Alain Goriely and Michael Tabor- Nonlinear Dynamics of Filaments IV: Spontaneous Looping of Twisted Elastic Rods

8/3/2019 Alain Goriely and Michael Tabor- Nonlinear Dynamics of Filaments IV: Spontaneous Looping of Twisted Elastic Rods

1/18

Nonlinear Dynamics of Filaments IV:Spontaneous Looping of Twisted Elastic Rods

Alain Goriely and Michael Tabor

University of Arizona, Program in Applied Mathematics ,Building #89, Tucson, AZ85721, USA

Universite Libre de Bruxelles, Departement de Mathematique, CP218/11050 Brussels, Belgium, e-mail: agoriel@ ulb.ac.be

Submitted to Proc. Roy. Soc. A

Abstract

Everyday experience shows that twisted elastic laments spontaneously form loops. We modelthe dynamics of this looping process as a sequence of bifurcations of the solutions to the Kirchhoff equation describing the evolution of thin elastic lament. The control parameter is taken to bethe initial twist density in a straight rod. The rst bifurcation occurs when the twisted straightrod deforms into a helix. This helix is an exact solution of the Kirchhoff equations whose stabilitycan be studied. The secondary bifurcation is reached when the helix itself becomes unstableand the localization of the post-bifurcation modes is demonstrated for these solutions. Finally,the tertiary bifurcation takes place when a loop forms at the middle of the rod and the loopingbecomes ineluctable. Emphasis is put on the dynamical character of the phenomena by studyingthe dispersion relation and deriving amplitude equations for the different congurations.

1 Introduction

Everybody has been faced at one point or another with the impossible tangles formed by coiledtelephone chords. The mechanism by which a telephone chord entangles with itself is a genericphenomenon encountered in many contexts. It is one of the most basic forms of instability encounteredwith elastic laments and is usually referred to as the writhing instability , i.e. , a change in spatialconguration of the lament to reduce the overall twist of the unstable structure. Consider thefollowing experiment: a straight elastic lament is held between ones ngers and twist is injectedby rotating one end while holding the other one xed. After a small amount of twist has beenadded (small rotation), the lament is no longer straight but assumes a helical form (with very smallradius). As the twist is increased the deformation of the lament tends to localize in the middle of therod and eventually a loop forms (see Fig. 1). Experimental studies of lament twisting (Thompson& Champneys, 1996) have shown that this sequence of bifurcations is qualitatively correct. It is thepurpose of this paper to study and describe this looping process as a sequence of dynamical instabilitieswithin the framework of elasticity theory.

The starting point of almost all analyses of twisted laments is the Kirchhoff equations. Thisset of partial differential equations describes the time and space evolution of a lament subjected toexternal stresses (induced by applied forces and moments at the ends). The classical analysis (Love,1892; Timoshenko & Gere, 1961) of the stationary equations suggests that a straight, twisted elasticlament deforms into a helix. However, these analyses are limited in many respects. First, they donot include or explain the dynamical phenomena triggering such an instability. Second, while theperiod of the helix can be easily computed by linear theory, its radius cannot be obtained. Third, it isnot known from this analysis whether the new helix is itself stable. Other analyses of the stationaryequation (Coyne, 1990) propose a completely different scenario for the looping by exhibiting a familyof stationary solutions (valid only for a restricted class of boundary conditions) deforming continuously

1


2/18

a

b

c

d

e

f

i n c r e a s e s

: twist

Figure 1: A schematic description of the sequence of bifurcation of a looping process

2


3/18

from a straight rod to a loop. We will come back to these solutions and their relationship with ourmodel in the nal discussion.

The analysis presented here departs radically from previous studies. We consider the dynamical instabilities as the main mechanism triggering the changes in lament conguration. The loopingprocess is then explained as a sequence of bifurcations of the solutions to the Kirchhoff equations inwhich the twist density is taken as a control parameter. The rst bifurcation occurs when the straight

twisted lament becomes unstable. It deforms to a helix whose radius can be computed by a nonlinearanalysis (Goriely & Tabor, 1996; Goriely & Tabor, 1997a; Goriely & Tabor, 1997b). It is then arguedthat this new helix is itself an exact solution to the Kirchhoff model and that its stability can in turnbe studied by recently developed methods (Goriely & Tabor, 1997c). The new physical parametersassociated with this helix (which are required for the associated stability analysis), such as the twistdensity, can be deduced from energetic considerations.

Another view of looping, which does not utilize Kirchhoff dynamics, has been proposed byRicca (Ricca, 1995) and is based on a consideration of the elastic energy of various space curvecongurations.

The secondary bifurcation occurs when the helix itself becomes unstable. The unstable mode of thehelix tends to localize the deformation at one point of the rod and forms a loop. Mode localization is awell-known experimental and theoretical phenomena for plates or laments under stress (Tvergaard &Needleman, 1980; Pomeau, 1981; Damil & Pottier-Ferry, 1986; Champneys & Thompson, 1997). Our

analysis provides a simple dynamical model explaining the tendency of stressed lamentary structuresto localize the deformations. Finally, the nonlinear analysis is used to compute the amplitude of thedeformation and the tertiary bifurcation is reached when the loop collapses onto itself.

