kirchhoff's problem of helical solutions of uniform rods and their
TRANSCRIPT
KIRCHHOFF’S PROBLEM OF HELICALSOLUTIONS OF UNIFORM RODS AND
THEIR STABILITY PROPERTIES
THESE N◦ 2717 (2003)
PRESENTEE A LA FACULTE SCIENCES DE BASE
SECTION DE MATHEMATIQUES
ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
POUR L’OBTENTION DU GRADE DE DOCTEUR ES SCIENCES
PAR
Nadia CHOUAIEB
Master of Science, University of New Jersey, Etats-Unis
de nationalite Tunisienne
acceptee sur proposition du jury:
Prof. John H. Maddocks, directeur de these
Prof. Stuart S. Antman, rapporteur
Prof. Alan Champneys, rapporteur
Prof. Tudor S. Ratiu, rapporteur
Prof. Gerhard Wanner, rapporteur
Lausanne, EPFL
2003
Abstract
It is shown that a uniform and hyperelastic, but otherwise arbitrary, non-
linear Cosserat rod has helices as the centerline of equilibrium configurations.
For anisotropic rods, and for each of the local two-parameter family of helical
centerlines corresponding to changes in the radius and pitch, there are a dis-
crete number, greater than or equal to two, of possible orientations of the cross-
section at equilibrium. The possible orientations are characterized by a pair of
finite-dimensional, dual variational principles involving point-wise values of the
strain-energy density and its conjugate function. For sufficiently short helical
segments, members of the two parameter family in this variational principle are
stable in the sense that they are local minima of the total elastic energy for the
corresponding boundary value problem (with, for example, clamped-clamped or
Dirichlet boundary conditions). For isotropic rods, the characterization of possi-
ble equilibrium configurations degenerates, and in replacement of multiple, two-
parameter families of helical equilibria, a single four-parameter family arises. The
four parameters correspond to the two of the helical centerlines, a one-parameter
family of possible cross-section orientations, and a one-parameter family of im-
posed twists. Stability properties can again be analyzed, and sufficiently short
helical segment with small enough twist are stable. The mathematical tools used
in the analysis are the non-canonical Hamiltonian formulation of the equilibrium
conditions, Euler-Poincare equations for both first and second variations of the
corresponding action, and conjugate point tests for isoperimetrically constrained
calculus of variations problems.
i
Version abregee
Dans cette these, nous montrons que toute tige de Cosserat, hyperelastique,
uniforme, mais de facon generale non lineaire, possede une famille de configura-
tions d’equilibre ayant un axe central helicoıdal. Dans le cas anisotrope, la tige
est identifiee par les parametres physiques de l’helice a savoir la torsion et la cour-
bure. Dans le cas isotrope, la caracterisation des configurations d’equilibre est
donnee par 4 parametres: deux parametres pour fixer l’axe helicoıdal, un troisieme
determine l’orientation de la section et un quatrieme pour exprimer la torsion
physique (twist) imposee. Les differentes orientations sont definies par deux
principes variationnels duaux appliques ponctuellement a la densite d’energie de
deformation et a sa conjuguee. Pour les systemes avec des conditions aux bords
prescrites (par exemple condition d’encastrement des deux extremites ou con-
ditions de Dirichlet) et pour une densite d’energie quadratique, nous montrons
que chaque configuration (helicoıdale) peut etre stable a condition de considerer
des tiges suffisamment courtes et avec suffisamment peu de torsion physique. Le
cadre mathematique general est celui des systemes hamiltoniens non canoniques.
Nous utilisons les equations d’Euler-Poincare pour les variations premiere et sec-
onde de la densite d’action, ainsi que la methode du point conjugue comme test
de stabilite du probleme variationel sans ou avec contraintes isoperimetriques.
ii
Dedication
This thesis is dedicated to my entire extended family with special thoughts to
my parents and my children Mohamed, Manel and yassin.
iii
iv
Acknowledgements
It is difficult to overstate my gratitude to my Ph.D. supervisor, Professor John
Maddocks. With his enthusiasm, his inspiration, and his great efforts to explain
things clearly and simply, I got the courage to finish this work. Throughout my
thesis-writing period, he provided encouragement, sound advice, good teaching,
and lots of good ideas. Professors Bernard Dacorogna, Stuart Antman, Alan
Champneys, Tudor Ratiu and Gerhard Wanner agreed to serve on my dissertation
committee. I wish to thank them all.
Special thanks go to my friend and loving husband, Maher Moakher who
has been a great source of strength all through this work, for proofreading the
manuscript and for the patience he had with me in the last months.
I am grateful to the past and present members of the Laboratory for Com-
putation and Visualization in Mathematics and Mechanics for their help and for
the nice scientific and social environments.
Lastly, and most importantly, I wish to thank my parents, Aziza Chouaieb
and Mohamed Chouaieb. They raised me, taught me, loved me and supported
me.
v
vi
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Version abregee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Space Curves, Adapted Frames and Helices . . . . . . . . . . . . 7
2.1. Frenet-Serret Frame of a Space Curve . . . . . . . . . . . . . . . . 7
2.2. Twist of a Framed Curve . . . . . . . . . . . . . . . . . . . . . . . 8
2.3. Adapted Frames of a Space Curve . . . . . . . . . . . . . . . . . . 9
2.4. Circular Helices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3. Cosserat Theory of Elastic Rods . . . . . . . . . . . . . . . . . . . 15
3.1. Configuration of a Cosserat Rod . . . . . . . . . . . . . . . . . . . 15
3.2. Kinematic Equations . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3. Coordinate-Free Equilibrium Equations . . . . . . . . . . . . . . . 17
3.4. Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4.1. Inextensible and Unshearable Rods . . . . . . . . . . . . . 20
3.4.2. Isotropic Rods . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.3. Uniform Rods . . . . . . . . . . . . . . . . . . . . . . . . . 21
vii
3.5. The Kirchhoff Kinetic Analogy . . . . . . . . . . . . . . . . . . . 22
4. A Hamiltonian Formulation for Equilibria of Hyperelastic Rods 25
4.1. Poisson Description of a Hamiltonian System . . . . . . . . . . . . 25
4.2. Relative Equilibria of a Hamiltonian System . . . . . . . . . . . . 28
4.2.1. Variational Characterization . . . . . . . . . . . . . . . . . 28
4.2.2. Example: Motion of a Heavy Rigid Body . . . . . . . . . . 29
4.3. Non-Canonical Formulation for Equilibria of Hyperelastic Rods . 31
4.4. Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5. Analysis of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 34
5. Relative Equilibria of Uniform, Hyperelastic Rods . . . . . . . . 35
5.1. Variational Characterization . . . . . . . . . . . . . . . . . . . . . 36
5.1.1. Closed-Form Expression for Relative Equilibria . . . . . . 37
5.1.2. Characterization of Relative Equilibria for Uniform Rods . 39
5.2. Recovering the Centerline of the Rod . . . . . . . . . . . . . . . . 41
5.2.1. Geometric Characterization . . . . . . . . . . . . . . . . . 46
5.2.2. Formulation of the Problem in Terms of the Stresses . . . 47
5.2.3. Dual Formulation of the Problem in Terms of the Strains . 47
5.2.4. Existence and Multiplicity . . . . . . . . . . . . . . . . . . 48
5.3. Example: Diagonal Quadratic Strain Energy . . . . . . . . . . . . 55
5.3.1. Inextensible and Unshearable Rod . . . . . . . . . . . . . . 56
5.3.2. Extensible and Shearable Rod . . . . . . . . . . . . . . . . 63
6. An Intrinsic Variational Formulation of the Non-canonical Ac-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.1. Extensible and Shearable Rods . . . . . . . . . . . . . . . . . . . 70
6.1.1. Derivation of the First Variation . . . . . . . . . . . . . . . 71
6.1.2. Derivation of the Second Variation . . . . . . . . . . . . . 74
viii
6.1.3. Change of Frame . . . . . . . . . . . . . . . . . . . . . . . 75
6.2. Inextensible and Unshearable Rods . . . . . . . . . . . . . . . . . 77
6.2.1. First Variation . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2.2. Derivation of the Second Variation and Linearized Con-
straints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7. Stability Analysis of Relative Equilibria . . . . . . . . . . . . . . 81
7.1. The Second Variation at a Relative Equilibrium . . . . . . . . . . 85
7.1.1. Extensible and Shearable Rod . . . . . . . . . . . . . . . . 85
7.1.2. Inextensible and Unshearable Rod . . . . . . . . . . . . . . 86
8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
ix
x
List of Figures
2.1. Right- and left-handed circular helices . . . . . . . . . . . . . . . 13
3.1. The Cosserat ribbon representation of the configuration of a rod . 17
5.1. Diagram in the (η1, η2)-parameter space . . . . . . . . . . . . . . . 43
5.2. Constrained extrema of an isotropic and quadratic strain energy . 50
5.3. Constrained extrema of an anisotropic and quadratic strain energy,
straight intrinsic shape (u1, u2) = (0, 0) . . . . . . . . . . . . . . . 51
5.4. Constrained extrema of an anisotropic and quadratic strain energy,
with (u1, u2) = (0.5, 0.2) . . . . . . . . . . . . . . . . . . . . . . . 51
5.5. Constrained extrema of an anisotropic and quadratic strain energy,
with (u1, u2) = (1, 0.8) . . . . . . . . . . . . . . . . . . . . . . . . 52
5.6. Constrained extrema of an isotropic and quadratic strain energy . 53
5.7. Anisotropic helices . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.8. Sheets of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.9. Isotropic helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.1. The determinant of the matrix M as a function of σ . . . . . . . 92
xi
xii
List of Tables
5.1. Solution for a diagonal quadratic strain energy . . . . . . . . . . . 67
xiii
xiv
1
Chapter 1
Introduction
The study of equilibrium configurations of elastic rods and their stability proper-
ties has a long history. The theory of pure bending of an elastic rod was initiated
by James Bernoulli (1705) (see [34, p. 2]) and was then followed by the work
of Euler (1727) and Daniel Bernoulli (1732) on the elastica (see [2, 34, 50]). In
1859, Kirchhoff [31] generalized the planar elastica theory to three-dimensional
rods. This theoretical problem has practical applications in different fields such
as structural mechanics, civil engineering, biochemistry and biology. One exam-
ple is the increasing interest in recent years to study the equilibrium structures
of polymers, bacterial fibers, and the supercoiled structure of DNA by using the
elastic rod as an idealized macroscopic model [20, 41].
Since the seminal work of the Cosserats [9], more sophisticated rod theories
that describe deformations of shear and extension, in addition to bending and
twisting, have been introduced. Of particular concern here is the special Cosserat
theory [2] which considers the rod as a curve with a triad of orthonormal directors
attached to each of its points. The addition of directors, which may be interpreted
as particular material directions associated with each cross section of the rod,
allows for the modeling of shearing, extension, bending and twisting effects. For
a detailed historical account of rod theories the reader is referred to [2, 13, 34, 50].
Kirchhoff observed that the equations that describe an inextensible, unshear-
able and uniform elastic rod in equilibrium are mathematically identical to Euler’s
equations describing the dynamics of a heavy top [34], in what is now known as
2
Kirchhoff’s kinetic analogy. The equilibrium configurations of an inextensible, un-
shearable and uniform rod are thus intimately related to the dynamics of spinning
tops. Using this analogy, Kirchhoff showed that an initially straight, inextensible,
unshearable, isotropic, and uniform rod admits helical solutions under the action
of appropriate forces and torques applied at its ends [34]. Kirchhoff’s theory rests
upon the constitutive assumption that the stress couple depends linearly upon
the curvature and the twist.
In 1974, using the director rod theory of the Cosserats, Antman [1], and
independently Whitman & DeSilva [55], generalized Kirchhoff’s problem of he-
lical rods to the case of an extensible and shearable, but still initially straight
and uniform, elastic rod. While Whitman and DeSilva used linear and isotropic
constitutive relations, Antman considered an isotropic but nonlinear constitu-
tive equations. Both studies found that the stretch is constant along the rod, and
that, similar to the classic Kirchhoff problem, the end forces and moments needed
to maintain the helical deformation are statically equivalent to a wrench acting
along the helical axis. For a general class of hyperelastic materials, Ericksen [15]
showed that certain invariance requirements characterizing a “uniform state” im-
ply that the solutions of the rod problem must be helical, but, as he observes, “it
is impossible to say much about the existence or multiplicity of these solutions
without introducing some assumptions concerning the form of the strain-energy
density function” [15, p. 376].
In this dissertation we reconsider Kirchhoff’s problem of helical solutions of
elastic rods using less restrictive assumptions on the constitutive relations. We
treat the case of a general extensible and shearable, but hyperelastic rod that is
uniform, accounting for both anisotropy due to stiffnesses and due to intrinsic
curvature. We also treat existence and multiplicity questions. As discussed in
Chapter 5 the main tool we use is the observation that relative equilibria of a non-
canonical Hamiltonian formulation of the equations governing equilibria of elastic
3
rods (in which arc-length is the time-like variable) are helices. By using variables
written in the rod director frame, the integrals of the Hamiltonian are quadratic
in the phase variables, which gives rise to two dual variational characterizations of
relative equilibria, one in terms of the strains and the other in terms of the stresses.
While both characterizations yield that the centerline is a helix, the formulation
in term of the strains permits a direct analysis of existence and multiplicity that is
particularly simple in the inextensible and unshearable case. We show that either
a discrete number of two parameter families or a single four parameter family of
equilibria arise depending on whether the rod is anisotropic or isotropic. The
parameters are intimately related to the curvature and the geometric torsion of
the helical centerline, and in the isotropic case, to the imposed twists and to the
cross-section orientation.
Once equilibrium configurations are found, it is also of interest to analyze their
stability properties as equilibria of the boundary value problem for the elastic rod
subject to prescribed boundary conditions. The study of stability of elastic rods
is a classic subject that started with the pioneering work of Euler (1744) on the
buckling of a straight rod under a vertical load. The literature on this subject
is very rich, and we shall cite only those articles that are directly related to the
present study.
Thompson & Champneys [49] studied the instability of a stretched and twisted
straight infinite rod subject to an axial force and a twisting moment. A first
bifurcation occurs and the rod deforms into a helix. Upon further increase of the
terminal loads, a secondary bifurcation is reached and the helical configuration
starts to deform locally.
Goriely and co-authers, in a series of papers [21, 22, 23, 24, 25] also considered
helical equilibria of infinite rods. They studied dynamics in both the isotropic
and anisotropic cases. Using a perturbation scheme involving linearized dynamics,
they gave various stability criteria.
4
For a finite length rod, Neukirch & Henderson [45] classified all possible equi-
librium configurations of a twisted elastic inextensible, unshearable, isotropic and
uniform rod under applied end loads and clamped boundary conditions. They
showed that helical configurations are in the set of solutions. They also discussed
the effect of the twist in the rod on the possible equilibria.
Recently, Manning & Hoffman [39] studied the stability of untwisted circular
equilibria for an intrinsically curved, inextensible, unshearable, anisotropic elastic
rods. In their study of the second variation, they used Maddocks formula [36]
giving the constrained stability index in terms of the unconstrained one. They
considered cases in which the stability index is determined explicitly in terms of
the rod parameters. For the buckling problem of a twisted elastic strut, Hoffman,
Manning, & Paffenroth [27] use the conjugate point theory to analyze the stability
of the strut. In this work the conjugate points are determined numerically.
In this thesis, we analyze the stability of helical equilibria with finite length
subject to clamped-clamped boundary conditions. We describe the rod as stable
if the second variation is non negative amongst variations satisfying the linearized
boundary conditions and constraints.
We show that the particular geometric characterization of relative equilibria
yields a self-adjoint operator associated with the second variation, evaluated at
relative equilibrium that has constant coefficients. In principle, this feature allows
explicit analysis of the parameter-dependent unconstrained or isoperimetrically
constrained variational problem. As an application we analyze the stability of
the particular case of linear constitutive relations.
We adopt the conjugate point method to analyze stability. For unconstrained
calculus of variations problems, the absence of a conjugate point on an extremal
(Jacobi’s condition) is a well-known necessary condition for the extremal to be a
local minimum [19]. For the constrained problem, Bolza [6] proposed a definition
5
of constrained conjugate points and derived a test analogous to Jacobi’s (uncon-
strained) condition for a constrained local minimum based on the absence of any
isoperimetric conjugate point.
Following the particular approach developed in [39] for minicircles, we apply
the general conjugate point method to analyze the second variation evaluated on
helical solution for a given set of parameters. Finding conjugate points in these
way involves finding roots of a certain determinant. The other approach that
is used here is to find conjugate points via a direct bifurcation analysis on the
linearized equilibrium equations. We identify the first bifurcation point with the
limiting length below which the rod is stable.
The thesis is organized as follows. In Chapter 2 we briefly review the differen-
tial geometry of space curves and discuss the twist of framed curves. We also give
some characterizations of circular helices. Chapter 3 contains a description of the
special Cosserat theory of rods. We discuss the kinematics and strain measures,
and give the equilibrium equations in coordinate-free form. We then review the
basic constitutive relations and constraints to be used in our analysis. Chapter 4
is devoted to the Hamiltonian formulation of equations governing equilibria of a
rod, and an analysis of the associated integrals. In Chapter 5 we give a complete
classification of the families of helical equilibria of rods as relative equilibria of
the associated Hamiltonian system. We discuss existence and multiplicity of rel-
ative equilibria of uniform rods, and illustrate the general theory with the study
of a particular case involving a general quadratic strain energy (corresponding to
linear stress-strain relations).
