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Abstract

Nonlinear phenomena are common in structural and solid mechanics; in general,

the corresponding system of equations cannot be solved analytically, and numerical

techniques are necessary. Furthermore, bifurcations of the solution are frequent;

they can appear either at the physical level (the system may have multiple equi-

librium configurations), or at the numerical level (related to the algorithm utilized

in calculating the solution). Unfortunately no algorithm can solve every nonlinear

system, and most of the time the recovery of solutions needs alternative iterative

techniques.

This thesis is concerned with finite element formulations and solution techniques

for structures undergoing large deformations. The two applications examined are

the steady state frictional rolling of tires and the postbuckling analysis of slender

structures.

A formulation for steady state rolling calculations is introduced, focusing on

the inclusion of frictional sliding conditions between a rolling tire and a flat road-

way. Algorithmically, it is seen that traditional return mapping strategies are often

ineffective for this problem even when frictional solutions exist; accordingly, an ap-

proach utilizing a global stick predictor is proposed to recover solutions to the sliding

contact problem. Numerical examples are presented, to demonstrate the effective-

ness of the approach advocated. Difficulties associated with enforcing frictional

conditions within such a framework are discussed. The interaction of frictional con-

iv

ditions with bifurcation phenomena is also studied in the case of adherent contact

conditions. Such phenomena are observed in the context of multiple solutions of

the discretized system, and are also manifested in the behavior of the iterative map

used to solve the nonlinear algebraic system of equations.

Another example of bifurcation is the buckling of slender structures with direct

application to solar sail booms. An interesting aspect in the boom design is that

postbuckled configurations are not avoided as is usually the case in structural de-

sign; instead, they are sometimes encouraged. In this context, the understanding

of the structural behavior after buckling is essential. Various structural systems

and loadings appropriate for the boom modeling are examined here. Natural fre-

quencies of vibration about highly–deflected equilibria are extracted, exposing the

high sensitivity that these structures have to minor changes in the geometry and

loading.

v

Dedicated to the memory of my mother

Ileana Carmina Slatineanu (1939–1999)

vi

Contents

Abstract iv

List of Figures xi

List of Tables xvii

1 Introduction 1

1.1 Frictional formulation and bifurcations in steady state rolling . . . . 3

1.2 Buckling and large deformation analysis of slender structures . . . . 7

1.3 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Nonlinear Problems and Bifurcations 17

2.1 Sources of nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Finite element formulations and solution techniques for nonlinearproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Bifurcations and associated numerical methodology . . . . . . . . . 22

2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.2 Continuation methods for finite element analysis beyond bi-furcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.3 Critical points. Methods of identification and characterization 29

2.3.4 Eigenvalue analysis . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.5 Nonlinear iterative maps. Convergence analysis of the Newton-Raphson algorithm . . . . . . . . . . . . . . . . . . . . . . . 34

vii

2.4 Applications considered in this thesis . . . . . . . . . . . . . . . . . 44

2.4.1 Static and dynamic buckling of slender structures . . . . . . 44

2.4.2 Standing waves and multiple solutions for rotating cylinders 45

2.4.3 Bifurcations of the Newton–Raphson nonlinear iterative map 46

3 Steady–State Frictional Rolling 51

3.1 Definition of the problem . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Finite element formulations . . . . . . . . . . . . . . . . . . . . . . 57

3.3.1 Constitutive law . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.2 Pressure and Dirichlet boundary conditions . . . . . . . . . 61

3.4 Finite element formulation for frictional contact . . . . . . . . . . . 65

3.4.1 Relative velocity measure for frictional sliding . . . . . . . . 68

3.4.2 Algorithmic treatment of the frictional conditions . . . . . . 70

3.4.3 Residual force vector and stiffness matrix . . . . . . . . . . . 72

3.4.4 Existence and uniqueness of solution for contact problems . 74

3.5 Alternative Iterative techniques . . . . . . . . . . . . . . . . . . . . 75

3.5.1 Augmented Lagrangians . . . . . . . . . . . . . . . . . . . . 75

3.5.2 Global stick predictor . . . . . . . . . . . . . . . . . . . . . . 78

3.6 Numerical examples. Verification . . . . . . . . . . . . . . . . . . . 79

3.6.1 Verification of the Mooney–Rivlin hyperelastic element . . . 80

3.6.2 Verification of the pressure loading formulation . . . . . . . 82

3.6.3 Critical points for rotating cylinders . . . . . . . . . . . . . . 84

3.7 Numerical examples. Frictional sliding calculations and algorithmicperformance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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3.7.1 Performance of the iterative technique . . . . . . . . . . . . 86

3.7.2 Comparison with other algorithms . . . . . . . . . . . . . . . 91

3.8 Numerical examples. Typical results on benchmark problems . . . . 93

3.9 Numerical examples. Bifurcations of the nonlinear iterative map . . 98

3.9.1 Example problems . . . . . . . . . . . . . . . . . . . . . . . 99

3.9.2 Eigenvalue analysis results . . . . . . . . . . . . . . . . . . . 104

3.9.3 Bifurcation of the iterative map. k–cycles . . . . . . . . . . 106

3.9.4 Bifurcation of the solution of the discretized problem . . . . 111

3.9.5 Mesh refinement study . . . . . . . . . . . . . . . . . . . . . 115

3.9.6 Influence of the ground velocity . . . . . . . . . . . . . . . . 116

3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4 Postbuckling Analysis of Slender Booms 120

4.1 Solar sailing. Structural configurations . . . . . . . . . . . . . . . . 120

4.2 Bifurcations and the concept of postbuckled configurations . . . . . 123

4.2.1 Buckling load for a cantilever beam loaded directly . . . . . 124

4.2.2 Buckling load for a cantilever beam under follower load . . . 125

4.2.3 Buckling load for a beam–cable system . . . . . . . . . . . . 125

4.3 Dynamic analysis of structures in postbuckled state . . . . . . . . . 130

4.3.1 Undamped free vibration analysis for beams in flexure . . . 130

4.3.2 Undamped free vibration analysis for beams in axial defor-mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.3.3 Vibration frequencies for the cantilever beam . . . . . . . . . 133

4.3.4 Free vibrations about highly deflected equilibria. Frequenciesfor the beam–cable system . . . . . . . . . . . . . . . . . . . 135

ix

4.4 Buckling under nonconservative forces . . . . . . . . . . . . . . . . 138

4.4.1 Differential equation of the beam . . . . . . . . . . . . . . . 139

4.4.2 Buckling load . . . . . . . . . . . . . . . . . . . . . . . . . . 140

4.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.5.1 Static and dynamic buckling of slender structures . . . . . . 141

4.5.2 Postbuckling dynamic characteristics . . . . . . . . . . . . . 150

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5 Conclusions and Future Work 163

5.1 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.1.1 Algorithmic Stabilization of Frictional Steady State RollingCalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.1.2 Comprehensive analysis of the interaction between bifurca-tions and the frictional finite element formulation for steadystate rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.1.3 Analysis techniques for slender structures in postbuckled orother large deformation configurations . . . . . . . . . . . . 166

5.2 Related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.3 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Bibliography 170

Biography 179

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List of Figures

1.1 Tire and finite element model of tire in contact with a flat surface. . 3

1.2 Small–scale test sail. Picture courtesy of D. Holland, Duke Univer-sity/NASA Langley. . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Continuation using natural parameterization. . . . . . . . . . . . . . 25

2.2 Continuation using natural parameterization and tangent predictor. 25

2.3 Typical “problematic” static loading path. . . . . . . . . . . . . . . 27

2.4 Schematic of the Modified Riks Algorithm. . . . . . . . . . . . . . . 28

2.5 Stability cases - schematic representation for an SDOF system. . . . 30

2.6 Basins of attraction of the cubic roots of unity. Shading convention:green (light grey)–basin of z1; red (medium grey)–basin of z2; black–basin of z3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.7 Periodic solutions of the cubic equation. . . . . . . . . . . . . . . . 48

2.8 Periodic solutions (•) of the cubic equation in relation to the basinsof attraction of the roots (¤). . . . . . . . . . . . . . . . . . . . . . 50

3.1 Notation for the steady state rolling contact problem. . . . . . . . . 53

3.2 Configurations for the pressure formulation. . . . . . . . . . . . . . 63

3.3 Node numbering convention for approximation of contact velocities. 69

3.4 Patch test for the Mooney–Rivlin element. . . . . . . . . . . . . . . 80

3.5 Verification of the centrifugation term included due to the ALE ref-erence frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.6 Convergence Rate: Energy norm versus Newton-Raphson iterationnumber for the centrifugation problem. . . . . . . . . . . . . . . . . 81

xi

3.7 Patch test for pressure loading. . . . . . . . . . . . . . . . . . . . . 83

3.8 Semilogarithmic plot of the relative energy norm, pressure loading. . 83

3.9 Different meshes used for the pure spinning problem. . . . . . . . . 84

3.10 Geometry of the test problem. . . . . . . . . . . . . . . . . . . . . . 87

3.11 Contour plot of σxy, adherent contact case (daN/mm2). . . . . . . . 87

3.12 Contour plot of σyy, adherent contact case (daN/mm2). . . . . . . . 87

3.13 Typical stalling of Newton-Raphson convergence for slip contact whenusing the local return map strategy for the friction, as measured byevolution of the energy norm. . . . . . . . . . . . . . . . . . . . . . 88

3.14 Contour plot: σxy, sliding contact case with µ = 0.3 (daN/mm2). . . 89

3.15 Contour plot: σyy, sliding contact case with µ = 0.3 (daN/mm2). . . 89

3.16 Tangential traction along the middle parallel on the contact patch. . 90

3.17 Tangential traction along the outer parallel on the contact patch. . 90

3.18 Contour plot of σyy daN/mm2 with our algorithm. . . . . . . . . . . 93

3.19 Contour plot of σyy daN/mm2 with Hu-Wriggers algorithm. . . . . . 93

3.20 Newton–Raphson convergence behavior for slip calculations with Hu-Wriggers algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.21 T322; d = 20 mm; Semilogarithmic plot for the convergence test;adherent contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.22 T322; d = 7 mm; Semilogarithmic plot for the convergence test;sliding contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.23 T322; d = 20mm; Contour plot, 3-3 stress component. . . . . . . . 96

3.24 T322; d = 7mm; Contour plot, 3-3 displacement component. . . . . 96

3.25 Meshing for Michelin problem T310. . . . . . . . . . . . . . . . . . 97

3.26 T310, 3-3 component of the stress, plot on the deformed configuration. 98

3.27 Simple disk; model 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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3.31 Idealized truck tire test problem. Discretizations with a) 16 and b)28 meridians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.32 Contour plot of the 2-2 stress [daN/mm2], simple disk, model 4. . . 102

3.33 Contour plot of the 1-1 stress [daN/mm2], simple disk, model 4. . . 102

3.34 Total normal reaction on contact patch; simple disk model 4. . . . . 103

3.35 Total tangential reaction on contact patch; simple disk model 4. . . 103

3.36 Horizontal tractions corresponding to multiple equilibria for idealizedtruck tire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.37 Vertical tractions corresponding to multiple equilibria for idealizedtruck tire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.38 Typical spectral signature for models M16 and M28 at an intermedi-ate road displacement before the critical point is reached. . . . . . . 105

3.39 Spectral signature for model M28 at a road displacement above thevalue of the first Hopf point. Top figure: right half of complex plane(i.e., eigenvalues with positive real part); bottom figure, zoom oneigenvalues with negative real part. . . . . . . . . . . . . . . . . . . 107

3.40 Energy norm levels for a stable 2-cycle obtained during Newton -Raphson iterations at d = 19.2 mm, for the simple disk problemwith model 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.41 Contour plot of tangential tractions (daN/mm2) for the first 2-periodicpoint corresponding to Newton–Raphson iterations at d = 19.2 mm;simple disk problem, model 4. . . . . . . . . . . . . . . . . . . . . . 109

3.42 Contour plot of tangential tractions (daN/mm2) for the second 2-periodic point corresponding to Newton–Raphson iterations at d =19.2 mm; simple disk problem, model 4. . . . . . . . . . . . . . . . 109

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3.43 Nonconverged configurations along the loading path obtained fromthe Newton–Raphson iterative map and the energies associated withthem: simple disk problem, model 4. . . . . . . . . . . . . . . . . . 110

3.44 Total normal reaction on the contact patch, simple disk problem,model 4. For steps where convergence was not obtained, no datapoint is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.45 Total tangential reaction on the contact patch, simple disk problem,model 4. For steps where convergence was not obtained, no datapoint is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.46 Contact patch for solution 1, simple disk problem, model 4. . . . . . 114

3.47 Contact patch for solution 2, simple disk problem, model 4. . . . . . 114

3.48 Contour plot of the tangential tractions for solution 1, simple disk,model 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.49 Contour plot of the tangential tractions for solution 2, simple disk,model 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.50 Total normal reactions obtained during simulations of 4 simple diskmodel problems. At values of d where a solution is not shown for aloading sequence, convergence was not obtained. . . . . . . . . . . . 115

3.51 Total tangential reactions obtained during simulations of 4 simpledisk model problems. At values of d where a solution is not shownfor a loading sequence, convergence was not obtained. . . . . . . . . 116

3.52 Influence of the ground velocity on the algorithmic behavior. . . . . 117

4.1 Structural configurations for solar sails (images created by BenjaminDiedrich, courtesy www.solarsails.info) . . . . . . . . . . . . . . . . 120

4.2 Sail attachment solutions. . . . . . . . . . . . . . . . . . . . . . . . 121

4.3 Isogrid configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.4 Structural systems for solar sail booms. . . . . . . . . . . . . . . . . 124

4.5 Solar Sail Configuration. . . . . . . . . . . . . . . . . . . . . . . . . 126

4.6 Beck’s problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

xiv

4.7 Load deflection diagram obtained with Riks’ method on baseline iso-grid model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

4.8 Deformed configuration at P = 5 kN. . . . . . . . . . . . . . . . . . 144

4.9 Dynamic analysis of Beck’s problem using ABAQUS. . . . . . . . . 144

4.10 Evolution of the deformation during a dynamic analysis with ABAQUS.145

4.11 Static analysis of Beck’s problem using ABAQUS; numerical dampingincluded as an attempt to control algorithmic instabilities. . . . . . 145

4.12 Dynamic analysis of Beck’s problem using ABAQUS; Numerical damp-ing included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.13 Equilibrium shapes for b/L =0.0167 and P/Pcr=0.00, 0.81, 1.01, 1.08,1.30, 1.66, and 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.14 Tension vs. horizontal tip deflection for two offsets. . . . . . . . . . 149

4.15 Tension vs. horizontal tip deflection for four offsets. . . . . . . . . 149

4.16 First bending mode of a slender isogrid, ω = 0.64 rad/s. . . . . . . 151

4.17 Third bending mode of a slender isogrid, ω = 11.23 rad/s. . . . . . 151

4.18 Variation of the square of the natural frequencies with the axial loading.151

4.19 Variation of the square of the natural frequencies with the slendernessratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.20 Buckling mode of a short isogrid. . . . . . . . . . . . . . . . . . . . 153

4.21 Vibration mode of a short isogrid. . . . . . . . . . . . . . . . . . . . 153

4.22 Variation of the square of the natural frequencies with the distancebetween supports. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.23 Comparisons on various bending mode frequencies; analysis of isogridand equivalent beam. . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.24 H3 FRF Sample experimental data P/Pcr = 1.12, b/L = 0.0167.Picture courtesy of D. Holland, Duke University . . . . . . . . . . . 156

4.25 Overlaid FRF’s for modal analysis (P/Pcr = 0.505, b/L = 0.0750).Picture courtesy of D. Holland, Duke University . . . . . . . . . . . 157

xv

4.26 Lowest four frequencies for b/L = 0.0167. . . . . . . . . . . . . . . . 158

4.27 Fundamental frequency for b/L = 0.0167. . . . . . . . . . . . . . . . 158

4.28 Lowest four frequencies for b/L = 0.0750. . . . . . . . . . . . . . . . 159

4.29 Fundamental frequency for b/L = 0.0750. . . . . . . . . . . . . . . . 159

4.30 First four vibration modes for b/L=0.0750 and P/Pcr = 0.505. . . . 160

4.31 Fourth vibration mode for b/L=0.0750 and P/Pcr = 0.505 for (a)FEA, (b) shooting, and (c) experiments. . . . . . . . . . . . . . . . 160

4.32 Frequencies for b/L = 0.0167 from 3–D finite element analysis. . . 161

xvi

List of Tables

2.1 Modified Riks Algorithm. . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 k–periodic solutions of the iterates for the cubic equation; k ≤ 4 . . 49

3.1 Augmented Lagrangian algorithm for frictional contact. . . . . . . . 77

3.2 Stick predictor algorithm for frictional contact. . . . . . . . . . . . . 78

3.3 Convergence sequence for the patch test . . . . . . . . . . . . . . . 81

3.4 Convergence sequence for the centrifugation problem . . . . . . . . 82

3.5 Convergence sequence for the pressure loading. . . . . . . . . . . . . 82

3.6 Iteration counts for the test problem in different frictional slip cases(iterations for the convergence of the stick predictor are not included);road surface displacement = 35 mm. . . . . . . . . . . . . . . . . . 89

3.7 Convergence results for T322; perfectly adherent contact . . . . . . 95

3.8 Stick predictor and slip step convergence sequence for problem T322. 96

3.9 Perfectly adherent contact convergence sequence for T310. . . . . . 98

3.10 Slip contact convergence sequence for T310. . . . . . . . . . . . . . 99

xvii

Acknowledgements

As I close this chapter of my academic life, I would like to thank those who, in one

way or another, made this day possible.

First and foremost, I thank my supervisor, Professor Tod Laursen, for his en-

couragement when research progressed well and his patience and help when it did

not. He has greatly influenced my professional development in many ways; I will

be forever grateful to him for promoting independent thinking and encouraging

originality.

I thank my co-supervisor, Professor Lawrence Virgin, for giving me the oppor-

tunity to collaborate closely with him and his research group during the last two

years. I am indebted to Professor Thomas Witelski for the countless hours we spent

discussing my research and for his valuable suggestions. I thank Professors John

Dolbow and Henri Gavin for their “open doors” and their willingness to help me

whenever I needed it. I am grateful to all of them for serving on my committee and

for their comments and insights that helped me improve the quality of this work.

I thank Yvonne Connelly for her help in editing the final draft of the dissertation.

I express my thanks to David Holland (Department of Mechanical Engineering

and Materials Science) for the fruitful collaboration we had this past year, and

to Professor Raymond Plaut, from Virginia Polytechnic Institute, for his help in

elaborating and submitting this collaborative work for publication.

During my first years in the PhD program, I was supported through a research

xviii

contract by Michelin America Research Corporation. This support, as well as col-

laboration of Mr. John Melson, Dr. Vasanti Gharpuray, Dr. Jean-Marc d’Harcourt,

and Dr. Ali Rezgui, are greatly appreciated. I also acknowledge the support re-

ceived from the In-Space Propulsion Technology Program, managed by NASA’s

Science Mission Directorate in Washington, D.C.

I wish to thank the staff working in the Vesic Library for their practical as-

sistance, particularly Ms. Linda Martinez. Remembering some of the very poor

references I gave her these past years, I sometimes suspect magic is involved in her

work; she never failed to help me.

I would also like to thank my friends and colleagues here at Duke for their pro-

fessional help and personal encouragements. Huidi Ji, Hashem Mourad, Bin Yang,

Natalia Hasler, Melek Kazezyilmaz, Gil Bohrer, Tae Yeon Kim, Anda Degeratu,

and Emma Buneci are only a few from a list that is too long to be included here

in its totality. They have all contributed to improving the quality of my thesis

and presentations, to broadening my research horizon, and to making my stay here

enjoyable.

Special thanks to John Mohan, not only for providing invaluable computer help

but also for being a supportive friend during these long years.

Others influenced me many years ago on the journey that I have taken to the

place where I now find myself. I wish to acknowledge here the contribution my

former professors made to my intellectual development.

I owe my interest in science, and the respect for the scholarly work, to my

mother. It is with regret that I cannot share this day with her, and with gratitude

for everything she taught me, that I dedicate this thesis to her memory.

xix

[...] for any computer algorithm there exist

nonlinear functions (infinitely continuously

differentiable, if you wish) perverse enough

to defeat the algorithm.

J.E. Dennis and R. Schnabel

xx

Chapter 1

Introduction

Nonlinear phenomena are widespread in many fields. In most cases, exact analyt-

ical solutions cannot be obtained; thus, numerical techniques must be developed.

Nonlinear problems in solid and structural mechanics, dynamics, fluid mechanics,

biomechanics, quantum mechanics, control theory, economics, and many other areas

of modern science are solved today by means of computational approaches.

In each of these fields, considerable effort has been put into creating mathemat-

ical models to accurately describe these phenomena and then into developing the

analysis and computational tools required to solve the associated nonlinear systems

of equations. Nonlinear problems are often subject to bifurcation phenomena (i.e.,

qualitative changes of the solutions when one or more parameters are varied), and

questions related to the existence and uniqueness of solutions also arise. Numeri-

cal techniques capable of handling bifurcation phenomena are therefore necessary

to identify the bifurcation points and characterize the solutions of the problem.

Moreover, the solution techniques must be efficient. As science evolves, specialists

are becoming better at understanding various phenomena and their mathematical

models are becoming increasingly complex by incorporating ever more parameters,

which results in increasingly large numerical models that require efficient algorithms.

Unfortunately, there are no generally applicable numerical techniques for nonlinear

1

CHAPTER 1. INTRODUCTION 2

problems that can answer all these issues, regardless of the problem to which they

are applied. The likelihood of success of a given numerical method when applied to

a specific nonlinear problem is strongly dependent upon the characteristics of that

problem.

The complexity of such problems explains the broad scope and the large extent

of the literature dedicated to topics in nonlinearity. Such sources range from books

introducing the general theory of numerical methods for nonlinear equations – Den-

nis and Schnabel (1983), Rheinboldt (1986), Ortega and Rheinboldt (1970) – to

books of “motivated mathematics” as Aubin (1998) characterized his textbook in-

spired by nonlinear problems in economics and game theory, to journal articles that

sometimes discuss a more general aspect but often are simply dedicated to a very

specific nonlinear problem, its mathematical model, and the appropriate numerical

techniques to solve it.

This thesis is concerned with nonlinear phenomena in solid and structural me-

chanics, particularly with finite element formulations for structures undergoing large

deformations, and with the study of related bifurcation phenomena. Since there

is no general method that can work regardless of the problem, the intent is to

rather concentrate on specific applications and develop the finite element formula-

tion and/or the analysis techniques that are efficient for that particular problem.

The two applications that are considered are the steady state frictional rolling of

tires and the postbuckling analysis of slender structures. The motivation for the

first lies in the wide use of steady state rolling models by the tire industry. The

purpose is to develop efficient algorithms for rolling tires in frictional sliding contact

with the ground. The motivation for the latter comes from applications in the space

industry, particularly large slender structures for solar sails.


1.1 Frictional formulation and bifurcations in steady

state rolling

Various computational aspects related to contact phenomena in general are dis-

cussed by Laursen (2002) and Wriggers (2002), with an overview of rolling contact

included in the latter. The classical problem involving steady state rolling contact

of a body against a flat surface is one of the most common contact problems, often

explored in engineering studies: Bordelon and Padula (1996), Ebbott et al. (1999),

Zheng (2003), Yavari et al. (1993), Wang et al. (1994), Kennedy and Padovan

(1987). In particular, analysis procedures in the tire industry often make use of

finite element formulations of this problem, considering the steady state rolling

of cylinders or disks in the context of large deformations. See, for instance, Faria

(1989), Le Tallec and Rahier (1994), Oden and Lin (1986), Padovan and Zeid (1980).

Figure 1.1: Tire and finite element model of tire in contact with a flat surface.

The left image in Figure 1.1 shows a standard automobile tire mounted and at


rest on the road surface while the image on the right depicts a typical finite element

model of a tire in contact with a flat surface.

Incorporation of frictional conditions into such models is particularly challenging

given the intricate dynamics of this seemingly straightforward structural system.

In such analyses, a common approach involves description of the kinematics of the

problem in what has been termed by some a particular application of the Arbitrary

Lagrangian Eulerian concept (Nackenhorst and Zastrau, 2001), where the frame

of reference is attached to the hub of the cylindrical wheel (assumed to move at

constant velocity under steady state conditions). Many aspects make this problem

more difficult than it may appear at first glance. At a minimum, tire rolling involves

nonlinear frictional contact, material and geometric nonlinearities, and pressure

loaded surfaces. Additionally, it is also known that bifurcation phenomena may

exist in many regimes of tire response, both for spinning tires and for tires in

contact with a rigid ground surface; see, for instance, the works by Oden and Lin

(1986), Oden and Rabier (1989), and Chatterjee et al. (1999).

One particularly troublesome aspect of recovering steady state frictional solu-

tions to the rolling contact problem resides in the robust algorithmic treatment of

the Coulomb conditions that may be assumed to govern sliding in such calculations.

The nonlinearity in this problem is due not only to the material properties but also

to the fact that the contact area and the distribution of the contact tractions are

not known beforehand. Frictional contact forces are nonconservative in the case

of sliding, which also introduces nonsymmetry into a consistently linearized algo-

rithm. In particular, it will be seen that the traditional “return map” strategies

for treatment of frictional contact, in which the global equilibrium equations are

iteratively solved while using a local trial state/return map update for the frictional


tractions, are not as effective in the solution of this problem as they are in other

frictional contact applications. Instead, we suggest the use of a global “adherent

stick predictor,” which is seen to produce a higher degree of robustness for recovery

of sliding solutions, not only within the numerical formulation proposed here but

also within frameworks proposed by others (Hu and Wriggers, 2002).

In general, inclusion of Coulomb friction laws in numerical analysis presents

significant numerical difficulties, the most relevant of which is that the existence

and uniqueness of solutions can only be proved under special hypotheses; see, for

instance, Han et al. (2001) and Chau et al. (2002), where uniqueness results for

the weak problem are established, as well as Han and Sofonea (2002), where, in

addition to such results, the continuous dependence of the solution on the data

and parameters is proved. Andersson and Klarbring (2001) prove a uniqueness

result for the discretized problem, a result that holds only for the case of small

deformations of a linear elastic body in frictional contact with a rigid obstacle. In

a two–dimensional unilateral contact problem, Doudoumis et al. (1994) establish a

sufficient criterion for uniqueness of the solution in the form of an upper limit for

the friction coefficient. More generally, however, such results cannot be obtained;

therefore, any finite element formulation including Coulombic friction may be open

to nonuniqueness. Examples of cases where solutions are not unique are widely

available in the literature. To mention only a few here, we cite the results of

Hassani et al. (2003), Hassani et al. (2004) and Ballard (1999).

Finite element formulations including frictional contact are subject to many nu-

merical challenges. In the adherent rolling contact problem however, the numerical

obstacles encountered appear to extend well beyond the norm for this difficult class

of problems. Nonlinear frictional contact, as well as material and geometric non-


linearities, are among the numerical challenges associated with this problem, which

make existence and uniqueness of solutions difficult if not impossible to prove for

the full range of parameters of practical interest. In fact, multiple solutions are ob-

tained in some of the examples to be examined in this work. The occurrence of limit

and bifurcation points is a common phenomenon in nonlinear structural mechanics

(Kouhia and Mikkola, 1998), and such points are known to exist in particular in

spinning cylinder problems even in the absence of contact conditions (Oden and

Lin, 1986). When such conditions are also included in the model, contact–induced

standing waves might appear.

