A post-buckling analysis of a constant or variable length spatial elastica constrained by a cylindrical wall is performed for a first time by adopting an optimal control methodology. Its application in a constrained buckling analysis is shown to be superior when compared to other numerical techniques, as the inclusion of the unilateral constraints is feasible without the need of any special treatment or approximation. Furthermore, the formulation is simple and the optimal configurations of the spatial elastica can be also obtained by considering the minimization condition of the Hamiltonian. We first present the optimal control formulation for the constrained buckling problem of a constant length spatial elastica, including its associated necessary optimality conditions that constitute the Pontryagin's minimum principle. This fundamental constrained buckling problem is used to validate the proposed methodology. The general buckling problem of a variable length spatial elastica is then analyzed that consists of two parts; (1) the solution of the optimal control problem that involves the inserted elastica inside the conduit and (2) the derivation of the buckling load by taking into account the generation of the configurational or Eshelby-like force at the insertion point of the sliding sleeve. A variety of examples are accordingly presented, where the effects of factors, such as the presence of uniform pressure, the clearance of the wall, and the torsional rigidity, on the buckling response of the spatial elastica, are investigated.

References

References
1.
Miller
,
J. T.
,
2014
, “
Mechanical Behavior of Elastic Rods Under Constraint
,”
Ph.D. thesis
, Massachusetts Institute of Technology, Cambridge, MA.https://dspace.mit.edu/handle/1721.1/88280
2.
Gao
,
G.
, and
Miska
,
S. Z.
,
2009
, “
Effects of Boundary Conditions and Friction on Static Buckling of Pipes in a Horizontal Well
,”
Soc. Pet. Eng. J.
,
14
(
4
), pp.
782
796
.
3.
Tang
,
W.
,
Wan
,
T. R.
,
Gould
,
D. A.
,
How
,
T.
, and
John
,
N. W.
,
2012
, “
A Stable and Real-Time Nonlinear Elastic Approach to Simulating Guidewire and Catheter Insertions Based on Cosserat Rod
,”
IEEE Trans. Biomed. Eng.
,
59
(8), pp.
2211
2218
.
4.
Wicks
,
N.
,
Wardle
,
B. L.
, and
Pafitis
,
D.
,
2008
, “
Horizontal Cylinder-in-Cylinder Buckling Under Compression and Torsion: Review and Application to Composite Drill Pipe
,”
Int. J. Mech. Sci.
,
50
(
3
), pp.
538
549
.
5.
Gao
,
D.-L.
, and
Huang
,
W.-J.
,
2015
, “
A Review of Down-Hole Tubular String Buckling in Well Engineering
,”
Pet. Sci.
,
12
(
3
), pp.
443
457
.
6.
Miller
,
J.
,
Su
,
T.
,
Dussan V
,
E.
,
Pabon
,
J.
,
Wicks
,
N.
,
Bertoldi
,
K.
, and
Reis
,
P.
,
2015
, “
Buckling-Induced Lock-Up of a Slender Rod Injected Into a Horizontal Cylinder
,”
Int. J. Solids Struct.
,
72
, pp.
153
164
.
7.
Miller
,
J.
,
Su
,
T.
,
Pabon
,
J.
,
Wicks
,
N.
,
Bertoldi
,
K.
, and
Reis
,
P.
,
2015
, “
Buckling of a Thin Elastic Rod Inside a Horizontal Cylindrical Constraint
,”
Extreme Mech. Lett.
,
3
, pp.
36
44
.
8.
Miller
,
J. T.
,
Mulcahy
,
C. G.
,
Pabon
,
J.
,
Wicks
,
N.
, and
Reis
,
P. M.
,
2015
, “
Extending the Reach of a Rod Injected Into a Cylinder Through Distributed Vibration
,”
ASME J. Appl. Mech.
,
82
(
2
), p.
021003
.
9.
Bernoulli
,
D.
,
1742
, “
The 26th Letter to Euler
,”
Correspondence Mathematique Et Physique
, Vol.
2
, p. h. Fuss, Saint Petersburg, Russia.
10.
Euler
,
L.
,
1744
,
Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accepti
(Additamentum),
apud Marcum-Michaelem Bousquet & Socios
,
Lausanne/Geneve, Switzerland
, Chap. 1.
11.
Goss
,
V. G. A.
,
2008
, “
The History of the Planar Elastica: Insights Into Mechanics and Scientific Method
,”
Sci. Educ.
,
18
(
8
), pp.
1057
1082
.
12.
Kirchhoff
,
G.
,
1859
, “
Uber Das Gleichgewicht Und Die Bewegung Eines Unendlich Dunnen Elastischen Stabes
,”
J. Fur Math. (Crelle)
,
56
(
56
), pp.
