On some frictional contact problems with velocity condition for elastic and visco-elastic materials

Previous Article
Planar quasilinear elliptic equations with right-hand side in $L(\log L)^{\delta}$
DCDS Home
This Issue
Next Article
Finite-time Lyapunov stability analysis of evolution variational inequalities

December 2011, 31(4): 1039-1051. doi: 10.3934/dcds.2011.31.1039

On some frictional contact problems with velocity condition for elastic and visco-elastic materials

Khalid Addi ^1, , Oanh Chau ^1, and Daniel Goeleven ^1,

1.	University of La Réunion, PIMENT EA4518, 97715 Saint-Denis Messag cedex 9 La Réunion, France, France, France

Received October 2009 Revised May 2010 Published September 2011

Abstract
Related Papers

We study the evolution of a class of quasistatic problems, which describe frictional contact between a body and a foundation. The constitutive law of the materials is elastic, or visco-elastic: with short or long memory, and the contact is modelled by a general subdifferential condition on the velocity. We derive weak formulations for the models and establish existence and uniqueness results. The proofs are based on evolution variational inequalities, in the framework of monotone operators and $fi$xed point methods. We show the approach of the viscoelastic solutions to the corresponding elastic solutions, when the viscosity tends to zero. Finally we also study the approach to short memory visco-elasticity when the long memory relaxation coefficients vanish.

Keywords: long and short memory, weak solution, quasistatic process, sub-differential contact condition, approaches to elasticity and short memory viscoelasticity., Elastic material, visco-elastic material, fixed point, variational inequality.

Mathematics Subject Classification: Primary: 74M15, 74M10; Secondary: 34G2.

Citation: Khalid Addi, Oanh Chau, Daniel Goeleven. On some frictional contact problems with velocity condition for elastic and visco-elastic materials. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1039-1051. doi: 10.3934/dcds.2011.31.1039

References:

[1]	L.-E. Andersson, A global existence result for a quasistatic contact problem with friction, Advances in Mathematical Sciences ans Applications, 5 (1995), 249-286.
[2]	B. Awbi, El. H. Essoufi and M. Sofonea, A viscoelastic contact problem with normal damped response and friction, Annales Polonici Mathematici, 75 (2000), 233-246.
[3]	V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Translated from the Romanian, Editura Academiei Republicii Socialiste România, Bucharest, Noordhoff International Publishing, Leiden, 1976.
[4]	N. Bourbaki, "Éléments de Mathématiques," Première Partie, Livre IV, Fonctions d'une variable réelle, Hermann, 1961.
[5]	H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175. doi: 10.5802/aif.280.
[6]	H. Brézis, Problèmes unilatéraux, J. Math. Pures et Appli, 51 (1972), 1-168.
[7]	F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, in "Nonlinear Functional Analysis" (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), 1-308, Amer. Math. Soc., Providence, RI, 1976.
[8]	O. Chau, R. Oujja and M. Rochdi, A mathematical analysis of a dynamical frictional contact model in thermoviscoelasticity, Discrete and Continuous Dynamical Systems, Series S, Volume 1 (2008), 61-70.
[9]	O. Chau, D. Motreanu and M. Sofonea, Quasistatic frictional problems for elastic and viscoelastic materials, Applications of Mathematics, 47 (2002), 341-360. doi: 10.1023/A:1021753722771.
[10]	O. Chau and M. Rochdi, On a dynamic bilateral contact problem with friction for viscoelastic materials, Int. J. of Appli. Math. and Mech, 2 (2006), 41-52.
[11]	P. G. Ciarlet, "Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988.
[12]	M. Cocu, E. Pratt and M. Raous, Formulation and approximation of quasistatic frictional contact, Int. J. Engn. Sci, 34 (1996), 783-798. doi: 10.1016/0020-7225(95)00121-2.
[13]	G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.
[14]	C. Eck and J. Jarušek, Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions, Mathematical models and Methods in Applied Sciences, 9 (1999), 11-34. doi: 10.1142/S0218202599000038.
[15]	H. Fraysse and J. M. Arnaudiès, "Cours de Mathématiques," Dunod, 1999.
[16]	D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, "Variational and Hemivariational Inequalities: Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics," Nonconvex Optimization and its Applications, 69, Kluwer Academic Publishers, Boston, MA, 2003.
[17]	I. Hlaváček, J. Haslinger, J. Necăs and J. Lovíšek, "Solution of Variational Inequalities in Mechanics," Applied Mathematical Sciences, 66, Springer-Verlag, New York, 1988.
[18]	W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis," Interdisciplinary Applied Mathematics, 9, Springer-Verlag, New York, 1999.
[19]	J. Jarušek, Dynamic contact problems with given friction for viscoelastic bodies, Czechoslovak Mathematical Journal, 46 (1996), 475-487.
[20]	N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," SIAM Studies in Applied Mathematics, 8, SIAM, Philadelphia, PA, 1988.
[21]	J. L. Lions and G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519. doi: 10.1002/cpa.3160200302.
[22]	J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlin. Anal., 11 (1987), 407-428. doi: 10.1016/0362-546X(87)90055-1.
[23]	J. Nečas and I. Hlavaček, "Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction," Elsevier, Amsterdam, 1981.
[24]	P. D. Panagiotopoulos, "Inequality Problems in Meechanical and Applications. Convex and Nonconvex Energy Functions," Birkhäuser Boston, Inc., Boston, MA, 1985.
[25]	P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering," Springer-Verlag, Berlin, 1993.
[26]	M. Raous, M. Jean and J. J. Moreau, eds., "Contact Mechanics," Plenum Press, New York, 1995.
[27]	M. Rochdi, M. Shillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction, Journal of Elasticity, 51 (1998), 105-126. doi: 10.1023/A:1007413119583.
[28]	E. Zeidler, "Nonlinear Functional Analysis and its Applications," Springer-Verlag, 1997.