In Chapter 6 we use Euler-Poincare equations to derive the equilibrium equa-
tions for Cosserat rods from Hamilton’s principle without the need to param-
eterize the rotation group. We then further use the Euler-Poincare apparatus
to derive the appropriate second variation. The final results presented comprise
the system of constant coefficient, second-order ordinary differential equations
6
which in principle can be solved analytically to yield an analysis of the stability
properties of helical solution for a large class of strain-energy density functions.
Nevertheless, the presence of many parameters and a large system of equations,
in the case of an extensible and shearable rod, or with the presence of the con-
straints in the inextensible and unshearable case, make the possibility of solving
the problem difficult. For a particular set of parameters, the problem simplifies
and the bifurcation analysis shows that for an integer number of turns of the
helical axis, the first bifurcation point arises at a particular relation between the
twist in the rod, and the torsion and the curvature of the helical axis.
7
Chapter 2
Space Curves, Adapted Frames and Helices
In this introductory chapter we gather background material on space curves and
framings that are needed in our study of helical configurations within the Cosserat
rod theory.
2.1 Frenet-Serret Frame of a Space Curve
A curve in space can be defined by a continuously differentiable vector-valued
function
r : I ⊂ R→ E3,
which maps some open interval I ∈ R into Euclidean 3-space E3. At each s ∈ I,
the vector r(s) gives the position vector from the origin O of the Euclidean space
E3 to the point of the curve specified by s. We assume that the curve r is a
regular curve, i.e., r′(s) 6= 0 for every s ∈ I. Furthermore, we assume, after
reparameterization if necessary, that the parameter s is an arc-length along the
curve. At any s ∈ I, the tangent to the curve is the vector τ = r′(s). Since s is
arc-length, τ is a unit vector. The curvature of the curve r(s) is the non-negative
scalar-valued function κ(s) defined by
τ ′ = r′′(s) = κν, (2.1)
where ν is a unit vector perpendicular to the tangent τ called the principal normal
to the curve. (Note that when τ ′(s) = 0, the curvature is necessarily zero, but
the principal normal is not well defined.) The binormal vector β is defined so
8
that {τ ,ν,β} is a right-handed orthogonal triad of unit vectors, i.e., β = τ × ν.
The triad {τ ,ν,β} 1, which is called the Frenet triad, defines an orthonormal
frame at every point s ∈ I along the the curve.
Differentiation of the relations β · β = 1 and β · τ = 0 yields β′ · β = 0 and
β′ · τ = 0, so that β′ must be in the direction of ν. We therefore introduce the
scalar-valued function τ(s) such that
β′ = −τν. (2.2)
This function, which is called the (geometric) torsion of the curve, gives the rate
of rotation of the principal normal about the tangent. Now differentiation of the
relations ν · ν = 1, ν · τ = 0 and ν · β = 0 implies
ν ′ = −κτ + τβ. (2.3)
The equations (2.1)–(2.3) giving the evolution in arc-length s of τ , ν and β along
the curve are called the Frenet-Serret equations. They can be written in compact
form as ν
β
τ
′
=
0 τ −κ
−τ 0 0
κ 0 0
ν
β
τ
, (2.4)
or as,
[ν β τ
]′=[ν β τ
]0 −τ κ
τ 0 0
−κ 0 0
. (2.5)
2.2 Twist of a Framed Curve
In order to discuss the (physical) twist of a rod, it is necessary to add to the
space curve r(s) an additional structure that will describe the orientation of the
1We here use the Greek {τ ,ν,β} to denote the Frenet frame instead of the more standardnotation {t,n, b} because in this dissertation we reserve the symbol n to denote the net forceacting across a cross-section of a Cosserat rod.
9
material points in the cross-section at s. To do that we introduce the notion of a
framed curve.
A framed curve is a space curve r(s) together with right-handed orthonormal
frame {d1(s),d2(s),d3(s)} that specifies an orientation at each s. Often, the
vector d3(s) is taken to be the tangent vector τ to the curve, the vector d1(s)
specifies a particular direction in the normal plane, and the third vector d2(s)
is defined such that {d1(s),d2(s),d3(s)} is orthonormal and right handed. But
here we will not necessary assume that d3(s) = τ .
As s varies, the orientation of the moving frame {d1(s),d2(s),d3(s)} changes
smoothly relative to a fixed frame {e1, e2, e3}, and the change can be expressed as
a three-dimensional rotation. The rate of rotation can be represented by a vector-
valued function u(s) called the Darboux vector. The evolution of the frame along
the curve is governed by the following differential equations
d′i = u× di, i = 1, 2, 3, or
d1
d2
d3
′
=
0 u3 −u2
−u3 0 u1
u2 −u1 0
d1
d2
d3
,where ui = u ·di, i = 1, 2, 3. The component u3 is called the twist of the material
curve about d3, which is, in general, different from the geometric torsion τ . In
the next section we will relate these two different notions.
2.3 Adapted Frames of a Space Curve
Given a space curve r(s), we call any right-handed orthonormal frame
{d1(s),d2(s),d3(s)} such that d3(s) = τ (s) an adapted frame. Any other adapted
frame {g1(s), g2(s), g3(s)} can be obtained from the frame {d1(s),d2(s),d3(s)}
by a rotation through an angle ϕ(s) about the vector d3(s), i.e.,
G = QD,
10
where
G = [g1 g2 g3], D = [d1 d2 d3],
and
Q =
cosϕ(s) − sinϕ(s) 0
sinϕ(s) cosϕ(s) 0
0 0 1
.Let u(s) be the Darboux vector associated with the frame {d1(s),d2(s),d3(s)},
such that
d′i = u× di, i = 1, 2, 3.
Then the Darboux vector w associated with the frame {g1(s), g2(s), g3(s)}, i.e.,
such that
g′i = w × gi, i = 1, 2, 3,
is related to u by
w = Qu+ ϕ′d3.
We note that u21 +u2
2 = w21 +w2
2 and w3−u3 = ϕ′, where ui = u·di and wi = w ·gifor i = 1, 2, 3.
The Frenet-Serret frame is a particular adapted frame with Darboux vector
w = κβ + ττ . Most importantly for our purposes we have that
u21 + u2
2 = κ2, and u3 = τ − ϕ′. (2.6)
2.4 Circular Helices
A circular helix is a curve on the surface of a circular cylinder that cuts lines
parallel to the axis at a constant angle. By cutting the cylinder along one of its
11
generators and flattening it out, the curve becomes a straight line. The shortest
path between two points on a cylinder not on the same generator is a fractional
turn of a helix. It is for this reason that squirrels chasing one another up and
around tree trunks follow helical paths.
Helices are fundamental curves that are ubiquitous in nature and in man-made
objects. They can be found on many different scales ranging from nanostructures,
such as α-helices in proteins, the DNA double helix, the collagen triple helix, and
carbon nano-tubes, to large structures such as horns of sheep, tendrils of plants,
screws, springs and helical staircases.
Pauling (see [18]) argued that the only configurations that can be adopted
by molecular chains compatible with equivalence are helical ones. Crane [10]
wrote “any structure that is straight or rod like is probably a structure having a
repetition along a screw axis”.
Here we quote the following statement by Galloway [18]: “The helix has a
strong claim to be nature’s favorite shape. It is adopted in the living world
at every anatomical and physiological level and exists as an almost universal
structural form”.
Indeed, helical shapes are special in the context of elastic rod theory in the
sense that, as we show later, they are relative equilibria of the Hamiltonian
system describing statics. Helical curves are the simplest non-trivial curves in
three-dimensional space. In the remainder of this chapter we review the basic
differential geometric facts of helical curves.
A helix has parametric equations
x = r cosσ, (2.7a)
y = r sinσ, (2.7b)
z = p σ. (2.7c)
where r is the radius of the helix and p is a constant giving the pitch, i.e., the
12
vertical separation along a generator of the helix loops. Arc-length is given by
ds =√r2 + p2 dσ.
The curvature of the helix is
κ =r
r2 + p2,
and the locus of the centers of curvature of the helix is another helix. The torsion
of a helix is given by
τ =p
r2 + p2.
When the pitch p goes to zero with constant radius the helix degenerates to a cir-
cle, and when the radius r goes to zero with constant pitch, the helix degenerates
to a straight line.
The last two relations can be inverted to give the radius and pitch as functions
of curvature and torsion
r =κ
κ2 + τ 2,
and
p =τ
κ2 + τ 2.
The pitch angle φ is the angle between the tangent to the helix and the generator,
it is also the angle between tangent to the helix and the Darboux vector of the
Frenet-Serret frame, and is given by
tanφ =κ
τ=r
p.
A circular helix can be right- or left-handed. A helix and its mirror image
have the same curvature, but opposite torsion and handedness (see Figure 2.1).
13
Figure 2.1: Tube visualization of two circular helices with the same curvature,κ = 1, but of opposite sign torsions τ = ±1
2π.
14
15
Chapter 3
Cosserat Theory of Elastic Rods
In this chapter we summarize the theory of elastic rods introduced by the Cosserat
brothers [9]. In the special Cosserat rod theory, the rod is allowed to experience
flexure, torsion, axial extension, and shear of cross sections with respect to the
axis. It is a generalization of Kirchhoff’s rod theory [31] for which the cross
sections are constrained to remain normal to the axis of the rod and the rod can
suffer neither axial extension nor shear. The latter is in turn a generalization of
the planar elastica theory of Euler & Bernoulli [16]. The Cosserat theory rests
upon the concept of a framed or directed curve, which is a curve together with
an orthonormal moving frame attached to each of its points. Their work was
forgotten for almost fifty years. In 1958 Ericksen and Truesdell [14] revived the
theory and complemented it by nonlinear constitutive equations. Thereafter a
considerable literature on Cosserat rods has appeared; for a historical survey and
comprehensive modern treatment see [2, Chap. VIII, IX].
3.1 Configuration of a Cosserat Rod
The configuration of a Cosserat rod is a smooth vector function r and a pair of
orthonormal vector functions d1, d2 of the single variable s with 0 ≤ s ≤ L. The
vector r(s) gives the position in Euclidean 3-space E3 of a material point on the
rod. The curve {r(s), s ∈ [0, L]} is called the centerline of the rod, and may (or
may not) be interpreted as the line of centroids of the cross sections of a slender
three-dimensional body in its deformed configuration. The material parameter
16
s is usually taken to be the arc-length parameter of the line of centroids of the
cross sections in its reference configuration. The unit vectors d1(s) and d2(s)
give information about the orientation of the material cross section at s in the
deformed configuration. Usually, d1(s) and d2(s) are taken to be the principal
directions of the material cross section. If we define the additional vector function
d3 by
d3(s) = d1(s)× d2(s),
we obtain at each s an orthonormal frame {d1(s),d2(s),d3(s)}. The unit vectors
di(s), i = 1, 2, 3 are called directors.
3.2 Kinematic Equations
The kinematics of the rod are described by two strain vectors v and u through
the relations
r′(s) = v(s), (3.1a)
d′i(s) = u(s)× di(s), i = 1, 2, 3, (3.1b)
where ′ denotes differentiation with respect to the independent variable s. The
components vk = v ·dk of the vector v with respect to the orthonormal basis {dk}
are the strain variables corresponding to the axial curve: v1 and v2 are associated
with transverse shearing and v3 is associated with stretching or compression.
The components uk = u · dk of the Darboux vector u are the strain variables
corresponding to the directors: u1 and u2 are associated with bending, and u3 is
associated with twisting.
17
Figure 3.1: A ribbon representation of the configuration of a rod. The centercurve is parameterized by undeformed arc-length s. The location of the materialpoint at s is given by its position r(s) with respect to the origin. The directorsd1, d2 and d3 describe the orientation of the cross-section of the rod at s, withd1 determining the ribbon. The components of r′ with respect to the di basisdetermine the strains of shear and extension.
3.3 Coordinate-Free Equilibrium Equations
The stresses acting across the material cross section at s are equivalent to a
resultant force n(s) and a resultant moment m(s) applied at the centroid r(s)
of this cross section. When the only external loads are couples and forces applied
at the ends of the rod, which is the case of interest here, balance of forces and
moments yields the equations
n′(s) = 0, (3.2a)
m′(s) + r′(s)× n(s) = 0. (3.2b)
Equation (3.2a) can be integrated
n(s) = n(0). (3.3)
18
Then, because n is constant, equation (3.2b) has the integral
m(s) + r(s)× n(s) = c, (3.4)
where c is a constant vector. If we take s = L in (3.3) and (3.4), relations between
the boundary conditions are obtained.
Equation (3.4) implies an integral that is independent of r which can be
obtained from the scalar product with n, namely
m · n = C1. (3.5)
3.4 Constitutive Relations
In order to complete the formulation the stresses n and m must be related to
the strains v and u via constitutive relations.
In this dissertation we assume the rod to be hyperelastic. That is there exists
a strain-energy density function W defined on a domain V(s) in R7 (see [2, p.
277])
W : V(s) → R+
(z,w, s) 7→ W (z,w, s)
such that
∂W
∂w(0, 0, s) = 0,
∂W
∂z(0, 0, s) = 0. (3.6)
The triples n and m of components of the resultant forces n and moments
m in the moving frame {d1,d2,d3} are then related to the triples u and v of
components of the strains in the same basis via the constitutive relations
m =∂W
∂w(u− u, v− v, s), (3.7a)
n =∂W
∂z(u− u, v− v, s), (3.7b)
19
where u(s) and v(s) are the strains in the unstressed reference configuration.
When s is arc-length in the reference (unstressed) configuration, one can take
v(s) = (0, 0, 1). If the rod is straight and untwisted in the reference configuration
then u(s) = (0, 0, 0). No particular strain energy density need be specified here,
but we assume that the function W is as often continuously differentiable as is
needed in the analysis. In addition, we require that it is both convex and coercive.
The function W is said to be coercive if
W (w, z)√|w|2 + |z|2
→∞ as |w|2 + |z|2 →∞.
To ensure that configurations preserve orientation i.e., a rod of positive length
cannot be compressed to zero length, and that a cross section can never be sheared
too severely, we require that
v3 = v · d3 > 0,
be satisfied at any configuration of the rod (see [1] and [48] for a detailed analysis
of this condition).
Because W is convex and coercive, the constitutive relations (3.7) can be
inverted to yield:
u =∂W ∗
∂m(m,n, s) + u, (3.8a)
v =∂W ∗
∂n(m,n, s) + v, (3.8b)
where W ∗(m,n, s) is the Legendre transform of W (w, z, s), i.e.,
W ∗(y) = supx{y · x−W (x)}. (3.9)
(See [19] for details on the Legendre transform.)
The main results of this dissertation use the general hyperelastic constitutive
equations (3.7) with no need to specify the strain-energy density W explicitly.
Detailed computations are given for some particular quadratic strain-energy den-
sity functions.
20
3.4.1 Inextensible and Unshearable Rods
The rod is said to be inextensible if
|r′| = 1,
and to be unshearable if
v1 = v · d1 = 0, v2 = v · d2 = 0.
Consequently, for an inextensible and unshearable rod, we rewrite equation (3.1a)
simply as
r′(s) = d3(s). (3.10)
In this case, the parameter s can be interpreted as the arc-length in any config-
uration.
For an inextensible and unshearable rod, the strain-energy density W is a
function of the strains u only. In such a case, only the moment m is given by a
constitutive relation
m =∂W
∂w(u− u, s). (3.11)
The force becomes a reactive parameter that is to be determined from the equi-
librium equations.
In the case where the strain-energy density W is quadratic in the strains u,
we obtain a linear constitutive relation of the form
m = K(s)(u− u), (3.12)
where K is a symmetric positive-definite matrix. When K is a constant, diagonal
matrix with equal bending rigidities, we recover the classic Kirchhoff theory of
elastic rods.
21
3.4.2 Isotropic Rods
In continuum mechanics, a material is called isotropic if it has no preferred ma-
terial directions. If the response of the material is the same for all directions
orthogonal to a given one, then the material is called transversely isotropic. In
the theory of Cosserat rods, the direction defined by vidi is distinct from others,
so that only transverse isotropy to vidi is a pertinent notion. In the following, we
suppress the adjective transversally and discuss isotropic or non isotropic rods.
For hyper-elastic rods, the strain-energy function W (w, z, s) will be called
isotropic if it is invariant under rotations about the axis defined by vidi. Under
the assumption that the shear strains v in the reference configuration are zero,
i.e., such that v = (0, 0, v3), the strain-energy density W (w, z, s) is isotropic if it
is invariant under rotations about the d3 axis, or, equivalently,
∂
∂αW (Q(α)w,Q(α)z, s) = 0, (3.13)
for all 0 ≤ α ≤ 2π, and w, z ∈ R3 with
Q(α) =
cosα sinα 0
− sinα cosα 0
0 0 1
. (3.14)
A rod is isotropic if (3.13) holds and if in addition, the bending strains (u1 =
u2 = 0) and the shear strains (v1 = v2 = 0) in the reference configuration vanish
[2].
In his work on symmetries of rods, Healey [26] defines isotropic strain-energy
functions to be ones that are invariant under both proper and improper rotations
(i.e., orthogonal transformations). He uses the terminology hemitropy to refer to
what is called isotropy here.
3.4.3 Uniform Rods
A Cosserat rod is called uniform if [2]:
22
• The material properties do not change along its length.
• The set of strains in the reference configuration, u, v are constant indepen-
dent of the arc-length s.
These assumptions are satisfied for a hyperelastic rod when the strain-energy
density has no explicit dependence on s and u, v are constant.
