Computational rod models have emerged as efficient tools to simulate the bending and twisting deformations of a variety of slender structures in engineering and biological applications. The dynamics of such deformations, however, strongly depends on the constitutive law in bending and torsion that, in general, may be nonlinear, and vary from material to material. Jacobian-based computational rod models require users to change the Jacobian if the functional form of the constitutive law is changed, and hence are not user-friendly. This paper presents a scheme that automatically modifies the Jacobian based on any user-defined constitutive law without requiring symbolic differentiation. The scheme is then used to simulate force-extension behavior of a coiled spring with a softening constitutive law.

References

References
1.
Simo
,
J. C.
,
Marsden
,
J. E.
, and
Krishnaprasad
,
P. S.
,
1988
, “
The Hamiltonian Structure of Nonlinear Elasticity: The Material and Convective Representations of Solids, Rods, and Plates
,”
Arch. Ration. Mech. Anal.
,
104
(
2
), pp.
125
183
.
2.
Goyal
,
S.
,
Perkins
,
N. C.
, and
Lee
,
C. L.
,
2005
, “
Nonlinear Dynamics and Loop Formation in Kirchhoff Rods With Implications to the Mechanics of DNA and Cables
,”
J. Comput. Phys.
,
209
(
1
), pp.
371
389
.
3.
Hwang
,
W.
,
2015
, “
Biofilament Dynamics: Line-to-Rod-Level Descriptions
,”
Multiscale Modeling in Biomechanics and Mechanobiology
,
S.
De
,
W.
Hwang
, and
E.
Kuhl
, eds.,
Springer
,
London
, pp.
63
83
.
4.
Neukirch
,
S.
, and
van der Heijden
,
G.
,
2002
, “
Geometry and Mechanics of Uniform n-Plies: From Engineering Ropes to Biological Filaments
,”
J. Elasticity
,
69
(
1/3
), pp.
41
72
.
5.
Klapper
,
I.
,
1996
, “
Biological Applications of the Dynamics of Twisted Elastic Rods
,”
J. Comput. Phys.
,
125
(
2
), pp.
325
337
.
6.
Lillian
,
T. D.
,
Goyal
,
S.
,
Kahn
,
J. D.
,
Meyhfer
,
E.
, and
Perkins
,
N.
,
2008
, “
Computational Analysis of Looping of a Large Family of Highly Bent {DNA} by Laci
,”
Biophys. J.
,
95
(
12
), pp.
5832
5842
.
7.
Hoffman
,
K. A.
,
2004
, “
Methods for Determining Stability in Continuum Elastic-Rod Models of DNA
,”
Philos. Trans. R. Soc. London A: Math., Phys. Eng. Sci.
,
362
(
1820
), pp.
1301
1315
.
8.
Goyal
,
S.
,
Perkins
,
N. C.
, and
Meiners
,
J. C.
,
2008
, “
Resolving the Sequence-Dependent Stiffness of DNA Using Cyclization Experiments and a Computational Rod Model
,”
ASME J. Comput. Nonlinear Dyn.
,
3
(
1
), p.
011003
.
9.
Sept
,
D.
, and
MacKintosh
,
F. C.
,
2010
, “
Microtubule Elasticity: Connecting All-Atom Simulations With Continuum Mechanics
,”
Phys. Rev. Lett.
,
104
(
1
), p.
018101
.
10.
Hawkins
,
T.
,
Mirigian
,
M.
,
Yasar
,
M. S.
, and
Ross
,
J. L.
,
2010
, “
Mechanics of Microtubules
,”
J. Biomech.
,
43
(
1
), pp.
23
30
.
11.
Hilfinger
,
A.
,
Chattopadhyay
,
A. K.
, and
Jülicher
,
F.
,
2009
, “
Nonlinear Dynamics of Cilia and Flagella
,”
Phys. Rev. E
,
79
(
5
), p.
051918
.
12.
Qin
,
Z.
,
Buehler
,
M. J.
, and
Kreplak
,
L.
,
2010
, “
A Multi-Scale Approach to Understand the Mechanobiology of Intermediate Filaments
,”
J. Biomech.
,
43
(
1
), pp.
15
22
.
13.
Goldstein
,
R. E.
, and
Goriely
,
A.
,
2006
, “
Dynamic Buckling of Morphoelastic Filaments
,”
Phys. Rev. E
,
74
(
1
), p.
010901
.
14.
Odijk
,
T.