As already mentioned, Thompson and Champneys have performed detailed experiments on thebuckling, localization and looping of straight twisted rod (Thompson & Champneys, 1996). Theyshow that the straight rod rst bifurcates into a helix. Then, the helical solution further localizesuntil a small dynamic jump occurs and a loop is formed. The analysis presented here is onlyslightly different than the one experimentally observed. The discrepancies between the sequence of events they describe in the paper and the one we use in ours are of two types: First a quantitativedifference, the rst bifurcation does not occur with the same wavelength (the famous one-turn-per-wave). The authors have indeed later shown that this puzzling discrepancy from the classical bucklingtheory results from the inuence of small initial curvature in the rod (Champneys, van der Heyden& Thompson, 1997). Therefore this discrepancy cannot appear in our analysis (since we never take

this effect into account and work only with the simplest possible hypothesis). The second differencecome from the observed continuous localization of the helical rod whereas we talk of a secondarybifurcation. One of the conclusion that we draw from our analysis is that for long enough rods, therst and secondary bifurcation are so close that they would actually be indistinguishable. After thesecondary bifurcation, our theory also predicts continuous localization until the tertiary bifurcationpoint is reached (equivalent to the dynamic jump of Thompson and Champneys). Therefore, webelieve that the sequence of events described in our paper is qualitatively consistent with the onedescribed in Thompson and Champneys.

The paper is organized as follows. In Section 2, we describe the general setup, the Kirchhoff modeland briey review the linear and nonlinear methods that will be used. Since our proposed model isquite elaborate and uses ideas from both linear and nonlinear stability theory we give a summary of the proposed three step process in Section 3 and then each of these steps is discussed in detail in thesections that follow. In Section 4 we study the primary bifurcation of the straight rod and obtain thepost-bifurcation helix. In Section 5, the linear and nonlinear analysis of the helix is performed andwe show the tendency to localization. In Section 6 we nd the tertiary bifurcation where the loopingrst occurs.

2 General setup

The derivation of the Kirchhoff model has been explained in detail in earlier paper (Goriely & Tabor,1997a; Goriely & Tabor, 1997b). Here, we review the most important aspects of the theory relevant tothe problem. We rst explain the kinematics of space-curves by introducing a director basis attachedto a general space-curve and then the dynamics of rods is described within the approximation of linearelasticity theory. The section ends with a brief review of the linear and nonlinear stability methods.

3


4/18

2.1 Spin and twist

Let X = X (s, t ) : R R R3 be a time dependent space curve, parameterized by the arc length s.

A director basis {d1 , d2 , d3}can be attached to the curve as follows: The vector d3(s, t ) = X (s, t ) isthe tangent vector of X at s (the prime denotes the s-derivative). The vectors {d1(s, t ), d2(s, t )}arechosen such that {d1 , d2 , d3}forms, a right-handed orthonormal triad ( d1 d2 = d3 , d2 d3 = d1). If d1 is along d3 , the director basis specializes to the well-know Frenet triad for which d1 is the normalvector and d2 the bi-normal vector. The curve X = X (s, t ) can be obtained by integrating the tangentvector: X (s, t ) =

s d3(s, t )ds.The space and time evolution of the director basis species the kinematics of the space-curve X :

di =3

j =1

K ij dj i = 1 , 2, 3, (1.a)

di =3

j =1

W ij dj i = 1 , 2, 3, (1.b)

where ( ) stands for the time derivative. The antisymmetry of W and K :

K =0 3 2

3 0 12 1 0, W =

0 3 23 0 12 1 0

, (2)

is a consequence of the orthonormality of the basis. The elements of K and W make up the componentsof the twist and spin vectors ; namely = 3i =1 i di and =

3i =1 i di .

2.2 The Kirchhoff model

The Kirchhoff model describes the space and time evolution of thin laments, i.e. laments whoselength is much greater than their radius and whose curvature is sufficiently large relative to the smalllength scales in the problem. In this approximation all physically relevant quantities characterizingthe three-dimensional elastic body are averaged over the cross sections attached to the central axis of the lament. This results in a one-dimensional theory.

The total force, F = F (s, t ), and the total moment, M = M (s, t ), are expanded in terms of the director basis; namely, F = 3i =1 f i di , M =

3i =1 M i di . The Kirchhoff model expresses the

conservation of linear and angular momentum together with the constitutive relationship of linearelasticity relating the moments to the strains as characterized by the twist vector. For a naturallystraight rod ( i.e. , no intrinsic curvature or twist) with circular cross-section the scaled equationsread (Dill, 1992; Coleman et al. , 1993):

F = d3 , (3.a)M + d3 F = d1 d1 + d2 d2 , (3.b)M = 1d1 + 2d2 + 3d3 , (3.c)

where = 1 / (1 + ) (where is the Poisson ratio) measures the ratio between bending and twistingcoefficients of the rod. These equations, together with (1) can be reduced to a set of 9 equations,second order in space and time, for the 9 unknowns ( ,,f ).

A constant twist of an elastic rod about its axis can be conveniently introduced by using theextra degree of freedom provided by the director basis. The vectors d1 and d2 can be chosen suchthat they rotate around d3 in the normal plane at a constant rate (with respect to the arc-length), ,independent of the space curve torsion. This axial twist is conveniently visualized as the twist of aribbon about the rod axis and, where appropriate, it is referred to as the ribbon twist . An equivalentway of introducing the twist is to dene a new set of variables rotating with the basis. That is,rather than using the variables {,,f }, a new set of variables {, , f }is obtained by introducing aconstant rotation along d3 and a translation of the vector . The rotated vectors are now:

4


5/18

= R., f = R.f, (4.a)

i =3

j =1

(R ij j 3 ) j (4.b)

di =3

j =1

R ij .dj , i = 1 , 2, 3, (4.c)

where

R =cos(s ) sin(s ) 0sin(s ) cos(s ) 0

0 0 1(5)

The force, spin and twist vector are form invariant with respect to the change of variables: = 3i =1 i di , =

3i =1 i di , F =

3i =1 f i di . The transformed Kirchhoff equations read:

F =d3 , (6.a)

M + d3 F = d1 d1 + d2

d2 , (6.b)M = 1 d1 + 2 d2 + ( 3 + )d3 , (6.c)

Note also that due to the translation on the last component of the curvature vector, we now have:. d3 = 3 +

The main advantage of this representation is that simple stationary lament congurations canbe conveniently expressed as constant solutions of the Kirchhoff equations. For instance, the straightrod with constant twist, , and axial force, f 3 , has:

= (0 , 0, 0), = (0 , 0, ) , f = (0 , 0, f 3) . (7)

which transform under (4) to:

= = (0 , 0, 0) , f = (0 , 0, f 3) . (8)

In the same way, a helical lament with twist = H and Frenet curvature and torsion F , F whichtakes the form (Goriely & Tabor, 1997c):

X H = F s

,

F

cos(s),F

sin(s) , 2 = 2F + 2F , (9)

with

= ( F sin(s ), F cos(s ), F + ) , (10.a)

= (0 , 0, 0) , (10.b)f = ( f 0 sin(s ), f 0 cos(s ),

F F

f 0) (10.c)

where f 0 = F F + F F , can also be written as a constant solution in terms of ( , f ), i.e.

= (0 , F , F ) , = (0 , 0, 0) , (11.a)f = (0 , F + F F ( 1), F ( F ( 1) + )) . (11.b)

We now drop the hats and consider system (6) as the main system of equations studied here.

5


6/18

2.3 Perturbation expansions

To study the stability of stationary solutions, we developed a perturbation method at the level of thedirector basis (Goriely & Tabor, 1996; Goriely & Tabor, 1997a; Goriely & Tabor, 1997b; Goriely &Tabor, 1997c) . The main idea is to expand all local quantities around the local stationary solutionsand close the system to each order in the perturbation parameter by demanding that the basis remainorthonormal up to a given order:

di = d(0)i + d

(1)i +

2d(2)i + ... i = 1 , 2, 3, (12)

The orthonormality condition di .dj = ij gives rise to a system of constraints which can be solved toeach order:

d(1)i =3

j =1

A(1)ij d(0)j , (13.a)

d(k )i =3

j =1A(k )ij + S

(k )ij d

(0)j , k > 1, (13.b)

where S (k ) is a symmetric matrix whose entries depend only on ( j )i with j < k and A(k ) is theantisymmetric matrix:

A(k ) =0 (k )3

(k )2

(k )3 0

(k )1

(k )2 (k )1 0

, (14)

Once the vector (k ) is known, it is an easy matter to obtain the perturbed solution by integratingthe tangent vector up to any given order:

X (s, t ) = s

ds3

i =1

3i + A3i +k

j =2

k (A(k )3i + S (k )3i ) d

(0)i + O(

k +1 ). (15)

Any local vector V = 3i =1 vi di can be expanded in terms of the perturbed basis; namelyV = V (0) + V (1) + 2V (2) + ... :

V (1) =i

v(1)i + ( A(1) .v(0) ) i d

(0)i . (16)

Hence we can express the rst order perturbation of the twist and spin matrix, i.e. K = K (0) +K (1) + ..., W = W (0) + W (1) + ... , where

K (1) =A (1)

s+ A(1) , K (0) , (17.a)

W (1) =A (1)

t + A(1) , W (0) . (17.b)

where [A, B ] = A.B B.A . Higher order perturbations can be obtained in terms of the lower orderterms.Using these equations, one can write the k-th order perturbation of Newtons equation (6.a)

and moments equation (6.b) in terms of { (k ) , f (k )}. The rst of this system ( k = 1), referredto as the dynamical variational equations , controls the stability of the stationary solutions withrespect to linear time-dependent modes. Higher-order modes provide better approximations of theperturbed solutions and are used to perform a nonlinear analysis. To emphasize the linear characterof these equations we rewrite them as a linear system of 6 equations for the 6-dimensional vector(0) = {(0) , f (0) }, (k ) = { (k ) , f (k )}, k > 0:

6


7/18

LE ((0) ).(1) = 0 , (18.a)LE ((0) ).(k ) = H k ((0) , (1) , . . . , (k 1) ), k > 1 (18.b)

where LE is a linear differential operator in s and t whose coefficients may depend on s throughthe unperturbed solution (0) = {(0) , f (0) }and H k are vector valued functions depending on thevariables ( (0) , . . . , (k 1) ).2.4 Linear stability analysis

The linear stability of stationary solutions is determined through the use of (18.a). It has a set of fundamental solutions which we label by the spatial mode number n, i.e.

(1)n = n et + i nsL , (19)

where n C 6 . The growth rate of this mode, = (n), is determined by the dispersion relation:

( , n ) = det( LE ) = 0 obtained by substituting (19) into (18.a). Typically is a very complicatedexpression which is best derived by symbolic manipulation. The threshold of instability is heraldedby a change in sign of (the real part of) and can be determined by examining the neutral curvescorresponding to the parameter values, for given n, for which = 0. These curves are thus solutionsof (0 , n ) = 0. In what follows all our statements concerning critical values of twist (or tension) atwhich a given conguration becomes unstable are based on their determination from the dispersionrelations.

Of course the linear analysis can only identify the initial instabilities as a function of the parameters.As these instabilities grow (exponentially) in time the linear approximation breaks down and anyfurther description of the bifurcation requires a nonlinear analysis.

2.5 Nonlinear analysis

The techniques of nonlinear analysis enable one to derive equations for the amplitude of a solutionclose to bifurcation. The distance from the bifurcation point is considered to be of the order of theperturbation itself and this relationship is used to introduce new, longer, space and time scales overwhich the solution varies. For the problems considered here, the twist is taken as the the control(or stress) parameter and one sets

2 = c . (20)where c is the critical value at which the bifurcation occurs.