3.5 The Kirchhoff Kinetic Analogy
The classic form of the Kirchhoff kinetic analogy is based on an inextensible and
unshearable rod that obeys the linear constitutive relation
m = Ku, (3.15)
where
m =
m1
m2
m3
, u =
u1
u2
u3
, K =
K1 0 0
0 K2 0
0 0 K3
.Here mi(s) are the components of the net moment with respect to the local frame
{d1(s),d2(s),d3(s)} which is attached to the rod at arc-length s; u1(s) and u2(s)
are the components of the curvature vector in the directions of d1 and d2; u3(s)
is the twist of the rod; K1 and K2 are the two bending rigidities in the directions
of d1 and d2, and K3 is the twisting rigidity. The equilibrium equations of the
rod written in the local frame are
m′ = u×m + k× n, (3.16a)
n′ = u× n, (3.16b)
where
n =
n1
n2
n3
, k =
0
0
1
,
23
ni are the components of the net force with respect to the local frame.
When the frame {di} is interpreted as a frame attached to a rigid body, s as
the time, K as the inertia matrix of the rigid body, n as the gravitational field,
m as the angular momentum of the body, u as the angular velocity, and k as the
coordinates in the body frame of the center of mass, the equilibrium equation
(3.16) are equivalent to the classic equations of motion of a heavy rigid body in a
uniform gravitational field tumbling about a fixed point. Two important special
cases in which the problem is completely integrable have been studied extensively.
A rigid body that is free to move with no external forces acting on it, and the case
of a symmetric (Lagrange) top. The last case, for which K1 = K2, corresponds
to an inextensible, unshearable, isotropic, and uniform Kirchhoff rod.
This kinetic analogy has been extended to rods naturally curved, but uniform,
by Larmor (1884) [34, 56]. The kinetic analogue in this case is a rigid body
spinning about a fixed point and carrying a flywheel rotating about an axis in
the body.
24
25
Chapter 4
A Hamiltonian Formulation for Equilibria of
Hyperelastic Rods
In this chapter we give the necessary background on finite-dimensional Hamil-
tonian systems that we will use in this dissertation. First, we give the Pois-
son description of a Hamiltonian system in terms of a Poisson bracket both in
coordinate-free form, and in terms of local coordinates. All definitions and results
presented here are standard, and can be found for example in Olver [46] or in
Marsden & Ratiu [42]. We then describe a non-canonical Hamiltonian formula-
tion for the statics of Cosserat elastic rods due to Kehrbaum and Maddocks [30].
Following Arnold [4], we use a reduced form of this Hamiltonian formulation.
4.1 Poisson Description of a Hamiltonian System
Let M be a smooth m-dimensional manifold called phase space. Often in appli-
cations M = R2n.
Definition 4.1.1 A Poisson bracket on M is an operator that assigns to each
pair of smooth real-valued functions F and G on M, another smooth real-valued
function {F,G} satisfying the following properties:
(i) Bilinearity
{aF + bG,H} = a{F,H}+ b{G,H},
for any constants a, b ∈ R,
26
(ii) Skew-Symmetry
{F,G} = −{G,F},
(iii) Jacobi Identity
{{F,G}, H}+ {{H,F}, G}+ {{G,H}, F} = 0,
(iv) Leibniz’ Rule
{F,GH} = {F,G}H +G{F,H}.
The manifoldM endowed with the Poisson bracket is called a Poisson manifold.
The above definition is coordinate-free. If z = (z1, . . . , zm) is a local coordi-
nate system on M, then it can be shown [46] that the Poisson bracket can be
written as
{F,G}(z) = ∇F (z) · J (z)∇G(z),
where J (z) is a skew-symmetric matrix, called the structure matrix, defined
by Ji,j(z) = {zi, zj}, 1 ≤ i, j ≤ m. If the underlying manifold M is even-
dimensional, the coordinate system z is called canonical if the structure matrix
is the standard symplectic matrix
J2n =
0 I
−I 0
,
where I and 0 are the identity and null matrices of dimension n.
Definition 4.1.2 Given a smooth function H :M→ R, a Hamiltonian system
is given by the system of m first-order ordinary differential equations
dz
dt= J (z)∇H(z). (4.1)
The function H is called the Hamiltonian of the Hamiltonian system (4.1).
27
The Hamiltonian function H could depend on a second variable, the time t,
but here we are concerned with time-independent, or autonomous Hamiltonian
systems. (And indeed for us, time will be arc-length along the rod.)
Definition 4.1.3 A function I :M→ R is an integral of (4.1) if it is constant
along solutions of (4.1).
The function I is thus an integral of (4.1) ifdI(z)
dt= 0 for all solutions z(t) of
(4.1). Using the chain rule we conclude that I is an integral of (4.1) if and only
if I Poisson commutes with H(z), i.e.,
{I,H}(z) = ∇I · J (z)∇H = 0.
Functions C :M→ R such that ∇C(z) is in the null space of J (z) are special
integrals called Casimirs. In a canonical formulation there are no Casimirs since
the simplectic matrix J2n is non-singular.
Here we prove the following proposition that is useful for our analysis. The
result is widely known, but here we give an explicit, elementary proof.
Proposition 4.1.4 Let z(s) be a solution of the Hamiltonian system
z′(s) = J(z(s))∇H(z(s)), (4.2)
and let I be an integral of the Hamiltonian system (4.2). Then, ∇I(z(s)) = 0 ∀s
if and only if ∇I(z(0)) = 0.
Proof. Let I(z(s)) be an integral of the Hamiltonian system, then
d
ds∇I(z(s)) = ∇2I(z(s))z′(s) = ∇2I(z(s))J(z(s))∇H(z(s)). (4.3)
Since I is an integral, it Poisson commutes with H, and hence the Poisson bracket
of I with H vanishes at all points in phase space
{I,H}(z) = ∇I(z) · J(z)∇H(z) = 0. (4.4)
28
The gradient of equation (4.4) implies
∇2IJ∇H + [∇(J∇H)]T∇I = 0, (4.5)
or
∇2IJ∇H = −[∇(J∇H)]T∇I. (4.6)
Using equation (4.6) in (4.3), we obtain a linear differential equation for ∇I
d
ds∇I = A(s)∇I, (4.7)
where A(s) = −[∇(J∇H(z(s))]T . Assuming smoothness of A(s), it follows from
uniqueness of solutions for (4.7) that ∇I(z(s)) = 0 for every s if and only if
∇I(z(0)) = 0.
4.2 Relative Equilibria of a Hamiltonian System
4.2.1 Variational Characterization
Consider the equation
∇H(ze) =r∑i=1
λi∇Ii(ze), (4.8)
where Ii, i = 1, . . . , r are independent integrals of the Hamiltonian system (4.1).
This equation identifies points ze in phase space where the gradients of the in-
tegrals are dependent. Of course (4.8) are the first-order conditions associated
with the (finite-dimensional) variational principle
Minimize H(z) subject to Ii(z) = Ci, i = 1, . . . , r, (4.9)
which has the associated “Lagrangian”
F (z;λi) ≡ H(z)−r∑i=1
λi Ii(z). (4.10)
29
(Here Lagrangian is used in the sense of the finite-dimensional theory of con-
strained optimization, it is not the Lagrangian conjugate to the given Hamilto-
nian.) Note that there is no Lagrange multiplier associated with H(z) in (4.10)
which is the condition that (4.10) is normal.
Points ze satisfying (4.8) are not equilibria of the dynamics, unless all the
integrals are Casimirs, but the orbit z(t) of (4.1) with initial data z(0) = ze is
special because, as proved in Proposition 4.1.4, it is trapped on the (critical) level
set of (4.10).
Definition 4.2.1 (see e.g. [35]) The orbit z(t) of the Hamiltonian system (4.1)
with initial data z(0) = ze, where ze is a point that satisfies (4.8), is called a
relative equilibrium of (4.1).
4.2.2 Example: Motion of a Heavy Rigid Body
We illustrate the above definition of relative equilibria with the case of a heavy
rigid body moving about a fixed point. After nondimensionalization, the equa-
tions of motion of the body in a moving frame are given by the non-canonical
Hamiltonian system
z′ = J(z)∇H(z), (4.11)
with
z ≡
m
k
∈ R6, J(z) =
m× k×
k× 0
, and H(z) = 12m · I−1 m + k · χ,
where m is the angular momentum, χ is the position of the center of mass, k is
the upward unit vector and I is the moment of inertia matrix, all expressed in the
body frame. Note that in the moving frame χ is constant but k is not. Here and
throughout this dissertation, for any triple p = (p1, p2, p3), we use the notation
30
p× to designate the matrix
p× =
0 −p3 p2
p3 0 −p1
−p2 p1 0
, (4.12)
so that p×z = p× z for all z ∈ R3.
Because |k| 6= 0, the nullspace of J is two dimensional. The corresponding
Casimirs are
C1(z) = m · k and C2(z) = 12k · k.
Because in this example the Hamiltonian and Casimir functions are quadratic in
m and k, the condition
∇H(ze) = λ1∇C1(ze) + λ2∇C2(ze),
is linear and hence can easily be solved. For this example, relative equilibria,
called the axes of Staude, are all steady, or permanent, rotations about the vertical
[35]. This example is typical in that the non-canonical structure (4.11) is obtained
from a canonical Hamiltonian system by changing to a moving coordinate system
and reducing by a symmetry action (in this case rotation about the vertical k).
The familiar case of a symmetric or Lagrange top is a degenerate limit of this
problem. In the Lagrange top the set of relative equilibria are spins about the
symmetry axis of the body. For an asymmetric body this single spin axis with
arbitrary rotation rate is unfolded to become a one-parameter family of allowable
steady-spin directions each with a specified rotation rate. We shall later see the
situation is analogous in the helical configurations of rods.
31
4.3 Non-Canonical Formulation for Equilibria of Hypere-
lastic Rods
In this section we consider the full equations governing the equilibrium conditions
of elastic rods rewritten entirely in terms of body variables, i.e., components with
respect to the moving frame {d1,d2,d3}. We remark that such components are
natural both physically and mathematically [2].
Assume that the force n 6= 0, let e3 ≡1
|n|n and let e1 ∈ E3 be a fixed unit
vector. For the formulation of the Hamiltonian system to be presented below,
we only need that e1 and n are non-collinear, but here we assume without loss
of generality, that e1 and e3 are orthonormal so that by taking e2 ≡ e3 × e1 we
obtain a fixed orthonormal frame {e1, e2, e3}.
We then rewrite the differential equations (3.2b) and (3.2a) in terms of body
coordinates to obtain
m′ = m× u + n× v, (4.13a)
n′ = n× u. (4.13b)
Similarly, we rewrite equation (3.1a) and the trivial equation e′1 = 0 in terms of
body coordinates
r′ = r× u + v, (4.14a)
e′1 = e1 × u. (4.14b)
In the work of Kehrbaum and Maddocks [30], it is shown that the equations
(4.13) and (4.14) yield a Hamiltonian system with
H(m,n, s) := W ∗(m,n, s) + m · u + n · v, (4.15)
32
and e1
r
m
n
′
= J12∇H(m,n, s), (4.16)
where J12 is the skew-symmetric operator defined by0 0 e×1 0
0 0 r× I
e×1 r× m× n×
0 −I n× 0
.
and I and 0 are the identity and the null matrices of dimension three. The
functional W ∗ is defined in (3.9), while v and u are the strains in the reference
configuration. Note that the relations (3.8) and (4.15) show that
∂H
∂m(m,n, s) = u, (4.17a)
∂H
∂n(m,n, s) = v. (4.17b)
Kehrbaum [30] showed that J12 satisfies the Jacobi identity, so that J12 gen-
erates a Poisson bracket [46, Proposition 6.8].
For the non-canonical formulation to be useful, it is important that the fixed
frame components of all phase variables can be determined from the phase vari-
ables e1, r,m and n. The rotation matrix relating the fixed frame {e1, e2, e3}
to the moving one {d1,d2,d3} can be constructed, for example, in terms of the
nine direction cosines relating the two frames. Then the triple p of body com-
ponents of any vector is related to the triple p of components in the fixed frame
{e1, e2, e3} through the relation p = RTp, or equivalently, p = Rp, where
R =
e1 · d1 e1 · d2 e1 · d3
e2 · d1 e2 · d2 e2 · d3
e3 · d1 e3 · d2 e3 · d3
. (4.18)
33
Alternatively, one can use a parameterization of the rotation group such as Euler
angles or Euler parameters to relate these two frames.
4.4 Order Reduction
The variables m and n in the twelve-dimensional ODE system (4.16) can be
decoupled from the other two variables e1 and r. Note that H does not depend
on e1 and r, and hence r and e1 are ignorable variables which can be found by
quadrature once the strains m and n are obtained. As a consequence, instead
of analyzing twelve equations, one can consider the reduced Hamiltonian system
consisting of only six equations given by
m′ = m× u + n× v, (4.19a)
n′ = n× u. (4.19b)
The associated Hamiltonian structure is given bym
n
′ = J6(m,n)∇H(m,n, s), (4.20)
where J6 is the skew-symmetric operator defined by
J6(m,n) =
m× n×
n× 0
. (4.21)
Note that when |n| 6= 0, the null space of J6 is the two dimensional space spanned
by n
m
and
0
n
.
The associated Casimirs are
C1 = 12n · n, and C2 = m · n.
34
The system (4.20)-(4.21) has been the subject of intensive study since at
least the work of Arnold [3] due to its interpretation as rigid body equations and
generalization in ideal fluid mechanics. In the context of rod problem, we refer
to the work of Mielke and Holmes [43].
4.5 Analysis of Integrals
In the work of [33], there is a detailed description of all integrals arising in the
Hamiltonian systems of ordinary differential equations describing the statics of
rods, together with their associated symmetries. That analysis shows that the
integrals arising for the extensible, shearable rod and for the inextensible and
unshearable rod are identical. These integrals can be classified as follow:
• Constitutive-independent integrals. The three components of force n in the
fixed basis {e1, e2, e3} are constants, due to the translational symmetry
of the Hamiltonian system in space. Similarly, the three components of
m + r × n in the fixed basis are constant as a consequence of rotational
symmetry about the three inertial axes.
We remark that for the reduced form of the non-canonical Hamiltonian
formulation, we will be concerned only with the integrals m · n and n · n
which are functions of m and n. The remaining integrals are used for
recovering the ignorable variables.
• Constitutive-dependent integrals that arise in each of the cases of a uniform
rod, and an isotropic rod. In fact, if we assume the rod to be translationally
symmetric in the arc-length s, then the Hamiltonian H given in (4.15) is
an integral. And finally, if the rod is transversally isotropic than one can
show that the twisting moment m3 = m · d3 is an integral.
35
Chapter 5
Relative Equilibria of Uniform, Hyperelastic
Rods
In this chapter we consider the six dimensional non-canonical Hamiltonian for-
mulation for the equilibrium conditions of a hyperelastic, uniform, Cosserat rod
introduced in Chapter 4. We demonstrate that all relative equilibria of the Hamil-
tonian system correspond to helical solutions of the rod subject to terminal loads.
We treat the problem for quite general, but uniform, constitutive relations with a
strain energy function that meets minimal conditions of convexity and coercivity
for large strains. Our analysis of the existence and multiplicity of such solutions
falls into the domain of finite-dimensional constrained minimization, and can be
written either in terms of the Hamiltonian variables, i.e., the internal forces and
moments (the stresses) or as a dual formulation in terms of the strains. The
dual formulation is apparently new and allows a simple analysis of the existence
and multiplicity of solutions. The analysis presented is for the general case of
an extensible, shearable rod, pointing as appropriate, to the inextensible and
unshearable case. In both cases, we consider in parallel the integrable and non-
integrable cases of an isotropic or anisotropic rod. We illustrate the analysis with
an explicit classification of these helical solutions for a quadratic and diagonal
strain energy function.
We will study the relative equilibria of the Hamiltonian system (4.15) which
admits three integrals: the Hamiltonian itself, I0 = H because we have a uniform
rod, the norm of the internal force I1 = 12n · n, and the moment in the direction
of the internal force, I2 = m · n. Moreover, in the case of an isotropic rod,
36
there is the additional integral of the moment in the direction of d3, I3 = m3 =
m · d3. It transpires that the cases of isotropic and anisotropic rods can be
treated simultaneously by introducing Lagrange multipliers corresponding to all
four integrals, but setting the one associated with m · d3 to zero in the case of
anisotropy.
5.1 Variational Characterization
We introduce the augmented Hamiltonian, F
F (m,n;λ0, λ1, λ2, λ3) = λ0 I0(m,n)− λ1 I1(n)− λ2 I2(m,n)− λ3 I3(m), (5.1)
where λ0, λ1, λ2 and λ3 are Lagrange multipliers associated with the objective
function
I0(m,n) ≡ H(m,n) = C0, (5.2a)
and the constraints
I1(m,n) ≡ 12n · n = 1
2C2
1 , (5.2b)
I2(m,n) ≡ m · n = C1C2, (5.2c)
I3(m,n) ≡ m · d3 = C3, (5.2d)
where C0, C1, C2 and C3 are constants. The particular forms of the constants
given in (5.2) emphasize the facts that I1 is nonnegative and I2 vanishes whenever
I1 vanishes.
We first show that other at trivial solutions with n = 0 or straight twisted
centerline, (5.1) is a normal problem, in the sense introduced in Chapter 4 which
is equivalent to having an expression for F as in (4.10) by taking λ0 6= 0.
Proposition 5.1.1 Critical points of F , except trivial ones with n = 0 or straight
twisted centerline, are critical points of F given by
F (m,n; λ1, λ2, λ3) = I0(m,n)− λ1 I1(n)− λ2 I2(m,n)− λ3 I3(m), (5.3)
37
where λ1 = λ1
λ0, λ2 = λ2
λ0and λ3 = λ3
λ0.