,
1998
, “
Microfibrillar Buckling Within Fibers Under Compression
,”
J. Chem. Phys.
,
108
(
16
), pp.
6923
6928
.
15.
Kumar
,
A.
,
Mukherjee
,
S.
,
Paci
,
J. T.
,
Chandraseker
,
K.
, and
Schatz
,
G. C.
,
2011
, “
A Rod Model for Three Dimensional Deformations of Single-Walled Carbon Nanotubes
,”
Int. J. Solids Struct.
,
48
(
20
), pp.
2849
2858
.
16.
Chen
,
Y.
,
Dorgan
,
B. L.
,
McIlroy
,
D. N.
, and
Eric Aston
,
D.
,
2006
, “
On the Importance of Boundary Conditions on Nanomechanical Bending Behavior and Elastic Modulus Determination of Silver Nanowires
,”
J. Appl. Phys.
,
100
(
10
), p. 104301.
17.
Calladine
,
C. R.
,
Drew
,
H. R.
,
Luisi
,
B. F.
, and
Travers
,
A. A.
,
2004
,
Understanding DNA, the Molecule and How It Works
,
Elsevier Academic Press
,
Amsterdam, The Netherlands
.
18.
Goyal
,
S.
,
Lillian
,
T.
,
Blumberg
,
S.
,
Meiners
,
J. C.
,
Meyhofer
,
E.
, and
Perkins
,
N. C.
,
2007
, “
Intrinsic Curvature of DNA Influences LacR-Mediated Looping
,”
Biophys. J.
,
93
(
12
), pp.
4342
4359
.
19.
Goyal
,
S.
, and
Perkins
,
N.
,
2008
, “
Looping Mechanics of Rods and DNA with Non-Homogeneous and Discontinuous Stiffness
,”
Int. J. Non-Linear Mech.
,
43
(
10
), pp.
1121
1129
.
20.
Goyal
,
S.
,
Perkins
,
N.
, and
Lee
,
C. L.
,
2008
, “
Non-Linear Dynamic Intertwining of Rods With Self-Contact
,”
Int. J. Non-Linear Mech.
,
43
(
1
), pp.
65
73
.
21.
Cloutier
,
T. E.
, and
Widom
,
J.
,
2004
, “
Spontaneous Sharp Bending of Double-Stranded DNA
,”
Mol. Cell
,
14
(
3
), pp.
355
362
.
22.
Wiggins
,
P. A.
,
Phillips
,
R.
, and
Nelson
,
P. C.
,
2005
, “
Exact Theory of Kinkable Elastic Polymers
,”
Phys. Rev. E
,
71
(
2
), p. 021909.
23.
Swati Verma
,
G. S.
, and
Palanthandalam-Madapusi
,
H. J.
,
2012
, “
Simulation Based Analysis of Constitutive Behavior of Microtubules
,”
Asian Conference on Mechanics of Functional Materials and Structures
, New Delhi, India, Dec. 5–8, pp.
679
681
.
24.
Fatehiboroujeni
,
S.
, and
Goyal
,
S.
,
2016
, “
Deriving Mechanical Properties of Microtubules From Molecular Simulations
,”
Biophys. J.
,
110
(
Suppl. 1
), p.
129A
.
25.
Fosdick
,
R. L.
, and
James
,
R. D.
,
1981
, “
The Elastica and the Problem of the Pure Bending for a Non-Convex Stored Energy Function
,”
J. Elasticity
,
11
(
2
), pp.
165
186
.
26.
Gupta
,
P.
, and
Kumar
,
A.
,
2017
, “
Effect of Material Nonlinearity on Spatial Buckling of Nanorods and Nanotubes
,”
J. Elasticity
,
126
(
2
), pp.
155
171
.
27.
Smith
,
M. L.
, and
Healey
,
T. J.
,
2008
, “
Predicting the Onset of DNA Supercoiling Using a Nonlinear Hemitropic Elastic Rod
,”
Int. J. Non-Linear Mech.
,
43
(
10
), pp.
1020
1028
.
28.
Haslach
,
H. W.
, Jr.
,
1985
, “
Post-Buckling Behavior of Columns With Non-Linear Constitutive Equations
,”
Int. J. Non-Linear Mech.
,
20
(
1
), pp.
53
67
.
29.
Baczynski
,
K. K.
,
2009
, “
Buckling Instabilities of Semiflexible Filaments in Biological Systems
,”
Ph.D. dissertation
, University of Potsdam, Potsdam, Germanyhttps://publishup.uni-potsdam.de/opus4-ubp/frontdoor/deliver/index/docId/3557/file/baczynski_diss.pdf.
