Waves on Elastic Rods

and Helical Spring Problems

by

Bojan Durickovic

A Dissertation Submitted to the Faculty of the

Graduate Interdisciplinary Programin Applied Mathematics

In Partial Fulfillment of the RequirementsFor the Degree of

Doctor of Philosophy

In the Graduate College

The University of Arizona

2 0 1 1

2

THE UNIVERSITY OF ARIZONA

GRADUATE COLLEGE

As members of the Dissertation Committee, we certify that we have read the disser-tation prepared by Bojan Durickovic entitled

Waves on Elastic Rodsand Helical Spring Problems

and recommend that it be accepted as fulfilling the dissertation requirement for theDegree of Doctor of Philosophy.

Date: May 20, 2011Alain Goriely

Date: May 20, 2011Michael Tabor

Date: May 20, 2011Andrew Hausrath

Final approval and acceptance of this dissertation is contingent uponthe candidate’s submission of the final copies of the dissertation tothe Graduate College.

I hereby certify that I have read this dissertation prepared undermy direction and recommend that it be accepted as fulfilling thedissertation requirement.

Date: May 20, 2011Alain Goriely

3

Statement by Author

This dissertation has been submitted in partial fulfillment of re-quirements for an advanced degree at The University of Arizona andis deposited in the University Library to be made available to bor-rowers under rules of the Library.

Brief quotations from this dissertation are allowable without spe-cial permission, provided that accurate acknowledgment of source ismade. Requests for permission for extended quotation from or repro-duction of this manuscript in whole or in part may be granted by thehead of the major department or the Dean of the Graduate Collegewhen in his or her judgment the proposed use of the material is in theinterests of scholarship. In all other instances, however, permissionmust be obtained from the author.

Signed: Bojan Durickovic

4

Acknowledgements

First and foremost, I would like to thank my advisor, Alain Goriely. In additionto my gratitude for his guidance, and both stimulating and enlightening discussions,I am very much indebted for his encouragement and support through some difficulttimes. Without him, completing this dissertation would have seemed (and thereforebe) an impossible task.

Thank you Michael Tabor, my Committee member, Program Head, and researchgroup chair, for having created such a wonderful atmosphere of learning, and for allthe suggestions for improvement of this dissertation. Thank you also to the Programstaff (Stacey Wiley in particular) for providing the best administrative support I canimagine. I feel privileged to have become a part of the University of Arizona AppliedMath family.

I also wish to thank John H. Maddocks for some insightful discussions aboutthe chapter on the Inverse Helical Spring Problem, my Committee member AndrewHausrath for helping me find my way in dealing with optimization problems, andSebastien Neukirch for having reviewed this dissertation with an incredibly keen eye,providing many comments that have resulted in substantial improvements of the workbefore you.

5

Table of Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Chapter 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 130.1. A brief historical overview of the theory of elastic rods . . . . . . . . 130.2. Elastica and localized planar waves . . . . . . . . . . . . . . . . . . . 160.3. Helical springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 1. Governing equations . . . . . . . . . . . . . . . . . . . . . 211.1. Special Cosserat rods . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.1.1. Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.1.2. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.1.3. Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.1.4. Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.1.5. Complete set of governing equations . . . . . . . . . . . . . . 301.1.6. Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . 311.1.7. First integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.1.8. Variational characterization of equilibria . . . . . . . . . . . . 321.1.9. First integral surfaces in strain space . . . . . . . . . . . . . . 34

1.2. Kirchhoff rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.2.1. Inextensibility and unshearability constraint . . . . . . . . . . 341.2.2. Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.2.3. Kirchhoff equations . . . . . . . . . . . . . . . . . . . . . . . . 371.2.4. Variational characterization of equilibria . . . . . . . . . . . . 371.2.5. First integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.2.6. Kirchhoff top analogy . . . . . . . . . . . . . . . . . . . . . . . 381.2.7. Diagonal case . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.2.8. Isotropic rods . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.3. Helical rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.3.1. Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.3.2. Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.4. Helical Kirchhoff rods . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.4.1. Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.4.2. Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.4.3. First integral surfaces in twist space . . . . . . . . . . . . . . . 46

Table of Contents — Continued

6

1.4.4. Helix hyperboloid . . . . . . . . . . . . . . . . . . . . . . . . . 47

Chapter 2. Compact planar waves . . . . . . . . . . . . . . . . . . . . 492.1. Traveling wave reduction . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.1.1. Equivalent static system . . . . . . . . . . . . . . . . . . . . . 502.2. Planar system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.2.1. Solitary waves and compact waves . . . . . . . . . . . . . . . . 532.2.2. Linear constitutive relation, solitary loop solution . . . . . . . 55

2.3. Compact planar waves . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3.1. Compact wave criterion . . . . . . . . . . . . . . . . . . . . . 572.3.2. General power-law strain-energy density . . . . . . . . . . . . 592.3.3. Quartic strain-energy density . . . . . . . . . . . . . . . . . . 61

2.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Chapter 3. Planar waves on heterogeneous rods . . . . . . . . . . 673.1. Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2. Multiple scales asymptotic expansion . . . . . . . . . . . . . . . . . . 68

3.2.1. O(ε−2) system . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2.2. O(ε−1) system . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.2.3. O(ε0) system . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.2.4. O(ε) system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.2.5. O(ε2) system . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.3. Homogenized traveling wave solution . . . . . . . . . . . . . . . . . . 843.3.1. Leading order solution . . . . . . . . . . . . . . . . . . . . . . 853.3.2. First-order correction . . . . . . . . . . . . . . . . . . . . . . . 863.3.3. Second-order correction . . . . . . . . . . . . . . . . . . . . . . 87

3.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Chapter 4. Helical Spring Problem . . . . . . . . . . . . . . . . . . . 944.1. Prescribed strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.1.1. Inextensible and unshearable case . . . . . . . . . . . . . . . . 964.2. Prescribed wrench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.2.1. Inextensible and unshearable case . . . . . . . . . . . . . . . . 974.3. Prescribed observables . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.3.1. Inextensible and unshearable case . . . . . . . . . . . . . . . . 99

Chapter 5. Inverse Helical Spring Problem . . . . . . . . . . . . . . 1015.1. Linear problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.2. Linear Kirchhoff problem . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3. Nonlinear problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Table of Contents — Continued

7

5.3.1. Solution for noisy data . . . . . . . . . . . . . . . . . . . . . . 1095.3.2. Examples of observables . . . . . . . . . . . . . . . . . . . . . 109

5.4. Variants of the IHSP . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.4.1. Known reference configuration . . . . . . . . . . . . . . . . . . 1115.4.2. Known reference curvature and torsion . . . . . . . . . . . . . 111

5.5. Nonlinear Kirchhoff problem . . . . . . . . . . . . . . . . . . . . . . . 1125.5.1. Examples of observables . . . . . . . . . . . . . . . . . . . . . 113

5.6. Special cases of the nonlinear Kirchhoff problem . . . . . . . . . . . . 1145.6.1. Diagonal case . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.6.2. Isotropic case . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.7. Numerical experiments with simulated data . . . . . . . . . . . . . . 1175.7.1. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.7.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Chapter 6. Overwinding helical springs . . . . . . . . . . . . . . . . 1246.1. Winding in twist-space . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.2. Geometric formulation of the problem . . . . . . . . . . . . . . . . . . 1256.3. Isotropic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.3.1. Behavior of force along the trajectory . . . . . . . . . . . . . . 1296.3.2. Overwinding criterion . . . . . . . . . . . . . . . . . . . . . . . 1306.3.3. Critical force . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.4. Anisotropic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.4.1. Anisotropic diagonal case . . . . . . . . . . . . . . . . . . . . 140

6.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Chapter 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.1. Planar waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.1.1. Quartic strain-energy denstiy . . . . . . . . . . . . . . . . . . 1447.1.2. Heterogeneous rod . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2. Helical springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.2.1. Direct and Inverse Helical Spring Problem . . . . . . . . . . . 1477.2.2. Overwinding helical springs . . . . . . . . . . . . . . . . . . . 149

Appendix A. Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8

List of Figures

Figure 1. Euler’s drawings of some of the nine species of the elastica.Species 6 is a periodic inflectional elastica, Species 8 a periodic non-inflectional one, and Species 7 is a limit between the two, also non-inflectional but not periodic, with exponential tails. . . . . . . . . . . . 14

Figure 2. A schematic representation of a single force molecule force spec-troscopy experiment with a dna molecule . . . . . . . . . . . . . . . . . 20

Figure 1.1. The director frame (red) and the Frenet frame (green) for asheared rod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Figure 1.2. The directors and the Frenet frame for a Kirchhoff rod. Notethat d2 is pointing along the stiffer axis of the cross section. . . . . . . . 35

Figure 1.3. Uniform anisotropic helical rods with different values of the reg-ister angle ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 1.4. The helix hyperboloid H(u) = 0 . . . . . . . . . . . . . . . . . . 48

Figure 2.1. The potential and the (κ, κ′) phase portrait with highlightedhomoclinic orbit for an intrinsically straight rod and a quadratic strain-energy density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Figure 2.2. The potential and the (y, y′) phase portrait with highlightedhomoclinic orbit for an intrinsically straight rod with a quartic strain-energy density (2.41). Scales: ymax := 64

a3, Vmin := −4608

a8, where a is the

characteristic length scale (2.45). . . . . . . . . . . . . . . . . . . . . . . 62Figure 2.3. Homoclinic solution for an intrinsically straight rod with quartic

strain-energy density (2.47). Scales: κmax := 4a, κ′max := 4

a2, a is the

characteristic length scale (2.45), and ` is the loop half-size, related to athrough (2.46). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Figure 2.4. Elastica with compact support (the strained part is highlighted)— the homoclinic solution for an intrinsically straight rod with quarticstrain-energy density (2.47). . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 3.1. Heterogeneous rod microstructure . . . . . . . . . . . . . . . . . 69Figure 3.2. Homogeneous rod solution (3.39) for the angle. (ξ0 = 0, ` = 10) 85Figure 3.3. First order correction (3.57) in terms of the angle. (ξ0 = 0, ` = 10) 86Figure 3.4. Numeric solution for the second order correction (3.79) in terms

of the angle. (ξ0 = 0, ` = 10, E1 = 1, E2 = 12, ρ1 = 0.8, ρ2 = 1, α = 0.2,

T = 1, and the wave speed c is given by (3.37)) . . . . . . . . . . . . . . 87Figure 3.5. Loop-like traveling wave solution for the homogenous rod, corre-

sponding to the solution (3.39) for the angle. (ξ0 = 0, ` = 10) . . . . . . 88Figure 3.6. First correction traveling wave solution (3.83). (ξ0 = 0, ` = 10) 90

List of Figures — Continued

9

Figure 3.7. Homogenized solution up to the first correction, φ = φ0 + εφ1,shown in the Cartesian plane for two different values of ε: ε = 3

8(solid

curve), and ε = 34

(dashed curve). The homogeneous solution (ε = 0) isshown in dotted. (ξ0 = 0, ` = 10) . . . . . . . . . . . . . . . . . . . . . . 91

Figure 3.8. Second correction to the traveling wave solution for the Cartesiancoordinates (3.84) (ξ0 = 0, ` = 10, E1 = 1, E2 = 1

2, ρ1 = 0.8, ρ2 = 1,

α = 0.2, T = 1, and the wave speed c is given by (3.37)) . . . . . . . . . 92Figure 3.9. Homogenized solution up to the second correction, φ = φ0 +

εφ1 + ε2φ2, shown in the Cartesian plane for two different values of ε:ε = 3

8(solid curve), and ε = 3

4(dashed curve). The homogeneous solution

(ε = 0) is shown in dotted. (ξ0 = 0, ` = 10, E1 = 1, E2 = 12, ρ1 = 0.8,

ρ2 = 1, α = 0.2, T = 1, and the wave speed c is given by (3.37)) . . . . . 93

Figure 4.1. A helical spring with one end fixed and a wrench—a force N(red) and a torque M (blue)—applied along the helix axis at the other end. 95

Figure 5.1. Inextensible and unshearable case with a full matrix Y (fiveunknown parameters). u = (0.4, 0.5, 0.7), exact values of parameters:Y = [[1, 0, 0.1]; [0, 2, 0.2]; [0.1, 0.2, 1.4]] . . . . . . . . . . . . . . . . . . . 122

Figure 5.2. Extensible and shearable case with a diagonal matrix Y (sixunknown parameters). u = (0.4, 0.5, 0.7), exact values of parameters:Y = diag(1, 1.1, 1.4, 0.5, 0.6, 0.7) . . . . . . . . . . . . . . . . . . . . . . 123

Figure 6.1. Zero axial moment ellipses E(κ, τ) = 0 (cf. (6.11)) correspodingto a subcritical (Γ = 1

2, solid curve), critical (Γ = 1, dashed curve), and

supercritical (Γ = 2, dotted curve) values of the elastic constant. A circleθ = θ is shown in red. The critical ellipse is a circle that shares a tangentline with the θ = θ circle at the reference point. A subcritical ellipse(Γ < 1) cuts the θ = θ circle between the reference point and the τ -axis. 131

Figure 6.2. Critical hyperbolas (6.17) for subcritical (Γ1 = 12) and supercrit-

ical (Γ2 = 2) values of the stiffness ratio. . . . . . . . . . . . . . . . . . 132Figure 6.3. Behavior of spring with increasing tensile force (Γ = 1

2, κ = 0.66,

τ = 0.066). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Figure 6.4. A trajectory for a tensile force and subcritical bend-to-twist stiff-

ness ratio (Γ = 12). The critical hyperbola (6.17) is shown in dashed. As

the force is increased from zero, the system moves from the reference point(κ, τ) so that the coiling angle is increased until it reaches the maximumvalue θc =

√κc2 + τc2. As the force is further increased, the helix starts

unwinding, reaching its initial value of the coiling angle θ at (κ0, τ0). Aninfinite force is needed to reach the point (0, τ). . . . . . . . . . . . . . 135

Figure 6.5. The coiling angle θ as a function of the applied force N , for Γ = 12

(solid curve), Γ = 1 (dashed), and Γ = 32

(dotted). (κ = 34, τ = 1

2) . . . . 136

List of Figures — Continued

10

Figure 6.6. Helix radius R as a function of the applied force N . (κ = 34,

τ = 12, Γ = 2

3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Figure 6.7. The critical coiling angle θc as a function of Γ, relative to thereference value θ =

√κ2 + τ 2. (κ = 3

4, τ = 1

2) . . . . . . . . . . . . . . . 138

Figure 6.8. The critical force Nc as a function of Γ. (κ = 34, τ = 1

2) . . . . . 139

Figure 6.9. The helix hyperboloid H(u) = 0 (yellow), the zero-axial-momentellipsoid E(u) = 0 (red), and the sphere |u| = θ (green) centered at theorigin through the reference point (purple). . . . . . . . . . . . . . . . . 139

Figure 6.10. The projection of the hyperbolic cylinder (6.31) (in blue) and thelevel sets on the ellipsoid (6.5) that are equidistant from the origin ontothe (u1, u2)-plane. The reference level set |u| = θ and the critical level set|u| = θc are shown in red. . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Figure 6.11. The curves in the parameter plane (β,Γ) that delimit the over-winding region (below the curves) in the diagonal anisotropic case, fordifferent values of the reference register angle ϕ. . . . . . . . . . . . . . . 143

11

List of Tables

Table 1.1. Kirchhoff top analogy . . . . . . . . . . . . . . . . . . . . . . . . 39

12

Abstract

This work1 examines problems in the statics and traveling wave propagation on uni-

form elastic rods with constant curvature and torsion, i.e. a straight rod and a helical

rod.

The first set of problems concerns planar traveling loop-like waves on intrinsically

straight rods. It is shown that loops with compact support can exist on homogeneous

rods with a nonlinear constitutive relation, where the strain-energy density contains

a quartic term. Next, the effect of heterogeneity in the material properties on the

shape of the loop is examined using a homogenization method.

The second set of problems deals with a system consisting of a helical spring with

a force and a torque applied along the helix axis. First, an overview is presented of

problems of finding the stresses given the strains, or vice-versa, assuming that the

elastic parameters of the spring are known. Then, the inverse problem is examined,

where both stresses and strains are measured, and optimal elastic parameters within

the linear consitutive model are sought. Various forms of measured strains are con-

sidered. Finally, the special problem with zero axial torque is considered, and criteria

when the spring overwinds with a tensile axial force applied are established.

1The full text of this dissertation is available at http://go.bojand.org/phd.pdf

13

Chapter 0

Introduction

0.1 A brief historical overview of the theory of elastic rods

The story of elasticity theory begins in the 17th century with two cornerstones of mod-

ern science: Hooke’s Law (discovered in 1660), and Newton’s Principia (published

in 1687). Combining Newton’s general principles of motion with Hooke’s constitu-

tive law provides—in principle—the necessary foundations for the development of the

theory of elasticity. The development of the theory of elastic rods started soon after,

but it would take three centuries and contributions by many of the most brilliant

scientists of their time to shape the exact general theory we have today.

In 1680, Mariotte formulated Hooke’s Law independently, and observed that the

resistance of a beam to flexure is due to some of its longitudinal filaments being

extended and some contracted [1, p. 2].

James Bernoulli made the first important step in the study of the elastica, a

planar infinitely thin rod, i.e. the curve that a centerline of a rod assumes in equi-

librium, by formulating his “Golden Theorem” (challenge published in 1691; solution

concealed for three years [2, p. 17]) laying down the differential equation that de-

termines the shape of the elastica which is essentially a relation of proportionality

between the curvature and the torque. This was the key to understanding that the

work done in bending the rod is proportional to the square of the curvature. In 1742,

Daniel Bernoulli suggested to Euler to solve the problem of minimizing the integral

of the square of the curvature. Euler then completely solved the problem of the (pla-

nar) elastica (published in 1744), and classified all the solutions into 9 “species” (see

Figure 1). Euler’s contribution to the theory of elastic rods did not end there: years

later, he was the first to write down the system of balance equations for a planar

14

rod, comprised of the tangential and normal components of the force balance, and

the (binormal) torque balance, from which all previously known equations for rods

and strings could be derived; he calculated the critical length for the buckling of a

column; and he was the first to recognize that for curved rods, the change of the

curvature plays the role that curvature does for straight rods. In 1770, Lagrange

followed up on Euler’s work and discovered higher buckling modes (Euler only found

the primary one), as well as the “strongest” shape of a column (i.e. one most resistant

to buckling).

(a) Species 6 (b) Species 7 (c) Species 8

Figure 1: Euler’s drawings of some of the nine species of the elastica. Species 6 is aperiodic inflectional elastica, Species 8 a periodic non-inflectional one, and Species 7is a limit between the two, also non-inflectional but not periodic, with exponentialtails.

A remarkable progress in the theory of elastic rods was achieved by Coulomb (1776).

He formulated a theory of rods with finite cross sections that is deduced from Hooke’s

Law and Mariotte’s assumption that resistance to bending stems from the rod’s lon-

gitudinal filaments’ resistance to extension and compression. Coulomb was the first

to consider resistance to torsion, and the first to consider shear, although not as an

15

elastic strain—it is Young who did this in the 1807 publication of his Lectures, where

he also introduced the elastic modulus. In 1815, Binet introduced the tangential

equation of the moments into the theory.

Love notes [1, p. 23] that the theory of the bending and twisting of thin rods

and wires—including the theory of helical springs—was for a long time developed

independently of the general equations of elasticity. However, the 19th century was

the golden age for the general theory of elasticity with the works of Navier, Saint-

Venant, and Cauchy, among others, and this progress was bound to eventually extend

to the theory of rods. Saint-Venant was the first to bring the problems of torsion

and flexure of beams under the general theory, using his semi-inverse method for

obtaining solutions. The first such problem he considered was that of the torsion

of a prism (1855) [3], then the problem of the cantilever. Saint-Venant built his

theory of torsion and flexure of beams on an assumption that the forces acting on the

terminal sections are distributed in a definite way. He made that assumption plausible

by enunciating the principle that bears his name, which states that deviations from

these distributions only produce a local effect near the ends of the rod. Saint-Venant

was also the first (1843) to note the necessity of introducing the register angle ϕ into

the theory [1, Art. 253].

A complete theory of a rod deforming in space subjected to a force and a torque

that is derived from the general theory is due to Kirchhoff (1859) [4]. His approach

consisted in dividing a rod into small segments, each undergoing small deformations

so that the general theory is applicable, and imposing certain continuity conditions

at the interfaces. Thus Kirchhoff’s theory is not restricted to an infinitesimal overall

deformation, which was the case with Saint-Venant’s. Kirchhoff also introduced the

strain-energy density in terms of the extension, components of curvature, and torsion,

and obtained equilibrium equations by varying the strain-energy density function.

What is most remarkable, and noted already in the original paper, is that the equa-

tions thus obtained are analogous to those of a heavy spinning top. This is referred

16

to as the Kirchhoff top analogy (cf. Section 1.2.6). Clebsch (1862) [5, 6] developed

further Kirchhoff’s theory, but it was an improvement in formalism rather than sub-

stance. A substantial improvement was carried out by Love in his seminal work

A Treatise on the Mathematical Theory of Elasticity (1st edition 1892, 2nd edition

1906 [1]). Love abandoned Kirchhoff’s and Clebsch’s approach of dividing the rod

into a discrete number of small prisms and treated the rod as a continuum of cross

sections, which are assumed to remain planar and normal to the centerline. The ori-

entation of the cross section about the centerline tangent is specified by the register

angle ϕ. He introduced the material twist dϕ/ ds, which, together with torsion τ ,

makes up the total twist u3 [1, Art. 253]. He classified the elastica into inflectional

and non-inflectional [1, Art. 263], rather than Euler’s nine species. As for modern

overviews of the Kirchhoff–Clebsch–Love theory, an account of the theory is given

in [7], while an exact derivation of the theory from three-dimensional elasticity can

be found in [8].

The final development in shaping the theory of rods into its present form was

done by the Cosserat brothers (1908) [9], who introduced the director theory—an

orthonormal frame fixed with respect to the material of the rod. The orientation

of the director frame specifies the orientation of the cross section, so that with the

directors all deformations can be described: bend, twist, stretch, and shear. The

Cosserats theory is not restricted to a linear constitutive relation: no constitutive

relation is specified in their work. Although the Cosserats’ made a significant step

towards a definite theory of rods, their work received no attention until Ericksen and

Truesdell brought it to light in 1957 [10].

0.2 Elastica and localized planar waves

In light of the Kirchhoff top analogy, the planar static Kirchhoff equations (i.e. the

flexure equations for an inextensible and unshearable rod, or the elastica equations)

17

are analogous to the pendulum. (Planar) elasticae have been extensively studied

and classified starting with the classical work of Euler (for modern follow-ups see

e.g. [11, 12]). According to Love’s classification, we distinguish inflectional from non-

inflectional elasticae. In the pendulum analogy, periodic inflectional elasticae (Euler’s

Species 6, cf. Figure 1a) correspond to oscillations about a stable equilibrium, while

the periodic non-inflectional elasticae (Species 8, cf. Figure 1c) are analogous to the

revolving pendulum. Delimiting the two behaviors is a homoclinic solution that cor-

responds to the pendulum released from the unstable equilibrium, and performing

one full revolution before reaching the unstable equilibrium again in infinite time. In

the rod picture, this homoclinic solution is a single loop of infinite extent connect-

ing asymptotically two straight states (Species 7, cf. Figure 1b) and can be simply

expressed in terms of the curvature by a sech function.

Due to the formal equivalence between the static and traveling wave Kirchhoff

equations, demonstrated in a general way in Chapter 2 (cf. Proposition 2.1) [13], these

loop-like solutions can travel along the rod with constant velocity, and represent the

propagation of localized flexural waves. The homoclinic static solution, which is a

wave traveling at the speed of sound, thus becomes a solitary wave. Like the well-

known Korteweg–de Vries solitons [14, 15], this is a localized wave with exponentially

decaying tails. As we show in Chapter 2, a nonlinear dispersive term in the equation

(brought about in the rod equation by a quartic term in the strain-energy density)

leads, as with the Korteweg–de Vries equation [16, 17], to the compactification of

the solitary wave, i.e. the exponential tails disappear and the disturbance is entirely

localized within a bounded region.

Homogenization. A classical problem in the theory of homogenization is to con-

sider longitudinal waves in a heterogeneous elastic medium with a periodic material

microstructure of two alternating homogeneous materials. For the case of small-

amplitude linear elastic waves, this problem has been analyzed using a homogeniza-

18

tion technique [18], based on a multiple scale expansion, introducing a fast length

variable on the scale of the microstructure. (The same authors also considered the

problem with a mutliple spatial and temporal scale expansion [19].) The leading order

balance yields the effective homogeneous material properties [20], which is a crude ap-

proximation considering that it does not exhibit the dispersive behavior characteristic

of the heterogeneous material, brought about by successive reflections on material in-

terfaces. Dispersion is then captured by higher order corrections. In Chapter 3, we

consider the effect of heterogeneity on loop elastica waves.

0.3 Helical springs

A helical spring is a classical example of elasticity theory. One often encounters

depictions of a coil as a symbolic representation of a one-dimensional elastic system,

e.g. as an illustration to Hooke’s law. This is not by chance: the omnipresence of

helices in nature (e.g. plant tendrils, bacteria flagella, dna molecules) is an indication

that this particular shape provides an efficient way of storing elastic energy for given

material properties, so that large stresses do not cause the breakage of the building

material. Rather than the rod being (exclusively) extended along the centerline as

in straight form, a helix extended along the helix axis deforms predominantly by

twisting. This was first observed by Binet in 1814, and in 1848 James Thomson

showed that a helical spring of infinitely small pitch angle has the same elastic response

as a torsion balance with the same rod straightened out, and he verified this result

experimentally [21]. As a consequence of this torsional behavior, a helical spring can

be modeled to a good approximation as an inextensible and unshearable rod.