Proof. If λ0 = 0, then the critical point of F given in (5.3) satisfy
λ1∇I1 + λ2∇I2 + λ3∇I3 = λ1
0
n
+ λ2
n
m
+ λ3
d3
0
= 0, (5.4)
for (λ1, λ2, λ3) 6= (0, 0, 0). For the anisotropic rod, λ3 = 0 so that the only possible
solutions of (5.4) are trivial ones with n = 0. For the isotropic rod with λ3 6= 0
the solutions of equation (5.4) are such that both n and m are parallel to d3
which correspond to straight twisted rods.
Therefore, non-trivial critical points of F are those of
F =1
λ0
F,
for given values of the constants λ1, λ2 and λ3 and we consider the trivial relative
equilibria separately.
5.1.1 Closed-Form Expression for Relative Equilibria
For brevity, we will omit the superposed bar on F and λi, i = 1, 2, 3. We will
denote by w(s) :=
w1(s)
w2(s)
w3(s)
the components of any vector w(s) with respect to
the moving orthonormal basis {d1,d2,d3}, and by w0 the value of w(s) at s = 0.
We denote by w, the shifted vector w := w − λ3d3 and by w, the shifted set of
components w := w − λ3d3 where d3 =
0
0
1
. Without loss of generality, the
moving frame in the reference configuration is chosen such that the third vector
d3 coincides with the tangent to the centerline of the rod, i.e.,
v =
0
0
1
.
38
The following analysis aims to give a characterization of relative equilibria for hy-
perelastic uniform rods independently of any constitutive relations. By definition,
a relative equilibrium, i.e., a critical point of F , satisfies
∇F =
∂F∂m
∂F∂n
=
∂H∂m− λ2
∂I2∂m− λ3
∂I3∂m
∂H∂n− λ1
∂I1∂n− λ2
∂I2∂n
=
0
0
. (5.5)
We claim that equation (5.5) can be solved in an essentially closed form for any
given hyperelastic constitutive relations for the inextensible, unshearable rod, and
in a slightly less explicit form, for an extensible and shearable rod. The solutions
are families of relative equilibria parameterized by the constants λ1, λ2 and λ3.
Equation (5.5) involves the gradients of all of the integrals of the Hamiltonian
system:
∇H =
u
v
, ∇I1 =
0
n
, ∇I2 =
n
m
and ∇I3 =
d3
0
.
The expression for the gradient of the Hamiltonian follows from (4.17). For the
case of an inextensible and unshearable rod we take the limit case given by v ≡ v.
Explicitly, solutions of equation (5.5) satisfyu
v
=
λ2I 0
λ1I λ2I
n
m
. (5.6)
Here I is the 3 × 3 identity matrix and the triple u is given by u =
u1
u2
u3 − λ3
.
Because the integrals are quadratic in the Hamiltonian variables m and n, the
matrix on the right hand side of equation (5.6) has constant entries. In the
following, we suppose that λ2 6= 0. The case λ2 = 0 can be analyzed as a limit
problem in which the strains are u = 0, and v = λ1n corresponding to straight
configuration. For λ2 6= 0 equation (5.5) is equivalent ton
m
=
µ2I 0
µ1I µ2I
u
v
. (5.7)
39
where µ1 and µ2 are real constants given by
µ1 = −λ1
λ22
, (5.8)
µ2 =1
λ2
. (5.9)
Therefore, for given constitutive relations, the variables of our problem can be
identified either as m and n taking into account the constraints (5.2b-d), or a
variational problem with u, v as variables and with the following set of constraints
12u · u = 1
2η2
1, (5.10a)
u · v = η2η1, (5.10b)
for some constants η1, η2 and λ3, inserted in u = u−λ3d3. The set of constraints
(5.10) is a simple consequence of equation (5.7) that relates the stresses to the
strains. A direct computation shows that the multipliers and the constraint
constants in each formulation are related byλ2 0
λ1 λ2
C21
C1C2
= µ2
η21
η1η2
, (5.11)
or, equivalently
λ2
C21
C1C2
=
µ2 0
µ1 µ2
η21
η1η2
. (5.12)
5.1.2 Characterization of Relative Equilibria for Uniform
Rods
Proposition 5.1.2 At relative equilibria, the components m(s) and n(s) of the
vectors m(s) and n(s) along the rod are simply related to their values at s = 0.
Explicitly we have:
m(s) = QT (λ3s)m0, (5.13a)
n(s) = QT (λ3s)n0, (5.13b)
40
where
Q(θ) =
cos(θ) − sin(θ) 0
sin(θ) cos(θ) 0
0 0 1
, (5.14)
is the matrix of rotation about d3 through an angle θ.
We remark that this result is implied by general consideration of the symmetries
generating the integrals I0, I1, I2, I3, but in this case the explicit computation is
straightforward, and will later be used to give more properties of relative equilibria
of a uniform rod.
Proof. As was shown in the previous section, relative equilibria are solutions of
the equation
∇H(z) = λ1
0
n
+ λ2
n
m
+ λ3
d3
0
, (5.15)
where z =
m
n
. Multiplying both sides of equation (5.15) by the 6× 6 matrix
J6 given in (4.21), and using the fact that 12n · n and m · n are Casimirs, we get
z′(s) = λ3J6
d3
0
= Dz(s), (5.16)
where D is the constant matrix D = −λ3
d×3 0
0 d×3
. Equation (5.16) can be
integrated to yield
z(s;λ3) =
QT (λ3s) 0
0 QT (λ3s)
z0, (5.17)
where the matrix Q(λ3) is given in (5.14).
41
Proposition 5.1.3 At relative equilibria of a uniform rod, the components of the
strains u and v are of the form:
u(s) = QT (λ3s)u0, (5.18a)
v(s) = QT (λ3s)v0, (5.18b)
where Q is given by (5.14).
Proof. These statements follow directly using equations (5.6) and (5.13).
Propositions 5.1.2 and 5.1.3 give an explicit characterization of the relative equi-
libria of uniform hyperelastic rod. For the anisotropic rod, with λ3 = 0, such
relative equilibria have the particular properties that the components of the mo-
ments, internal forces and strains in the director frame are constants along the
rods. Moreover, in all cases the norms of these quantities are constant along the
rod.
5.2 Recovering the Centerline of the Rod
Theorem 5.2.1 The configuration of the centerline of the rod associated with a
relative equilibrium is a circular helix.
Proof. In the previous section, the relative equilibria were shown to have constant
norm of the strains. In particular, if we take |v| = λ for some constant λ, then
we can consider a general rod that undergoes shear and extension in addition to
bending and twist. The case of inextensible and unshearable rod is obtained by
letting λ = 1. Following Ilyukhin [28], it has been shown in [30] that cylindrical
coordinates (ρ, φ, ξ) relative to the fixed basis {e1, e2, e3} for the centerline r,
i.e.,
r · e1 = ρ cosφ, (5.19a)
r · e2 = ρ sinφ, (5.19b)
r · e3 = ξ, (5.19c)
42
can be defined through the relations
ρ2 =1
|n|4|m× n|2, (5.20a)
φ′ =1
|n|3ρ2{|n|2m · v− (m · n)n · v}, (5.20b)
ξ′ = r′ · n
|n|. (5.20c)
For a relative equilibrium, if we take into account the expressions of n and of m
given in (5.7), and the constraints (5.10), the coordinates of the centerline become
explicit functions of the strain v and the parameters η1 and η2:
ρ =1
|η1|
√|v|2 − η2
2, (5.21a)
φ′ = η1, (5.21b)
ξ′ = ±η2. (5.21c)
By the fact that we have |v| ≡ λ which is a constant, we deduce that the radius
ρ of the centerline is constant.
Let σ(s) be the arc length parameter of the deformed axis:
dσ(s)
ds= |v| ≡ λ.
If we choose s and σ to vanish at the same point, then σ = λs. The angle φ is
given by
φ =η1
λσ + φ0.
The centerline is then given by
r(σ) = ρ[cos(
η1
λσ + φ0)e1 + sin(
η1
λσ + φ0)e2
]+ (±η2
λσ + z0)e3, (5.22)
where φ0, the value of the angle φ at s = 0, and z0 are constants of integration
which locate the helix in space.
Equation (5.22) shows that the centerline is a circular helix (possibly degen-
erate) of radius ρ =1
|η1|
√|v|2 − η2
2 and pitch p = ±η2
η1along the e3 axis. Note
43
that ρ and η2 cannot vanish at the same time since we impose r′ ·d3 > 0, but we
can distinguish the following cases depending on the parameters.
(i) Deformations with a straight centerline |v| = ±η2: The helix degen-
erates into a twisted straight rod. Taking into account the constraints in
the system (5.10), we see that u is parallel to v.
(ii) Deformation with a circular centerline η2 = 0: The helix degenerate
to a twisted circle. In this case, u is perpendicular to v.
(iii) Deformations with a helical centerline |v| 6= η2: We have a non-
degenerate helix.
More explicit expressions for all these solutions can be given when the constitutive
relations are specified.
Figure 5.1: Diagram in (η1, η2)-parameter space showing the different classes ofhelical solutions.
Figure 5.1 shows the different classes of helical solutions in the (η1, η2)-parameter
space, we indicate the limits of the domain of such possible solutions with straight
44
lines which is precise when the rod is inextensible and unshearable. In the exten-
sible and shearable case, the right limits are given by the more complicated curves
|v(η1, η2)| ± η2 = 0, whose specific form depends on the constitutive relations.
We remark here, that the limit case of η1 = 0 and η2 = 0 correspond to a
straight centerline with vanishing force in the rod but possibly with moments. In
the following, we ignore the different degenerate cases since they require special
considerations.
Proposition 5.2.2 Non-degenerate relative equilibria of a rod are non-degenerate
helices and the components of internal force and moment in the Frenet basis are
constants. The vectors n and m are both orthogonal to the principal normal ν
of the helical centerline. Moreover the direction of the force n is along the axis
of the helical centerline
Proof. The Frenet basis {τ ,ν,β} of the centerline (5.22) is given by
τ = −ρη1
λ(sin(
η1σ
λ+ φ0) e1 − cos(
η1σ
λ+ φ0) e2) +
η2
λe3, (5.23a)
ν = − cos(η1σ
λ+ φ0) e1 − sin(
η1σ
λ+ φ0) e2, (5.23b)
β =η2
λ(sin(
η1σ
λ+ φ0) e1 − cos(
η1σ
λ+ φ0) e2) +
ρη1
λe3. (5.23c)
Note that the curvature and the torsion of the helix are given by
κ =ρη2
1
λ2=|η1|λ2
√(λ2 − η2
2), (5.24a)
τ =η1η2
λ2, (5.24b)
then we get the relation between the curvature and the torsion
κ2 + τ 2 =η2
1
λ2.
We note that for the inextensible and unshearable rod λ = 1 and the curvature
and the torsion are uniquely determined from the values of η1 and η2. However for
the shearable and extensible rod, the curvature and the torsion involve not only
45
the constants η1 and η2 but also the constitutive relations through the constant
λ.
The expressions of the moment vector m and internal forces n for a relative
equilibrium are:
m = ρ|C1|(− sin(η1σ
λ+ φ0) e1 + cos(
η1σ
λ+ φ0)) e2 + C2 e3,
n = |C1|e3,
where C1 and C2 are the constants introduced in (5.2) that are related to η1 and
η2 through the relations given in (5.12). By the fact that λ2 = ρ2η21 + η2
2 and
hence |η2
λ| < 1 and |ρη1
λ| < 1, we define an angle α by
cosα =η2
λ,
sinα =ρη1
λ.
Therefore we deduce that
m · τ = |C1|ρ sinα + C2 cosα, (5.25a)
m · ν = 0, (5.25b)
m · β = −|C1|ρ cosα + C2 sinα, (5.25c)
and that
n · τ = |C1| cosα, (5.26a)
n · ν = 0, (5.26b)
n · β = |C1| sinα. (5.26c)
Relations (5.25) and (5.26) show that for both the inextensible and unshearable
rod and for the extensible and shearable one, the components of the internal
forces and moments in the Frenet basis for relative equilibria are constants with
the special expressions given in (5.25) and (5.26).
46
5.2.1 Geometric Characterization
In this section, we want to give a geometric characterization of relative equilibria
for a general rod. We will show that the rotation tensor R that takes the fixed
frame {e1, e2, e3} to the moving one {d1,d2,d3} is given by
R ≡ Π(η1s)R0Q(λ3s), (5.27)
where Π(η1s) is a matrix which depends only on the given constant η1, and Q is
the rotation matrix given in (5.14). Equation (5.22) shows that
r′(s) = Π(η1s)r′0,
where
Π(θ) =
cos(θ) − sin(θ) 0
sin(θ) cos(θ) 0
0 0 1
. (5.28)
On the other hand, by definition of the strain v we have
v(s) = RT (s)r′(s). (5.29)
Then using (5.18b) we get
QT (λ3s)v0 = RTΠ(η1s)r′0. (5.30)
If we replace v0 by RT0 r′0 then we get
R ≡ Π(η1s)R0Q(λ3s).
In summary, we gave so far an analysis of some properties of the relative equi-
libria of a hyperelastic rod that are independent of the details of the constitutive
relations. We showed that the associated centerline is a helix, and that the stresses
and the strains are completely determined by their values at a given point s = 0.
In the following two subsections, we give two dual formulations characterizing
relative equilibria for hyperelastic uniform rods where we introduce explicitly the
constitutive relations.
47
5.2.2 Formulation of the Problem in Terms of the Stresses
The first and most evident finite-dimensional variational problem is given by
Minimize H(z) subject to Ii(z) = Ci, i = 1, . . . , 3, (5.31)
where the expression of H is given in (4.15). For given intrinsic strains u and v,
the strains can be related to the internal forces and moment using the constitutive
relation,
u =∂W ∗
∂m(m,n) + u, (5.32a)
v =∂W ∗
∂n(m,n) + v, (5.32b)
where W ∗(m,n) is the Legendre transform of the strain energy. Therefore equa-
tion (5.6) can be rewritten as∂W ∗
∂m(m,n) + ˆu
∂W ∗
∂n(m,n) + v
=
λ2I 0
λ1I λ2I
n
m
. (5.33)
5.2.3 Dual Formulation of the Problem in Terms of the
Strains
A dual formulation to the previous one can be developed in terms of the strains
u and v in the following way: For a hyperelastic extensible and shearable rod,
the triples m and n are related to the strains u and v through the constitutive
relations
m =∂W
∂u(u− u, v− v), (5.34a)
n =∂W
∂v(u− u, v− v). (5.34b)
Hence equation (5.7) becomes∂W∂v
(u− u, v− v)
∂W∂u
(u− u, v− v)
=
µ2I 0
µ1I µ2I
u
v
. (5.35)
48
Solutions of equation (5.35) for the general case of an extensible and shearable
rod, are critical points of the functional
W (u− u, v− v) (5.36)
subject to the constraints
12u · u = 1
2η2
1, (5.37a)
u · v = η1η2. (5.37b)
For a hyperelastic, inextensible and unshearable rod, only the triple m is related
to the strains u through the constitutive relation
m =∂W
∂w(u− u). (5.38)
In this case, v ≡ v and n is a reactive parameter that is not given by a constitutive
relation, but rather is determined from the equilibrium equations. Therefore,
equation (5.7) becomes
∂W
∂u(u− u) = µ1u + µ2v, (5.39)
which is equivalent to minimizing W (u− u) subject to the following constraints
12u · u = 1
2η2
1, (5.40a)
u · v = η1η2. (5.40b)
5.2.4 Existence and Multiplicity
We use the dual formulation to prove the existence and to analyze the multi-
plicity of relative equilibria for uniform rods. The parameters introduced in this
formulation are closely related to some physical quantities such as the curvature
and the torsion of the centerline of the rod.
49
Proposition 5.2.3 For any continuous, convex and coercive strain energy den-
sity W , there exist at least two extrema of the energy density W that satisfy the
constraints (5.37).
Proof.
• inextensible and unshearable case: In the particular case of an inexten-
sible and unshearable rod, the strains v in a deformed configuration coincide
with the strains v in the reference configuration. The set of constraints in
terms of the strains is given in (5.40). The discussion of existence and of
multiplicity of extrema can be done in our case in two different ways:
– Knowing explicitly the strain energy function, we can write necessary
conditions in terms of the two Lagrange multipliers that yield solutions,
or
– by observing that with v =
0
0
1
, the second equation in (5.40) yields
that u3 = λ3+η1η2, and hence u3 is a given constant. Then the remain-
ing constraint in (5.40) can be parameterized by a single parameter ξ
through
u1 = η1
√1− η2
2 cos ξ,
u2 = η1
√1− η2
2 sin ξ.
Therefore, the objective functionW (ui−ui) restricted to the constraint
set (5.40) becomes a 2π-periodic function of the single variable ξ. The
problem then reduces to finding extrema of
W (ξ) := W (η1
√1− η2
2 cos ξ − u1, η1
√1− η2
2 sin ξ − u2, η2, λ3),
(5.42)
50
where ξ lies in [0, 2π]. The new problem (5.42) admits at least two
solutions because the function W is a continuous real-valued, peri-
odic function of a real variable and hence achieves its maximum and
its minimum. Moreover, if we consider the case of an isotropic rod,
then W is a constant function independent of the angle ξ (by defini-
tion of isotropy), which makes any angle ξ a solution of the problem.
This gives rise to an additional freedom in the parameterization of an
isotropic rod. Whereas for an anisotropic rods, the number of extrema
varies depending on the explicit form of the function W .