Stretched time and space scales appropriate to the problem at hand are introduced:

t0 = t, s 0 = s, (21)t1 = t, s1 = s. (22)

and taking into account the expansion in the bifurcation parameter and the new scales, one seekssolutions of the full system ( c.f. equations (18) order by order in :

0( 0) : E ((0) ; s0 , t 0) = 0 (23.a)

0( 1) : L(0)E ((0) ).(1) = 0 (23.b)

0( 2) : L(0)E ((0) ).(2) + L(1)E (

(0) ).(1) = H 2((1) ) (23.c)

0( 3) : L(0)E ((0) ).(3) + L(1)E (

(0) ).(2) + L(1)E ((0) ).(1) = H 3((1) , (2) ) (23.d)

...

7


8/18

where now the ( i ) are functions of the stretched variables, i.e. ( i ) = ( i ) (s0 , s1 ,...,t 0 , t 1 , .. ) andLE = i L

( i )E is the expansion of the linear operator in terms of the new variables ( s i , t i ). These

linear solutions ( c.f. (19)) are written as a superposition of the neutral modes: for example, to O( ),(1) will involve terms of the form X n (s1 , t 1)n ei

n c s 0L where X n (s1 , t 1) represents the slowly varying

amplitude of the unstable mode n = n c . A nonlinear amplitude equation for this slowly varyingamplitude arises as a condition (the Fredholm alternative) that the solutions ( i ) remain bounded.

Typically one nds this condition at O(3) and its derivation involves, at least in the cases consideredhere, massive symbolic manipulation.

In the case of a straight rod, discussed in the next section, the nonlinear amplitude equation takesthe form of a (system of) nonlinear partial differential equations and the amplitude(s) can be thoughtof as the envelope of a small packet of unstable wave numbers centered around the unstable mode. Inother cases, such as the helical problem considered later, we are only interested in the amplitude of anisolated mode and in this case, as will be shown explicitly, one derives a simpler nonlinear ordinarydifferential equation.

3 Summary of looping mechanism

The sequence of bifurcations that constitutes our model of spontaneous looping are, in summary, as

follows:

Primary Bifurcation(a) Starting with a straight rod, with arbitrary twist, S , and tension 2 , linear stability analysisis used to show that it becomes unstable (for a given ) at a critical twist value S = 1 and thenbifurcates into a helix.

(b) The precise geometrical form of the helix is identied by using nonlinear analysis to determine itsamplitude.

(c) Using energy considerations the redistribution of the straight rod twist, S , into twist and torsionof the new helix is determined. The helix is then specied by its curvature F , torsion F and twist

H .

Secondary Bifurcation(a) A linear stability analysis of the helix obtained in step (a) is carried out and the critical twistvalue, H = 2 , at which it becomes unstable is determined.

(b) The (linear) post-bifurcation solutions are constructed and a one-loop solution, of arbitrary am-plitude B , is identied.

(c) A nonlinear analysis of the unstable helical mode found in step (b) is used to determine the am-plitude of the loop.

Tertiary Bifurcation(a) A simple criteria is developed to determine the critical B value, Bc , at which the loop will ip (i.e. the onset of looping).

(b) The twist value, 3 , at which this loop amplitude equals Bc is found.

The various special values of the twist, H , 2 , 3 , can all be related back to the initial twist density, S , injected into the straight rod. Thus, in what follows, the term control parameter refers to thevalue of S .

8


9/18

Process Parameters Denition of parameters

Initial straight S Twist density in straight rodtwisted rod 2 Tension in straight rod(Fig 1a) 1 Value of S where the rod is unstablePrimary bifurcation F , F Frenet curvature and torsion of helixStraight rod helix H Axial twist(Fig 1b) A Amplitude of the helixSecondary bifurcation 2 Value of S where the helix is unstableHelix becomes linearly B Amplitude of the rst mode of deformationunstable (Fig 1c) of the helixTertiary bifurcation 3 Value of S where the loop collapsesLooping occurs B c Critical value of the amplitude for looping(Fig 1d)

Table 1: The different bifurcations and the new parameters introduced at each step

4 Primary bifurcation: Instability of a straight twisted rod

We consider a stationary rod of length L, simply supported, with tension T = 2 along the x-axis andtwist S . Rather than considering general boundary conditions, we just consider the case in which thetangents at the ends can assume any values while the tension along the x-axis is kept constant. Weassume that these boundary conditions are maintained throughout the sequence of bifurcation andthat the unique control parameter is the twist S . The effect of other type of boundary conditionswill be further discussed in the nal section. The stationary solution is given by:

(0) = 0, 0, 0, 0, 0, 2 . (24)

4.1 Linear analysis

Using the techniques of linear stability analysis we are able to derive the dispersion relations for thiscase (Goriely & Tabor, 1996; Goriely & Tabor, 1997b) and obtain the neutral curve from the relation(0 , n ; S ) = 0, where

( n; S ) = L2 S 2 n2 L2 S 2( 1) L22 n22

L2 S 2( 2)2n2 (25)From this one can deduce that the rst instability occurs for the critical mode

n c =L(2 )

(26)

with critical twist value 1 =

2

, (27)

This condition for the rst instability is the classical buckling condition that can be foundin (Timoshenko & Gere, 1961) but written in re-scaled variables. For, S bigger than, but closeto, 1 , the straight rod bifurcates to the helix:

X H (s) = s,A

cos s,A

sin s . (28)