30.
Avril
,
S.
,
Bonnet
,
M.
,
Bretelle
,
A.-S.
,
Grédiac
,
M.
,
Hild
,
F.
,
Ienny
,
P.
,
Latourte
,
F.
,
Lemosse
,
D.
,
Pagano
,
S.
,
Pagnacco
,
E.
, and
Pierron
,
F.
,
2008
, “
Overview of Identification Methods of Mechanical Parameters Based on Full-Field Measurements
,”
Exp. Mech.
,
48
(
4
), pp.
381
402
.
31.
Bonnet
,
M.
, and
Constantinescu
,
A.
,
2005
, “
Inverse Problems in Elasticity
,”
Inverse Probl.
,
21
(
2
), p.
R1
.
32.
Hinkle
,
A. R.
,
Goyal
,
S.
, and
Palanthandalam-Madapusi
,
H. J.
,
2009
, “
An Estimation Method of a Constitutive-Law for the Rod Model of DNA Using Discrete-Structure Simulations
,”
ASME
Paper No. DETC2009–87763.
33.
Palanthandalam-Madapusi
,
H. J.
, and
Goyal
,
S.
,
2011
, “
Robust Estimation of Nonlinear Constitutive Law From Static Equilibrium Data for Modeling the Mechanics of DNA
,”
Automatica
,
47
(
6
), pp.
1175
1182
.
34.
Goyal
,
S.
,
2006
, “
A Dynamic Rod Model to Simulate Mechanics of Cables and DNA
,”
Ph.D. dissertation
, University of Michigan, Ann Arbor, MIhttp://hdl.handle.net/2027.42/126037.
35.
Neidinger
,
R. D.
,
2010
, “
Introduction to Automatic Differentiation and MATLAB Object-Oriented Programming
,”
SIAM Rev.
,
52
(
3
), pp.
545
563
.
36.
Kirchhoff
,
G.
,
1859
, “
Uber Das Gleichgewicht Und Die Bewegung Eines Unendlich Dunnen Elastischen Stabes
,”
J. Reine Angew. Math. (Crelle)
,
1859
(
56
), pp.
285
343
.
37.
Chung
,
J.
, and
Hulbert
,
G. M.
,
1993
, “
A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation—The Generalized-Alpha Method
,”
ASME J. Appl. Mech.
,
60
(
2
), pp.
371
375
.
38.
Bottasso
,
C. L.
, and
Borri
,
M.
,
1998
, “
Integrating Finite Rotations
,”
Comput. Methods Appl. Mech. Eng.
,
164
(
3–4
), pp.
307
331
.
39.
Simo
,
J.
, and
Vu-Quoc
,
L.
,
1986
, “
A Three-Dimensional Finite-Strain Rod Model—Part II: Computational Aspects
,”
Comput. Methods Appl. Mech. Eng.
,
58
(
1
), pp.
79
116
.
40.
Zhang
,
Z.
,
Qi Zhaohui
,
W. Z.
, and
Huiqing
,
F.
,
2015
, “
A Spatial Euler-Bernoulli Beam Element for Rigid-Flexible Coupling Dynamic Analysis of Flexible Structures
,”
J. Shock Vib.
,
2015
, p. 208127.
41.
Cardona
,
A.
, and
Geradin
,
M.
,
1988
, “
A Beam Finite Element Non-Linear Theory With Finite Rotations
,”
Int. J. Numer. Methods Eng.
,
26
(
11
), pp.
2403
2438
.
42.
Fan
,
W.
, and
Zhu
,
W. D.
,
2018
, “
An Accurate Singularity-Free Geometrically Exact Beam Formulation Using Euler Parameters
,”
Nonlinear Dyn.
,
91
(
2
), pp.
1095
1112
.
43.
Sun
,
Y.
,
Leonard
,
J. W.
, and
Chiou
,
R. B.
,
1994
, “
Simulation of Unsteady Oceanic Cable Deployment by Direct Integration With Suppression
,”
Ocean Eng.
,
21
(
3
), pp.
243
256
.
44.
Maghsoodi
,
A.
,
Chatterjee
,
A.
,
Andricioaei
,
I.
, and
Perkins
,
N. C.
,
2016
, “
A First Model of the Dynamics of the Bacteriophage t4 Injection Machinery
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
4
), p.
041026
.
You do not currently have access to this content.