There is more to the helical shape than the spring: entropical arguments have

been suggested as the driving force for helix formation within the cell [22]. Thus, at

the molecular level, entropy may be the underlying principle behind the statement

“the helix is the Nature’s preferred shape,” rather than elasticity. A geometric study

19

of the density of packing of helices can be found in [23].

Helical springs have been man-made since the 15th century. Robert Hooke sup-

posedly considered small deformations of a helical spring when he formulated what

is historically the first law of elasticity. Hooke’s Law is thus a first approximation to

the solution to the problem of finding the force applied along the helix axis at the

end of the spring (the other end being fixed) that is necessary to maintain the spring

in a state with a given extension in the axial direction. (A better approximation for

the force is given by the product of the elongation of the spring and the square of the

secant of the pitch angle [21].) In Chapter 4 we consider the problem in a general

way, the only assumption being that the constitutive relation is linear.

By the second half of the 19th century, a description of an inextensible and un-

shearable isotropic helical spring was known that relates the axial wrench to the

strains. Such a formula can be found in the first edition of Thomson and Tait’s Trea-

tise on Natural Philosophy, published in 1867 [24, Art. 605], where the strains are

expressed in terms of the helix radius and pitch angle. Essentially the same formula

is found in Love’s Treatise [1, Art. 271] (1906).

The theory of helical springs somewhat fell out of fashion in the 20th century, but

a revived interest in the past few decades is partly due to advances in experimental

methods where a mechanical force is applied to a macromolecule and the resulting

strain is measured, generally referred to as single molecule force spectroscopy [25],

and partly to a new field of nanosprings [26]. A vast amout of data has been collected

in experiments with dna molecules [27–34] (see Figure 21). Theoretical models used

to describe the observed behaviors of dna molecules has had limited success [35].

This has inspired the work presented in Chapter 5 [36] and Chapter 6 [37], where

problems with a simple helical spring system are posed in the most general way

within the linear constitutive model. The system considered here is an ideal elastic

spring. Other effects would need to be taken into account for a more realistic model

1Reprinted from [33]

20

of dna, most notably thermal fluctuations [38].

Figure 2: A schematic representation of a single force molecule force spectroscopyexperiment with a dna molecule

One effect observed with dna is that it overwinds when pulled [39, 33]. The

authors describe this as “contradict[ing] the intuition that dna should lengthen as

it is unwound and get shorter with overwinding” [39] and comment that “simple

physical intuition predicts that dna should unwind under tension” [33]. It was also

observed that at a critical force, dna reverts from overwinding to unwinding [33].

This behavior is known (but perhaps forgotten?) to be exhibited by simple helical

springs (Miller 1902 [21]), and can be derived from the rod model, but it cannot be

captured by an elastic energy that is quadratic in the elongation of the spring and

the coiling angle, which is encountered in some of the cited experimental papers.

21

Chapter 1

Governing equations

This chapter is an overview of the theory of elastic rods that subsequent chapters are

based upon.

1.1 Special Cosserat rods

We describe an elastic rod, also referred to here simply as a rod, as a special Cosserat

rod [9, 40]. This is a model of the three-dimensional rod where all physical quantities

are averaged over the cross sections, thereby reducing it to a one-dimensional object

with some additional parameters representing the orientation of the cross sections.

Deformations of the cross sections are neglected.

1.1.1 Geometry

We consider a rod whose length in the stress-free configuration (henceforth: the

reference configuration) is L. The configuration of a rod is given by its centerline,

which is a curve in the three-dimensional Euclidean space E3 parametrized by the arc

length in the reference configuration, r : [0, L] 3 s 7→ r(s) ∈ E3, and by a right

orthonormal frame d1,d2,d3 : [0, L] → R3, (where R3 is the three dimensional real

vector space) associated with every point on the curve, which captures the orientation

of the material cross section at that point. The vectors d1(s),d2(s),d3(s) are called

the directors, and we refer to the basis they form as the director basis. Following

Antman [41], the director basis coordinates of a vector are denoted using sans-serif

fonts, e.g. a = a1d1 + a2d2 + a3d3, and the corresponding matrix representations

using bold sans-serif fonts, e.g. a = (a1, a2, a3) ≡ [a1, a2, a3]T . We will be referring

to director basis coordinates also as local coordinates.

22

Frenet frame The Frenet–Serret basis (henceforth called the Frenet frame) at a

given point of a curve is the orthonormal triple (ν,β, τ ), where τ is the tangent

vector, ν is the principal normal vector that is the unit vector along the derivative

of τ , and β := τ × ν is the binormal vector.

Centerline The Frenet curvature, (hereinafter simply curvature) of a curve is a

measure of a deviation of the curve from the straight line:

κ := |τ ′| , (1.1)

where (·)′ denotes the derivative with respect to s. The curvature is positive by

definition. The torsion is defined as

τ := −β′ · ν . (1.2)

A fundamental theorem of geometry states that a curve in space is uniquely deter-

mined up to a rigid body motion by the curvature and the torsion (cf. e.g. [42]). As

rigid body motion of the rod as a whole will be of no interest here, the centerline will

be assumed given by the curvature κ(s) and the torsion τ(s).

Frenet–Serret equations The Frenet–Serret equations relate the derivatives τ ′, ν ′, β′

with the basis vectors τ , ν, β, providing the equations of evolution of the Frenet

frame along the curve. The Frenet–Serret equations areνβτ

′ = 0 τ −κ−τ 0 0κ 0 0

νβτ

. (1.3)

Denoting

Ω = κβ + ττ , (1.4)

the Frenet–Serret equations can also be written as

ν ′ = Ω× ν , β′ = Ω× β , τ ′ = Ω× τ . (1.5)

23

The vector Ω is called the Darboux vector. In (1.4), κ and τ are scalar quan-

tities independent of parametrization, while β and τ are vectors that flip direction

under the inversion s 7→ −s. Therefore, the Darboux vector is also dependent on

parametrization: Darboux vectors Ω, Ω corresponding to opposite parametrizations

are related by Ω = −Ω.

Cross sections The cross section plane is defined as the normal plane to the center-

line tangent in the reference configuration at a given point r(s). The term material

cross section is used designate the collection of material points of the rod lying in

a cross section plane. In the special Cosserat rod model, deformations of material

cross sections are neglected, i.e. the material cross sections are treated as planar rigid

bodies. A cross section plane in a deformed configuration is the plane containing the

material cross section.

The cross section plane and the orientation of the material cross section in this

plane are specified by the vectors d1 and d2, which we choose so as to point along the

principal axes of area of the cross section. Therefore, we assume that in the reference

configuration, d3 ≡ d1 × d2 is a vector tangent to the centerline.

Shear A deformed configuration that preserves the orthogonality between the tan-

gent vector and the cross sections is said to be shearless, while shear denotes the

deformation involving the deviation of d3 from the tangent vector v := r′. A sheared

rod with directors is shown in Figure 1.1.

Shear has two degrees of freedom, and can be specified by two angular parameters

determining the orientation of the cross section plane. These parameters may be

chosen as the shear amplitude angle

α := ∠(d3,v) , (1.6)

where ∠(·, ·) denotes the angle between two vectors, and the shear orientation

angle, which represents the orientation about the vector v of the d3 vector on the

24

Figure 1.1: The director frame (red) and the Frenet frame (green) for a sheared rod.

cone given by the angle α.

Twist equations The configuration of a rod can be obtained from the following system

of ordinary differential equations (odes)

r′ = v , (1.7a)

d′i = u× di , i = 1, 2, 3 , (1.7b)

where u,v : [0, L] → R3 are vector fields we refer to as the twist vector and the

stretch vector, respectively. Equations (1.7b), called the twist equations, are a

generalization of the Frenet–Serret equations (1.5), where the twist vector u is the

analogue of the Darboux vector Ω. Due to the analogy with a spinning top (cf. Sec-

tion 1.2.6), where the independent variable is time, equations (1.7) are sometimes

referred to as the kinematic equations.

Integrating the equations (1.7) with known twist and stretch vector functions, we

obtain the configuration of the rod (r,d1,d2). The initial conditions in this integration

are given by the position and orientation of the rod in space. Therefore, an alternative

25

representation of the configuration of the rod, one that removes rigid body motion

from consideration, is provided by the twist and stretch vectors u and v. From here

on, the term configuration of the rod will be used to designate the pair of functions

(u(s),v(s)). A configuration is thus defined up to a rigid body motion of the rod.

Stretch The norm of the stretch vector is called the stretch ν.

ν := |v| ≡ |r′| (1.8)

The direction of the stretch vector v is tangential to the centerline by definition (1.7a),

thus

v ≡ ντ . (1.9)

Since the spatial parameter s is defined as the arc length in the reference config-

uration, the stretch in the reference configuration ν is unity by definition:

ν ≡ 1 . (1.10)

In a deformed configuration, a value of the stretch that is larger than unity means

that the rod is locally extended, while the stretch is less than one when it is locally

compressed. For a configuration to retain physical meaning, the stretch must be

strictly positive, ν(s) > 0, ∀s ∈ [0, L].

Reference configuration As the shear is defined with respect to the reference config-

uration, we have used the freedom to choose the cross sections so that the reference

configuration is shearless. Therefore, we assume, without loss of generality, that the

reference twist vector u and the reference stretch vector v functions have the

following director basis coordinates:

u = (κ sin ϕ, κ cos ϕ, τ + ϕ′) , (1.11)

v = (0, 0, 1) . (1.12)

We denote quantities associated with the reference configuration by hats. A naturally

straight rod has zero reference twist, u ≡ 0.

26

1.1.2 Kinematics

Spin vector Shifting to a time-dependent picture, as the directors di = di(s, t), i =

1, 2, 3 preserve orthonormality in time, the evolution of the director basis in time is

given by equations analogous to the twist equations (1.7b):

di = w × di , i = 1, 2, 3 , (1.13)

where the time derivative is denoted with a dot,

˙(·) :=∂

∂t(·) ,

and the vector w is called the spin vector. Equations (1.13) will be referred to as

the spin equations.

Compatibility relations The twist vector and the spin vector are related in the follow-

ing way. Differentiating (1.7b) with respect to time, and (1.13) with respect to arc

length s, (and assuming the order of differentiation can be interchanged) we obtain

the following constraint

w′ − u = u×w . (1.14)

1.1.3 Mechanics

We now derive balance equations in terms of one spatial variable by integrating over

cross sections. As a result we obtain six scalar equations (three linear and three

angular), which are the basis of both the Kirchhoff–Clebsch–Love theory and the

special Cosserat theory. For a modern account of the detailed derivation from three-

dimensional elasticity, see e.g. [7].

The position of a material point in the rod is given by the radius vector

x = r + X1d1 + X2d2 , (1.15)

27

where r = r(s, t) is the point on the centerline that is the centroid of the corresponding

cross section, and X1,X2 are the local coordinates of the material point witin the cross

section, i.e. coordinates in the basis (d1(s, t),d2(s, t)) with origin at r(s, t).

Conservation of linear momentum The (infinitesimal) force acting on an infinitesi-

mally thin slice of the rod of width ds and cross section A with area A is

dn(s, t) +

∫∫A(s,t)

F(s, t) dX1 dX2 ds =

∫∫A(s,t)

ρx dX1 dX2 ds , (1.16)

where ρ is the mass density of the rod, and F is the body force per unit volume

element. Integrating over the cross section A, in the left-hand side we obtain a per-

unit-length body force f , while in the right-hand side, the X1 and X2 terms from (1.15)

dissapear by the center of mass property, and we have

n′ + f = ρAr . (1.17)

Differentiating with respect to s and noting that r′ ≡ v, this equation becomes

n′′ + f ′ = ρAv . (1.18)

We will refer to (1.18) (and equations derived therefrom) as the force equation.

The tension in the rod is the projection of the force onto the tangent vector,

T := n · τ .

Conservation of angular momentum Similarly, the conservation of angular momen-

tum yields the following equation

m′ + r′ × n +

∫∫A

L dX1 dX2 =

∫∫Aρ r× r dX1 dX2 , (1.19)

where L is the body couple per unit volume. This yields

m′ + v × n + ` = ρI2 d1 × d1 + ρI1 d2 × d2 , (1.20)

28

where ` is the body couple per unit length, and

I1 =

∫∫AX1

2 dX1 dX2 , I2 =

∫∫AX2

2 dX1 dX2 (1.21)

are the two principal moments of area of the cross section. We will refer to (1.20)

(and equations derived therefrom) as the moment equation.

The stresses acting on the cross section at r(s) are averaged over the cross section

into a force vector n(s) and a moment of force vector m(s) (henceforth referred

to as the moment vector for brevity).

Static equations Setting all the time derivatives to zero and disregarding the effects

of gravity and other body loads yields the following static balance

n′ = 0 , (1.22a)

m′ + v × n = 0 . (1.22b)

Shifting to the director frame of reference, the balance equations become

n′ + u× n = 0 , (1.23a)

m′ + u×m + v × n = 0 , (1.23b)

or, expressed in block-matrix form[mn

]′=

[m× n×

n× 0

] [uv

], (1.24)

where (·)× : R3 → R3×3 is the skew-symmetric matrix associated with the vector a =

(a1, a2, a3)

a× :=

0 −a1 a2a1 0 −a3−a2 a3 0

. (1.25)

In order to abbreviate the notation, we denote the column 6-vectors

f :=

[mn

], x :=

[uv

], (1.26)

29

and we refer to them as the stress vector and the strain vector,1 respectively. The

matrix that relates the two 6-vectors in the balance equation (1.24) is denoted

J(f) :=

[m× n×

n× 0

]. (1.27)

Note, that the two 3-blocks in both f and x have different dimensions:

JmK = force× length , JnK = force , JuK = 1/length , JvK = 1 . (1.28)

1.1.4 Elasticity

The remaining equation that closes the system is a constitutive relation, which relates

the stresses to the strains, f = f(x).

A rod is called hyperelastic if there exists a strain-energy density function

W : R6 → R+ such that

f = Wx(x− x) , (1.29)

where the vector subscript denotes the gradient. Note that the argument to the

strain-energy density funciton W is not a vector, but the local coordinates of a vec-

tor. Henceforth, it will aways be assumed that the rod is hyperelastic, so that equa-

tion (1.29) is the constitutive relation in its most general form. We assume that

the strain-energy density W is a convex, coercive function, with W (0) = 0.

Legendre transform Because the strain-energy density is a convex and coercive func-

tion, relation (1.29) can be inverted

x = W ∗f (f) + x , (1.30)

where W ∗ is the Legendre transform of W

W ∗(y) = supx∈R6

yTx−W (x)

. (1.31)

1 Strictly speaking, the term strains denotes the deviation from the state at rest, i.e. u− u andv− v, but many authors refer to absolute twist and stretch vectors u and v—instead of their valuesrelative to the reference configuration u and v—as strains, and we will also use the term in the loosersense.

30

Quadratic strain-energy density When the strain-energy density is a quadratic func-

tion,

W (x) =1

2xTKx , (1.32)

where K is a 6 × 6 symmetric positive definite matrix called the stiffness matrix,

the constitutive relation is linear:

f = K(x− x) , (1.33)

and the conjugate form of the constitutive relation (1.30) simplifies to

x = Yf + x . (1.34)

where Y = K−1 is the compliance matrix (also symmetric and positive definite).

We denote the 3× 3 blocks in the K matrix as follows:

K =

[K CCT D

]. (1.35)

The diagonal blocks K and D are symmetric and positive definite, and the coupling

block C is such that the matrix K is positive definite. Note that, due to different

dimensions of the first three and last three components in the 6-dimensional stress

vector f as well as in the 6-dimensional strain vector x (cf. (1.28)), elements of the

three blocks K, D, and C have different dimensions.

Orienting the d1 and d2 vectors along principal axes of the cross section results in

K12 = 0 . (1.36)

The matrix K thus contains twenty independent elastic constants.

1.1.5 Complete set of governing equations

The kinematic equations (1.7), the balance laws (1.23), and the constitutive rela-

tion (1.29) constitute the system of governing equations for a special Cosserat rod.

31

These represent a complete system of 12 equations for the director basis coordinates

u, v, m, and n, which can be written as

f ′ = J(f)x , f = Wx(x− x) , (1.37)

where J(f) is the 6× 6 matrix given by (1.27).

The unknowns in these equations are coordinates in a basis that is itself a priori

unknown. However, once a solution is obtained for u and v, the director basis and

the centerline can be reconstructed from the kinematic equations (1.7). Then, the

actual solution vectors u, v, m, and n can be formed.

1.1.6 Hamiltonian formulation

The governing equations for the special Cosserat rod as expressed in terms of the

director basis coordinates (1.23) can be cast as a non-canonical Hamiltonian sys-

tem [43, 44], with Hamiltonian function:

H(m,n) := W ∗(m,n) + mT u + nT v = W ∗(f) + fT x (1.38)

In view of the constitutive relation (1.30), the gradient of this Hamiltonian is simply:

∇H = ∇W ∗(f) + x = x . (1.39)

Equation (1.24) therefore gives the Hamiltonian structure:

f ′ = J(f)∇H , (1.40)

where the structure matrix J(f) is given by (1.27).

1.1.7 First integrals

A special Cosserat rod has the following first integrals:

I1(m,n) :=1

2nTn (1.41a)

I2(m,n) := mTn (1.41b)

32

Their gradients are

∇I1 =

[0n

]=

[0 00 I

]f , ∇I2 =

[nm

]=

[0 II 0

]f , (1.42)

where I is the 3× 3 identity matrix.

Note that ∇I1 and ∇I2 are in the null space of the structure matrix J, i.e. they

are Casimir functions [45]. Moreover, I1 and I2 are the only Casimir functions for the

given system, as the null space of J is spanned by ∇I1 and ∇I2.

Force vector integral The force equation (1.22a) yields a vector constant of motion,

which, without loss of generality, we choose in the ez direction:

n = Nez = const . (1.43)

1.1.8 Variational characterization of equilibria

A relative equilibrium is a solution to the variational problem (cf. [44])

Minimize H(f) (1.44a)

subject to the constraints

I1(f) =1

2C1

2 , I2(f) = C2C1 , (1.44b)

where C1 and C2 are constants. This form of the constraints was chosen to emphasize

the fact that I1 is non-negative, and that I2 is zero when I1 is zero.

The first integrals (1.41) can be expressed in terms of 6D vectors as follows:

nTn = fT[0 00 I

]f , mTn = fT

[0 I0 0

]f (1.45)

The variational constraints (1.44b) are thus

2I1(f) = fT[0 00 I

]f = C1

2 , I2(f) = fT[0 I0 0

]f = C1C2 . (1.46)

33

A relative equilibrium is characterized by

∇H = λ1∇I1 + λ2∇I2 , (1.47)

where λ1, λ2 are Lagrange multipliers associated with the constraints (1.44b). Us-

ing (1.39) and (1.42) this yields

x =

[0 λ2Iλ2I λ1I

]f . (1.48)

If λ2 = 0, the twist vector u identically vanishes, and the centerline is a straight

line. We discard this case since we consider helical-shaped rods. For λ2 6= 0, the

relation (1.48) can be inverted

f =

[µ1I µ2Iµ2I 0

]x , (1.49)

and the µ’s are related to the λ’s by

µ1 = − λ1

λ22 , µ2 =

1

λ2. (1.50)

Note that the second block-component of (1.49) implies that µ2 is the coefficient of

proportionality between the force n and the twist vector u.

Dual variational problem Equation (1.49) is a first-order necessary condition for the

dual problem:

Minimize W (x− x) (1.51a)

subject to constraints

uTu = η12 , uTv = η1η2 , (1.51b)

which can also be expressed in terms of 6D strain vectors as

xT[I 00 0

]x = η1

2 , xT[0 I0 0

]x = η1η2 . (1.51c)

The relation between the constants η1, η2 and C1, C2 is found from (1.49) to be

λ2

[C1

2

C1C2

]=

[µ2 0µ1 µ2

] [η1

2

η1η2

]. (1.52)

34

Finally, we note that the multipliers µ1 and µ2 (which are the Lagrange multipli-

ers associated with constraints (1.51b)) can be expressed in terms of the constraint

constants C1, C2, η1, η2 of the two dual problems:

µ1 =η1C2 − η2C1

η12, (1.53a)

µ2 =C1

η1. (1.53b)

For future reference, we denote this map as:

µ : (C1, C2, η1, η2) 7→ (µ1, µ2) =

(η1C2 − η2C1

η12,C1

η1

). (1.54)

1.1.9 First integral surfaces in strain space

The constitutive relation (1.33) turns the constraints (1.46) into a pair of equations

for the strains:

J1(x) := (x− x)TK

[0 00 I

]K(x− x) = C1

2 , (1.55a)

J2(x) := (x− x)TK

[0 I0 0

]K(x− x) = C1C2 . (1.55b)

These are two quadrics in the six-dimensional strain space x.

1.2 Kirchhoff rods

1.2.1 Inextensibility and unshearability constraint

An elastic rod is said to be inextensible if the stretch ν is unity at every point,

ν := |v| = |r′| ≡ 1, and it is unshearable if d3 points along the tangent vector, i.e. if

v1 ≡ 0, v2 ≡ 0. The inextensibility and unshearability assumptions are encapsulated

in the following constraint:

d3 ≡ r′ . (1.56)

A special Cosserat rod satisfying these assumptions is called a Kirchhoff rod.

35

For a Kirchhoff rod2, the director basis is related to the Frenet frame by a rotation

through the register angle [46] ϕ about the tangent vector τ ≡ d3 (see Figure 1.2).[d1

d2

]=

[cosϕ sinϕ− sinϕ cosϕ

] [νβ

](1.57)

Unlike the Frenet frame, the orientation of which is a geometric property of the

Figure 1.2: The directors and the Frenet frame for a Kirchhoff rod. Note that d2 ispointing along the stiffer axis of the cross section.

centerline, the director basis is fixed with respect to the material cross-section of the

rod. The evolution of the director basis along the centerline is given by the twist

equations (1.7b). The director basis coordinates of the twist vector are related to the

Frenet curvature κ and torsion τ of the centerline, and the register angle ϕ as follows:

u = (κ sinϕ, κ cosϕ, τ + ϕ′) . (1.58)

The Kirchhoff rod model thus represents the three-dimensional rod as a strip, which

is a geometric object consisting of a curve and an angular parameter.

2This is true in the more general case of an unshearable but extensible special Cosserat rod.

36

Equations (1.7b) are a generalization of the Frenet–Serret equations. The magni-

tude of the projection of u onto the normal plane is the curvature κ. A configuration

is said to be uniform (and we speak of pure flexure) if there is no twisting of the

rod about its axis, i.e. if ϕ′ ≡ 0. For pure flexure, the tangential component of the

twist vector u is the torsion τ , and the director basis is fixed with respect to the

Frenet basis.

In the Kirchhoff rod model, the configuration of the rod is entirely described by

the twist vector u, since, by the inextensibility and unshearability condition (1.56),

the stretch vector does not deviate from its reference value:

v ≡ r′ ≡ d3 ≡ v =⇒ v ≡ (0, 0, 1) . (1.59)

The inextensibility condition also ensures that the centerline parameter s, chosen

to be the arc length in the reference configuration, remains the arc length in any

deformed configuration.

1.2.2 Elasticity

Due to the inextensibility and unshearability constraint (1.56), the force is an un-

known in the system, and a constitutive relation for the moment of force, relating

the moment vector to the strains, closes the system. As the strains are now entirely

given by the twist vector u, the constitutive relation is

m = Wu(u− u) , (1.60)

where W : R3 → R+ is the strain-energy density, which is now a function of the

twist vector coordinates only. We use the same symbol as for the strain-energy density

in the extensible and shearable case (cf. (1.29)), but the two are clearly discernable

by (the dimesion of) their arguments. The simplest case is that of a quadratic strain-

energy density,

W (u) =1

2uTKu , (1.61)

37

where K is the 3× 3 stiffness matrix (symmetric, positive-definite), which yields a

linear constitutive relation:

m = K(u− u) . (1.62)

The conjugate constitutive relation (1.34) reduces to:

u = Ym + u . (1.63)

where Y = K−1 is the 3× 3 compliance matrix.

1.2.3 Kirchhoff equations

Equations (1.7b), (1.22), and (1.60) are collectively called the Kirchhoff equations

for an inextensible and unshearable elastic rod.

1.2.4 Variational characterization of equilibria

A solution to the Kirchhoff equations is also a solution to the variational prob-

lem (1.51), which, with the inextensibility and unshearability constraint (1.56), now

reads:

Minimize W (u− u) (1.64a)

subject to constraints

uTu = η12 , uTd3 ≡ u3 = η1η2 . (1.64b)

The equilibrium equation (1.49) provides a relation between the stresses and the

strains via the Lagrange multipliers µ1 and µ2 associated with constraints (1.64b):

m = µ1u + µ2d3 , (1.65)

n = µ2u . (1.66)

38

1.2.5 First integrals

In addition to the first integrals in the extensible and shearable case (cf. Section 1.1.7),

a Kirchhoff rod admits the following first integral that represents a local form of

energy:

H := u ·Wu(u− u)−W (u− u) + n3 = const , (1.67)

where n3 = Nez · d3 is the (generally nonconstant) tension, i.e. the d3 coordinate of

the constant force vector. We will refer to H as the energy integral. Note that if

u = u, then the energy integral is equal to the tension, H = n3.

If the strain-energy density W is a homogenous function with degree of homo-

geneity k, then

H = (k − 1)W (u− u) + uWu(u− u) + n3 . (1.68)

1.2.6 Kirchhoff top analogy

The analogy between the static Kirchhoff equations and the heavy spinning top was

first discovered by Gustav Kirchhoff [4]. For an intrinsically straight rod (u = 0),

the formal analogy of the equations is complete, and the correspondance of various

variables is listed in Table 1.1. Note that the constitutive relation corresponds to the

definition of the angular momentum of the top as the gradient of the angular kinetic

energy, thus the analogy is only valid for a linear constitutive relation. However, the

analogy is not complete between two problems, since the spinning top is an initial

value problem, while the elastic rod is a boundary value problem.