The linear analysis does not specify the amplitude A of the new solution. However, a nonlinearanalysis can be performed (Goriely & Tabor, 1997b) yielding an amplitude equation consisting of asystem of nonlinear partial differential equations coupling the amplitude of the unstable mode to thelament twist density. The stationary solution of this amplitude equation gives the amplitude A as afunction of S , namely:

9


10/18

A2 = 2( S 1)

. (29)

4.3 Resummation

The remarkable feature of the rst bifurcation is that the new solution (28) is an exact solution of the Kirchhoff equations. Indeed, as was shown explicitly in the rst section, it is well-know thatthe Kirchhoff system sustains helicoidal solutions. This property can also be seen at the level of the perturbation expansion. Thus, if we carry on the perturbation analysis to higher-order in andsolve the subsequent linear system (18), it can be observed that the effect of higher-order corrections(2) , (3) , . . . is to provide correction to the radius of the helix without changing the overall form of thesolution (that is, higher-orders do not introduce higher-order harmonics). Since the second orderin provides correction of the radius of order A2 and, as shown below, the radius A is small at thesecond bifurcation, the rst order approximation of the helix is sufficient for our analysis. Therefore,to study the stability of the perturbed helicoidal solution, one can study the stability of the new exactstationary solutions by taking the zeroth-order solution to be the helix itself with parameters givenby the nonlinear analysis of the straight rod.

In order to study the stability of the helix (28), we rst re-write it in the standard form (9):

X H = F s, F

cos(s), F

sin(s) , 2 = 2F +

2F , (30)

where s is the arc-length along X H (as opposed to s which is the arc length along the original straightrod). The identication of (30) with (28) gives:

F =

1 + A2, F =

A1 + A2

, (31.a)

=

1 + A2 , s =s

1 + A2 , (31.b)

The tangential force is determined from the relation f (H )3 = 2 cos , where is the pitch angle of the helix (tan = F

F ). Hence

f (H )3 =2

1 + A2 . (32)So far, we have computed the geometrical parameters of the new helical (space) curve. However,

our elastic lament also has an imposed twist. In the deformation from straight rod to helical lament,the twist density changes. In order to nd the exact new twist density corresponding to the helicallament, we compute the energy for both structures.

The total energy of the system is:

H= L

0ds 21 +

22 + ( 3 + )

2 + 2 cos , (33)

where, the two rst terms in the integral represent the elastic energy due to curvature effects, the

third one the elastic energy due to torsion and twist and the last one, the potential energy due toexternal constraints.For the straight rod and the helices, we have, respectively:

S = (0 , 0, 0), = S , (34.a)H = (0 , F , F ), = H , (34.b)

where H is the unknown ribbon twist superimposed on the helical curve of Frenet torsion F andcurvature F . The ribbon twist is the actual twist of the rod: it represents the rotation in the cross-section of the director basis with respect to the Frenet basis. The energies of the straight rod andhelix are, respectively:

10


11/18

HS = L( 2S + 2), (35.a)HH = L 2F + ( F + H )2 + . (35.b)

Assuming conservation of energy and equating (35.a) and (35.b), the helical ribbon twist is found tobe

H = F 2S + ( ) 2F

, (36)

where the choice of sign is yet to be determined.There is a degeneracy in the limit where the radius of a helix shrinks to zero. Indeed, in the limit

A 0, both torsion F and twist H contribute to the total twist of the straight rod S . Formally,a twisted straight rod can be written as a helix with zero radius, zero twist and nite torsion or ahelix with zero radius, zero torsion and nite twist or any combination of the two. Therefore, in orderto relate the ribbon twist H to the twist S as A 0 we introduce the pseudo-torsion 0 and theresidual twist 0 of a straight rod as the limits:

0 = limA 0

F , (37.a)

0 = limA 0

H . (37.b)

The twist of the straight rod, viewed as the limit of the helix with vanishing radius, is simply S = 0 + 0 , and in general, we have:

.d 3 = F + H , (38)

for all A 0.In the limit A 0, the twist density goes to 1 and the residual twist is 0 = 1 0 = (2 )

.The sign determination in (36) can now be found by demanding that H

0 as the radius vanishes.

Thus

H = F + 2S + ( ) 2F

(39)

We now have all the parameters of the helix as a function of the initial twist density S . Indeed,the radius A is a function of S (see (29)), and the Frenet curvature and torsion are function of Athrough (31.a). Finally the ribbon twist is expressed in terms of all the other parameters and S .Therefore, the parameters ( F , F , H ) describing the helical lament are know in terms of S andthe stability of the helix as a function of the control parameter S and the number of helical turnN = L2 can now be analyzed.

5 Secondary bifurcation: Instability of the helix

An extensive study of the stability of helices has been given in (Goriely & Tabor, 1997c). We basethe following analysis on this work and specialize it to the family of helices, obtained in the previoussection, with one free parameter (the control parameter S ). The stationary solution is written as(c.f. equations (11)):

(0) = 0, F , F , 0, F H + F F ( 1), F H + 2F ( 1) . (40)where the helical ribbon twist, H , the curvature and the torsion should be thought of in terms of itsrelationship to the control parameter S .

