Contact in Structural Mechanics

Anh Le van is Professor of Structural Mechanics in the Faculty of Science and Technology, University of Nantes, France. His research at the Research Institute in Civil and Mechanical Engineering (GeM) focuses on membrane structures and, more specifically, on contact and bifurcation problems in these structures.

• Preface ix1 Introduction to Contact Problems in Structural Mechanics 11.1 Solving a contact problem numerically via the penalty method 31.2 Numerical solution of a contact problem using the multiplier method 81.2.1 Preliminaries: problems with equality constraints 81.2.2 Problems with inequality constraints 101.3 Numerical solution of a contact problem by the augmented Lagrangian method 151.4 Book synopsis 212 Contact Kinematics 232.1 Motions and strains 232.2 Potential contact surfaces 252.3 Normal contact kinematics 262.4 Variation of kinematic quantities with respect to time 292.5 Tangential contact kinematics – Relative velocity 343 Sthenics of Contact 373.1 Stresses in bodies 373.2 Contact stress vector 384 The Constitutive Law 394.1 Hyperelastic materials 394.2 Elastoplastic materials with isotropic hardening 415 Contact Laws 455.1 Normal contact law 455.2 Tangential contact law 476 Strong Formulation of the Contact Problem 516.1 Field equations 516.2 Boundary conditions 526.3 Initial conditions 536.4 Remarks 537 Weak Formulation of the Contact Problem 557.1 Transforming the contact laws into equalities 557.2 Preliminary ideas for the weak form 597.3 Weak form of the contact problem 607.4 Equivalence between the strong and the weak forms 627.5 Final remarks 668 Matrix Equations of the Contact Problem 698.1 Introduction 698.2 Meshes 708.3 Matrix notation in finite elements 728.4 The element nodal vectors 738.5 Interpolation of positions, displacements and virtual velocities 758.5.1 Interpolation on the contactor surface 758.5.2 Interpolation on the target surface 758.6 Interpolation of multipliers 768.6.1 Definition of the vector λ 768.6.2 Interpolation of λ 788.6.3 Interpolation of λ∗ 788.7 Discretization of the element virtual contact power (P∗contact)e(1) 788.7.1 Explicit expressions for {Φe(1)contact}, {Φe(2)contact} and {Re(1)Λ } in the three cases: algorithmic gap, algorithmic slip and algorithmic stick 858.8 System of matrix equations for the contact problem 888.8.1 Global nodal vectors 888.8.2 Discretization of the classical terms 908.8.3 Assembly of element virtual contact powers 918.8.4 System of matrix equations 948.9 Abnormal contact stresses 968.9.1 First cause of abnormal contact stresses 968.9.2 Second cause of abnormal contact stresses 988.9.3 Third cause of abnormal contact stresses 988.10 Projection calculation: contact detection 998.11 Discrete expression of the slip VTΔt 1018.12 Physical units 1068.13 Chapter summary 1079 Solution of the Quasi-static Contact Problem 1099.1 System of equations for the static contact problem 1099.2 Incremental loop initialization: the vectors U0, Λ0 1119.3 Calculation of step n ≥ 1: calculating Un , Λn 1119.3.1 Principle of the iterative Newton–Raphson scheme 1119.3.2 Tangent matrix 1139.3.3 Block matrix inversion 1149.3.4 Iterative loop initialization: the vectors U0n and Λ0n 1159.4 Solution algorithm 1159.5 Calculation method for the tangent matrix 1179.5.1 Direct method 1179.5.2 Indirect method 1189.5.3 Restriction to the contact tangent matrix 1219.6 Calculation of the contact tangent matrix 1239.6.1 Variations of the arguments of the functional P∗contact 1239.6.2 Calculation of the variation δP∗contact 1269.6.3 Calculation of the variation (δP∗contact)e(1) 1279.6.4 Discretization of the variation (δP∗contact)e(1) – Element contact tangent matrix [Kecontact] 1339.6.5 Discretization of the variation δP∗contact – Contact tangent matrix [Kcontact] 1359.6.6 Explicit expression for the element contact tangent matrix [Kecontact] 1389.6.7 [Kecontact] in the case of the algorithmic gap at the considered integration point 1449.6.8 [Kecontact] in the case of algorithmic contact with slip at the considered integration point 1449.6.9 [Kecontact] in the case of algorithmic contact with stick at the considered integration point 1469.6.10 Symmetry of the contact tangent matrix [Kcontact] 1479.7 Particular case of two non-contacting bodies 1489.8 Particular case of the frictionless problem 1499.8.1 Algorithmic gap case at the considered integration point 1509.8.2 Algorithmic contact with slip case at the considered integration point 1529.9 Solution via the arc-length method 1529.10 Physical units 1549.11 Summary of the chapter 15510 Numerical Examples of Quasi-static Contact 15710.1 Contact patch test 15710.2 Hertzian contact problem 15910.2.1 Frictionless contact case 16010.2.2 Case of frictional contact with μ = 0.3 16310.3 Rolling disk 16710.4 Contact between two beams 17110.4.1 Dead load 17110.4.2 Follower load 17610.5 Contact of two pressurized membranes 17610.5.1 Centered membranes 18010.5.2 Membranes Staggered Along X 18210.6 Extrusion of an elastoplastic cylinder 18410.7 Interference fit problem 18910.7.1 Abnormal contact stresses 19210.7.2 Influence of the mesh 19410.8 Conclusion 19411 Solution of the Dynamic Contact Problem 19711.1 A brief review of the computational methods in dynamic contact 19711.2 Solution of the dynamic contact problem via Newmark’s algorithm 20011.2.1 Initializing the incremental loop: the vectors U0 , V0 , A0 and Λ0 20211.2.2 Calculation for a step n ≥ 1: calculating Un , Vn , An , Λn 20211.2.3 Initializing the iterative loop: the vectors U0n, V0n, A0n, Λ0n 20711.3 Solution algorithm 20811.4 Summary 21012 Numerical Examples of Dynamic Contact 21312.1 Impact of two elastic rods 21312.1.1 Analytical solution 21412.1.2 Numerical applications 21712.1.3 Numerical solution 21812.2 Disk impacting a rigid plane 22012.2.1 Frictionless case 22212.2.2 Case with friction μ = 0.3 22412.3 Disk falling into a funnel 22812.3.1 Frictionless case 23112.3.2 Case with friction μ = 0.4 23412.4 Final remarks 236Appendix A: Variations of Kinematic Quantities 239References 247Index 257