Figure 5.2: Intersection of the cylinder u21 + u2
2 = c2 with convex strain-energysurface W (u1, u2) = 1
2[K1(u1 − u1)2 +K2(u2 − u2)2] with K1 = K2 = 1 and u1 =
u2 = 0. In this isotropic case, due to the continuous symmetry, the intersectioncurve is a circle centered at (u1, u2) = (0, 0) of all points with same strain-energyvalue.
• Extensible and Shearable Rods:
The analysis of existence and multiplicity of solutions for the case of ex-
tensible and shearable rod is more complicated due to the fact that the
minimization is constrained on a sub-manifold of dimension four of R6.
51
Figure 5.3: Intersection of the cylinder u21 + u2
2 = c2 with convex strain-energysurface W (u1, u2) = 1
2[K1(u1−u1)2 +K2(u2−u2)2] with K1 = 1, K2 = 2 and u1 =
u2 = 0. In this anisotropic but symmetric case, due to the discrete symmetry,the intersection curve has two minima with same strain-energy value and twomaxima same strain-energy value.
Figure 5.4: Intersection of the cylinder u21 + u2
2 = c2 with strain-energy surfaceW (u1, u2) = 1
2[K1(u1 − u1)2 + K2(u2 − u2)2] with K1 = 1, K2 = 2, u1 = 0
and u2 = 0.2. In this anisotropic case the intersection curve has two maximawith same strain-energy value and two minima with different values of the strain-energy.
52
Figure 5.5: Intersection of the cylinder u21 + u2
2 = c2 with strain-energy surfaceW (u1, u2) = 1
2[K1(u1 − u1)2 + K2(u2 − u2)2] with K1 = 1, K2 = 2, u1 = 1 and
u2 = 0.8. In this anisotropic case the intersection curve has one minimum andone maximum.
The problem is to find critical points of the functional
W (u− u, v− v)
subject to the constraints
12u · u = 1
2η2
1,
u · v = η1η2.
The objective function W in R6 is convex and coercive so it certainly has
a minimum on the constraint set. We can discuss multiplicity of solutions
via a more detailed argument. If we reconsider the set of equations
∂W
∂u(u− u, v− v) = µ1u + µ2v, (5.44a)
∂W
∂v(u− u, v− v) = µ2u, (5.44b)
then for each u, if we consider equation (5.44b), we deduce by convexity of
the strain energy density W , with respect to the strains v, the existence of a
unique minimizer v(u) of W . Once we have v(u) then the problem reduces
53
Figure 5.6: Intersection of the sphere u · u = η21 with the surface u · g(u) = α
where g(u) =[u1
A1, u2
A2, u3
A3
]T, A1 = 10, A2 = 6, A3 = 4, η1 = 1, α = −η1η2
µ2∈[
1A3, 1A1
]. The intersection of these two sets consists of two smooth closed curves
(two diametrically opposite points in the degenerate cases α = 1A1
and α = 1A3
).
54
to a minimization of a continuous function W := W (u, v(u, µ2u)), u, v) on a
set defined by the intersection of a sphere of radius η1 with the set u ·v(u) =
η1η2. We note that, the Implicit Function Theorem, ensures the existence
of a neighborhood B of u ∈ R3 in which v(u) is a differentiable function
therefore the level set of u ·v(u) on the surface of the sphere is topologically
equivalent to closed loops. Therefore the continuous function W achieves
its maximum and minimum.
In conclusion, we have proved that, for the inextensible and unshearable rod
there exist at least two relative equilibria of each value of the constants η1, η2
and λ3. For the isotropic rod, the solution is parameterized by four constants
namely, η1, η2, λ3 and the free angle ξ allowing different possible orientations
of the cross-section. For the anisotropic rod there exists a discrete number of
family of solutions parameterized by two constants, namely η1 and η2. In the
inextensible and unshearable rod the geometrical torsion of the helix is given
by η1η2, the curvature is given by |η1|√
1− η22 and λ3 expresses the difference
between the geometric torsion of the centerline and the physical twist of the rod.
For the extensible and shearable rod, we proved the existence of such possible
helical solutions. The multiplicity can not be analyzed generally unless we know
explicitly the strain energy function.
We also note that for the anisotropic rod, the presence of intrinsic curvature
affects the multiplicity of possible families of solutions.
We first illustrate our discussion by showing in Figs. 5.2-5.5, the geometri-
cal description of the set of possible extrema for the case of a diagonal linear
constitutive relations, then we make the direct calculations giving explicitly the
expressions of the strains as a function of the parameters µ1 and µ2, the different
rigidities of the rod and the intrinsic strains.
55
5.3 Example: Diagonal Quadratic Strain Energy
To illustrate our previous analysis, we give here a complete classification of rel-
ative equilibria for a quadratic strain energy. To simplify the problem further
we eventually restrict to the diagonal case. In general the constitutive relations
relate the components with respect to the orthonormal basis {d1,d2,d3} of the
net force n =
n1
n2
n3
and of the net moment m =
m1
m2
m3
to the strains u =
u1
u2
u3
and v =
v1
v2
v3
linearly through a 6× 6 stiffness matrix
n
m
=
A G
GT K
v− v
u− u
. (5.45)
Here K and A are symmetric positive-definite 3 × 3 matrices. The 3 × 3 matrix
G is such that the 6× 6 partitioned matrix
K =
A G
GT K
(5.46)
is positive definite. For the diagonal case, the matrix G ≡ 0 and the constitutive
relations reduce to n
m
=
A 0
0 K
v− v
u− u
, (5.47)
with K and A diagonal, positive-definite 3× 3 matrices. For the uniform rod, the
intrinsic strain vectors u and v are given constants of the form
u =
u1
u2
u3
; v =
0
0
1
.
56
In the following we discuss solutions for the strains u and v of the system of
equations (5.35) and (5.47). We consider first the inextensible and unshearable
rod where we distinguish separately the two main cases: the isotropic rod where
its helical solutions are described by four parameters, and the anisotropic rod
whose multiple solution families are each described by only two parameters. The
analysis of the case of an extensible and shearable rod will provide an illustration
of the proof of multiplicity of solutions introduced in the previous section.
5.3.1 Inextensible and Unshearable Rod
In this particular case we have v ≡ v, therefore for given Lagrange multipliers µ1,
µ2 and λ3, and for a given triplet of intrinsic strains u we will solve the system
of equations given by (5.35) and (5.47) only for the strains u. We will show
that for the anisotropic case, two or four families of solutions arise due to the
existence of two distinct principle directions of bending. In the isotropic case the
non-degenerate helices belong to only one family.
• Anisotropic Rod
Recall that an anisotropic rod may have two principal directions of bending
and that the strains in the reference configuration associated with bending
need not vanish as is the case for the isotropic rod. The stiffness matrix K
and the triples u can be taken as
K =
K1 0 0
0 K2 0
0 0 K3
, u =
u1
u2
u3
,
where in general we have K1 6= K2 6= K3 6= K1. The simplified system of
equations to solve for the triplet of strains (u1, u2, u3) for given constants
57
Figure 5.7: Two possible cross-section orientations for helical configurations ofan anisotropic rod.
µ1, µ2, λ3, u1,u2 and u3 is given byK1 − µ1 0 0
0 K2 − µ1 0
0 0 K3 − µ1
u1
u2
u3
=
K1u1
K2u2
K3u3 + µ2
. (5.48)
We will discuss the solvability of equation (5.48) depending on the value of
the components u1 and u2 of the strains in the reference configuration, and
the value of the Lagrange multipliers.
(1) If u1 6= 0 and u2 6= 0: then equation (5.48) admits solutions when
(1.1) µ1 6= K1, µ1 6= K2, µ1 6= K3: in which case
u1 =K1u1
K1 − µ1
,
u2 =K2u2
K2 − µ1
,
u3 =K3u3 + µ2
K3 − µ1
.
58
(1.2) If u3 6= 0, µ2 = −K3u3, µ1 = K3: Then
u1 =K1u1
K1 −K3
,
u2 =K2u2
K2 −K3
,
and u3 is free.
(2) If u1 = 0, u2 6= 0: Solutions exist if
(2.1) µ1 = K1:
u1 = u,
u2 =K2u2
K2 −K1
,
u3 =K3u3 + µ2
K3 −K1
,
with freedom in one of the bending strains.
(2.2) If µ1 6= K1:
u1 = 0,
u2 =K2u2
K2 − µ1
,
u3 =K3u3 + µ2
K3 − µ1
.
(3) If u1 6= 0, u2 = 0: Solutions exist if
(3.1) µ1 = K2:
u1 =K1u1
K1 −K2
,
u2 = u,
u3 =K3u3 + µ2
K3 −K1
,
with freedom in one of the bending strains.
59
(3.2) If µ1 6= K2:
u1 =K1u1
K2 − µ1
,
u2 = 0,
u3 =K3u3 + µ2
K3 − µ1
.
(4) If u1 = 0, u2 = 0: then
(4.1) For µ1 = K1:
u1 = u,
u2 = 0,
u3 =K3u3 + µ2
K3 −K1
,
for an arbitrary u.
(4.2) For µ1 = K2:
u1 = 0,
u2 = u,
u3 =K3u3 + µ2
K3 −K2
.
(4.3) For µ1 6= K1 and µ1 6= K2:
u1 = 0,
u2 = 0,
u3 =K3u3 + µ2
K3 − µ1
.
This solution which corresponds to a twisted straight rod.
• Isotropic Rod
The isotropic rod is a particular case of the anisotropic one in the sense
that we have equal stiffnesses of bending and we do not consider intrinsic
60
bending. The matrix of stiffness and the strains configuration are given by
K =
K1 0 0
0 K1 0
0 0 K3
, u =
0
0
u3
. (5.51)
Then we haveK1 − µ1 0 0
0 K1 − µ1 0
0 0 K3 − µ1
u1
u2
u3 − λ3
=
0
0
K3(u3 − λ3) + µ2
.
(5.52)
Suppose that K1 6= K3, which is the case in practical problems, then we
have the following discussion about the possible values of µ1 that yield a
solution u for (5.52).
(1) For µ1 = K1: In this case, we have freedom in the two values of the
strains, so that the bending strains (u1, u2) can take any values in R2,
while the twist u3 is given by
u3 = λ3 +1
K3 −K1
(K3(u3 − λ3) + µ2).
This expression of the twist generalizes the fact that for the twisted
circular rod, the end force is uniquely determined by the twist to the
case of helical rods.
In terms of the values of the constraints η1, η2 and λ3 we write the
solution as
u1 = u cos ξ,
u2 = u sin ξ,
u3 = λ3 + η2,
where ξ is an arbitrary constant in [0, 2π] and u2 = η21 − η1η
22. The
set of admissible values of η1 and η2 is {(η1, η2) ∈ R2 | η21 − η2
1η22 ≥ 0}.
61
Figure 5.8: Two projections of a multi-sheeted surface that represents the Hamil-tonian, evaluated along the families of critical points of the isotropic rod, as afunction of the Lagrange multipliers λ1 = −µ1
µ22
and λ2 = 1µ2
. The sheet with a
parabola as a base corresponds to non-degenerate helices.
This family of solutions corresponds to a sheet of helices of curvature
u and torsion η2. Note that depending on the values of parameters η1
and η1η2 the helix can degenerates into a straight line or into a circle.
However, these two degeneracies cannot occur simultaneously, i.e., the
axis of the rod cannot degenerate into a point.
(2) For µ1 6= K1:
u1 = 0,
u2 = 0,
u3 = λ3 ± η1.
In terms of the constraints, this solution, which corresponds to a
twisted straight line, requires that
η2 = ±1.
62
Figure 5.9: Circular helix with symmetrical cross-section.
63
5.3.2 Extensible and Shearable Rod
• Anisotropic case
The anisotropic rod has two distinct principal directions of bending or two
distinct directions of shear deformations. In this case, the matrix of stiffness
is given by
K =
K1 0 0
0 K2 0
0 0 K3
, A =
A1 0 0
0 A2 0
0 0 A3
and u =
u1
u2
u3
.
(5.53)
The particular structure of the system of equations to solve can be decoupled
so that we find the shear strains as a function of the bending and twist
strains then we solve for the remaining variables. The idea is to minimize
the energy about v for u fixed then solve for u. We have, then the two
following equations to solve:v1
v2
v3
= µ2
1A1
0 0
0 1A2
0
0 0 1A3
u1
u2
u3
+
0
0
1
,
andK1 − µ1 − µ2
2
A10 0
0 K2 − µ1 − µ22
A20
0 0 K3 − µ1 − µ22
A3
u1
u2
u3
=
K1u1
K2u2
K3u3 + µ2
.
(5.54)
The first equation of this system give an explicit expression of the strains
v as a function of the strains u. The discussion of the solvability of the
second equation in the system (5.54) for the strains u will follows as for
the inextensible and unshearable case. We consider special values for the
64
intrinsic strains then we discuss depending on the values of the Lagrange
multipliers µ1 and µ2.
(1) If u1 6= 0 and u2 6= 0: In this case (5.54) admits solutions when
(1.1) K1 − µ1 − µ22
A16= 0, K2 − µ1 − µ2
2
A26= 0, K3 − µ1 − µ2
2
A36= 0: Solu-
tions of (5.54) are given by
u1 =K1u1
K1 − µ1 − µ22
A1
,
u2 =K2u2
K2 − µ1 − µ22
A2
,
u3 =K3u3 + µ2
K3 − µ1 − µ22
A3
.
(1.2) If u3 6= 0, µ2 = −K3u3, µ1 = K3 − K23 u3
2
A3:
u1 =K1u1
K1 −K3 +K2
3 u32
A3− K2
3 u32
A1
,
u2 =K2u2
K2 −K3 +K2
3 u32
A3− K2
3 u32
A2
.
(2) If u1 = 0, u2 = 0:
(2.1) If K1 − µ1 − µ22
A1= 0: The solution is given by
u1 = u,
u2 = 0,
u3 =K3u3 + µ2
K3 −K1 +µ2
2
A1− µ2
2
A3
.
(2.2) If K2 − µ1 − µ22
A2= 0:
u1 = 0,
u2 = u,
u3 =K3u3 + µ2
K3 −K2 +µ2
2
A2− µ2
2
A3
.
65
(2.3) If K1 − µ1 − µ22
A16= 0 and K2 − µ1 − µ2
2
A26= 0: The solution is given
by
u1 = 0,
u2 = 0.
(3) If u1 = 0, u2 6= 0: Solutions of (5.54) exist if
(3.1) If K1 − µ1 +µ2
2
A1= 0: The solution is given by
u1 = u,
u2 =K2u2
K2 −K1 +µ2
2
A1− µ2
2
A2
,
u3 =K3u3 + µ2
K3 −K1 +µ2
2
A1− µ2
2
A3
.
(3.2) If K1 − µ1 − µ22
A16= 0: The solution is given by
u1 = 0,
u2 =K2u2
K2 −K1 − µ22
A2+
µ22
A1
,
u3 =K3u3 + µ2
K3 −K1 − µ22
A3+
µ22
A1
.
• Isotropic case
For the isotropic rod we have
K =
K1 0 0
0 K1 0
0 0 K3
, A =
A1 0 0
0 A1 0
0 0 A3
and u =
0
0
u3
. (5.57)
Then the system of equations to be solved is equivalent tov1
v2
v3
= µ2
1A1
0 0
0 1A2
0
0 0 1A3
u1
u2
u3 − λ3
+
0
0
1
,
66
andK1 − µ1 − µ2
2
A10 0
0 K1 − µ1 − µ22
A10
0 0 K3 − µ1 − µ22
A3
u1
u2
u3 − λ3
=
0
0
K3(u3 − λ3) + µ2
.
(5.58)
Suppose that K1 6= K3 and A1 6= A3, then we have the following discussion
about the possible values of µ1 that yield a solution u = (u1, u2, u3) for
(5.58).
(1) µ1 = K1 − µ22
A1: We recover a case of a helix with curvature u and a
twist u3 with
u3 = λ3 +K3(u3 − λ3) + µ2
K3 −K1 +µ2
2
A1− 1
µ22A3
.
In terms of the constants η1, η2 and λ3 we have:
u1 = u cos ξ,
u2 = u sin ξ,
u3 = λ3 + η1,
where ξ lies in [0 2π] and the condition u · v = η2 gives an expression
of the magnitude of the force as a function of the twist.
(2) µ1 6= K1 − µ22
A1: There exist solutions given by
u1 = 0,
u2 = 0,
u3 = λ3 ± η1.
This last solution is also a twisted straight rod.
67
Inexte
nsi
ble
&U
nsh
eara
ble
Exte
nsi
ble
&Sheara
ble
Anisotropic
u16=
0u
26=
0u
1=
K1u
1
K1−µ
1
u2
=K
2u
2
K2−µ
1
u3
=K
3u
3+µ
2
K3−µ
1
u1
=K
1u
1
K1−µ
1−
µ2 2
A1
u2
=K
2u
2
K2−µ
1−
µ2 2
A2
u3
=K
3u
3+µ
2
K3−µ
1−
µ2 2
A3
u1
=0
u26=
0u
1=u
u2
=K
2u
2
K2−K
1
u3
=K
3u
3+µ
2
K3−K
1
u1
=u
u2
=K
2u
2
K2−K
1+
µ2 2
A1−
µ2 2
A2
u3
=K
3u
3+µ
2
K3−K
1+
µ2 2
A1−
µ2 2
A3
u16=
0u
2=
0u
1=
K1u
1
K1−K
2
u2
=u
u3
=K
3u
3+µ
2
K3−K
2
u1
=K
1u
1
K1−K
2−
µ2 2
A2
+µ
2 2
A1
u2
=u
u3
=K
3u
3+µ
2
K3−K
2−
µ2 2
A2
+µ
2 2
A3
u1
=u
u2
=0
u3
=K
3u
3+µ
2
K3−K
1
u1
=u
u2
=0
u3
=K
3u
3+µ
2
K3−K
1−
µ2 2
A3
+µ
2 2
A1
u1
=0
u2
=0
oror
u1
=0
u2
=u
u3
=K
3u
3+µ
2
K3−K
2
u1
=0
u2
=u
u3
=K
3u
3+µ
2
K3−K
2−
µ2 2
A3
+µ
2 2
A2
Isot.
u1
=0
u2
=0
u1
=u
cosξ
u2
=u
sinξ
u3
=λ
3+K
3(u
3−λ
3)
K3−K
1
u1
=u
cosξ
u2
=u
sinξ,
u3
=λ
3+
K3(u
3−λ
3)
K3−K
1+
µ2 2
A1−
µ2 2
A3
Tab
le5.