1.2.7 Diagonal case

If there is no coupling between bending and twisting in the case of a linear constitutive

relation (1.62), i.e. if the stiffness matrix K is diagonal, the elastic properties are

usually expressed in terms of the asymmetry coefficient β and the twist-to-bend

39

Table 1.1: Kirchhoff top analogy

quantity elastic rod spinning tops arc length time

(d1,d2,d3) basis attached to rod basis attached to rigid bodyd3 unit tangent vector unit vector from FP to CofMn force −mgm moment of force angular momentumu twist vector angular velocity vectorτ torsion angular velocity about top axis

EI1, EI2 principal bending stiffnesses principal moments of inertia ⊥ d3

µJ torsional stiffness principal moment of inertia about d3

W strain-energy density angular kinetic energym = Wu constitutive relation angular momentum definition

stiffness ratio Γ, defined as

β :=K22

K11

, Γ :=K33

K11

. (1.69)

As K11 ≤ K22 by convention (set by the choice of orientation the director basis in the

cross section plane), it follows that β ≥ 1. By scaling out K11, the stiffness matrix K

becomes

K = diag(1, β,Γ). (1.70)

1.2.8 Isotropic rods

A Kirchhoff rod is said to be isotropic if the strain-energy density function W is

invariant under rotations in the cross section plane, i.e. when it is independent of the

register angle ϕ

W (κ cosϕ, κ sinϕ, τ) = W (0, κ, τ) , ∀ϕ , ∀κ, τ . (1.71)

In case of a quadratic strain-energy density (1.61), this invariance can only be fulfilled

by a diagonal stiffness matrix K containing only two elastic parameters: the bending

stiffness K11, and the torsional stiffness (or twisting stiffness) K33.

K = diag(K11,K11,K33) (1.72)

40

As one of the parameters can be scaled out, the dimensionless stiffness matrix is

expressed in terms of the twist-to-bend stiffness ratio (1.69)

K = diag(1, 1,Γ) . (1.73)

In a uniform rod with an isotropic strain-energy density, we are free to rotate the

director basis in the reference configuration about the tangent vector (because of the

isotropy, any direction in the rod cross section is a principal axis). We can therefore

choose to orient the director basis so as to coincide with the Frenet frame, in which

case u = (0, κ, τ).

1.3 Helical rods

A helical rod is one whose centerline is a uniform helix, i.e. a curve with constant

curvature κ and torsion τ (here referred to simply as a helix). We denote the helix

axis by z, with a unit vector ez pointing in the direction of increasing parameter s.

1.3.1 Geometry

A uniform helical rod is a helical rod with constant register angle ϕ, constant

stretch ν, and constant shear (i.e. both the shear amplitude angle and the shear

orientation angle are constant).

In a uniform helical rod, the director basis coordinates of both the twist vector

and the stretch vector are constant, x = (u, v) = const. Therefore, the configuration

of a uniform helical rod is given by six geometric parameters.

Curvature and torsion as functions of the strain vectors For a uniform helical rod,

the tangential component of the twist vector is the torsion τ . However, if the rod is

sheared, this is not the same as the d3 component u3, since the d3 vector is normal

to the sheared cross section. Therefore, the following equations

κ =√

u12 + u22 , τ = u3 , (1.74)

41

are true only if there is no shear, i.e. if d3 = τ . In general, the curvature and the

torsion cannot be extracted from the twist vector alone, but information about the

tangent vector τ = v/ |v| is also necessary:

κ = |u|

√1−

(u · v|u| |v|

)2

, (1.75a)

τ =u · v|v|

. (1.75b)

Direction of the twist vector For a helix, the twist vector u is along the helix axis,

and points in the direction of increasing parameter s for a right-handed helix, and

in the opposite direction for a left-handed one. To account for the effect of the helix

handedness, we introduce the chirality coefficient σ,

σ := sgn τ ≡

+1 , right-handed helix

−1 , left-handed helix. (1.76)

Thus,

u = uzez , (1.77)

where the sign of uz is given by the chirality

uz = σ |u| . (1.78)

Alternatively, this provides an expression for a unit vector along the helix axis

pointing in the direction of increasing parameter s, in terms of the twist vector:

ez =σ

|u|u . (1.79)

Helix angle The helix angle ψ is defined as the angle between the helix axis and the

tangent vector. For a uniform helix, it is uniquely determined by the curvature and

torsion

ψ := ∠(ez, τ ) = arctanκ

τ. (1.80)

This follows directly from the fact that the torsion τ is the projection of u = uzez

onto the tangent vector τ , while the curvature κ is the orthogonal complement. The

complement of the helix angle is the pitch angle.

42

The projection of the stretch vector onto the helix axis is

vz = |v| cosψ = σu · v|u|

. (1.81)

Variational constraints The constraints for the dual variational problem (1.51b) are

given by the projections of the strain vectors onto the helix axis:

uz ≡ η1 , vz ≡ η2 . (1.82)

1.3.2 Mechanics

Since x = const, the constitutive relation implies that the director basis coordinates of

the stress vectors are also constant, f = (m,n) = const. Therefore, the conservation

laws (1.23) become algebraic equations for the director basis coordinates of the stresses

and the strains:

u× n = 0 , (1.83a)

u×m + v × n = 0 . (1.83b)

Direction of the force vector Equation (1.83a) implies that the force vector n is

collinear with the twist vector u,

n = µ2u , (1.84)

(the coefficient of proportionality µ2 has already been seen in (1.49)) i.e. the force is

along the helix axis z

n = Nez . (1.85)

Relation (1.84) can be put in scalar form via (1.78):

N = σµ2 |u| . (1.86)

43

Direction of the moment vector From (1.83b) and the fact that n and u are collinear,

we conclude that the moment vector lies in the plane spanned by u and v, i.e. by

the helix axis and the tangent vector. The moment vector makes an angle γ with the

z-axis. Equation (1.83b) implies

m⊥ =|N ||u|

v⊥ =|N | |v| sinψ|u|

, (1.87)

where the subscript ⊥ denotes the magnitude of the projection onto the plane per-

pendicular to the helix axis. Therefore,

tan γ =m⊥mz

=|N | |v| sinψmz |u|

. (1.88)

Coefficient µ2 between force and twist Equation (1.87) provides a way of evaluating

the absolute value of the Lagrange multiplier µ2:

|µ2| =|N ||u|

=m⊥v⊥

=

√|u|2 |m|2 − (u ·m)2

κ |v|. (1.89)

Alternatively, in case of a linear constitutive relation (1.33), µ2 can be found by

projecting the force part of the constitutive relation onto the helix axis (1.79),

N =σ

|u|uT(CT (u− u) + D(v − v)

), (1.90)

where C and D are blocks in the stiffness matrix K (cf. (1.35)). Hence, (1.86) yields

µ2 =N

σ |u|=

1

|u|2uT(CT (u− u) + D(v − v)

). (1.91)

Axial wrench A wrench is a pair of a torque and a force acting along a common axis.

By axial wrench we term a wrench acting along the helix axis, denoted (M,N).

Whereas the total force vector is along the helix axis (cf. (1.85)), i.e. equal to the

applied force, the applied axial torque only gives the projection of the total moment

onto the helix axis:

N = ± |n| , M := mz = |m| cos γ . (1.92)

44

Comparing with equations (1.44b) and (1.41), we have

N ≡ C1 , M ≡ C2 . (1.93)

Therefore, the applied wrench uniquely determines the constraints for the variational

problem (1.44).

The special feature of an axial wrench is that a helical rod under an axial wrench

retains a helical shape.

1.4 Helical Kirchhoff rods

1.4.1 Geometry

A uniform helical Kirchhoff rod is one with constant register angle ϕ, i.e. one

whose material orientation of the cross section follows the geometry of the centerline.

As it has been shown that non-uniform helical solutions of Kirchhoff equations are

highly atypical (cf. [46, Supporting Text]), we consider from this point on only uniform

helical solutions, for which all three director basis coordinates of the twist vector are

constant, u = const, and the third component is the torsion, u3 = τ . Helical rods

with different values of the register angle are shown in Figure 1.3. As a result of the

director basis coordinates of the strains being constant, the local coordinates of the

stresses are also constant,

n = const , m = const . (1.94)

The helix angle (1.80) in terms of the twist vector coordinates u is

ψ = arctanκ

τ= arctan

√u12 + u22

u3. (1.95)

As the stretch vector v is the (unit) tangent vector τ , its projection onto the helix

axis is the cosine of the helix angle:

vz = cosψ . (1.96)

45

(a) ϕ = 0 (b) ϕ = π4 (c) ϕ = π

2

Figure 1.3: Uniform anisotropic helical rods with different values of the register an-gle ϕ

1.4.2 Mechanics

As v ≡ v ≡ (0, 0, 1), the angular momentum balance (1.83b) in coordinate form reads:

u2m3 − u3m2 − n2 = 0 , (1.97a)

u3m1 − u1m3 + n1 = 0 , (1.97b)

u1m2 − u2m1 = 0 . (1.97c)

Direction of the moment vector If the twist vector coordinates are known, the angle γ

(cf. (1.88)) that the moment vector makes with the helix axis is given by

cos γ = σuTm

|u| |m|= σ

uTWu(u− u)

|u| |Wu(u− u)|. (1.98)

46

1.4.3 First integral surfaces in twist space

The first integrals (1.41) in the extensible and shearable case can be viewed as surfaces

in strain space (cf. Section 1.1.9). With Kirchhoff rods, the first integrals cannot be

evaluated directly using the constitutive relation since the force is now an unknown

variable. However, in the case of helical rods, we can exploit the collinearity of the

force n with the twist vector u (cf. (1.84)) in order to view the first integrals (1.41)

as surfaces in u-space. The coefficient of proportionality µ2 can be found either

from (1.97a) or (1.97b):

µ2 =n1u1

=u1m3 − u3m1

u1(1.99a)

=n2u2

=u2m3 − u3m2

u2. (1.99b)

As the curvature of a helix is strictly positive, κ =√u12 + u22 > 0, at least one

of (1.99a), (1.99b) is well defined. We will be assuming henceforth that u2 6= 0, and

use (1.99b), noting that (1.99a) is to be used instead should that assumption not

hold.

Eliminating the moment m using the constitutive relation (1.60),

mi = ∂iW (u− u) , i = 1, 2, 3 , (1.100)

the coefficient µ2 reads explicitly in terms of u:

µ2 =u2∂3W (u− u)− u3∂2W (u− u)

u2≡ (u×Wu(u− u))1

u2(1.101)

The equations of the integral surfaces (1.41) in u-space are thus

N2 = µ22 uTu =

((u×Wu(u− u))1

u2

)2

uTu (1.102a)

NM = µ2 uTm =

(u×Wu(u− u))1u2

uTWu(u− u) , (1.102b)

For a linear constitutive relation (1.62), multiplying both equations by the denom-

inator, equations (1.102a) and (1.102b) yield a sextic and a quartic, respectively:

47

S1(u;N) := (u× K(u− u))12 uTu−N2u2

2 = 0 , (1.103a)

S2(u;N,M) := (u× K(u− u))1 uTK(u− u)−NMu2 = 0 . (1.103b)

Alternatively, instead of surfaces (1.103) that correspond to the first integrals

I1 = 12N2 and I2 = MN , it may be more convenient to consider surfaces of the form

N (u) = N and M(u) = M . The force as a function of the twist vector is obtained

from (1.86) and (1.101):

N (u) :=σ |u|u2

(u×Wu(u− u))1 , (1.104)

while the axial moment is obtained by projecting the constitutive relation (1.60) onto

the helix axis unit vector ez (1.79):

M(u) :=σ

|u|uTWu(u− u) . (1.105)

1.4.4 Helix hyperboloid

The following is based upon the characterization of helical equilibria presented in [46].

A helical equilibrium of a Kirchhoff rod lies at the intersection of three surfaces in

u-space: the two integral surfaces S1(u;N) = 0 and S2(u;N,M) = 0 (cf. (1.103)), and

a third surface, defined by the tangential component of the moment equation (1.97c):

(u×Wu(u− u))3 = u1∂2W (u− u)− u2∂1W (u− u) = 0 . (1.106)

Unlike the integral surfaces, the shape of which depends on the applied wrench (N ,

M), i.e. on the values assigned to the two integrals I1 and I2, the third surface is

generic insofar as it is independent of boundary conditions, and characterized solely

by the constitutive relation. For a linear constitutive relation, this surface in the most

general form is a hyperboloid,

H(u) := (u× K(u− u))3 = 0 , (1.107)

48

(a) view from above (b) view from the side with iso-curves N (u) =const (blue) and M(u) = const (red)

Figure 1.4: The helix hyperboloid H(u) = 0

referred to henceforth as the helix hyperboloid, and is shown in Figure 1.4.

The following properties of the helix hyperboloid hold:

• It is a hyperboloid of one sheet, except in the degenerate cases listed below.

• It contains the u3 axis, as (u1 = 0, u2 = 0) is always a solution.

• It is independent of the torsional stiffness K33, it is determined by the bending

stiffnesses K11,K22 and the coupling terms K13,K23 alone.

• For an isotropic rod, it degenerates to a plane.

• In the diagonal case (i.e. with no bend-twist coupling) it degenerates to a hy-

perbolic cylinder.

49

Chapter 2

Compact planar waves

In this chapter, we consider planar waves on a homogeneous, inextensible and un-

shearable rod of infinite length with a nonlinear constitutive relation. We look for

homoclinic solutions that correspond to a loop-like elastica. In the case of a linear

constitutive relation, this is a solitary wave whose shape is a single loop of infinite

extent that straightens out exponentially on both ends (Euler’s Species 7 elastica,

cf. Figure 1b). In the pendulum analogy, this corresponds to the pendulum released

from the unstable equilibrium, and performing one full revolution before reaching the

unstable equilibrium again in infinite time. Adding a quartic term to the strain-energy

density, leads to a singular ode. The effect of the quartic term at the speed of sound,

as we show below, is the compactification of the solitary wave, i.e. the homoclinic

solution has compact support. A similar effect of a non-linear constitutive relation in

shear waves in the bulk of solids was demonstrated by Destrade and Saccomandi [47].

2.1 Traveling wave reduction

We assume that a force n = Nez (cf. (1.43)) is applied to the rod at infinity, hence

N is the tension at infinity, and ez is the asymptotic value of the tangent vector,

τ (s = ±∞) = ez. Denoting the traveling wave variable by χ = s− ct (c is the wave

speed) and the corresponding derivatives with primes, the conservation laws (1.18)

and (1.20) in the traveling wave system become

n′′ = c2ρAd′′3 , (2.1a)

m′ + d3 × n = c2 (ρI1u1d1 + ρI2u2d2 + (ρI1 + ρI2)u3d3)′ . (2.1b)

50

The system (1.7b), (2.1), together with the constitutive relation (1.29) represents a

system of 18 equations for 18 unknowns (u,n,m,d1,d2,d3). However, all unknown

functions can be expressed in terms of the twist vector u: the force n through u and

first integrals, the moment m explicitly via the constitutive relation, and the director

basis vectors (d1,d2,d3) from u and the boundary conditions. We will therefore refer

to u(χ) as the solution of the traveling wave system.

2.1.1 Equivalent static system

A remarkable property of the Kirchhoff equations is that the form of the traveling

wave system is formally equivalent to that of a static system (c = 0), as described by

the following proposition.

Proposition 2.1. u(χ = s − ct) is a traveling wave solution of the Kirchhoff equa-

tions, i.e. solution of the system (1.7b), (2.1), and (1.60), if and only if u is a solution

of the equivalent static system, comprised of (1.7b) and

n′ = 0 , (2.2a)

m′ + d3 × n = 0 , (2.2b)

m = Wu(u− u) , (2.2c)

where

n := n− c2ρAd3 =: n− Tcd3 , (2.3a)

m := m− c2 (ρI1 u1d1 + ρI2 u2d2 + (ρI1 + ρI2)u3d3) , (2.3b)

W (u) := W (u)− c2

2

(ρI1 u12 + ρI2 u22 + (ρI1 + ρI2)u32

)=: W (u)− Zc(u) , (2.3c)

are respectively the effective force, effective moment, and effective strain-energy den-

sity in the equivalent static system.

Proof. Trivial by substitution of (2.3) into (2.2).

51

Note that in the equivalent static system, the force is reduced by a tension Tc,

while the strain-energy density is reduced by a term Zc quadratic in the strains. Both

Tc and Zc are proportional to the square of the wave speed.

The transformation of the force by Tc (2.3a) implies that the boundary conditions

in the equivalent static problem also need to be transformed via (2.3a). Thus, the

tension at infinity is

N = N − Tc . (2.4)

If the rod in the wave system is under a tensile (rather than compressive), i.e. positive

force, the rod in the equivalent static system can be under tension, no tension, or

compression, depending on whether the wave speed is below, equal to, or above the

threshold speed

cT =

√N

ρA, (2.5)

respectively.

The transformation of the strain-energy density by Zc (2.3c) is of particular inter-

est as it allows the adjustment of the quadratic term in the equivalent static strain-

energy density W by choosing a suitable wave speed c. If the strain-energy density

W of the original traveling wave system contains quadratic and higher order terms,

this transformation makes it possible to cancel the quadratic terms, and unravel the

effect of higher-order terms. In two dimensions, the quadratic term in W can be

canceled completely by Zc (this occurs when c equals the speed of sound for the rod

material). However, in three dimensions, a particular condition relating geometric

properties of the rod to its elastic properties must be satisfied to cancel completely

the quadratic terms (the torsional stiffness would have to be equal to the sum of the

bending stiffnesses).

Proposition 2.1 justifies us in turning our attention to static systems exclusively.

Henceforth, we consider the system (1.7b), (2.2), with the arc length s as the inde-

pendent variable, and we drop the tildes hereafter.

52

2.2 Planar system

In two dimensions, the static Kirchhoff equations (1.7b), (2.2) simplify to a single

ode as follows. We confine the rod to the (x, z) plane of a fixed laboratory frame of

reference (x, y, z), with basis (ex, ey, ez), by pointing the binormal vector along the

(constant) y-axis by setting d2 := ey. Doing so, we have ensured that the director

basis is a continuous function of s. This is generally not true of the Frenet basis,

where the normal and binormal vectors discontinously change direction at inflection

points, thus β = ±d2, where the sign depends on whether the rod centerline in the

(x, z)-plane is convex or concave.

The only non-zero component of the twist vector is now in the binormal direction,

u = κd2, and the strain-energy density is a function of one scalar variable, W (u) ≡

w(κ), Wu ≡(dwdκ

)d2. Note that we denote the binormal component u2 of u by the

same symbol as the curvature: κ. Strictly speaking, however, the Frenet curvature

is positive, κ = |u2|, but as the director basis is not identical to the Frenet basis,

the continuity of d2 implies that the curvature κ must be allowed to take on negative

values (it changes sign at inflection points). In the remainder of this chapter, therefore,

we replace the definition (1.1) by the signed curvature

κ := θ′ , (2.6)

where

θ(s) := ∠(ez,d3(s)) (2.7)

is the angle that the tangent vector makes with the z-axis of the laboratory frame.

The effective strain-energy density of the equivalent static system (2.3c) reduces

to

w(κ) = w(κ)− c2ρI22

κ2 . (2.8)

We do not impose the requirement of non-intersection in the plane. This can be

interpreted as a rod with an infinitesimally small cross-section where non-neighboring

53

parts are stacked on top of each other (with no non-local interactions) if their Carte-

sian coordinates coincide.

There are two possible odes describing the static planar Kirchhoff rod: one using

the angle, and the other using the curvature as the dependent variable.

Angle formulation. The tension at infinity first integral N (cf. (1.43)) can be used

to eliminate the force from (2.2b), yielding(dw

dκ(κ− κ)

)′= N sin θ , (2.9)

Since κ ≡ θ′, equation (2.9) is a second-order ode in θ.

For a quadratic strain-energy density, equation (2.9) is the pendulum equation,

with its well-known solutions, none of which has compact support. For all initial value

problems of the pendulum equation the solution exists and is unique. Therefore, by

contradiction, it is not possible to stitch together parts of different solutions in order

to obtain a compact wave.

Even though equation (2.9) is the simplest equation to integrate, we turn our

attention to an alternative formulation that leads to a potential system.

Curvature formulation. Eliminating n1 from the normal component of the force

equation using the the binormal component of the moment equation, and using the

energy integral (1.67), we have:(dw

dκ(κ− κ)

)′′= κ

(H + w(κ− κ)− κdw

dκ(κ− κ)

). (2.10)

2.2.1 Solitary waves and compact waves

A solitary wave is a solution for which the strains and their derivatives asymptoti-

cally vanish on both ends. For equation (2.10), the solitary wave condition is

κ(s)→ κ(s) , κ′(s)→ κ′(s) as s→ ±∞ . (2.11)

54

A compact wave is a solitary wave with compact support [−`, `], i.e. a wave in

which the intrinsic state is reached with a finite value ±` of the independent variable

s (“in finite time”) rather than approaching it asymptotically (exponentially). By

analogy with dynamical systems, it is often easier to think of the variable s as a time

and picture the solution as evolving in time rather than space. We (ab)use the word

“time” to refer to the independent variable of the reduced dynamical system.

For equation (2.10), the compact wave condition is

κ(s) = κ(s) , κ′(s) = κ′(s) , ∀s ∈ R \ (−`, `) , 0 < ` <∞ . (2.12)

In other words, a compact wave is a solution u(s) that differs from the intrinsic state

u(s) only over a bounded set of values of its argument. We will assume that the rod

is infinite, so that a compact wave is composed of three parts: two semi-infinite parts

with zero strain bridged by a finite part with non-zero strain. Thus

u(s) =

u(s) , s ∈ R \ [−`, `]v(s) , s ∈ [−`, `]

, (2.13)

where v 6≡ u. At the boundaries between the three regions, continuity of u is required:

v(−`) = u(−`) , v(`) = u(`) . (2.14)

If the derivative u′ is continuous as well, the solution is a classical solution, otherwise

it is a weak solution of the Kirchhoff equations.

The problem of finding a compact wave is therefore a boundary value problem for

v subject to boundary conditions (2.14), where the length of the interval 2` is left

unspecified (` is a parameter to be determined from the equations). The boundaries

are at points (u(±`), 0) in phase space, and v corresponds to an orbit connecting these

two points. If the reference twist vector is constant (u = const), the two boundary

points coincide, and v corresponds to a homoclinic orbit with (u, 0) as the homoclinic

point.

55

Energy integral and tension. In case of either solitary or compact waves, the

solution is unstrained at infinity, κ(±∞) = κ(±∞). Therefore, the energy inte-

gral (1.67) is numerically equal to the tension at infinity, i.e.

H = N . (2.15)

2.2.2 Linear constitutive relation, solitary loop solution

We first consider planar waves on a straight rod (κ ≡ 0) with a quadratic strain-energy

density. In our equivalent static formulation, the effective strain-energy density (2.8)

is also quadratic:

w(κ) =α

2κ2 , α := (E − c2ρ)I2 . (2.16)

The effective stiffness vanishes when the wave speed equals the speed of sound

c0 :=√E/ρ . (2.17)

The curvature equation (2.10) is then the unforced and undamped Duffing equation:

κ′′ − ω02κ+

1

2κ3 = 0 , ω0

2 :=Hα. (2.18)

This is a potential system

κ′′ = −dV

ds, (2.19a)

V (κ) = −ω02

2κ2 +

1

8κ4 , (2.19b)

where V is a double-well potential for ω02 > 0 and a single-well potential otherwise.

We look for solutions that are in the form of a single loop on an intrinsically

straight rod. Therefore, we have null Dirichlet boundary conditions for the curva-

ture at infinity. A non-trivial solution is clearly only possible with a double-well

potential V , hence

ω02 > 0 =⇒

H > 0 & c < c0 , or

H < 0 & c > c0 .(2.20)

56

As H is equal to the value of the tension at infinity (cf. (2.15)), loop solutions exist

for subsonic waves if the tension is positive, and for supersonic waves if the rod is

under compression.

-2Ω0 - 2 Ω0 2 Ω0 2Ω0

Κ

-Ω0

4

2

V HΚL

(a) Potential.

-2Ω0 - 2 Ω0 2 Ω0 2Ω0

Κ

-Ω02

Ω02

Κ¢

(b) Phase portrait (κ, κ′) with highlighted ho-moclinic orbit.

Figure 2.1: The potential and the (κ, κ′) phase portrait with highlighted homoclinicorbit for an intrinsically straight rod and a quadratic strain-energy density.

The loop-like solution is the homoclinic orbit (for the fixed point at the origin)

in this double-well potential. The homoclinic orbit corresponds to zero energy of the

potential system, i.e.1

2κ′

2+ V (κ) = E = 0 (2.21)

This equation can be integrated for the curvature, yielding the following solution:

κ(s) = 2ω0 sech (ω0(s− s0)) , (2.22)

where s0 is an integration constant that corresponds to the position of the center of

the loop.

57

2.3 Compact planar waves

2.3.1 Compact wave criterion

We now examine the conditions under which equation (2.10) admits solutions with

compact support. We assume that w is a power function (excluding the quadratic

case, examined in Section 2.2.2 above), and that the intrinsic curvature is constant

(κ = const).

The general question is the following. Given a boundary value problem for an ode,

what are the conditions under which the solution’s orbit in phase space is traced in

finite time? We only consider homoclinic orbits with the homoclinic point at the

origin. We assume that the orbit itself has a finite length, and we consider separately

the boundaries and the interior of the orbit. More precisely, the orbit is traced in

finite time if and only if

1. the orbit leaves any neighborhood of the origin in finite time,

2. the orbit spends a finite time in any neighborhood of any point in the interior

of the orbit.

We refer to these properties as the finite-time property at the boundaries, and the

finite-time property in the interior, respectively. The necessary and sufficient condi-

tion for the former is given by the following lemma.

