Abstract

We present a methodology to simulate the mechanics of knots in elastic rods using geometrically nonlinear, full three-dimensional (3D) finite element analysis. We focus on the mechanical behavior of knots in tight configurations, for which the full 3D deformation must be taken into account. To setup the topology of our knotted structures, we apply a sequence of prescribed displacement steps to the centerline of an initially straight rod that is meshed with 3D solid elements. Self-contact is enforced with a normal penalty force combined with Coulomb friction. As test cases, we investigate both overhand and figure-of-eight knots. Our simulations are validated with precision model experiments, combining rod fabrication and X-ray tomography. Even if the focus is given to the methods, our results reveal that 3D deformation of tight elastic knots is central to their mechanical response. These findings contrast to a previous analysis of loose knots, for which 1D centerline-based rod theories sufficed for a predictive understanding. Our method serves as a robust framework to access complex mechanical behavior of tightly knotted structures that are not readily available through experiments nor existing reduced-order theories.

References

References
1.
Daily-Diamond
,
C. A.
,
Gregg
,
C. E.
, and
O’Reilly
,
O. M.
,
2017
, “
The Roles of Impact and Inertia in the Failure of a Shoelace Knot
,”
Proc. Math. Phys. Eng.
,
473
(
2200
), p.
20160770
. 10.1098/rspa.2016.0770
2.
Zimmer
,
C. A.
,
Thacker
,
J. G.
,
Powell
,
D. M.
,
Bellian
,
K. T.
,
Becker
,
D. G.
,
Rodeheaver
,
G. T.
, and
Edlich
,
R. F.
,
1991
, “
Influence of Knot Configuration and Tying Technique on the Mechanical Performance of Sutures
,”
J. Emerg. Med.
,
9
(
3
), pp.
107
113
. 10.1016/0736-4679(91)90313-5
3.
Uehara
,
H.
,
Kimura
,
H.
,
Aoyama
,
A.
,
Yamanobe
,
T.
, and
Komoto
,
T.
,
2007
, “
Effects of Knot Characteristics on Tensile Breaking of a Polymeric Monofilament
,”
New J. Phys.
,
9
(
3
), p.
65
. 10.1088/1367-2630/9/3/065
4.
Calhoun
,
P.
,
2016
,
Advanced Surgical Knot Tying
.
Independently Published
.
5.
Pieranski
,
P.
,
Kasas
,
S.
,
Dietler
,
G.
,
Dubochet
,
J.
, and
Stasiak
,
A.
,
2001
, “
Localization of Breakage Points in Knotted Strings
,”
New J. Phys.
,
3
(
1
), p.
10
. 10.1088/1367-2630/3/1/310
6.
Patil
,
V. P.
,
Sandt
,
J. D.
,
Kolle
,
M.
, and
Dunkel
,
J.
,
2020
, “
Topological Mechanics of Knots and Tangles
,”
Science
,
367
(
6473
), pp.
71
75
. 10.1126/science.aaz0135
7.
Konyukhov
,
A.
, and
Schweizerhof
,
K.
,
2010
, “
Geometrically Exact Covariant Approach for Contact Between Curves
,”
Comput. Methods Appl. Mech. Eng.
,
199
(
37–40
), pp.
2510
2531
. 10.1016/j.cma.2010.04.012
8.
Durville
,
D.
,
2012
, “
Contact-friction Modeling Within Elastic Beam Assemblies: An Application to Knot Tightening
,”
Comput. Mech.
,
49
(
6
), pp.
687
707
. 10.1007/s00466-012-0683-0
9.
Qwam Alden
,
A. Y.
,
Geeslin
,
A. G.
, and
Gustafson
,
P. A.
,
2018
, “
Validation of a Finite Element Model of the Mechanical Performance of Surgical Knots of Varying Topology
,”
ASME IMECE
, Vol.
52026
,
American Society of Mechanical Engineers
,
Tampa, FL
, p.
V003T04A055
.
10.
Maddocks
,
J. H.
, and
Keller
,
J. B.
,
1987
, “
Ropes in Equilibrium
,”
J. Appl. Math.
,
47
(
6
), pp.
1185
1200
. 10.1137/0147080
11.
Katritch
,
V.
,
Bednar
,
J.
,
Michoud
,
D.
,
Scharein
,
R. G.
,
Dubochet
,
J.
, and
Stasiak
,
A.
,
1996
, “
Geometry and Physics of Knots
,”
Nature
,
384
(
6605
), pp.
142
145
. 10.1038/384142a0
12.
Grosberg
,
A. Y.
,
Feigel
,
A.
, and
Rabin
,
Y.
,
1996
, “
Flory-Type Theory of a Knotted Ring Polymer
,”
Phys. Rev. E
,
54
(
6
), p.
6618
. 10.1103/PhysRevE.54.6618
13.
Gonzalez
,
O.
, and
Maddocks
,
J. H.
,
1999
, “
Global Curvature, Thickness, and the Ideal Shapes of Knots
,”
Proc. Natl. Acad. Sci. U.S.A.
,
96
(
9
), pp.
4769
4773
. 10.1073/pnas.96.9.4769
14.
Audoly
,
B.
,
Clauvelin
,
N.
, and
Neukirch
,
S.
,
2007
, “
Elastic Knots
,”
Phys. Rev. Lett.
,
99
(
16
), p.
164301
. 10.1103/PhysRevLett.99.164301
15.
Clauvelin
,
N.
,
Audoly
,
B.
, and
Neukirch
,
S.
,
2009
, “
Matched Asymptotic Expansions for Twisted Elastic Knots: A Self-Contact Problem With Non-Trivial Contact Topology
,”
J. Mech. Phys. Solids
,
57
(
9
), pp.
1623
1656
. 10.1016/j.jmps.2009.05.004
16.
Jawed
,
M.
,
Dieleman
,
P.
,
Audoly
,
B.
, and
Reis
,
P.
,
2015
, “
Untangling the Mechanics and Topology in the Frictional Response of Long Overhand Elastic Knots
,”
Phys. Rev. Lett.
,
115
(
11
), p.
118302
. 10.1103/physrevlett.115.118302
17.
Grandgeorge
,
P.
,
Baek
,
C.
,
Singh
,
H.
,
Johanns
,
P.
,
Sano
,
T. G.
,
Flynn
,
A.
,
Maddocks
,
J. H.
, and
Reis
,
P. M.
,
2020
, “
Mechanics of two Filaments in Tight contact: The Orthogonal Clasp
,”
arXiv preprint arXiv:2010.08773
.
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