Matched asymptotic expansions for twisted elastic knots: A self-contact problem with non-trivial contact topology

Journal of the Mechanics and Physics of Solids

Volumn 57, Issue 9, 2009, Pages 1623-1656

  Clauvelin, N ^a,b   Audoly, B ^a,b   Neukirch, S ^a,b  

^a SORBONNE UNIVERSITÉ   (France)

^b INSTITUT JEAN LE ROND D ALEMBERT   (France)

Author keywords

Asymptotic analysis; Contact mechanics; Elastic material; Knots; Rods and cables

Indexed keywords

APRIORI; CONTACT MECHANICS; CONTACT PROBLEM; CONTACT REGIONS; ELASTIC MATERIAL; ELASTIC ROD; EQUILIBRIUM SOLUTIONS; GEOMETRICALLY NONLINEAR; KIRCHHOFF EQUATION; KNOTS; LOADING PARAMETERS; LOOSE KNOT; MATCHED ASYMPTOTIC EXPANSION; NON-TRIVIAL; POINTS OF CONTACT; PULLING FORCE; ROD MODEL; RODS AND CABLES; SQUARE ROOTS; TWISTING MOMENT;

CABLES; ELASTICITY; MECHANICS; TOPOLOGY;

ASYMPTOTIC ANALYSIS;

EID: 68049085591     PISSN: 00225096     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmps.2009.05.004     Document Type: Article

References (29)

1. Arai Y., Yasuda R., Akashi K.-I., Harada Y., Miyata H., Kinosita K., and Itoh H. Tying a molecular knot with optical tweezers. Nature 399 (1999) 446-448
2. Audoly B., Clauvelin N., and Neukirch S. Elastic knots. Physical Review Letters 99 (2007) 164301
3. Audoly B., and Pomeau Y. Elasticity and Geometry: From Hair Curls to the Nonlinear Response of Shells (2009), Oxford University Press, Oxford
4. Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., Grinspun, E., 2008. Discrete elastic rods. ACM Transactions on Graphics (SIGGRAPH) 27, 1-12.
5. Bourgat J.-F., and Le Tallec S. Modélisation et calcul des grands déplacements de tuyaux élastiques en flexion torsion. Journal de Mécanique Théorique et Appliquée 7 (1988) 379-408
6. Cantarella, J., Piatek, M., Rawdon, E., 2005. Visualizing the tightening of knots. In: Visualization, VIS 05, IEEE, pp. 575-582.
7. Coleman B., and Swigon D. Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. Journal of Elasticity 60 (2000) 173-221
8. Gallotti R., and Pierre-Louis O. Stiff knots. Physical Review E 75 (2007) 031801
9. Goriely A., and Neukirch S. The mechanics of climbing and attachment in twining plants. Physical Review Letters 97 (2006) 184302
10. Hinch E.J. Perturbation Methods. Cambridge Texts in Applied Mathematics (1991), Cambridge University Press, Cambridge
11. Katritch V., Bednar J., Michoud D., Scharein R.G., Dubochet J., and Stasiak A. Geometry and physics of knots. Nature 384 (1996) 142-145
12. Landau L.D., and Lifshitz E.M. Theory of Elasticity (Course of Theoretical Physics). second ed (1981), Pergamon Press, Oxford
13. 3. Journal of the London Mathematical Society 30 (1984) 512-520
14. Maddocks J.H., and Dichmann D.J. Conservation laws in the dynamics of rods. Journal of Elasticity 34 (1994) 83-96
15. Michell J.H. On the stability of a bent and twisted wire. Messenger of Mathematics 11 (1890) 181-184
16. Neukirch S. Extracting DNA twist rigidity from experimental supercoiling data. Physical Review Letters 93 (2004) 198107
17. Pieranski P., Kasas S., Dietler G., Dubochet J., and Stasiak A. Localization of breakage points in knotted strings. New Journal of Physics 3 (2001) 10
18. Pieranski P., Przybyl S., and Stasiak A. Tight open knots. European Physical Journal E 6 (2001) 123-128
19. Plaut R.H., Suherman S., Dillard D.A., Williams B.E., and Watson L.T. Deflections and buckling of a bent elastica in contact with a flat surface. International Journal of Solids and Structures 36 (1999) 1209-1229
20. Saitta A.M., Soper P.D., Wasserman E., and Klein M.L. Influence of a knot on the strength of a polymer strand. Nature 399 (1999) 46-48
21. Schuricht F., and von der Mosel H. Euler-Lagrange equations for nonlinearly elastic rods with self-contact. Archive for Rational Mechanics and Analysis 168 (2003) 35-82
22. Seemann W. Deformation of an elastic helix in contact with a rigid cylinder. Archive of Applied Mechanics 67 (1996) 117-139
23. Stasiak, A., Katritch, V., Kauffman, L.H. (Eds.), 1998. Ideal Knots. World Scientific, Singapore.
24. Steigmann D.J., and Faulkner M.G. Variational theory for spatial rods. Journal of Elasticity 33 (1993) 1-26
25. Tabor M., and Klapper I. The dynamics of knots and curves (part I). Nonlinear Science Today 4 (1994) 7-13
26. Tong L., Gattass R.R., Ashcom J.B., He S., Lou J., Shen M., Maxwell I., and Mazur E. Subwavelength-diameter silica wires for low-loss optical wave guiding. Nature 426 (2003) 816-819
27. van der Heijden G.H.M., Neukirch S., Goss V.G.A., and Thompson J.M.T. Instability and self-contact phenomena in the writhing of clamped rods. International Journal of Mechanical Sciences 45 (2003) 161-196
28. van der Heijden G.H.M., and Thompson J.M.T. Helical and localised buckling in twisted rods: a unified analysis of the symmetric case. Nonlinear Dynamics 21 (2000) 71-99
29. von der Mosel H. Elastic knots in Euclidean 3-space. Annales de l'Institut Henri Poincare C-Non-Linear Analysis 16 (1999) 137-166