Examples are available in the literature on bifurcations and standing wave phe-

nomena in the case of pure spinning cylinders and of spinning cylinders in contact

with a rigid surface. Chatterjee et al. (1999) present an experiment in which the

behavior of a rolling tire in contact with a rigid cylindrical surface is studied. The

experiment described in this work identifies a critical value of the rolling speed above

which a standing wave configuration is obtained. Oden and Lin (1986) present bi-

furcation analyses of the free spinning cylinder and of the rotating cylinder in both

frictionless and frictional contact with a flat rigid surface. Their work shows that

in a free spinning analysis, for small values of the angular velocity ω, the solution

is unique and radially symmetric. For ω above a critical value, however, branch-

ing of the solution is encountered and manifests itself in the emergence of standing

waves. The number of the peaks in the standing wave solution is exactly half of the

number of elements used for discretization in the circumferential direction, which

suggests that a discrete model is only able to provide a finite number of frequency

components. Furthermore, these authors show that refining the mesh induces an

increase in the number of the wave peaks without significantly changing the location


of the critical point. When analyzing the case with frictionless contact, the authors

observed the same behavior as in the case of pure spinning with bifurcation points

that remain practically unchanged. The wavelets are almost equally distributed

along the circumference and not merely concentrated in the region neighboring the

contact zone. In a later publication, Oden and Rabier (1989) concentrated their

attention on the spectral properties of the linearized operator and tried to analyze

the nature of the bifurcations of the steady state solution and the stability of the

branches emerging after the bifurcation.

Despite so much attention having been given to bifurcation phenomena in the

steady state rolling problem, the bifurcations that were examined referred almost

exclusively to the standing wave solutions emerging when the angular velocity is

gradually increased. Little attention has been given to a more obscure circumstance

when bifurcations are related not to the physical problem examined but are man-

ifested instead at the numerical level. The present work finds that they are quite

important in explaining some of the numerical difficulties related to the use of iter-

ative solution methods in connection with the frictional finite element formulation.

1.2 Buckling and large deformation analysis of

slender structures

Solar sails are a form of satellite propulsion technology that utilizes the photon fluxes

as propulsion force. Composed of large flat membranes (very thin film surfaces),

and supported by ultralight structures (inflatable rigidizable booms), solar sails

need to satisfy several requirements in order to represent an efficient propulsion

alternative. First, they need a very large area of film surface in contact with the


sunlight, combined with restrictive geometrical constraints to ensure their correct

orientation (generally requiring that the membrane is in tension at all times). The

whole structure then must be very light and easy to deploy once transported into

space. Last but not least, durability of the ensemble is an important characteristic.

These are very large and slender structural elements. A glimpse of the difficulties

faced in their design is caught in Figure 1.2 which shows a small–scale sail (two

orders of magnitude smaller that the expected sail size) used for ground testing.

The deformed configuration shown here is due only to gravitational loading.

Figure 1.2: Small–scale test sail. Picture courtesy of D. Holland, Duke Univer-sity/NASA Langley.

Impressive advances in materials science, specifically the development of new

materials, have made solar sails a viable alternative propulsion technology for space


travel. As an example of the material characteristics that are desirable in these ap-

plications, we give here the properties of Kevlar: tensile modulus of 83 to 186 GPa,

tensile strength of 3.6 to 4.1 GPa, and density 1.44 g/cm3. Other materials that are

considered are Mylar and Kapton. Thicknesses for the sail membrane of the order

of microns are typical in these applications. For efficiency (i.e., to maximize the

transfer of momentum), the surface of the membrane facing the sun must have high

reflection indices. The other face should dissipate space charges and temperature

and should provide radiation protection. Given the proximity to the sun during

space travel, some film materials may not be adequate since their melting points

may be lower than the actual temperatures experienced. Another important issue

is the vacuum of space. At extremely low pressures, polymers may decompose and

metals sublimate (lose molecules).

There are different environmental effects and desirable physical, optical, and

electrical properties that must be considered when analyzing the performance of

sail membranes. Environmental effects include charge particles, meteoroid impacts,

and solar intensity. Radiation exposure should be maintained below the damage

limits for the sail film. The travel velocity increases in the proximity of the sun,

the vehicle has increased efficiency which is obviously a desired effect. However, the

consequences of the environment in such conditions include a significant increase

in the temperature and in the radiation level. Therefore, the film material has

to be thermally stable and highly resistant to radiation. The design of thin films

should take into account electromagnetic and particle radiation which potentially

can remove structural material. Impact with a meteor could cause damage in various

ranges, from superficial degradation to total puncture of the sail film. Since all

spacecraft are likely to be impacted by some type of space debris, the structure


should be designed to successfully resist debris up to at least a certain size.

One of the challenging aspects of designing a solar sail involves packaging and

deployment procedures (Jenkins, 2001), both for the solar sail film and for the

supporting structure. The packaging scheme has to be consistent with the deploy-

ment requirements and should aim for a minimum stowed volume, preferably with

no residual air. Deployment must be stable, ideally insensitive to small perturba-

tions and imperfections, and passively controllable. Stability of the deployment is a

very important condition since the structure will undergo a significant configuration

transformation during this procedure. As shown by Wang and Johnson (2002), in

the case of inflatable solar sail booms, a dynamic deployment analysis is required

to investigate the behavior. Testing prototypes for deployment in space are very

expensive, and ground testing does not provide sufficient accuracy for simulating

deployment in space. Performing the testing in a vacuum chamber simulates the

vacuum of space. The effect of gravity on the deployment dynamics however can-

not be eliminated in ground testing. This is why a computational simulation of the

inflation deployment procedure is necessary. Wang and Johnson (2002) present an

analysis performed with a nonlinear finite element formulation that uses a control

volume (CV) approach and an ideal gas law. To include the effect of the inflation

gas inertia in the simulation, the Arbitrary Lagrangian Eulerian Method is used.

Usually, explicit integration algorithms are employed, but these require small time

steps. Since the model describes a process with slow inflation rates, this is compu-

tationally expensive (a number of the order of one million time steps is necessary to

simulate the process). There are many factors that can affect the inflation behavior.

The report by Wang and Johnson (2002) studies the effect of the residual air still

left inside the structure during packaging, as well as the the effect of gravity and


inflation rate on the dynamical inflation deployment.

A critical factor for the future use of Gossamer structures (i.e., large, inflatable

structures) is their dynamic response to self–generated and environmental loads. As

can be seen in the work of Pappa et al. (2003), structural dynamics and vibration

control technologies are going to be of major importance for their performance.

The maximum dynamic response that ultralight inflatable structures are going to

experience will be during deployment and operation in space. No standardized

ground experiments exist (at least not yet) for simulating their space structural

dynamic response, hence the importance of having reliable numerical models.

There are many other variables involved in a structural dynamic study of solar

sails. Like any other nonlinear dynamic system, these structures are subject to

bifurcations. Phenomena such as buckling of the booms and localized membrane

wrinkling should be taken into account as do Yang et al. (2004), Mansson and

Soderqvist (2003), and Johnston (2002). Accurate prediction of the structural be-

havior through finite element analysis is a key requirement for the development of

solar sail technology. It is also a very complex task due to geometrical nonlinearities

and localized buckling/wrinkling, as well as to numerical stability issues.

The primary component of the support structure are the slender booms that can

be modeled as very long slender beams carrying various types of loads. The goal

of the application considered in this thesis is to identify the appropriate analysis

methods for these slender beams, which should be capable not only of identifying

the load capacity but also of accurately describing the structural behavior in the

domain of very large deformations. To this end, various structural systems have been

considered. Different boom and solar sail designs are discussed by Lichodziejewski

et al. (2003), Lin et al. (2002), and Greschik and Mikulas (2002). The nature


of the buckling phenomena and the postbuckling behavior is dependent on the

configuration which may include loads of constant direction, follower–type loads,

and loads of other variable orientation.

Buckling and vibration of a cantilevered column subjected to a load that passes

through a point on the column’s axis (e.g., the base) has been treated by various

researchers. Critical loads were obtained analytically by Timoshenko and Gere

(1961) and experimentally by Willems (1966). Frequencies about the straight

equilibrium configuration were computed by Huang et al. (1967), Anderson and

Done (1971), Sugiyama et al. (1983), and Xiong et al. (1989), with experiments

included in the works by Huang et al. (1967) and Xiong et al. (1989). Postbuckled

equilibrium shapes based on an elastica analysis were obtained by Mladenov and

Sugiyama (1983), Willems (1966), Huang et al. (1967), Anderson and Done (1971),

Sugiyama et al. (1983), Xiong et al. (1989), Mladenov and Sugiyama (1983).

Dmitriyuk (1992) related this problem to that of a column subjected to a tangential

load at its tip, while Tabarrok and Xiong (1989) discussed variational principles.

Chaudry and Rogers (1992) determined equilibrium shapes for the case in which

there was no offset at the base but where a cable was attached eccentrically at the

tip of the column. They were interested in the use of shape–memory–alloy actuators

to control the shapes of beams. Tomski et al. (1998) studied a related problem in

which a rigid link was attached to the beam tip and passed through a fixed point on

the beam axis. In that problem, the bending moment is not zero at the beam tip.

In other related problems, a load that passes through a point on the column was

considered by Dmitriyuk (1992), and a cable passing through a point on the column

and then through the base was analyzed by Chaudry and Rogers (1992) as well as

Sugiyama and Masuyoshi (2003). Beam buckling under nonconservative loading is a


problem whose difficulty has been recognized by Beck (1952) in his classic paper on

dynamic aspects of buckling of a cantilever beam under follower tip load. Various

other nonconservative structural systems were analyzed by Bolotin (1963).

1.3 Structure of the thesis

The body of the thesis starts with Chapter 2 where a background on nonlinear prob-

lems, bifurcation phenomena, and associated numerical techniques is presented. The

chapter describes the general form of a nonlinear problem, the sources of nonlinear-

ities, and the finite element approach for problems in nonlinear solid and structural

mechanics. The concept of bifurcations is then introduced, and the difficulties

with numerical methods for solving problems where bifurcations are possible are

described. The chapter ends with a detailed description of the specific bifurcation

problems that are considered in this thesis, encompassing both physical bifurcations

(buckling and standing wave solutions) and numerically induced behavior (bifurca-

tions in the iterative map).

Chapter 3 focuses on the steady state frictional rolling contact problem. One

of the primary extensions of the current work – beyond earlier efforts mentioned in

Section 1.1 – is the inclusion of a modified iterative algorithm for Coulombic friction

that allows for successful solution of problems within a larger range of friction coef-

ficients. Moreover, it will be shown that this technique is effective not only for the

specific sliding formulation included in our code, but is also useful for some other

formulations from the literature. The chapter begins with a detailed description

of the implemented finite element formulation. The general tools that were devel-

oped and implemented via user elements in FEAP (Taylor, 2003) consist of (1)

a three–dimensional, eight–node finite element (formulated in finite deformations


using a Mooney–Rivlin constitutive law for the material, with incompressibility han-

dled through a Q1P0 approach, i.e., using reduced integration on the volumetric

terms); (2) inclusion of the pressure loading based on the algorithm by Simo et al.

(1991), with exact integration and consistent Newton-Raphson linearization; and,

(3) formulation for contact with sliding (with Coulomb friction law). Section 3.5 in-

troduces alternative iterative techniques that extend the applicability of the sliding

formulation beyond the range of parameters that was effectively covered previously.

The chapter ends with several sections containing numerical examples, the first

of which cover code testing and verification (see Section 3.6), as well as typical

results obtained on representative problems (Section 3.8). Examples comparing

the frictional formulation with other formulations from the literature are included

in Section 3.7. These examples show a very good qualitative and quantitative

agreement and, more importantly, they prove that the alternative techniques are

necessary and efficient not only for our formulation but for other steady state for-

mulations as well.

The robustness of finite element formulations for the steady state rolling of pneu-

matic tires under adherent rolling conditions is also analyzed. Such formulations are

widely utilized in the tire industry, in large part because contrary to time–stepping

techniques, the steady-state rolling approach allows us to refine the mesh only in

the region of the tire–road interface, eliminating the need for a fine mesh over the

entire domain and/or the need for remeshing as the simulation proceeds (see, for

example, the simple but representative meshes presented later in Figures 3.27 to

3.33). Section 3.9 provides a thorough examination of the algorithmic behavior that

can be observed when using such formulations. A number of interesting patholo-

gies, associated with both the equilibrium states of the discretized system, as well


as the iterative maps used to obtain these states, are observed, and to some degree,

characterized. The most important pathologies that are analyzed include (1) the

failure of the iterative algorithm to always find the root (the iterative map converges

instead to a periodic k–cycle), and (2) recovery of multiple solutions for some par-

ticular combinations of parameters. Throughout, it is important to emphasize that

all of these pathologies are associated with the presence of frictional conditions on

the tire–road interface; i.e., if one assumes frictionless response within the steady

state framework, our numerical investigations have not revealed any of the difficul-

ties that will be discussed. Building upon contributions listed in Section 1.1, this

work examines for the first time the specific influence of frictional interaction upon

the steady state rolling problem. We identify limit and bifurcation points of the

rolling cylinder in adherent contact with a rigid surface and examine the relationship

of these points to the development of effective iterative schemes for the frictional

rolling problem. We further investigate the nature of the bifurcation points that

are encountered and the effect of the numerical discretization on the stability and

convergence of algorithms used to compute steady state equilibria. A combination

of techniques such as eigenvalue analysis and study of the bifurcations of the nu-

merical maps are employed in order to assess the performance of the algorithm.

These techniques allowed us to identify regions in the parameter space where the

algorithm has a robust behavior.

Chapter 4 addresses the postbuckling behavior of slender structures with direct

application to solar sail booms. Different sail configurations, attachment solutions,

and beam designs are described. The concept of postbuckled configurations is then

introduced, and three main structural systems are considered: (1) a cantilever beam

subjected to compressive loads of constant direction; (2) a cantilever beam subjected


to nonconservative (follower) loading; and (3) a cantilever beam loaded by an angled

cable passing through a fixed point.

First, we analyze the buckling of slender beams of different design and under

various types of loading. The appropriateness of some simplifying assumptions, and

the use of simpler structural models, are examined. For the first of the three sys-

tems, various beam designs are considered, complex designs are analyzed by using

at first detailed models; simplified models are then suggested, and their accuracy

is studied. Postbuckling loading paths are followed by making use of continuation

methods, up to highly–deflected configurations. For the second system, dynamic

buckling analysis is performed, and various options for numerical stabilization are

examined. In the study of the third system, a new feature is added to the pre-

vious literature (Section 1.2) treating similar configurations: the end of the cable

away from the beam tip is offset from the beam’s axis. Second, vibrations about

the highly–deformed equilibrium states are investigated for the first and third sys-

tem. Geometric nonlinearities are considered, and the dynamic properties of slender

beams are shown to be highly sensitive to small changes in the axial (mainly com-

pressive) loading. ABAQUS (2003) is applied in a finite element analysis of these

problems for the numerical examples included in Section 4.5. In some cases, the

finite element results were compared with other numerical results (for instance, the

third system is also studied analytically, as an elastica, and numerical solutions

are obtained with a shooting method), and with experimental data (where avail-

able). Equilibrium shapes and vibration frequencies and modes are presented and

compared.

The last chapter introduces the concluding remarks and possible directions for

future study.

Chapter 2

Nonlinear Problems andBifurcations

The general form of a system of equations associated with a nonlinear problem is

N (x) = 0, (2.1)

where N is a nonlinear operator and x is the vector of unknowns. All applications

discussed in this thesis correspond to problems in structural and solid mechanics.

In this context, Equation (2.1) represents the dependence of the displacement field

at equilibrium, x, on the loading applied to the system.

Typically, the size of a problem is an important issue because some nonlinear

problems can be very expensive to solve; in general, the algorithm will require

repeated evaluations of complicated nonlinear functions and/or of their derivatives.

Even though one hopes to be able to solve most small problems, sometimes a two

variable system may be too difficult.

17

CHAPTER 2. NONLINEAR PROBLEMS AND BIFURCATIONS 18

2.1 Sources of nonlinearity

Many problems in computational mechanics contain nonlinearities, which can arise

from various sources:

• Material nonlinearities, where the material parameters are a function of the

solution – in our case the displacement field. Examples here include nonlinear

elasticity and visco-elasticity, plasticity and visco-plasticity. In some of these

cases, not only do we have a load vector that is a function of the unknowns,

but it also matters how the system reaches that load level (i.e., the problem

is history–dependent).

• Geometric nonlinearities, where the large deformations require consideration

of the deformed geometry.

• Nonlinear boundary conditions, where the boundary condition value and/or

orientation depend upon the solution of the problem.

The most general problems can present all these types of nonlinearity simultane-

ously. This is indeed the case of one of the applications considered in this thesis,

the steady state frictional rolling problem, which is presented in detail in Chapter 3.

The second application examined in this thesis, the postbuckling analysis of slender

booms for space applications (Chapter 4), can also present in the most general case

all types of nonlinearities. However, the numerical examples considered there ex-

amine only the case of linear elastic materials, and, as a result, only nonlinearities

from large deformation response and from boundary conditions are included in the

analysis.


2.2 Finite element formulations and solution tech-

niques for nonlinear problems

In this thesis, we are concerned with finite element formulations for parameter–

dependent nonlinear problems, as well as with the associated numerical techniques

utilized for the solution of such problems. In recent years, the computational sci-

ences have witnessed an incredible development, with efficient algorithms having

been implemented for various classes of nonlinear problems.

There are many ways to solve nonlinear problems; indeed, it sometimes seems

that there are too many possible schemes from which to choose. This indicates

that a generally applicable method in solving such problems does not exist, and

the fact that the success of most of the available techniques is dependent upon the

particular properties of the nonlinear problem to which they are applied. What is

not possible is to have an algorithm that will solve every nonlinear problem. Even

though there are many approaches, some facts remain true: in general, nonlinear

problems are solved using an iterative scheme, and usually an iterative algorithm

cannot answer general questions about the existence and uniqueness of the solution

of a given problem. At best, it can report the finding of one solution or the inability

of converging to a root in a given (finite) number of iterations.

Consider the general nonlinear problem described by Equation (2.1), and assume

also that the problem is parameter–dependent. The corresponding finite element

formulation can be expressed in the general schematic form

R(x,λ) = 0; R : Rn+k → Rn, (2.2)

where R may be thought of as the residual (here, an out–of–balance force), x is


the vector of unknowns – nodal displacements in our case (the dimension n is the

number of degrees of freedom in the system) – and λ is the vector of generalized

loading parameters (with dimension k).

A widely used method for solving the system of nonlinear equations (2.2) is the

classical Newton–Raphson method, where the “load” (i.e., λ) is often applied incre-

mentally. The residual R can be expressed as the difference between the internal

and external force vectors,

R(x,λ) = F ext(x(λ))− F int(x(λ)). (2.3)

At each increment in λ, the iterative method seeks to determine the configuration

x equaling F ext(x(λ)) and F int(x(λ)). The Newton–Raphson method involves the

linearization of the system (2.2). The consistent tangent DxR is defined by its

elements, Kij;

Kij =∂Ri

∂xj

, (2.4)

and the algorithm will advance the solution from iteration p to iteration p + 1:

xp+1 = xp − (DxR(xp))−1R(xp). (2.5)

where Dx is the linearization operator. The sequence (xp)∞p=1 is the discrete Newton–

Raphson trajectory of the initial iterate x0. If this sequence is convergent, that is,

if ∃ limp→∞ xp, then the iterative algorithm was successful in obtaining a solution,

and the initial iterate x0 was in the basin of attraction of this solution.

It can be shown that, under specific conditions, such a successful sequence of

iterates has an asymptotically quadratic convergence rate; once an iterate gets suf-

ficiently close to the root, convergence is obtained very quickly. This is one of the


strengths that makes this method so useful. Caution is advised, however, against its

naive use, since it has some undesirable characteristics. Unless the problem is only

mildly nonlinear, and we have reasonable starting guesses, we are not guaranteed

convergence.

Unfortunately, in most cases, very little a priori information about the solution

is available, and the iteration given in Equation (2.5) often fails since poor “initial

guesses” are likely to be used. Moreover, even though what appears to be a good

initial iterate might be chosen, if the basin of attraction of the solution is very small,

the method may still fail. A more detailed analysis of the behavior of this iterative

approach is presented in Section 2.3.5.

In the case of the steady state frictional rolling application (described in detail

in Chapter 3), some of these problems are clearly affecting the convergence behavior

of this method. Based on specific properties of the problem, alternative iterative

strategies are developed that are proven to be advantageous for this solution tech-

nique not only for the specific finite element formulation proposed here, but also in

the case of some alternate formulations presented in the literature (see Section 3.5).

In the case of the postbuckling analysis of slender beams (Chapter 4), the stan-

dard (i.e., “load–incrementation”) technique fails (for reasons that will be described

later), and the numerical stabilization is obtained by making use of alternate path–

following procedures.


2.3 Bifurcations and associated numerical method-

ology

2.3.1 Definition

It is not unusual in nonlinear mechanics problems to observe bifurcations of the

solution. A bifurcation is a topological (i.e., qualitative) change of the solution of a

parameter–dependent problem under the variation of one or more controlled system

parameters. The particular values of the parameters at which these changes occur

are called bifurcation points .

In this thesis, we are concerned with two types of bifurcations, different in nature,

but equally important for the finite element solution of nonlinear problems: (1)

physical bifurcations, where the system that is analyzed can have multiple solutions

and/or changes in the stability of a solution branch, and (2) numerical bifurcations,

where the instabilities are numerical in nature and related to the behavior of the

iterative method used to compute the solution.

The typical example of physical bifurcation in structural engineering is the buck-

ling of structures, where the parameter that is varied is the load. This problem has

been extensively studied in the literature. For a general overview of different meth-

ods and structural systems that have been studied see, for instance, Timoshenko

and Gere (1961), Simitses (1976), and Bolotin (1963). The examples discussed in

Chapter 4 fall into this category; they analyze the postbuckling behavior of slen-

der structures for space applications. From the finite element analysis perspective,

there are several numerical aspects that are important:

• Identification and characterization of the bifurcation points


• Use of appropriate continuation methods that are capable of following the

loading path beyond the bifurcation

• Robustness of the numerical method.

In some simulations, the iterative methods used to compute the solution fail to

converge to a unique solution, with the nonlinear map instead settling into periodic

orbits. An example of when this occurs (presented in Chapter 3) is the numerical

modeling of steady state rolling contact; here the frictional formulation incorporated

into the model interacts with bifurcations of the iterative map, and the numerical

model is shown to have multiple solutions. The nonlinear system of equations is a

discretization of the original system, and the question appears whether or not the

numerically identified branches correspond to equivalent phenomena in the original

problem.

Whether or not the bifurcations are physical or a consequence of the use of an

iterative map, the numerical techniques used to identify them are similar. After

all, an iterative map can be easily viewed as a dynamical system, and results from

the general study of such systems can be applied. A detailed description of the

numerical techniques for the detection of bifurcation points of dynamical systems

and for the continuation of equilibria is found in Govaerts (2000).

2.3.2 Continuation methods for finite element analysis be-

yond bifurcations

Consider the problem of finding a solution path of the parameter–dependent system

from Equation (2.2),

R(x,λ) = 0, (2.6)


where x ∈ Rn is the vector of unknowns and λ ∈ Rk is the vector of parameters. If

a system has many parameters (i.e., if k À 1), achieving a general understanding

of the global behavior under the variation of all parameters seems a hopeless task.

Consequently, the specific applications introduced in Chapters 3 and 4 consider only

the dependence on one parameter at a time.

A continuation method is a numerical technique to obtain points along a solution

branch of Equation (2.6). Suppose that, for a given set of parameters (λ0), we have

already obtained a solution, (x0, λ0). The objective is to find other points – (x1,

λ1), (x2, λ2)... – that are also solutions of Equation (2.6).

The historically prevalent method, known to engineers as the incremental–load

method (the parameter being the structural loading), relies on the natural param-

eterization of the solution space. The solution path is parameterized by the same

parameters λ that describe the nonlinear problem, and successive solution configu-

rations are sought by fixing λ = λ1 and solving (for x) the equation

R(x,λ1) = 0 (2.7)

by means of Newton’s method, one of its variants, or by other iterative methods.

An initial iterate for x might be x0. Geometrically, this choice is equivalent to

a predictor step that first approximates the path by a straight line, followed by

a corrector step that iteratively solves the system in the subspace defined by the

hyperplane λ = λ1 (see Figure 2.1).

One can also apply a predictor along the tangent line, still making use of the

same natural parameterization; the corresponding predictor–corrector scheme is

presented in Figure 2.2.

The natural parameterization is not an optimal one when special events (i.e.,


λ

x ( x 1 , λ 1 )

( x 0 , λ 0 )

( x 2 , λ 2 )

Figure 2.1: Continuation using natural parameterization.

λ

x

( x 1 , λ 1 )

( x 0 , λ 0 )

Figure 2.2: Continuation using natural parameterization and tangent predictor.

limit or bifurcation points) are encountered along the solution path. In some in-

stances, even though λ seems like a satisfactory choice as parameterization for the

path, a very small increment ||∆λ|| may be necessary in order for the iterative

algorithm to converge, and thus a poor algorithmic performance may be recorded.

Alternate parameterizations are necessary in these cases; in general, a suitable op-


tion is the consideration of the arclength as parameter for the curve. Sometimes,

pseudoarclength methods are used. They usually consist of a predictor that mea-

sures arclengths along the tangent, thus approximating the “real” arclength (along

the path), and a corrector step for which there exist several choices (i.e., a solution

is sought in the hyperplane passing through the end of the predictor step, a solution

is sought in a hyperplane orthogonal to the tangent vector, a solution is sought that

minimizes the distance from the endpoint of the predictor step, et cetera). In addi-

tion to these predictor–corrector methods, another class of continuation techniques

includes piecewise linear methods. The interested reader can find detailed descrip-

tions, and the underlying mathematical theory associated with various continuation

methods in Allgower and Georg (1990). Rheinboldt and co-workers contributed ex-

tensively to the literature dedicated to the solution of nonlinear problems and,

in particular, to continuation methods. Rheinboldt (2000) provides a historical

overview to continuation methods and the mathematical sources that strengthened

their development. Rheinboldt (1980) analyses a particular steplength algorithm

used in an Euler-predictor–Newton-corrector continuation scheme.

An important feature of any continuation algorithm is the control of the step

size. Large steps may either cause failure (the method will not converge) or lead

to convergence to a point on a different branch, while excessively small steps can

result in much unnecessary work.

Modified Riks Algorithm

Consider the case of the load-displacement path presented in Figure 2.3. This path

exhibits some features that make this problem impossible to solve with a standard

incremental approach because the load and the displacement have local maxima


and minima as the solution evolves. For this type of unstable behavior, several

methods have been proposed in the literature. The most successful seems to be

the algorithm presented by Riks (1979); a modified version was applied for the

postbuckling analysis of slender space structures (numerical examples presented in

Section 4.5).

λ

x

Figure 2.3: Typical “problematic” static loading path.

We describe here this method as it is implemented in ABAQUS (2003). The

basic iterative algorithm is still Newton’s method (with the risk of a limited basin

of convergence); therefore, the increment size has to be limited to ensure conver-

gence. It is assumed that all loads are proportional (the problem is defined by a

scalar parameter). The predictor step is a pseudoarclength step along the tangent,

with size limited by the standard convergence–dependent automatic incrementation

algorithm for static calculations in ABAQUS (scaling down accordingly the tangent


vi calculated in the predictor step). The corrector step searches for the solution in

the hyperplane that passes through the point describing the configuration at the

end of the predictor step and is orthogonal to the tangent vector vi. A simplified

geometrical representation of the algorithm is shown in Figure 2.4, and a step–by–

step description is given in Table 2.1. The corrector updates are performed in the

directions represented by the dashed lines in the figure. The equilibrium search is

always perpendicular to the last tangent rather than to the tangent at the beginning

of the corrector step (as in the standard Riks algorithm). This update was included

in ABAQUS primarily for facilitating the use of the method in plasticity problems.

λ

x

(x0 , λ0)

(x i , λi)

p i

v0

v i

(x i+1 , λi+1)

Figure 2.4: Schematic of the Modified Riks Algorithm.


Table 2.1: Modified Riks Algorithm.

1. Initialize load step:(x0, λ0) = (x∗, λ∗) ; converged solutionForm force vector and tangent stiffness

2. Solve for v0 (i.e., the predictor step)3. Calculate increment size and scale v0 accordingly4. Initialize corrector step:

i = 1(xi, λi) = (x0, λ0) + v0

5. Iterate in the corrector step:Form force vector and tangent stiffness at (xi, λi)IF equilibrium THEN

GO TO 8ELSE:

solve for vi

project residuals onto the load subspace (i.e., get pi)scale vi such that the solution advances in the hyperplane normal to vi−1.