Lemma 2.1. Consider the equation

(zn)′′ = P (z) , (2.23)

where P (z) =∑k

j=0 ajzj is a polynomial, and n > 1, subject to the null Dirichlet

boundary conditions

z(−`) = z(`) = 0 , (2.24)

58

where ` is a parameter to be determined from the equation. Let m0 be the multiplicity

of the root z = 0 in P (z). Then ` is finite (i.e. the solution satisfies the finite-time

property at the boundaries) if and only if

n 6= m0 ≥ n− 2 . (2.25)

Proof. Equation (2.23) is a potential system in terms of y := zn:

y′′ = −dV

dy(y) , (2.26a)

V (y) = −∫ y

0

P (η1n ) dη . (2.26b)

The Dirichlet boundary conditions (2.24) imply that both y and y′ = nzn−1z′ are zero

at the boundaries. In order to satisfy these boundary conditions, the energy E of the

potential system must be set to zero

1

2(y′)2 + V (y) = E = 0 . (2.27)

Solving for z′ yields

z′ =2√n

√√√√ k∑j=0

ajj + n

zj−n+2 . (2.28)

Expanding the root around z = 0, the derivative z′ becomes, to first order:

z′ ∝ zm0−n

2+1 , (2.29)

where m0 is the multiplicity of the root z = 0 in P (z). We first impose the regularity

restriction: in order for the orbit to remain bounded, the power of z in (2.29) must

be non-negative:

m0 ≥ n− 2 . (2.30)

The critical value of n for which the behavior is exponential—and cannot satisfy

the boundary conditions (2.24)—is m0 = n, while all other values (m0 6= n) satis-

fying (2.30) yield solutions that converge to z = 0 polynomially, which is consistent

with the boundary conditions.

59

The finite-time properties in terms of the original equation translate directly into

finite-time properties of the potential system because the transformation y(s) = zn(s)

does not depend on the ‘time’ s explicitly. The finite-time property in the interior

is guaranteed if V is a coercive function with a finite depth and has no quadratic

extrema.

2.3.2 General power-law strain-energy density

We now apply Lemma 2.1 to equation (2.10) with a general homogenous strain-energy

density leading to a nonlinear constitutive relation,

w(x) =α

kxk , k ∈ 2N \ 2 . (2.31)

Only even values of k are considered, since odd values do not correspond to a stable

unstressed state. The variables z in the prototype equation (2.23), y in the corre-

sponding potential system (2.26), and the curvature κ are related by

κ− κ = z = y1n = y

1k−1 . (2.32)

Note that y is proportioinal to the moment of force. Thus we have a potential

system (2.26) in terms of the moment.

The key feature of equation (2.10) in the case of nonlinear elasticity (k > 2) is its

singularity: the left-hand side is:

α((κ− κ)k−1

)′′= α

((k − 1)(κ− κ)k−2κ′

)′. (2.33)

Near the boundaries, the curvature κ approaches the intrinsic curvature κ, thus the

function multiplying the highest derivative in (2.10) approaches zero and reaches zero

for a finite value of the independent variable. The right hand side of (2.10) must also

equal zero for κ = κ, thus

κH = 0 . (2.34)

60

Therefore, in the intrinsically curved case (κ 6= 0), the energy integral H must be

equal to zero, otherwise the boundary conditions cannot be reached, not even asymp-

totically.

It is interesting to note that, as we have a singular equation (cf. (2.33)) that can

be cast as a potential system (2.26), it is the nonlinear transformation (2.32) between

the two that carries the singularity.

Potential. Applying Lemma 2.1 to the curvature equation (2.10) for the power

function strain-energy density (2.31), in the left-hand side we identify n = k − 1,

while the right hand side is

P (z) =Hακ+Hαz − zk−1

(κ2 + zκ(2− 1

k) + z2(1− 1

k)

). (2.35)

The potential V (y) = −∫ y0P (η

1k−1 ) dη derived from (2.35) is a double-well potential.

Condition (2.34) ensures that the origin is at a local extremum of V .

Note on zero H and intrinsically curved rods. If H = 0, meaning that there is

no tension in the non-strained boundary parts (−∞,−`), (`,∞), and that the force

is compressive in the strained part [−`, `] (n3 < 0, cf. (1.68)), the origin is a local

minimum of V (y) for arbitrary even k. The origin is therefore a fixed point, and the

only solution is the trivial one: κ ≡ κ. Therefore, there are no compact wave solutions

for H = 0, implying that no compact waves are possible on intrinsically curved rods

κ 6= 0 (cf. (2.34)). Henceforth, we assume that H 6= 0, and consider the intrinsically

straight case exclusively:

H 6= 0 , κ = 0 , (2.36)

in which case the origin is at the central local maximum of the double-well potential V .

Behavior near the boundaries. With κ = 0, (2.35) reduces to:

P (z) =Hαz − (1− 1

k)zk+1 . (2.37)

61

The coefficient of the linear term is Hα6= 0, yielding m0 = 1, and the criterion (2.25)

is satisfied for k ≤ 4, i.e. k = 4.

Behavior away from the boundaries. For an intrinsicaly straight rod, the po-

tential is

V (y) = −∫ y

0

P (η1

k−1 ) dη =k − 1

2k2y

kk−1

((k − 1)y

kk−1 − 2k

Hα

). (2.38)

This double-well potential is a coercive function, and has a finite depth. It is straight-

forward to verify that it satisfies finite-time requirement in the interior for an arbitrary

power k > 2.

2.3.3 Quartic strain-energy density

A quartic system. Consider a linearly elastic rod, with strain-energy density

w0(x) = A2x2, where A is the flexural rigidity, that is the product of the Young’s

modulus by the cross-section second moment of area. If this strain-energy density is

perturbed by a quartic term, w(x) = w0(x) + α4x4, the quartic term can be brought

to light by means of Proposition 2.1: a wave propagating on this rod with a suitably

chosen speed is equivalent to a purely quartic-w static system, the quadratic term

being canceled by Zc (cf. (2.3c)). The critical speed at which this occurs, cS =√E/ρ,

is the speed of sound in the material of the rod.

Potential system. For the quartic strain-energy density

w(x) =α

4x4 , (2.39)

the polynomial (2.35) is

P (z) =Hαz − 3

4z5 . (2.40)

62

Applying Lemma 2.1, we have m0 = 1, n = 3, and (2.25) becomes 3 6= 1 ≥ 1. The

potential (2.38) is

V (y) =3

8y4/3

(3

4y4/3 − 2

Hα

), (2.41)

and is shown in Figure 2.2a. Again, this is a double-well potential with the origin

sitting at the local maximum. The homoclinic orbit is shown in Figure 2.2b.

-1 -2-

3

4 2-

3

4 1

y

ymax

-1.0

-0.5

0.5

V

Vmin¤

(a) Potential.

-1 -2-3

4 2-3

4 1

y

ymax

y¢

(b) Phase portrait (y, y′) with highlighted ho-moclinic orbit.

Figure 2.2: The potential and the (y, y′) phase portrait with highlighted homoclinicorbit for an intrinsically straight rod with a quartic strain-energy density (2.41).Scales: ymax := 64

a3, Vmin := −4608

a8, where a is the characteristic length scale (2.45).

Integrating the curvature equation. Rather than integrating the potential sys-

tem, it is more convenient to go back to the curvature equation (2.10) with the quartic

strain-energy density (2.39), (ακ3

)′′= κ

(H− 3α

4κ4), (2.42)

which can be solved exactly. Multiplying through by (κ3)′ ≡ 3κ2κ′, the resulting

equation can be integrated to yield

α

2

[(κ3)′

]2=

3H

4κ4 − 9α

32κ8 + C , (2.43)

63

where C is an integration constant, which corresponds to the energy in the potential

system for y, C ≡ E. The only value of the integration constant which yields solutions

compatible with the null Dirichlet boundary conditions is E = 0. Moreover, any non-

zero value of E implies a blow-up of the derivative at the origin, and the only value

leading to a finite jump in κ′ at the boundaries is E = 0. With the integration

constant set to zero, dividing both sides of (2.43) by 9α2κ4 leads to

(κ′)2 =1

6

Hα− 1

16κ4 , (2.44)

The solution of this equation can be expressed in terms of the Jacobi elliptic sine

function:

κ(s) =4

asn(s/a;−1) , a := 2

4

√6α

H. (2.45)

The Jacobi sn function is periodic with period 4K(−1), where K is the complete ellip-

tic integral of the first kind. In order to construct a compact wave of the form (2.13)

with u ≡ 0, we need a half-period between two consecutive zeros. The size of the

support of the compact wave solution is thus

2` = 2aK(−1) ≈ 10.49a . (2.46)

Therefore, the compact wave solution is given by:

κ(s) =

4a

sn(s/a;−1) s ∈ [0, 2aK(−1)]

0 otherwise(2.47)

(The support is now [0, 2`], which is equivalent to [−`, `] up to a phase that is the

integration constant in (2.45), which we have set to zero for simplicity.) The solu-

tion (2.47), as well as its integrated form (the angle θ) are shown in Figure 2.3, both

in phase portraits, and in the explicit form. The shape of the rod corresponding to

this curvature is shown in Figure Figure 2.4.

We have obtained a one-parameter family of compact wave solutions, where the

characteristic length a is a function of the ratio of the elastic constant α to the energy

integral H.

64

Κ

Κ¢

(a) Phase portrait (κ, κ′) with highlighted zero-energy orbit. This is the sole orbit that reachesthe κ′-axis.

2

s

Κmax

2

Κmax

Κ

(b) Homoclinic solution κ(s).

Π 2 ΠΘ

Κmax

2

Κmax

Θ¢

(c) Homoclinic solution in angle phase space(θ, θ′).

2

s

Π

2 Π

Θ

(d) Homoclinic solution θ(s).

Figure 2.3: Homoclinic solution for an intrinsically straight rod with quartic strain-energy density (2.47). Scales: κmax := 4

a, κ′max := 4

a2, a is the characteristic length

scale (2.45), and ` is the loop half-size, related to a through (2.46).

The following properties of the exact solution (2.45) follow from the properties of

the sn function.

• The graph of sn passes through the origin with unit slope, so (the jump in) the

derivative of the curvature (2.47) at the boundary is 4a2

. Solution (2.47) is a

weak solution.

65

Figure 2.4: Elastica with compact support (the strained part is highlighted) — thehomoclinic solution for an intrinsically straight rod with quartic strain-energy den-sity (2.47).

• The sn function reaches a maximum at s = K(−1) with value equal to one.

Therefore, the maximum value of the curvature (2.45) is

κmax = κ(s = aK(−1)) =4

a. (2.48)

• The derivative of sn can also be expressed in terms of Jacobi elliptic functions:

κ′(s) =4

a2cn(s/a;−1) dn(s/a;−1) ; (2.49)

thus the phase portrait can be drawn using exact values (2.45), (2.49) (see Fig-

ure 2.3a).

2.4 Conclusion

Unlike the linearly elastic rod, a static Kirchhoff rod with quartic or higher order

strain-energy density is described by a singular ode (cf. (2.33)). This singularity

results in non-uniqueness of solutions. The singular point is the fixed point κ = κ.

Therefore, if the homoclinic orbit can reach the fixed point in finite time, then we can

stitch two solutions together, e.g. the homoclinic solution with the trivial solution

66

κ = κ. By doing this at both ends of the homoclinic solution, we have constructed a

solution with compact support. Thus, the central point in establishing the criterion

for a compactly supported solution is examining the necessary criteria for traversing

the homoclinic orbit in finite time. We have shown that this is not possible for

intrinsically curved rods, while naturally straight rods do have this property if the

strain-energy density is quartic or has a quartic leading-order term (since it is only

the leading order term that matters for the criterion (2.25)).

The analogy between static and traveling wave equations demonstrated in Propo-

sition 2.1 implies that this static result can be made to travel. A wave on a rod

that has a strain-energy density function with both a quadratic and a quartic term

translates into an equivalent static system with an effective strain-energy where the

quadratic term is reduced by Zc (cf. (2.3c)). If the wave speed is such that Zc cancels

out the quadratic term, we are left with a quartic effective strain-energy density. The

wave speed at which this occurs is the speed of sound in the rod material. There-

fore, a rod with a strain-energy density that has a leading quadratic term and a

non-vanishing quartic term admits a sonic traveling wave solution with a compact

support.

67

Chapter 3

Planar waves on heterogeneous rods

We consider an inextensible and unshearable elastic rod of infinite length, that is

straight in a stress-free state. The rod is assumed to obey a linear constitutive

relation, but we abandon the homogeneity assumption: the rod is now heterogeneous,

i.e. its material properties vary along the centerline. The goal of this chapter is to

investigate the effect of such heterogeneities in localized flexural waves on a straight

Kirchhoff rod.

3.1 Governing equations

Rather than starting with balance equations in the director basis (cf. (1.18) and (1.20))

as in the previous chapter, we use here an alternative formulation set in a fixed frame.

We assume that the rod is confined to the (x, y)-plane and ignore the possible effect

of self-contact. Let (x, y) be the coordinates of a point of the rod centerline, (F,G)

the coordinates of the force acting at that point, and Φ the angle the tangent vector

at (x, y) makes with the x-axis. The dynamics of the rod is then governed by the

following system of equations (cf. e.g. [11] or [48]):

ρAxtt = Fs , (3.1a)

ρAytt = Gs , (3.1b)

ρIΦtt = (EIΦs)s +G cos Φ− F sin Φ , (3.1c)

xs = cos Φ , (3.1d)

ys = sin Φ , (3.1e)

where A and I are the cross-section area and second moment of area, ρ is the (mass)

density, and E is the Young modulus. We eliminate x and y from the first two

68

equations by differentiating them with respect to s and using the last two equa-

tions differentiated twice with respect to t. Thus we obtain the following system for

(F,G,Φ):

A (cos Φ)tt =

(Fsρ

)s

, (3.2a)

A (sin Φ)tt =

(Gs

ρ

)s

, (3.2b)

ρIΦtt = (EIΦs)s +G cos Φ− F sin Φ . (3.2c)

We assume that the rod is uniform with a circular cross-section of radius R, hence

A = πR2 , I =πR4

4. (3.3)

The scaling:

[s] =R

2, [t] = 1s , [F ] = [G] = πR2 [E] , [E] =

R2

4

[ρ]

[t]2, (3.4)

yields the following non-dimensionalized system (all variables are now dimensionless,

but are denoted by the same symbol as their dimensional counterparts):

(cos Φ)tt =

(Fsρ

)s

, (3.5a)

(sin Φ)tt =

(Gs

ρ

)s

, (3.5b)

ρΦtt = (EΦs)s +G cos Φ− F sin Φ . (3.5c)

3.2 Multiple scales asymptotic expansion

We consider a rod with variable material properties on a small scale, so that regions

of two different constant properties alternate periodically (see Figure 3.1). We denote

the length of the unit cell by ε. This cell is composed of two subdomains with lengths

αε and (1− α)ε, densities ρa and ρb, and Young moduli Ea and Eb, respectively.

69

ε

αε

a b

Figure 3.1: Heterogeneous rod microstructure

Assuming that the solution to the system (3.5) is essentially constant over a unit

cell, i.e. that ε is a small parameter with respect to a characteristic length of the

solution, we introduce a fast length scale s:

s =s

ε, (3.6)

and proceed with a standard multiple scale analysis (see e.g. [49]). The material

parameters are functions of the fast arc length s only,

ρ ≡ ρ(s) :=

ρa , s ∈ [0, α)

ρb , s ∈ [α, 1), (3.7a)

E ≡ E(s) :=

Ea , s ∈ [0, α)

Eb , s ∈ [α, 1). (3.7b)

The periodic structure of the rod induces periodicity in terms of the fast arc length

variable s with period 1 (size of the unit cell in terms of s) in the dependent variables

in the system,

F ≡ F (s, s, t) , G ≡ G(s, s, t) , Φ ≡ Φ(s, s, t) . (3.8)

With the addition of the fast arc length variable, the spatial differential operator

needs to be modified:

∂s 7→ ∂s +1

ε∂s , (3.9)

70

and the system is:

(cos Φ)tt =

(Fs + 1

εFs

ρ

)s

+1

ε

(Fs + 1

εFs

ρ

)s

(3.10a)

(sin Φ)tt =

(Gs + 1

εGs

ρ

)s

+1

ε

(Gs + 1

εGs

ρ

)s

(3.10b)

ρΦtt =

(E

(Φs +

1

εΦs

))s

+1

ε

(E

(Φs +

1

εΦs

))s

+G cos Φ− F sin Φ (3.10c)

We formally expand the variables in ε,

F =∞∑i=0

εiFi(s, s, t) , G =∞∑i=0

εiGi(s, s, t) , Φ =∞∑i=0

εiΦi(s, s, t) , (3.11)

and expand the trigonometric functions on the left-hand sides about Φ0, e.g.:

cos Φ = cos Φ0 +∞∑i=1

cos(i) Φ0

i!

(∞∑j=1

εjΦj

)i

. (3.12)

3.2.1 O(ε−2) system

Collecting terms in the system (3.10) expanded via (3.11) and (3.12), the lowest order

O(ε−2) yields the following system: (F0,s

ρ

)s

= 0 , (3.13a)(G0,s

ρ

)s

= 0 , (3.13b)

(EΦ0,s)s = 0 . (3.13c)

Multiplying (3.13a) by F0 and integrating by parts with respect to s,

F0F0,s

ρ

∣∣∣∣10

−∫ 1

0

F0,s2

ρds = 0 , (3.14)

the first term vanishes by periodicity of F0, and, as expected, we conclude that F0,s ≡

0, i.e. that F0 is a function of s and t only. Equations (3.13b) and (3.13c) yield

analogous results for G0 and Φ0, thus

F0 ≡ f0(s, t) , G0 ≡ g0(s, t) , Φ0 ≡ φ0(s, t) . (3.15)

71

We consistently denote variables that do not explicitly depend on the fast arc length

s with lowercase letters, and reserve uppercase letters for variables that depend on

the rod microstructure.

3.2.2 O(ε−1) system

The next order of ε in the system (3.10) expanded via (3.11) is O(ε−1):

F0,ss

ρ+

(F0,s + F1,s

ρ

)s

= 0 , (3.16a)

G0,ss

ρ+

(G0,s +G1,s

ρ

)s

= 0 , (3.16b)

EΦ0,ss + (EΦ0,s + EΦ1,s)s = 0 . (3.16c)

We solve equation (3.16a) for F1 using the following ansatz (cf. [18, 19]):

F1(s, s, t) = f1(s, t) +K(s)f0,s(s, t) , (3.17)

where

f1(s, t) := 〈F1〉 =⇒ 〈K〉 = 0 , (3.18)

where the operator 〈·〉 averages over the unit cell:

〈·〉 : ϕ 7→∫ 1

0

ϕ(x) dx. (3.19)

Equation (3.16a) then implies the following ode for K:(1 +Ks

ρ

)s

= 0 . (3.20)

Recall that ρ is a piecewise-constant function (cf. (3.7a)). Integrating (3.20) over

each subdomain, we obtain affine functions that we denote Ka and Kb, respectively.

The four integration constants (two on each subdomain) are found from the following

conditions:

(a) continuity (in terms of s) of F1,

72

(b) periodicity (in terms of s) of F1,

(c) the normalization condition (3.18),

(d) continuity of the parenthesized expression in (3.20), which we term the validation

condition.

Conditions (a) and (b) imply continuity and periodicity of K, respectively, yielding

Ka(α) = Kb(α) and Ka(0) = Kb(1). Note that the validation condition (d) implies

differentiability, since the derivative of the expression vanishes on both intervals, hence

on both sides of the point s = α. The four conditions yield the following solution

K(s) =

Ka(s) := (1−α)(ρa−ρb)

αρa+(1−α)ρb

(s− α

2

), s ∈ [0, α)

Kb(s) := α(ρa−ρb)αρa+(1−α)ρb

(1+α2− s), s ∈ [α, 1)

. (3.21)

By symmetry, equation (3.16b) yields

G1(s, s, t) = g1(s, t) +K(s)g0,s(s, t) , (3.22)

where K is also given by (3.21), and g1(s, t) ≡ 〈G1〉.

The solution to (3.16c) is entirely analogous. The ansatz

Φ1(s, s, t) = φ1(s, t) + L(s)φ0,s(s, t) , (3.23)

with the normalization condition

〈Φ1〉 = φ1(s, t) =⇒ 〈L〉 = 0 , (3.24)

yields the following ode for L:

(E (1 + Ls))s = 0 (3.25)

with the following solution:

L(s) =

La(s) := (1−α)(Eb−Ea)

(1−α)Ea+αEb

(s− α

2

), s ∈ [0, α)

Lb(s) := α(Eb−Ea)(1−α)Ea+αEb

(1+α2− s), s ∈ [α, 1)

. (3.26)

73

For future reference, we note that the equations (3.20) and (3.25) along with the

validation conditions imply that the differentiated expressions are constant over the

unit cell. We can evaluate these constants using the solutions (3.21) and (3.26) for

K and L, respectively:

1 +Ks

ρ≡ 1

αρa + (1− α)ρb= 〈ρ〉−1 =: ρ−1h , (3.27a)

E(1 + Ls) ≡EaEb

(1− α)Ea + αEb=⟨E−1

⟩−1=: Eh . (3.27b)

3.2.3 O(ε0) system

Next, we consider the system of order O(ε0):

(cos Φ0)tt =

(F0,s + F1,s

ρ

)s

+

(F1,s + F2,s

ρ

)s

(3.28a)

(sin Φ0)tt =

(G0,s +G1,s

ρ

)s

+

(G1,s +G2,s

ρ

)s

(3.28b)

ρΦ0,tt = (E (Φ0,s + Φ1,s))s + (E (Φ1,s + Φ2,s))s +G0 cos Φ0 − F0 sin Φ0 (3.28c)

Using the ansatz (3.17), (3.22), (3.23) for F1, G1,Φ1, as well as identities (3.27), the

system (3.28) becomes:

(cosφ0)tt =f0,ssρh

+

(f1,s +Kf0,ss + F2,s

ρ

)s

(3.29a)

(sinφ0)tt =g0,ssρh

+

(g1,s +Kg0,ss +G2,s

ρ

)s

(3.29b)

ρφ0,tt = Ehφ0,ss + (E (φ1,s + Lφ0,ss + Φ2,s))s + g0 cosφ0 − f0 sinφ0 (3.29c)

Averaged O(ε0) system We apply the averaging operator 〈·〉 (3.19) on the system (3.29).

We note that 〈ϕs〉 ≡ 0 for any function ϕ periodic on a unit cell, hence the second

terms on the right hand sides of (3.29) all vanish when averaged. The O(ε0) balance

74

is thus

(cosφ0)tt =f0,ssρh

(3.30a)

(sinφ0)tt =g0,ssρh

(3.30b)

ρhφ0,tt = Ehφ0,ss + g0 cosφ0 − f0 sinφ0 (3.30c)

This is a system describing the homogenized behavior of the heterogeneous rod in

the leading order approximation. It has the form of a system of equations describing

a homogeneous rod (cf. (3.5)), where the constant material properties are the bulk

density ρh, and Eh, which is one-half of the harmonic average of the Young modulus

(cf. (3.27); compare with [50]). Up to now, the analysis of the system was general.

We now focus on the localized flexural waves in order to understand the effect on

the microstructure in their characteristics. To do so, we solve the system (3.30) by a

traveling wave reduction:

ξ = s− ct , ∂t 7→ −c∂ξ , ∂s 7→ ∂ξ . (3.31)

The reduced system is

c2ρh (cosφ0)ξξ = f0,ξξ (3.32a)

c2ρh (sinφ0)ξξ = g0,ξξ (3.32b)(c2ρh − Eh

)φ0,ξξ = g0 cosφ0 − f0 sinφ0 (3.32c)

The solutions we are looking for are loops on a straight rod with a tension T applied

at infinity. We thus impose the following boundary conditions at infinity for the force,

limξ→±∞

f0 = T , limξ→±∞

g0 = 0 , (3.33)

while the boundary conditions for the angle for a single loop are

limξ→−∞

φ0 = 0 , limξ→+∞

φ0 = 2π . (3.34)

75

Integrating twice equations (3.32a) and (3.32b) subject to the above boundary con-

ditions, we have

f0 = c2ρh (cosφ0 − 1) + T , (3.35a)

g0 = c2ρh sinφ0 . (3.35b)

Now we can eliminate f0 and g0 from (3.32c):

φ0,ξξ =1

`2sinφ0 , (3.36)

where

`2 :=Eh − c2ρhT − c2ρh

. (3.37)

As expected from the Kirchhoff analogy, equation (3.36) is the pendulum equation

where the tangent angle plays the role of the angle the pendulum makes with the

vertical and the arc length corresponds to time [12]. The boundary conditions (3.34)

correspond to a homoclinic orbit, with φ0 = 0 mod 2π as the homoclinic point. We

therefore conclude that `2 > 0 (a negative value of `2 would have φ0 = π for a

homoclinic point), which implies the following condition on the wave speed:

c2 ∈ R+ \ [c12, c2

2] , (3.38a)

c12 := min

(Ehρh,T

ρh

), c2

2 := max

(Ehρh,T

ρh

). (3.38b)

Note that ` → 0 when the wave speed approaches c0 :=√Eh/ρh (speed of sound

in a homogeneous material with Young modulus Eh and density ρh), and ` → ∞

for c2 → T/ρh. With zero tension, the parameter ` ≡√

1− c20/c2 is an increasing

function of the wave speed c, and (3.38) yields a lower bound c0 for the wave speed.

Therefore, we have 0 < ` < 1, where ` → 0 for c → c0, and ` → 1 for c → ∞. The

solution to (3.36) with boundary conditions (3.34) is the well known homoclinic orbit

of the pendulum (cf. [12]),

φ0(ξ) = 4 arctan eξ−ξ0` , (3.39)

76

where ξ0 is an integration constant that corresponds to the position of the midpoint of

the loop. The parameter ` can now be identified as the characteristic size of the loop.