1:A
llpo
ssib
lest
rain
solu
tion
sfo
rth
edi
agon
alqu
adra
tic
stra
inen
ergy
for
give
nin
trin
sic
stra
ins
(u1,u
2,u
3)
and
give
nL
agra
nge
mu
ltip
lier
s(µ
1,µ
2,λ
3).
68
69
Chapter 6
An Intrinsic Variational Formulation of the
Non-canonical Action
In this section we develop a variational formulation describing families of equi-
librium configurations of a hyperelastic rod which is held fixed at both ends.
In deriving this variational formulation we use the approach of Euler-Poincare
equations [3, 42].
The energy functional is given by
J(r, r′,R,R′) =
∫ L
0
W(r, r′,R,R′) ds, (6.1)
where W is the strain-energy of the rod. It is a function of the centerline r
of the rod, the rotation tensor R that takes the fixed frame {e1, e2, e3} into the
moving one {d1,d2,d3}, and their first derivatives r′ andR′. Note that, as in this
dissertation we are concerned with uniform rods, we consider only the case where
W is independent of the variable s. The only restriction we place on W is that it
must be invariant under rigid body displacements. Such invariance restricts the
strain-energy density function W to be independent of r, and that its dependence
on R is only through the strains of the rod. The boundary conditions that we
consider here are those of a clamped-clamped rod, i.e., a rod for which positions
in space and orientation with respect to the fixed basis {e1, e2, e3} at both ends
are prescribed,
r(0) = 0, r(L) = rL, (6.2a)
R(0) = R0, R(L) = RL. (6.2b)
70
A necessary condition for J to have an extremum is that its first variation
vanishes for admissible variations of r and R, i.e., for variations that respect the
boundary conditions, and which respect the fact that R is a rotation.
To derive the first and the second variation, we consider for any scalar, vector
or tensor field variable, say ξ(s), a smooth one-parameter family of curves ξε(s)
such that ξ0(s) = ξ(s), and we denote by δξ the derivative of ξε with respect to
the parameter ε evaluated at ε = 0,
δξ =d
dεξε
∣∣∣∣ε=0
.
Similarly, we denote by δ2ξ the second derivative of ξε with respect to the pa-
rameter ε evaluated at ε = 0,
δ2ξ =d2
dε2ξε
∣∣∣∣ε=0
.
In the following sections, we consider in parallel the unconstrained case of
a shearable and extensible rod, and the case of an inextensible and unshearable
rod. In the inextensible and unshearable case dependence on r can be suppressed
provided only that boundary conditions on r are replaced by isoperimetric integral
side conditions. In both cases we first compute the first variation that yields the
equilibrium conditions, then we derive expressions for the second variation at
equilibrium.
6.1 Extensible and Shearable Rods
In the case of an extensible and shearable rod, the function W depends on R
through its dependence on the strains u and v in the following way
v =RTr′, (6.3a)
u×=RTR′. (6.3b)
Accordingly, we introduce the reduced density function
W(v− v,u− u) := W (RTr′,RTR′),
71
where v and u are the strains in the reference configuration. Then the strain-
energy density function associated with the rod becomes
J(v,u) =
∫ L
0
W(v− v,u− u) ds. (6.4)
6.1.1 Derivation of the First Variation
At equilibrium the rod is assumed to satisfy the principle of stationary action
expressed in the following variational statement
δJ =
∫ L
0
{∂W∂v· δv +
∂W∂u· δu}ds = 0. (6.5)
To compute the variation (6.5) we assume explicitly that the functions r, R and
their derivatives r′, R′ are sufficiently smooth and differentiable.
AsRRT = I, we have δRRT = −RδRT , i.e., δRRT is a 3×3 skew-symmetric
matrix so that there exists a vector δθ such that
δθ× = δRRT . (6.6)
By using δθ as a variation of the rotation R we avoid introducing a parameteri-
zation of SO(3). We will show that vanishing of the first variation of the energy
function yields the equilibrium equations in the fixed frame as the Euler-Poincare
equations, rather than the canonical Euler-Lagrange equations that would arise
if a parameterization of SO(3) was invoked.
The first variation of the strains expressed in (6.3),is given by
δv = δRTr′ +RT δr′, (6.7a)
δu× = δRTR′ +RT δR′. (6.7b)
We have
(Rδu)×w =R(δu×RTw)
= R(δu×RTw)
= Rδu×RTw,
72
which shows that (Rδu)× ≡ Rδu ×RT . As we also have R′RT = −RR′T , and
δRRT = −RδRT equation (6.7b) implies that
(Rδu)× = δRR′T
+ δR′RT .
On the other hand, by differentiation of (6.6) we get
δθ×′= δRR′
T+ δR′RT , (6.8)
which shows that
δθ×′= (Rδu)×. (6.9)
We can interchange the operations × and ′ in (6.9) and in (6.7a), then the first
variations of the strains u and v can be expressed as functions of the independent
variables δr and δθ as follows
δv =RT (δr′ + r′ × δθ), (6.10a)
δu =RT δθ′. (6.10b)
Hence by substituting the expression (6.10) for the first variation of the strains
in (6.5), and after integration by parts we find that∫ L
0
{−(R∂W∂v
)′· δr −
[(R∂W∂u
)′+ r′ ×
(R∂W∂v
)]· δθ}ds = 0, (6.11)
must hold for every admissible variation δr and δθ. The set of admissible varia-
tions h =
δrδθ
is
A ≡ {h ∈ R6 : h(0) = 0 = h(L)}.
Then equation (6.11) gives(R∂W∂v
)′= 0, (6.12a)(
R∂W∂u
)′+ r′ ×
(R∂W∂v
)= 0. (6.12b)
73
By using the constitutive hypothesis
n = Rn = R∂W∂v
, (6.13a)
m = Rm = R∂W∂u
, (6.13b)
relating the forces and moments to the strains, equations (6.12) yield the equi-
librium equations written in the fixed frame
n′ = 0, (6.14a)
m′ + r′ × n = 0. (6.14b)
Equations (6.12), give the necessary conditions that a critical point of the func-
tion J should satisfy. They were obtained by expressing the variation in the
rotation matrix R through the relation δθ× = δRRT .
If instead, we take the variation of the identityRTR = I, we obtain δ(RT )R =
−RT δR, and hence we conclude that RT δR is a skew-symmetric matrix. Let
δψ be the corresponding axial vector, i.e., such that
δψ× = RT δR.
Then we have the relation δψ× = RT δθ×R, or equivalently, δψ = RT δθ.
The first variation of the strains can then be expressed as
δv =RT δr′ + v× δψ, (6.15a)
δu = u× δψ + δψ′, (6.15b)
and, the first variation of the functional J has as Euler-Poincare conditions(∂W∂v
)′+ u× ∂W
∂v= 0, (6.16a)(
∂W∂u
)′+ u× ∂W
∂u+ v×
(∂W∂v
)= 0, (6.16b)
74
which after substitution of the constitutive equations (6.13) gives directly the
equilibrium equations written in the moving frame
n′ + u× n = 0, (6.17a)
m′ + u×m + v× n = 0. (6.17b)
6.1.2 Derivation of the Second Variation
Classification of the extrema given by the first-order conditions requires second-
order information. In this section, we calculate the second variation of the func-
tional (6.4).
By direct calculation from (6.5) we get
δ2J =
∫ L
0
{δv · Wvvδv + 2δu · Wuvδv +Wv · δ2v + δu · Wuuδu +Wu · δ2u
}ds.
(6.18)
In the following, using the chain rule, we give explicit formulas for the second
variation (6.18) in terms of the independent variables, (δr, δθ).
Using (6.6), variation of (6.10a) yields
δ2v = RT (δr′ × δθ) + δRT (δr′ + r′ × δθ)
= RT (δr′ × δθ)−RT δθ×(δr′ + r′ × δθ)
= RT [δr′ × δθ − δθ × r′ − δθ(r′ × δθ)].
Similarly, variation of (6.10b) yields
δ2u = δRT δθ′ = −RT (δθ × δθ)′.
In summary, the second variations of u and v are given by
δ2v=RT [(2δr′ + r′ × δθ)× δθ], (6.19a)
δ2u=RT (δθ′ × δθ). (6.19b)
75
Using the constitutive hypothesis given in (6.13) to express the quantities Wu
and Wv as a function of the net force n and the net moment m, the expression
(6.18) of the second variation can be written as
δ2J [h] = 12
∫ L
0
{h′ · Ph′ + h ·Ch′ + h′ ·CTh+ h ·Qh} ds, (6.20)
where h =
δrδθ
∈ R6 is an admissible variation. The coefficients P , C and
Q are 6× 6 matrices written in the fixed frame and are in principle s-dependent
through their dependence on the equilibrium at which they are evaluated. Ex-
plicitly we have
P =
RTWvvR RTWuvR
RTWvuR RTWuuR
, (6.21a)
C =
0 0
n× − r′×RWvvRT −r′×RWuvR
T + 12m×
, (6.21b)
and
Q =
0 0
0 −r′×RWvvRTr′× + 1
2(n×r′× + r′×n×)
. (6.21c)
6.1.3 Change of Frame
The matrices P , C andQ written in the fixed frame are related to the correspond-
ing matrices P, C and Q written in the moving frame, by a proper orthogonal
transformation. We here show that each block of P , C and Q is an explicit
function of the strains u and v, the net force n and the moment m and that
the same block of P, C and Q is given by the same function evaluated at the
corresponding body variables, i.e., u, v, n and m. The matrices P , C and Q in
76
(6.21) can be rewritten as
P =
RWvvRT RWuvR
T
RWvuRT RWuuR
T
= R
Wvv Wuv
Wvu Wuu
RT = RPRT , (6.22a)
C =
0 −r′×RWvvRT
n× −r′×RWuvRT + 1
2m×
(6.22b)
= R
0 −v×Wvv
n× −v×Wuv + 12m×
RT = RCRT , (6.22c)
and
Q =
0 0
0 −r′×RWvvRTr′× + 1
2(n×r′× + r′×n×)
(6.22d)
= R
0 0
0 −v×Wvvv× + 12(n×v× + v×n×)
RT = RQRT . (6.22e)
where R is the proper orthogonal transformation in SO(6) given by
R =
R 0
0 R
.
An alternate form of the expression of the second variation (6.20) is achieved
after integration by parts:
δ2J [h] = 12
∫ L
0
hT · Sh ds, (6.23)
where S is the self-adjoint second-order vector differential operator:
Sh ≡ − d
ds
[Ph′ +CTh
]+Ch′ +Qh. (6.24)
The utility of giving the expressions of the coefficients of the second order
operator (6.24) in different frames will be the ability to choose an appropriate
77
frame in which these coefficients are simpler. In particular for relative equilibria,
we will show that there is a special frame in which the coefficient matrices become
constants, and hence the analysis of (6.24) will be easier.
6.2 Inextensible and Unshearable Rods
For inextensible and unshearable, uniform rods, the strain-energy density W de-
pends on the tensor R only through its dependence on the strains u defined by
the relation u× = RTR′. The total energy functional is given by
J =
∫ L
0
W(u− u) ds (6.25)
which is to be minimized subject to the boundary conditions (6.2). In addi-
tion the inextensibility and unshearability condition restricts d3, the director
perpendicular to the cross-section, to be aligned with the tangent vector to the
centerline:
r′(s) ≡ dr(s)
ds= d3(s). (6.26)
Note that the given functional W (6.25) does not depend on r and r′. In ad-
dition, the boundary conditions (6.2) on r and the constraint (6.26), imply the
isoperimetric constraint on R∫ L
0
d3(R) ds :=
∫ L
0
Re3 ds =
∫ L
0
r′ ds = rL − r0. (6.27)
The isoperimetric condition (6.27) and the freedom in choice of initial condition in
(6.26) regarded as a quadrature for r(s), imply that a centerline satisfying (6.26)
can always be reconstructed if the isoperimetric constraint (6.27) is satisfied.
Thus, the variational problem of an inextensible and unshearable rod can be
formulated as an isoperimetrically constrained calculus of variations problem on
R only. We construct the associated functional
J =
∫ L
0
{W(u− u)− d3 · λ} ds, (6.28)
78
where λ is a constant vector of multipliers, and the variable r′ has been explicitly
eliminated. According to the usual multiplier rule, constrained critical points y of
J subject to the constraint (6.27) correspond to standard extrema (y,λc) of the
new functional J , and the multiplier λc is determined by the integral constraints
(6.27).
6.2.1 First Variation
The first variation of (6.28) is given by∫ L
0
{∂W∂u· δu− λ · δd3
}ds. (6.29)
In the expansion of (6.29) we need expressions of the variations of the strains u
and the variation of the vector d3. The expression of the variation of the strains
u is given as before by equation (6.10b), while the expression of the variation of
the vector d3 is given by the variation of the tensor R in the e3 direction. In
fact, by definition we have d3 = Re3. Then
δd3 =δ(Re3) = (δR)e3
=δθ×Re3 = δθ×d3. (6.30)
By direct substitution of the expression for the first variation of d3 and the
expression for the first variation of u, followed by integration by parts, equation
(6.29) becomes ∫ L
0
((R∂W∂u
)′− λ× d3
)· δθ ds = 0, (6.31)
which must hold for every variation δθ that vanishes at the end points. This
variational statement implies
m′ + d3 × λ = 0,
which is the balance of moments equation, with the identification λ = n, the
unknown force, and of the moments m = Rm with m is given by constitutive
relations.
79
Alternatively, using the variation δψ of the rotation R we have
δd3 = R(δψ × e3).
Substitution of the above and (6.15) into (6.29) followed by integration by parts
yields ∫ L
0
[(∂W∂u
)′+ u×
(∂W∂u
)− (RTλ)× e3
]· δψ ds = 0, (6.32)
which must hold for every variation δψ that vanishes at the end points. This
variational statement implies
m′ + u×m + d3 ×RTλ = 0,
which is the balance of moments equation in the moving frame, with the identi-
fication RTλ = n, the body components of the unknown force and m given by
constitutive relations.
6.2.2 Derivation of the Second Variation and Linearized
Constraints
The second variation δ2J , and its associated operator S, take the unconstrained
forms (6.20) and (6.24) but with matrix coefficients of dimension 3× 3 and with
the matrix Q having an additional term associated with the Lagrange multiplier
term. Moreover, the admissible variations θ ∈ R3 must satisfy the linearized
constraints ∫ L
0
δd3 ds = 0, (6.33)
giving rise to the admissible set of constrained variations,
Acons ≡{ζ ∈ R3 : ζ(0) = 0, ζ(L) = 0, and
∫ L
0
d3 × ζ ds = 0, i = 1, 2, 3
},
Explicitly we have
δ2J [ζ] = 12
∫ L
0
{δu · Wuuδu +Wu · δ2u + λ · δ2d3} ds, (6.34)
80
where ζ = δu ∈ R3 is a smooth variation in the strain u that vanishes at the ends
of the rod and satisfies the linearized constraint (6.33).
We now derive the expression of (6.34) in terms of δθ. Using the expression
(6.19b) for the second variation of u
δ2u = −RT (δθ × δθ′),
and by deriving the second variation of d3 from from the expression of the first
variation of d3 given in (6.30) and which yields to
δ2d3 = δθ×δθ×d3,
then, the second variation formula given in (6.34) becomes
δ2J [θ] = 12
∫ L
0
{δθ′ · P δθ′ + δθ ·Cδθ′ + δθ′ ·CT δθ + δθ ·Qδθ
}ds, (6.35)
where P , C and Q are 3 × 3 matrices, evaluated at the equilibrium under con-
sideration, given by the formulas
P = RWuuRT = RPRT , (6.36a)
C = 12m× = 1
2Rm×RT = RCRT , (6.36b)
and
Q = −12
(n×e3
× + e3×n×
)= −1
2R(n×d3
× + d3×n×
)RT = RQRT . (6.36c)
The linearized constraints are given by the integral∫ L
0
δθ × d3 = 0. (6.37)
Equation (6.37) is a vector representation of a system of three equations.
81
Chapter 7
Stability Analysis of Relative Equilibria
Our motivation for developing the second variation in Chapter 6 is to be able
to analyze stability properties of relative equilibria of uniform rods subject to
clamped-clamped boundary conditions. We will describe a given equilibrium y
as stable if
δ2J [y,h] = 12
∫ L
0
hT · S[y]h > 0, ∀h ∈ R6, h(0) = 0, h(L) = 0. (7.1)
Where S and h were first introduced in the previous chapter, equation (6.24).