END IF6. Update for the next iteration

(xi+1, λi+1) = (xi, λi) + vi;i = i + 1

7. GO TO 58. Corrector step converged

update: (x∗, λ∗) = (xi, λi)i = 0

9. Check if analysis is over:IF (λ∗ > λmax) THEN

STOPELSE:

GO TO 1

2.3.3 Critical points. Methods of identification and charac-

terization

Standard incremental approaches may behave poorly (lose the optimal convergence

rate or even diverge) in the vicinity of critical points. Therefore, it is of great interest


to be able to identify these critical points and also to determine their nature (limit

points, bifurcation points). Convergence problems usually appear at or after these

values, and stabilizing methods and/or different algorithms may be necessary to

overcome them.

An equilibrium configuration (x,λ) is a regular point of the solution path if

detK|(x,λ) 6= 0 with K the consistent tangent stiffness; otherwise, it is called a

singular/critical/stability point. A schematic representation (for a single degree of

freedom system) of these different stability cases is presented in Figure 2.5. Critical

points can either be limit points or bifurcation points. Schematized representations

of these cases are shown in Figure 2.5 b) and c).

load parameter

a) Monotonically increasing

all equilibrium configurations

are regular points

u

load parameter

b) One singular point (limit point)

on an equilibrium path

load parameter

c ) Bifurcation point

Equilibrium path splits, more than one

configuration exists for λ > λcr

uu

limit point

bifurcation point

λcrλcr

Figure 2.5: Stability cases - schematic representation for an SDOF system.

Limit points are configurations on the solution path where the Jacobian is sin-

gular and the vector ∂R∂λ

does not belong to the space spanned by the column vectors

of the Jacobian. Limit points are characterized by the fact that equilibrium config-

urations do not exist in their neighborhood for values of the load greater than the

critical value (Riks, 1972). This is not usually true for bifurcation points. In the

neighborhood of a bifurcation point, a system can have multiple possible equilib-


rium configurations with different stability properties. There are several types of

bifurcations – examples include asymmetric, stable symmetric, unstable symmetric

– that can occur at limit or turning points, or they can be multiple limit point

bifurcations.

There are two classes of methods that can be utilized to compute the critical

points. An indirect method indicates a critical point with the help of a detecting

parameter that is monitored while tracing the equilibrium path in an incremental

manner. For instance, the parameter can be chosen to be either the determinant

of the tangent stiffness matrix or its smallest eigenvalue. In a direct method, the

critical point is included as an unknown in the system of equations, thus obtaining

an extended system with additional unknowns whose solution will directly give the

location and nature of the point. For instance, Wriggers et al. (1988) and Wriggers

and Simo (1990) consider the eigenvector equation Kψ = 0 as the stability point

condition, while Planinc and Saje (1999) present a method that uses the determinant

of the stiffness matrix. Rheinboldt (1978) introduces an iterative method that

identifies simple bifurcation points and solution points on the secondary curve in

a computationally efficient manner. In a more recent paper, Fink and Rheinboldt

(1985) present a mathematical framework for the study of the bifurcations of the

solution of parameterized equations.

2.3.4 Eigenvalue analysis

The method used to locate the critical points in the applications discussed in this

thesis is an indirect one. Using an incremental approach, we start from an unloaded

configuration and gradually apply the load to our structure while monitoring the

eigenvalue signature of the stiffness matrix along the equilibrium path.


The equilibrium path is unique and stable (see Figure 2.5 a) as long as we have

a positive definite Jacobian, detK > 0. When a critical state is encountered, it

may be identified by the value of the controlled parameter for which the determi-

nant becomes null for the first time (i.e., when the smallest real eigenvalue becomes

negative). Since an incremental approach can accurately obtain the critical points

only with a refinement of the incremental procedure, and since any eigenvalue anal-

ysis is very sensitive to perturbations when the matrix is nearly singular, standard

methods will only return approximate values.

Once a point on an equilibrium path is obtained, one can also examine the

stability of an equilibrium configuration. This is a local property and concerns the

behavior of the equilibrium path in the neighborhood of that configuration. For a

conservative (self adjoint) system, the equilibrium is stable if the matrix is positive

definite, neutrally stable if the matrix is positive semidefinite, and unstable if the

matrix is not definite.

One can identify the new unstable modes that appear during a time step in

the following manner. Denote by U the number of real eigenvalues that become

negative during the time step [tn, tn+1] and by V the number of solution branches

after the possible critical point that belongs to this interval. U coincides with the

number of new unstable modes, and, according to Kouhia and Mikkola (1998), one

can determine the range for the number of possible solution branches emanating

from a critical point: for a symmetric system

U ≤ V ≤ 1

2(3U − 1), (2.8)

and for a nonsymmetric one

1 ≤ V ≤ 2U − 1. (2.9)


For a nonsymmetric system, the eigenvalues might also be complex valued (pairs

of complex conjugate eigenvalues since the matrix is real). Along with bifurcations

corresponding to singularities of the stiffness matrix, other types of critical states

(associated with the presence in the spectrum of K of purely imaginary eigenvalues)

can be identified. These states are mathematically characterized as local Hopf

bifurcations (Hale and Kocak, 1991). For a nonconservative loading, the loss of

stability of the system might manifest itself not only by the system evolving toward a

different equilibrium state but also by the system presenting an unbounded motion.

This may be manifested numerically as a divergence situation.

The mathematical eigenvalue problem has been studied extensively, and much

work has been devoted to providing efficient eigenvalue extraction methods; for an

overview of various methods see Wilkinson (1965). The eigenvalue problems asso-

ciated with finite element calculations involve usually very large but sparse/banded

matrices, and only a small number of eigenpairs are of interest. Moreover, in many

cases, the matrices that are involved are symmetric, which further simplifies the

calculation. For the numerical examples presented in Chapter 4, the eigenvalue

extraction is performed in ABAQUS, using Lanczos or subspace iteration for sym-

metric problems and a subspace projection method for complex eigenproblems. In

the case of the steady state rolling problem (Chapter 3), the eigenvalue extraction

is carried out in Matlab. Even though only extraction of the eigenpairs with the

smallest real component is required for locating the bifurcations, a calculation is

performed that extracts the full spectrum for a better understanding of its charac-

teristics.


2.3.5 Nonlinear iterative maps. Convergence analysis of the

Newton-Raphson algorithm

When solutions of Equation (2.3) are searched using an iterative scheme, the idea is

to recast this equation into a form involving a new function, g(x), such that the set

of fixed points of the map g coincides with the roots of Equation (2.2). Solving the

initial system of nonlinear equations therefore becomes equivalent with obtaining

the fixed points of the function g, i.e., obtaining the roots of the equation

g(x) = x, (2.10)

and this is done via an iterative approach described by the scheme

xi+1 = g(xi). (2.11)

A successful sequence of iterations will converge toward a fixed point x; i.e.,

limi→∞

gi(x0) = x, (2.12)

where by gi we denote the i–fold composite of the map g. If the convergent sequence

gi(x0)i exists, x is called an asymptotically stable fixed point of the map, with

the initial iterate thus belonging to the basin of attraction of the stable fixed point.

Let us recall some definitions and properties related to the r- and q-order of

convergence for the iterative sequence (Dennis and Schnabel, 1983) that will be

useful in the following discussions.


Definition 1 (LINEAR CONVERGENCE)

Let (an) be a sequence of positive real numbers. We say that

1. (an) converges r-linearly if

lim supn−→∞

n√

an < 1 (2.13)

2. (an) converges q-linearly if

lim supn−→∞

an+1

an

< 1. (2.14)

Observe that if an is either r- or q-linear convergent, then

limn→∞

an = 0.

Note also that the q-linear convergence is stronger than the r-linear convergence

since

lim supn−→∞

n√

an ≤ lim supn−→∞

an+1

an

. (2.15)

This last observation applies to higher orders of convergence as well.

Definition 2 (SUPERLINEAR CONVERGENCE)

Let (an) be a sequence of positive real numbers. We say that

1. (an) converges r-superlinearly if

lim supn−→∞

n√

an = 0 (2.16)


2. (an) converges q-superlinearly if

lim supn−→∞

an+1

an

= 0. (2.17)

Definition 3 (ORDERS OF CONVERGENCE)

Let (an) be a sequence of positive real numbers.

1. The r-order of convergence of (an) is defined by

R((an)) = lim infn−→∞

n√| ln an| (2.18)

2. The q-order of convergence of (an) is defined by

Q((an)) = lim infn−→∞

| ln an+1|| ln an| . (2.19)

An important result for the convergence of iterative maps is given by the fol-

lowing theorem (Dennis and Schnabel, 1983).

Theorem 1 (CONTRACTIVE MAPPING THEOREM)

Let G : D → D , with D a closed subset of Rn. If for some norm ‖ · ‖, there

exists α ∈ [0, 1) such that

‖G(x)−G(y)‖ ≤ α‖x− y‖ (2.20)

then:

1. there exists a unique x∗ ∈ D such that G(x∗) = x∗;


2. for any x0 ∈ D, the sequence xk generated by xk+1 = G(xk), k=0,1, ...,

remains in D and converges q-linearly to x∗ with constant α;

3. for any η ≥ ‖G(x0)− x0‖,

‖xk − x∗‖ ≤ ηαk

1− α. (2.21)

This theorem states sufficient conditions under which an iterative map converges

to a fixed point. The result is weak in that it only proves a q-linear rate of conver-

gence, while the consistent Newton–Raphson method that will be extensively used

in this thesis, if convergent, has a q-quadratic convergence rate. Even though weak

with respect to convergence rate characterization, this theorem is important since

it makes no assumptions about the specific method that is used and therefore is

applicable to any iterative technique. More convergence results and discussion of

various iterative techniques can be found in Ortega and Rheinboldt (1970).

If a map has a fixed point, and the iterative sequence converges to it, then its

stability can sometimes be characterized by studying the linearization of the map

at the fixed point, x → Dg(x)x, where Dg(x) is the Jacobian matrix

Dg(x) =

∂g1

∂x1(x) . . . ∂g1

∂xn(x)

. . . . . . . . .

∂gn

∂x1(x) . . . ∂gn

∂xn(x)

. (2.22)

A fixed point x is said to be hyperbolic if the Jacobian matrix at x has no

eigenvalues of modulus one. The stability of such a point is easy to determine. If

all eigenvalues have moduli less than one, the point is asymptotically stable; if one

or more eigenvalues have modulus greater than one, the point is unstable.


Next, we recall some other definitions regarding elements and characteristics of

an iterative map that might be useful (Hale and Kocak, 1991).

The set of initial iterates that converge to a fixed point of the map is called the

basin of attraction of that fixed point.

A positive orbit of the map, starting at point x0, is the sequence of images of that

point under successive compositions of the map, γ+(x0) = x0, g(x0), ..., gk(x0), ....Similarly with a fixed point, we say x is a k–periodic point if x = gk(x), where

gk is the k–fold composite of the map g. Like fixed points, a k–periodic point

can be stable, neutral, or unstable according to whether or not the spectral radius

of the k–fold composite of the map is less, equal to, or greater than one (Kim

and Feldstein, 1997). By this definition, an equilibrium state of (2.2) is a period–

one solution (Guttalu, 1996). It is obvious that if x is a k–periodic point, so are

gi(x) for all i = 1..k − 1, and an orbit of the map starting at x is called a k–

cycle. Stability of a k–periodic point implies stability of the k–cycle. An iterative

map often exhibits such stable k–cycles; numerical examples of this behavior are

presented in Section 3.9.

A point y is called an ω limit point of the positive orbit γ+(x0) if there is a

sequence of integers, ni, such that the subsequence gni(x0) converges to it,

limi→∞

gni(x0) = y.

Similar with the analysis presented in Section 2.3.4, critical points of the map

can be identified and characterized if the set of eigenvalues of the Jacobian matrix

given in Equation (2.22) is known. Through an elementary linear transformation

(with a nonsingular transformation matrix, P ), any n×n matrix can be put in the


Jordan normal form

AJ = P−1AP =

λ1 0 0 0 0 0 0 0 0 0

0. . . 0 0 0 0 0 0 0 0

0 0 λk 0 0 0 0 0 0 0

0 0 0. . . 0 0 0 0 0 0

0 0 0 0 λm 1 0 0 0 0

0 0 0 0 0. . . 1 0 0 0

0 0 0 0 0 0 λm 0 0 0

0 0 0 0 0 0 0. . . 0 0

0 0 0 0 0 0 0 0 α −β

0 0 0 0 0 0 0 0 β α

(2.23)

thus exposing its eigenvalues. This normal form matrix can have on the diagonal

different types of blocks:

1. 1× 1 for simple real eigenvalues or multiple eigenvalues with equal geometric

and algebraic multiplicity

2. Jordan blocks

λm 1 0

0. . . 1

0 0 λm

corresponding to multiple eigenvalues with

different geometric and algebraic multiplicity; or

3. 2× 2 blocks

α −β

β α

corresponding to complex conjugate pairs of eigenval-

ues.

An in–depth study of an iterative map will be carried out in the application

presented in Chapter 3. In relation to this, of high interest are the occurrences of


blocks of type 3 in the linearized map, blocks that can be written as

A3 =√

α2 + β2

cos θ − sin θ

sin θ cos θ

. (2.24)

The action of the linearized map A3 is in this case to rotate the vector with an

angle θ and scale it with the factor λ =√

α2 + β2. Nonhyperbolic points (corre-

sponding to λ=1) have a more complex stability behavior. The linear map under-

goes a bifurcation at these points in the sense that closed orbits invariant under the

map do not exist for λ 6= 1 but do exist for λ = 1. The orbit in the case of the

linear map lies on the circle, and it is periodic if θ2π∈ Q, and dense if θ

2π∈ R \Q.

In the nonlinear case, the equivalent is called an Andronov-Hopf bifurcation.

When the eigenvalues move across the unit circle, a closed invariant curve (orbits

starting on any point on this curve remain on the curve) appears, which encloses

the fixed point. Therefore, if a current iterate happens to fall on one of these curves,

convergence will not be obtained through the iterative algorithm.

The simplest nonlinear map illustrating the birth of an invariant closed orbit

(local bifurcation near a fixed point) is the nonlinear planar map

r

θ

→

λr − r3

θ + ω

, (2.25)

whose linearization falls under the form of the blocks of type 3 from Equation (2.23).

For λ > 1 the map has an invariant circle of radius r =√

λ− 1 and the iterates of

the map are rotations of angle ω. When λ passes 1, the asymptotically stable fixed

point becomes unstable and an invariant stable circle appears.

A more general result is proven in the Poincare-Andronov-Hopf theorem (Hale


and Kocak, 1991) by transforming locally any nonlinear map into a “canonical”

form similar to the example above.

Application: The Newton–Raphson scheme

In Section 2.2, we briefly introduced the Newton–Raphson technique. Recall that,

if this iterative scheme is utilized, at each iteration the linearized system

K(xi) ·∆x = F ext(xi)− F int(xi) (2.26)

is solved, and the unknowns are updated in the usual manner via xi+1 = xi + ∆x.

For the linearized system to be a consistent Newton–Raphson iteration, we require

that

Ki = K(xi) =∂(F int − F ext)

∂x

∣∣∣x=xi

. (2.27)

If the Jacobian is not singular, the function g (defining the Newton–Raphson

iterative map) may be identified from Equation (2.26) as

g(xi) = xi + (Ki)−1[F ext(xi)− F int(xi)

], (2.28)

and will be viewed as a dynamical system for the purpose of the analysis of the algo-

rithm. Given the wide use of this technique, whether or not we use it in conjunction

with a continuation method for a parameter–dependent problem, or just by itself

for obtaining the solution of a nonlinear equation, its convergence properties are

extremely important.

As early as 1818, Fourier was the first to give sufficient conditions for the conver-

gence of Newton’s method. The rigorous proof of his statements was given later by

Darboux. Their work was subsequently published in (Fourier and Darboux, 1890).


A traditional result about Newton’s method used today is a theorem that proves its

q-quadratic local convergence, but for doing so, it assumes at least three key points:

(1) the system has a root (i.e., the map has a fixed point); (2) the initial iterate is

sufficiently close to it; and, (3) the Jacobian J is nonsingular and Lipschitz contin-

uous in the neighborhood of the root. The theorem and its proof are detailed by

Dennis and Schnabel (1983). The basin of convergence estimated by this theorem

is the open ball centered at the root and with radius evaluated as the bound in

the direction of the strongest nonlinearity of the function. In directions where the

function is only mildly nonlinear, the actual basin of convergence extends farther

than this worst–case scenario estimate. Along with the quadratic convergence rate,

Newton’s method has another important advantage; it is exact for affine compo-

nents (i.e., it will converge in one step for linear functions), thus working well on

problems with mild nonlinearities. A different convergence result for this method

was proved by Kantorovich (1948) and is stated in Dennis and Schnabel (1983) as

follows:

Theorem 2 (KANTOROVICH THEOREM)

Let r > 0, x0 ∈ Rn, F : Rn −→ Rn, and assume that F is continuously

differentiable in B(x0, r) – the open ball centered at x0 and of radius r. Assume

that J ∈ Lipγ(B(x0, r)), J(x0) is nonsingular, and (∃)β, η ≥ 0 such that

‖J(x0)‖ ≤ β, ‖J(x0)−1F (x0)‖ ≤ η.

Define α = βηγ. If α ≤ 12

and r ≥ r0 ≡ 1−√1−2αβγ

, then the sequence xk generated

by

xk+1 = xk − J(xk)−1F (xk), k=0, 1, ...,


is well defined and converges to x∗, a unique zero of F in the closure of B(x0, r0).

If α < 12, then x∗ is the unique zero of F in the closure of B(x0, r1), where

r1 ≡ min [r, 1+√

1−2αβγ

] and

‖xk − x∗‖ ≤ (2α)2k η

α, k = 0, 1, .... (2.29)

This result is weaker than the aforementioned one as far as the convergence rate

is concerned (only r-quadratic convergence is proved), but the assumptions are less

limiting. For instance, it does not require the existence of the solution; it actually

proves it under the assumed conditions on the function and its Jacobian. Other

results on Newton’s method convergence can be found in Potra and Rheinboldt

(1986).

In general, the effectiveness of Newton’s method in finding the root depends on

the existence of a nonsingular Jacobian and on the proximity of the initial iterate to

the actual solution. This immediately suggests some disadvantages of the method;

it is not globally convergent; to be consistent, it requires evaluation of the Jacobian

at each step; and, a linearized system (that may be ill–conditioned), must be solved

at each step. Various modifications of this method exist (Quasi-Newton, Descent,

Line Search, Secant, Broyden) that can either make it globally convergent or, at

least, can extend its radius of convergence. In making such modifications, however,

the price of a less–optimal convergence rate often has to be paid.

Convergence of the xi iterates to a fixed point represents the simplest possible

algorithmic behavior of the map, and non–fixed periodic or dense (and asymptotic

to periodic) orbits are also typical for maps. For iterative maps that are used in

solving nonlinear systems, the periodic solutions are of great interest. Even though

divergence is not encountered, their presence still implies a failure of the root–finding


method.

Gelman and Rheinboldt (1989) propose an algorithm, based on simple recur-

rence formulas, that computes closed curves invariant under a map. Similarly with

Kantorovich’s theorem, criteria for the convergence toward a periodic orbit of the

map can be obtained. This problem is the object of a paper, by Ocken (1998) that

identifies sufficient conditions for Newton’s method starting at an initial point x0

to converge to an attracting k–periodic orbit with k ≥ 2.

2.4 Applications considered in this thesis

2.4.1 Static and dynamic buckling of slender structures

While buckling phenomena are characteristic of many engineering systems, they

are probably of the greatest importance in the case of slender structures, where

buckling loads are in the range of the expected loads.

The first theoretical study of unstable structural systems was done by Leonhard

Euler (1744), who analyzed the case of a slender column under compressive loading.

Now, more than two centuries later, there is a vast literature dedicated to this topic.

Several methods exist for obtaining the critical conditions; see, for instance, Simitses

(1976), Bazant and Cedolin (1991), and Ziegler (1968). The classical method reduces

the analysis to an eigenvalue problem. Another method, the dynamic (or kinetic)

approach, relies on the equations governing small free vibrations about some static

equilibrium configuration. If the loading is conservative (i.e., forces can be derived

from a potential), an approach that is equivalent to the kinetic approach is known

as the potential energy method. If the load is either explicitly time–dependent or

nonconservative, the only method that can be used is the dynamic approach.


2.4.2 Standing waves and multiple solutions for rotating

cylinders

Examples are presented in the literature of bifurcations and standing wave phe-

nomena in the case of pure spinning cylinders as well as in the case of the spinning

cylinders in contact with a rigid surface; see, for instance, Oden and Lin (1986) and

Chatterjee et al. (1999). The results presented in these studies show that standing

waves appear in tires at critical speeds above the rolling speeds that are charac-

teristic for regular cars and that only high–speed vehicles can be affected by such

phenomena.

Chatterjee et al. (1999) present experimental data for spinning tires in contact

with a cylindrical surface. They show the occurrence of standing waves after some

critical value of the angular velocity and reveal properties of the deformed kinemat-

ics after the onset of standing waves. It is shown, for instance, that radial directions

remain radial in the standing wave configuration.

Oden and Lin (1986) also present an analysis of the free spinning case. Their

variational formulation is not limited to motions possessing radial symmetry, which

would obscure a very rich class of problems in the bifurcation theory. They show that

for small values of the parameter ω, the solution is unique and radially symmetric.

For a critical value, branching of the solution is encountered (loss of the rank in

the Jacobian matrix). Near this value, they observe a number of 24 standing waves

(and this number is exactly half of the number of meridians they are using for the

discretization). Furthermore, they show that refining the mesh induces an increase

in the number of the wave peaks without significantly changing the location of

the critical point. When analyzing the case with frictionless contact, the same

behavior is observed as in the case of pure spinning, and the bifurcation points are


practically the same. Standing wave solutions are obtained by the authors from

their numerical simulation not only for the pure spinning of cylinders, but also for

the case of a rotating cylinder in frictionless contact with a flat rigid surface. It is

shown that the wavelets are almost equally distributed along the circumference and

not merely concentrated in the region neighboring the contact zone.

In a later publication, Oden and Rabier (1989) develop mathematical tools that

are not standard in bifurcation analysis. They study and characterize the spectral

properties of the linearized operator. The spectrum of the operator contains isolated

eigenvalues, and they concentrate their attention on the neighborhood of every

isolated eigenvalue with finite multiplicity. The conclusion of their study is that

bifurcations of the steady state solution are not due to sudden instability of the

trivial branch, and no proof is obtained to conclude whether or not the stability is

lost at any point.

2.4.3 Bifurcations of the Newton–Raphson nonlinear itera-

tive map

The behavior of the Newton–Raphson map applied for the solution of the steady

state rolling problem will be examined in detail via numerical examples presented in

Section 3.9. As a preview, we include here a very interesting, although suprisingly

simple example showing the behavioral complexity of Newton’s map. Consider

the case of the simple cubic equation z3 − 1 = 0 with z ∈ C. Even though this

is a low–dimensional system (two components, the real and imaginary part of a

complex variable) described by a simple analytical function (a cubic polynomial),

many features that make Newton’s map behavior very intricate are present here.

The map given by the use of Newton’s method for the solution of this equation can


be written as

zi+1 = g(zi) = zi −(

z3i − 1

3z2i

). (2.30)

There are three solutions, the three cubic roots of unity, that are also fixed points

Figure 2.6: Basins of attraction of the cubic roots of unity. Shading convention:green (light grey)–basin of z1; red (medium grey)–basin of z2; black–basin of z3.

of the map g, z1 = 1, z2 = −12

+ i√

32

, z3 = −12− i

√3

2, and one would expect the

convergence behavior of the map to be simple. And indeed, it has been shown by

Epureanu and Greenside (1998) that there are very few points in the complex plane

that are not good choices as initial iterates and result in nonconvergent sequences.

But this does not mean the problem is indeed simple. Even though convergent

almost everywhere, the behavior is more complex than one may expect. The basins

of attraction of the roots are fractal in nature (see Figure 2.6); their boundaries

are not one– or two–dimensional but present instead a complex geometry with the

same features apparent at all scales. In the figure, green (lighter–grey) identifies


-1.5 -1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

Re

Im

rootsperiod-2period-3period-4

Figure 2.7: Periodic solutions of the cubic equation.

the basin of attraction of z1, red (medium–grey) corresponds to z2, and black to z3.

Furthermore, we can show that k–periodic solutions exist, and their number

increases with k. They are computed (using Mathematica) by numerically solving

the equation gk(x) = x, and their values are listed (for k ≤ 4) in Table 2.2 and

graphically represented in the complex plane in Figure 2.7. There are 3 period–1

solutions (the roots), 6 period–2, 24 period–3, and 48 period–4 solutions.

At first glance, it seems that the periodic points of period k > 2 follow the

fractal boundaries. An overlap of the fractal basins of attraction of the roots and

the periodic points from Table 2.2 is presented in Figure 2.8.


Table 2.2: k–periodic solutions of the iterates for the cubic equation; k ≤ 4

k periodic solutions k periodic solutions1 -0.5 + 0.866025 i 4 0.209612 + 0.74025 i1 -0.5 - 0.866025 i 4 0.209612 - 0.74025 i1 1. 4 -0.745881 + 0.188596 i

4 -0.745881 - 0.188596 i2 0.538609 + 0.417204 i 4 0.536269+ 0.551654 i2 0.538609 - 0.417204 i 4 0.536269 - 0.551654 i2 -0.630614 + 0.257847 i 4 0.145278 + 0.652224 i2 -0.630614 - 0.257847 i 4 0.145278 -0.652224 i2 0.0920053 + 0.675051 i 4 -0.637481 + 0.200297 i2 0.0920053 - 0.675051 i 4 -0.637481- 0.200297 i

4 0.492203 + 0.451926 i3 -0.709489 +0.226356 i 4 0.39615 + 0.492027 i3 -0.709489 -0.226356 i 4 0.492203 - 0.451926i3 0.158714 + 0.727614 i 4 0.39615 - 0.492027 i3 0.158714 - 0.727614 i 4 -0.624183 + 0.0970627 i3 0.550775 + 0.501258 i 4 -0.624183 - 0.0970627 i3 0.550775 - 0.501258 i 4 0.228033 + 0.58909 i3 -1.32285 + 0.241745 i 4 0.228033 - 0.58909 i3 -1.32285 - 0.241745 i 4 0.175923 +0.586937 i3 0.452069 + 1.2665 i 4 0.175923 -0.586937 i3 0.452069 - 1.2665 i 4 0.420341 + 0.445822 i3 0.870784 + 1.02475 i 4 0.420341 - 0.445822 i3 0.870784 - 1.02475 i 4 -0.596263 + 0.141115 i3 0.187183 + 0.582854 i 4 -0.596263 - 0.141115 i3 0.187183 - 0.582854 i 4 0.187401 + 0.560509 i3 -0.598358 + 0.129321 i 4 0.187401 - 0.560509 i3 -0.598358 - 0.129321 i 4 -0.579116 +0.117961 i3 0.411175 + 0.453532 i 4 -0.579116 - 0.117961 i3 0.411175 - 0.453532 i 4 0.391715 + 0.442549i3 0.423636 + 0.264192 i 4 0.391715 - 0.442549 i3 0.423636 - 0.264192 i 4 -0.486479 + 0.23957 i3 -0.440615 + 0.234783 i 4 -0.486479 - 0.23957 i3 -0.440615 - 0.234783 i 4 0.450713 + 0.301518 i3 0.0169794 + 0.498975 i 4 0.450713 -0.301518 i3 0.0169794 - 0.498975 i 4 -0.465624 + 0.169726 i

4 -0.465624 - 0.169726 i4 0.035766 + 0.541088 i 4 0.379799 +0.318379 i4 0.035766 - 0.541088 i 4 0.379799 - 0.318379 i4 0.0858252 + 0.488105 i 4 -0.339805 + 0.198246 i4 0.0858252 - 0.488105 i 4 -0.339805 - 0.198246 i4 0.341588 + 0.195157 i 4 -0.00178333 + 0.393402 i4 0.341588 - 0.195157 i 4 -0.00178333 - 0.393402 i

Figure 2.8: Periodic solutions (•) of the cubic equation in relation to the basinsof attraction of the roots (¤).

