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≠

• Length L :

• Same formula for D :

−=

t

m

td

41

16 2

• Same solution manifold : ( )( )max2 cos12 θ+= tm

]2

1;

2

1[,,,,

2

+−∈===Γ==L

Ss

L

Dd

EI

GJ

EI

LTt

EI

LMm

• because we consider only part of the homoclinic :

• R depends on Γ whereas d and m2 do not.

+

Γ=

t

mArcCos

mR

24

∞<R

[Heijden, Thompson]

Very long rod : post-buckling surfacepost-buckling surface

1 2 3 4 5m

1

2

3

4

5

6

7

8t

• Semi-finite correction.

• tangency between const. R and const. D curves

• Stability limit :

- corresponds to

- more stable than in the infinite case.

( )

+Γ+Γ−=

t

ttm

4

214

22

2

• Special curve :

2max

πθ >

π2=R

• No instability for : π2<R

unstable stable

0.5 1 1.5 2 2.5 3 3.5 4 d

2

4

6

8

t

Very long rod : stability for constant stability for constant RR experiments experiments

• Stability changes at folds in D.

• We tune D : conjugate parameter is T.

• Index [Maddocks] ; potential V [Thompson] :

Level curves of :

−+−

Γ=

1614

161

2 22

0

tdArcCos

tdtR

• Special path : π2=R

• Curves with have no foldπ2<R

ddt

Γ+=

482

Lift-off Loss of stability :

0R

2 4 6 8 10 12R

1

2

3

4

5

6

7

8m

Very long rod : stability for constant stability for constant DD experiments experiments

• Stability changes at folds in R

• We tune R, conjugate parameter is - m

−±+

Γ=±

02

0

164224

dm

dmArcCos

mR

Level curves :

π22

+

Γ

−Γ

=m

ArcTanm

R

• At low D : jump to contact.

• At large D : quasistatic to planar elastica.

Γ=

2MILd

Loss of stability :

0d

Finite length rod : homoclinic

• Buckling under compression.

• Same equations as before but more (free) parameters.

• System is still integrable.

• No homoclinic trajectory in phase space (generically).

• Equivalent Wrench (M, F) no longer // rig.

• Shear force and bending moment at the clamps.

θ

θ

Finite length rod : buckling modes (clamped)buckling modes (clamped)

−=−

−

− tmSinttm

mCostmCos 4

2

14

24

2

1 222

−4 −2 2 4mz

−4

−3

−2

−1

1

2

3

4t

∞

1st

2nd

3rd

2

2

=

πL

B

Tt

π2

L

B

Mm z

z =

Finite length rod : planar modes (clamped)planar modes (clamped)

−4 −2 2 4 6t

1

2

3

4

mx0

∞

1st inflex

3rd inflex

1st non-inflex

2nd non-inflex

3rd non-inflex

2

2

=

πL

B

Tt

π20

0

L

B

Mm x

x =

Finite length rod : where is the post-buckling surface ?where is the post-buckling surface ?

01

23

4mz

−4

−2

02

4t

0

1

2

3

4

mx0

01

23

4mz

−4

−2

02

4t

buckling 1st mode

non-inflex 1st mode

inflex 1st mode

• we consider :

- non trivial initial conditions :

- free parameters :

• Global Representation Space [Domokos]

<=> unique configuration.

Boundary conditions :

• 2D solution manifold in 4D space.

• All the non-closed configurations

are s-symmetric (because ).

Finite length rod : post-buckling surfacepost-buckling surface

z

y

x

yxxyz

xzzxzy

yzzx

dz

dy

dx

dmdxtdytd

dmdmdytd

dmdxtd

3

3

3

30333

33033

333

=′

=′=′

+−−=′

+−=′

−=′

0θ

0,, xz mtm

( )00 ,,, θxz mtm

=

=

xz

y

dzdx

d

33

3 0

• Boundary value problem (BVP).

• Centre line of rod : 6D system :

00 =ym

Finite length rod : sliding without rotationsliding without rotation

LIFT OFF

BIRD CAGE

0.2 0.4 0.6 0.8 1 1.2 1.4D

−4

−3

−2

−1

1

2

3

t

R=7 rad .

0.2 0.4 0.6 0.8 1 1.2 1.4D

−3

−2

−1

1

t

R=6 rad .

0.2 0.4 0.6 0.8 1 1.2 1.4D

−2.5

−2

−1.5

−1

−0.5

0.5t

R=1 rad .

0.2 0.4 0.6 0.8 1 1.2 1.4D

−2

−1

1

2

3

t

R=10 rad .

Finite length rod : 3 typical shapes3 typical shapes

0 < D < 1 D = 1 1 < D < 0

closed revertedopened

Finite length rod : rotation without slidingrotation without sliding

Same fold as before (jump to contact).

New fold leading to hysteresis

D0=0.5 D0=0.8

D0=0.9

2π

-2π

R

M

WX

YZ

O

D0=0.6

D0=0.7 D0=0.99−2 π 2 π

R

−1

1

m3

Finite length rod : connection between 1st and 2nd buckled modesconnection between 1st and 2nd buckled modes

−4−2

02

4

mz0

12

34

t

0

1

2

3

4

mx0

−4−2

02

4

mz

0

1

2

3

4

mx0

D = 1.1planar 1st mode inflex planar 2nd mode non-inflex

−2 π −π π 2 πR

−0.5

0.5

1

m3

Finite length rod : how to change how to change Γ

Γ2

Γ1

∆ ∆

Comparison of the 3 models for the jump to contactjump to contact

DêD∗

2 π

4 π

6 π

8 π

R

∞=L

∞≤L

finiteL

Finite length rod : overall stabilityoverall stability

1 2D

2 π

4 π

6 π

8 π

R

contact

hysteresis hysteresis

Planar elastica : connections between connections between inflexinflex. paths. paths

05

10t

0

πθ

0

2

4

6

mx0

05

10t

0

πθ