THESE` Pour obtenir le grade de DOCTEUR DE L’UNIVERSITE´ DE GRENOBLE Specialit´ e´ : Mathematiques´ et Informatique Arretˆ e´ ministerial´ : 7 Aoutˆ 2006 Present´ ee´ par Alejandro Blumentals These` dirigee´ par Bernard Brogliato et codirigee´ par Florence Bertails-Descoubes prepar´ ee´ au sein Laboratoire Jean Kuntzmann et de Ecole´ Doctorale Mathematiques,´ Sciences et technologies de l’information, Informatique Numerical modelling of thin elastic solids in contact These` soutenue publiquement le 03 Juillet 2017, devant le jury compose´ de : Jean Claude Leon´ Professeur, Grenoble INP, President´ Olivier Br ¨uls Professeur, Universite´ de Liege,` Rapporteur Remco Leine Professeur, Universite´ de Stuttgart, Rapporteur David Dureissex Professeur, INSA Lyon, Examinateur Bernard Brogliato Directeur de recherche, Inria Rhoneˆ Alpes, Directeur de these` Florence Bertails-Descoubes Chargee´ de recherche, Inria Rhoneˆ Alpes, Co-Directrice de these` Acknowledgements I would first like to thank my thesis advisers Bernard Brogliato and Florence Bertails-Descoubes for their guidance during this endeavour. Bernard, I couldn't think of a better person with whom to learn mechanics with. And Florence, thank you for always having your door open for me and for always fuelling my scientific curiosity. I would also like to thank the members of my thesis committee for kindly accepting to evaluate my work and for their patient reading of my dissertation. Thanks to my teachers and colleagues at Grenoble-INP Ensimag for motivating me to pursue scientific research and for making possible the wonderful teaching experience I had. Thanks to Valentin Sonneville, Joachim Linn and Holger Lang for all the instructive exchanges we had concerning rods and shells. Thanks to all the members of our team at Inria, past and present, for the many interactions we had and for their help. Thanks Jerome, Pierre-Brice, Guillaume, Arnaud, Vincent and Dimitar: our many discussions, scientific and other, certainly shaped the course of my thesis. And thanks to Olivier, Romain, Gilles, Alexandre, Nestor, Nahuel, Jan and to all of those who made these years at Inria a real pleasure. I would like to thank my family and friends for always encouraging me. And last but not least I would like to thank Jana, whose love and support is what makes my work possible. 2 Contents General Introduction 5 I Lagrangian Mechanics of systems subject to sliding Coulomb's friction in bilateral and unilateral constraints 7 1 Introduction 8 2 Solvability of the contact problem for frictionless systems 13 2.1 Bilaterally constrained systems . 13 2.2 Unilaterally constrained systems . 17 2.3 Unilaterally/bilaterally constrained systems . 20 3 Lagrangian Mechanics of systems subject to unilateral and bilateral con- straints with sliding Coulomb's friction 24 3.1 Bilaterally constrained systems . 25 3.1.1 All frictional bilateral constraints . 26 3.1.2 Mixed frictional/frictionless contacts . 31 3.2 Unilaterally constrained systems . 33 3.3 Unilaterally/bilaterally constrained systems . 34 3.4 Conclusion . 38 II The statics of elastic rods from an Optimal Control point of view 41 4 Introduction 42 5 An Optimal Control Interpretation of Rod Statics 44 5.1 The continuum description of planar rods . 45 5.1.1 Planar Kirchhoff rods . 45 5.1.2 Planar Cosserat rods . 48 5.2 The continuum description of spatial rods . 49 5.2.1 Spatial Kirchhoff rods . 49 5.2.2 Spatial Cosserat rods . 51 5.3 The Kirchhoff Kinetic analogy . 53 5.4 Conclusion . 54 6 A Hamilton-Pontryagin Approach for the statics of Kirchhoff rods 55 6.1 Indirect Methods . 56 6.1.1 Derivation of the Kirchhoff Equations for rod statics . 57 3 6.1.2 Kirchhoff Indirect single shooting experiments . 60 6.1.3 Indirect methods for Euler's Elastica with fixed-fixed boundary conditions. 61 6.2 Discrete Kinematics . 66 6.2.1 On the choice of the integrator to discretize the Kinematics. 67 6.2.2 Runge-Kutta-Munthe-Kaas methods . 69 6.3 Direct Methods and the derivation of Strain based and mixed discretizations . 76 6.3.1 Direct Single Shooting . 76 6.3.2 Direct Multiple Shooting . 80 6.3.3 Numerical evaluation . 85 6.4 Quasi static experiments . 87 6.4.1 Instability when varyingκ ¯1........................... 87 6.4.2 Circle Packing . 87 6.4.3 Plectoneme Formation . 89 III Strain based dynamics of developable shells 96 7 Introduction 97 8 The Strain based approach to the mechanics of shells 100 8.1 The first and second fundamental forms . 100 8.2 The fundamental theorem of surface theory . 103 8.3 Koiter Shells . 105 8.4 Strain based methods for shells . 106 9 Strain based dynamics of developable shells 108 9.1 Strain based kinematics of developable shells . 108 9.2 Dynamics of a developable shell patch with spatially constant curvatures . 