Some general remarks are in order. First because the problem at hand is in
one space dimension, and the energy functional is strictly convex in the deriva-
tives of the unknowns, the analysis of whether or not a critical point is a local
minimum of the potential can be addressed solely by a study of the second vari-
ation (7.1). Second, we do not enter into a discussion of the delicate issues of
whether or not an equilibrium that is a local minimum of the potential is in fact a
Lyapunov stable solution of any associated dynamics of Caflisch and Maddocks [7]
for example. Instead we adopt the so-called static criterion involving the type of
critical point of the potential realized by a particular equilibrium. This property
is certainly closely related to dynamics stability properties, although the precise
connections are not simple. Third, the Dirichlet boundary conditions assumed
in (7.1) correspond to so-called hard-loading devices where both ends of the rod
are completely prescribed. This case is probably the simplest amongst a wider
class of physical pertinent cases. However the appropriate generalization of the
conjugate point theory necessary to consider more general boundary conditions
82
are not entirely straightforward and will not be considered here.
A necessary condition for (7.1) to be satisfied is Legendre’s strengthened con-
dition [6, 19], which requires that the matrix P in the expression (6.24) of the
operator S, or in its explicit form (6.21-a), be positive definite. We note that as
the entries of the matrix P correspond to the second derivative of a strictly con-
vex strain energy function with respect to the strains u and v, then P is always
positive definite at any equilibrium and therefore that Legendre’s strengthened
condition is always satisfied. Given Legendre’s strengthened condition, condition
(7.1) is equivalent to Jacobi’s necessary condition, namely that y has no conju-
gate point in [0, L] [40, 47]. The standard method for finding conjugate points
in the absence of isoperimetric constraints, which can be used for the extensible
and shearable case, is based on solving the initial value problem
S[y]h = 0, h(0) = 0, (7.2)
which admits a 6-parameter family solutions. A conjugate point arises at s = σ,
0 < σ < L, when there is a solution h 6≡ 0 in this family satisfying h(σ) = 0.
For the inextensible and unshearable problem, due to the additional isoperimetric
constraint one must use an appropriate isoperimetric conjugate point method for
testing the stability. In the following, we describe briefly, two approaches which
can be used for detecting conjugate points in the isoperimetric case. The first
method consists in analyzing a 6×6 constrained stability matrix, and the second
one involves a bifurcation analysis.
Explicitly, the first method consists in first finding a general solution to the
homogeneous initial value problem:
Sζ = 0, ζ(0) = 0,
which includes p undetermined constants:
ζ = c1ζ1 + c2ζ2 + · · ·+ cpζp.
83
For an isoperimetric constraints of the type∫ L
0
gi = 0, i = 1, . . . , n,
the variations ζ must satisfy the linearized constraints∫ L
0
ζ · δgi ds = 0, i = 1, . . . , n,
thus, for each i, we find a single solution to the inhomogeneous initial value
problem
Sζi = T i, ζi(0) = 0,
where the T i ≡ δgi is associated with the linearized constraints are taken to be
linearly independent (as function of s) on every interval (0, σ) for 0 < σ < L. For
example in the rod problem as the linearized isoperimetric constraint is given by∫ L0d×3 h ds then T i = d3 × ei, i = 1, 2, 3.
Now, we have the function
ζ =
p∑i=1
ciζi −n∑i=1
ciζi
as a general solution to
Sζ = −n∑i=1
ciT i, ζ(0) = 0.
A conjugate point arises at s = σ if there exist constants ci and ci such that ζ(s)
satisfies the boundary conditions at s = σ,
p∑j=1
cjζj(σ)−n∑j=1
cj ζj(σ) = 0, (7.3)
along with the linearized constraints,∫ σ
0
[p∑j=1
cjζj(s)−n∑j=1
cj ζj(s)
]TT i(s) ds = 0, i = 1, . . . , n. (7.4)
84
Equations (7.3) and (7.4) can be written in matrix form:
M(σ)
c
−c
≡A(σ) A(σ)
F (σ) F (σ)
c
−c
= 0, (7.5)
where
c ≡ [c1 . . . cp]T ∈ Rp,
c ≡ [c1 . . . cn]T ∈ Rn,
A(σ) ≡[ζ1(σ) . . . ζp(σ)
]∈ Rp×p,
A(σ) ≡[ζ1(σ) . . . ζn(σ)
]∈ Rp×n,
F (σ) ≡ {fij(σ)} =
{∫ σ
0
ζj(τ)TT i(τ)dτ
}∈ Rn×p,
F (σ) ≡{fij(σ)
}=
{∫ σ
0
ζj(τ)TT i(τ)dτ
}∈ Rn×n.
Equation (7.5), has a nontrivial solution only at values of σ; 0 < σ < L for which
the 6× 6 constrained stability matrixM(σ) has determinant zero. The vanishing
of detM is Bolza’s ([6], p. 220) definition of an isoperimetric conjugate point. In
direct analogy with the unconstrained case, Bolza’s treatment uses the absence
of a constrained conjugate point to rewrite the integrand in the second variation
as a complete square, and thus prove its positivity.
In the second method, we develop the Lagrangian (or variational) form of
the bifurcation analysis used in [33] for the particular circular equilibrium. We
begin with defining a bifurcation point. A necessary condition for there to be a
bifurcation is that the linearized Euler-Lagrange equations or Jacobi equations
are singular. That is, there is a nontrivial solution to the system
Sζ +n∑i=1
λiT i = 0, (7.6a)∫ L
0
ζ · T i ds = 0, i = 1, 2, 3, ζ(0) = 0, ζ(L) = 0. (7.6b)
where λi are constants. Following Maddocks [36], the bifurcation points are
classified as either type 1 bifurcations, if λ = 0, or type 2 bifurcations, if λ 6= 0.
85
The type 1 bifurcations occur at non-trivial solutions to the homogeneous system,
defined by setting λi = 0 in (7.6). A type 2 bifurcation point can be found by
first finding nonhomogeneous solution X i of (7.6) with λ 6= 0 then the matrix
U(L) = {SX i,Xj} = {T i,Xj} becomes singular at such points. Then we have a
type 2 bifurcation point at L = L1 if the determinant of U(L) vanishes at L = L1.
Conjugate point theory reveals that for any L < L1, the rod is stable.
7.1 The Second Variation at a Relative Equilibrium
In this section we compute the second variation at a relative equilibrium. We
first show that there exists a basis in which the second-order operator associated
with a general strain-energy function has constant coefficients. In principle, this
allows an explicit analysis of the second variation in both the extensible and
shearable, and the inextensible and unshearable problems. In Chapter 5, we
gave a geometric characterization of a given relative equilibrium in terms of the
parameters (η1, η2, λ3), which shows that the expressions of the tensor R, the
components of the moment m, of the internal force n and of the strains u and v
in the moving frame (d1,d2,d3) associated to a helical rod are given by
R(s) = Π(η1s)R0Q(λ3s), (7.7a)
m(s) = QT (λ3s)m0, n(s) = QT (λ3s)n0, (7.7b)
u(s) = QT (λ3s)u0, v(s) = QT (λ3s)v0, (7.7c)
where the zero subscript implies evaluation at s = 0, Π is given in (5.28) and Q
is given in (5.14).
7.1.1 Extensible and Shearable Rod
The case of an extensible and shearable rod is an unconstrained problem. The
coefficients of the second order operator are 6×6 matrices. The explicit expression
86
of the second order operator evaluated at a relative equilibrium is determined by
substituting (7.7) into (6.20), then the second variation for the extensible and
shearable rod, is given by
δ2J [ζ] =1
2
∫ L
0
{ζ ′ · Pζ ′ + ζ · Cζ ′ + ζ ′ · C
Tζ + ζ · Qζ
}ds, (7.8)
where ζ ∈ R6 is given by
ζ =
RT0 ΠT (η1s) 0
0 RT0 ΠT (η1s)
h. (7.9)
The matrices P, C and Q are given by
P = Q(λ3s)PQ(λ3s)T = P0, (7.10a)
C = −u×P + C0, (7.10b)
Q = −u×Pu× − u×CT0 + C0u× + Q0. (7.10c)
The matrix u×0 is a 3 × 3 skew matrix with axis given by the shifted strain u
evaluated at s = 0. Recall that for our solution we denote by u, the triple
[u1, u2, u3 − λ3], where u1 and u2 are the bending strains and u3 is the twist.
The matrices P, C and Q depend only on the strains of the rod evaluated at
s = 0. The justification of the expression of P is based on the fact that for an
anisotropic rod, the strains are constants so that P is constant. For the isotropic
case, relations (7.7) involving the strains shows that P = Q(λ3s)TP0Q(λ3s).
Therefore, the operator S associated with the second variation (7.11) has always
constant coefficients. The stability analysis in the extensible and shearable can be
done by using the standard conjugate points by solving the initial value problem
(7.2).
7.1.2 Inextensible and Unshearable Rod
For the inextensible and unshearable rod, the coefficients of the second order
operator are 3 × 3 matrices. The difficulty in solving this case comes from the
87
presence of an isoperimetric constraint. In this section we first show, as was
done for the previous case, that the second-order operator associated with the
second variation of the isoperimetric case has constant coefficients. Using the
same notation as for the extensible and shearable case, we obtain the following
expression of the second variation
δ2J [ζ] = 12
∫ L
0
{ζ ′ · Pζ ′ + ζ · Cζ ′ + ζ ′ · C
Tζ + ζ · Qζ
}ds, (7.11)
where the variation is expressed by ζ = RT0 ΠT (η1s)δθ. The components ζ1, ζ2
and ζ3 measure perturbations in the direction of the principal normal, the binor-
mal and the tangent to the helix, respectively.
The 3× 3 matrices C and Q in (7.11) are given by
C = −u×0 P0 + C0, (7.12a)
Q = −u×0 P0u×0 − u×0 CT0 + C0u×0 + Q0, (7.12b)
with u×0 being the skew matrix with axis in the direction of the strains at s = 0.
By the same argument used in the previous section, these matrices are constant.
As an application, we analyze the stability of a relative equilibrium for an in-
extensible and unshearable rod where the strain-energy function is quadratic and
diagonal. The equilibria in this case were described in detail in Chapter 5. There,
we showed that in the anisotropic case, the existence of a non degenerate helix
depends essentially on the intrinsic strains of bending and upon a parameter, µ1,
which corresponds to one of the Lagrange multipliers of the dual formulation. In
this treatment, we regard the constants η1, η2 and λ3 as given which is equivalent
to regard, the curvature, torsion and twist as given.
In order to simplify notation, we denote by u1, u2 and u3 the strains at s = 0
(omitting the zero subscript), and we introduce the constants γ, κ, τ and b for
given η1, η2 and λ3, for given stiffnesses K1, K2 and K3 and for given intrinsic
88
twist u3:
γ =K3
K1
, (7.13a)
κ2 = η21(1− η2
2), (7.13b)
τ = η1η2, (7.13c)
u3 = τ + λ3, (7.13d)
b =K3(u3 − u3)− 2τ(K1 +K2)
µ1
, (7.13e)
ω2 = b2 + κ2. (7.13f)
The physical interpretation of these constants is as follows: κ is the curvature, τ
is the torsion of the helix, γ is the ratio of the rigidities of twisting and bending,
u3 is the physical twist in the rod, and b and ω are two constants which will
appear frequently in the expression of the second variation.
In our analysis of the second variation for the inextensible and unshearable
rod, we will consider only non-degenerate helices for which both κ and τ are
non zero. We assume, without loss of generality, that for the anisotropic rod,
K1 < K2.
As was shown in Chapter 5 a helical solution exists if µ1 = K1 or if µ1 = K2
for the intrinsically straight but possibly twisted rod. If the rod is intrinsically
curved, the strains u1 and u2 are given
u1 =K1
(K1 − µ1)u1, (7.14a)
u2 =K2
(K2 − µ1)u2. (7.14b)
The value of µ1 in (7.14) can be found by solving the equation u21+u2
2 = η21−η1η2,
for given η1 and η2, which gives rise to, depending on the values of the intrinsic
bending, u1 and u2, to at most four possible values of µ1.
For the isotropic rod, K1 = K2, u1 = 0, u2 = 0 and the only possible non-
degenerate helices have µ1 = K1.
89
An evaluation of twice the second variation at a relative equilibrium for an
inextensible and unshearable rod, yields the following expression
(K1 − µ1)
∫ L
0
{u2
κζ ′1 −
u1
κζ ′2 +
τu1
κζ1 +
τu2
κζ2 − u2ζ3
}2
ds (7.15a)
+ (K2 − µ1)
∫ L
0
{u1
κζ ′1 +
u2
κζ ′2 +
τu2
κζ1 −
τu1
κζ2 + u1ζ3
}2
ds (7.15b)
+ µ1
∫ L
0
{ζ ′1
2 − ω2ζ21 + (ζ ′2 − bζ1)2
}ds (7.15c)
+K3
∫ L
0
{ζ ′3 − κζ1}2ds. (7.15d)
The variation vector ζ should satisfy the homogeneous boundary conditions
ζ(0) = ζ(L) = 0 and the following linearized constraints∫ L
0
ζTT 1 ds = 0, (7.16a)∫ L
0
ζTT 2 ds = 0, (7.16b)∫ L
0
ζTT 3 ds = 0, (7.16c)
with
T 1 =
τ cos(η1s)
κ sin(η1s)
0
, T 2 =
τ sin(η1s)
κ cos(η1s)
0
, T 3 =
1
0
0
.
In the following we discuss the stability properties of the isotropic case using the
isoperimetric conjugate point method introduced at the beginning of this chapter.
• Isotropic rod
We begin our analysis of the second variation, with the isotropic case which
corresponds to the single value of µ1 = K1 = K2. Hence, the expression of
the second variation (7.15) simplifies to
δ2J [ζ] = K1
∫ L
0
{ζ ′1
2 − ω2ζ21 + (ζ ′2 − b ζ1)2 + γ(ζ ′3 + κζ1)2
}ds. (7.17)
90
We remark that when ζ1(s) = 0 δ2J [ζ] > 0, so we need to consider only the
case where ζ1(s) 6= 0. We note also that for L sufficiently short∫ L
0
{ζ ′1
2 − ω2ζ21
}ds > 0,
so that, as other terms are non-negative, δ2J [ζ] > 0. Therefore we want to
locate the smallest L for which δ2J [ζ] = 0, which corresponds to the first
conjugate point or equivalently to the first bifurcation point.
In this particular case, we will use the special isoperimetric conjugate point
method introduced at the beginning of this chapter and which consists
on building a 6 × 6 matrix for which the smallest root of its determinant
corresponds to the first conjugate point. Then we will develop a bifurcation
analysis of first and second type.
As was observed before, the method of isoperimetric conjugate point re-
quires the construction of a 6×6 matrix for which the determinant vanishes
at a conjugate point. We give here such a matrix in closed form. However
we have not been able to obtain the roots of its determinant analytically.
We first start by giving the homogeneous solution that satisfies the initial
value problem
Sζ = 0, ζ(0) = 0 and which is given by a system of three second order
differential equations
ζ ′′3 − κ ζ ′1 = 0, (7.18a)
ζ ′′2 − b ζ ′1 = 0, (7.18b)
ζ ′′1 + b ζ ′2 + γκ ζ ′3 − (γ − 1) κ2ζ1 = 0. (7.18c)
with ζ =
ζ1
ζ2
ζ3
and ζ(0) = 0. The system (7.18) gives rise to the following
91
three possible independent solutions
ζ1 =
sin(ωs)
− bω
(cos(ωs)− 1)
− κω
(cos(ωs)− 1)
, ζ2 =
1− cos(ωs)
− bω
(ωs− sin(ωs)) + ω2
bx
κω
(ωs− sin(ωs))
, ζ3 =
0
−γκs
bs
.
On the other hand, the nonhomogeneous solution for which Sζ = T i, ζ(0) =
0, taking into account of the constraint (7.16) gives rise to another set of
three independent solutions
ζ1 =
sin(η1s)
τ(bη1−κτ)η1(τη1−bκ)
cos(η1s) + κ2γη1bs
− κη1
(cos(η1s)− 1)
, ζ2 =
1− cos(η1s)
τ(bη1+κτ)η1(τη1+bκ)
sin(η1s)− κ2γbs
κη1
(η1s− sin(η1s))
, ζ3 =
0
0
s
.
Therefore the general solution to the homogeneous initial value problem:
Sζ = −3∑i=1
ciT i, ζ(0) = 0
is given by the function
ζ =3∑i=1
ciζi −3∑i=1
ciζi.
A conjugate point arises if the following system admits non-trivial solutions
M(σ)
c
−c
≡A(σ) A(σ)
F (σ) F (σ)
c
−c
= 0, (7.19)
The blocks A and A are already built. The expressions for F and F are
not given here, because they are complicated.
The expression of the determinant ofM(σ) can be found using the symbolic
software Maple c© but the roots are more complicated to find explicitly. We
give in Fig. 7.1, a plot of the determinant as a function of σ for the indicated
values of the parameters (7.13).
92
Figure 7.1: Plot of the determinant of the constrained matrix M as a functionof σ for a particular choice of the twist u3 = 20, η1 = 2π, torsion τ =
√2π.
The intersection of the curve with the abscissa axis corresponds to a root of thedeterminant of the matrix M and hence to a conjugate point.
The second possible approach that permits localization of conjugate points
consists of developing a bifurcation analysis. Recall that the second varia-
tion for the isotropic rod is given by
δ2J [ζ] = K1
∫ L
0
{ζ ′1
2 − ω2ζ21 + (ζ ′2 − b ζ1)2 + γ(ζ ′3 + κζ1)2
}ds, (7.20)
This analysis involves solutions of the following Euler-Lagrange equations
ζ ′′3 − κ ζ ′1 = 0, (7.21a)
ζ ′′2 − b ζ ′1 = λ1κ sin(η1s) + λ2κ cos(η1s), (7.21b)
ζ ′′1 + b ζ ′2 + γκ ζ ′3 − (γ − 1) κ2ζ1 = λ1τ cos(η1s) + λ2τ sin(η1s) + λ3,
(7.21c)
subject to two boundary conditions ζ(0) = ζ(L) = 0 and the linearized
constraints given in (7.16).
