Google Scholar 

  11. Mordukhovich, B.: Variational Analysis and Generalized Differentiation II. Springer Verlag, Berlin (2006)

    Book  Google Scholar 

  12. Moreau, J.: Proximité et dualité dans un espace Hilbertien. Bulletin de la Sociéte Mathématique de France 93, 273–299 (1965)

    Article  MathSciNet  Google Scholar 

  13. Rockafellar, R.T., Wets, R.J-B.: Variational Analysis, volume 317 of A Series of Comprehensive Studies in Mathematics. Springer (1998)

  14. Rudin, Walter: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc., New York (1991)

    Google Scholar 

  15. Schuricht, F.: A variational approach to obstacle problems for shearable nonlinearly elastic rods. Arch. Ration. Mech. Anal. 140, 103–159 (1997)

    Article  MathSciNet  Google Scholar 

  16. Schuricht, F.: Variational approach to contact problems in nonlinear elasticity. Calc. Var. 15, 433–449 (2002)

    Article  MathSciNet  Google Scholar 

  17. Schuricht, F.: Nonlinear analysis and Applications to Physical Sciences. Springer, Berlin (2004)

    MATH  Google Scholar 

  18. Schuricht, F., von der Mosel, H.: Euler-Lagrange equations for nonlinearly elastic rods with self-contact. Arch. Ration. Mech. Anal. 168, 35–82 (2003)

    Article  MathSciNet  Google Scholar 

  19. Seidman, T.I.: Normal cones to infinite intersections. Nonlinear Anal. 72, 3911–3917 (2010)

    Article  MathSciNet  Google Scholar 

  20. Yosida, K.: Functional Analysis. Springer-Verlag, Berlin (1964)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors thank Stuart Antman and Boris Mordukhovich for fruitful conversations.

Funding

The authors did not receive support from any organization for the submitted work.

Author information

Authors and Affiliations

  1. University of Maryland, Baltimore County, Baltimore, MD, 21250, USA

    Kathleen A. Hoffman & Thomas I. Seidman

Authors
  1. Kathleen A. Hoffman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas I. Seidman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kathleen A. Hoffman.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Communicated by Irena Lasiecka.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hoffman, K.A., Seidman, T.I. Approximation of an Elastic Rod with Self-Contact. J Optim Theory Appl 192, 1001–1021 (2022). https://doi.org/10.1007/s10957-022-02002-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-022-02002-5

Keywords

Mathematics Subject Classification

Search

Navigation