Finite Finite size effects on twisted rods stability size effects on twisted rods stability Sébastien Neukirch Sébastien Neukirch (joint work with G. van (joint work with G. van der der Heijden Heijden & J.M.T. Thompson) & J.M.T. Thompson) Elasticity of twisted rods supercoiling of DNA plasmids carbon nanotubes Applications sub marine cables Climbing plants
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Finite size effects on twisted rods stability - ida.upmc.fr · • Semi -finite correction. • tangency between const . R and const . D curves • Stability limit : - corresponds
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Finite Finite size effects on twisted rods stabilitysize effects on twisted rods stabilitySébastien NeukirchSébastien Neukirch (joint work with G. van (joint work with G. van derder HeijdenHeijden & J.M.T. Thompson) & J.M.T. Thompson)
Elasticity of twisted rods
supercoiling of DNA plasmids
carbon nanotubesApplications
sub marine cables Climbing plants
2 types of experiments
Hypothesis
• no shear
• no extensibility
• no intrinsic curvature
• no gravitation
• rotation without sliding (D constant)
• sliding without rotation (constant R)
Equilibrium equations
u
GJ
EI
EI
M
duddS
d
dRdS
d
dFMdS
d
FdS
d
ii
rr
rrr
rr
rrr
r
=
×=
=
×=
=
00
00
00
0
3
3
)(
)(
)(
)(
)(
)(
)(
3
2
1
Su
Sd
Sd
Sd
SR
SM
SF
r
r
r
r
r
r
r
• Boundary conditions :
• 1 independent variable S : ODEs
• Static-Dynamic Kirhhoff analogy : spinning top <=> spatial elasticaelastica
7 unknowns
)()()( 333 BdAdBdkBArrrr
==
integrable
21 EIEI =
7 equations
What do we want to get ?
• All the static configurations of the rodfor the clamped boundary conditions.
• Stability of these configurations underthe 2 typical experiments.
Systems of ODEs with :
initials conditions
boundary conditions
In the parameters space ( L , EI , F, M , ...) there will be a ‘n-D’ solution manifold.
Define an index I [K. Hoffman]:I = 0 : stable,I = 1 : unstable,I > 1 : more unstable.
Number of negative eigenvalues of the second order differentialoperator of the constrained variational problem.
3 different models
∞=L
∞≤L
finiteL
homoclinic trajectory
homoclinic trajectory without fixed point
other trajectories in phase space
A
C
B
• A & B are much easier than C because less parameters
• we will compare, as we go A → B → C, :
1 - how stability changes,2 - how new solutions appear.
• Reduction to an equivalent oscillator (2D)
• Spatial localisation of the deformation.
• Applied force and moment // rig axis.
• Buckling :
• D end-shortening
• Parameters space :
• Solution manifold :
Rod of infinite length (van der Heijden - Champneys - Thompson)
TM 42 =
−=
T
M
TD
41
16 2
∞=R (as soon as M > 0)
( )( )max2 cos12 θ+= TM
Rigid Loading (R,D) :
max,, θTM
Rod of infinite length : sliding without rotationsliding without rotation
E
F
G
H
2max
πθ =1 2 3 4 5
M1
2
3
4
5
6
7
8T
unstable stable
this is a projection !
Very long rod : constconst. R. R const const. M. M≠