An Error Estimate for the Signorini Problem with Coulomb Friction Approximated by Finite Elements

The present paper is concerned with the unilateral contact model and the Coulomb friction law in linear elastostatics. We consider a mixed formulation in which the unknowns are the displacement field and the normal and tangential constraints on the contact area. The chosen finite element method involves continuous elements of degree one and continuous piecewise affine multipliers on the contact zone. A convenient discrete contact and friction condition is introduced in order to perform a convergence study. We finally obtain a first a priori error estimate under the assumptions ensuring the uniqueness of the solution to the continuous problem.

[1]  Faker Ben Belgacem,et al.  Numerical Simulation of Some Variational Inequalities Arisen from Unilateral Contact Problems by the Finite Element Methods , 2000, SIAM J. Numer. Anal..

[2]  J. Lions,et al.  Les inéquations en mécanique et en physique , 1973 .

[3]  A. Klarbring,et al.  FRICTIONAL CONTACT PROBLEMS WITH NORMAL COMPLIANCE , 1988 .

[4]  J. T. Oden,et al.  Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws , 1987 .

[5]  Christof Eck,et al.  Unilateral Contact Problems: Variational Methods and Existence Theorems , 2005 .

[6]  C. Bernardi,et al.  A New Nonconforming Approach to Domain Decomposition : The Mortar Element Method , 1994 .

[7]  Patrice Coorevits,et al.  Mixed finite element methods for unilateral problems: convergence analysis and numerical studies , 2002, Math. Comput..

[8]  Jaroslav Haslinger,et al.  Numerical methods for unilateral problems in solid mechanics , 1996 .

[9]  F. B. Belgacem,et al.  EXTENSION OF THE MORTAR FINITE ELEMENT METHOD TO A VARIATIONAL INEQUALITY MODELING UNILATERAL CONTACT , 1999 .

[10]  Yves Renard,et al.  Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers , 2006 .

[11]  Meir Shillor,et al.  On friction problems with normal compliance , 1989 .

[12]  Remi Rocca,et al.  Numerical Analysis of Quasi-Static Unilateral Contact Problems with Local Friction , 2001, SIAM J. Numer. Anal..

[13]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[14]  Yves Renard,et al.  A uniqueness criterion for the Signorini problem with Coulomb friction , 2006 .

[15]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[16]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[17]  Yves Renard.  A Uniqueness Criterion for the Signorini Problem with Coulomb Friction , 2006, SIAM J. Math. Anal..

[18]  V. Thomée,et al.  The stability in _{} and ¹_{} of the ₂-projection onto finite element function spaces , 1987 .

[19]  Vladimir Maz’ya,et al.  Theory of multipliers in spaces of differentiable functions , 1983 .

[20]  Faker Ben Belgacem,et al.  Hybrid finite element methods for the Signorini problem , 2003, Math. Comput..

[21]  L. Andersson,et al.  Existence Results for Quasistatic Contact Problems with Coulomb Friction , 2000 .

[22]  Peter Wriggers,et al.  Computational Contact Mechanics , 2002 .

[23]  Patrick Laborde,et al.  Fixed point strategies for elastostatic frictional contact problems , 2008 .

[24]  W. Han,et al.  Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity , 2002 .

[25]  T. Laursen.  Computational Contact and Impact Mechanics , 2003 .

[26]  M. Cocu,et al.  Existence of solutions of Signorini problems with friction , 1984 .

[27]  W. Han,et al.  Contact problems in elasticity , 2002 .

[28]  J. Oden,et al.  Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods , 1987 .

[29]  J. Oden,et al.  Contact problems in elasticity , 1988 .

[30]  D. Adams,et al.  Review: V. G. Maz′ya and T. O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions , 1986 .

[31]  J. Jarusek,et al.  EXISTENCE RESULTS FOR THE STATIC CONTACT PROBLEM WITH COULOMB FRICTION , 1998 .

[32]  Faker Ben Belgacem,et al.  The Mortar finite element method with Lagrange multipliers , 1999, Numerische Mathematik.

[33]  J. Oden,et al.  On some existence and uniqueness results in contact problems with nonlocal friction , 1982 .

[34]  Patrick Hild,et al.  A mixed finite element method and solution multiplicity for Coulomb frictional contact , 2003 .

[35]  Barbara I. Wohlmuth,et al.  An Optimal A Priori Error Estimate for Nonlinear Multibody Contact Problems , 2005, SIAM J. Numer. Anal..

[36]  Jaroslav Haslinger,et al.  Approximation of the signorini problem with friction, obeying the coulomb law , 1983 .

[37]  Patrick Hild,et al.  Non‐unique slipping in the coulomb friction model in two‐dimensional linear elasticity , 2004 .

[38]  José Barros-Neto,et al.  Problèmes aux limites non homogènes , 1966 .

[39]  W. Ostachowicz,et al.  Mixed finite element method for contact problems , 1984 .

[40]  Patrick Hild,et al.  Multiple solutions of stick and separation type in the Signorini model with Coulomb friction , 2005 .

[41]  Ricardo H. Nochetto,et al.  Residual type a posteriori error estimates for elliptic obstacle problems , 2000, Numerische Mathematik.

[42]  J. T. Oden,et al.  Models and computational methods for dynamic friction phenomena , 1984 .

[43]  Jaroslav Haslinger,et al.  On the approximation of the Signorini problem with friction , 1984 .