The solution (3.39) is shown in Figure 3.2. The shape of the planar rod corresponding

to this tangent angle is a single loop that straightens out exponentially on the two

ends, and is shown in Figure 3.5.

Ansatz for F2, G2,Φ2 We now go back to the O(ε0) system (3.29). The averaged

balance (3.30) implies (f1,s +Kf0,ss + F2,s

ρ

)s

= 0 (3.40a)(g1,s +Kg0,ss +G2,s

ρ

)s

= 0 (3.40b)

(E (φ1,s + Lφ0,ss + Φ2,s))s = (ρ− ρh)φ0,tt (3.40c)

Following the ansatz in (3.17), we decompose F2, G2, and Φ2 as follows

F2(s, s, t) = f2(s, t) +K(s)f1,s(s, t) +M(s)f0,ss(s, t) , (3.41a)

G2(s, s, t) = g2(s, t) +K(s)g1,s(s, t) +M(s)g0,ss(s, t) , (3.41b)

Φ2(s, s, t) = φ2(s, t) + L(s)φ1,s(s, t) +N(s)φ0,ss(s, t) , (3.41c)

subject to the following normalization

〈M〉 = 0 , (3.42a)

〈N〉 = 0 , (3.42b)

which is equivalent to

〈F2〉 = f2(s, t) , 〈G2〉 = g2(s, t) , 〈Φ2〉 = φ2(s, t) . (3.43)

Equation (3.40a) becomes:(1 +Ks

ρ

)s

f1,s(s, t) +

(K +Ms

ρ

)s

f0,ss(s, t) = 0 (3.44)

77

The first term vanishes by (3.20), leaving(K +Ms

ρ

)s

= 0 . (3.45)

We solve this equation for M analogously to (3.20), by integrating over the two

subdomains separately. Recall that on each subdomain ρ is constant (cf. (3.7a)),

while K is an affine function (cf. (3.21)). We thus obtain two quadratic functions Ma,

Mb, where the integration constants are obtained from:

(a) continuity (in terms of s) of F2,

(b) periodicity (in terms of s) of F2,

(c) the normalization condition (3.42a)

(d) the validation condition: continuity of the parenthesized expression in (3.45).

The solution for M is

M(s) =

Ma(s) := − (1−α)(ρa−ρb)2ρh

(s2 − αs+ α(2α−1)

6

), s ∈ [0, α)

Mb(s) := α(ρa−ρb)2ρh

(s2 − (1 + α)s+ 2α2+3α+1

6

), s ∈ [α, 1)

(3.46)

Similarly, we find N from (3.40c),

(ρ− ρh)φ0,tt = (E (L+Ns))s φ0,ss . (3.47)

Noting that φ0 satisfies the wave equation with speed c, this leads to

(E (L+Ns))s = c2 (ρ− ρh) . (3.48)

Integrating the equation on the two subdomains where E and ρ are constants, as L is

affine function on each, we obtain two quadratic functions Na and Nb. The constants

of integration are obtained by imposing the same four conditions as above. We note,

however, that the validation condition, i.e. the continuity of (E (L+Ns)) here does

78

not implies differentiability, as the right-hand side of (3.48) has a jump at the material

interface s = α. Therefore, the solution obtained for N is a weak solution.

N(s) =

Na(s) := na2s

2 + na1s+ na0 , s ∈ [0, α)

Nb(s) := nb2s2 + nb1s+ nb0 , s ∈ [α, 1)

, (3.49)

where the n’s are constants that depend on the parameters α, ρa, ρb, Ea, Eb, and the

wave speed c. (Henceforth, we omit the explicit expressions for the coefficients as

they become rather cumbersome at this point.)

3.2.4 O(ε) system

The next order system we consider is O(ε):

− (Φ1 sin Φ0)tt =

(F1,s + F2,s

ρ

)s

+

(F2,s + F3,s

ρ

)s

(3.50a)

(Φ1 cos Φ0)tt =

(G1,s +G2,s

ρ

)s

+

(G2,s +G3,s

ρ

)s

(3.50b)

ρΦ1,tt = (E (Φ1,s + Φ2,s))s + (E (Φ2,s + Φ3,s))s +G1 cos Φ0

−G0Φ1 sin Φ0 − F1 sin Φ0 − F0Φ1 cos Φ0 (3.50c)

Applying the ansatz for (F1, G1,Φ1) and (F2, G2,Φ2), as well as the identity (3.27a)

and the following one obtained from (3.45) and the solutions for K (3.21) and

M (3.46):K +Ms

ρ≡ 0 , (3.51)

the system (3.50) becomes:

− ((φ1 + Lφ2,s) sinφ0)tt =f1,ssρh

+

(f2,s +Kf1,ss +Mf0,sss + F3,s

ρ

)s

(3.52a)

((φ1 + Lφ2,s) cosφ0)tt =g1,ssρh

+

(g2,s +Kg1,ss +Mg0,sss +G3,s

ρ

)s

(3.52b)

ρ(φ1,tt + Lφ0,stt) = Ehφ1,ss + E(L+Ns)Φ0,sss

+ (E (φ2,s + Lφ1,ss +Nφ0,sss + Φ3,s))s

+ (g1 +Kg0,s) cosφ0 − g0(φ1 + Lφ0,s) sinφ0

− (f1 +Kf0,s) sinφ0 − f0(φ1 + Lφ0,s) cosφ0 (3.52c)

79

Averaged O(ε) system Averaging the system (3.52) over the unit cell, and using the

following identities:

〈E (L+Ns)〉 = 0 , (3.53a)

〈ρL〉 = 0 , (3.53b)

we obtain the O(ε) balance:

−ρh (φ1 sinφ0)tt = f1,ss (3.54a)

ρh (φ1 cosφ0)tt = g1,ss (3.54b)

ρhφ1,tt = Ehφ1,ss − φ1 (g0 sinφ0 + f0 cosφ0)

+ g1 cosφ0 − f1 sinφ0 (3.54c)

As the zeroth-order solution (f0, g0, φ0) is known (cf. (3.35a), (3.35b), (3.39)), this

is a system of equations for (f1, g1, φ1). Applying a traveling wave reduction, and

integrating the first two equations with zero boundary conditions at infinity, we have

f1 = −c2ρhφ1 sinφ0 (3.55a)

g1 = c2ρhφ1 cosφ0 (3.55b)

Eliminating f1, g1 from the reduced third equation yields

φ1,ξξ =1

`2φ1 cosφ0 , (3.56)

where ` is given by (3.37). The solution of (3.56) satisfying the boundary conditions

is given by

φ1(ξ) = φ0,ξ(ξ) =2

`sech

ξ − ξ0`

, (3.57)

where φ0 is the zeroth-order solution (3.39). This is the first correction in the homog-

enized solution, one that captures (the leading order of) the difference with respect

to the homogeneous system behavior.

80

Ansatz for F3, G3,Φ3 We now go back to the O(ε) system (3.50), with the following

ansatz for F3, G3,Φ3:

F3(s, s, t) = f3(s, t) +K(s)f2,s(s, t) +M(s)f1,ss(s, t) + P (s)f0,sss(s, t) (3.58a)

G3(s, s, t) = g3(s, t) +K(s)g2,s(s, t) +M(s)g1,ss(s, t) + P (s)g0,sss(s, t) (3.58b)

Φ3(s, s, t) = φ3(s, t) + L(s)φ2,s(s, t) +N(s)φ1,ss(s, t) +Q(s)φ0,sss(s, t) , (3.58c)

where P and Q satisfy the normalization conditions

〈P 〉 = 0 , 〈Q〉 = 0 , (3.59a)

i.e.

f3(s, t) = 〈F3〉 , g3(s, t) = 〈G3〉 , φ3(s, t) = 〈Φ3〉 . (3.60)

Using the ansatz for (F1, G1,Φ1) and (F2, G2,Φ2), as well as the solution (3.57) and

the identities (3.27), and (3.53), the system (3.50) becomes:

L (cosφ0)stt =

(M + Ps

ρ

)s

f0,sss (3.61a)

L (sinφ0)stt =

(M + Ps

ρ

)s

g0,sss (3.61b)

ρLφ0,stt = [E (L+Ns) + (E (N +Qs))s]φ0,sss

+ c2 〈ρ〉 (K − L (1− cosφ0))φ0,s (3.61c)

Transforming the left-hand side with the help of the averaged O(1) system (3.30),

each of the first two equations reduces to:

〈ρ〉−1 L =

(M + Ps

ρ

)s

(3.62)

As L is an affine function, and M is a quadratic one, the solution for P is a cubic

function each subdomain,

P (s) =

Pa(s) := pa3s

3 + pa2s2 + pa1s+ pa0 , s ∈ [0, α)

Pb(s) := pb3s3 + pb2s

2 + pb1s+ pb0 , s ∈ [α, 1), (3.63)

81

where the coefficients p are determined by the usual conditions: continuity, peri-

odicity, normalization and validation, and are functions of the material parameters

Ea, Eb, ρa, ρb, α, and the wave speed c.

Since φ0 satisfies the wave equation with wave speed c, equation (3.61c) yields:

[c2ρL− E (L+Ns)− (E (N +Qs))s

]φ0,sss =

c2ρh (K cos 2φ0 − L (1− cosφ0))φ0,s (3.64)

This equation does not yield an ode for Q because of the extra term on the right-

hand side. We therefore amend the ansatz (3.58c) for Φ3 so as to cancel this term

out. The modified ansatz is

Φ3(s, s, t) = φ3 + Lφ2,s +Nφ1,ss +Qφ0,sss +H(s, s, t) , (3.65)

H being a function such that

(EHs)s = Kχ(s, t) + Lλ(s, t) , (3.66)

where

χ(s, t) := c2ρhφ0,s cos 2φ0 (3.67a)

λ(s, t) := −c2ρhφ0,s (1− cosφ0) (3.67b)

Moreover, we impose the following normalization condition on H:

〈H〉 = 0 , (3.68)

so that φ3 = 〈Φ3〉 still holds, and we require H to be continuous and periodic. Anal-

ogously to the validation conditions seen above, we require EHs to be continuous as

well. We obtain H by integrating equation (3.66) twice over each of the two subdo-

mains, and determine the integration constants from the aforementioned conditions

82

on H. It is clear that the s-dependence of H is cubic. Note that integrating equa-

tion (3.66) with respect to s leaves χ and λ intact, therefore the form of the solution

for H is

H(s, s, t) = HK(s)χ(s, t) +HL(s)λ(s, t) (3.69)

The new ansatz (3.65) cancels out the last two terms in the right-hand side

of (3.64), yielding the following ode for Q:

(E (N +Qs))s =(c2ρ− E

)L− ENs (3.70)

As previously, Q is obtained by integrating twice over each of the two subdomains,

and the integration constants are found the usual way. As L is an affine function,

and N is quadratic, the resulting function Q is a cubic on each subdomain:

Q(s) =

Qa(s) := qa3s

3 + qa2s2 + qa1s+ qa0 , s ∈ [0, α)

Qb(s) := qb3s3 + qb2s

2 + qb1s+ qb0 , s ∈ [α, 1). (3.71)

3.2.5 O(ε2) system

Collected terms of order ε2 yield the following system:(−Φ2 sin Φ0 −

1

2Φ1

2 cos Φ0

)tt

=

(F2,s + F3,s

ρ

)s

+

(F3,s + F4,s

ρ

)s

(3.72a)(Φ2 cos Φ0 −

1

2Φ1

2 sin Φ0

)tt

=

(G2,s +G3,s

ρ

)s

+

(G3,s +G4,s

ρ

)s

(3.72b)

ρΦ2,tt = E (Φ2,s + Φ3,s)s + (E (Φ3,s + Φ4,s))s

+G0

(−Φ2 sin Φ0 −

1

2Φ1

2 cos Φ0

)−G1Φ1 sin Φ0 +G2 cos Φ0

−F0

(Φ2 cos Φ0 −

1

2Φ1

2 sin Φ0

)− F1Φ1 cos Φ0 − F2 sin Φ0 (3.72c)

83

Averaged O(ε2) system Using the ansatz expressions and the known identities, the

averaged system (3.72) becomes

− (φ2 sinφ0)tt = ρ−1h f2,ss +

⟨M + Ps

ρ

⟩f0,ssss +

1

2

⟨L2⟩ (φ0,s

2 cosφ0

)tt

(3.73a)

(φ2 cosφ0)tt = ρ−1h g2,ss +

⟨M + Ps

ρ

⟩g0,ssss +

1

2

⟨L2⟩ (φ0,s

2 sinφ0

)tt

(3.73b)

ρhφ2,tt = Ehφ2,ss − c2 〈ρ〉φ2 (1− cosφ0)

− f2 sinφ0 + g2 cosφ0 −c2ρh 〈L2〉

2φ0,s

2 sinφ0

+ 〈E (N +Qs)〉φ0,ssss − 〈ρN〉φ0,sstt

+ 〈EHK,s〉χs + 〈EHL,s〉λs (3.73c)

This is a system for (f2, g2, φ2), which we solve using a traveling wave reduction.

Integrating twice the reduced first two equations with zero boundary conditions at

infinity yields:

f2 = −c2ρhφ2 sinφ0 − ρh⟨M + Ps

ρ

⟩f0,ξξ −

c2

2ρh⟨L2⟩φ0,s

2 cosφ0 (3.74a)

g2 = c2ρhφ2 cosφ0 − ρh⟨M + Ps

ρ

⟩g0,ξξ −

c2

2ρh⟨L2⟩φ0,s

2 sinφ0 (3.74b)

Eliminating f2 and g2 from the reduced equation (3.73c) yields the following ode for

φ2(ξ):

φ2,ξξ −1

`2φ2 cosφ0 = ψ(ξ) , (3.75)

where ` is given by (3.37), and ψ is

ψ(ξ) =1

`2

(〈E (N +Qs)〉

c2ρh− 〈ρN〉

ρh

)φ0,ξξξξ

+1

`2

(〈EHK,s〉 − 〈EHL,s〉 (1− cosφ0)− ρh

⟨M + Ps

ρ

⟩)φ0,ξξ

− 1

`2

(〈EHL,s〉+

〈L2〉2

)φ0,ξ

2 sinφ0 (3.76)

The homogeneous part of equation (3.75) is the same as the equation for φ1 (3.56).

One solution (satisfying the boundary conditions for φ1) was found to be (3.57):

φ(1)2,hom = φ0,ξ . (3.77)

84

The other solution, found by a reduction φ2(ξ) = u(ξ)φ0,ξ(ξ) of the homogeneous

equation, is:

φ(2)2,hom = φ0,ξ

∫1

φ0,ξ2(ξ)

dξ . (3.78)

A particular solution of the equation (3.75) is now obtained by variation of parameters

φ2,part = φ(1)2,hom

∫W1(x)

W (x)dx+ φ

(2)2,hom

∫W2(x)

W (x)dx , (3.79)

where W is the Wronskian determinant for the homogeneous basis (3.77), (3.78), and

W1(ξ) := −ψ(ξ)φ(2)2,hom , W2(ξ) := ψ(ξ)φ

(1)2,hom . (3.80)

The explicit form of the particular solution φ2 is far too complex to be reproduced

here, but it is found to satisfy null boundary conditions at infinity.

3.3 Homogenized traveling wave solution

Using a homogenization approach, we have obtained a leading order solution φ0 and

the first correction φ1 in terms of the angle variable,

φ0(ξ) = 4 arctan eξ−ξ0` , (3.39)

φ1(ξ) = φ0,ξ(ξ) =2

`sech

ξ − ξ0`

, (3.57)

as well as the second correction φ2, given by (3.79). Whereas φ0 and φ1 depend

on the material parameters, the tension T , and the wave speed c only through the

characteristic length ` (3.37), the second-order correction explicitly depends on all

material parameters Ea, Eb, ρa, ρb, α, and the wave speed c (note the dependence

of ψ on various averaged quantities in (3.76), each being an expression involving the

material parameters).

The graphs of the three solutions are shown in Figures Figure 3.2, Figure 3.3,

and Figure 3.4. The effect of the first correction is to increase the angle φ within a

localized region, which coincides with the extent of the loop (compare with Figure 3.2).

85

-40 -20 20 40Ξ

Π

2 Π

Φ0

Figure 3.2: Homogeneous rod solution (3.39) for the angle. (ξ0 = 0, ` = 10)

In order to view the solution in terms of the shape of the rod in the (x, y) plane,

rather than integrating the x and y equations (3.1d) and (3.1e) for the combined

angle φ = φ0 + εφ1 + ε2φ2, we carry out the same multiple scale expansion as seen

above for these two equations.

3.3.1 Leading order solution

Collected terms of order O(ε0) for the Cartesian coordinates system and averaging

over the unit cell yields:

x0,s = cosφ0 , (3.81a)

y0,s = sinφ0 . (3.81b)

The solution of the zero-order system (3.81) is the well-known loop solution on a

homogeneous rod, depicted in Figure 3.5.

86

-40 -20 20 40Ξ

0.05

0.10

0.15

0.20

Φ1

Figure 3.3: First order correction (3.57) in terms of the angle. (ξ0 = 0, ` = 10)

3.3.2 First-order correction

Collecting terms of order O(ε) and applying the averaging operator yields:

x1,s = −φ1 sinφ0 , (3.82a)

y1,s = φ1 cosφ0 , (3.82b)

This gives the following solution in terms of the Cartesian coordinates:

x1(ξ) = cos(

4 arctan eξ−ξ0`

)− 1 , (3.83a)

y1(ξ) = sin(

4 arctan eξ−ξ0`

), (3.83b)

where the integration constants have been set by null Dirichlet boundary conditions

at infinity. The graphs of the two coordinate solutions are shown in Figure 3.6. The

effect of the first correction in the x direction is to move the homogeneous solution

loop to the left (see 3.6a). In the y direction, the “front part” of the loop, i.e. the

part corresponding to values of the independent variable ξ < ξ0, experiences a shift

upwards, while the ξ > ξ0 part shifts downwards (see 3.6b). This effect is illustrated

in Figure 3.7, where relatively large values of ε have been used in order to accentu-

ate the effect. The first-order correction conserves the arc length of the loop, since

87

-4 -2 2 4Ξ

-0.4

-0.2

0.2

0.4

Φ2

Figure 3.4: Numeric solution for the second order correction (3.79) in terms of theangle. (ξ0 = 0, ` = 10, E1 = 1, E2 = 1

2, ρ1 = 0.8, ρ2 = 1, α = 0.2, T = 1, and the

wave speed c is given by (3.37))

limξ→∞ x1(ξ) = limξ→−∞ x1(ξ).

3.3.3 Second-order correction

The O(ε2) system yields:

x2,s = −1

2φ1

2 cosφ0 − φ2 sinφ0 , (3.84a)

y2,s = −1

2φ1

2 sinφ0 + φ2 cosφ0 , (3.84b)

While the leading-order and first-order systems are solved analytically above, the

solutions we obtained for the the second-order system are numerical. Integrating the

coordinates x2 and y2 from (3.84) yields solutions depicted in Figures 3.8a and 3.8b

The arc length of the loop is altered by the second correction: note the shift in the

x-coordinate versus the arc length variable ξ in 3.8a. In the horizontal direction, the

loop gets stretched out (cf. 3.8a), while in the vertical direction, the tails get slightly

pushed upwards, while the central part of the loop is significantly pulled downwards

(cf. 3.8b). The effect of the second correction on the overall shape of the loop is

88

-20 -10 10 20x

1

y

Figure 3.5: Loop-like traveling wave solution for the homogenous rod, correspondingto the solution (3.39) for the angle. (ξ0 = 0, ` = 10)

shown in Figure 3.9 for different values of ε (compare with corresponding values of ε

in Figure 3.7).

3.4 Conclusion

Using multiple scale homogenization to second order in the space variable for the

localized loop-like traveling wave solution on a heterogeneous rod where the hetero-

geneity is made of periodically alternating regions of two different material properties,

we have studied the effect of the heterogeneity on the shape of the loop. The leading

order balance yielded the effective homogeneous system with the solution in the form

of the standard Euler homoclinic elastica.

An analytic solution for the first order correction was found, both in terms of the

angle variable (3.57) and in terms of the Cartesian coordinates (3.83). The effect of

the first order correction is to shift the loop in the opposite direction from the wave

propagation, to lower the front part of the loop and raise the back part of the loop, as

can be seen in Figure 3.6. The effect of the first correction is localized to the extent

of the loop, and decays exponentially in the tails of the loop, hence the length of the

loop is preserved and no shift is introduced. It is interesting to note that, although

89

there are several parameters in play, namely the material parameters Ea, Eb, ρa, ρb,

α, the tension T , and the wave speed c, both the leading order solution and the first

order correction explicitly depend only on one parameter, the characteristic length `,

which is a function of all of the above parameters (cf. (3.37)).

The second order correction φ2 has a far more complex form, too complex to be

given here explicitly (cf. (3.76), (3.77), (3.78), (3.79), (3.80)). Thus the equations for

the Cartesian coordinates of the second correction (3.84) were integrated numerically,

and the solutions are shown in Figure 3.8. Its effect is quite different from the first

correction: the vertical shift qualitatively looks like the horizontal shift in the first

correction, while in the horizontal direction, the rear (left) tail is shifted forward

(to the right), while the front (right) tail is exponentially unaffected, thus the loop

increases in length. Unlike the leading order solution and first correction, the second

correction depends explicitly on all parameters.

90

-40 -20 20 40Ξ

-2.0

-1.5

-1.0

-0.5

x1

(a) x1(ξ)

-40 -20 20 40Ξ

-1.0

-0.5

0.5

1.0

y1

(b) y1(ξ)

Figure 3.6: First correction traveling wave solution (3.83). (ξ0 = 0, ` = 10)

91

-20 -10 10 20x

5

10

15

20

y

Figure 3.7: Homogenized solution up to the first correction, φ = φ0 + εφ1, shownin the Cartesian plane for two different values of ε: ε = 3

8(solid curve), and ε = 3

4

(dashed curve). The homogeneous solution (ε = 0) is shown in dotted. (ξ0 = 0,` = 10)

92

-20 -10 10 20Ξ

0.02

0.04

0.06

0.08

0.10

0.12

0.14

x2

(a) x2(ξ)

-20 -10 10 20Ξ

-0.8

-0.6

-0.4

-0.2

y2

(b) y2(ξ)

Figure 3.8: Second correction to the traveling wave solution for the Cartesian coor-dinates (3.84) (ξ0 = 0, ` = 10, E1 = 1, E2 = 1

2, ρ1 = 0.8, ρ2 = 1, α = 0.2, T = 1, and

the wave speed c is given by (3.37))

93

-20 -10 10 20x

5

10

15

20

y

Figure 3.9: Homogenized solution up to the second correction, φ = φ0 + εφ1 + ε2φ2,shown in the Cartesian plane for two different values of ε: ε = 3

8(solid curve), and

ε = 34

(dashed curve). The homogeneous solution (ε = 0) is shown in dotted. (ξ0 = 0,` = 10, E1 = 1, E2 = 1

2, ρ1 = 0.8, ρ2 = 1, α = 0.2, T = 1, and the wave speed c is

given by (3.37))

94

Chapter 4

Helical Spring Problem

In this and the following chapter, we consider a system consisting of a helical spring

whose one end is fixed, while a wrench (a force and a torque acting along the same

axis) is applied to the other end along the helix axis (see Figure 4.1) A similar

system may have been considered by Hooke in discovering his Law, which solves

the problem of finding the force (the torque being zero) that maintains the spring at

a given elongation. This is a simple variant of what is here referred to as the Helical

Spring Problem (abbreviated hsp, or Direct Helical Spring Problem as opposed to

the Inverse Helical Spring Problem considered in the next chapter), which consists

in finding the stresses (typically the axial wrench) necessary to maintain the spring

in a state described by some given strains, or vice-versa, finding the strains given

the stresses. We assume that the elastic parameters of the spring are known. No

assumptions are made about the elastic properties other than a linear constitutive

relation (i.e. we assume that the rod is anisotropic with all possible couplings, and

consider the inextensible and unshearable, as well as the general—extensible and

shearable—case), and different possibilities for measured data are explored.

Note that our use of the terms direct problem and inverse problem in the context

helical springs differs from the use of those terms in the context of the elastica, where

a direct problem is one of finding the stresses applied at the ends for a given shape

of the elastica, the inverse problem consists in finding a shape for given boundary

stresses, while a semi-inverse problem of the elastica (introduced by Saint-Venant)

is that of finding the shape that satisfies certain assumptions. Thus, we might call

the Direct Helical Spring Problem a semi-inverse problem in the sense of the elastica,

since a helical shape of the solution is assumed.

95

Figure 4.1: A helical spring with one end fixed and a wrench—a force N (red) and atorque M (blue)—applied along the helix axis at the other end.

The results of this chapter are undoubtedly known to those familiar with the

field, but, to the best of the author’s knowledge, they have not been systematically

presented as here.

4.1 Helical Spring Problem for Prescribed Strains

We begin by examining the simplest case, where the measured data includes complete

information about the strains, and the corresponding stresses are sought.

Problem 4.1 (hsp for Prescribed Strains). Let the elastic parameters K and the

reference configuration u of a helical rod be known. Given a configuration x = (u, v),

96

find the axial wrench (M,N) that needs to be applied in order to maintain the rod in

the configuration x.

The solution for this problem is straightforward:

Solution. Expanding the constitutive relation (1.33) into 3D vector form,

m = K(u− u) + C(v − v) (4.1a)

n = CT (u− u) + D(v − v), (4.1b)

where K, D, and C are 3× 3 blocks in the 6× 6 matrix K (cf. (1.35)), and projecting

the two equations onto the z-axis (1.79), we obtain an explicit solution to Problem 4.1

M =σ

|u|uT (K(u− u) + C(v − v)) (4.2a)

N =σ

|u|uT(CT (u− u) + D(v − v)

). (4.2b)

4.1.1 Inextensible and unshearable case

Problem 4.2 (Kirchhoff hsp for Prescribed Strains). Let the elastic parameters K

and the reference configuration u of an inextensible and unshearable helical rod be

known. Given a configuration u, find the axial wrench (M,N) that needs to be applied

in order to maintain the rod in this configuration.