Chapter 3

Steady–State Frictional Rolling

The inclusion of frictional calculations in the steady state analysis of rolling tires

represents a very challenging aspect of contact mechanics. Not only do we deal with

a problem presenting strong nonlinearities, but we also face the incertitude related

to the existence and uniqueness of solutions. This chapter introduces the general

formulation of the problem and the corresponding implementation of all terms.

The major results include the formulation of a alternative iterative technique that

improves significantly the algorithmic performance and a study of the interaction

between the finite element formulation and bifurcations in the iterative map. The

latter is a key aspect to explaining some numerical difficulties and to identifying

the domain of robust algorithmic behavior.

3.1 Definition of the problem

The physical problem under consideration is the steady state rolling of a deformable

cylindrical body that is rolling at constant angular velocity ω and is in contact with

a flat rigid surface (see Figure 3.1). To make it applicable to the study of rolling

tires, the internal pressure p is considered, and Dirichlet boundary conditions are

enforced on the portion of the tire in contact with the rim.

51

CHAPTER 3. STEADY–STATE FRICTIONAL ROLLING 52

Consistent with the steady state assumption, an observer riding along with the

axle, without rotating with it, will see the same deformed configuration at all times.

Following Le Tallec and Rahier (1994), we denote by X0 the undeformed, stationary

configuration of the material points at some previous time, and by X we denote their

coordinates in the translating (Arbitrary Lagrangian Eulerian) reference frame. We

then denote by x the current position (as observed in the ALE frame), which obeys:

x = ϕ(X) = X + U (X), (3.1)

where ϕ(X) is the deformation map referred to the ALE frame, and U is likewise

the displacement in this frame. For the steady state rolling case, we may then write

the reference coordinates X of a material particle originally at X0 as

X = Q(t)X0, (3.2)

where Q(t) is a proper orthogonal tensor representing a constant velocity rotation

about some axis.

3.2 Variational formulation

With this kinematic framework in place, the virtual work principle can be written

as

0 = G0(ϕ0,∗ϕ0) :=

∫

Ω0

ρ0A0 · ∗ϕ0 dΩ +

∫

Ω0

[F 0S0] : [Grad0∗ϕ0] dΩ

−∫

Ω0

f 0 · ∗ϕ0 dΩ−∫

∂Ω0

t0 · ∗ϕ0 dΩ

(3.3)

which must hold for all admissible variations∗ϕ0. A denotes the material acceler-

ation, F is the deformation gradient tensor, S is the second Piola Kirchoff stress


initial (undeformed) configuration

V g

d

x

y

z

p

n

ω

Ω

X0

X

U (X)

x

Figure 3.1: Notation for the steady state rolling contact problem.

tensor, f the body force vector and t is the surface traction. A subscript 0 on a

quantity indicates that it is referred to the undeformed stationary configuration.

Since the finite element approximation will be applied in the moving reference

frame (denoted by the lack of subscripts 0), the virtual work expression in (3.3) is

of limited use to us in the current context. Accordingly, we represent each of the

terms in (3.3) with respect to the domain Ω.

Beginning with the inertial term, and noting that the material acceleration A0

is the second time derivative of the position of the particle holding X0 fixed, we

may use the notation ddt

(•) to denote such a material time derivative of a generic

quantity (•), and compute first the material velocity V (X):

V =d

dt[x] = X +

d

dtU (X(X0))

= X +∂U

∂XX = (I +

∂U

∂X)X.

(3.4)


Time differentiation of (3.2) gives

X = QX0 = QQT X

= WX,

(3.5)

where W := QQT is a (constant) skew symmetric tensor.

Denoting the angular velocity of the rotating cylinder by ω, we follow the nota-

tion of Le Tallec and Rahier (1994), and define Π and P such that

ωΠ := QQT

P := −Π2.

(3.6)

Using these notations, we may write (3.5) as

X = ωΠX. (3.7)

The material acceleration expressed in the ALE frame (A) is obtained by taking

another material time derivative of (3.4):

A =d

dt[V ] =

d

dt

[I +

∂U

∂X]X

= ω2ΠΠX + ω2 ∂

∂X

[∂U

∂XΠX

]ΠX

= −ω2PX + ω2 ∂

∂X

[∂U

∂XΠX

]ΠX.

(3.8)

We may now transform the first term of (3.3) to its corresponding representation


in the ALE frame

∫

Ω0

ρ0A0 · ∗ϕ0 dΩ =

∫

Ω

ρ0A · ∗ϕ dΩ (change of variables, with Jacobian = 1)

= −∫

Ω

ω2ρ0[PX] · ∗ϕ dΩ

+

∫

Ω

ω2ρ0

(∂

∂X

[∂U

∂XΠX

]ΠX

)· ∗ϕ dΩ

= −∫

Ω


+

∫

∂Ω

ω2ρ0

([∂U

∂XΠXΠX

]· ∗ϕ

)ΠX ·N dΓ

−∫

Ω

ω2ρ0

(∂U

∂XΠX

)·Div

[ΠX ⊗ ∗

ϕ]

dΩ(int. by parts).

(3.9)

The surface integral in (3.9) vanishes since ΠX is collinear with X, which is, in

turn, orthogonal to the normal N on the surface of the cylinder. Considering the

integrand in the last term of (3.9), we find that

Div[ΠX ⊗ ∗

ϕ]

= (Π : I)∗ϕ +

∂∗ϕ

∂XΠX

=∂∗ϕ

∂XΠX,

(3.10)

where the first term disappears since Π is skew. We may then substitute the above

into (3.9) to obtain the final expression for the inertial virtual work:

∫

Ω0

ρ0A0 · ∗ϕ0 dΩ = −∫

Ω


−∫

Ω

ω2ρ0

(∂U

∂XΠX

)·(

∂∗ϕ

∂XΠX

)dΩ.

(3.11)

Turning next to the internal virtual work term in (3.3), one may proceed by

considering simple tensorial transformations between the stationary frame (Ω0) and

the ALE frame (Ω). Explicitly, one may note that the second Piola-Kirchhoff stress


S in the ALE frame is given by a simple push-forward of the analogous stress in

the stationary frame

S = QS0QT . (3.12)

Employing a change of variables, one then finds

∫

Ω0

[F 0S0] : [Grad0∗ϕ0] dΩ =

∫

Ω

[F 0QT SQ] : [Grad0

∗ϕ] dΩ. (3.13)

Note that F = F 0QT (chain rule) and by the same reasoning, one may also conclude

that Grad0∗ϕ = (Grad

∗ϕ)Q. Substituting these results in Equation (3.13), yields

∫

Ω0

[F 0S0] : [Grad0∗ϕ0] dΩ =

∫

Ω

[FSQ] : [(Grad∗ϕ)Q] dΩ

=

∫

Ω

[FS] : [(Grad∗ϕ)] dΩ,

(3.14)

which follows because of the orthogonality of Q.

Similar transformations can be applied to the surface traction and body force

terms in (3.3). Applying these transformations, making use of the results in (3.11)

and (3.14), and converting all terms to integrals in the ALE frame (i.e., without

subscripts 0) yields the following virtual work expression (Le Tallec and Rahier,

1994):

0 = G(ϕ,∗ϕ) :=

∫

Ω

[FS] : [Grad∗ϕ] dΩ−

∫

Ω


−∫

Ω

ω2ρ0

(∂U

∂XΠX

)·(

∂∗ϕ

∂XΠX

)dΩ−

∫

Ω

f · ∗ϕ dΩ−∫

∂Ω

t · ∗ϕ dΩ,

(3.15)

which must hold for all admissible variations∗ϕ, defined over the closure of Ω,

Ω = Ω⋃

∂Ω.


3.3 Finite element formulations

Components of the numerical strategy used to approximate the global system (whose

variational form is detailed in Section 3.2) are described in this section.

3.3.1 Constitutive law

Although many choices are possible, this formulation considers a hyperelastic mate-

rial with a Mooney-Rivlin constitutive law, which many specialists agree is appro-

priate for describing the rubber behavior. The stored energy function considered

has the general form

W = W vol + W dev, (3.16)

where

W vol :=1

2κ(J − 1)2 (3.17)

and

W dev =1

2µ

[(1/2 + β)(I1 − 3) + (1/2− β)(I2 − 3)

]. (3.18)

In Equations (3.17) and (3.18), κ is the bulk modulus, µ is the shear modulus,

and β is an additional constitutive parameter. Also, I1 and I2 denote “modified”

invariants of the Cauchy-Green tensor, C = F T F , and J = detF . Some variations

of these expressions exist in the literature, in particular a different choice for the

volumetric term:

W vol :=1

2κ(ln I3)

2. (3.19)


The modified invariants I1 and I2 are expressed in terms of the usual invariants

I1 := trC = CKK

I2 :=1

2

[I21 − trC2

]=

1

2

[(CKK)2 − CKMCMK

]

I3 := detC,

(3.20)

and are defined such that they include only deviatoric effects.

We may now obtain expressions for the second Piola-Kirchhoff stress by differ-

entiating the stored energy function in the usual manner,

S =∂W

∂C. (3.21)

Starting with the deviatoric part of the stress (3.18), we obtain:

SdevIJ = µ(1/2 + β)

[δIJ − 1

I3

C−1JI

]+ µ(1/2− β)

[I1δIJ − CIJ − 2

I3

C−1JI

]. (3.22)

Similarly, the volumetric part of the stress is determined via:

SvolIJ = κ(J − 1)JC−1

IJ .. (3.23)

Finally, computation of the stiffness requires that the consistent algorithmic

moduli be recovered. Taking the deviatoric part first, we obtain the final expression

for the deviatoric moduli as follows :

CdevIJKL = 2µ(1/2− β) [δIJδKL − 1/2(δIKδJL + δILδJK)]

+ 2µ(3/2− β)1

I3

[C−1

IJ C−1KL + 1/2(C−1

IL C−1JK + C−1

IKC−1JL)

].

(3.24)


The volumetric moduli may be computed in a similar fashion, with their final ex-

pression being:

CvolIJKL = κJ(2J − 1)C−1

IJ C−1KL − κJ(J − 1)

[C−1

IL C−1JK + C−1

IKC−1JL

]. (3.25)

Element stiffness and force vector

With the Mooney–Rivlin stress–strain relations and associated tangent moduli, the

internal element stress vector and stiffness matrix for the finite element formula-

tion may be computed in the standard manner, but with additional rotatory terms

(indicated by the presence of ω2) included as per Equation (3.15). An eight–noded

three–dimensional element with linear shape functions was utilized. When J = 1,

an isochoric (volume-preserving) condition prevails. In practice, this incompress-

ibility condition will be represented by making κ large and by underintegrating the

volumetric terms in order to alleviate potential locking.

One may consider an elemental expression of virtual work generated by an ele-

ment of volume Ωe (using indicial notation as above):

Ge(ϕ,∗ϕ) =

∫

Ωe

[FiISIJ

∗ϕi,J − (fi + ρ0ω

2PiIXI)∗ϕi − ω2ρ0Ui,IΠIJXJ

∗ϕi,KΠKLXL

]dΩ.

(3.26)

One may now use element expansions for the variations∗ϕi and the Lagrangian

displacements Uj:

∗ϕi =

nen∑a=1

Na(Xe)cia

Uj =nen∑

b=1

Nb(Xe)djb,

(3.27)

where the cia are arbitrary constants (in accordance with the arbitrary nature of


the weighting functions). Substituting these into (3.26) gives

Ge(ϕ,∗ϕ) =

nen∑a=1

nen∑

b=1

∫

Ωe

[FiISIJNa,J cia − (fi + ρ0ω

2PiIXI)Nacia

− δijω2ρ0Nb,IdjbΠIJXJNa,KciaΠKLXL

]dΩ.

(3.28)

If we index by p the element equation number corresponding to element node num-

ber a and degree of freedom index i, we may readily extract the element contribution

to the right–hand side of the global equations via

f inte

p =

∫

Ωe

[FiISIJNa,J − (fi + ρ0ω

2PiIXI)Na

−nen∑

b=1

δijω2ρ0Nb,IdjbΠIJXJNa,KΠKLXL

]dΩ,

(3.29)

where SIJ = SvolIJ + Sdev

IJ , with SvolIJ given in (3.23) and Sdev

IJ given in (3.22). Note

that in our implementation, full quadrature was applied to all terms except those

involving SvolIJ , where reduced integration was applied (i.e., one point quadrature in

the case of an eight–node element).

Finally, the stiffness may be exposed by considering the directional derivative

of the elemental virtual work in (3.26), in the direction of a change of element

displacements ∆U . Performing this operation gives

DGe(ϕ,∗ϕ) ·∆U =

∫

Ωe

[∗ϕi,IFiJCIJKLFjK∆Uj,L +

∗ϕi,ISIJ∆Ui,J

− ω2ρ0∆Ui,IΠIJXJ∗ϕi,KΠKLXL

]dΩ.

(3.30)

The element stiffness can be extracted by again considering p to be the element

equation number corresponding to indices i and a, and introducing q as the index


corresponding to j and b. One then obtains

kepq =

∫

Ωe

[Na,IFiJCIJKLFjKNb,L + δijNa,ISIJNb,J

− δijω2ρ0Nb,IΠIJXJNa,KΠKLXL

]dΩ.

(3.31)

In (3.31), it is to be noted that CIJKL = CvolIJKL +Cdev

IJKL, where CvolIJKL is as given in

(3.25) and CdevIJKL is as given in (3.24). Again, as with the internal force vector, full

quadrature is applied to all terms except those containing CvolIJKL, where reduced

one–point integration is used.

3.3.2 Pressure and Dirichlet boundary conditions

The inner surfaces of a rolling tire are typically subjected to pressure loading. The

treatment of the pressure–loaded surfaces used here is that first presented by Simo

et al. (1991). Specifically, we consider the case where the (Piola) surface tractions

t in (3.15) are produced by a pressure loading, which must remain normal to the

surface even as it (finitely) deforms and which is specified on current (not reference)

areas. As before, we denote by Ω the reference configuration of the body. The nota-

tion ∂pΩ ⊂ ∂Ω is used for that portion of the surface subjected to pressure loading.

Although (3.15) considers the virtual work expression as written in reference coor-

dinates, the character of pressure loading itself makes it more advantageous to write

the pressure virtual work in current coordinates. Writing the Cauchy traction as s,

and the spatial variations w :=∗ϕ ϕ−1, the virtual work of the pressure loads can

be written through a simple coordinate transformation as

∫

∂pΩ

t · ∗ϕ dΩ =

∫

ϕ(∂pΩ)

s ·w da. (3.32)


We then consider a loading of the form:

s = pn, (3.33)

where p is a spatially constant pressure in the current implementation and n is the

unit normal to the deformed surface. The pressure virtual work then takes the form

∫

∂pΩ

t · ∗ϕ dΩ =

∫

ϕ(∂pΩ)

pn ·w da. (3.34)

Parameterization of the pressure loading surface

A finite element implementation of (3.34) may be accomplished by subdividing the

integral into element subintegrals via

∫

ϕ(∂pΩ)

pn ·w da =∑

e

∫

ϕ(∂pΩe)

pn ·w da, (3.35)

where the summation is done over all elements e that have at least one pressure–

loaded surface. To discuss the implementation, therefore, it suffices to consider the

representation of just one of the element surface integrals indicated in (3.35), which

we will denote by Gep(∗ϕ,ϕ) via

Gep(∗ϕ,ϕ) :=

∫

ϕ(∂pΩe)

pn ·w da. (3.36)

We consider the parameterization of a single element surface through coordinates

(ξ1, ξ2), with each ranging between −1 and 1 in the usual isoparametric manner.

As before, ϕ denotes the mapping from the reference configuration to the current

configuration, and Γe and γe are used to indicate the mapping of the parent domain


N

n

Currentconfiguration

Parent domain

Referenceconfiguration γeΓe

Figure 3.2: Configurations for the pressure formulation.

into the reference and current configurations (as presented in Figure 3.2). One then

has:

Γe(ξ1, ξ2) = X (3.37)

and

γe = ϕ Γe(∂pΩ). (3.38)

The product of the unit (spatial) normal and an elemental spatial area is:

nda = (∂γe

∂ξ1× ∂γe

∂ξ2)dξ1dξ2

= (Dϕ Γe)Γe,1 × (Dϕ Γe)Γe

,2dξ1dξ2,

(3.39)

which is the term needed for (3.35).

Stiffness and force vector for the pressure loading

With the parameterization and the mappings presented in the previous section, we

can now deal with the pressure–loading term by performing all integrations in the


parent domain:

Gep(∗ϕ, ϕ) =

∫

¤p

(∂γe

∂ξ1× ∂γe

∂ξ2

)·wdξ1dξ2. (3.40)

A linearization of (3.40) is needed to expose the contribution to the stiffness resulting

from the pressure loading; accordingly, the directional derivative of Gep(∗ϕ,ϕ) in the

direction of ∆U is given by

DGep(∗ϕ,ϕ) ·∆U =

∫

¤pw · [(∆U Γe),1 × γe

,2 + γe,1 × (∆U Γe),2]]dξ1dξ2.

(3.41)

If we denote by Na the shape functions, the finite element implementation re-

quires the calculation of the residual force vector f e and the stiffness matrix ke as

follows:

f ep =

∫

ϕ(∂p(Ωe))

pNa(ξ1, ξ2)nida (3.42)

where p is the equation number corresponding to degree of freedom number i and

local node number a (note that a = 1, . . . , 4 for a bilinear pressure face). Defining

m =∂γ

∂ξ1× ∂γ

∂ξ2=

4∑a=1

4∑

b=1

Na,1Nb,2xa × xb, (3.43)

which can be evaluated analytically, one may substitute this into (3.42) to obtain

the residual:

f ep =

∫

ϕ(∂p(Ωe))

pNamidξ1dξ2. (3.44)

The stiffness matrix is likewise extracted from (3.41):

kepq =

∫

ϕ(∂p(Ωe))

pNa

∑c

(Nc,1Nb,2 −Nc,2Nb,1)αcijdξ1dξ2, (3.45)


where the three–by–three matrices αc are defined such that they satisfy

xc × h = αch for all vectors h (3.46)

(i.e., each skew symmetric αc has xc as its axial vector).

Note that since the pressure loading is not conservative, the contribution to the

global stiffness matrix obtained from this term is not symmetric (in fact, it happens

to be skew symmetric in this case). Exact integration of (3.44) and (3.45) is per-

formed in the parent domain so that no quadrature need be employed. Linearization

is exact and no geometric approximation is made.

Dirichlet boundary conditions

Dirichlet conditions must be placed on some portion of the tire for the solution to

be well-posed. Although the choice of boundary conditions made here is somewhat

simplistic, in this study that portion of the tire in contact with the rim is assumed

to be rigidly held. This condition is enforced by restraining the displacements of all

the nodes that are in contact with the rim.

3.4 Finite element formulation for frictional con-

tact

With the above theoretical framework in place, we now turn our attention to the

frictional conditions prevailing on the tire/roadway interface. This is included in

our formulation in (3.15) by including in∫

∂Ωt · ∗ϕ dΩ the following contact virtual


work:

Gc(∗ϕ,ϕ) =

∫

∂cΩ

t · ∗ϕ dΩ (3.47)

where t are surface tractions produced by a contact constraint and ∂cΩ ⊂ ∂Ω is

the subset of ∂Ω that is in contact with the ground, henceforth referred to as “the

contact patch.” As in any contact problem, the geometry of the contact patch and

the pressure distribution acting on it are not known a priori, which contributes

substantially to the difficulty of the problem.

The contact tractions can be resolved into their normal and tangential compo-

nents with respect to the ground surface, which converts (3.47) to:

Gc(∗ϕ,ϕ) =

∫

∂cΩ

tN · ∗ϕ dΩ +

∫

∂cΩ

tT · ∗ϕ dΩ (3.48)

where tN and tT represent the tractions in the normal and tangential directions,

respectively.

The contact in the normal direction is formulated in the classical manner in

terms of Kuhn–Tucker conditions relating a gap function and the contact pressure.

Defining the gap function as

g = (ϕ(X)−Xr) · n, (3.49)

with Xr representing an arbitrary point on the flat rigid contact surface and n

denoting the unit normal to the rigid surface, the Kuhn–Tucker conditions may be


stated as

g(X) ≥ 0 (3.50)

tN(X) · n ≤ 0 (3.51)

(tN(X) · n)g(X) = 0 (3.52)

for all X ∈ ∂cΩ. Implementation of these conditions can be readily accomplished

using either a penalty or an augmented Lagrangian approach; the interested reader

may consult Laursen (2002) for more details. In this work, we present results

corresponding primarily to penalty treatments, although both alternatives have

been implemented and tested.

Frictional effects are included using Coulomb’s law. Denoting the coefficient of

friction on the contact surface by µ, the friction law can be incorporated into the

model by appending to the Kuhn–Tucker conditions the following conditions, also

to be satisfied for all X ∈ ∂cΩ:

Φ := ‖tT‖ − µ‖tN‖ ≤ 0 (3.53)

tT = −αvT (3.54)

α ≥ 0 (3.55)

αΦ = 0, (3.56)

where Equation (3.53) requires the norm of the tangential traction to be bounded

by the product of the friction coefficient and the contact pressure. Equations (3.54)

and (3.55) force the tangential traction tT to oppose the tangential relative velocity

vT , while (3.56) only allows slip to occur when ‖tT‖ = µ‖tN‖.


3.4.1 Relative velocity measure for frictional sliding

The tangential velocity of a point X ∈ ∂cΩ relative to the roadway can be com-

puted as a consequence of the steady state kinematic assumptions. To begin, Equa-

tion (3.4) gives the material velocity

V = (I +∂U

∂X)X. (3.57)

Denoting by ω the axial vector of W = QQT , we may rewrite (3.5) as

X = ω ×X, (3.58)

which makes clear the interpretation of ω as the angular velocity vector associated

with the rotating body.

The term I+ ∂U∂X

in (3.57) is the deformation gradient associated with the moving

frame; i.e.,

F :=∂

∂X[x(X)] = I +

∂U

∂X. (3.59)

Substitution of (3.58) and (3.59) into (3.57) gives an exact expression for the ma-

terial velocity of a point currently at X

V = F [ω ×X] . (3.60)

Finally, with vg denoting the velocity of the ground relative to the moving frame,

an expression for the velocity of a particle relative to the roadway is given by

vrel = V − vg = F [ω ×X]− vg. (3.61)


Equation (3.61) is valid for any angular velocity vector ω. In the event that

this vector is aligned with the axis of the cylinder, we can further simplify this

expression by noting that

ω ×X = R0ωT , (3.62)

where R0 is the undeformed outer radius of the cylinder and T is the unit tangent

vector (in the plane normal to the symmetry axis) to the outer surface of the

undeformed tire. Without any numerical approximation, then, one may write the

relative velocity as

vrel = R0ωFT − vg. (3.63)

vg

ω

N(N − 1)(i + 1)i(i− 1)21

θi,i−1

Figure 3.3: Node numbering convention for approximation of contact velocities.

From the standpoint of implementation, there are many conceivable ways of ap-

proximating vrel. Here, this relative velocity is numerically approximated by using a

backward difference scheme along parallels of nodes in contact (see Figure 3.3 for an

illustration of the node indexing). It is important to note that this approximation is


only valid for structured meshes containing rows of nodes aligned in the circumfer-

ential direction about the tire; in the event that such a nodal arrangement does not

exist, an alternative approximation must be utilized. Explicitly, the approximation

of T i, based at node i on the periphery, can be given as

T i ≈ X i −X i−1

‖X i −X i−1‖ , (3.64)

which can be used to approximate FT in (3.63) as follows:

FT (X i) ≈ 1

‖X i −X i−1‖F (X i −X i−1) =1

‖X i −X i−1‖(xi − xi−1)

≈ 1

R0θi,i−1

(xi − xi−1).

(3.65)

We may then substitute this result into (3.63), and resolve the relative velocity into

the tangent plane to the roadway, to give the following numerical approximation to

the tangential relative velocity at node i:

vTi≈ [I − n⊗ n]vrel(X i) = [I − n⊗ n]

ω

θi,i−1

(xi − xi−1)− vg

. (3.66)

3.4.2 Algorithmic treatment of the frictional conditions

We consider here a penalty regularization of the frictional problem, introducing a

tangential penalty εT that can be different from the normal penalty εN . In dis-

cretized form, the virtual work can be summarized as:

∑

Mooney−Rivlin elements e

Ge(∗ϕ, ϕ)−

∑

pressure faces e

Gep(∗ϕ,ϕ)

+N∑

i=1

Aigi

εN

δgi =N∑

i=1

AitTi· δxi,

(3.67)


where the frictional traction tTiis evaluated at all i = 1, . . . , N on the contact patch

(and for all circumferential parallels of nodes) such that it satisfies Equations (3.50)

to (3.54). The Ai represent the surface Jacobians (tributary areas) associated with

each contacting node.

The traction force is set to zero at the first node in contact, and no specific

traction condition is enforced at the exit from contact. There is some arbitrariness

involved in imposing such a condition; namely, there is no reason why one cannot

allow for a non–zero frictional force for node 1. It can be argued that some entry

condition is needed to ensure that the problem is well–posed, but there is no ap-

parent justification for the choice of imposing this condition on the first node in

contact. An alternative choice was considered as part of our numerical testing (zero

traction force on all nodes outside the contact area only), and this did not affect

the behavior of the algorithm.

A classical approach to enforcing the Coulomb conditions would be to invoke a

pointwise return map strategy for frictional traction calculations. In this approach,

a “trial” tangential stress would be computed as

ttrialT = −vT

εT

(3.68)

and, if necessary, corrected (scaled) such that (3.53) is satisfied via

tT =

ttrialT if ‖ttrial

T ‖ ≤ µ‖tN‖

µ‖tN‖ ttrialT

‖ttrialT ‖ otherwise

(3.69)

In a traditional implementation (corresponding, for example, to the manner in which

elastoplasticity theories are typically implemented in nonlinear mechanics), evalua-


tions of (3.68) and (3.69) would take place for each contacting point at each global

equilibrium iteration. In the current context, however, numerical experimentation

proved this to be a poor algorithmic choice for the problems studied. A more robust

approach involved the use of a global “stick predictor.” This approach will be in-

troduced in section 3.5, where various alternative iterative techniques are analyzed.

3.4.3 Residual force vector and stiffness matrix

We present below the two cases included in our formulation, the perfectly adherent

contact and the sliding contact.

The adherent (no-slip) friction case

The contact force vector f ci and corresponding stiffness matrix kc

i are assembled

in terms of the degrees of freedom associated with nodes i and i − 1 as shown in

Figure 3.3, such that the following equalities hold:

δΦTi f c

i = fTi· δxi − Ai

gi

εN

δgi (3.70)

and

δΦTi kc

i∆Φi =∂

∂Φi

δΦT

i f ci

·∆Φi, (3.71)

where the contact “element” degree of freedom vectors δΦi and ∆Φi (containing

variations and incremental displacements of nodes i and i − 1, respectively) are

arranged as

δΦi =

δxi

δxi−1

, ∆Φi =

∆ui

∆ui−1

. (3.72)

With these definitions, the contact force assembled for each node i, and the


corresponding contribution to the stiffness, are given by

f ci =

f trialTi

+ Aigi

εNn

0

(3.73)

and

kci =

Aiω

εT θi,i−1

I3 − n⊗ n −I3 + n⊗ n

O3 O3

+

Ai

εN

n⊗ n O3

O3 O3

, (3.74)

where I3 is the 3× 3 identity matrix, O3 is the 3× 3 zero matrix, and n is the unit

normal to the road surface.