11


12/18

-6

-5

-4

-3

-2

-1

0

1

1 2 3 4 5

= 1

1 < < 2 = 2

> 2

n: mode

: growth rate ( x 10 -6 )

Figure 2: Solutions of the dispersion relation for 2 as a function of n with increasing values of S = 0 .2267, 0.2675, 0.2687, 0.2695. =3/4, = 1 / 10, N = 5

5.1 Linear analysis

The fundamental mode solutions to the variational equations (18.a) are of the form:

(1)n = n e

n t + i nsN, (41)

where n C 6 and n denotes the mode number which, due to the choice of boundary conditions, is an

integer between 1 and N . The explicit form of the linear operator LE as a function of the curvature,torsion, twist and tension is given in (Goriely & Tabor, 1997c) and from this the dispersion relations,( , n ; H ) = 0 can be obtained in the usual way. A typical plot of them is shown on Fig. 2 forincreasing value of H .The neutral curve is determined from (0 , n ; H ) = 0, where

(0 , n ; H ) = ( 1)( 2) 2F + 2( 2) H F + 2 2H + 2(1 n2

N 2) (42)

For given n, one can then read off the the value of H at which the instability occurs. In general,different modes n can become unstable; however, within the family of helices parameterized by S ,

the mode n = 1 is always the rst unstable mode as the control parameter is increased. It is this casethat corresponds to our secondary bifurcation.Let 2 be the critical value at which new solutions appear, i.e. (0 , 1; 2) = 0. This last

relation is transcendental in S ; therefore, the exact value of 2 as a function of S cannot beobtained. Nevertheless, by expanding all the parameters to rst order in ( S 1), a remarkablygood approximation of 2 is found:

2 = 1 +4

4N 2 + (3 + 16) N + 4(43)

The approximation obtained by expanding the parameters to second-order in ( S 1) only slightlyimproves this last result. The difference 2 1 is always very small and decreases to zero as the

12


13/18

a

b

Figure 3: The deformation of a helix due to the unstable mode n = 1, a) to rst-order, b) to secondorder. The parameters are: = 3 / 4, = 1 / 10, N = 5, K = 20 and s runs from 80 to 80+100

number of loops N increases. In the limit of an innite long rod, 1 2 . Therefore, the delay of thebifurcation is only due to the discretization of modes (induced by the boundary conditions).At this point, it is of interest to reect on the classical results of elasticity theory. The instabilityof the twisted innite (or nite) straight rod is a well-known result obtained by many authors (see

(Love, 1892; Timoshenko & Gere, 1961) for instance). However,the stability of the new helix obtainedafter bifurcation has, to the best of our knowledge, never been investigated (or even questioned). Thecondition (43) gives the instability threshold of the twisted helix obtained after the rst bifurcation of a straight rod. It is different from the delayed bifurcation condition obtained for a nite straight rod .The derivation of this new condition stems from our determination of stability properties directly fromthe dynamical Kirchhoff equations. However, it is likely that in real, well-controlled, experiments onthe bifurcation of rods (where the number of loops N may be large) the twisted rod could jump pastthe secondary bifurcation (see Thompson and Champneys (1996) for recent experimental data) andthe helical solution might then not be observed.

Since the mode n = 1 is the rst unstable mode, the effect of the instability is to localize thesolution at one point. Indeed, to rst-order the explicit form of the bifurcated solution is:

x1(s, t ) = P s 2NKR 1

n F cos(

nsN

), (44.a)

x2(s, t ) = R cos(s) K

2 1n N

sin(n N

N s) +

2 + 1n + N

sin(n + N

N s) , (44.b)

x3(s, t ) = R sin(s) K

2 1n N

cos(n N

N s)

2 + 1n + N

cos(n + N

N s) (44.c)

where K = Bet and

1 = n3 F 2F 5 (n2

N 2)24n2 + ( n2

N 2)2N 22(2

2F ) + N 42(N 2 + n2)

2 = Nn 442F F 4(n2 N 2)(2 ) + 2 N 5n42 (45)To second-order the approximate solution is much closer to the exact solution since the arc-length

is conserved to order O( 4). The shape of the bifurcated solution is shown on Fig. 3 to rst and secondorder in .


The unstable modes of the helix are discretized due to the boundary conditions. Therefore, a nonlinearanalysis can only take into account the temporal evolution of these discrete modes. At the rstinstability, there is only one unstable mode, namely n = 1. Thus, we seek to derive an equationdescribing the temporal evolution of the mode amplitude B . The corresponding stationary solution of

13


14/18

this amplitude equation will give us a relation between the amplitude and the control parameter. Asdiscussed in Section 2, the main idea behind a (weakly) nonlinear analysis is to look at the solutionnear threshold and introduce new scales (in this case, a new time scale) proportional to the distanceto the bifurcation point and let the arbitrary amplitude, B , vary on this time scale. If the system isclose enough to the secondary bifurcation, the difference between the twist density and the criticaltwist 2 is proportional to the perturbation parameter itself, namely:

2 = S 2 (46)The new, longer, time scale is t1 = t. The choice of this new scale can be justied by expanding the

dispersion relation in power of (see (Goriely & Tabor, 1997b)). Taking into account the possibility of the solutions varying on these (independent) different scales, one can now solve the linear system (18).To rst order one obtain the linear solution:

(1) = B (t1)1eisN + c.c. (47)

where c.c. stands for the complex conjugate and 1C 6 is specied such that 1 .1 = 1.