Solution. The solution is explicitly given by equations (1.104)

N = N (u) =σ |u|u2

(u× K(u− u))1 , (4.3)

and (1.105)

M =M(u) =σ

|u|uTK(u− u) . (4.4)

It is assumed in (4.3) that u2 6= 0. Otherwise, the Lagrange multiplier µ2 in (1.86)

should be evaluated via (1.99a).

97

4.2 Helical Spring Problem for Prescribed Wrench

Equations (4.2) are linear in the elastic parameters K, but nonlinear in the strain

variable u. Therefore, the reverse problem, that of finding the configuration x for a

given wrench (M,N), is less trivial, and the solution may not be unique.

Problem 4.3 (hsp for Prescribed Wrench). Let the elastic parameters K and the

reference configuration u of a helical rod be known. Given a wrench (M,N) applied

along the helix axis, find the corresponding configuration x = (u, v).

Solution. The applied wrench (M,N) provides the constraints I1(f) = 12N2, I2(f) =

NM (1.44b) for the variational problem of minimizing the Hamiltonian H(f) (1.44a),

where H is given by (1.38) with a quadratic strain-energy density, W ∗(f) = 12fTYf:

H(f) =1

2fTYf + fT x , (4.5)

and Y = K−1. Once a solution fmin to this variational problem is found, the configura-

tion is obtained from the conjugate constitutive relation (1.34), x(M,N) = Yfmin + x.

4.2.1 Inextensible and unshearable case

Problem 4.4 (Kirchhoff hsp for Prescribed Wrench). Let the elastic parameters K

and the reference configuration u of an inextensible and unshearable elastic helical rod

be known. Given a wrench (M,N) applied along the helical axis, find the corresponding

configuration u.

Solution. In the inextensible and unshearable case, in addition to the two constraints

provided by the applied wrench, N (u) = N (1.104) andM(u) = M (1.105), we have

a third, generic constraint given by the helix hyperboloid H(u) = 0 (cf. (1.107)).

These three constraint surfaces in twist-space have more than one intersection, but

98

imposing the following additional constraint yields a unique solution for the twist

vector u:

u3 u3 > 0 , (4.6)

This condition is an expression of the fact that we do not consider the possibility of

inversion of chirality due to the self-contact barrier.

4.3 Helical Spring Problem for Prescribed Observables

This is a generalization of Problem 4.1, where it is assumed that, instead of being

provided with complete information about the strains (i.e. all six components of

the strain vector x), the measured data includes less than six parameters describing

the geometry of the spring. We denote such a k-tuple of geometric parameters as

q = (q1, . . . , qk), and we refer to its components as observables.

Observables are functions of the strains. For example, the curvature of the helix

is an observable defined by the function κ(x) =√u12 + u22. In practice, the choice of

observables qj is driven by the experimental setup. For example, q could be composed

of the helix radius, the pitch, and the total height, q = (R,P, Z).

Problem 4.5 (hsp for Prescribed Observables). Let the elastic parameters K and

the reference configuration u of a helical rod be known, and let Q be a k-dimensional

vector function of the strain vector,

Q : x 7→ q = (Q1(x), . . . , Qk(x)) . (4.7)

Given a k-vector q = (q1, . . . , qk), find the axial wrench (M,N) that needs to be

applied in order to maintain the rod in a configuration x such that

Q(x) = q . (4.8)

Unlike Problem 4.1, where the strain vector x is given, what is given here is a

constraint on the strain vector, Q(x) = q (k scalar constraints).

99

Solution. A measured k-tuple q = (q1, . . . , qk) corresponds to a (6− k)–dimensional

manifold in the 6-dimensional strain space that is at the intersection of k iso-surfaces

Qj(x) = qj, j = 1, . . . , k. On the other hand, the wrench (M,N) —the solution

to the problem— defines two quadrics in the x-space J1(x) = N2 and J2(x) = MN

(cf. (1.55)), which intersect on a 4-dimensional manifold that we refer to as the wrench-

surface. In order to satisfy the constraint, the wrench-surface must intersect the

surface Q(x) = q.

The solution procedure therefore consists in fitting the wrench-surface, which is

specified by the unknown wrench (M,N), so that it intersects the surface Q(x) =

q, which is given by the measured data q. Assuming that the variational con-

straints (1.51b) are set by the observables q, i.e. that the functions

Uz : q 7→ uz , Vz : q 7→ vz (4.9)

are known, we can find the strain vector x that solves the variational problem (1.51)

with the additional constraint (4.8), i.e. we solve the following variational problem

for x:

Minimize W (x− x) =1

2(x− x)TK(x− x) (4.10)

subject to these constraints:

xT[I 00 0

]x = Uz

2(q) , xT[0 I0 0

]x = Uz(q)Vz(q) , Q(x) = q . (4.11)

Once the optimal strains xmin are found, the wrench (M,N) is obtained by solving

Problem 4.1, i.e. from equations (4.2).

4.3.1 Inextensible and unshearable case

Problem 4.6 (Kirchhoff hsp for Prescribed Observables). Let the elastic parame-

ters K and the reference configuration u of an inextensible and unshearable helical rod

be known, and let Q be a k-dimensional vector function of the strain vector,

Q : u 7→ q = (Q1(u), . . . , Qk(u)) . (4.12)

100

Given a k-vector q = (q1, . . . , qk), find the axial wrench (M,N) that needs to be

applied in order to maintain the rod in a configuration u such that

Q(u) = q . (4.13)

Solution. As in the extensible and shearable case, we assume that the constraints for

the dual variational problem (1.64b) can be evaluated from the observables, i.e. that

the functions (4.9) are known.

uTu = Uz2(q) , u3 = Uz(q)Vz(q) (4.14)

These two constraints define a circle in twist-space. Intersecting the circle with the

helix hyperboloid H(u) = 0 (cf. (1.107)) yields a solution for the strains u. The

wrench is then found by Problem 4.2, i.e. it is given by equations (4.4) and (4.3).

101

Chapter 5

Inverse Helical Spring Problem

In the Helical Spring Problem examined in the previous chapter, the input and the

output were the stresses and the strains, while the elastic parameters describing the

material properties of the spring—the stiffness matrix and the reference configura-

tion u—were assumed to be known. The inverse problem consists in finding the elastic

parameters given measurements of both the stresses and the strains.

As a rule, inverse problems are more challenging than direct problems. In the

present problem, this is illustrated by the fact that there are 23 unknown parameters

in the most general case, while the system that we are measuring only has two degrees

of freedom (e.g. the force and the torque applied along the helix axis).

In essence, the inverse problem consists in fitting the constitutive equation (1.34),

where the unknown parameters are the elements of the compliance matrix Y as well as

the three reference coordinates u, and the measured data includes the wrench (M,N)

and some parameters describing the geometry of the rod. However, since the stresses

and the strains figuring in the constitutive relation are not explicitly given by the

data, the fitting process is not straigtforward.

Representation of elastic parameters. In this chapter, we use the compliance

matrix Y = K−1 to represent the elastic parameters of the spring rather than the

stiffness matrix K. This presents some advantages, but also some drawbacks, e.g. the

fact that Y has an extra parameter as compared with K: as the directors were oriented

along the principal axes so as to yield K12 = 0 (cf. (1.36)), the inverted matrix

Y = K−1 does not have that property, i.e. in general we have Y12 6= 0. There are two

approaches to deal with this issue: one is to fit for 21 parameters Yij that are not all

independent, which would have for a consequence that the fitted result for K would

102

have a small but non-zero (1, 2) component. The second approach is to redefine the

directors so that d1 and d2 point along the principal axes of the compliance matrix

Y, not the stiffness matrix K. We have adopted the latter approach. Thus, we again

have 20 independent parameters in the general case:

Y =

Y11 0 Y13 Y14 Y15 Y16

0 Y22 Y23 Y24 Y25 Y26

Y13 Y23 Y33 Y34 Y35 Y36

Y14 Y24 Y34 Y44 Y45 Y46

Y15 Y25 Y35 Y45 Y55 Y56

Y16 Y26 Y36 Y46 Y56 Y66

, (5.1)

and five parameters in the inextensible and unshearable case:

Y =

Y11 0 Y13

0 Y22 Y23

Y13 Y23 Y33

. (5.2)

Elastic model. For any given data set, regardless of the true constitutive rela-

tion that the rod is governed by, we can choose to fit it with either the extensible

and shearable, or the inextensible and unshearable rod model. Moreover, we can fit

the data with a general compliance matrix, or assume a more specific form, such as

isotropic, diagonal, or other. The collection of assumptions made about the consti-

tutive relation (and the compliance matrix) is referred to as an elastic model. The

elastic model thus determines the number of unknown elastic parameters.

5.1 Linear problem

Ideally, the measured geometric parameters are all six components of the strain vec-

tor x.

Problem 5.1 (Linear Inverse hsp). Find the elastic parameters Y (5.1) and the

reference configuration u of a helical rod given a set of measurements

δ =

(M (i), N (i), x(i)) | i = 1, . . . , N

(5.3)

103

for the strain vector x = (u, v) and the corresponding wrench (M,N).

Solution. By substituting the stresses from (1.49) into the constitutive relation (1.34),

and denoting

M :=

[µ1I µ2Iµ2I 0

](5.4)

(this is a block matrix where I is the 3× 3 identity matrix), we obtain the following

equation

(I− YM)x = x . (5.5)

Here, Y and x are the unknown parameters, I is the 6×6 identity matrix, x is measured

directly, and the matrix M (5.4) is given by the Lagrange multipliers µ1 and µ2. The

multipliers µ1 and µ2, in turn, are evaluated from the four variational constraints C1,

C2, η1, and η2 (cf. (1.53a), (1.53b)), which have been identified as N , M , uz, and vz,

respectively (cf. (1.93), (1.82)). While the wrench (M,N) is explicitly given in the

data, uz and vz can be evaluated from the configuration point x via (1.78) and (1.81),

which we rewrite in coordinate form:

uz = σ |u| , vz =uTv

|u|. (5.6)

Equation (5.5) is linear in the parameters Y and u, and can be rewritten in the

form

Ap = b , (5.7)

where A = A(µ1, µ2, x) is a 6 × 23 matrix, b = b(x) is a 6-dimensional column

vector, and p is a 23-dimensional column vector whose components are the unknown

parameters. Each data point d(i) = (M (i), N (i), x(i)) ∈ δ yields a 6 × 23 system

A(i)p = b(i). The fitting problem can therefore be formulated as a simple (linear)

least squares problem, with an explicit optimal solution for the parameters. For N

data points we evaluate the matrices A(i) and b(i), and combine them into matrices

104

with 6N rows:

Aδ :=

A(1)

A(2)

...

A(N)

, bδ :=

b(1)

b(2)

...

b(N)

, (5.8)

which yield a concrete system

Aδp = bδ . (5.9)

With a minimum of N = 4 data points, the system (5.9) is overdetermined, and the

parameters p can be found by a simple least-squares problem minimizing |Aδp− bδ|.

The explicit solution for the optimal parameters is given by

popt = (ATδ Aδ)

−1ATδ bδ . (5.10)

5.2 Linear Kirchhoff problem

Problem 5.2 (Kirchhoff Linear Inverse hsp). Find the elastic parameters Y (5.2)

and the reference configuration u of an inextensible and unshearable helical rod given

a set of measurements δ =

(M (i), N (i),u(i)) | i = 1, . . . , N

for the local coordinates

of the twist vector u = (u1, u2, u3) and the corresponding wrench (M,N).

Solution. Eliminating the moment from the constitutive relation (1.63) using (1.65),

we obtain

(I− µ1Y)u = (µ2Yd3 + u) . (5.11)

Equation (5.11) comprises three scalar equations for the 8 parameters Y and u that are

linear in these parameters. This system can therefore be rewritten in the form (5.7),

105

where

A =

1 0 0 µ1u1 µ1u3 + µ2 0 0 00 1 0 0 0 µ1u2 µ1u3 + µ2 00 0 1 0 µ1u1 0 µ1u2 µ1u3 + µ2

, (5.12a)

p =[u1 u2 u3 Y11 Y13 Y22 Y23 Y33

]T, (5.12b)

b =[u1 u2 u3

]T. (5.12c)

Each data point d(i) = (M (i), N (i), u1(i), u2

(i), u3(i)) ∈ δ now yields a 3 × 8 system

A(i)p = b(i). For N data points, we evaluate the 3N × 8 matrix Aδ, and the 3N -

dimensional column matrix bδ via (5.8), which yield a concrete system (5.9). The

system is overdetermined with a minimum of N = 3 data points, and the explicit

solution to the least-squares problem is again given by (5.10).

5.3 Nonlinear problem

Let us now assume that the coordinates of the configuration vector x are not all

measured, directly or indirectly. Instead, the data has the form (M,N,q), where

q = (q1, . . . , qk) are some k parameters describing the geometry of the spring, that

we refer to as observables (introduced in Section 4.3). Henceforth, whenever we

refer to observables q, we assume that the underlying function Q (4.7), defining the

observables q in terms of the strain vector, q = Q(x), is known.

The wrench (M,N) determines the constraints (1.44b) for the variational prob-

lem (1.44) (cf. (1.93)). In order for the data to provide sufficient information about

the system, the constraints (1.51b) for the dual variational problem(1.51) must also

be set by the data, since knowing all four constraints for the two dual variational

problems determines the Lagrange multipliers for both problems (cf. (1.53)). As

the constraints (1.51b) are given by the axial projections of the strains uz and vz

(cf. (1.82)), we impose the following assumption on the observables q: that there

exists a map Z (composed of two scalar functions Uz and Vz)

Z : q 7→ (Uz(q), Vz(q)) (5.13)

106

such that the following diagram commutes:

x

Q

pz // (uz, vz)

qZ

;;(5.14)

where pz is the projection operator onto the z-axis, defined by (5.6). We say that

the maps Q (4.7) and Z (5.13) define a data model. Note that, as uz and vz are

independent quantities, the map Z must be such that the rank of its Jacobian matrix

is

rank(J(Z)) = 2 . (5.15)

Therefore, the number of observables must be k ≥ 2.

The linear problem considered in Section 5.1 above can be viewed as a trivial

special case, where the observables are the strain vector, q = x, so that the map Q

is the identity, and Z is the axial projection pz.

Problem 5.3 (Inverse hsp). Find the elastic parameters Y and the reference configu-

ration u of an elastic helical rod, given a set of measurements δ =d(i) | i = 1, . . . , N

,

d(i) = (M (i), N (i),q(i)) for the observables q = (q1, . . . , qk) and the corresponding

wrench (M,N).

Solution. We assume that the observables q represent a data model in the sense

above, i.e. that maps Q and Z are known functions such that the diagram (5.14)

commutes. For each data point d(i) it is then possible to evaluate the Lagrange

multipliers µ(i)1 and µ

(i)2 as

(µ1(i), µ2

(i)) = µ(N (i),M (i), Uz(q(i)), Vz(q

(i))) , (5.16)

where µ is a generic function given by (1.54). The Lagrange multipliers µ(i)1 and µ

(i)2 ,

in turn, yield the strain vector x(i) by solving the equilibrium equation (5.5),

x(i) = (I− YM(i))−1x , (5.17)

107

provided the unknown parameters Y and u. Thus, we have a point in strain-space

x(i) that is uniquely determined by the data point d(i), and a choice for the unknown

parameters Y and u. On the other hand, x(i) is required to satisfy the geometric

constraint Q(x(i)) = q(i), and this is an equation that provides a basis for constructing

an objective function: the left hand side is an estimate for the observables based on

the constitutive relation for the given measured data point, while the right hand side

is directly measured.

In order to make the outline above more explicit, let us denote the right hand side

in (5.17) as a function of the multipliers µ1 and µ2 and the unknown parameters:

X(µ1, µ2;Y, u) := (I− YM)−1[ud3

], (5.18)

where d3 = (0, 0, 1) is the director basis representation of the d3 vector. For each

data point d(i) = (M (i), N (i),q(i)), i ∈ 1, . . . , N, we obtain k scalar equations

q(i)j = Qj

(X(µ1

(i), µ2(i);Y, u

)), j = 1, . . . , k , (5.19)

where the explicit expression for the Lagrange multipliers in terms of the data is given

by (5.16).

We evaluate the residual r (a k-dimensional vector) corresponding to a given

data point d(i), i ∈ 1, . . . , N by subtracting the two sides of the equation (5.19)

r(i)(Y, u) =(r(i)1 , . . . , r

(i)k

):= q(i) −Q

(X(µ1

(i), µ2(i);Y, u

)). (5.20)

The evaluation of the residual is schematically depicted in the following diagram:

q

Z

xQoo

(uz, vz)µ(N,M,·,·)

// (µ1, µ2)

X(·,·;Y,u)

OO (5.21)

Starting from measured values for the observables, we evaluate the constraints of

the dual variational problem (uz, vz) by applying the map Z (5.13). Together with

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the measured wrench (M,N), these are all four variational constraints, allowing the

evaluation of the Lagrange multipliers µ1 and µ2 via the map µ (1.54). In the next

step, functional dependence on the parameters Y, u is picked up in obtaining the

configuration vector x from the µ1, µ2 multipliers (cf. (5.18)). Finally, by mapping the

obtained strain vector x through Q (4.7), we obtain an estimate for the observables

based on the constitutive relation and the unknown parameters. The residual is

obtained by subtracting this end-result after a full cycle in the diagram from the

measured values for the observables that the cycle started with.

We define the vector objective function for the data set δ as the component-

wise root of the sum over the data points of the squares of the residuals:

Rδ(Y, u) :=

√√√√ N∑

i=1

∣∣∣r(i)1 (Y, u)∣∣∣2, . . . ,

√√√√ N∑i=1

∣∣∣r(i)k (Y, u)∣∣∣2 . (5.22)

In order to formulate an ordinary (i.e. scalar) optimization problem, we need

a norm to scalarize the vector objective function. Under our assumption that there

is no noise in the data, equation (5.19) is exact provided that the system obeys

the constitutive law (1.34) exactly and that the values of the parameters are exact.

Therefore, the optimal value of the parameters corresponds not only to a minimum,

but to a root in each coordinate of the vector objective function. Hence, any norm

on the residual yields the optimal solution. However, the choice of a norm becomes

a nontrivial matter when noise is present in the data.

After nondimensionalizing the coordinates of the residual with some appropriate

characteristic scales (note that different geometric parameters qj may have different

physical dimensions!), we use a Euclidean norm to scalarize the vector objective

function. Defining the (scalar) objective function as

Rδ(Y, u) := |Rδ(Y, u)| ≡

√√√√ N∑i=1

|r(i)(Y, u)|22 , (5.23)

the solution to Problem 5.3 is the absolute minimizer (and root) of Rδ.

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5.3.1 Solution for noisy data

Nonlinear regression. In order to solve Problem 5.3 with noise in the data, non-

linear regression methods need to be employed. We consider equation (5.19),

q = Q (X (µ1, µ2;Y, u)) , (5.24)

as the basis for the nonlinear regression analysis. This, however, is not an explicit

equation for q, as the multipliers µ1, µ2 depend on q (cf. (5.16)). Therefore, this is a

case of nonlinear regression with implicit functional relationships (cf. [51, 52]),

g(M,N,q;Y, u) = 0 . (5.25)

Solving this problem, however, poses a computational challenge, as it involves evalu-

ating a very complex function and its Jacobian.

Least squares. A more pragmatic method of finding the solution by least squares

of the residuals consists in minimizing the scalar objective function (5.23). This

approach is not as rigorous as the implicit nonlinear regression insofar as the residuals

are obtained by subtracting two quantities that both contain noise. However, this

solution is substantially easier from a computational point of view, and numeric

simulations presented below produced satisfactory results in some cases (but not

in the fully general problem as stated in Problem 5.3).

5.3.2 Examples of observables

Minimal set of observables (uz, vz). The simplest choice for the observables is

q = (uz, vz), as these quantities appear directly in the equations. This is the set of ob-

servables holding the minimum amount of information, and constitutes the minimal

data model, where the map Z is the identity, and the map Q is given by (5.6).

The quantities uz and vz can be measured indirectly e.g. from data on the following

extensive quantities for data: centerline length L, the cumulative angle Θ (total

110

number of helical turns times 2π, cf. (6.1)), and the helix height Z, provided that the

centerline length in the unstressed state L is known:

uz =Θ

L, vz = ν cosψ =

L

Lcosψ =

Z

L. (5.26)

Curvature κ, torsion τ , stretch ν. The next data model according to the amount

of information it encapsulates is obtained by adding the helix angle ψ to the minimal

data model (uz, vz). The helix angle is a natural geometric parameter to add in this

context, as it amounts to resolving uz ≡ σ |u| into curvature κ = |uz| sinψ and torsion

τ = uz cosψ. Moreover, vz can be expressed in terms of the stretch ν and the helix

angle ψ as vz = ν cosψ. The three preceding equations provide a one-to-one mapping

of (uz, vz, ψ) to (κ, τ, ν). On the other hand, the inverse mapping is given by

uz =√κ2 + τ 2 (5.27a)

vz =ντ√κ2 + τ 2

(5.27b)

ψ = arctanκ

τ. (5.27c)

Therefore, the two data models (uz, vz, ψ) and (κ, τ, ν) are equivalent (in the sense

that they contain the same information). However, as the curvature, the torsion and

the stretch may be easier to measure directly than (uz, vz, ψ), we consider the former

the representative one.

The defining maps Q (4.7) for the data model (κ, τ, ν) follow from (1.75) and (1.8):

κ = |u|

√1−

(uTv

|u| |v|

)2

, (5.28a)

τ =uTv

|v|, (5.28b)

ν = |v| , (5.28c)

while the components of the mapping Z (5.13), Uz and Vz are given by (5.27a)

and (5.27b), respectively. This function Z can be verified to satisfy the criterion (5.15).

111

5.4 Variants of the IHSP

So far, we have treated the reference configuration u as an unknown parameter. How-

ever, it may well be that information—complete or partial—on the reference configu-

ration can be obtained. This information is very valuable, as numerical experiments

presented below have shown that it can reduce the computation time by several orders

of magnitude, and sometimes makes the difference between intractable and tractable

problems.

5.4.1 Known reference configuration

If the reference configuration u is known completely (i.e. all three coordinates are

known), then u in Problem 5.3 simply changes the role from an unknown parameter

to a known constant.

Problem 5.4 (Inverse hsp with known u). Find the elastic parameters Y (5.1) of

an extensible and shearable elastic helical rod with known reference configuration u,

given a set of measurements δ =

(M (i), N (i),q(i)) | i = 1, . . . , N

for the observables

q = (q1, . . . , qk) and the corresponding wrench (M,N).

The solution is entirely analogous, the only necessary modification being removing

u from the list of unknown parameters.

5.4.2 Known reference curvature and torsion

It is conceivable that the reference curvature κ and torsion τ can be measured, while

the reference register angle ϕ cannot, because the anisotropy in the elastic response

may not be accompanied by a detectable geometric anisotropy of the cross section.

This partial information about the reference configuration can be used to reduce the

number of unknown parameters, and the problem becomes:

112

Problem 5.5 (Inverse hsp with known κ, τ). Find the elastic parameters Y (5.1) and

the reference register angle ϕ of an extensible and shearable elastic helical rod with

known reference curvature κ and reference torsion τ , given a set of measurements

δ =

(M (i), N (i),q(i)) | i = 1, . . . , N

for the observables q = (q1, . . . , qk) and the

corresponding wrench (M,N).

All the necessary modifications to the formulae in the solution to Problem 5.3 are:

(i) replacing the list of unknown parameters with (Y, ϕ), and (ii) using the following

expression for u

u = (κ sin ϕ, κ cos ϕ, τ) , (5.29)

where κ and τ are set to their constant values, while ϕ is an unknown parameter.

5.5 Nonlinear Kirchhoff problem

As the configuration space is now three dimensional, the observables q are now defined

in terms of the corresponding function

Q : u 7→ q = (Q1(u), . . . , Qk(u)) . (4.12)

On the other hand, the requirements on the map Z : q 7→ (η1, η2) ≡ (uz, vz) remain

the same.

Problem 5.6 (Kirchhoff Inverse hsp). Find the elastic parameters Y (5.2) and the

reference configuration u of an inextensible and unshearable elastic helical rod, given

a set of measurements δ =d(i) | i = 1, . . . , N

, d(i) = (M (i), N (i),q(i)) for the ob-

servables q = (q1, . . . , qk) and the corresponding wrench (M,N).

Solution. Solving (5.11) for u, to obtain the analogue of equation (5.17),

u = (I− µ1Y)−1 (µ2Yd3 + u) , (5.30)

113

we denote the right hand side as

U(µ1, µ2;Y, u) := (I− µ1Y)−1 (µ2Yd3 + u) . (5.31)

The Lagrange multipliers µ1 and µ2 are evaluated as in the extensible and shearable

case (cf. (5.16)) The residual is defined analogously to (5.20) by replacing X with U:

r(i)(Y, u) := q(i) −Q(U(µ1

(i), µ2(i);Y, u

)). (5.32)

The vector and scalar objective functions Rδ and Rδ are constructed from the residu-

als (5.32) in the same way as they are from the residuals (5.20) in the extensible and

shearable case (cf. (5.22) and (5.23)). The solution to Problem 5.6 is the absolute

minimizer and root of the scalar objective function Rδ.

5.5.1 Examples of observables

Minimal data model. The minimal set of observables is still 2-dimensional, corre-

sponding to the two constraints of the dual variational problem (1.51a). The value of

the variational constraint η2 = vz is related to the helix angle vz = cosψ. This data

model is clearly equivalent to the one composed of the curvature κ and the torsion τ .