285
313
.
13.
Lazarus
,
A.
,
Miller
,
J. T.
, and
Reis
,
P. M.
,
2013
, “
Continuation of Equilibria and Stability of Slender Elastic Rods Using an Asymptotic Numerical Method
,”
J. Mech. Phys. Solids
,
61
(
8
), pp.
1712
1736
.
14.
Davies
,
M. A.
, and
Moon
,
F. C.
,
1993
, “
3D Spatial Chaos in the Elastica and the Spinning Top: Kirchhoff Analogy
,”
Chaos
,
3
(
1
), pp.
93
99
.
15.
Maddocks
,
J.
,
1984
, “
Stability of Nonlinearly Elastic Rods
,”
Arch. Ration. Mech. Anal.
,
85
(
4
), pp.
311
354
.
16.
Majumdar
,
A.
,
Prior
,
C.
, and
Goriely
,
A.
,
2012
, “
Stability Estimates for a Twisted Rod Under Terminal Loads: A Three-Dimensional Study
,”
J. Elasticity
,
109
(
1
), pp.
75
93
.
17.
van der Heijden
,
G.
,
Neukirch
,
S.
,
Goss
,
V.
, and
Thompson
,
J.
,
2003
, “
Instability and Self-Contact Phenomena in the Writhing of Clamped Rods
,”
Int. J. Mech. Sci.
,
45
(
1
), pp.
161
196
.
18.
Manning
,
R. S.
,
Rogers
,
K. A.
, and
Maddocks
,
J. H.
,
1998
, “
Isoperimetric Conjugate Points With Application to the Stability of DNA Minicircles
,”
Proc. R. Soc. A
,
454
(
1980
), pp.
3047
3074
.
19.
Cherstvy
,
A. G.
,
2011
, “
Torque-Induced Deformations of Charged Elastic DNA Rods: Thin Helices, Loops, and Precursors of DNA Supercoiling
,”
J. Biol. Phys.
,
37
(
2
), pp.
227
238
.
20.
Love
,
A. E. H.
,
2013
,
A Treatise on the Mathematical Theory of Elasticity
,
Cambridge University Press
,
Cambridge, UK
.
21.
Bigoni, D., ed.,
2016
,
Extremely Deformable Structures
,
Springer
,
Wien, Austria
.
22.
Domokos
,
G.
,
Holmes
,
P.
, and
Royce
,
B.
,
1997
, “
Constrained Euler Buckling
,”
J. Nonlinear Sci.
,
7
(
3
), pp.
281
314
.
23.
Ro
,
W.-C.
,
Chen
,
J.-S.
, and
Hong
,
S.-Y.
,
2010
, “
Vibration and Stability of a Constrained Elastica With Variable Length
,”
Int. J. Solids Struct.
,
47
(
16
), pp.
2143
2154
.
24.
Liakou
,
A.
, and
Detournay
,
E.
,
2018
, “
Constrained Buckling of Variable Length Elastica: Solution by Geometrical Segmentation
,”
Int. J. Non-Linear Mech.
,
99
, pp. 204–217.
25.
Denoel
,
V.
, and
Detournay
,
E.
,
2011
, “
Eulerian Formulation of Constrained Elastica
,”
Int. J. Solids Struct.
,
48
(
3–4
), pp.
625
636
.
26.
Thompson
,
J. M. T.
,
Silveira
,
M.
,
van der Heijden
,
G. H. M.
, and
Wiercigroch
,
M.
,
2012
, “
Helical Post-Buckling of a Rod in a Cylinder: With Applications to Drill-Strings
,”
Proc. R. Soc. A
,
468
(
2142
), pp.
1591
1614
.
27.
Thompson
,
J.
, and
van der Heijden
,
G.
,
2013
, “
A Graphical Criterion for the Instability of Elastic Equilibria Under Multiple Loads: With Applications to Drill-Strings
,”
Int. J. Mech. Sci.
,
68
, pp.
160
170
.
28.
Manning
,
R. S.
, and
Bulman
,
G. B.
,
2005
, “
Stability of an Elastic Rod Buckling Into a Soft Wall
,”
Proc. R. Soc. A
,
461
(
2060
), pp.
2423
2450
.
29.
Fang
,
J.
,
Li
,
S.-Y.
, and
Chen
,
J.-S.
,
2013
, “
On a Compressed Spatial Elastica Constrained Inside a Tube
,”
Acta Mech.
,
224
(
11
), pp.
2635
2647
.
30.
Liakou
,
A.
,
2018
, “
Application of Optimal Control Method in Buckling Analysis of Constrained Elastica Problems
,”
Int. J. Solids Struct.
