r5.03.52 Contact elements derived from a continuous hybrid formulation

Summary:

This document describes how friction contact elements are derived from a continuous hybrid formulation of contact problems between solids (2D or 3D) in large transformations and specifies the resolution strategy used.

An original formulation called a stabilized Lagrangian formulation is used, which makes it possible to find the classic cases of the literature (Lagrangian, augmented Lagrangian, penalization) by a careful choice of its parameters.

This formulation is available in command DEFI_CONTACT under the name “CONTINUE”.

r5.03.52 Table of contents

r5.03.52 Table of illustrations

• 1. Introduction
• 2. Cinematics
  • 2.1. Solid state kinematics
  • 2.2. Kinematic non-interpenetration condition
  • 2.3. Variations in kinematic quantities
• 3. Laws of contact and friction
  • 3.1. Contact law
  • 3.2. Law of friction
  • 3.3. Contact-friction law
• 4. Hybrid continuous touch/friction formulation
  • 4.1. Spaces
  • 4.2. Principle of virtual work
  • 4.3. Mixed variational inequality
  • 4.4. Variational formulation penalized
  • 4.5. Dynamic — Writing in speed/impulse
• 5. Algorithms for solving the nonlinear system
  • 5.1. Incremental formulation
  • 5.2. Temporal discretization for Coulomb’s law
  • 5.3. Calculation parameters
  • 5.4. Linearizations of terms
• 6. Discretization of the mixed variational formulation
  • 6.1. Theoretical elements
  • 6.2. Contact elements
  • 6.3. Preparing for discretization
  • 6.4. Resulting discretized system
• 7. Management of incompatibilities in contact processing
  • 7.1. Redundancy between boundary conditions and contact and friction conditions
• 8. Cycling during the Newton process
  • 8.1. Problem
  • 8.2. Detection
  • 8.3. Treatment
• 9. Features and verification
• 10. Bibliography
• 11. Appendix A: Calculation of Jacobian terms
  • 11.1. Equilibrium equation
  • 11.2. Contact law
  • 11.3. Law of friction
• 12. Appendix B: second member vector
  • 12.1. Balance terms
  • 12.2. Terms of Signorin’s law
  • 12.3. Coulomb’s law terms
• 13. Appendix C: tangent matrix
  • 13.1. Preliminary matrix quantities
  • 13.2. Quantities for balance
  • 13.3. Quantities for the law of contact
  • 13.4. Quantities for the law of friction
  • 13.5. Writing infinitesimal sliding: GRAND_GLIS