We use the latter (κ, τ) as the standard one. It is defined by the following maps:

Q : u 7→ (κ, τ) =(√

u12 + u12, u3)

(5.33a)

Z : (κ, τ) 7→ (η1, η2) =

(√κ2 + τ 2,

τ√κ2 + τ 2

)(5.33b)

The explicit form of the Lagrange multipliers µ1 and µ2 in terms of the data (cf. (5.16))

is:

µ1 =M(κ2 + τ 2)−Nτ√

(κ2 + τ 2)5, (5.34a)

µ2 =N√

κ2 + τ 2. (5.34b)

114

General data model. Since the number of observables is 2 ≤ k ≤ 3, any con-

ceiveable form of data in the inextensible and unshearable case is equivalent either to

the full 3-dimensional twist vector u or to (κ, τ). As in the former case, the problem

becomes Problem 5.2, which is explicitly solvable by linear least-squares, the minimal

data model q = (κ, τ) remains the only case of interest for Kirchhoff rods.

5.6 Special cases of the nonlinear Kirchhoff problem

5.6.1 Diagonal case

In the diagonal case Y = diag(Y11,Y22,Y33), the function U (5.31) is easy to evaluate

explicitly:

U1(µ1, µ2;Y, u) =u1

1− µ1Y11

(5.35a)

U2(µ1, µ2;Y, u) =u2

1− µ1Y22

(5.35b)

U3(µ1, µ2;Y, u) =u3 + µ2Y33

1− µ1Y33

. (5.35c)

The residual is given by:

r(i)κ (Y11,Y22,Y33, u1, u2, u3) = κ(i) −

√u1

2

(1− µ(i)1 Y11)2

+u2

2

(1− µ(i)1 Y22)2

(5.36a)

r(i)τ (Y11,Y22,Y33, u1, u2, u3) = τ (i) − u3 + µ(i)2 Y33

1− µ(i)1 Y33

, (5.36b)

where µ1 and µ2 are evaluated from the data via (5.34). The torsional coefficient Y33

is uncoupled from the two bending coefficients Y11 and Y22 and can be fitted inde-

pendently by minimizing the sum of the squares of the torsional residuals r(i)τ , while

the bending residuals r(i)κ yield fitted values for Y11 and Y22.

Two minima When the reference register angle ϕ is π/4 (i.e. |u1| = |u2|), the objective

function is degenerate, there are two different parameter values with the same objec-

tive function value equal to the absolute minimizer. (We assume here that Y11 6= Y22.

115

The case when they are equal is examined in Section 5.6.2 below.) This degeneracy

originates from the symmetry Y11 ↔ Y22 in the residual function. When ϕ deviates

from π/4, the symmetry is broken: one relative minimum rises above the other, and

their locations are no longer mirror images of each other in the Y11 = Y22 plane.

Assuming that no noise is present in the data, the optimal solution is given by the

absolute minimum.

5.6.2 Isotropic case

Finally, we consider the simplest situation of an isotropic rod (cf. Section 1.2.8). In

the isotropic case, the elastic properties of the rod are characterized by only two

constants, the bending rigidity K11 = 1/Y11, and the torsional rigidity K33 =

1/Y33, i.e. where the compliance matrix is (cf. (1.72))

Y = diag(Y11,Y11,Y33) . (5.37)

The dimension of the problem is thus reduced: only four unknown parameters remain,

(κ, τ ,Y11,Y33).

The isotropic case is degenerate in the sense that we obtain a linear problem for

a 2D data model (κ, τ): equation (5.11) yields

(1− µ1Y11)u1 = u1 (5.38a)

(1− µ1Y11)u2 = u2 (5.38b)

(1− µ1Y33)u3 = µ2Y33 + u3 (5.38c)

Now, as u1 ≡ 0 implies that u1 ≡ 0 (the coefficient cannot be zero since the multi-

plier µ1 varies from one datapoint to another, while the parameter Y11 is constant),

the register angle ϕ in any configuration is equal to the reference value ϕ. This means

that the register angle provides no information; all the information is contained in

the curvature and the torsion: (κ, τ) is at the same time the minimal data model and

116

the complete configuration information. The above system of equations is equivalent

to:

(1− µ1Y11)κ = κ , (5.39a)

(1− µ1Y33)τ = µ2Y33 + τ . (5.39b)

Reference coordinates κ, τ are unknown parameters Equations (5.39) are linear in all

four unknown parameters Y11,Y33, κ, τ , and can be rewritten as

Ap = b , (5.40)

where

A =

[1 0 µ1κ 00 1 0 µ1τ + µ2

], (5.41a)

p =[κ τ Y11 Y33

]T, (5.41b)

b =[κ τ

]T. (5.41c)

Each data point d(i) = (M (i), N (i), κ(i), τ (i)) yields a system A(i)p = b(i). The entire

data set δ =d(i) | i = 1, . . . , N

combines into a 2N × 4 system Aδp = bδ. The

optimal solution to the least-squares problem is given by

popt =(ATδ Aδ

)−1ATδ bδ . (5.42)

Reference coordinates κ, τ are known constants In this case, (5.39) is a system of two

equations for two unknowns Y11,Y33, which can be solved explicitly:

Y11 =κ− κµ1κ

(5.43a)

Y33 =τ − τ

µ1τ + µ2

(5.43b)

A single data point (κ, τ) is sufficient to evaluate the elastic parameters Y11 and Y33.

A special case of this problem where the applied torque is zero was considered by

Miller (1902), and he obtained approximate expressions for Y11 and Y33 [21, eqs. (11)

and (12)]1 that are valid for small changes in the coiling angle.

1In Miller’s notation, Y11 = 1B , Y33 = 1

A .

117

5.7 Numerical experiments with simulated data

The algorithms presented in this chapter for the solution of the nonlinear problem

were implemented and run on simulated data using a standard library optimizer for

minimizing the objective function.

5.7.1 Methods

Generating simulated data A method of generating simulated data that does not in-

volve solving the direct problem, was based on the idea of using the multipliers µ1

and µ2, instead of supplying the geometric parameters q as input. The multipli-

ers µ1 and µ2 are convenient for parametrizing the data points, as a choice of values

for (µ1, µ2) uniquely determines both the wrench (M,N) and the observables q. A

data point was generated by selecting normal random values for µ1 and µ2 with zero

mean and standard deviation σ = 0.1, and ensuring that the wrench is such that

−1 ≤M ≤ 1, and 0 ≤ N ≤ 1 (the rod is always under tension). The generated data

sets contained 20,000 data points each. The data models used were q = (κ, τ) for the

inextensible and unshearable case, and q = (κ, τ, ν) for the extensible and shearable

one.

When noise was applied, it was done on the geometric parameters q only, while

the values of the wrench (M,N) were left exact. For each exact (i.e. noise-free) data

set, fourteen noisy data sets were generated by adding Gaussian noise with standard

deviations in the range (10−14, 10−13, . . . , 10−1).

Fitting A modification of the Levenberg-Marquardt algorithm [53] (implemented in

SciPy2) was used for minimizing the (scalarized) objective function. Each run of the

optimizer would either yield a (local) minimum, or would fail to find a solution. A

series of fits (typically 100) was run for each data set, with a different initial value of

2scipy.optimize.leastsq package, http://scipy.org

118

parameters each time. The best fit among them was taken to be the one with the

lowest value of the objective function.

The choice of initial values for the parameters was an important aspect due to the

existence of multiple local minima that are not easily discernable near the degenerate

case described in Section 5.6.1. The selection process of seeds for parameter values

ensured that the two halfs of the parameter-space separated by the plane Y11 = Y22

were equally well explored. The seeds were then randomly perturbed.

Choice of elastic model Even though we have a choice in the elastic model to use,

the ultimate test of the algorithm is given by its output on the elastic model that

matches the constitutive relation that was used to generate the data. If the elastic

model does not match the true constitutive relation, e.g. if the true compliance matrix

is a full matrix, while a diagonal form is chosen to fit the data, the effect of the extra

terms that are unaccounted for in the elastic model can be viewed as a sort of a

systematic noise in the data. The results presented below are obtained for matching

elastic models.

5.7.2 Results

Number of data points With noise-free data sets, three data points were sufficient

to obtain the right parameter values close to within numerical precision. Even with

the lowest level of noise, however, this no longer the case, and a significantly higher

number of data points was necessary to obtain a fit. All 20,000 points were used for

noisy data sets.

Variant of the Inverse hsp The fitting algorithm proved itself most effective in the

case where u is known, i.e. the unknown parameters are comprised of the elements

of the Y matrix only. All the results presented below are of that case, i.e. these are

119

results for Problem 5.4 and its inextensible and unshearable counterpart, which is the

analogous modification of Problem 5.6.

The algorithm produced some results—albeit not very consistent—in the case

where u is partially known with reference curvature and torsion being known con-

stants, while the register angle was the one additional unknown parameter to the

elements of the compliance matrix.

As for the case where u is an unknown parameter (i.e. three unknown scalar

parameters (u1, u2, u3) in addition to the elastic parameters Yij), it was apparently

computationally intractable: the machines that were used to run the implementa-

tion of the fitting algorithm failed to return a result, even in the inextensible and

unshearable case (8 uknown parameters).

Inextensible and unshearable case In the inextensible and unshearable case, with ei-

ther the diagonal or the full-matrix elastic model and known reference configuration

u, virtually all runs (over 99.9% of them) returned a local minimum. Figure 5.1

shows a typical set of results for one data set with various levels of noise. The

true value of the compliance matrix was a full matrix, and five elastic parameters

(Y11,Y13,Y22,Y23,Y33) were fitted for.

The fit is a good one if the best fit is at the true global minimum (i.e. the minimum

that corresponds to the global one in the noise-free case). This was achieveable

with noise levels up to a certain threshold. The fits revealed that this threshhold

corresponds to the (appropriately normalized) value of the objective function at the

secondary minimum. This is seen in Figure 5.1a, where the value of the objective

function at the secondary minimum (shown in red) lies above the value of the objective

function at the true global minimum (blue) up to a certain level of noise (here: 10−6).

Above the threshold, the two local minima are indistinguishable: the values of the

objective function at the two minima are very close to each other, and the difference

(which may have either sign) is governed by the noise. Therefore, the best fit returned

120

by the optimizer may or may not correspond to the global minimum of the objective

function. In Figure 5.1b, for example, at noise level 10−3, the optimizer picked the

secondary minimum as the best fit. This can be seen by the fact that the error in

the fitted parameters coincides with the distance between the two minima (the red

dashed line). Below the threshold, however, the optimizer consistently returns the

global minimum, and the error in the fitted parameters scales well with noise (see

Figure 5.1b). Results from other data sets support the intuitively clear notion that the

threshold is higher for systems that exhibit a larger degree of asymmetry (i.e. where

ϕ deviates significantly from π4).

Extensible and shearable case The diagonal extensible and shearable case (6 unknown

parameters) produced as reliable results as the full-matrix inextensible and shearable

case (5 uknown parameters) (see Figure 5.2). The full-matrix extensible and shearable

case (20 unknown parameters), however, was computationally intractable.

5.8 Conclusion

We have examined the inverse problem of finding spring parameters from measure-

ments of both the stresses (axial wrench: force N and torque M) and the strains. If

the strains are measured in full detail, i.e. if information on all three or six components

of the strain vector(s) is available in the inextensible/unshearable or general case, re-

spectively, the problem reduces to an overdetermined system of linear equations, and

the optimal parameters are found by a simple least-squares procedure. However, if

we do not dispose of data on all components of the strains, but only fewer than the

maximum number of parameters describing the geometry of the spring, the solution

to the problem is not straightforward. The algorithm presented here is conceptually

simple, and the objective function was constructed in what may be the most straight-

forward way, but computational challenges on actual data in the general case were

such that it failed to produce results. However, it did produce satisfactory results

121

in the following cases, both with known reference configuration u, provided that the

noise level in the data was sufficiently low:

• the diagonal and full-matrix inextensible and unshearable case,

• the diagonal extensible and shearable case.

For noise levels above a certain threshold, an ambiguity was arising due to the fact

that the noise has brought two local minima to the same level so that the true global

minimum of the objective function could not be differentiated from a local minimum.

122

10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

noise in data

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

valu

e o

f obje

ctiv

e f

unct

ion a

t m

inim

a(n

orm

aliz

ed p

er

data

poin

t)

dataset: ktfull (20000 data points); #fits for given noise: 100

true global minimumtrue local minimum

(a) Values of the objective function at the two minima vs. noise

10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

noise in data

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

rela

tive e

rror

of

the f

itte

d p

ara

ms

w.r

.t. tr

ue v

alu

e

dataset: ktfull (20000 data points); #fits for given noise: 100

rel err of fitted paramsrel err of loc_min w.r.t. glob_min

(b) Error in the fitted parameters vs. noise. The red dashed line showsthe distance between the two minima.

Figure 5.1: Inextensible and unshearable case with a full matrix Y (five un-known parameters). u = (0.4, 0.5, 0.7), exact values of parameters: Y =[[1, 0, 0.1]; [0, 2, 0.2]; [0.1, 0.2, 1.4]]

123

10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2

noise in data

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

valu

e o

f obje

ctiv

e f

unct

ion a

t m

inim

a(n

orm

aliz

ed p

er

data

poin

t)

dataset: 6kts (20000 data points); #fits for given noise: 100

true global minimumtrue local minimum

(a) Values of the objective function at the two minima vs. noise

10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2

noise in data

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

rela

tive e

rror

of

the f

itte

d p

ara

ms

w.r

.t. tr

ue v

alu

e

dataset: 6kts (20000 data points); #fits for given noise: 100

rel err of fitted paramsrel err of loc_min w.r.t. glob_min

(b) Error in the fitted parameters vs. noise. The red dashed line showsthe distance between the two minima.

Figure 5.2: Extensible and shearable case with a diagonal matrix Y (six un-known parameters). u = (0.4, 0.5, 0.7), exact values of parameters: Y =diag(1, 1.1, 1.4, 0.5, 0.6, 0.7)

124

Chapter 6

Overwinding helical springs

In this chapter, we consider a system consisting of a helical spring with one end

fixed, while an axial force is applied on the other end, in such a way that the end is

free to rotate about the helix axis. This is a special case of the system considered

in the preceding two chapters and depicted in Figure 4.1 where the axial moment

of force is zero. As the force is applied, the free end rotates about the helix axis

either in the coiling direction (the spring winds, i.e. the number of helical turns

is increased) or in the opposite uncoiling direction (the spring unwinds). In this

chapter, we investigate the conditions under which one or the other behavior occurs.

More specifically, we want to answer the following questions:

1. When does the spring overwind, i.e. under what conditions does it wind as a

small tensile axial force is applied to it at rest?

2. Is there a critical value of the force Nc > 0 for which the behavior reverts from

winding to unwinding, or vice-versa?

This problem was considered in the isotropic case by Miller in 1902 [21]. We

address here the problem in full generality, assuming only a linear constitutive rela-

tion (1.62) within the Kirchhoff rod model (cf. Section 1.4).

It is interesting to point out that winding/unwinding on pulling can be prevented

by tightly intertwining two or more helices at a maximum pitch angle, and that this

fact is a consequence of helix geometry and not material properties [54]. This is the

mechanism that makes ropes inextensible.

125

6.1 Winding in twist-space

The cumulative angle that a point rotates by about the helix axis as it traverses the

helix centerline from bottom to top (i.e. the number of helical turns multiplied by 2π)

is proportional to the norm of the twist vector, and to the length of the centerline L:

Θ := L |u| . (6.1)

We define the coiling angle θ as the cumulative angle Θ per unit centerline length,

i.e.

θ :=Θ

L= |u| =

√κ2 + τ 2 . (6.2)

The coiling angle is equal to the distance from the origin in the (κ, τ) plane or in the

three-dimensional u-space.1 Therefore, the spring is winding if the configuration point

in u-space moves away from the origin, and it is unwinding if the point decreases its

distance from the origin.

Having worded the winding in terms of the twist vector, it becomes apparent

that the problem at hand can be solved by the Kirchhoff Helical Spring Problem for

Prescribed Wrench (Problem 4.4), where the wrench consists of the force only (the

moment is zero). However, an explicit solution to the Kirchhoff hsp is not necessary to

answer the questions raised above: conclusions can be drawn by geometric arguments

in twist space.

6.2 Geometric formulation of the problem

The helical equilibria are points that lie at the intersection of the helix hyperboloid

H(u) = 0 (cf. (1.107)) and the two first integral surfaces S1(u;N) = 0 and S2(u;N,M)

(cf. (1.103)), or, equivalently, the force and moment surfaces N (u) = N andM(u) =

1 Note that we do not include the coiling direction in the definition (6.2), and define θ to bepositive. The coiling direction is equal to the chirality σ (cf. (1.76)), so the signed coiling angleis σθ.

126

M given by (1.104) and (1.105), respectively

N (u) =σ |u|u2

ν(u) = N (6.3a)

M(u) =σ

|u|uTK(u− u) = M , (6.3b)

where ν(u) is the numerator in µ2 as expressed in (1.99b) by

ν(u) := µ2u2 ≡ (u× K(u− u))1 . (6.4)

In the problem at hand, there is no applied torque, M = 0. The moment sur-

face (6.3b) becomes an ellipsoid:

E(u) := uTK(u− u) = 0 , (6.5)

the zero-axial-moment ellipsoid. This is another generic surface, independent of

the force applied. Therefore, the intersection of the hyperboloid H(u) = 0 and the

ellipsoid E(u) = 0 represents the set of all helical equilibria with zero axial moment M .

This is a curve in u-space that can be parametrized by the applied axial force N ,

and that will be referred to as the trajectory.2 The arguments presented below

do not rely on the parametrization in terms of the force, but an important piece of

information is the direction along the trajectory that corresponds to increasing force.

As ν(u) = 0 the gradient of the force at the reference point u is proportional to the

gradient of ν:

∇N (u) = σ|u|u2∇ν(u) = σ

|u|u2

K13u2K23u2 − K22u3K33u2 − K23u3

. (6.6)

Note that the trajectory always contains the reference point, which is the solution

for no applied force. The fact that the helix hyperboloid contains the u3 axis and that

the ellipsoid (6.5) cuts the u3-axis (at the origin and a second point) implies that the

2 This is an abuse of the word, as the curve is not specified by initial conditions of any kind, butis entirely defined by the values of the stiffness matrix K and the reference point u. It can, however,be thought of as a trajectory that the configuration point traces in u-space as the force N is variedquasi-statically.

127

trajectory intersects the u3 axis, but this intersection would correspond to an infinite

force N : as (u1, u2) → (0, 0), the coefficient µ2 blows up (cf. (1.99) or (1.89)). The

u3-intercept is found from the ellipsoid equation (6.5) E(u∞) = 0, u∞ := (0, 0, u3∞)

to be

u3∞ :=

K13u1 + K23u2 + K33u3K33

. (6.7)

Therefore, the trajectory connects the reference point u with the point u∞, the latter

being a limit that is never reached for finite values of the force N .

6.3 Isotropic case

The simplest case to consider is the one with a diagonal stiffness matrix symmetric in

the cross section plane, K = diag(K11,K11,K33) (1.72). Of the two elastic coefficients,

one can be scaled out, so we set

K11 7→ 1 , K33 7→ Γ :=K33

K11

, (6.8)

where Γ is the twist-to-bend stiffness ratio (1.69).

Because of the isotropy in the cross-section plane, the u-space reduces to a plane:

as any plane containing the u3 axis is an invariant plane for the K matrix, the system

is confined to the plane containing the reference point u. In other words, the register

angle is constant along the trajectory and we can choose our frame so as to set it to

zero. By orienting the d1 and d2 vectors so as to set u1 to zero, the system is confined

to the (u2, u3) ≡ (κ, τ) plane. Moreover, this freedom in the orientation of the director

basis in the cross-section plane allows us to assume without loss of generality that

u2 ≡ κ > 0 , (6.9)

where whe have discarded the possibility of zero curvature, since it would require an

infinite force.

The system is thus confined to the right (κ, τ) semi-plane. Furthermore, the

self-contact barrier prevents it from switching chirality, i.e. from reaching the κ-axis.

128

Therefore, as an axial force is applied to a helical spring at rest (κ, τ), it will remain

in the same quadrant in the curvature-torsion plane.

The constitutive relation becomes:[m2

m3

]=

[κ− κ

Γ(τ − τ)

](6.10)

The ellipsoid (6.5) reduces to an ellipse:

E(κ, τ) := κ(κ− κ) + Γτ(τ − τ) = 0 , (6.11)

while the helix hyperboloid degenerates to a plane, which we have effectively used

already. Therefore, the ellipse E(κ, τ) = 0 is the trajectory. Note that the ellipse

contains the origin and the reference configuration (κ, τ) as two diametrically opposite

points, and that it can be brought to canonical form:(κ− κ

2

)2a2

+

(τ − τ

2

)2a2/Γ

= 1 , (6.12)

where a :=√κ2 + Γτ 2/2. The major axis is horizontal if Γ > 1, and vertical if Γ < 1.

For Γ = 1, the trajectory is a circle.

Inverse problem. As a sidenote and extension to Section 5.6.2, the inverse problem

of finding the elastic constant Γ given a measurement of the curvature and torsion is

trivial in the isotropic case, and requires no information about the force in the case

when the axial torque is zero: its solution is given by solving (6.11) for Γ:

Γ = −κ(κ− κ)

τ(τ − τ). (6.13)

(This equation, in a form in terms of the pitch angle and the coiling angle, was given

in Miller’s 1902 paper [21, eq. (5)].) Geometrically, the inverse problem consists in

fitting an ellipse (with horizontal and vertical axes) given three points: two known di-

ametrically opposite points (the origin and the reference point), and a third, measured

point.

129

6.3.1 Behavior of force along the trajectory

Direction of increasing force at u. The direction of increasing force along the

ellipse at the reference point is given by the vector (6.6), which, in the isotropic case,

becomes

∇N (κ, τ) = σ

√κ2 + τ 2

κ

[−τΓκ

]. (6.14)

For a right-handed helix, this points in the counter-clockwise direction, and in the

clockwise direction for a left-handed helix. In either case, at the reference point the

force increases towards the τ -axis.

Force is monotonous on the trajectory. This fact seems intuitively clear on

physical grounds, and can be verified by considering the elastic energy W (u − u),

which is an increasing function of the distance from the reference point in twist space,

|u− u|. It is not difficult to verify (by an argument analogous to the one presented

below in the qualitative analysis for critical force) that the energy stored in the spring

has a minimum at the reference point, and that it increases as the point u moves away

from the reference point on the ellipse in either direction. This statement holds true

on the entire part of the ellipse lying in the same quadrant as the reference point, as

the distance |u− u| cannot reach a maximum in this quadrant.

On the other hand, the energy is input into the system through the work done by

the axial force. The path on which the axial force N acts (per unit centerline length)

is given by the change in height of the helix,

Z − Z = cosψ − cos ψ , (6.15)

where ψ is the helix angle (cf. (1.80)). In the (κ, τ) plane, this is the angle that the

radius vector makes with the vertical (i.e. the complement of the polar angle). It

is clear that ψ increases monotonously from 0 to π2

as the ellipse is traced from the

130

point (0, τ) on the τ -axis to the point (κ, 0) on the κ-axis. Therefore, the force in∫ Z

Z

N(Z) dZ = W (u− u) (6.16)

is increasing (and positive) from the reference point to the τ -axis and decreasing (and

negative) from the reference point to the κ-axis.

The force blows up to infinity as the curvature approaches zero. All configurations

with an arbitrary tensile force applied N ∈ (0,∞) hence lie on the part of the ellipse

between (κ, τ) and (0, τ).

6.3.2 Overwinding criterion

The first question we want to answer is: For what values of Γ does the spring wind

when a tensile force is applied at rest?

Geometric approach. In the geometric picture, this question translates into:

When does the trajectory at the reference point move away from the origin? Or:

when does the tangent vector to the trajectory at the reference point in the direction

of increasing force have a positive radial component? The answer to the question lies

in the shape of the ellipse (6.12): the overwinding criterion is that the major axis of

the ellipse is vertical, i.e. Γ < 1, as shown graphically in Figure 6.1.

Analytic approach. The criterion for the critical point when the spring starts

unwinding is obtained by equating the slopes of the tangent lines to the ellipse and

the circle centered at the origin. This yields the following hyperbola:

2(1− Γ)κτ + Γτκ− κτ = 0 . (6.17)

Intersecting this hyperbola with the trajectory yields the critical values (κc, τc), as

shown in Figure 6.2. The hyperbola (6.17) contains the origin and the center of the

ellipse (κ/2, τ /2), and has asymptotes parallel to the coordinate axes.

131

HΚ`,Τ` L

G<1

G=1

G>1

Κ` Κ

Τ`

Τ

Figure 6.1: Zero axial moment ellipses E(κ, τ) = 0 (cf. (6.11)) correspoding to asubcritical (Γ = 1

2, solid curve), critical (Γ = 1, dashed curve), and supercritical

(Γ = 2, dotted curve) values of the elastic constant. A circle θ = θ is shown inred. The critical ellipse is a circle that shares a tangent line with the θ = θ circle atthe reference point. A subcritical ellipse (Γ < 1) cuts the θ = θ circle between thereference point and the τ -axis.

Clearly, the hyperbola intersects the ellipse (6.11) at some point in the quadrant

containing the reference point (the first quadrant for a right-handed helix). The

question of overwinding reduces to whether this point of intersection is located in the

part corresponding to a tensile force (between the reference point and the τ -axis) or

to a compressive force (between the reference point and the κ-axis). The answer to

this question is governed by the concavity or convexity of the hyperbola, which can be

checked by evaluating the second derivative τ ′′(κ) (obtained by implicit differentiation

132

HΚ`,Τ`L

HΚc1,Τc1L

HΚc2,Τc2L

Κ`

2Κ`

Τ`

2

Τ`

Figure 6.2: Critical hyperbolas (6.17) for subcritical (Γ1 = 12) and supercritical (Γ2 =

2) values of the stiffness ratio.

of the hyperbola equation (6.17)) at the center of the ellipse:

τ ′′|(κ,τ)=(κ/2,τ/2) = 41− Γ

Γ2

τ

κ2, (6.18)

the sign of which is given by the sign of σ(1−Γ). This provides an alternate, analytical

way of obtaining the overwinding criterion.