The Coulomb frictional slip case

The contact force vector and stiffness matrix in the case of slip contact are readily

expressed by first defining ntr, the normalized trial force vector:

ntr =f trial

Ti

||f trialTi

|| . (3.75)

Expressions for f ci and kc

i , analogous to those given above for the adherent case,

may be written as

f ci =

Aiµgi

εNntr + Ai

gi

εNn

0

(3.76)


and

kci =

Aiµgi

εN ||f trialTi

||ω

εT θi,i−1

I3 − n⊗ n −I3 + n⊗ n

O3 O3

+Ai

εN

n⊗ n O3

O3 O3

− Aiµ

εN

ntr ⊗ n O3

O3 O3

− Aiµgi

εN ||f trialTi

||

ntr ⊗ ntr − (ntr · n)ntr ⊗ n −ntr ⊗ ntr + (ntr · n)ntr ⊗ n

O3 O3

.

(3.77)

3.4.4 Existence and uniqueness of solution for contact prob-

lems

Proving the existence and uniqueness of the solution is usually impossible for prob-

lems involving friction. Even if the system one tries to analyze is sufficiently simple

to allow for an analytical approach, the analysis is usually conducted in a manner

dependent upon the range of the friction coefficient.

Andersson and Klarbring (2001) give a very simple counterexample to existence

and uniqueness for quasistatic problems. In their example, a system composed of a

single particle, having two degrees of freedom, is in contact with a rigid surface and is

in a current state characterized by tT = µtN (i.e. on the boundary between slip and

stick). From this configuration, there are only two possible continuous evolutions

the particle may have while still remaining in contact: (1) towards a stick state

(the particle is not allowed to move in the tangential direction) or (2) towards a

slip state. They show that nonuniqueness or nonexistence can be expected for some

characteristics of the entries in the stiffness matrix.


3.5 Alternative Iterative techniques

As it will be shown in Section 3.7, the algorithm presented in Section 3.4 fails to

converge for some of the cases in which we have interest (i.e., it does not converge

for the full useful range of values of the friction coefficients). Various alternative

iterative techniques were investigated and some are introduced in this section.

3.5.1 Augmented Lagrangians

Enforcing the constraints by use of penalization has some obvious advantages: the

method is simple and easy to implement, it does not introduce additional un-

knowns, and it can be physically interpreted. However, it also has disadvantages

since the constraints are exactly enforced only in the limit of infinite penalty values

(1/ε → ∞) and the system is affected by ill–conditioning when penalty values are

increased.

The augmented Lagrangian method as introduced for contact formulations by

Simo and Laursen (1992) and Simo and Laursen (1993), tries to bring the advantages

of the Lagrange multipliers technique into a penalty formulation. This approach

can deal with increased ill–conditioning of the governing equations and can, at the

same time, get closer to the “exact” enforcement of the constraints even with softer

penalties, thus correcting underpenalized solutions.

Assuming that fT is additively decomposed into its penalty and Lagrange mul-

tiplier parts, the Kuhn-Tucker conditions for Coulomb friction are reformulated as


follows:

tN = 〈λN +g

εN

〉 (3.78)

Φ = ‖tT‖ − µtN ≤ 0 (3.79)

1

εT

(vTn+1 − ξ∂

∂tT

Φn+1) = (tTn+1 − tTn −∆λT ) (3.80)

ξ ≥ 0 (3.81)

ξφ = 0 (3.82)

with < • > denoting the Macauley bracket, < x >= (x + |x|)/2.

Using the return mapping scheme, and a backward Euler scheme to integrate

the above, we get:

tNn+1 = 〈λNn+1 +1

εN

g(un+1)〉 (3.83)

tTn+1 = tTn + ∆λT +1

εT

(vT −∆ξ

ttrialTn+1

‖ttrialTn+1

‖

)(3.84)

where

ttrialTn+1

= tTn + ∆λT +1

εT

vT (3.85)

∆λT = λTn+1 − λTn (3.86)

and the consistency parameter, ∆ξ is given by:

∆ξ =

0 if φ ≤ 0

εT φ otherwise.

(3.87)


Table 3.1: Augmented Lagrangian algorithm for frictional contact.

1. Initialize:

set λ(0)N = 〈λN + g

εN〉 from the last time/load step

∆λ(0)T = 0

k = 0

2. Solve for u(k)n+1

3. Check for constraint satisfaction:

IF g(u(k)n+1) ≤ TOL AND ‖vTn+1‖ ≤ TOL

for all nodes in contact THENCONVERGE. EXIT

ELSEAUGMENT for all nodes in contact:

λ(k+1)N with equation (3.88)

∆λ(k+1)T with equation (3.89)

k ← k + 1GO TO 2

END IF

The update formulas for the normal and tangential multipliers are given by:

λ(k+1)N = 〈λ(k)

N +g

εN

〉 (3.88)

∆λ(k+1)T = ∆λ

(k)T +

1

εT

(v

(k)T −∆ξ(k)

ttrial,kTn+1

‖ttrial,kTn+1

‖

). (3.89)

Algorithmically, this is represented by Uzawa’s method, in which the Lagrange

multipliers are iteratively updated in an “outer” loop, with Newton-Raphson iter-

ation taking place in an “inner” loop where the multipliers are held constant. The

algorithm can be represented schematically as shown in Table 3.1.

Use of this approach seems to improve the behavior in the case of a perfectly

adherent contact, where the constraints can be enforced with softer penalties. How-

ever, no improvement is observed in the convergence of the slip calculation.


3.5.2 Global stick predictor

When using a global “stick predictor,” one first enforces the frictional constraints

using an adherent model (i.e., one corresponding to an infinite coefficient of friction).

Once global convergence is obtained in this model, the no-slip condition is relaxed on

the boundary (by using a finite coefficient of friction) and equilibrium iterations are

performed until convergence is obtained. The procedure is described in Table 3.2.

Table 3.2: Stick predictor algorithm for frictional contact.

1. Initialize:set µ ←∞ (i.e. a very large value, say 1010)set penalties

2. Perform global equilibrium solve for ϕstick, the adherent solution,using (3.68) with tT = ttrial

T to define frictional tractions3. Check for constraint satisfaction:

IF (g(ϕstick) ≤ TOL1 and vT (ϕstick) ≤ TOL2for all nodes in contact) THENpenalties adequate, GOTO 4

ELSEincrease penalties or perform augmentations of multipliers,GO TO 2

END IF4. Allow for sliding:

set µ to its physical value5. Perform global equilibrium solve for ϕslip

using ϕstick as initial Newton-Raphson iterate;Eqs. (3.68) and (3.69) with physical µ now govern sliding

Even though the literature on steady state rolling calculations does not lack

algorithms for frictional contact, some of these earlier formulations are subject to the

same sort of convergence difficulties we encountered in our work before implementing

the “global stick predictor” approach. Not only is this global stick predictor strategy

a very successful alternative iterative technique for the contact formulation that we


use but it also proves to greatly improve the performance of some other formulations.

This is shown in Section 3.7 with the aid of numerical examples, one of which

includes a direct comparison with earlier algorithms.

It is true that there is a legitimate question about existence and uniqueness

of solutions for this class of problems. However, when a solution exists, the stick

predictor approach has a definitive beneficial effect; its use is not restricted to the

specific contact algorithm presented in this thesis but can be extended to earlier

formulations as well.

3.6 Numerical examples. Verification

The formulation described in this chapter was implemented numerically, and its

performance on various problems representative of hyperelastic steady state rolling

and the comparison with other formulations from the literature are described in

this section. The implementation was done within the framework of F inite Element

Analysis P rogram (Taylor, 2003).

All convergence results are quoted in terms of an “energy norm” criterion, which

in the context of a Newton-Raphson iteration scheme is defined as :

relative energy norm =Ri ·∆di

R0 ·∆d0 . (3.90)

In this expression, di refers to the search direction calculated at Newton-Raphson

iteration level i, Ri refers to the residual that produces this search direction, and

the superscripts 0 in the denominator denote the respective quantities in the first

Newton-Raphson iteration in a load step. It is to be noted that all operators

mentioned in this chapter (the Mooney–Rivlin continuum, the pressure loading,


and the contact operators) are consistently linearized, such that when convergence

is reported, it is achieved quadratically and to machine precision.

3.6.1 Verification of the Mooney–Rivlin hyperelastic ele-

ment

Patch test

The patch test considered is an eight–element problem with 27 nodes (see Fig-

ure 3.4). On the exterior surfaces, a linear displacement field is applied and the

displacements of the center node (free degrees of freedom) are compared to the

analytically computed ones. The material properties are: µ = 12.0106 daN/mm2,

β = 0.4219, ρ = 0.104 · 10−9daN·s2/mm4, ω = 0, and κ = 100daN/mm2.

DISPLACEMENT 1Min = 0.00E+00Max = 1.00E-01

1.43E-02

2.86E-02

4.29E-02

5.71E-02

7.14E-02

8.57E-02

Current ViewMin = 0.00E+00X = 0.00E+00Y = 5.00E+00Z = 1.00E+01Max = 1.00E-01X = 1.10E+01Y = 0.00E+00Z = 1.00E+01

Time = 1.00E+01Time = 1.00E+01

Figure 3.4: Patch test for the Mooney–Rivlin element.

Although all constant strain patch tests were performed, the only example de-

picted here corresponds to uniaxial straining in the x−direction. For this problem,


convergence is obtained in only three iterations, which is consistent with an opti-

mal (quadratic) convergence rate (see Table 3.3). In all cases, the numerical results

confirm that the formulation successfully passes the patch test.

iteration 1 1.000000000000000E+00iteration 2 1.355999042672062E-14iteration 3 1.237642229382411E-30

Table 3.3: Convergence sequence for the patch test

Verification of the centrifugation term

In the patch tests, the centrifugal terms were ignored by setting ω = 0. Therefore,

we consider for these terms a test problem consisting in the spinning (with a large

angular velocity, ω = 10000 rad/s) of an axisymmetric body with 16 elements

distributed in eight sectors, and having 48 nodes for a total of 144 degrees of freedom

(Figure 3.5). The material properties are: β = 0.4219, µ = 12.0106daN/mm2,

ρ = 0.104 · 10−9daN·s2/mm4, and κ = 100daN/mm2.

S T R E S S 3Min = -1.47E-01Max = 2.30E+00

2.02E-01

5.52E-01

9.01E-01

1.25E+00

1.60E+00

1.95E+00

Current ViewMin = -1.47E-01X = 3.00E+01Y = 0.00E+00Z = 1.00E+01Max = 2.30E+00X = 7.07E+00Y = 7.07E+00Z =-1.00E+01

Time = 1.00E+00Time = 1.00E+00Time = 1.00E+00

Figure 3.5: Verification of the centrifu-gation term included due to the ALE ref-erence frame.

1 2 3 4 510

30

1025

1020

1015

1010

105

100

Convergence rate for the centrifugation problem

iterations

Rel

ativ

e en

ergy

nor

m

Figure 3.6: Convergence Rate: Energynorm versus Newton-Raphson iterationnumber for the centrifugation problem.


The convergence sequence as presented in Table 3.4 and Figure 3.6; again, it is

consistent with an optimal (i.e., quadratic) convergence rate.

iteration 1 1.000000000000000E+00iteration 2 2.487892003497592E-03iteration 3 1.444479776627728E-08iteration 4 1.001912537943468E-15iteration 5 1.163968140188026E-26

Table 3.4: Convergence sequence for the centrifugation problem

3.6.2 Verification of the pressure loading formulation

Patch test

To test the pressure formulation, a single eight–node element is considered, with a

pressure loading applied on one surface. The material properties are: β = 0.4219,

µ = 12.0106 daN/mm2, ρ = 0.104·10−9daN·s2/mm4, ω = 0, and κ = 100 daN/mm2.

The formulation passes the patch test successfully and a linear displacement field

is recovered, as seen in Figure 3.7. The numerical results obtained for the stress

and the displacement fields are verified to be the expected ones. The convergence

sequence for the pressure loading formulation is presented in Table 3.5.

iteration 1 1.000000000000000E+00iteration 2 1.271563180100964E-01iteration 3 4.606970997136403E-04iteration 4 2.342232039917357E-08iteration 5 6.474582803603215E-18

Table 3.5: Convergence sequence for the pressure loading.

A semilogarithmic plot of the convergence rate is shown in Figure 3.8. Quadratic

convergence is obtained as indicated.


DISPLACEMENT 3Min = -7.18E-01Max = 0.00E+00

-6.15E-01

-5.13E-01

-4.10E-01

-3.08E-01

-2.05E-01

-1.03E-01

DISPLACEMENT 3Min = -1.44E+00Max = 0.00E+00

-1.23E+00

-1.03E+00

-8.22E-01

-6.16E-01

-4.11E-01

-2.05E-01

DISPLACEMENT 3Min = -2.15E+00Max = 0.00E+00

-1.85E+00

-1.54E+00

-1.23E+00

-9.23E-01

-6.15E-01

-3.08E-01

Figure 3.7: Patch test for pressure loading.

1 2 3 4 510

18

1016

1014

1012

1010

108

106

104

102

100

Convergence rate for the pressure loading

iterations

Rel

ativ

e en

ergy

nor

m

Figure 3.8: Semilogarithmic plot of the relative energy norm, pressure loading.


It should also be noted that the pressurization terms have been validated on

nonplanar element surfaces as well. These tests included comparisons with calcula-

tions performed with Mathematica, which showed that, indeed, the pressure terms

were correctly implemented.

3.6.3 Critical points for rotating cylinders

Since identification of the bifurcation points along the loading path is of interest

in some of the numerical simulations, we tested our approach for their identifica-

tion with a simple spinning simulation for which data for qualitative comparison

exist in the literature. The problem considered is a small cylinder discretized with

a coarse mesh: eight meridians and one element wide (presented in Figure 3.9).

The cylinder’s outer radius is R = 30 cm, the width is w = 20 cm, and the in-

ner radius is r = 10 cm. The material properties are as follows: β = 0.4219,

µ = 12.0106 daN/mm2, ρ = 1.0409 · 10−10 da N·s2/mm4, and k = 100 daN/mm2.

8 meridians 24 meridians 48 meridians

Figure 3.9: Different meshes used for the pure spinning problem.

Since the interest in this case lies in identifying critical points in the pure

spinning problem, the angular velocity is considered a parameter and it is grad-


ually increased from zero (corresponding to an stable equilibrium state, the unde-

formed configuration) until some eigenvalues become negative (occurrence of critical

points). We identified the first critical point around the value ω = 1341.4 rad/s, at

which two of the stiffness matrix eigenvalues become negative. As discussed in Sec-

tion 2.3.4, we can conclude that this corresponds to a bifurcation point. Moreover,

in this case U = 2 and from Equation (2.8) we obtain that the number of solution

branches after this point is between two and four. The same analysis performed

on more refined meshes (24 and 48 meridians) identifies the first critical point at

an approximate value of ω ' 1310 rad/s and 1300 rad/s, respectively. A similar

eigenvalue analysis is performed for an idealized truck tire with the same outer ra-

dius, also for two different meshes: one with 16 meridians and a more refined one

with 28 meridians. For both meshes we found that the first critical point appears

at ωcr ' 1400 rad/s. As the mesh is refined, the differences in ωcr are decreasing.

Since the solution of the eigenvalue problem is very sensitive to small pertur-

bations, we can only accept the above values of ωcr as approximate. A spatial

convergence study is performed on the cylinder of outer radius 20 cm, with refine-

ment in circumferential, radial, and transverse directions. The value of ωcr does not

seem to depend on the discretization (mesh) once we have a sufficiently refined one.

However, the number of eigenvalues that change sign increases with the refinement

in the circumferential direction, suggesting an increase in the number of the solution

branches with the circumferential refinement. This observation is consistent with

the conclusion from Oden and Lin (1986): after the branching, the problem has

an infinite number of standing wave solutions but a discretized model can capture

only a finite number since the number of wave peaks is limited by the number of

circumferential elements.


It is obvious that these values are much higher than the angular velocity that

can correspond to a normal range of interest for tires. However, these examples

confirm that the approach adopted is suitable for the identification of critical points

and will be further used for problems with other parameters as well.

3.7 Numerical examples. Frictional sliding calcu-

lations and algorithmic performance

3.7.1 Performance of the iterative technique

The problem presented here corresponds to the rolling of a small disk with an outer

radius of 200 mm and an inner radius of 66 mm. The disk spins with an angular

velocity ω = 48.7 rad/s. The contact with the ground is enforced by gradually

bringing the roadway toward the tire in seven loading steps of ∆d = 5 mm each.

The rolling velocity is taken to be vg = 10150 mm/s. The finite element model

used to describe the problem has two elements through the thickness and is di-

vided into 30 circumferential sectors with the mesh refined in the contact region;

a typical deformed geometry is presented in Figure 3.10. The material proper-

ties for the Mooney-Rivlin material are: β = 0.4219, µ = 12.0106 daN/mm2,

κ = 100 daN/mm2, and ρ = 0.104·10−9 daN·s2/mm4. Penalties of εT = εN = 10−8

are used to enforce the contact conditions.

Considering first the adherent contact solution obtained by enforcing a rela-

tive displacement of 35 mm of the road relative to the hub of the tire, one may

consult Figures 3.11 and 3.12 to find contour plots of σxy and σyy (note that the

the y-direction is the vertical and the x-direction is the rolling direction). Global


1

2

3

Figure 3.10: Geometry of the test problem.

equilibrium convergence is obtained without difficulty in this case.

S T R E S S 4Min = -2.30E+00Max = 2.34E+00

-1.63E+00

-9.71E-01

-3.08E-01

3.55E-01

1.02E+00

1.68E+00

Current ViewMin = -2.30E+00X = 6.49E+01Y =-1.65E+02Z = 6.36E-14

Max = 2.34E+00X =-6.61E+01Y =-1.65E+02Z = 6.36E-14

Time = 3.50E+01Time = 3.50E+01

Figure 3.11: Contour plot of σxy, ad-herent contact case (daN/mm2).

S T R E S S 2Min = -1.18E+01Max = 1.08E+00

-9.93E+00

-8.09E+00

-6.26E+00

-4.42E+00

-2.59E+00

-7.51E-01

Current ViewMin = -1.18E+01X = 1.64E-13Y =-6.60E+01Z =-6.60E+01Max = 1.08E+00X = 4.04E-15Y = 6.60E+01Z =-6.60E+01

Time = 3.50E+01

Figure 3.12: Contour plot of σyy, ad-herent contact case (daN/mm2).

We now consider the recovery of sliding solutions for this geometry, testing the

sliding formulation for various values of µ. When sliding occurs on any portion

of the tire-roadway interface, convergence is generally difficult to obtain with the

pointwise return map algorithm. In such cases, it is typical for the energy norm

to drop quickly in the first few iterations and then present oscillations of reduced

amplitude around some small value (10−12), which is nevertheless considerably larger


than tolerances associated with machine precision (often approximately 10−20 or

smaller). A sequence of energy norms obtained using the traditional local return

map strategy (for a friction coefficient of µ = 0.3) is presented in Figure 3.13.

100

101

102

10-12

10-10

10-8

10-6

10-4

10-2

100

Convergence test for the slip contact

iterations

Rel

ativ

e en

ergy

nor

m

Figure 3.13: Typical stalling of Newton-Raphson convergence for slip contactwhen using the local return map strategy for the friction, as measured by evolutionof the energy norm.

Use of a stick predictor, on the other hand, as described in Section 3.5 and in

Table 3.2, produces convergence to machine precision in this problem for all friction

coefficients. Although additional iterations are needed to get the final solution

(since two full equilibrium iteration loops are involved, one for the adherent and

one for the slipping solution), an optimal convergence rate can be observed; usually

the increase in the number of iterations is not dramatic (see Table 3.6).

Stick–slip oscillations are not observed when using the stick predictor; the only

algorithmic oscillations that appear are manifested in the form of nodes going in

and out of contact. Their occurrence is dependent on the discretization. However,


we found that in most cases, a solution can be obtained at that level of the loading

if using a different mesh (either slightly rotated or with a different element size).

µ # slip iterations µ # slip iterations0.25 5 0.3 40.35 4 0.4 40.45 3 0.5 30.55 3 0.6 30.7 4 0.8 40.9 4 1.0 41.25 4 1.5 41.8 3 >2.0 0 (no slip)

Table 3.6: Iteration counts for the test problem in different frictional slip cases(iterations for the convergence of the stick predictor are not included); road surfacedisplacement = 35 mm.

For the particular case of µ = 0.3 and for a road displacement of 35 mm, contour

plots of the xy and yy traction fields are presented in Figures 3.14 and 3.15.

S T R E S S 4Min = -2.30E+00Max = 2.34E+00

-1.64E+00

-9.74E-01

-3.11E-01

3.53E-01

1.02E+00

1.68E+00

Current ViewMin = -2.30E+00X = 6.49E+01Y =-1.65E+02Z =-2.56E-13

Max = 2.34E+00X =-6.60E+01Y =-1.65E+02Z =-2.56E-13

Time = 3.50E+01Time = 3.50E+01

Figure 3.14: Contour plot: σxy, slidingcontact case with µ = 0.3 (daN/mm2).

S T R E S S 2Min = -1.18E+01Max = 1.09E+00

-9.93E+00

-8.09E+00

-6.26E+00

-4.42E+00

-2.58E+00

-7.50E-01

Current ViewMin = -1.18E+01X = 1.64E-13Y =-6.60E+01Z =-6.60E+01Max = 1.09E+00X = 4.04E-15Y = 6.60E+01Z =-6.60E+01

Time = 3.50E+01Time = 3.50E+01

Figure 3.15: Contour plot: σyy, slidingcontact case with µ = 0.3 (daN/mm2).

In Figures 3.16 and 3.17 we present the tangential tractions along parallels of

nodes on the contact patch for friction coefficients µ ∈ [0.3, 0.7]. As expected,

the difference between the solutions is observed only in the exit part of the contact

patch region where the slip tends to localize when it occurs. The plots show the


contact tractions only; any node outside the contact area has zero contact-traction

values associated with it.

-100 -80 -60 -40 -20 0 20 40 60 80 100-40

-20

0

20

40

60

80Tractions on the middle parallel

daN

/mm

2

x coordinate on the contact patch (mm)

adherentµ = 0.3µ = 0.4µ = 0.5µ = 0.6µ = 0.7

entry into contact exit from contact

Figure 3.16: Tangential traction alongthe middle parallel on the contact patch.

-100 -80 -60 -40 -20 0 20 40 60 80 100-120

-100

-80

-60

-40

-20

0

20

x coordinate on the contact patch (mm)

daN

/mm

2

Tractions on the outer parallel

adherentµ = 0.3µ = 0.4µ = 0.5µ = 0.6µ = 0.7

entry into contact

exit from contact

Figure 3.17: Tangential traction alongthe outer parallel on the contact patch.

A kick-out instability (i.e., a large localized tangential traction) is observed in

the exit from the contact area. This is always present in the solution and it is

accentuated by mesh refinement. Although an increase in convergence difficulties

can also be related to the mesh refinement, a direct connection between this kick-out

instability and the overall convergence behavior cannot be established. It was found

that if such difficulties exist for a mesh, a converged solution can still be obtained on

a mesh having the same refinement but slightly rotated with respect to the original

one. A minor increase in the required number of iterations to reach convergence

was noted for finer meshes, but the asymptotic quadratic convergence rate was not

affected. Moreover, if convergence difficulties appear, they are in general associated

with nodes oscillating in and out of contact not only at the exit from the contact

but also in the entry into the contact area, where no large localized tractions are

observed.


3.7.2 Comparison with other algorithms

The formulation presented previously in this chapter uses the same kinematic de-

scription adopted by many formulations for steady state rolling (Le Tallec and

Rahier, 1994). The contact constraints are enforced in the normal direction (via an

impenetrability condition imposed through a penalty approach on the gap function)

and in the tangential direction by relating the frictional stress to the relative ve-

locity vrel, and imposing the slip condition via Coulomb’s friction law. This is also

similar in many aspects to approaches presented in the literature; see for instance

the work by Oden and Lin (1986), Faria (1989), Faria et al. (1989), Bass (1987),

and Hu and Wriggers (2002).

From the standpoint of the actual implementation, the algorithm proposed in

this thesis differs first by the choice made in the approximation of vrel, for which

we use a backward difference scheme along parallels of nodes in contact, and by

the choice of enforcing the condition directly at the nodes instead of using Gauss

integration points on the contact surface. The most important difference, however,

lies in the use of a global stick predictor where this proves useful rather than the

traditional return map, which sometimes fails. We compared our results based on

this implementation with results presented in the above–mentioned papers and we

have seen a good qualitative agreement on similar problems. Results also exist

in the literature for cases where the same type of formulations are used for the

contact constraints, but the material is described by a viscoelastic constitutive law

(Le Tallec and Rahier, 1994). For these cases, the presence of dissipation in the

model is numerically helpful, and some of the computational difficulties that might

appear when using a hyperelastic model are not likely to be present in those cases.

Most results presented in the earlier literature were obtained on very coarse


meshes. Fewer convergence problems can be exposed on such problems. There is

also a tendency in those papers to use very small friction coefficients which make

the problem much easier; for instance, Bass (1987) uses in his frictional calculation

µ = 0.13, a value that is much lower than the range of friction coefficients that are

relevant for tire rolling (µ ∈ [0.3, 1.5]).

A very good qualitative and quantitative agreement was observed when repro-

ducing a numerical simulation presented by Hu and Wriggers (2002). Even though

the example presented here simulates a two–dimensional model, we found that the

degree of mesh refinement that was used is more appropriate than were the much

coarser meshes presented in the earlier literature. Enforcing the appropriate con-

straints to simulate the two–dimensional behavior for this problem and ignoring the

pressure loading made clear that this is one of the cases that present no particular

difficulty from a numerical point of view. For this combination of geometry, mate-

rial parameters, angular velocity, road displacement, and relative road velocity, a

converged solution can be obtained directly in a slip calculation using our imple-

mentation. And the convergence is reached in a very small number of iterations for

all road displacement levels up to the largest one shown there.

For comparison purposes, we have implemented in FEAP the formulation for

contact conditions introduced by Hu and Wriggers (2002). We present here a com-

parison of the results obtained with this implementation and with the one described

in this thesis for an elementary rolling calculation. In the problem considered, the

outer radius is 50.8 mm, the inner radius is 25.4 mm, the angular velocity is ω = 10

rad/s, and the total road displacement is 14 mm. The rolling velocity is taken to

be vg = 457.2 mm/s. The material properties are: β = 0.3, µ = 13.8951 daN/mm2,

and ρ = 0.3848 · 10−9 daN·s2/mm4. Penalties of εT = εN = 10−8 are used to enforce


the contact conditions. The two sets of results show a very good agreement (com-

parison in σyy is presented in Figures 3.18 and 3.19). The largest relative error in the

value of the nodal reactions in the contact area is 0.729% in the normal reactions,

0.304% in the tangential reactions, and 0.476% in the lateral ones. Similarly, the

nodal displacements compare very well; the largest error recorded in this problem

is 0.607% in the normal direction and 0.277% in the tangential one.

S T R E S S 2Min = -3.60E+01Max = 4.84E+00

-3.01E+01

-2.43E+01

-1.85E+01

-1.26E+01

-6.82E+00

-9.88E-01

Current ViewMin = -3.60E+01X =-7.23E-14Y =-2.54E+01Z =-1.00E+00Max = 4.84E+00X = 2.67E+01Y =-4.12E+01Z =-1.00E+00

Time = 7.00E+00Time = 7.00E+00

Figure 3.18: Contour plot of σyy

daN/mm2 with our algorithm.

S T R E S S 2Min = -3.60E+01Max = 4.84E+00

-3.01E+01

-2.43E+01

-1.85E+01

-1.26E+01

-6.81E+00

-9.87E-01

Current ViewMin = -3.60E+01X =-7.23E-14Y =-2.54E+01Z =-1.00E+00Max = 4.84E+00X = 2.67E+01Y =-4.12E+01Z = 0.00E+00

Time = 7.00E+00Time = 7.00E+00

Figure 3.19: Contour plot of σyy

daN/mm2 with Hu-Wriggers algorithm.

Using this contact formulation for the same test problem presented in Sec-

tion 3.8, the same type of numerical pathologies we first observed in our formulation

became apparent: the energy norm drops quickly in the first few iterations and then

settles at small values but never drops below tolerances that should be seen in a

machine precision converged solution (see Figure 3.20).

