r5.03.54 Contact in small slips with X- FEM

Summary:

This document presents a new approach for dealing with contact problems in small slips with the eXtended Finite Element Method (X- FEM) [bib1]. We consider the continuous hybrid formulation of contact problems between solids [bib2] and the resolution strategy is similar to that already implemented in Code_Aster for the classical finite element framework [bib3]. A new type of mixed contact element is introduced, specific to the X- FEM frame.

The approach is implemented in Code_Aster in 2D and 3D, and treats both interfaces completely cut by a crack as well as interfaces with a crack background. It can be used with the STAT_NON_LINE [U4.51.03] command. Coulomb friction is taken into account.

• 1. Introduction
• 2. Strong formulation of the rubbing contact problem
  • 2.1. Formulation of the equations of the general problem
  • 2.2. Laws of contact
  • 2.3. Laws of friction
  • 2.4. Cohesive laws
• 3. Mixed variational formulation
  • 3.1. Formalism common to regularized cohesive and to touch-friction
  • 3.2. Augmented Lagrangian method
  • 3.3. Penalized method
  • 3.4. Formulation for a cohesive regularized law
  • 3.5. Formulation for a cohesive mixed law
• 4. EF discretizations
  • 4.1. Contact multipliers
  • 4.2. Friction semi-multipliers
  • 4.3. Finite contact element
  • 4.4. Setting inactive degrees of freedom to zero
  • 4.5. Calculation of the normal to the facet at the integration points
  • 4.6. Packaging for the penalized method
• 5. Resolution strategy
  • 5.1. Resolution algorithm
  • 5.2. Criteria for stopping the loop on contact statuses
  • 5.3. Criteria for stopping the loop on friction thresholds
  • 5.4. Writing the formulation during a Newton’s iteration
  • 5.5. Linearization of the problem
  • 5.6. Basic terms of rubbing contact
• 6. LBB condition
  • 6.1. Description of the Moës” algorithm for linear and quadratic elements (algo1)
  • 6.2. Description of the modified algorithm for linear and quadratic elements (algo2)
  • 6.3. Description of the algorithm for quadratic elements (algo 3):
  • 6.4. Imposed relationships between friction semi-multipliers
  • 6.5. Notes on the relationships imposed by algorithm 1 or 2
• 7. Bibliography
• 8. Description of the versions