Type 1 Bifurcation Points
93
First type bifurcation points are solutions of equation (7.21) with λ1 = λ2 =
λ3 = 0 that satisfy the Dirichlet boundary conditions and the linearized
constraints given in (7.16). The system (7.21) reduces to
ζ ′′3 − κ ζ ′1 = 0, (7.22a)
ζ ′′2 − b ζ ′1 = 0, (7.22b)
ζ ′′1 + b ζ ′2 + γκ ζ ′3 − (γ − 1) κ2ζ1 = 0. (7.22c)
By taking into account the constraint∫ L
0ζ1 = 0 and the boundary con-
ditions, we can give the solution of the first two equations in (7.22) as a
function of ζ1
ζ2(s) = b
∫ s
0
ζ1(σ)dσ,
ζ3(s) = κ
∫ s
0
ζ1(σ)dσ.
A straight forward computation shows that the system of three equations
(7.22) then reduces to the single equation for ζ1
ζ ′′1 + ω2ζ1 = 0, (7.23)
with homogeneous boundary conditions, and subject to the following set of
constraints ∫ L
0
ζ1 ds = 0, (7.24)
(τ +bκ
η1
)
∫ L
0
ζ1 cos(η1s) ds = 0, (7.25)
(τ − bκ
η1
)
∫ L
0
ζ1 sin(η1s) ds = 0. (7.26)
The general solution of (7.23) is given by ζ1 = sinωs. In order to satisfy
the boundary conditions and the linearized constraints we find that
L =2pπ
ω, (7.27)
L =2qπ
η1
, (7.28)
94
where p and q are integers. The relation L = 2qπη1
shows that type I bifurca-
tion can occur only at an integer number of turns of the helical centerline
of the rod. Then the first type 1 bifurcation is given by L = 2pπω
, with
ω2 = κ2 + (γ(u3 − u3)− 2τ)2, κ is the curvature, τ is the torsion, u3 is the
twist and u3 is the intrinsic twist. The relations given in (7.28) with the
relation κ2 + τ 2 = η21 restrict the possibility of detection of first type bi-
furcation points to only a class of rods with very special relations between
the parameters given by the curvature, torsion of the centerline and the
physical twist.
Type 2 Bifurcation Points
A type 2 bifurcation point can be found by constructing a matrix U(L) =
{SX i,Xj} = {T i,Xj}. Then we need to find the solutions Xi of the
system (7.21) that satisfies a homogeneous boundary conditions and the lin-
earized constraints (7.16). By taking into account the constraint∫ L
0X1(σ) dσ =
0, with the boundary conditions, the solutions of (7.21) for X2 and X3 are
given as a function of X1 by
X2(s) = b
∫ L
0
X1(σ) dσ +λ1κ
η21
[sin(η1s)−
sin(η1L)
Ls
]+
λ2κ
η21
[cos(η1s)− 1− cos(η1L)− 1
Ls
],
X3(s) = κ
∫ L
0
X1(σ) dσ.
Then the system of equations (7.21) reduces to one equation
−X ′′1 − ω2X1 =(τ +bκ
η1
)λ1 cos(η1s) + (τ − bκ
η1
)λ2 sin(η1s) +
{λ3 −bκ
η21L
(λ1 sin(η1L)− λ2(cos(η1L) − 1))}, (7.29)
for which solution X1, must satisfy the homogeneous boundary conditions
95
and the following set of constraints∫ L
0
X1 ds = 0, (7.30a)
(τ +bκ
η1
)
∫ L
0
cos(η1s)X1 ds +
λ1κ
η21
∫ L
0
{cos2(η1s)−sin(η1L)
Lcos(η1s)} ds−
λ2κ
η21
∫ L
0
{cos(η1s) sin(η1s) +cos(η1L) − 1
Lcos(η1s)} ds = 0, (7.30b)
(τ − bκ
η1
)
∫ L
0
sin(η1s)X1 ds−
λ1κ
η21
∫ L
0
{cos(η1s) sin(η1s)−sin(η1L)
Lsin(η1s)} ds +
λ2κ
η21
∫ L
0
{sin2(η1s) +cos(η1L)− 1
Lsin(η1s)} ds = 0. (7.30c)
The solution X1 of equation (7.29) that satisfies the boundary condition is
given in terms of the three constants λ1, λ2 and λ3 as
1
η1ω2L sin(ωL){(ω2ALη1 sin(ωs)(cos(η1L) − cos(ωL)) +
ω2ALη1 sin(ωL)(cos(ωs)− cos(η1s)) + sin(ωL))λ1 +
κb sin(η1L)[sin(ωL) + sin(ωs) cos(ωL)− cos(ωs)− sin(ωs)]}+
1
η1ω2 sin(ωL)L(−ω2BLη1(sin(η1s) sin(ωL) + sin(ωs)
sin(η1L)) + κb cos(η1L)(sin(ωs) cos(ωL) + sin(ωL))−
κb(sin(ωs) cos(ωL) + cos(ωs) sin(ωL)− sin(ωL)−
cos(ωs) cos(η1L) sin(ωL)− sin(ωs) cos(η1L) + sin(ωs))λ2)−
λ3
sin(ωL)ω2{sin[ω(s− L)] + sin(ωL)− sin(ωs)},
where A = (τ + bκη1
)(ω2 − η21) and B = (τ − bκ
η1)(ω2 − η2
1). If we substitute
this solution in the system of constraints given in (7.30), then we will get
96
three homogeneous equations in λ1, λ2 and λ3 expressed by the equation
Uλ = 0, (7.31)
where λ =
λ1
λ2
λ3
and U is the 3× 3 matrix given on the next page.
Equation (7.31) will have non trivial solution only if the matrix U becomes
singular.
97
U(L
)=
ωA
2[(S
2−
2)co
s(ωL
)+
2C]
sin(ωL
)
+κ
2[2S
2−L
(CS−η 1L
)]
2Lη
3 1
+
[ L(η
2 1−ω
2)−
(ω2
+η
2 1)CS
η 1
] A2 2
ωSAB
[1−C
cos(ωL
)]
sin(ωL
)
+[κ
2−η
2 1(ω
2+η
2 1)AB
]S2
2η3 1
+κ
2(C−
1)S
Lη
3 1
{ (C+
1)[1−
cos(ωL
)]
ωsi
n(ωL
)−S η 1
} A
ωSAB
[1−C
cos(ωL
)]
sin(ωL
)
+[κ
2−η
2 1(ω
2+η
2 1)AB
]S2
2η3 1
+κ
2(C−
1)S
Lη
3 1
−ωB
2S
2co
s(ωL
)
sin(ωL
)
+κ
2[2
(C−
1)2
+L
(CS−η 1L
)]
2Lη
3 1
+
[ L(η
2 1−ω
2)
+(ω
2+η
2 1)CS
η 1
] B2 2
{ S[1−
cos(ωL
)]
ωsi
n(ωL
)+
(C−
1)
η 1
} B
{ (C+
1)[1−
cos(ωL
)]
ωsi
n(ωL
)−S η 1
} A
{ S[1−
cos(ωL
)]
ωsi
n(ωL
)+
(C−
1)
η 1
} B2[
1−
cos(ωL
)]
ω3
sin(ωL
)−
L ω2
,
wher
e
τ=η 1η 2,
κ=√ η
2 1−τ
2,
µ1
=K
1,
b=K
3
K1
u3−
2τ,
ω=√b2
+κ
2,
C=
cos(η 1L
),S
=si
n(η
1L
),
A=
(ω2−η
2 1)(τ
+bκ η 1
),B
=(ω
2−η
2 1)(τ−bκ η 1
).
The
free
par
amet
ers
areη 1
,η 2
andu
3.
98
• Anisotropic case
For the anisotropic case, we will distinguish three cases in the analysis of
the second variation (7.15). First we consider an anisotropic rod which is
intrinsically straight, then we have to analyze two cases: the case of a rod
which bent in the soft direction and the case of a rod which is bent in the
stiff direction. The third case follows from considering an anisotropic rod
which may have an intrinsic strains of bending. For an anisotropic rod
which is initially straight we have either
µ1 = K1, u1 = κ, u2 = 0, u3 = τ, (7.32)
or,
µ1 = K2, u1 = 0, u2 = κ, u3 = τ. (7.33)
For the case given in (7.32) the rod is deformed in the plane of the minimum
bending stiffness. The expression (7.15) of twice the second variation, where
we take µ1 = K1, u1 = κ, u2 = 0 simplifies to
K1
∫ L
0
{ζ ′1
2+ ζ ′2
2 − b(ζ ′2ζ1 − ζ ′1ζ2)− {ζ ′1 − τζ2 − κζ3}2 − κ2ζ12}ds
+K2
∫ L
0
{ζ ′1 − τζ2 − κζ3}2ds+K3
∫ L
0
{ζ ′3 − κζ1}2ds.
For the case given in (7.33) the rod is deformed in the plane of the maximum
bending stiffness and the expression of twice the second variation (7.15) for
which we take µ1 = K2, u1 = 0, u2 = κ reduces to
K2
∫ L
0
{ζ ′1
2+ ζ ′2
2 − b(ζ ′2ζ1 − ζ ′1ζ2)− {ζ ′1 + τζ2 − κζ3}2 − κ2ζ12}ds
+K1
∫ L
0
{ζ ′1 + τζ2 − κζ3}2ds+K3
∫ L
0
{ζ ′3 − κζ1}2ds.
If we consider an intrinsically curved rod, twice the second variation in the
anisotropic case takes the complete form (7.15) with the strains u1 and u2
99
are related the the intrinsic curvature through the relations
u1 =K1
K1 − µ1
u1,
u2 =K2
K2 − µ1
u2,
u3 = τ,
for the given intrinsic curvatures u1 and u2, and for given constants η1 and
η2; the constant µ1 can be one of the roots of the equation u21+u2
2 = η21−η1η2:
K21
(K1 − µ1)2u2
1 +K2
(K2 − µ1)2u2
2 − η21 − η1η2 = 0
giving rise to at most four possible values of the constant µ1. We restrict
our analysis of the anisotropic case by just giving general remarks about
this stability properties:
– It can immediately be seen that for µ1 ≤ K1, (7.15a) and (7.15b) are
both positive and then the associated equilibrium is stable whenever
the equilibria of the concomitant isotropic rod is stable. Here con-
comitant is used in the same sense as in [37] where it is defined as
the isotropic rod with rigidity of bending equal to µ1 especially for the
initially straight rod this will be the soft direction.
– The variation ζ3, along the tangent to the helix has to satisfy only the
homogeneous boundary conditions. Furthermore, it can be demon-
strated that twice the second variation evaluated on the functions
ζ1 ≡ 0, ζ2 ≡ 0
is ∫ L
0
{K3ζ
′3
2 − [(µ1 −K2)u21 + (µ1 −K1)u2
2]ζ23
}ds. (7.36)
The solution ζ3 is not involved in the linearized constraints then solu-
tion of (7.36) has to satisfy only the homogeneous boundary conditions.
100
Such solution exists for sufficiently long rod with parameters that make
the quantity [(µ1 − K2)u21 + (µ1 − K1)u2
2] positive which proves that
for this class of rods, such as the case of an anisotropic rod which is
bent in the stiff direction and for which µ1 = K2 > K1, long enough
rods are unstable.
We conclude, in the end of this section, that for the analysis of the differ-
ent cases of an inextensible and unshearable rod, all the functions necessary for
an explicit conjugate point analysis are in principle analytically computable be-
cause the systems of second order differential equations are constant coefficients.
However the computations, for an arbitrary values of the parameters becomes
complicated at the stage of finding the roots of the determinant either for the
6 × 6 matrix of the 3 × 3 one (referring here the simplest case of an isotropic
case). At this stage, for given parameter values, conjugate points can easily be
found numerically.
101
Chapter 8
Conclusion
The departure point for this thesis is the non-canonical Hamiltonian formulation
of the equations governing the equilibria of special Cosserat rods. We considered
the case of a uniform hyperelastic rod that can undergo bending and twist along
with either deformations of extension and shear, or is inextensible and unshear-
able. In this formulation, three or four integrals arise depending on whether or
not the rod is isotropic. In all cases, the Hamiltonian H, the moment m · n,
in the direction of the internal force and the norm of the internal force |n| are
conserved quantities. In the case of an isotropic rod, m · d3, the moment in the
direction of d3, is the fourth conserved quantity.
In any Hamiltonian system, relative equilibria are special solutions which
evolve on critical sets of the integrals. In the case of the Hamiltonian system
governing uniform rods, relative equilibria are shown to correspond to helical
configurations of the rod.
At relative equilibrium, the components of the internal moment, internal force
and strains in the Frenet frame of the centerline are constant along the rod. We
also found that the presence of the additional symmetry associated with isotropy,
is the only case where a difference between the physical twist of the rod and the
geometrical torsion of the centerline arises. This excess twist is exactly given by
a Lagrange multiplier associated with the additional integral of isotropy.
Two finite-dimensional dual variational principles, involving point-wise values
of the strain-energy density and its conjugate function, are used: one is in terms
of the strains and the other in terms of the stresses. The stress formulation is
102
standard from the point of view of Hamiltonian systems, but the dual formulation
in terms of the strains only arises because all integrals other than the Hamilto-
nian are quadratic, and the matrix of Lagrange multipliers is, in an appropriate
sense, invertible. The dual formulation permits a simple analysis of the existence
and multiplicity of relative equilibria. For the anisotropic rod, relative equilib-
ria correspond to two-parameter (radius and pitch) families of helical centerlines
with a discrete number, greater than or equal to two, of possible orientations
of the cross-section at equilibrium. For isotropic rods, the possible equilibrium
configurations are members of a single four-parameter family: two for the radius
and the pitch of the helical centerline, one for the imposed twist, and one for the
possible cross-section orientations. We do not believe that this dual variational
principle has been discussed in the context of rods. For the Kirchhoff analogue
of rigid body dynamics, a related, but distinct, dual variational characterization
of relative equilibria has previously been used in [54].
Our analysis demonstrates that, by considering the four integrals, it is possible
to treat the cases of isotropic and anisotropic rods within the same framework.
The case of anisotropic rod is obtained merely by setting to zero the Lagrange
multiplier associated with the last integral. We illustrated our analysis by giving
a complete classification of relative equilibria for a diagonal, quadratic strain-
energy density. We discussed the effects of shear and extension, and the influence
of the intrinsic curvature on the possible solutions and their multiplicities. For
the different cases, we gave the range of possible values of the parameters that
yield a helical centerline.
In the second part of the dissertation, stability properties of elastic rod equi-
libria were analyzed. We used a variational formulation of the two point boundary
value problem describing equilibrium configurations of a hyperelastic rod of finite
length which is held fixed at its both ends. In the case of an inextensible and
103
unshearable rod, the constrained problem is reformulated as an isoperimetric cal-
culus of variation problem, with the constraints enforcing boundary conditions on
the centerline of the rod. For both the constrained and unconstrained problems,
we used Euler-Poincare equations to derive the first and second variations of the
strain energy density in the vicinity of a given equilibrium in a coordinate-free
formulation. In particular, by using a variational formulation of an action which
is invariant under rigid body displacement, the equations governing the statics of
hyperelastic rods can be derived, and the equilibria can be classified in terms of
the second variation as local minima or not, without using a parameterization to
describe the moving frame {di}. This formulation has the advantage over oth-
ers, such as those involving Euler angles, or Euler parameters, that there is no
artificial coordinate singularity nor parameterization-induced constraints. The
analysis of the first variation has previously appeared in the literature, for exam-
ple see [42, 4], but the derivation of the second variation has not apparently been
considered before. The straightforward computations of the second variations
makes the analysis of the stability properties of relative equilibrium in principle
possible for a large class of strain energy functions. In fact, we proved that, when
evaluated at relative equilibrium, the self-adjoint operator associated with the
second variation, is a large system of second-order ODEs with constant coeffi-
cients which can in principle be analyzed completely. We present an analysis of
stability using an isoperimetric conjugate point method on both the isotropic and
anisotropic cases. We do not give an analytic expression for the limit length of
stability of the rod for the analyzed cases. The main difficulty is due to the pres-
ence of many parameters, making the expressions in the last step of the analysis
very complicated. Nevertheless the methods presented allow a simple stability
classification of any given helical equilibrium of a rod with given constitutive
relations. We showed that the stability of the rod is controlled by the different
parameters of the relative equilibrium under consideration, that sufficiently short
104
helical segments are stable, and that long enough rods can be destabilized by the
presence of the twist.
105
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Vita
Nadia Chouaıeb
Personal Data:
Tunisian citizen.
Married and mother of three children.
Education & Diplomas
1985 “Baccalaureat Math-Sciences”, Habib Maazoun Secondary School, Sfax,Tunisia.
1985-91 Attended the “Ecole Nationale d’Ingenieurs de Tunis” (ENIT), Uni-versity of Tunis II, Tunisia.
1991 “Ingenieur Diplome” in Civil Engineering, ENIT.
1992 “Diplome des Etudes Approfondies” in Applied Mechanics, ENIT.
1993-95 Engineer at “Ecole Polytechnique de Tunis”, La Marsa, Tunisia.
1996-99 Graduate work in Mechanics, Rutgers, The State University of NewJersey.
1999 M.S. in Mechanics, Rutgers University.
1999- Assistant in the Chair of Applied Analysis, at EPFL.