110 9.2.1 Kinematics of a developable shell patch with spatially constant curvatures 110 9.2.2 Dynamics of the developable shell patch with spatially constant curvatures113 9.2.3 Numerical solution of the equations of motion . 115 9.2.4 Implementation . 115 9.2.5 Difficulties . 118 9.3 Extension of the model . 118 9.3.1 Explicit solutions to the compatibility conditions . 118 9.3.2 Dynamics of a clothoidal shell patch . 120 9.4 Conclusion . 121 Perspectives 123 A Convex Analysis and Complementarity theory 125 A.1 Some convex analysis and complementarity theory tools . 125 A.2 Theorem 3.1.7 in [1] (excerpts) . 126 A.3 Theorem 3.8.6 in [1] . 126 A.4 Theorems 2.8 and 2.11 in [2] . 126 B KKT system: solvability and solution uniqueness 127 C Lie Groups 131 C.1 Differential Geometry of SO(3) and SE(3) . 131 4 General Introduction Many objects around us, either natural or man-made, are slender deformable objects. Curve- like objects such as industrial cables, helicopter blades, plant stems and hair can be modelled as thin elastic rods. While surface-like objects such as paper, boat sails, leaves and cloth can be modelled as thin elastic plates and shells. The numerical study of the mechanical response of such structures is of the utmost importance in many applications of mechanical and civil engineering, bio-mechanics, computer graphics and other fields. In this dissertation we address several problems related to the numerical simulation of slender elastic structures subject to contact and friction constraints. This thesis is organized in three parts and each part can be read independently. We have, however, in numerous occasions throughout this manuscript, underlined the similarities between the problems considered in each part. Throughout this thesis we shall treat rods, plates and shells as finite dimensional multibody systems. When performing extensive numerical simulations of the dynamics of multibody sys- tems subject to Coulomb's friction, it is common to encounter configurations where the Coulomb friction solver fails to compute the contact forces. While these failures may sometimes be due to implementation errors or to numerical ill-conditioning, it is also often the case that they are due to Painlev´eparadoxes. Quite simply, in some configurations there may exist no contact force which can prevent the system from violating its contact constraints. This happens even for very simple mechanical systems, like that of a rigid rod sliding against a frictional plane, as Paul Painlev´efirst showed. In Part I of this manuscript we analyze the contact problem (whose unknowns are the accelerations and the contact forces) and we derive computable upper bounds on the friction coefficients at each contact, such that this problem is well-posed and Painlev´eparadoxes are avoided. The obtained results apply not only to flexible systems, but to any general multibody system. To approximate a rod or a shell as a multibody system, a spatial discretization procedure needs to be carried out. In Parts II and III we deal with the problem of how to carry out such spatial discretizations. Thin rod-like structures which may easily bend and twist but hardly stretch and shear can be modelled as Kirchhoff rods. In the second part of this manuscript we consider the problem of computing the stable static equilibria of Kirchhoff rods subject to different boundary conditions and to frictionless contact constraints. We formulate the problem as an Optimal Control Problem, where the strains of the rod are interpreted as control variables and the position and orientation of the rod are interpreted as state variables. This leads us to a new understanding of the finite element discretization of Kirchhoff rods developed in [3] and to the proposal of new spatial discretization schemes for Kirchhoff rods. The proposed schemes are either of the strain-based type, where the main degrees of freedom are the strains of the rod, or of the mixed type, where the main degrees of freedom are both the strains and the generalized displacements. Similarly to the case of Kirchhoff rods, thin surface-like structures such as paper can hardly 5 stretch or shear at all. One of the advantages of the strain based approach is that the no extension and no shear constraints of the Kirchhoff rod are handled intrinsically, without the need of stiff repulsion forces, or of further algebraic constraints on the degrees of freedom. In Part III of this manuscript we propose an extension of this approach to model the dynamics of inextensible and unshearable shells. We restrict our study to the case of a shell patch with a developable mid-surface. We use as primary degrees of freedom the components of the second fundamental form of the shell's mid-surface. This also leads to an intrinsic handling of the no shear and no extension constraints of the shell. To the best of our knowledge the strain based simulation of developable shells is a problem that had not been treated before.