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A Variational Rod Model with a Singular Nonlocal Potential

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Abstract

The classical theory of elastic rods does not account for the possibility that large deformations may involve distinct points along the rod occupying the same physical space. We develop an elastic rod model with a pairwise repulsive potential such that, if two non-adjacent points along the rod are close in physical space, there is an energy barrier which prevents contact while for points nearby along the rod the potential is describable classically. This framework is developed to prove the existence of minimizers within each homotopy class, where the idea of topological homotopy of a curve is generalized to elastic rods as framed curves. Finally, the relevant first-order optimality conditions are derived and used to investigate the regularity of minimizers.

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References

  1. Alexander J.C., Antman S.S.: The ambiguous twist of Love. Quart. Appl. Math. XL, 83–92 (1982)

    MathSciNet  Google Scholar 

  2. Antman S.S.: Equilibrium states of nonlinearly elastic rods. J. Math. Anal. Appl. 23, 459–470 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  3. Antman S.S.: Existence of solutions of the equilibrium equations for nonlinearly elastic rings and arches. Indiana Univ. Math. J. 20, 281–302 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. Antman S.S.: Existence and nonuniqueness of axis-symetric equilibrium states of nonlinearly elastic shells. Arch. Rat. Mech. Anal. 40, 329–372 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  5. Antman S.S.: Ordinary differential equations of nonlinear elasticity II: existence and regularity theory for conservative boundary value problems. Arch. Rat. Mech. Anal. 61, 353–393 (1976)

    MathSciNet  MATH  Google Scholar 

  6. Antman S.S.: Nonlinear Problems in Elasticity. 2nd edition. Springer, New York (2005)

    Google Scholar 

  7. Ball J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)

    Article  MATH  Google Scholar 

  8. Ball J.M., Marsden J.E.: Quasiconvexity at the boundary, positivity of the second variation, and elastic stability. Arch. Ration. Mech. Anal. 86, 251–277 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  9. Banavar J.R., Gonzalez O., Maddocks J.H., Maritan A.: Self-interactions of strands and sheets. J. Stat. Phys. 110, 35–50 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ciarlet P.G., Nečas J.: Injectivity and self-contact in nonlinear elasticity. Arch Ration. Mech. Anal. 97, 171–188 (1987)

    Article  Google Scholar 

  11. Coleman B.D., Swigon D.: Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. J. Elast. 60, 171–221 (2000)

    Article  MathSciNet  Google Scholar 

  12. Coleman B.D., Swigon D., Tobias I.: Elastic stability of DNA configurations: II. supercoiled plasmids with self-contact. Phys. Rev. E 61, 759–770 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  13. Delrow J.J., Gebe J.A., Schurr J.M.: Comparison of hard cylinder and screened Coulomb interaction in the modeling of supercoiled DNAs. Biophysics 42, 455–470 (1997)

    Google Scholar 

  14. Dichmann, D.J., Li, Y.W., Maddocks, J.H.: Hamiltonian formulations and symmetries in rod mechanics. Mathematical Approaches to Biomolecular Structure and Dynamics, Vol. 82 (Eds. Mesirov, J.P., Schulten, K. and Sumners, D.W.). IMA Volumes in Mathematics and its Applications. Springer, New York, 71–113, 1996

  15. Emmrich E., Weckner O.: On the well-posedness of the linear peridynamic model and its convengence toward the Navier equation of linear elasticity. Commun. Math. Sci. 5(4), 851–864 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Fukuhara, S.: A Fete of Topology, chapter Energy of a Knot. Academic Press, New York, 1988

  17. Gonzalez O., Maddocks J.H.: Global curvature, thickness and the ideal shape of knots. PNAS 96(9), 4769–4773 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Gonzalez O., Maddocks J.H., Schuricht F., von der Mosel H.: Global curvature and self-contact of nonlinearly elastic curves and rods. Calc. Var. Partial Differ. Equ. 14, 29–68 (2002)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Habeck D., Schuricht F.: Contact between nonlinearly elastic bodies. Proc. Roy. Soc. Edinburgh A 136, 1239–1266 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hoffman K.A., Manning R.S.: Stability of elastic rods with repulsive potentials. SIAM Math. Anal. 41, 465–494 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hoffman, K.A., Seidman, T.I.: A variational characterization of a hyperelastic rod with an impenetrable tube, in preparation

  24. Katritch V., Bednar J., Michoud D., Scharein R.G., Dubochet J.U., Stasiak A.: Geometry and physics of knots. Nature 384, 142–145 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  25. Klapper I.: Biological applications of the dynamics of twisted elastic rods. J. Comput. Phys. 125, 325–337 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Krasnosel’skii M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, New York (1964)

    Google Scholar 

  27. Kunin, I.A.: Elastic Media with Microstructure, Vol. I/II. Springer, New York, 1982/83

  28. Kusner, R.B., Sullivan, J.M.: Möbius-invariant Knot Energies. Ideal Knots, Vol. 19 of Ser. on Knots and Everything. (Eds Stasiak, Katritch, Kauffman) World Scientific, Singapore, 315–352, 1998

  29. Lehoucq R.B., Silling S.A.: Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 184, 1566–1577 (2007)

    MathSciNet  Google Scholar 

  30. O’Hara J.: Energy of a knot. Topology 30, 241–247 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rogula D.: Nonlocal Theory of Material Media. Springer, New York (1982)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Schuricht, F.: Nonlinear analysis and applications to physical sciences. Chapter Contact Problems in Nonlinear Elasticity. Modeling, Analysis, and Application Springer, New York, 91–133, 2004

  34. Schuricht F.: A new mathematical foundation for contact interations in continuum physics. Arch. Ration. Mech. Anal. 185, 495–551 (2007)

    Article  MathSciNet  Google Scholar 

  35. 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  MATH  Google Scholar 

  36. Silling S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech Phys. Solids 48, 175–209 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. Simon J.: Energy functions for polygonal knots. J Knot Theory Ramif. 3, 299–320 (1994)

    Article  MATH  Google Scholar 

  38. Starostin, E.L.: Closed loops of a thin elastic rod and its symmetric shapes with self-contacts. Proc. 16th IMACS World Congress, Lausanne Switzerland, 21–25 August 2000

  39. Starostin E.L.: Symmetric equilibria of a thin elastic rod with self-contacts. Phil. Trans. Roy. Soc. A 362, 1317–1334 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Strzelecki P., von der Mosel H.: On rectifiable curves with L p bounds on global curvature: self-avoidance, regularity, and minimizing knots. Math. Z. 257, 107–130 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. vander Heijden G., Neukirch S., Goss V.G.A., Thompson J.M.T.: Instability and self-contact phenomena in the writhing of clamped rods. Int. J. Mech. Sci. 45, 161–196 (2003)

    Article  MATH  Google Scholar 

  42. Vologodskii A., Cozzarelli N.: Modeling of long-range electrostatic interactions in DNA. Biopolymers 35, 289–296 (1995)

    Article  Google Scholar 

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Authors and Affiliations

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

    Kathleen A. Hoffman & Thomas I. Seidman

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  1. Kathleen A. Hoffman
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  2. Thomas I. Seidman
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Correspondence to Kathleen A. Hoffman.

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Communicated by S. Antman

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Hoffman, K.A., Seidman, T.I. A Variational Rod Model with a Singular Nonlocal Potential. Arch Rational Mech Anal 200, 255–284 (2011). https://doi.org/10.1007/s00205-010-0368-9

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  • DOI: https://doi.org/10.1007/s00205-010-0368-9

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