3.8 Numerical examples. Typical results on bench-

mark problems

Some other typical results on representative problems are presented here. The data

for the following problems was provided by Michelin America Research Corpora-


100

101

102

10-12

10-10

10-8

10-6

10-4

10-2

100

iterations

Re

lati

ve

en

erg

y n

orm

Figure 3.20: Newton–Raphson convergence behavior for slip calculations withHu-Wriggers algorithm.

tion. For the Mooney-Rivlin element: β = 0.4219, µ = 12.0106 daN/mm2,

ρ = 0.104 · 10−9 daN· s2/mm4 , ω = 48.7rad/s, and κ = 100 daN/mm2; for the

contact, a ground velocity of vg = 9990 mm/s.

Model problem Michelin-T322

The Michelin tire model T322 has 16 axisymmetric sections, each containing 21

nodes and 10 elements. The model has a total of 160 Mooney–Rivlin elements, 63

contact elements, and 96 pressure loaded surfaces. This is the first larger test prob-

lem we used and the first time we introduced all elements together (Mooney–Rivlin,

contact, pressure loading). In the case of the perfectly adherent contact, we present

in Table 3.7 and in Figure 3.21 the convergence results for a total displacement of


d = 20 mm.

iteration relative energy1 1.000000000000000E+002 1.813085843693777E-013 1.317496137792380E-014 6.040352316784298E-035 2.159093105130913E-046 1.290696407999472E-107 2.147872617814868E-158 1.756555493736664E-22

Table 3.7: Convergence results for T322; perfectly adherent contact

1 2 3 4 5 6 7 810

−25

10−20

10−15

10−10

10−5

100

Perfectly adherent contact

Rel

ativ

e en

ergy

nor

m

iterations

Figure 3.21: T322; d = 20 mm;Semilogarithmic plot for the conver-gence test; adherent contact.

1 2 3 4 5 6 710

-30

10-20

10-10

100

Convergence of the stick predictor

Re

lati

ve

en

erg

y n

orm

1 2 3 4 5 610

-30

10-20

10-10

100

Convergence of the slip contact step

Re

lati

ve

en

erg

y n

orm

iterations

Figure 3.22: T322; d = 7 mm; Semilog-arithmic plot for the convergence test;sliding contact.

The stick predictor approach performed satisfactorily here. The convergence

results for both the stick predictor step and the slip step are presented in Figure

3.22, for a total displacement equal to d = 7 mm (larger displacements enforced in

the stick predictor do not manifest slip nodes).

Although we observe some struggle in the convergence sequence in the first slip

iterations, we can see a quadratic behavior when approaching the solution in the


iterations stick predictor step slip contact step1 1.000000000000000E+00 7.145786129428396E-062 6.526842301882080E-02 4.955238988526497E-053 3.447147155331904E-02 1.932596766289169E-094 3.172171378147155E-03 7.297466035804401E-125 1.879619665309791E-07 5.668579139519221E-166 4.843107915332872E-14 3.475028030129545E-217 2.457085601064657E-22 -

Table 3.8: Stick predictor and slip step convergence sequence for problem T322.

slip contact step. Plots of the vertical component of the stress and of displacement

are presented in Figures 3.23 and 3.24.

S T R E S S 3Min = -3.31E+00Max = 2.24E-01

-2.81E+00

-2.30E+00

-1.80E+00

-1.29E+00

-7.85E-01

-2.80E-01

Current ViewMin = -3.31E+00X =-5.15E+00Y =-9.88E+01Z =-1.93E+02Max = 2.24E-01X =-8.66E+01Y =-1.00E+02Z = 5.00E+01

Time = 7.00E+00Time = 7.00E+00

Figure 3.23: T322; d = 20mm; Con-tour plot, 3-3 stress component.

DISPLACEMENT 3Min = -5.13E-01Max = 7.00E+00

5.60E-01

1.63E+00

2.71E+00

3.78E+00

4.85E+00

5.93E+00

Current ViewMin = -5.13E-01X = 1.27E+02Y =-9.87E+01Z =-1.54E+02Max = 7.00E+00X =-4.86E+00Y = 3.43E+01Z =-1.93E+02

Time = 7.00E+00Time = 7.00E+00

Figure 3.24: T322; d = 7mm; Contourplot, 3-3 displacement component.

Model problem Michelin-T310

The Michelin tire model T310 has 28 axisymmetric sections, each containing 165

nodes and 128 elements. The entire model has a total of 165 x 28 = 4620 nodes

(4620 x 3 = 13860 degrees of freedom) and 128 x 28 = 3584 Mooney-Rivlin elements,

with ω = 35.00 rad/s, κ = 100 daN/mm2, and four different sets of other material

properties:


1

2

3

Figure 3.25: Meshing for Michelin problem T310.

1. β = 1.1665 µ = 10.004 daN/mm2 ρ = 1.1900 · 10−10 daN·s2/mm4

2. β = 0.25618 µ = 27.348d aN/mm2 ρ = 2.0000 · 10−10 daN·s2/mm4

3. β = 0.43407 µ = 0.091 daN/mm2 ρ = 1.0900 · 10−10 daN·s2/mm4

4. β = 0.43023 µ = 0.172 daN/mm2 ρ = 1 · 10−13daN·s2/mm4.

The ground velocity is vg = −9999.90mm/s. If we use only half of the tire, the

problem reduces in size to 7140 degrees of freedom and 64 x 28 = 1792 Mooney–

Rivlin elements. To this we have to add 171 “pressure elements” and 392 “contact

elements.”

For small displacements we obtain some slip nodes. For larger displacements no

slip is observed. Figure 3.26 presents a plot of the stress component 3-3, for the

adherent contact case, for a total displacement of d = 6 mm.

For a displacement d = 2 mm, the convergence sequence, for both the stick

predictor step and the slip contact (in two cases, friction coefficients µ = 0.5 and

µ = 1.0) is presented in Table 3.10.


iter load step 1 load step 2 load step 3 load step 4 load step 51 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+002 9.182E-02 1.098E-05 1.922E-03 7.404E-03 9.324E-033 1.922E-11 1.585 2.304E-04 9.277E-05 3.082E-034 1.181E-13 6.921E-16 9.471E-08 7.007E-07 4.062E-055 9.250E-18 2.928E-26 1.250E-14 5.992E-13 4.116E-096 - - 5.120E-24 1.562E-20 3.0569E-17

Table 3.9: Perfectly adherent contact convergence sequence for T310.

S T R E S S 3Min = -7.57E-02Max = 6.99E-02

-5.49E-02

-3.41E-02

-1.33E-02

7.48E-03

2.83E-02

4.91E-02

Current ViewMin = -7.57E-02X =-7.18E+01Y = 8.28E+01Z =-2.08E+02Max = 6.99E-02X = 0.00E+00Y = 8.20E+01Z =-2.21E+02

Time = 6.00E+00Time = 6.00E+00

12

3

Figure 3.26: T310, 3-3 component of the stress, plot on the deformed configura-tion.

3.9 Numerical examples. Bifurcations of the non-

linear iterative map

As shown in Section 3.7, an alternative technique was found that greatly improves

the algorithmic behavior for the sliding rolling calculations. In some cases, an

intermediate nonconverged iterate, obtained from an adherent calculation, can be


iter stick predictor slip contact µ = 0.5 slip contact µ = 1.01 1.000000000000000E+00 3.015039733758444E-14 6.295935237781108E-142 1.037230630326389E-03 9.411214838838839E-14 1.968505091513040E-133 5.693676029971924E-03 1.192040428760630E-17 2.502612016929389E-174 4.021830711794489E-09 7.125977763960221E-23 5.735906443002834E-245 1.637084843403346E-10 - -6 1.326978801687815E-15 - -7 8.904231192836397E-24 - -

Table 3.10: Slip contact convergence sequence for T310.

successfully utilized as initial iterate (i.e., the “predictor”) in the slip analysis and

convergence is obtained. However, in many cases, the success of this technique

relies upon the recovery of an adherent solution that is then used as a “predictor.”

Through extensive simulations, we revealed some of the difficulties associated with

the recovery of such solutions. This section introduces numerical examples analyzing

the interaction between frictional formulations and bifurcations in the iterative map

used in the solution search. This interaction explains to a large extent the numerical

difficulties and helps in identifying the domain of robust algorithmic behavior. The

examples refer exclusively to adherent contact and no sliding is considered.

3.9.1 Example problems

Two example problems are used to demonstrate the performance of the iterative

map used in the solution of the frictional rolling problems. In the first, a small

rubber disk of outer radius 200 mm and inner radius 66 mm is used as a simplified

idealization for a tire (see Figures 3.27 to 3.30, which will subsequently be referred

to as simple disk models 1–4). As can be seen from the figure, these meshes allow

us to investigate the effect of mesh refinement in the footprint region. Starting with

the “basic” mesh, numbered 1, with ten nodes (equally spaced) in contact on each


parallel, two elements along the radius and two elements wide, the mesh was refined

by proportionally increasing the number of elements up to mesh number 4, which

has 40 nodes along one parallel of the contact patch, eight elements along the radius

and eight elements wide. The problem is solved using an angular velocity of ω =

48.7 rad/s, a rolling velocity of vg = 10170 mm/s, material properties β = 0.4219,

µ = 12.0106 daN/mm2, ρ = 0.104 · 10−9 daN ·s2/mm4, and κ = 100 daN/mm2.

Penalties of εN = εT = 10−8 were used in simulation of this problem, with adherent

contact assumed and an internal pressure set to p = 0.03 daN/mm2.

1

2

3

Figure 3.27: Simple disk; model 1.

1

2

3


Another problem studied, representing a highly idealized truck tire, is depicted

in Figure 3.31, with the left and right portions of the figure presenting again varying

degrees of refinement of this problem. In this simulation, the material properties

for the Mooney–Rivlin material are as follows: β = −0.39473, µ = 0.73 daN/mm2,

κ = 100 daN/mm2. The density is ρ = 0.104 · 10−9 daN · s2/mm4, the

angular velocity ω = 48.7 rad/s, and the ground velocity is prescribed to be

vg = − 9999 mm/s. The penalties for both the tangential and the normal con-


1

2

3


1

2

3


straint in this problem are taken as εN = εT = 10−7, and the internal pressure is

p = 0.03 daN/mm2.

(a) (b)

Figure 3.31: Idealized truck tire test problem. Discretizations with a) 16 and b)28 meridians.

An example result obtained for the simple disk problem is presented in Fig-

ures 3.32 and 3.33 for the most refined mesh (model 4) with d = 20 mm. In many


cases such as this, the algorithm presented provides a reliable and stable solution.

However, for some combinations of road displacement d, ground velocity vg, and

angular velocity ω, we encounter Newton–Raphson iterative sequences that are un-

able to reach the solution and settle into a cycle, periodically visiting a finite set of

configurations. Multiple converged solutions are obtained for some combinations of

parameters. All these difficulties are analyzed in the following examples.

1

2

3

S T R E S S 2Min = -1.09E+01Max = 9.78E+00

-7.97E+00

-5.02E+00

-2.06E+00

9.03E-01

3.86E+00

6.82E+00

Current ViewMin = -1.09E+01X = 6.90E+00Y =-6.56E+01Z = 6.60E+01Max = 9.78E+00X = 6.60E+01Y = 0.00E+00Z = 6.60E+01

Time = 2.00E+01Time = 2.00E+01

Figure 3.32: Contour plot of the 2-2stress [daN/mm2], simple disk, model 4.

S T R E S S 1Min = -7.62E+00Max = 1.10E+01

-4.96E+00

-2.31E+00

3.48E-01

3.00E+00

5.66E+00

8.32E+00

Current ViewMin = -7.62E+00X =-3.30E+01Y =-5.72E+01Z = 6.60E+01Max = 1.10E+01X = 3.30E+01Y =-5.72E+01Z = 6.60E+01

Time = 2.00E+01Time = 2.00E+01

Figure 3.33: Contour plot of the 1-1stress [daN/mm2], simple disk, model 4.

As a preview of the types of difficulties encountered, we present in Figures 3.34

and 3.35 the total reactions on the contact patch for model 4 of the simple disk

problem, in the case where a loading sequence of 300 load steps of ∆d = 0.1 mm is

imposed. For some displacements along this sequence, convergence is not reached.

This behavior is discussed in Section 3.9.3. As seen in Figure 3.34, the normal

reaction exhibits a smooth monotonic increase with the increase in displacement.

However, the tangential component appears to be discontinuous with plateaus and

jumps (as shown in Figure 3.35). Each jump separates states having a different

contact patch geometry (i.e., a different set of nodes in contact). Moreover, the plot

of the tangential component brings up another unusual behavior manifested in the

form of “negative” jumps. If the shape of the contact patch is examined before and


after such a jump, we can see that there are actually nodes appearing to go out of

contact at that point (even though the roadway has been forced further up into the

tire). This is usually not observed in contact problems when loading is applied in a

monotonic incremental manner. A possible explanation for such behavior is given

in Section 3.9.4.

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10x 10

4 Total normal reactions on the contact patch

displacement (mm)

Σ R

y (d

aN)

Figure 3.34: Total normal reaction oncontact patch; simple disk model 4.

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5x 10

4 Total tangential reactions on the contact patch

displacement (mm)

Σ R

x (d

aN)

Figure 3.35: Total tangential reactionon contact patch; simple disk model 4.

The idealized truck tire problem, using the coarser mesh depicted in the left of

Figure 3.31, allows us another opportunity to examine the pathologies occasionally

encountered with this formulation of the problem – in this case, multiple equilibria.

Many analyses were performed with different incremental loading paths, and two

of them resulted in two different solutions corresponding to the same loading level.

Results in the form of the contour plots of the tractions on the contact patch are

presented in Figures 3.36 and 3.37. In the first case (plots at the top), the total

load was applied in ten equal steps and in the other, the same load was applied in

one step.

The two solutions are very close to each other, with the difference coming only

from the two nodes that are clearly identified on the images. The solution depicted


Rolling direction

Figure 3.36: Horizontal tractions cor-responding to multiple equilibria for ide-alized truck tire.

Rolling direction

Figure 3.37: Vertical tractions corre-sponding to multiple equilibria for ide-alized truck tire.

in the top images is an equilibrium configuration with those two nodes in contact,

while the configuration corresponding to the solution from the lower images has the

two nodes out of contact.

3.9.2 Eigenvalue analysis results

Since the multiple solutions observed above suggest the occurrence of bifurcation

points along the loading path, the method presented in Section 2.3.4 is employed

to identify the bifurcation points on the loading path suggested by the presence of

these multiple solutions. The method relies on monitoring the spectral signature of

the stiffness matrix at equilibrium positions along the loading path.

We present and discuss results obtained for the idealized truck tire depicted in

Figure 3.31; subsequently, we will refer to the mesh on the left as test problem

M16, and the more refined mesh on the right as M28. In these two problems,

as well as others we have studied, the progression of the eigenvalue signature with


increasing road displacement d tended to follow a similar pattern. For loading levels

corresponding to small values of the road displacement d, the spectrum contains

only eigenvalues with positive real components and null or insignificant imaginary

components. This is precisely what one would expect, as for small loads the system

is very nearly a self-adjoint system. For intermediate values of d, one obtains a

spectral signature with a concentration of eigenvalues on or close to the real axis,

accompanied in this case by a set of eigenvalues that all have approximately the

same real part and large imaginary parts (see Figure 3.38 for a result typical of

models M16 and M28). The occurrence and general location of these imaginary

eigenvalues does not seem to depend strongly on the size of the elements for this

and the other problems studied.

100

102

104

106

108

1010

2

1.5

1

0.5

0

0.5

1

1.5

2x 10

6 Model M16, displacement d = 8 ( d / R = 0.01 )

Real component

Imag

inar

y co

mpo

nent

Figure 3.38: Typical spectral signature for models M16 and M28 at an interme-diate road displacement before the critical point is reached.

At some value of the displacement (denoted subsequently as the critical displace-

ment), the eigenvalues start crossing the complex axis, and eigenvalues with negative


real parts appear in the spectrum. The value of the critical displacement displays a

strong dependence on the discretization and decreases when refining the mesh. For

refined meshes, several critical points tend to be concentrated in a small interval

in which the stiffness matrix is obviously ill–conditioned and therefore converged

solutions are hard to obtain.

In models M16 and M28, the refined model M28 presents critical points earlier

than did M16. In both cases, the first critical point appears in the form of a

singular stiffness matrix with a single real eigenvalue crossing the zero threshold.

Above this value of the road displacement, the spectral signature in the right half

of the complex domain remains similar to signatures prior to the critical point as

presented in Figure 3.38. Figure 3.39 displays a post–critical spectral image for

model M28, with the top portion of the figure depicting the half of the complex

plane corresponding to positive real components, and the bottom portion depicting

a zoom of those eigenvalues that have crossed over the imaginary axis. This behavior

confirms the presence of Hopf points. The multiple equilibria presented for this

problem in Section 3.9.1 correspond to a value of the road displacement above the

first Hopf point.

3.9.3 Bifurcation of the iterative map. k–cycles

As mentioned earlier, for many of the problems studied using this formulation, we

were unable to obtain convergence in some steps but obtained instead stable pe-

riodic cycles under the iterative map. None of the states visited under the map

satisfy a machine–precision convergence tolerance but, in fact, the energies associ-

ated with many of these unsuccessful iterations are quite small. Furthermore, in

such situations, the iterations never diverge, the displacements remain bounded,


100

102

104

106

108

1010

1012

1

0.5

0

0.5

1x 10

7 Model M28, displacement d = 7 ( d / R = 0.035 )

-103

-102

-101

6000

4000

2000

0

2000

4000

6000

Real component

Imag

inar

y co

mpo

nent

Figure 3.39: Spectral signature for model M28 at a road displacement above thevalue of the first Hopf point. Top figure: right half of complex plane (i.e., eigenvalueswith positive real part); bottom figure, zoom on eigenvalues with negative real part.

and the Newton–Raphson iterations settle into a stable k–cycle.

We present below results from a case we encountered when analyzing the sim-

ple disk in an adherent calculation using the most refined mesh (model 4) with a

ground velocity vg = 10000 mm/s in the x−direction. In this example, the cycle

has a periodicity of 2; the map iterates indefinitely between two distinct configura-

tions, visiting one of them at every other iteration as shown in Figure 3.40. This

is most likely a situation where the two configurations are not in the basin of at-

traction of any stable branches of the solution of the discretized problem (if such a

stable solution even exists). The condition number of the stiffness matrix is normal

for both configurations (i.e., similar to condition numbers obtained for converged

solutions around this level of deformations). This behavior is very different from

what is encountered when ill–conditioning or singularity of the stiffness matrix are


contributory factors (in the vicinity of critical points, for instance).

0 5 10 15 20 25 30 350.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35x 10

3 Nonconvergent iterations; vg = 10000 mm/s; d = 19.2 mm

Iterations

Rel

ativ

e en

ergy

nor

m

Figure 3.40: Energy norm levels for a stable 2-cycle obtained during Newton -Raphson iterations at d = 19.2 mm, for the simple disk problem with model 4.

In examining the traction fields corresponding to the two states, we observe

no significant differences as far as the normal components are concerned. The

only noticeable difference appears in the tangential fields (see the contour plot of

tangential tractions presented in Figures 3.41 and 3.42). The two configurations

differ only slightly, with the discrepancy coming from one of them having two more

nodes in contact than does the other. It this case, these are the two leading nodes

on the most lateral parallels. Similar situations can be observed in some other cases

with trailing nodes oscillating in and out of contact during the iterations. In some

sense, it is the same qualitative difference that is observed in the case of the multiple

solutions.

This behavior is not unique to this problem and parameter set, and k–cycles

appeared often in our study. The example presented here involves the occurrence of


a 2–cycle; however, k–cycles (with k = 3, 4, 5) were also obtained in some other cases

for adherent contact. k–cycles with larger k’s seem to be characteristic for larger

load values and occur more often on the coarser meshes than on the refined ones. We

also encountered situations where, for a given d on a particular loading sequence,

a stable k–cycle appeared, while a different loading sequence produced a period

one solution (equilibrium) at the same load level. From these observations, we

can conclude that the the iterative map bifurcates since its behavior is qualitatively

dependent on the range of the road displacement.

Rolling direction

-40 -30 -20 -10 0 10 20

-60

-40

-20

0

20

40

60

x

z

0

1

2

3

4

5

6

7

8

9

Figure 3.41: Contour plot of tan-gential tractions (daN/mm2) for thefirst 2-periodic point corresponding toNewton–Raphson iterations at d = 19.2mm; simple disk problem, model 4.

-40 -30 -20 -10 0 10 20

-60

-40

-20

0

20

40

60

x

z

0

1

2

3

4

5

6

7

8

9

Figure 3.42: Contour plot of tangen-tial tractions (daN/mm2) for the sec-ond 2-periodic point corresponding toNewton–Raphson iterations at d = 19.2mm; simple disk problem, model 4.

To better describe this dependency, we looked at the outcome of the Newton–

Raphson iterations for d ∈ [0, 30] mm applied in a number of different road dis-

placement increments ∆d in the model 4 problem. These results may be seen in

Figure 3.43, where the different configurations of nonconverged steps are represented

by their relative energy norm and in the abscissa we have the variation of the pa-


rameter d. For small values of d, a stable fixed point (i.e., “period–1” solution) is

obtained that represents the equilibrium configuration (note that since only non-

converged states are shown, regions with a stable fixed point are indicated by no

data points being present at the d in question). As d is increased, the solution bifur-

cates (by either disappearing or becoming unstable), and a stable periodic solution

takes its place or coexists with it. In some other cases, nonperiodic (and probably

dense) stable orbits may appear. We can also see that a period–1 solution recovers

its stability along certain intervals of the parameter d. In this application, the first

period–k solution with k 6= 1 was identified at d = 8.8 mm. It is also worth noting

5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

101

d (mm)

rela

tiv

e e

ne

rgy

no

rm

Periodic configurations in nonconverged steps for different loading sequences

∆ d = 0.1

∆ d = 0.2

∆ d = 0.4

∆ d = 0.5

∆ d = 0.6

∆ d = 1.0

∆ d = 5.0

Figure 3.43: Nonconverged configurations along the loading path obtained fromthe Newton–Raphson iterative map and the energies associated with them: simpledisk problem, model 4.


that the road displacements at which the k−cycles occur tend to be the same for

a variety of road displacement increments. Similar analysis on less–refined meshes

guided us to a very interesting observation. Large relative energy norms (sometimes

on the order of 104) that appeared for some configurations in the periodic cycles

never led to a divergence situation and the periodic solution obtained under the

map kept its stability.

Finally, periodic orbits have been shown to exist in some other similar applica-

tions presented in the literature. For example, Narayanan and Sekar (1996) analyzed

the case of two cylinders (one rigid, one flexible) in rolling contact. The presence

of stable and unstable period–2, 3, and 4 solutions coexisting with stable or unsta-

ble period–1 (fixed point) solutions was proven for some ranges of the bifurcation

parameter.

3.9.4 Bifurcation of the solution of the discretized problem

As we have seen, the frictional rolling problem exhibits bifurcations in the Newton–

Raphson map used to locate equilibrium states, and when convergence is obtained,

one may often identify multiple solutions for the same discretized problem. To

further investigate the solution bifurcations that can occur along an equilibrium

path, we present a test performed on model 4 for the simple disk, using a ground

velocity of 10170 mm/s. The total loading of d = 30 mm was applied in four different

loading sequences (300 steps x 0.1 mm, 60 steps x 0.5 mm, 30 steps x 1 mm, and 15

steps x 2 mm); all sequences contained some steps whose solutions under the map

were stable k–cycles. The plots given in Figures 3.44 and 3.45 present the total

reactions on the contact patch corresponding to displacements d = 2i mm with

i = 1, .., 15.


0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10x 10

4 Total normal (y) reactions on the contact patch

d (mm)

Σ R

y∆ d = 2

∆ d = 1

∆ d = 0.5

∆ d = 0.1

Figure 3.44: Total normal reaction on the contact patch, simple disk problem,model 4. For steps where convergence was not obtained, no data point is shown.

Depending on the loading sequence, for equivalent road displacement we obtain

two or more solutions that each has a different set of nodes in contact. Solutions

are in general close to each other, and if we examine the reactions on the contact

surface, the differences in the normal components are insignificant (see Figure 3.44),

with the only noticeable difference appearing in the tangential component (shown

in Figure 3.45). It is this component that is the most sensitive to the discretization.

This difference is quite small and comes from the difference in size of the contact

area, whose boundary is slightly shifted. In this particular case, the largest difference

that can be observed was between two solutions obtained at d = 20 mm, one along


0 5 10 15 20 25 300.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4x 10

4 Total tangential (x) force on the contact patch

d (mm)

Σ R

x

∆ d = 2

∆ d = 1

∆ d = 0.5

∆ d = 0.1

Figure 3.45: Total tangential reaction on the contact patch, simple disk problem,model 4. For steps where convergence was not obtained, no data point is shown.

the loading sequence with ∆d = 2 mm and one with ∆d = 0.1 mm. The shapes of

the contact patches obtained for these two cases are presented in Figures 3.46 and

3.47. We will denote by “solution 1” the solution at d = 20 mm obtained during a

loading sequence with ∆ d = 2 mm and by “solution 2” the solution at d = 20 mm

obtained during a loading sequence with ∆d = 0.1 mm. The contour plot of the the

tangential tractions corresponding to these solutions are presented in Figures 3.48

and 3.49.

We may now recall the case introduced earlier where nodes appeared to go out of

contact along an incremental monotonic loading path. Having proved that multiple


−100 −80 −60 −40 −20 0 20−80

−60

−40

−20

0

20

40

60

80Nodes in contact

x

z

Figure 3.46: Contact patch for solution1, simple disk problem, model 4.

−100 −80 −60 −40 −20 0 20−80

−60

−40

−20

0

20

40

60

80Nodes in contact

x

z

Figure 3.47: Contact patch for solution2, simple disk problem, model 4.

Rolling direction

−80 −60 −40 −20 0 20 40

−60

−40

−20

0

20

40

60

x

z

0

2

4

6

8

10

12

14

Figure 3.48: Contour plot of the tan-gential tractions for solution 1, simpledisk, model 4.

−80 −60 −40 −20 0 20 40

−60

−40

−20

0

20

40

60

x

z

0

2

4

6

8

10

12

14

Figure 3.49: Contour plot of the tan-gential tractions for solution 2, simpledisk, model 4.

numerical solutions associated with a given discretized problem exist, we can now

suggest a justification for what first appeared to be inexplicable. If multiple numer-

ical solutions exist, these negative jumps may be manifestations of the numerical

solution jumping between different branches. To ensure that the algorithm stays on

one branch along the whole loading sequence, we probably have to run the problem


with a small displacement increment for each load step. This is particularly chal-

lenging since the different minima seem to be close to each other so the load step

has to be indeed very small (in our experience, often impractically so).

3.9.5 Mesh refinement study

0 50 100 150 200 250 3000

2

4

6

8

10

12

14x 10

4 Total normal force on the contact patch

load step

Σ R

y

mesh 1mesh 2mesh 3mesh 4

Figure 3.50: Total normal reactions obtained during simulations of 4simple disk model problems. At values of d where a solution is not shownfor a loading sequence, convergence was not obtained.

For a ground velocity of vg = 10170 mm/s, the evolution of the normal and

tangential reaction along the loading path are presented in Figures 3.50 and 3.51

for the four simple disk models introduced in Section 3.9.3 (see Figures 3.27–3.30).

While a monotonic variation in the normal reaction can be observed as the mesh


0 50 100 150 200 250 300−0.5

0

0.5

1

1.5

2

2.5x 10

4 Total tangential (x) force on the contact patch

load step

Σ R

x

mesh 1mesh 2mesh 3mesh 4

Figure 3.51: Total tangential reactions obtained during simulations of4 simple disk model problems. At values of d where a solution is notshown for a loading sequence, convergence was not obtained.

is refined, the tangential reactions exhibit the discontinuities emanating from the

frictional formulation and do not display a clear monotonic variation when refining

the discretization. However, in both the normal and tangential components, we can

observe that the difference between two consecutive refinements decreases with the

refinement.