• If Γ < 1, the hyperbola intersects the ellipse between the reference point and

the τ -axis. The critical force is positive (tensile). The spring overwinds.

• If Γ > 1, the hyperbola intersects the ellipse between the reference point and

the κ-axis. The critical force is negative (compressive). The spring does not

overwind, but it winds when compressed until the critical value of the force.

133

6.3.3 Critical force

The second question we ask is: Is there a critical value of the applied force N for

which the behavior reverts from winding to unwinding, or vice-versa?

Geometric approach. This question is also best answered graphically by looking

at Figure 6.1:

• If the major axis of the ellipse is horizontal (i.e. for Γ > 1), the trajectory

from the zero-force point (κ, τ) to the infinite-force limit (0, τ) is monotonously

decreasing its distance from the origin. This is clear as the tangent vector to

the ellipse has a negative radial component (towards the origin) going from the

reference point to the τ -axis. The coiling angle is thus monotonously decreasing,

and the spring keeps unwinding as the force is increased.

• If the major axis of the ellipse is vertical (i.e. for Γ < 1), the trajectory starts

from the zero-force point (κ, τ) by moving away from the origin, reaches a

maximum distance and then monotonously decreases its distance from the origin

until the infinite-force limit point (0, τ), as seen in Figure 6.4. The coiling angle

thus increases from the reference value θ (spring at rest, see Figure 6.3a) to the

maximum value θc (Figure 6.3b), and then decreases back to the initial value θ

at the point (κ0, τ0) (Figure 6.3c), and further down (Figure 6.3d) to the limit

value θ∞ = τ . The graph of this behavior is shown in Figure 6.5, which does

not show the asymptote κ− θ in order to emphasize the critical behavior.

The critical behavior is not observed in the helix radius: it monotonously decreases

as the force is increased (see Figure 6.6).

Analytic approach. The critical point (κc, τc) lies at the intersection of the el-

lipse (6.11) and the critical hyperbola (6.17) in the quadrant containing the reference

134

(a) Spring at rest

(b) Critical point - maximum over-winding

(c) Return point (d) Unwinding

Figure 6.3: Behavior of spring with increasing tensile force (Γ = 12, κ = 0.66, τ =

0.066).

135

Θ=Θc

Θ=Θ`

HΚ`,Τ`L

HΚc,ΤcLHΚ0,Τ0L

Κ` Κ

Τ`

Τ

Figure 6.4: A trajectory for a tensile force and subcritical bend-to-twist stiffness ratio(Γ = 1

2). The critical hyperbola (6.17) is shown in dashed. As the force is increased

from zero, the system moves from the reference point (κ, τ) so that the coiling angle isincreased until it reaches the maximum value θc =

√κc2 + τc2. As the force is further

increased, the helix starts unwinding, reaching its initial value of the coiling angle θat (κ0, τ0). An infinite force is needed to reach the point (0, τ).

point. Solving the system for the critical point, we can find the actual value of the

critical force from (6.3a) as Nc = N ((0, κc, τc)).

The dependence of the critical coiling angle θc =√κc2 + τc2 and the corresponding

critical force Nc on the elastic constant Γ is shown in Figure 6.7 and Figure 6.8,

respectively.

The results of this section were given, in a different form, by Miller [21]. He

assumed that Γ < 1, as this is true of “normal” materials, i.e. those for which the

Poisson ratio is 0 < η < 12

(see e.g. [6, Art. 3]).

Finally, we observe that overwinding provides a simple test whether Γ is less than

136

G<1

G=1

G>1

0.2 0.4 0.6 0.8 1.0 1.2 1.4N

Π

60

Π

180

-Π

60

-Π

180

Θ - Θ`

Figure 6.5: The coiling angle θ as a function of the applied force N , for Γ = 12

(solidcurve), Γ = 1 (dashed), and Γ = 3

2(dotted). (κ = 3

4, τ = 1

2)

or greater than unity: if the helical spring winds when pulled from rest, then Γ < 1;

if it winds when compressed from rest, then Γ > 1. However, this test is only valid if

the rod is isotropic.

6.4 Anisotropic case

In the general case of an anisotropic rod the stage extends to the 3-dimensional u-

space. The trajectory, which was the ellipse (6.11) in the isotropic case, is now the

intersection of the zero-axial-moment ellipsoid E(u) = 0 (6.5) and the helix hyper-

boloid H(u) = 0 (1.107). If there is no bend-twist coupling, the ellipsoid axes are

parallel to the coordinate axes (this special case is considered in the following sec-

tion), but if the stiffness matrix has non-zero off-diagonal terms, this is no longer the

case (see Figure 6.9, where the zero-axial-moment ellipsoid is rotated so that its axes

are no longer parallel to the coordinate axes).

We obtain the overwinding criterion by considering the tangent vector to the

trajectory at the reference point in the direction of increasing force N . The spring

winds if this vector has a positive radial component.

137

0.5 1.0 1.5 2.0N

0.4

0.5

0.6

0.7

0.8

0.9

R

Figure 6.6: Helix radius R as a function of the applied force N . (κ = 34, τ = 1

2, Γ = 2

3)

A tangent vector to the trajectory at a point u is given by ∇E(u) × ∇H(u),

where ∇E(u) and ∇H(u) are normal vectors to the ellipsoid (6.5) and the hyper-

boloid (1.107), respectively. At the reference point u, these two vectors are given by

simple expressions:

∇E(u) = Ku , (6.19a)

∇H(u) =

−K11u2K22u1

K23u1 − K13u2

(6.19b)

The tangent vector ∇E(u) × ∇H(u), needs to be corrected for the sign in order to

point in the direction of increasing force N . The trajectory tangent at the reference

point pointing in the right direction is thus:

t := sgn (∇N (u) · (∇E(u)×∇H(u))) (∇E(u)×∇H(u)) , (6.20)

where ∇N (u) is given by (6.6). Due to continuity of the two normal vectors and the

monotonicity of the force N on the trajectory, the sign correction factor evaluated at

the reference point,

χ := sgn (∇N (u) · (∇E(u)×∇H(u))) , (6.21)

138

0.4 0.5 0.6 0.7 0.8 0.9 1.0G

Π

180

Π

60

Π

30

Θc - Θ`

Figure 6.7: The critical coiling angle θc as a function of Γ, relative to the referencevalue θ =

√κ2 + τ 2. (κ = 3

4, τ = 1

2)

remains the same throughout the trajectory. Explicit expressions (6.6) and (6.19)

yield

χ = σ detK ≡ σ , (6.22)

by positive definiteness of the stiffness matrix. Thus the tangent vector pointing in

the direction of increasing force at an arbitrary point of the trajectory u is given by

t(u) := σ∇E(u)×∇H(u) , (6.23)

where general expressions of the normal vectors to the ellipsoid and hyperboloid at

an arbitrary point u are given by:

∇E(u) = K(2u− u) (6.24a)

∇H(u) =

−K11u2 + K22(u2 − u2) + K23(u3 − u3)K22u1 − K11(u1 − u1)− K13(u3 − u3)

K23u1 − K13u2

. (6.24b)

Winding criterion. The winding criterion at an arbitrary point u is now obtained

by checking the sign of the radial component of the tangent vector t(u), which is

given by the sign of the dot product of t(u) and the radius vector u. The spring is

139

0.4 0.5 0.6 0.7 0.8 0.9 1.0G

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Nc

Figure 6.8: The critical force Nc as a function of Γ. (κ = 34, τ = 1

2)

(a) All three surfaces (b) Helix hyperboloid and intersections withthe other two surfaces

Figure 6.9: The helix hyperboloid H(u) = 0 (yellow), the zero-axial-moment ellipsoidE(u) = 0 (red), and the sphere |u| = θ (green) centered at the origin through thereference point (purple).

winding as the force is increased from the point u if

uT t(u) > 0 , (6.25)

140

or, more explicitly,

σ uT (∇E(u)×∇H(u)) > 0 , (6.26)

where ∇E(u) and ∇H(u) are given by (6.24).

Overwinding criterion. This is just the special case of the winding criterion (6.25)

for u = u, in which case the explicit expressions are particularly simple (cf. (6.19)).

The overwinding criterion is thus

σ uT (Ku×∇H(u)) > 0 . (6.27)

where ∇H(u) is given by (6.19b).

Critical point. The critical point can be found from the condition that the tangent

vector has no radial component, i.e.

uT (∇E(u)×∇H(u)) = 0 . (6.28)

However, rather than solving the system (6.28), E(u) = 0 (6.5), H(u) = 0 (1.107), in

concrete numerical cases it may be more efficient to maximize the distance from the

origin |u| over the trajectory (i.e. subject to the constraints E(u) = 0 and H(u) = 0).

6.4.1 Anisotropic diagonal case

In this case, the stiffness matrix contains three elastic parameters K = diag(K11,K22,K33),

one of which can be scaled out:

K11 7→ 1 , K22 7→ β :=K22

K11

≥ 1 , K33 7→ Γ :=K33

K11

, (6.29)

where β is the asymmetry coefficient, and Γ is the twist-to-bend stiffness ratio (1.69).

The diagonal form of the stiffness matrix has for a consequence that the axes of

the ellipsoid E(u) = 0 (6.5) are parallel to the coordinate axes. The ellipsoid can be

141

reduced to canonical form:(u1 − u1

2

)2a2

+

(u2 − u2

2

)2a2/β

+

(u3 − u3

2

)2a2/Γ

= 1 , (6.30)

where a := 12

√u1

2 + βu22 + Γu3

2 is the equatorial radius along the u1-axis. As β ≥ 1

by assumption, the other equatorial radius (along the u2-axis) is lesser than a.

In the diagonal case, the helix hyperboloid H(u) = 0 (1.107) degenerates into a

hyperbolic cylinder :

u2(u1 − u1)− βu1(u2 − u2) = 0 , (6.31)

with translational symmetry in the u3 direction. Note that it is independent of the

twist-to-bend stiffness ratio Γ, and is specified by the asymmetry coefficient β alone.

In the (u1, u2) plane, (6.31) is a hyperbola that can be put in the following form

when β > 1 (u1 +

u1β − 1

)(u2 −

βu2β − 1

)=

βu1u2(β − 1)2

, (6.32)

which makes it apparent that its asymptotes are parallel to the coordinate axes,

with the center located at(− u1β−1 ,

βu2β−1

). The vertical asymptote lies in the opposite

(vertical) semi-plane from the reference point, while the horizontal asymptote is in the

same (horizontal) semi-plane and above the reference point (u1, u2). The hyperbola

connects the origin with the reference point and its graph is concave on (0, u1). For

β = 1, the hyperbola degenerates to a line, and this is the already seen isotropic case.

In Figure 6.10, the hyperbola is superimposed with the (u1, u2)-projection of the level

sets on the ellipsoid that are equidistant from the origin.

Overwinding criterion. The criterion (6.27) in the diagonal case reduces to:

Γ <u1

2 + u22

u12 + 1

βu2

2 . (6.33)

The right-hand side of the inequality only depends on the reference point through the

register angle ϕ = arctan u1u2

:

Γ <β

cos2 ϕ+ β sin2 ϕ. (6.34)

142

0.1 0.2 0.3 0.4 0.5 0.6 0.7u1

0.2

0.4

0.6

0.8

u2

Figure 6.10: The projection of the hyperbolic cylinder (6.31) (in blue) and the levelsets on the ellipsoid (6.5) that are equidistant from the origin onto the (u1, u2)-plane.The reference level set |u| = θ and the critical level set |u| = θc are shown in red.

For a given asymmetry coefficient β > 1, the critical value of Γ is also greater than

unity. Overwinding occurs for subcritical values of Γ. The critical curve in the (β,Γ)-

plane is shown in Figure 6.11.

Critical point. In the diagonal case, the winding criterion (6.25) is never satisfied

at the limit point u∞:

u∞T t(u∞) = −(u12 + βu2

2) |u3| < 0 . (6.35)

The spring is always unwinding for large enough force N , i.e. as the trajectory ap-

proaches the u3-axis at u3∞ (cf. (6.7)). Therefore, as long as the spring is overwinding,

the coiling angle will reach a maximum between u and u∞. If the spring is not over-

winding, the coiling angle decreases monotonously from u to u∞.

143

j`

=0

j`

=Π

4

j`

=Π

2

1.2 1.4 1.6 1.8 2.0Β

0.5

1.0

1.5

2.0

G

Figure 6.11: The curves in the parameter plane (β,Γ) that delimit the overwindingregion (below the curves) in the diagonal anisotropic case, for different values of thereference register angle ϕ.

6.5 Conclusion

We considered a helical spring whose end is free to rotate about the helix axis, and

examined the direction of this rotation as an axial force is applied. In the isotropic

case, we recovered the results of Miller [21] with two alternative approaches, a geo-

metric an an analytic one. The results state that, if the torsional stiffness is lesser

than the bending stiffness, (i.e. for Γ < 1) and for a tensile force, the spring will over-

wind up to a critical force, then unwind. Otherwise (Γ ≥ 1), the spring will unwind

monotonously as the force is increased. In the anisotropic diagonal case, the behavior

is qualitatively the same, but the critical value of Γ depends on the asymmetry coeffi-

cient β and the reference register angle ϕ (see Figure 6.11). In the general anisotropic

case, no general statements can be made without specifying the stiffness matrix K,

but we have derived an overwinding criterion in the form of inequality (6.27).

144

Chapter 7

Conclusion

Two classes of problems have been presented: one dealing with planar waves on

intrinsically straight rods, and the other with helical springs deformed by a wrench

applied along the helix axis.

7.1 Planar waves

We have considered a planar localized wave on a rod. As it was shown that traveling

wave equations are equivalent to static equations, static solutions for the elastica

can travel along the rod unaltered. Thus, a localized wave on a rod has the shape of

Euler’s Species 7 elastica, cf. Figure 1b. It is the homoclinic solution that corresponds

in the pendulum analogy to a pendulum released from the unstable equilibrium and

making one full revolution before reaching the unstable equilibrium again in infinite

time. We examined the following two effects on this solution: (a) the addition of a

quartic term to a quadratic strain-energy density, and (b) the effect of heterogeneity

in the rod material.

7.1.1 Quartic strain-energy denstiy

In a classical homoclinic solution, the fixed point is reached in infinite time. Therefore,

there is no possibility of combining the homoclinic solution with any other solution.

By contrast, if a fixed point can be reached in a finite time, different solutions can

be combined on a bounded time interval. A finite-time homoclinic orbit combines

features of an infinite-time homoclinic orbit and a periodic orbit: like an infinite-time

homoclinic orbit, it has a fixed point at both ends, and like a periodic orbit, it only

takes a finite time for a complete cycle. Having reached the fixed point at either end,

145

based on the non-uniqueness property, the solution can go for another round along

the homoclinic orbit (a periodic solution), or stay at the fixed point indefinitely (a

compact support solution).

We have shown in Chapter 2 that a quartic strain-energy density leads to a finite-

time homoclinic solution. This is always true for intrinsically straight rods, while

intrinsically curved rods do not exhibit this type of behavior.

The obtained results hold for homogenous strain-energy densities of degree 4, but

they are also applicable to non-homogenous ones with leading power 4, since the finite-

time criterion (2.25) only depends on the lowest power appearing in the strain-energy

density.

Although all results have been reached in a static framework, Proposition 2.1

provides a broader context: a static solution for a system with strain-energy densityW

corresponds to a wave traveling at the speed of sound (which is the speed at which the

quadratic term of the effective strain-energy density is canceled, cf. (2.16) and (2.17))

on a rod whose strain-energy density combines W with a quadratic term (cf. (2.3c)).

Thus, each static solution with compact support also represents a compact wave

traveling at the speed of sound. Similarly, compact waves in nonlinear Klein–Gordon

equations are possible only if the waves move at the speed of sound [55, 56].

The significance of this result lies in the fact that no artificial construction was

required to obtain a solution in the form of a localized wave with a compact support,

merely a strain-energy density with a quartic term, which corresponds to real-world

systems better than a linear constitutive relation. The limitation in relating this

sonic compact wave to real elastic rods lies in its planar nature: we have neglected

the issue of self-intersection. It would therefore be of interest to extend this result to

three dimensions.

146

7.1.2 Heterogeneous rod

Next, in Chapter 3, we have considered, within the linear constitutive model, the

effect of a heterogeneity consisting of a periodic microstructure of two different ma-

terial properties on the localized loop waves. The rod was assumed inextensible and

unshearable, and we used a multiple scales homogenization technique.

The leading order balance (3.30) is a system describing a homogeneous rod, and

allowed us to identify the effective material properties. In the case of the density ρ,

it is just the bulk density ρh = 〈ρ〉, which is an intuitively clear result, but as far as

the Young modulus is concerned, the effective value is Eh = 〈E−1〉−1.

Two lowest-order corrections have been found and shown to distort the homogeneous-

type loop, breaking its symmetry. The effect of the corrections is shown in Figure 3.9.

From the effect of the second-order boundary conditions, we can infer that an initially

travelling loop on a rod with periodic micro-structure would eventually deform and

loose its travelling wave structure as expected due to dispersion effects.

Although longitudinal waves in bulk materials with an analogous heterogeneous

microstructure have been studied, this is the first result for transverse waves on an

elastic rod. A further study of dispersion effects would provide a better insight into

the effect of heterogeneity on loop waves.

7.2 Helical springs

The second class of problems considered in this dissertation concerns a system con-

sisting of a helical spring with one end fixed and a wrench (a force and a torque)

applied along the helix axis at the other end (see Figure 4.1). This is a simple sys-

tem that—in the isotropic case—was closely studied in the 19th century, and the

treatment of which can be found in such classical works as Lord Kelvin’s and Tait’s

Treatise on Natural Philosophy [24] and Love’s Treatise on the Mathematical Theory

of Elasticity [1]. We turned to this system in full generality, assuming nothing more

147

than a linear constitutive relation within the Special Cosserat rod model and the

Kirchhof rod model.

7.2.1 Direct and Inverse Helical Spring Problem

First we considered what is here referred to as the Direct Helical Spring Problem,

which is about relating the stresses to the strains for the helical spring system, where

the stresses are given by the axial force and torque (two scalar variables), while the

strains are abstracted to any geometric parameters describing the configuration of

the spring that satisfy certain compatibility conditions, e.g. the radius and the pitch

of the helix. We refer to such generalized geometric parameters as observables. An

arbitrary number of observables is considered, but a minimum of two is required,

and as we assume that the observables are independent, the maximum number of

observables is equal to the number of strains in the rod model used, i.e. three for the

inextensible and unshearable, and six for the extensible and shearable. As the sought

relations between stresses and strains cannot be made explicit in the most general

case, the results are presented in the form of problems of the kind “given the strains,

find the stresses,” or vice versa.

Next, we considered the inverse problem, which consists in finding the elastic

parameters of the spring given measurements of both stresses and strains. This is an

important problem, as the estimation of elastic parameters for some systems (most

notably on the (macro)molecular scale) is sometimes only possible by taking into

account the helix geometry.

In the ideal case when all components of the strains are measured (i.e. all three

components of the twist vector u, and, in the extensible and shearable case, also all

three components of the stretch vector v), the equations are linear in the unknown

parameters—even in the case where the reference configuration u is considered an

unknown parameter—and the optimal parameters are found by solving a simple least-

148

squares problem, with an explicit solution.

In laboratory conditions, however, it is not always possible to obtain a complete

information on the strains from measurements. For example, shear may not be ac-

cessible to measurement, but even working within the inextensible and unshearable

model, the register angle ϕ may not be measurable. With this in mind, we have

tackled the problem of finding the elastic parameters given the measurements of the

axial wrench (i.e. the force N and the torque M) and some generalized geometric

parameters describing the geometry of the spring (observables, introduced in the di-

rect problem). We outlined a general procedure of constructing an objective function

both for the inextensible and unshearable case and the general one.

We carried out numerical experiments on simulated data, using an implementation

of the Levenberg–Marquardt algorithm for minimizing the objective function. The

simulated data consisted of the wrench and the observables, where the observables

were the curvature κ and the torsion τ in the inextensible and unshearable case, while

in the extensible and shearable case a third observable was added—the stretch ν. The

experiments showed that even though the procedure is conceptually simple, in some

cases the computational difficulties are such that solutions could not be obtained

on the machines that were on our disposal. In particular, this is the case when all

three components of the reference twist vector are treated as unknown parameters,

and in the fully general extensible and shearable case (i.e. full 6 × 6 matrix K; 20

unknown parameters). However, our algorithm produced good results in the general

inextensible and unshearable case (full 3× 3 matrix K; 5 unknown parameters) with

known reference configuration u, assuming the level of noise in the data was relatively

low (of order 10−7 or lower). Good results were also obtained in the diagonal extensible

and shearable case (diagonal 6× 6 matrix K; 6 unknown parameters) with known u.

The inextensible and unshearable model is far more relevant than the extensible

and shearable, as extension along the centerline can be neglected as compared with

twisting and bending. For practical applications, the requirement that the reference

149

configuration must be known may be a serious limitation. As results for the problem

with unknown u appeared uncomputable, we considered the case where u is partially

known, assuming that the reference curvature and torsion are known, but the reference

register angle is not. Among various possibilities for partial information about u this

is a realistic scenario in practice. The results obtained were not always consistent,

and far less accurate than in the case when the register angle is known. Improving the

algorithm in this regard would, in view of the potential applications, be a valuable

extension of the present work.

7.2.2 Overwinding helical springs

Finally, we looked at a special case of the helical spring system (Figure 4.1) with zero

axial torque, i.e. where the end is free to rotate about the helix axis, and considered

a simple question: when the spring is pulled on (i.e. for positive axial force), in which

direction will it turn, winding or unwinding? We treated the rod as inextensible and

unshearable, and the first integrals viewed as surfaces in the strain space of twist

vectors u enabled us to formulate the problem in terms of simple geometry.

In the isotropic case, the spring will wind if its twist-to-bend stiffness ratio Γ

is below the critical value of unity, Γ < 1. Moreover, the asymptotic behavior for

a large applied force is unwinding. If the value of the twist-to-bend stiffness ratio

is subcritical, Γ < 1, the spring will wind as the force is increased from rest until

a maximum value of the coiling angle, and then unwind as the force is increased

to arbitrarily large values. For a supercritical value Γ > 1, the spring will unwind

monotonously as the force is increased.

In the anisotropic diagonal case, we have obtained a winding criterion in terms

of the two elastic constants, the twist-to-bend stiffness ratio Γ and the asymmetry

coefficient β. The effect of the anisotropy is to increase the critical value of Γ for

reference register angles ϕ that are less than π/2, while the asymmetry of the rod has

150

no effect for ϕ = π/2. Qualitatively, the behavior is the same as in the isotropic case.

In the general case of a stiffness matrix with coupling terms there are few general

statements that can be made about the behavior of the spring as the force is increased.

In addition to the winding-then-unwinding and monotonous unwinding behaviors seen

in the diagonal cases coupling terms may lead to monotonous winding. However,

criteria for winding at rest, winding at an arbitrary applied force, and the critical

point of reverting from winding to unwinding have been presented, which yield definite

answers given concrete values of the stiffness matrix elements.

The work presented in Chapter 6 generalizes the results of Miller [21], and shows

that it is rather straightforward to derive using geometric arguments in twist-space the

winding-then-unwinding behavior of a helical spring—labeled as “counterintuitive” in

some recent papers on dna experiments—from the Kirchhoff–Clebsch–Love theory of

rods as presented in Love’s frequently quoted Treatise [1].

151

Appendix A

Index

asymmetry coefficient, 38, 140

axial wrench, 43

bending stiffness, 39

best fit, 118

binormal vector, 22

centerline, 21

coiling angle, 125

coiling direction, 124

compact wave, 54

compliance matrix, 30

Kirchhoff rod, 37

configuration of a rod, 25

constitutive relation, 29

Kirchhoff rod, 36

cross section plane, 23

cumulative angle, 109

curvature, 22, 35

Darboux vector, 23

data model, 106

director basis, 21

directors, 21

elastic model, 102

elastic rod, see rod

elastica, 13

energy integral, 38

equivalent static system, 50

force equation, 27

force vector, 28

Frenet curvature, 22

Frenet frame, 22

Frenet–Serret basis, see Frenet frame

Frenet–Serret equations, 22

helical rod, 40

helical spring, 18

helix, 40

helix angle, 41

Kirchhoff rod, 44

helix hyperboloid, 48

hyperelastic rod, 29

inextensible rod, 34

isotropic Kirchhoff rod, 39, 127

kinematic equations, 24

Kirchhoff equations, 37

Kirchhoff rod, 34

152

local coordinates, 21

material cross section, 23

minimal data model, 109

moment equation, 28

moment of force vector, see moment

vector

moment vector, 28

observable, 98, 105

overwinding spring, 124

pitch angle, 41

principal moments of area, 28

principal normal vector, 22

pure flexure, 36

reference configuration, 21

reference stretch vector, 25

reference twist vector, 25

register angle, 35

rod, 21

shear, 23

shear amplitude angle, 23

shear orientation angle, 23

shearless configuration, 23

signed curvature, 52

solitary wave, 53

spin equations, 26

spin vector, 26

stiffness matrix, 30

Kirchhoff rod, 37

strain vector, 29

strain-energy density, 29

Kirchhoff rod, 36

strains, 29

stress vector, 29

stretch, 25

stretch vector, 24

strip, 35

tangent vector, 22

tension, 27

top analogy, 38

torsion, 22, 35

torsional stiffness, 39

trajectory, 126

twist equations, 24

twist vector, 24

twist-to-bend stiffness ratio, 39, 127, 140

twisting stiffness, see torsional stiffness

uncoiling direction, 124

uniform configuration, 36

uniform helical rod, 40

Kirchhoff, 44

unshearable rod, 34

unwinding spring, 124

winding spring, 124

153

wrench, 43, 94

zero-axial-moment ellipsoid, 126

154

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