, in press.
31.
Liberzon
,
D.
,
2012
,
Calculus of Variations and Optimal Control Theory: A Concise Introduction
,
Princeton University Press
, Oxford,
UK
.
32.
Maurer
,
H.
,
1979
, “
Differential Stability in Optimal Control Problems
,”
Appl. Math. Optim.
,
5
(
1
), pp.
283
295
.
33.
Goyal
,
S.
,
Perkins
,
N. C.
, and
Lee
,
C. L.
,
2008
, “
Non-Linear Dynamic Intertwining of Rods With Self-Contact
,”
Int. J. Non-Linear Mech.
,
43
(
1
), pp.
65
73
.
34.
Maurer
,
H.
, and
Mittelmann
,
H. D.
,
1991
, “
The Non-Linear Beam Via Optimal Control With Bounded State Variables
,”
Optim. Control Appl. Methods
,
12
(
1
), pp.
19
31
.
35.
Chen
,
J.-S.
, and
Fang
,
J.
,
2013
, “
Deformation Sequence of a Constrained Spatial Buckled Beam Under Edge Thrust
,”
Int. J. Non-Linear Mech.
,
55
, pp.
98
101
.
36.
Bryson
,
A.
,
2016
,
Applied Optimal Control: Optimization, Estimation and Control
,
CRC Press
, Boca Raton, FL.
37.
Ross
,
M. I.
,
2015
,
A Primer on Pontryagin's Principle in Optimal Control
,
Collegiate Publishers
, Oceanside, CA.
38.
Nocedal
,
J.
,
2006
,
Numerical Optimization
,
Springer
,
New York
.
39.
Falugi
,
P.
, and
van Wyk
,
K. E.
,
2010
, “
Imperial College London Optimal Control Software User Guide (ICLOCS)
,” Imperial College London, London.
40.
Wachter
,
A.
, and
Biegler
,
L. T.
,
2005
, “
On the Implementation of an Interior-Point Filter Line-Search Algorithm for Large-Scale Nonlinear Programming
,”
Math. Program.
,
106
(
1
), pp.
25
57
.
41.
Biegler
,
L.
, and
Zavala
,
V.
,
2009
, “
Large-Scale Nonlinear Programming Using IPOPT: An Integrating Framework for Enterprise-Wide Dynamic Optimization
,”
Comput. Chem. Eng.
,
33
(
3
), pp.
575
582
.
42.
Bigoni
,
D.
,
Corso
,
F. D.
,
Bosi
,
F.
, and
Misseroni
,
D.
,
2015
, “
Eshelby-Like Forces Acting on Elastic Structures: Theoretical and Experimental Proof
,”
Mech. Mater.
,
80
(Pt. B), pp.
368
374
.
43.
Bigoni
,
D.
,
Dal Corso
,
F.
, and
Misseroni
,
D.
,
Bosi
,
F.
,
2014
, “
Torsional Locomotion
,”
Proc. R. Soc. A
,
470
(
2171
), p. 20140599.
44.
Bosi
,
F.
,
Misseroni
,
D.
,
Corso
,
F.
, and
Bigoni
,
D.
,
2015
, “
Development of Configurational Forces During the Injection of an Elastic Rod
,”
Extreme Mech. Lett.
,
4
, pp.
83
88
.
45.
Hartl
,
R. F.
,
Sethi
,
S. P.
, and
Vickson
,
R. G.
,
1995
, “
A Survey of the Maximum Principles for Optimal Control Problems With State Constraints
,”
SIAM Rev.
,
37
(
2
), pp.
181
218
.
46.
Dill
,
E. H.
,
1992
, “
Kirchhoff's Theory of Rods
,”
Arch. Hist. Exact Sci.
,
44
(
1
), pp.
1
23
.
47.
Lee
,
A. A.
,
Le Gouellec
,
C.
, and
Vella
,
D.
,
2015
, “
The Role of Extensibility in the Birth of a Ruck in a Rug
,”
Extreme Mech. Lett.
,
5
, pp.
81
87
.
48.
Peterson
,
K.
, and
Manning
,
R.
,
2010
, “
Ineffective Perturbations in a Planar Elastica
,”
Involve
,
2
(
5
), pp.
559
580
.
49.
Weiss
,
H.
,
2002
, “
Dynamics of Geometrically Nonlinear Rods: I. Mechanical Models and Equations of Motion
,”
Nonlinear Dyn.
,
30
(
4
), pp.
357
381
.
50.
Atanackovic
,
T.
,
1997
,
Stability Theory of Elastic Rods
,
World Scientific Publishing
,
London
.
You do not currently have access to this content.