An alternative formulation for quasi-static frictional and cohesive contact problems

P. Areias, A. Pinto Da Costa, T. Rabczuk, F. J.M. Queirós De Melo, D. Dias-Da-Costa, Mourad Bezzeghoud

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


It is known by Engineering practitioners that quasi-static contact problems with friction and cohesive laws often present convergence difficulties in Newton iteration. These are commonly attributed to the non-smoothness of the equilibrium system. However, non-uniqueness of solutions is often an obstacle for convergence. We discuss these conditions in detail and present a general algorithm for 3D which is shown to have quadratic convergence in the Newton-Raphson iteration even for parts of the domain where multiple solutions exist. Chen-Mangasarian replacement functions remove the non-smoothness corresponding to both the stick-slip and normal complementarity conditions. Contrasting with Augmented Lagrangian methods, second-order updating is performed for all degrees-of-freedom. Stick condition is automatically selected by the algorithm for regions with multiple solutions. The resulting Jacobian determinant is independent of the friction coefficient, at the expense of an increased number of nodal degrees-of-freedom. Aspects such as a dedicated pivoting for constrained problems are also of crucial importance for a successful solution finding. The resulting 3D mixed formulation, with 7 degrees-of-freedom in each node (displacement components, friction multiplier, friction force components and normal force) is tested with representative numerical examples (both contact with friction and cohesive force), which show remarkable robustness and generality.

Original languageEnglish
Pages (from-to)807-824
Number of pages18
JournalComputational Mechanics
Issue number4
Publication statusPublished - 2014 Apr

Bibliographical note

Funding Information:
The authors gratefully acknowledge financing from the “Fundação para a Ciência e a Tecnologia” under the Project PTDC/EME-PME/108751 and the Program COMPETE FCOMP-01-0124-FEDER-010267.


  • Cohesive law
  • Complementarity
  • Friction

ASJC Scopus subject areas

  • Computational Mechanics
  • Ocean Engineering
  • Mechanical Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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