PENALTY METHOD FOR UNILATERAL CONTACT PROBLEM WITH COULOMB’S FRICTION FOR LOCKING MATERIAL
الموضوعات : فصلنامه ریاضیSalah Bourichi 1 , El Essoufi 2
1 - University Hassan I, FSTS, Department of Mathematics and Informatics
Morocco
2 -
الکلمات المفتاحية: Finite Element, Locking material, Unilateral contact, Coulomb’s friction, variational inequality, penalty method,
ملخص المقالة :
In this work, we study a unilateral contact problem with non local friction of Coulombbetween a locking material and a rigid foundation. In the first step , we present the mathematicalmodel for a static process, we establish the variational formulation in the form of a variationalinequality and we prove the existence and uniqueness of the solution. In the second step, usingthe penalty method we introduce the penalized problem numerical in the form of variationalequality where we replace the law behavior and the law contact of Sigorini . The we show theconvergence of the continuous penalty solution as the penalty parameter n tends towards infinity.Then, the analysis of the finite element discretized penalty method is carried out.
F.Demengel and P.Suquet; On Locking materials Acta Applicandae Mathematicae 6
(1986), 185-21 I. 185 1986 by D. Reidel Publishing Company.
F.Demengel; Displacements bounded deformation and measures stress, published in annals
of superior school of Pise (1972).
F. Demengel; Relaxation and existence for locking materials problems , to be published in
Rairo.
W.Prager; On Ideal Locking Materials, Trans. Soc. Rheology 1 (1957) 169-175.
W.Prager; Unilateral Constraints in Mechanics of Continua , Estratto dagli atti del Simposio
Lagrangiano. Accademia delle Science di Torino, 1964, pp. 1-11.
W.Prager; Elastic Solids of Limited Compressibility, Proc. 9th Int. Congress AppL Mech.,
Vol. 5, Brussels, 1958, pp. 205-211.
P. G. Ciarlet;The finite element method for elliptic problems, North-Holland, Amsterdam,
G. Duvaut, J.-L. Lions; Les in´equations en m´ecanique et en physique, Dunod, Paris, 1972.
R. Glowinski; Numerical Methods for Nonlinear Variational Problems. Springer,
NewYork, 1984. Contact problems in elasticity: a study of variational inequalities and
finite element methods, volume 8 of SIAM Studies in Applied Mathematics, Society for
Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.
J. Neˇcas and I. Hlav´aˇcek; Mathematical Theory of Elastic and Elastico-Plastic Bodies:
An Introduction, Elsevier Scientific Publishing Company, Amsterdam, Oxford, New York,
G. Strang, G. Fix; An Analysis of the Finite Element Method, Prentice-Hall, Englewood
Cliffs, NJ 1973.
R. Temam; Probl`eme math´ematiques en plasticit´e, Gauthier-Villars, Paris (1983).
A.Ern, J.L.Guermound;
Theory and Pratice of finite element, Appl.Math.Sci,vol.159.Springer-Verlag New York,
P.Hild Y.Renard An improved a priori error analysis for finite element approximations of
Signorin’s, SIAM J. Numer. Anal. 50(2012)2400-2419
R.Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1979.
R.Temam, R. and G.Strang; Duality and Relaxation in the Variational Problems of Plasticity,
J. Mecanique. 19 (1980) 493-528.