The second-order solution can be found in the same way by solving (18.b) with k = 2:

(2) = 0 + 1eisN + 2e2i

sN + c.c., (48)

In order to nd a condition on the amplitude B (t1) we demand that the solutions to the third-order system (18.b) with k = 3 remains bounded. Thus, we apply the Fredholm alternative to thesystem (18.b): here this consists of integrating H 3 against all neutral solutions of the adjoint operatorLE :

L

0 1 .H 3((0) , (1) , (2) )ds = 0 , (49)

where 1 is the adjoint solution to (1)1 (i.e. L

E . 1 = 0) . This compatibility condition gives rise to a

differential equation for B as a function of t1 :

2Bt 21

= B (c1 c3|B |2) (50)where c1 and c3 are

c1 = 2 ( 1)2 F 2 + 2 c ( 1) F + 2 c 2 [(2 ) F c ]

2 4 + 3 F 2 + 4 c + 2 2 c F + 2 c 2 + 2 2 4 + 5 12 3 + 23 2 18 F 4 + 46 2 c + 8 4 c 36 c 3 18 c F 3

+ 3 22 8 + 23 2 c 2 + 3 2 + 12 4 c 2 36 3 c 2 + 6 + 2 2 6 2 F 2+ c 6 2 12 2 c 2 + 8 3 c 2 8 + 4 62 F +2 2 c 2 + 2 4 c 4 + 3 2 c 22 + 2 2

1(51)

c3 = 2 2 4 + 3 F 2 + 2 4 c + 2 2 c F + 2 2 2 2 + 2 2 2 3 9 + 17 10 2 F 2 + 2 10 12 + 4 2 F 3 2 2 2 c 2 + 2 3 c 2 + 2

( 1)3 F 3 + 3 c ( 1)

2 F 2 + 3 2 c 2 ( 1) F + 3 c 3 [( 2) F + c ] 2 4 + 5 12 3 + 23 2 18 F 4 + 46 2 c + 8 4 c 36 c 3 18 c F 3

+ 3 22 8 + 23 2 c 2 + 3 2 + 12 4 c 2 36 3 c 2 + 6 + 2 2 6 2 F 2+ 2 6 2 12 2 c 2 + 8 3 c 2 8 + 4 62 F +2 2 c 2 + 2 4 c 4 + 3 2 c 22 + 2 2

1(52)

14


15/18

0.26

0.265

0.27

0.275

0.28

0.285

2 4 6 8 10

3

2

1

N

Figure 4: The primary, secondary and tertiary bifurcation as a function of the parameters for N = 2to 10 with = 3 / 4, = 1 / 10

where c = H ( 2), that is the critical value of the helical ribbon twist (as given by (39) with S = 2)at which the helix becomes unstable with respect to the rst unstable mode n = 1.

All the parameters involve in the coefficients of the amplitude equation depends on the controlparameter S . The stationary solution of (50) B 2stat = c1 /c 3 gives a relation between the amplitudeand the initial twist density.

6 Tertiary bifurcation

The effect of the instability is to locate the perturbation at one point of the rod (chosen here to bethe middle point). Eventually, as the amplitude B is varied, a critical point is reached where theprojection of the solution in the x y plane becomes multivalued. The situation is schematicallydepicted on Fig. 1.d. The critical point at which the loop becomes perpendicular to the x-axis is anew bifurcation point where the loop is about to ip over on itself. The critical value of the amplitudeB c at which the loop centered at s = sc will ip corresponds to the condition:

x1(sc ; Bc ) = 0 , x1 (sc ; B c ) = 0 , (53)

This ipping condition is somewhat similar to the one used in Coynes analysis (Coyne, 1990) of the static Kirchhoff model with the main difference that the parameter used to characterize theFlipping in Coynes paper is the end displacements. The ipping condition (53) can be solvedanalytically using the second order solution. However, the explicit form is too involved to be usefuland is not given here.

It is now possible to nd the value of the tertiary bifurcation by nding the value of S = 3 suchthat B stat = B c . A plot of the sequence of bifurcation is shown on Fig. 4 for different values of N .Fig. 5 shows the lament as 3 is varied.

15


16/18

i n c r e a s e s

a

b

c

d

e

f

g

Figure 5: A sequence of helical laments for varying values of S , a) S < 1 , b) 1 < S < 2 , c) 2 < S > 3 , d) S = 3 , e) S > 3 . The parameters are = 3 / 4, = 1 / 10, N = 5, 1 = 0 .2667, 2 = 0 , 2684, 3 = 0 .27376,and S = 2 + with 103 =0.1, 1.5, 3, 4.9, 5.36, 5.4.

16


17/18

7 Discussion and conclusions

7.1 The dynamical picture

In the previous section, we studied the looping process as a sequence of bifurcations as the controlparameter is varied from S < 1 to S > 3 . This sequence of congurations can be obtained byconsidering that the system attains its stationary state for every small change of the control parameter.In order to obtain these congurations we rely on two assumptions: First, that in a real system, thepresence of damping will allow the system to relax in time (otherwise, none of the stationary solutioncould be obtained as the system is conservative). Second, that the change in the control parameteroccurs on a much smaller time scale than the relaxation time; thereby giving rise to a sequence of quasi-stationary conguration. However, these assumptions are not necessary for the basic phenomenaof looping. Indeed, we chose to present the sequence of bifurcations as quasi-stationary for the sake of simplicity. The real problem of looping consists in bringing a system to a region of instability wherelooping takes place, that is (in our setting) by suddenly varying the twist density of a stable straightrod to a value S > 3 . Then, the dynamics of the system will directly take the system from a straightrod to a loop without passing through the intermediary steps (this situation is actually closer to theeveryday experience of playing with telephone chords). The sequence of congurations of Fig. 5 canthen be seen as a dynamical sequence.

In this analysis, we have used the twist density as a control parameter. This is only one of thepossible choices. We could have used, equivalently, the tension (think of suddenly decreasing thetension at the end of a straight twisted rod to produce the same dynamics) or the intrinsic twistdensity. This last choice might be relevant to some problems occurring in biology where lamentsmay grow with no twist but a large intrinsic twist decit: effectively creating a twist density resultingin the formation of a loop (for a remarkable example of this biological phenomena in the growth of bacterial lament, see (Mendelson, 1978; Mendelson, 1990)). In any cases, the dynamics obtained byusing different parameters as control parameters are equivalent and it is likely that in any real systemchanges in all parameters will actually occur simultaneously.