3.9.6 Influence of the ground velocity

As might be expected, the ground velocity vg has a strong influence on the algo-

rithm’s ability to converge to an equilibrium solution. This is intuitively reasonable

since under the assumed condition of steady state rolling the angular velocity ω


and the ground velocity vg should not be entirely independent but rather related

through the rolling radius of the tire under steady state conditions (which of course

is also unknown). One might expect, therefore, that inauspicious choices for vg may

not even be physically consistent with the steady state assumption. Accordingly,

we study here the influence of the ground velocity on the ability of the algorithm to

recover solutions during loading paths characterized by the same incremental road

displacement ∆d. For the simple disk problem considered previously, a plot of the

percentage of nonconvergent load steps (for models 1 and 2) as a function of the

ground velocity is presented in Figure 3.52. The initial count of nonconverged steps

was done on an incremental sequence employing 300 load steps of ∆d = 0.1 mm.

1.02 1.025 1.03 1.035 1.04 1.045 1.050

5

10

15

20

25Convergence dependence on ground velocity

% n

onco

nver

ged

step

s

vg / v

0

mesh 1mesh 2

Figure 3.52: Influence of the ground velocity on the algorithmic behavior.

The coarsest mesh (model 1) presents a large interval of the ground velocity

(vg ∈[10050, 10200] mm/s) for which convergence was obtained easily. The repre-

sentation in the plot was normalized and has vg/v0 in the abscissa, where we denoted

by v0 the velocity of a point on the outer circumference of the cylinder assumed


rigid, v0 = ωR = 9740 mm/s. Refining the mesh was shown to have an unwanted

effect on the algorithmic behavior, as a higher percentage of nonconverged steps

along a loading sequence are observed in those cases. However, a “good interval”

for vg can still be observed on the plot for model 2. In both problems, most of

the nonconverged steps correspond to loading levels for which converged solutions

can be obtained either via another loading sequence or simply by applying the full

displacement in one step.

As might be expected, there are values of the ground velocity for which a steady

state solution is unlikely to exist. As stated above, it makes no sense to look for

such a solution in regimes of strong acceleration or braking. Indeed, when the

ground velocity is either increased or decreased significantly, reaching a period–1

solution is more and more unlikely. Even the coarser meshes display a large number

of nonconverged states in this instance. Eventually, the numerical behavior will

evolve toward divergence situations when very large or very small values of vg are

used.

3.10 Summary

In this chapter, a formulation for steady state frictional rolling calculation has

been presented. The frictional formulation exhibits some numerical (convergence)

difficulties when sliding is included, and an alternative technique, the use of a

“global stick predictor,” is suggested. This technique not only insures convergence

for a wider range of the friction coefficients within this formulation but it can also

be used with similar success to improve the convergence behavior of some other

algorithms for rolling friction. In general, either very small or very large coefficients

of friction do not pose particular problems. For small values of the coefficient of


friction, the problem is easy and convergence can be obtained directly, but this

is of little use to us; the coefficients characterizing the tire–road surface are not

usually in this range. Very large coefficients characterize a state where sliding never

happens. It is between these limits where the usefulness of the predictor approach

is demonstrated.

Furthermore, since in some cases an adherent solution must be used as initial

iterate for the corrector step, it is of interest to characterize the convergence prop-

erties of the numerical method utilized. Extensive testing on adherent simulations

pointed out other algorithmic difficulties associated with steady state rolling calcu-

lations; for some combinations of parameters, convergence cannot easily be obtained

under the adherent contact assumption. Exploring this issue brought up a strong

connection between the frictional formulation and bifurcations in the iterative map

used to compute the solution. This interaction has been investigated and the do-

main of robust algorithmic behavior identified.

Chapter 4

Postbuckling Analysis of SlenderBooms

4.1 Solar sailing. Structural configurations

Solar sails offer the prospect of an effective propulsion system for deep space ex-

ploration using the energy from photon fluxes. Advances in lightweight materials,

especially, have led to consideration of various geometries and configurations de-

signed to harness the sun photons. Square, circular, or heligyro configurations have

been proposed (see Figure 4.1).

(a) Square Sail (b) Circular Sail (c) Heligyro

Figure 4.1: Structural configurations for solar sails (images created by BenjaminDiedrich, courtesy www.solarsails.info)

In order to provide an appropriate surface to capture this solar effect, the sail

must be supported in its deployed (or operational) state in the same way that

120

CHAPTER 4. POSTBUCKLING ANALYSIS OF SLENDER BOOMS 121

a kite is supported by a relatively rigid framework. The key component of the

support structure for solar sails are the slender booms, which are able to carry the

axial and lateral loads to which they are subjected and can maintain the geometric

configuration of the sail. Given the need to minimize weight, these booms tend

to be somewhat slender, and hence buckling and vibration problems become an

important issue. Different structural configurations include the possibility of using

inflatable members and various composite materials. There are also different choices

available for the attachment of the sail membrane to the booms. Some of the

possible solutions and their advantages and disadvantages are discussed by Murphy

and Murphey (2002) and are shown in Figure 4.2. Essentially, the determinant

factor influencing the static and dynamic postbuckling behavior of these slender

structural elements is the way in which the load is transferred from the membrane

to the boom.

a) Four point suspension b) Five point suspension c) Separate quadrants d) Continuous connection e) Stripped architecture

Figure 4.2: Sail attachment solutions.

The work described in this chapter focuses on some aspects of the static and dy-

namic analysis of such structures; corresponding numerical examples are presented

in Section 4.5.1. Much of this work is based on finite element analysis considering

complicated effects associated with large deflections, geometric imperfections, and

transient dynamics. Since experimental testing of full–scale models that includes

the characteristics of the environment in which they will be used (no gravity and


practically no damping) is almost impossible, we need to achieve high confidence in

the numerical models we use.

H = P cr / 100

x

y

z

P

Figure 4.3: Isogrid configuration.

In this thesis, two different struc-

tural designs of the booms are analyzed.

The first design considered is, from the

numerical model point of view, just a

regular beam. The second is the iso-

grid configuration, practically a three–

dimensional truss–like structure. For a

detailed description of the geometry, see

AIAA 2003-4659 (Lichodziejewski et al.,

2003) and AIAA 2002-1297 (Lin et al.,

2002). In the case of the isogrid boom, high modulus fibers are oriented longitudi-

nally and designed to absorb the compressive loads, while the ones oriented laterally

absorb the inflation loads and stabilize the cross section. The fibers are impregnated

with a Sub Tg resin to rigidize the structure after deployment. There are different

isogrid configurations that have been considered for the solar sail booms. The mesh

for the particular configuration that was used in this study was generated according

to the geometry described in AIAA 2002-1297 (Lin et al., 2002) and is shown in

Figure 4.3. The baseline model considered is a cantilever beam that has 16 circum-

ferential bays (distributed uniformly on a circle of diameter 17.78 cm) and a helix

that wraps around; the individual bars have a diameter of 5 mm. The total length

corresponds to 100 full helices; i.e., L = 32.04 m, the structure thus having a very

high slenderness ratio.


4.2 Bifurcations and the concept of postbuckled

configurations

The main structural role of the booms is to create an appropriate support structure

that will maintain the correct deployed configuration, with the membrane in ten-

sion at all times. Other functional requirements follow from the required general

characteristics of the structure itself. The booms must be very light and have a

reduced launch volume; i.e., they must have the ability to be folded. Consequently,

the material must be able to withstand high strain rates and must have a good

shape memory function in order for the deployment sequence to be successful.

In a standard “structural engineering” approach, we frequently assume that

buckling represents structural failure. What is somehow unusual about the solar

sail booms is that in this case we rely on a totally different design concept. Not

only do we deal with buckled configurations that are accepted as operational but

sometimes we design the structure to buckle. Buckled booms have many advantages

in this case. First, they allow for configurations in which the sail is guaranteed to

be in tension, thus satisfying the geometric requirement. Then, if the booms are

allowed to buckle, they can be made slender and therefore lighter, which is favorable

not only for vehicle efficiency but also for the costs associated with launching the

structure into space. And last but not least, in the vicinity of buckling, a structure

has lower natural frequencies, thus favoring structural control.

In anticipation of such expected configurations, the buckling of booms for various

structural configurations is considered. The simplest structural model for the booms

one can consider is the cantilever beam. Studying the effect of the attachment to the

sail requires the analysis of various type of loads. Figure 4.4 introduces the three


structural systems and corresponding types of loads that are studied in this thesis:

a) direct loading with a force of constant orientation, b) follower force loading, and

c) indirect loading via a cable.

x

y

P

L

yL

yx

x

y

P

L

yL

yx

θ

x

y

s

L

P

A

B

C

b

a

a) Cantilever Beam b) Follower Load c) Beam-Cable System

Figure 4.4: Structural systems for solar sail booms.

4.2.1 Buckling load for a cantilever beam loaded directly

In the case of the Bernoulli–Euler cantilever beam loaded directly at the free end by

a force of constant orientation, the buckling load is (Timoshenko and Gere, 1961)

Pcr =π2EI

4L2, (4.1)


and the deformed configuration may be obtained along the loading path beyond

the bifurcation point by making use of Riks’ continuation method (as presented in

Section 2.3.2).

4.2.2 Buckling load for a cantilever beam under follower

load

In this case, the follower load is nonconservative and the loss of stability is dynamic.

A detailed analysis for nonconservative loads is presented in Section 4.4; for the

particular case shown in Figure 4.4b, the vertical component of the buckling load is

P vcr =

20.1EI

L2. (4.2)

As it will be shown later, postbuckled configurations in this case can only be deter-

mined using a dynamic analysis.

4.2.3 Buckling load for a beam–cable system

The previous models present two extreme cases, but let us now consider an inter-

mediate situation where the force changes direction without remaining tangent to

the free end of the beam but instead passes through a fixed point. Such a situation

is relevant for some solar sail configurations; it arises, for instance, in cases similar

to the one depicted in Figure 4.5. A simplified system that can capture this type of

loading is a cantilever beam loaded by a cable attached to the free end and passing

through a fixed point, as presented in Figure 4.4c.

Analytical solutions can be obtained in this case as well if some simplifications

are considered: the beam (of length L, modulus of elasticity E, minimum moment


Figure 4.5: Solar Sail Configuration.

of inertia I with respect to the plane of bending, and mass per unit length m) is

assumed inextensible, shear deformations are ignored, and the behavior is assumed

two–dimensional. Points on the beam have coordinates x(s, t) and y(s, t), where s

is the arc length and t is time. The rotation is denoted θ(s, t). The internal forces in

the beam parallel to the −x and −y axes on a positive face, respectively, are Pv(s, t)

and Ph(s, t). The beam is modeled as an elastica, with bending moment M(s, t)

proportional to the curvature. The bottom of the cable is attached at point C in

Figure 4.4c, which has coordinates (x, y) = (a, b). (In the numerical results, a will

be 0.0375L and b will be denoted the offset.) The cable remains straight and the

tension P (t) in the cable has vertical component Pv(L, t) and horizontal component


Ph(L, t) due to dynamic equilibrium at its upper end B. The cable is assumed to

act as a spring with stiffness k.

Based on the geometry, moment-curvature relation, and dynamic equilibrium,

and neglecting damping, the governing equations for the beam are

∂x∂s

= cos θ

∂y∂s

= sin θ

∂θ∂s

= MEI

∂M∂s

= Ph cos θ − Pv sin θ

∂Pv

∂s= −m∂2x

∂t2

∂Ph

∂s= −m∂2y

∂t2.

(4.3)

The boundary conditions are

x(0, t) = 0

y(0, t) = 0

θ(0, t) = 0

M(L, t) = 0

Pv(L,t)Ph(L,t)

= x(L,t)−ay(L,t)−b

,

(4.4)

with the last one representing the condition that the cable remains straight.

The subscript e is used to denote equilibrium values. In equilibrium, the in-

ternal force components Pve and Phe are constant along the beam. Starting from


Equations (4.3), the variables xe(s), ye(s), θe(s), and Me(s) satisfy the equations

dxe

ds= cos θe

dye

ds= sin θe

dθe

ds= Me

EI

dMe

ds= Phe cos θe − Pve sin θe.

(4.5)

The associated boundary conditions are

xe(0) = 0

ye(0) = 0

θe(0) = 0

Me(L) = 0

Pve

Phe= xe(L)−a

ye(L)−b.

(4.6)

This type of loading is conservative. The differential equation for the cross-

sectional rotation can be written as

EI∂2θe

∂s2− Phe cos θe + Pve sin θe = 0, (4.7)

with the boundary conditions θe(0) = 0 and ∂θe

∂s(L) = 0. The total energy corre-

sponding to this system is

U =1

2EI

∫ L

0

(∂θe

∂s)2ds + Phe

∫ L

0

sin θeds + Pve

∫ L

0

cos θeds, (4.8)

and its first variation can be determined as

δU =1

2EI

∫ L

0

2∂θe

∂s

∂δθe

∂sds + Phe

∫ L

0

cos θeδθeds− Pve

∫ L

0

sin θeδθeds. (4.9)


Using integration by parts, the previous equation can be written as

δU =

[EI

∂θe

∂sδθe

]L

0

− EI

∫ L

0

∂2θe

∂s2δθeds + Phe

∫ L

0

cos θeδθeds

− Pve

∫ L

0

sin θeδθeds

= EI∂θe

∂s(L)δθe(L)− EI

∂θe

∂s(0)δθe(0)

−∫ L

0

[EI

∂2θe

∂s2− Phe cos θe + Pve sin θe

]δUds,

(4.10)

where the first two terms are zero as a consequence of the boundary conditions, and

the last integral is zero since it follows from Equation (4.7) that the integrand is

null everywhere.

This boundary value problem can be solved numerically using a shooting method

(Burden and Faires, 1997). In such an approach, the Equations (4.5) are solved as

an initial value problem using the Runge–Kutta method with an assumed value of

Pve. The last condition in Equations (4.6) can be replaced by

Me(0) = Pveb− Phea (4.11)

which was obtained using moment equilibrium about point A in Figure 4.4c, or by

manipulation and integration in Equations (4.5) and (4.6). The parameter Pve is

then updated using Newton’s method until the condition Me(L) = 0 is satisfied

with sufficient accuracy.

Solutions of this simplified analytical system, obtained based on the above de-

scribed technique as well as experimental data (both provided by Holland, 2005),

are used as comparison for the finite element results. Since the loading is conser-

vative, it suffices to use a static large–deflection analysis. Given the nature of the


problem, numerical instabilities may, however, affect the calculation; therefore, the

postbuckling configurations are determined using Riks’ continuation method (as

described in Section 2.3.2) instead of the standard incremental load approach.

For all these structural configurations, for booms with simple beam cross–section

or for isogrid designs, and for various types of loads (both conservative and non-

conservative), numerical examples that calculate the buckling load and follow the

loading path up to highly deflected equilibria are introduced in Section 4.5.1.

4.3 Dynamic analysis of structures in postbuck-

led state

The prediction of the dynamic response of space structures in postbuckled and/or

large deflection configurations is an important aspect of an adequate structural de-

sign. Furthermore, for effective control, the frequencies of vibration about highly

deflected equilibria need to be known. For the structural designs that are analyzed

in this thesis, these frequencies usually have been determined by finite element

calculations. Wherever possible, we have compared the results with estimates ob-

tained on simplified calculations, and this section introduces some of the analytical

techniques used to this end.

4.3.1 Undamped free vibration analysis for beams in flexure

Consider the case of a beam with constant cross–section and distributed mass

(EI(x) = EI, and m(x) = m). The undamped free–vibration equation corre-


sponding to this system, including the effect of the axial force N , is

∂4v(x, t)

∂x4+

N

EI

∂2v(x, t)

∂x2+

m

EI

∂2v(x, t)

∂t2= 0. (4.12)

A solution can be obtained by separation of variables

v(x, t) = φ(x)A(t), (4.13)

which assumes a motion of characteristic shape φ(x) and time–dependent amplitude

A(t). Substituting this assumed motion into Equation (4.12), and adopting the (•)′

and ˙(•) notations to indicate derivatives with respect to x and t respectively, results

in

φiv(x)

φ(x)+

N

EI

φii(x)

φ(x)= − m

EI

A(t)

A(t)= k4, (4.14)

which then yields two ordinary differential equations

A(t) + ω2A(t) = 0 (4.15)

φiv(x) + α2φii − k4φ(x) = 0, (4.16)

where the notations ω2 = k4EIm

and α2 = NEI

were used. This system of independent

equations can be solved in the usual manner (Walter, 1998), and the general solution

is

φ(x) = F1 cos δx + F2 sin δx + F3 cosh λx + F4 sinh λx (4.17)

A(t) = A1 cos ωt + A2 sin ωt, (4.18)


where

δ =

√(k4 +

α4

4

)1/2

+α2

2, (4.19)

λ =

√(k4 +

α4

4

)1/2

− α2

2. (4.20)

The constants of integration are obtained by making use of problem–specific bound-

ary conditions. The harmonic characteristic of the free vibration is not affected by

the presence of the axial force as can be seen from the expression of the time–

dependent component of the solution. However, the axial force has an important

influence in both the frequencies and the mode shapes, and this influence is mani-

fested in the position–dependent component of the solution.

4.3.2 Undamped free vibration analysis for beams in axial

deformation

The free vibration equation of motion for an unloaded beam is

∂2u(x, t)

∂x2− m

EA

∂2u(x, t)

∂t2= 0, (4.21)

and we asume the solution

u(x, t) = φ(x)U(t). (4.22)

The equation can now be written as

φii(x)

φ(x)=

m

EA

U(t)

U(t)= −k2, (4.23)


and denoting by

ω2 = k2EA

m, (4.24)

it can be solved in the usual manner; the following general solution is obtained:

φ(x) = C1 cos kx + C2 sin kx (4.25)

U(t) = U1 cos ωt + U2 sin ωt, (4.26)

with constants determined by making use of specific boundary conditions.

4.3.3 Vibration frequencies for the cantilever beam

Bending modes of the unloaded beam

The natural frequencies for the bending modes of a cantilevered beam with N = 0

(i.e., λ = 0 and δ = 0) can be computed based on the general solution given in

Section 4.3.1 with boundary conditions

φ(0) = 0

φ′(0) = 0

M(L) = EIφ′′(L) = 0

V (L) = EIφ′′′(L) = 0.

(4.27)

where M denotes the bending moment and V the shear force. Utilizing the first two

equations from (4.27), the Equation (4.17) yields F3 = −F1 and F4 = −F2, while

using these, and writing the last two conditions from (4.27) in matrix form results


into

(cos kL + cosh kL) (sin kL + sinh kL)

(sin kL− sinh kL) (cos kL + cosh kL)

F1

F2

=

0

0

. (4.28)

For this system to have a nontrivial solution (i.e., F 21 + F 2

2 6= 0), the determinant

has to be null, which reduces to

cos kL = − 1

cosh kL(4.29)

that can be solved for values of kL. The three lowest natural frequencies thus

obtained are

ω1 = (1.875)2

√EI

mL4ω2 = (4.694)2

√EI

mL4ω3 = (7.855)2

√EI

mL4, (4.30)

and based on the approximate expression

(kL)n =π

2(2n− 1) (∀)n ≥ 4, (4.31)

the higher frequencies can be obtained with similar accuracy:

ωn = (kL)2n

√EI

mL4. (4.32)

The corresponding mode shapes are

φ(x) = F1

[cos kx− cosh kx− cos kL + cosh kL

sin kL + sinh kL(sin kx− sinh kx)

], (4.33)

(∀) kL solutions of Equation (4.29).


Axial modes of the unloaded beam

In a similar manner, one can obtain the natural frequencies corresponding to the

axial modes by making use of the boundary conditions

φ(0) = 0 (4.34)

N(L) = EAφ′(L) = 0, (4.35)

and the general solution given in Section 4.3.2. In this case, the natural frequencies

are

ωn =π

2(2n− 1)

√EA

mL2∀n ≥ 1, (4.36)

and the corresponding mode shapes are

φn(x) = C2 sin[π

2(2n− 1)

x

L

]. (4.37)

4.3.4 Free vibrations about highly deflected equilibria. Fre-

quencies for the beam–cable system

The previous sections illustrated standard analytical techniques to obtain the vi-

bration modes in some simple cases. There are, however, cases where closed form

expressions cannot be obtained. If complicated boundary conditions are involved,

if vibrations about highly deflected equilibria are sought, or if one wishes to include

other effects, simple analytical solutions do not exist and so numerical approaches

are used instead. Most of the numerical examples included in Section 4.5 are solved

by means of finite element calculations. However, when this was possible, results

obtained based on simplified analytical systems were used for comparisons in order


to produce a better understanding of the influence some of the common simplifying

assumptions may have on the results.

For instance, for the beam–cable system, in a manner similar to that presented

in Section 4.2.3, a dynamic analysis is performed on the same simplified analyti-

cal system. Small undamped vibrations about the equilibrium configurations are

considered, and the variables can be written

x(s, t) = xe(s) + xd(s) sin ωt

y(s, t) = ye(s) + yd(s) sin ωt

θ(s, t) = θe(s) + θd(s) sin ωt

M(s, t) = Me(s) + Md(s) sin ωt

Pv(s, t) = Pve + Pvd(s) sin ωt

Ph(s, t) = Phe + Phd(s) sin ωt,

(4.38)

where the subscript d denotes a dynamic amplitude, and ω is the circular frequency.

Equations (4.38) are substituted into Equations (4.3) and (4.4), and the resulting

system is linearized in the dynamic variables.

The governing equations along the beam are found to be

dxd

ds= −θd sin θe

dyd

ds= θd cos θe

dθd

ds= Md

EI

dMd

ds= Phd cos θe − Pvd sin θe − θd(Phe sin θe + Pve cos θe)

dPhd

ds= mω2xd

dPvd

ds= mω2yd.

(4.39)


The boundary conditions for the dynamic variables are

xd(0) = 0

yd(0) = 0

θd(0) = 0

Md(L) = 0,

(4.40)

plus the following conditions at s = L:

Pvd(ye − b) + Pveyd = Phd(xe − a) + Phexd (4.41)

PvePvd + PhePhd√P 2

ve + P 2he

=k[(xe − a)xd + (ye − b)yd]√

(xe − a)2 + (ye − b)2. (4.42)

Equation (4.41) is obtained from the last condition in Equations (4.6), while Equa-

tion (4.42) follows from the cable stiffness relation

√P 2

h + P 2v −

√P 2

he + P 2ve = k(

√(x− a)2 + (y − b)2 −

√(xe − a)2 + (ye − b)2).

(4.43)

A shooting method was applied by Holland et al. (2005), using Equations (4.39)–

(4.42), and also the equilibrium equations and their solutions since equilibrium

variables are involved in the dynamic boundary value problem. The quantities a,

b, L, EI, m, and k were specified. Since the vibration amplitude is arbitrary (but

small), the initial condition Md(0) could be specified, for example. The frequency ω

and the initial conditions Pvd(0) and Phd(0) were updated until Equations (4.41) and

(4.42) and the last condition in Equations (4.40) were satisfied. To obtain different

frequencies and modes, the initial guess for ω was chosen in different ranges. The

results obtained based on this approach are used in Section 4.5.2 for comparison.


4.4 Buckling under nonconservative forces

x

y

P

Ph

P

L

d T

yL

ξ + δξ

ξ

yx

v

Figure 4.6: Beck’s problem.

If the loading on a system is nonconserva-

tive, the loss of stability may not occur via a

static buckling (the system evolving toward

another equilibrium state) but by the sys-

tem going into an unbounded motion. The

buckling load in the case of a loading with

conservative forces can be evaluated stati-

cally, but for most nonconservative forces a

dynamic analysis is necessary. To be able

to analyze the case of a follower load, for

instance, dynamic effects must be consid-

ered, stability being essentially a dynamic

concept.

The problem of the cantilever beam un-

der a tip follower force was first dealt with in

a dynamic context by Max von Beck (1952).

He proved that buckling in this case has dy-

namic effects and that not including those

in an analysis will result in an erroneous conclusion. The vertical component of the

force at buckling is approximately eight times higher than the value corresponding

to buckling under a force keeping a constant orientation. In a more general context,

the problem of buckling under nonconservative loading was also studied by Bolotin

(1963).


4.4.1 Differential equation of the beam

Consider a cantilever beam with bending stiffness EI, mass density ρ, length L

and cross–sectional area A (all assumed constant over the length of the beam). It

is noted here that the bending analysis neglects transverse shear deformation and

rotary inertia. The cartesian coordinate x is along the undeformed elastic axis of

the beam, and we denote by y(ξ, t) the deflection at time t of the section at x = ξ

(see Figure 4.6).

The tip force can be resolved into its vertical and horizontal components (P v

and P h) acting at the section at x = L, and the elementary inertia force at a current

section can be calculated via

dT (ξ) = ρy(ξ, t)Adξ (4.44)

The bending moment at current section (x) produced by the action of the inertia

forces is

Mi(x) =

∫ L

x

ρy(ξ, t)A(ξ − x)dξ, (4.45)

and the differential equation of the beam can be written as

EI∂4y

∂x4+ P v ∂2y

∂x2+ ρAy = 0. (4.46)

The boundary conditions associated with this problem are

y(0, t) = 0,∂y

∂x|(0,t) = 0,

∂2y

∂x2|(l,t) = 0,

∂3y

∂x3|(l,t) = 0. (4.47)


4.4.2 Buckling load

Assuming the solution can be expressed as y(x, t) = A0eλxeiωt, and using the nota-

tions p = P v/EI and a = ρA/EI, we obtain the characteristic equation

λ4 + pλ2 − aω2 = 0 (4.48)

that has the solutions ±λ1 and ±iλ2, where

λ1 =

√√4aω2 + p2 − p

2(4.49)

λ2 =

√√4aω2 + p2 + p

2. (4.50)

The spatial component of the general solution of Equation (4.46) thus obtained

is

Y (x) = A cosh(λ1x) + B sinh(λ1x) + C cos(λ2x) + D sin(λ2x). (4.51)

Using the boundary conditions from Equations (4.47), and requiring existence of a

nontrivial solution, we obtain

det

1 0 1 0

0 λ1 0 λ2

λ21 cosh(λ1l) λ2

1 sinh(λ1l) −λ22 cos(λ2l) −λ2

2 sin(λ2l)

λ31 sinh(λ1l) λ3

1 cosh(λ1l λ32 sin(λ2l) −λ3

2 cos(λ2l)

= 0, (4.52)

which can be recast into

(p2 + 2aω2) + 2aω2 cosh(λ1l) cos(λ2l) + p√

aω2 sinh(λ1l) sin(λ2l) = 0, (4.53)


representing a curve in the p− ω plane whose local maxima are the eigenvalues of

the buckling problem. The smallest of these eigenvalues, i.e., the dynamic buckling

load is obtained (using Mathematica) from Equation (4.53) and is

P vcr =

20.1EI

L2. (4.54)

A comment should be made here on the indispensability of including the time

component in the solution. It can be seen that if the dynamic component would

have been ignored (i.e. ω = 0) from Equation (4.53), we would obtain p = 0,

the trivial solution only. Since this would appear to be the only solution and no

neighboring equilibria could be found, we would mistakenly have concluded that

the trivial solution never loses stability.

4.5 Numerical examples

4.5.1 Static and dynamic buckling of slender structures

Cantilever beam. Isogrid configuration

Various static and dynamic analyses are performed (using the finite element anal-

ysis software ABAQUS) on both the standard beam and the isogrid configuration.

In both cases, first-order, three-dimensional Timoshenko (shear-flexible) beam el-

ements (B31) are used for discretization. These elements are formulated for large

strains and large rotations, allow for transverse shear deformation, and are efficient

for thick as well as slender beams. The discretizations utilized for both configura-

tions are proved (via a spatial convergence study) to offer the desired accuracy.