7.2 Why do loops forms at the middle of the rod?

One commonly observes that the loops created by twisting the ends of a rod usually form in the middleof the rod. This is due to the choice of boundary conditions. Indeed for the sake of simplicity we haveconsider that the ends are xed in space but the tangents are free. Doing so, all the points of thestrings are actually equivalent and the loop can form at any point. However, a simple argument canshow why loops form at the middle for different boundary conditions. If the ends are held clamped(that is the tangents at the ends are constrained along the axis), the rst bifurcation does not leadto a helix but rather (as shown in (Goriely & Tabor, 1997b)) to a helix modulated by a envelope-likeshape (so that the radius of the modulated helix goes to zero at the ends). The maximal amplitudeof this solution is reached at the middle. We have seen here that the secondary instability is triggeredby increasing the radius of the helix (controlled by the initial twist density). Therefore, the secondaryinstability for a twisted rods with clamped ends will be rst triggered at the middle of the stringwhere the amplitude of the deformation is maximal.

7.3 Conclusions

We have now completed our picture of the looping as a dynamical process: a phenomena triggeredby a sequence of instabilities for different congurations of the lament. The rst bifurcation occurswhen a pulled, twisted straight rod becomes unstable. The linear analysis predicts that it deforms toa helix. The radius of such a helix can be computed via a nonlinear analysis (Goriely & Tabor, 1996;Goriely & Tabor, 1997b) while its twist density can be obtained by energetics consideration. Theimportant feature of the bifurcated solution is that it is an exact solution of the model. Therefore,its stability can be easily studied by the same method used to study the stability of the straight rod.The linear stability of the helix shows that it rapidly becomes unstable, given rise to a secondarybifurcation. Moreover, the instability tends to localize the deformation of the rod at one point. Theamplitude of this localization can be obtained by performing a one-mode amplitude expansion. Thisnonlinear analysis provides a relationship between the amplitude of the deformation and the control

17


18/18

parameters. The tertiary bifurcation is reached when a loop is formed at the middle and looping takesplace.

The dynamical mechanism proposed here differs in many respects from previous analysis andexploits a series of results on the linear and nonlinear stability of laments. These techniques arequite general and should be applicable to other problems involving dynamical changes in form.

Acknowledgments This work is supported by DOE grant DE-FG03-93-ER25174 and FlinnFoundations Biomathematics and Dynamics Initiative program (ID#048-1000206-94).

References

Champneys, A. R., & Thompson, J. M. T. 1997. A multiplicity of localised buckling modes fortwisted rod equations. Proc. Roy. Soc. London A , 452 , 24672491.

Champneys, A. R., van der Heiden, G. H. M. & Thompson, J. M. T. 1997. Spatially complexlocalisation after one-twist-per-wave equilibria in twisted rod circular rods with initial curvature.To be published in Phil. Trans. Roy. Soc. A .

Coleman, B. D., Dill, E. H., Lembo, M., Lu, Z., & Tobias, I. 1993. On the dynamics of rodsin the theory of Kirchhoff and Clebsch. Arch. Rational Mech. Anal. , 121 , 339359.

Coyne, J. 1990. Analysis of the formation and elimination of loops in twisted cable. IEE Journal of Oceanic Engineering , 15 , 7283.

Damil, N., & Pottier-Ferry, M. 1986. Wavelength selection in the postbuckling of a longrectangular plate. Int. J. Solids Structures , 22 , 511526.

Dill, E. H. 1992. Kirchhoffs theory of rods. Arch. Hist. Exact. Sci. , 44 , 223.

Goriely, A., & Tabor, M. 1996. New amplitude equations for thin elastic rods. Phys. Rev. Lett. ,77 , 35373540.

Goriely, A., & Tabor, M. 1997a. Nonlinear dynamics of laments I: Dynamical instabilities.Physica D , 105 , 2044.

Goriely, A., & Tabor, M. 1997c. Nonlinear dynamics of laments II: Nonlinear Analysis. Physica D, 105 , 4561.

Goriely, A., & Tabor, M. 1997b. Nonlinear dynamics of laments III: Instabilities of helical rods.Proc. Roy. Soc. London (A) .

Love, A. E. H. 1892. A treatise on the mathematical theory of elasticity . Cambridge UniversityPress, Cambridge.

Mendelson, N. H. 1978. Helical Bacillus subtilis macrobers: morphogenesis of a bacterialmulticellular macroorganism. Proc. Natl. Acad. Sci. USA , 75 , 24722482.

Mendelson, N. H. 1990. Bacterial macrobers: the morphogenesis of complex multicellular bacterialforms. Sci. Progress Oxford , 74 , 425441.

Pomeau, Y. 1981. Nonlinear pattern selection in a problem of elasticity. J. Physique Lettres , 42 ,L1L4.

Ricca, R. L. 1995. The energy spectrum of a twisted exible string under elastic relaxation. J. Phys.A, 28 , 23352352.

Thompson, J. M. T., & Champneys, A. R. 1996. From helix to localized writhing in the torsionalpost-buckling of elastic rods. Proc. Roy. Soc. London A , 452 , 117138.

Timoshenko, S. P., & Gere, J. M. 1961. Theory of elastic stability . Mc Graw-Hill, New York.

Tvergaard, V., & Needleman, A. 1980. On the localization of buckling patterns. J. Appl. Mech. ,47 , 613619.

Documents

Alain Goriely and Michael Tabor- Nonlinear Dynamics of Filaments IV: Spontaneous Looping of Twisted Elastic Rods