As an example, in the case of the baseline model in the eigenvalue problem, a


buckling load of Pcr = 2.4736 kN is obtained. The same configuration (see Fig-

ure 4.3) is then loaded with a transverse load of 1% of the buckling load and a large

deformation static analysis is performed with the aid of Riks (1979) continuation

method. The algorithm performs very well, and the structure is loaded up to ap-

proximately twice the critical load. Figure 4.7 presents the load–deflection diagram

obtained from this analysis (u1, u2, and u3 are the deflections at the free end in

the x, y, and z directions), while the deformed configuration corresponding to the

maximum compressive load (5 kN) is shown in Figure 4.8. These deformations are

excessive, extending well beyond the expected range of design configurations for a

solar sail boom, and are included here mainly to assess the capabilities of the finite

element software.

Dynamic buckling analysis of cantilever beam under follower force

A dynamic analysis is performed with ABAQUS for Beck’s problem and shows an

oscillatory response after buckling (as seen in Figure 4.9). In Figure 4.10, a set

of screen snapshots at different times present the evolution of the deformed shape

during this analysis. A spatial convergence study performed in this case exposed

a surprising effect: the number of elements required in this analysis is much larger

than what proves to be more than sufficient in the case of the static analysis. All

subsequent results are obtained based on these very refined discretizations.

From the numerical point of view, one can can sometimes simplify the analysis

of instabilities by using either a static arclength algorithm or a numerically more

efficient approach, namely a static analysis with numerical damping included (to

avoid algorithmic instabilities). However, these methods do not perform well when

the instabilities are dynamic in nature. For instance, in the case of Beck’s problem,


-40 -30 -20 -10 0 10 20 300

1

2

3

4

5

6

deflection of a node in the symmetry plane (m)

P

(kN

) Riks Analysis Isogrid Boom, 100 Helices

u1

u2

u3

Figure 4.7: Load deflection diagram obtained with Riks’ method on baseline iso-grid model.

the Riks (arclength) algorithm is not able to follow the loading path beyond the

buckling. Including numerical damping in the model makes the analysis more stable,

but the artificial viscous forces included in the system are locally significant and

alter the results (see Figure 4.11). Even though the numerical damping is very small

(characterized by a coefficient of 10−6), the buckling load predicted using this type

of analysis is almost twice the exact value. We conclude that in cases like this, a

dynamical analysis is required.

Since the analysis failed soon after the onset of the oscillatory behavior, the


deformed configuration

reference configuration

Figure 4.8: Deformed configuration at P = 5 kN.

-2 -1 0 1 2 3 40

5

10

15

20

25

30

35

forc

e m

ag

nit

ud

e [

kN

]

displacements [m]

δ1

δ2

δ3

Figure 4.9: Dynamic analysis of Beck’s problem using ABAQUS.


t = 27.04 t = 28.40 t = 29.12 t = 29.81 t = 30.25 t = 30.68 t = 30.82 t = 30.84

Figure 4.10: Evolution of the deformation during a dynamic analysis withABAQUS.

-4 -3 -2 -1 0 1 2 3 455

55.5

56

56.5

57

57.5

58

58.5

59

59.5

60

forc

e m

ag

nit

ud

e [

kN

]

displacements [m]

δ1

δ2

δ3

Figure 4.11: Static analysis of Beck’s problem using ABAQUS; numerical dampingincluded as an attempt to control algorithmic instabilities.


inclusion of the numerical damping in the dynamic simulation was also tested but

did not seem to offer any advantages. Results were strongly dependent on the

amount of damping, and the analysis did not extend much above the failure limit

for the case with zero damping as seen in Figure 4.12. It appears indeed, that local

effects are too important in this case and inclusion of any artificial damping factors

is not beneficial.

-10 -8 -6 -4 -2 0 2 4 60

5

10

15

20

25

30

35

forc

e m

ag

nit

ud

e (

kN

)

transverse displacement (m)

no damping1 %5 %

Figure 4.12: Dynamic analysis of Beck’s problem using ABAQUS; Numericaldamping included.

Large deformation analysis of the beam–cable system. Comparison with

analytical and experimental data

For the finite element simulations of the beam–cable system, the beam is discretized

using first–order, shear–flexible (Timoshenko) three–dimensional beam elements

(B31). Recall that these elements are formulated for large strains and large ro-


tations, use linear interpolation, and allow for “transverse shear strain” (i.e., the

cross–section does not necessarily remain normal to the beam centerline). However,

while the axial strain can be arbitrarily large, due to assumptions in the formulation

only a “moderately large” torsional strain is modeled accurately. In ABAQUS, this

beam element is formulated to be efficient for thin beams (for which the Euler–

Bernoulli theory is accurate) as well as for thick beams.

The discretization of the beam has 1,000 such elements, which is sufficient for the

desired accuracy in both the static and the dynamic calculations (as confirmed by

a spatial convergence study). The material and geometrical properties utilized for

the beam are consistent with the values used in the shooting method and were mea-

sured experimentally where possible. The cable is assumed to deform only by axial

stretching and is modeled with a single two–node, linear interpolation truss element

(T3D2) having three degrees of freedom per node. Due to the intrinsic restrictions

of any finite element simulation, in order to apply a load that strictly follows the di-

rection of the cable, the material of the cable is defined to be temperature–sensitive

and the desired axial load is applied indirectly via a change in temperature.

If the cable passes through the base of the cantilevered beam and the tension

in the cable is increased, the beam remains straight until buckling occurs when P

reaches the critical value (Timoshenko and Gere, 1961)

Pcr =π2EI

L2. (4.55)

As the beam begins to buckle, the moment at the base (as well as at the tip) is

zero, so the effective length is L as for a pinned–pinned column. If the load were

to act vertically instead of passing through the base, the effective length would be

2L, and the critical load would be one–fourth the value in Equation (4.55). In both


cases the loading is conservative.

Figure 4.13: Equilibrium shapes for b/L =0.0167 and P/Pcr=0.00, 0.81, 1.01,1.08, 1.30, 1.66, and 2.2.

In all numerical examples corresponding to this structural system, a = 0.0375L

(this is the offset used in experiments also). Figure 4.13 depicts a sequence of

equilibrium configurations for the offset b/L = 0.0167 as P is increased. The beam

bends as soon as the cable tension is nonzero.

Figures 4.14(a) and 4.14(b) depict equilibrium paths for offsets b/L = 0.0167 and

0.075, respectively. The abscissa is the normalized horizontal deflection at the tip

of the beam. Experimental values are denoted by open squares, numerical solutions

from the elastica analysis are shown as continuous curves, and results from the finite

element analysis are given by the dashed curves. Good agreement is observed, with

a better match exhibited for the case with a larger offset depicted in Figure 4.14(b).

In Figure 4.15, additional results from the finite element analysis are drawn.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

2

2.5P

/ P

cr

ye(L) / L

ShootingFEAExperiment

(a) b/L = 0.0167

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

2

2.5

P /

Pcr

ye(L) / L


(b) b/L = 0.0750

Figure 4.14: Tension vs. horizontal tip deflection for two offsets.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

2

2.5

P /

Pcr

ye(L)/L

b/L = 0.0525b/L = 0.0394b/L = 0.0263b/L = 0.0131

Figure 4.15: Tension vs. horizontal tip deflection for four offsets.


The normalized offsets used are b/L = 0.0131, 0.0263, 0.0394, and 0.0525. The

maximum tip deflection decreases as the offset increases. In the finite element

analysis, the load is applied indirectly via a temperature variation. This is necessary

in order to make sure that the direction of the load (always following the direction

of the cable) is preserved. In some sense this is also helpful numerically: it has the

advantage of creating a path-following procedure that is neither force–controlled

nor displacement–controlled, but rather an ”indirect” arc-length approach which in

itself is more stable and appropriate for path–following in problems with bifurcations

and critical points along the path.

4.5.2 Postbuckling dynamic characteristics

Cantilever beam. Isogrid design and simple beam

The dynamic analysis confirms the expectation that isogrid beams with high slender-

ness ratios present the same dynamic behavior as the long beams. The lower modes

are bending modes as expected, and two of them (corresponding to the unloaded

beam) are shown in Figures 4.16 and 4.17. They appear in pairs corresponding to

the two principal directions of the cross–section.

The next step of the study analyzes the variation of the natural frequencies

with the axial load (which was increased from zero to approximately twice the

critical load). As can be seen in Figure 4.18, the lower bending modes present

the same behavior (the red lines correspond to modes obtained during an analysis

with a transversal load of H = Pcr/100, and the blue dashed lines correspond to

the geometrically perfect system under pure axial loading). Since the transverse

load represents a break in the system’s symmetry, it is not surprising that the pair

separates after buckling in that case. For the symmetric case, the modes remain


Figure 4.16: First bending mode of aslender isogrid, ω = 0.64 rad/s.

Figure 4.17: Third bending mode of aslender isogrid, ω = 11.23 rad/s.

-2 -1 0 1 2 3 4 5-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Axial Force (kN)

ω 2

Bending mode 1

-2 -1 0 1 2 3 4 58

10

12

14

16

18

20

Axial Force (kN)

ω 2

Bending mode 2

-2 -1 0 1 2 3 4 570

80

90

100

110

120

130

140

Axial Force (kN)

ω 2

Bending mode 3

-2 -1 0 1 2 3 4 5360

380

400

420

440

460

480

500

Axial Force (kN)

ω 2

Bending mode 4

Figure 4.18: Variation of the square of the natural frequencies with the axialloading.


identical in the two principal directions – the solution obtained in this case being

the unstable branch (i.e., the non-buckled configuration). Figure 4.19 presents the

expected decrease of the natural frequencies with the slenderness ratio of the beam

defined as

L

r=

L√I/A

, (4.56)

where L is the length of the beam, I and A are the cross–sectional inertia moment

and area, and r is the radius of gyration.

In all dynamic analyses that are performed, the occurrence of longitudinal and

circumferential modes is also observed. For large overall dimensions (i.e., very long

slender booms) these modes are not important (they represent higher modes). How-

ever, the sail attachments provide constraints and thus may considerably shorten

the effective lengths. In the case of the very small slenderness ratios (very stocky

beams), the behavior is considerably different. Both the buckled configuration (see

Figure 4.20) and the lowest vibration modes (Figure 4.21) have a significant cir-

cumferential component.

Figure 4.22 shows the variation of the natural frequencies with the distance

between the supports and the nature of these supports. We considered two cases

with free or restrained rotation at the supports. Since the model with intermediate

supports is motivated by the designs with intermediate sail attachments, the more

realistic one is probably the free–rotation version. The effect of the lateral supports

on all modes is studied and, as expected, longitudinal modes are not affected.

For long slender isogrid booms, the use of an equivalent beam model for the

analysis leads to an increased numerical efficiency and is capable of recovering most


0 200 400 600 800 1000 120010

-4

10-2

100

102

104

106

108

1010

First bending modes

ω

L/r

Figure 4.19: Variation of the square of the natural frequencies with the slendernessratio.

1

2

3

Figure 4.20: Buckling mode of a shortisogrid.

1

2

3

Figure 4.21: Vibration mode of a shortisogrid.

of the results obtained on the complex isogrid model. Both the evaluation of the

buckling load as well as the modal analysis return satisfactory results based on


Figure 4.22: Variation of the square of the natural frequencies with the distancebetween supports.

this simplified model. A comparison of the results is shown in Figure 4.23 for the

vibration mode pairs 1, 2, and 7.

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

axial load [kN]

ω 2

isogrid mode 1isogrid mode 2eq beam, mode 1eq beam mode 2

(a) First bending mode.

0 1 2 3 48

9

10

11

12

13

14

15

16

17

axial load [kN]

ω 2


(b) Second bending mode.

0 1 2 3 45200

5300

5400

5500

5600

5700

5800

axial load [kN]

ω 2


(c) Seventh bending mode.

Figure 4.23: Comparisons on various bending mode frequencies; analysis of isogridand equivalent beam.


A very good agreement is obtained for lower modes and relatively low axial

loading. Some differences are observed in the higher modes and in the postbuckled

state, but even in this case the relative error is small and the behavior is quite well

predicted by a very simple model that recovers the lowest (bending) modes with

good accuracy.

Another application that is of interest considers the case of long slender booms

with intermediate supports. There are several choices for the suspension of the sail

(Lichodziejewski et al., 2003). A stripped architecture or continuous connections

bring additional sail attachment points, and the numerical model has to take into

account the effect of these lateral constraints. The buckling load and the natural

frequencies corresponding to bending modes for this case increase with the increase

in the number of lateral supports (i.e., with the reduction of the effective length),

while the longitudinal modes remain, as expected, unchanged by any modification

in the lateral conditions.

Finite element analysis of the dynamic properties of the beam–cable

system. Comparison with experimental and analytical data

The experimental FFT data were collected by David Holland (Duke University, De-

partment of Mechanical Engineering and Materials Science) from the accelerometer

and laser vibrometer using 3200 lines between 0-100 Hz, for a ∆f of 0.03125 Hz.

For each load level, six averages were taken using 50% data overlap and Hanning

windowing on both the laser and accelerometer data. The natural frequencies were

then estimated from the peaks of the H3 Frequency Response (root–mean–squared

transfer function) using the laser as the signal and the accelerometer as the refer-

ence. An example of the frequency response data is shown in Figure 4.24. Similar


with the static calculations introduced earlier, the offsets are: a = 28.6 mm (which

is the value that is used for this parameter in all examples), and two b values,

12.7 mm and 57.1 mm. When normalized with respect to the length of the beam,

these offset values translate to approximately a/L = 0.0375, b/L = 0.0167, and

b/L = 0.0750, respectively.

Frequency R esponse H3(Laser,Accel) - C urrent (Magnitude)

W orking : t7 : Input : F FT Analyzer

0 10 20 30 40 50 60 70 80 90 100

-40

-30

-20

-10

0

10

20

30

40

[Hz]

[dB /1.00 (m/s)/(m/s)]

0 10 20 30 40 50 60 70 80 90 100

-40

-30

-20

-10

0

10

20

30

40

[Hz]

[dB

/1.0

0 (

m/s

)/(m

/s)]

Figure 4.24: H3 FRF Sample experimental data P/Pcr = 1.12, b/L = 0.0167.Picture courtesy of D. Holland, Duke University

The first four frequencies for small vibrations about the equilibrium configura-

tion are plotted in Figure 4.26 for the offset b/L = 0.0167 and a cable stiffness of

k = 11.67 kN/m. The frequency is normalized by the fundamental natural frequency

3.516L2

√EI/m of a cantilever with no cable and no applied load (k = 0 and P = 0).

Again, the continuous curves correspond to the elastica analysis, the dashed curves

to finite element results, and the open squares to experimental data. The data in


Hz

0.0 10 20 30 40 50 60 70 80 90 100

Lo

g M

ag

nitu

de

(m

/s/m

/s)

10E-3

0.1

1

10

Figure 4.25: Overlaid FRF’s for modal analysis (P/Pcr = 0.505, b/L = 0.0750).Picture courtesy of D. Holland, Duke University

Figure 4.24 correspond to experimental points in Figure 4.26 at P/Pcr = 1.12.

The results for the fundamental frequency ω1 are depicted in Figure 4.27 with

a magnified frequency axis. As the tension in the cable is increased past P = Pcr,

the fundamental frequency tends to decrease and then increase, while the other

frequencies in Figure 4.26 tend to do the opposite.

Similar results for offset b/L = 0.075 are presented in Figures 4.28 and 4.29. For

this larger offset, the frequencies shown are higher for low values of tension and then

become comparable to those for the lower offset. Again, when P is near Pcr the

fundamental frequency exhibits a minimum and each of the next three frequencies

exhibits a maximum. The experimental points corresponding to a loading value of

P/Pcr = 0.505 in Figure 4.28 are obtained from data in Figure 4.25.

For the offset b/L = 0.075, vibration modes are sketched in Figures 4.30 and


0 10 20 30 40 50 600

0.5

1

1.5

P /

Pcr

ω / ω1,P=k=0


Figure 4.26: Lowest four frequencies for b/L = 0.0167.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

P /

Pcr

ω1 / ω

1,P=k=0


Figure 4.27: Fundamental frequency for b/L = 0.0167.


0 10 20 30 40 50 600

0.5

1

1.5

P /

Pcr

ω / ω1,P=k=0


Figure 4.28: Lowest four frequencies for b/L = 0.0750.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

P /

Pcr

ω1 / ω

1,P=k=0


Figure 4.29: Fundamental frequency for b/L = 0.0750.


(a) (b) (c) (d)

Figure 4.30: First four vibration modes for b/L=0.0750 and P/Pcr = 0.505.

a) b) c)

Figure 4.31: Fourth vibration mode for b/L=0.0750 and P/Pcr = 0.505 for (a)FEA, (b) shooting, and (c) experiments.

4.31, with the equilibrium shape of the beam also shown. Figure 4.30 shows the

first four modes obtained from the finite element analysis. Except for the second

mode, the tip of the beam sways considerably during vibration. A comparison of

the fourth mode obtained from (a) finite element analysis, (b) elastica analysis, and

(c) experiments is seen in Figure 4.31. The correlation between the results is good.

Finally, a full three–dimensional finite element analysis is conducted. The bend-

ing stiffness of the beam is 30 times greater in the strong direction than in the


0 100 200 300 400 500 600 700 8000

0.5

1

1.5

2

2.5

P/

Pcr

ω [rad/s]

out-of-plane frequencies

in-plane frequencies

Figure 4.32: Frequencies for b/L = 0.0167 from 3–D finite element analysis.

weak direction. However, the out–of–plane modes appear quite early in the mode

sequence, especially for larger axial loads. This can be explained by the additional

constraint imposed by the cable in the plane in which most of the deformation oc-

curs, a constraint that increases the structural stiffness in that plane. Figure 4.32

presents results from the three-dimensional analysis for offset b/L = 0.0167. The

larger dots are associated with out–of–plane modes. As the cable tension is increased

from zero, the corresponding frequencies tend to decrease, and the second and third

of these sets of frequencies intersect curves associated with in–plane modes. At high

loads, frequencies for in–plane and out–of–plane vibration modes alternate.


4.6 Summary

In this chapter, several structural systems (geometry and loading) appropriate for

the modeling of solar sail booms are investigated. Finite element analysis techniques

for the postbuckling analysis are explored and their performance for various cases is

assessed. Since buckled configurations are likely to be frequent for these structures,

particular attention is given to path–following techniques capable of following the

loading path beyond bifurcation events and up to largely deflected configurations.

Since structural control for these structures is also an important aspect, natural

frequencies and mode shapes for vibration about highly deflected equilibria are

extracted. This analysis exposed the high sensitivity of the structural behavior to

slight changes in the geometry and loading, confirming these structures to be at

risk for evolving toward static or dynamic unstable behavior. Simplified models for

complex designs are also considered, and their limitations and domain of validity

are assessed. Wherever possible, comparison with experimental data or results of

simplified analytical approaches is included.

Chapter 5

Conclusions and Future Work

This thesis analyzes aspects related to nonlinear finite element formulations for

structures undergoing large deformations, and numerical techniques for the solution

of the associated system of nonlinear equations. Two applications are considered,

the steady state frictional rolling of tires and the postbuckling analysis of slender

structural elements.

5.1 Main contributions

5.1.1 Algorithmic Stabilization of Frictional Steady State

Rolling Calculations

It is seen that the seemingly easy problem of rolling tires in frictional contact with

the ground possesses some challenging features that make the corresponding nu-

merical formulation troublesome at best.

A finite element framework for the three–dimensional analysis of steady state

rolling of inflated tires was developed. The main features of the formulation include:

use of the Mooney–Rivlin material constitutive law, modeling of the pressure load-

ing and inclusion of frictional effects with sliding capabilities. Sliding conditions

are enforced via a stick predictor approach in cases where a direct slip calculation

163

CHAPTER 5. CONCLUSIONS AND FUTURE WORK 164

fails to reach convergence to machine precision. Earlier approaches generally use a

traditional return map. In contrast, the global predictor allows for robust sliding

calculations with optimal convergence rates in most cases (as long as an adherent

solution exists). In some simulations a fully converged stick predictor step is not

absolutely necessary, and a state characterized by a reasonably small relative energy

norm is acceptable as an initial iterate for determination of a slip solution. Some

problems (for instance the ones corresponding to small coefficients of friction) are

“easy” enough and a direct slip calculation can return the solution. However, we

found that more realistic values of the coefficient of friction make the problem fall

into the category that requires stabilization for convergence. The global stick pre-

dictor provides this stabilization with no significant increase in the numerical costs

while also facilitating the robust use of a consistent Newton–Raphson linearization

which ensures an optimal convergence rate. In conclusion, the use of the global stick

predictor extends the set of problems that can be solved using Coulombic friction

over a broader range of friction coefficients. It was found also that its use is not

restricted to, nor only required by the particular frictional formulation that we have

implemented. It can be applied successfully to other formulations, and extends their

domain of applicability as well (as shown in the direct comparison of algorithmic

performances presented in Section 3.7).

5.1.2 Comprehensive analysis of the interaction between bi-

furcations and the frictional finite element formula-

tion for steady state rolling

Since many frictional formulations for steady state rolling seem to be subject to

some numerical pathologies, the interaction of these formulations with the algorith-


mic behavior of the iterative method used for numerical solution of the nonlinear

system was studied. The finite element formulation used in this study presupposes

a steady state rolling condition; it is of no surprise indeed that there exists a very

specific range of the ground velocity for which this algorithm has a robust behavior.

Numerical difficulties have been seen to affect the algorithmic behavior once we try

to solve for a steady state solution in ranges of the rolling velocity that correspond

to braking or accelerating. An interesting fact associated with this observation is

that occurrence of periodic stable solutions associated with the nonlinear iterative

Newton–Raphson map is observed. These solutions often coexist with period–one

solutions which seem to have limited basins of attraction, therefore requiring a

good initial iterate in order to be recovered by the root finding method. Although

not unusual for Newton–Raphson maps, the bifurcations seem to be particularly

important in these applications, perhaps much more than in other cases.

Moreover, multiple numerical solutions are shown to exist. The local minima

for these solutions are close to each other, thus making difficult to follow one of the

solutions along an incremental loading path. Jumps from one solution branch to

other seem to be quite frequent.

Solutions of engineering utility can often be recovered by using the sort of steady

state frictional description described in this work, but there is significant evidence

that stable, well–posed solutions in the steady state framework can be elusive for

many parameter combinations. Although one can argue physically that certain

parameter sets may be inconsistent with steady state solutions (particularly with

respect to ω and vg), it remains unclear whether the difficulties identified in this

study are associated with the physical problem itself, the numerical discretization,

or both.


5.1.3 Analysis techniques for slender structures in post-

buckled or other large deformation configurations

The analysis of a system beyond a bifurcation event is a very challenging problem in

structural mechanics and unlike other cases where buckling is not acceptable (being

considered a structural failure), in the case of the solar sail booms it is actually

desirable providing we have a supercritical bifurcation (i.e., postbuckled stiffness

is present). Here, calculating the buckling/limit load is no longer sufficient; since

we expect postbuckled configurations, the analysis has to be able also to follow

the structural behavior up to and beyond the bifurcation. From the computational

perspective, the choice of the appropriate path–following technique is essential and

in the case of the solar sail booms, the very large slenderness ratios make the

problem even more difficult. Several configurations and types of loading relevant

for the boom modeling have been identified and studied, and it has been shown

that these very slender structures are incredibly sensitive to slight changes in the

loading or the geometry. Moreover, due to the nature of the application, there

exist very strict constraints to be satisfied by these structures; in particular, they

have to maintain the correct geometry at all times. Structural control is therefore

important and to this end, the identification of the dynamic properties of the system

(the natural frequencies of vibration) is necessary.

Since large geometric nonlinearities and bifurcation phenomena are expected,

most of the effort related to this application was dedicated to the analysis of the

structural behavior in the domain of the large deformations. The performance of

various path–following techniques was analyzed, and stabilization techniques have

been explored. Given the large size of detailed finite element models, the use of

simplified models was examined, and their domain of validity and limitations were


identified. Natural frequencies and corresponding vibration modes at high levels

of loading were computed for all structural configurations and types of loading

analyzed.

5.2 Related problems

Although dedicated to a specific problem, conclusions and techniques developed

here may be useful for some other applications too. The modeling of the interfacial

behavior related to frictional contact is important not only in the case of the prob-

lem presented in Chapter 3. It is also crucial in calculations of the wear of railroad

tracks, in the choice of materials and treatment of surfaces to reduce the friction

(ice–skating) or to maximize it (tire–road for braking). The correct identification

of the adherent–sliding limit is also necessary for determining safe speed limit for

given road surfaces/conditions (icing). Sliding phenomena are significant in other

mechanical systems or civil engineering structures: simply supported bridges may

have sliding supports, protective films may be removed by a sliding action thus ini-

tiating corrosive wear. Stability problems including contact constraints also appear

some other applications, for instance in sheet metal forming or in the drilling of deep

holes. And last but not least, the tangential slip conditions are for some frictional

formulations equivalent to an elasto-plastic split of the operator. Therefore, it is

to be expected that some numerical difficulties in computational plasticity may be

similar in nature to those observed here.

Techniques developed for the postbuckling analysis of booms are not restricted

to space applications. They are also valid for the analysis of other structures that

are very flexible from design (slender bridges, antennas, flexible infrastructures), or

for any existing structure that suffered significant stiffness degradation due to cyclic


loading, thermal cycles or loading that induced irreversible deformations.

5.3 Future directions

Numerical difficulties associated with the inclusion of frictional effects into steady

state rolling calculations were observed not only in the context of the formulation

proposed in this thesis but have also plagued earlier formulations. Stabilization

techniques are most of the times necessary, and some alternatives were investigated

in this thesis. One that was proposed in this work proved to be quite effective,

others might be developed and brought to high level of performance or effectiveness.

However, since the steady state assumption is made, it is to be expected that such a

formulation can only be appropriate for some combination of the parameters (rolling

and angular velocity are most important here) where this assumption is accurate.

This limitation, the observed numerical difficulties, and the legitimate question

whether or not the steady state frictional contact problem is well–posed, suggest

that, even though computationally more expensive, a better approach would be to

consider the full dynamic setup for this type of analysis.

Many algorithmic challenges characterize the dynamic stability analysis and the

study of buckling of structures subjected to large deformations. In particular, the

buckling under nonconservative forces where loss of stability might occur through

divergence but often times occurs through flutter (essentially a spatio-temporal

instability) needs special consideration for at least two reasons. First, standard

static analyses for structures under certain nonconservative loadings are, at best ,

subject to troublesome numerical instabilities that make computations very difficult.

Secondly, an even more alarming situation happens when not including the time

component in the formulation leads to an erroneous evaluation of the buckling load.


Some simple structures under specific nonconservative forces (like for instance a

cantilever beam under follower tangential load) have been studied analytically and

identified to present a dynamic loss of stability. However, if we are interested

in evaluating the buckling load numerically, we need to know before starting a

finite element simulation what type of instability is likely to occur, and use the

correct analysis (static or dynamic). Systematic computational methods capable of

predicting such phenomena are essential for applications in engineering structural

analysis and for optimum structural design.

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Biography

Ilinca Stanciulescu-Panea was born on January 20, 1972 in Bucharest, Romania. She

received a B.Eng and a M.A.Sc from the Technical University of Civil Engineering

in 1995, and 1996 respectively, and a B.S in Applied Mathematics from Bucharest

University in 2000. During her studies she was awarded the Merit Scholarship from

the Romanian Government (1990 - 1996), Tempus Scholarships (Spring 1995 for the

Senior Year Project, and 1995-1996 for graduate studies at ENPC, Paris), French

Government Scholarship (Summer 1994), and University of Florence Scholarship

(Summer School in Mathematics, Perugia, Italy, 1998).

Before joining the PhD Program at Duke University, she worked as a lecturer

and junior researcher (1996-2000) in the Department of Strength of Materials of the

Technical University of Civil Engineering (T.U.C.E.) in Bucharest, Romania. Dur-

ing that time she taught several classes (Strength of Materials, Elasticity Theory,

Finite Element Analysis, Nonlinear Analysis of Structures), participated in various

research projects, and co-authored a textbook, Post-Elastic Analysis of Structures.

She has also worked as a structural design engineer (full time in 1995 before joining

the faculty at T.U.C.E., and part–time thereafter).

179

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