show all references

References:

[1]	L.-E. Andersson, A global existence result for a quasistatic contact problem with friction, Advances in Mathematical Sciences ans Applications, 5 (1995), 249-286.
[2]	B. Awbi, El. H. Essoufi and M. Sofonea, A viscoelastic contact problem with normal damped response and friction, Annales Polonici Mathematici, 75 (2000), 233-246.
[3]	V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Translated from the Romanian, Editura Academiei Republicii Socialiste România, Bucharest, Noordhoff International Publishing, Leiden, 1976.
[4]	N. Bourbaki, "Éléments de Mathématiques," Première Partie, Livre IV, Fonctions d'une variable réelle, Hermann, 1961.
[5]	H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175. doi: 10.5802/aif.280.
[6]	H. Brézis, Problèmes unilatéraux, J. Math. Pures et Appli, 51 (1972), 1-168.
[7]	F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, in "Nonlinear Functional Analysis" (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), 1-308, Amer. Math. Soc., Providence, RI, 1976.
[8]	O. Chau, R. Oujja and M. Rochdi, A mathematical analysis of a dynamical frictional contact model in thermoviscoelasticity, Discrete and Continuous Dynamical Systems, Series S, Volume 1 (2008), 61-70.
[9]	O. Chau, D. Motreanu and M. Sofonea, Quasistatic frictional problems for elastic and viscoelastic materials, Applications of Mathematics, 47 (2002), 341-360. doi: 10.1023/A:1021753722771.
[10]	O. Chau and M. Rochdi, On a dynamic bilateral contact problem with friction for viscoelastic materials, Int. J. of Appli. Math. and Mech, 2 (2006), 41-52.
[11]	P. G. Ciarlet, "Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988.
[12]	M. Cocu, E. Pratt and M. Raous, Formulation and approximation of quasistatic frictional contact, Int. J. Engn. Sci, 34 (1996), 783-798. doi: 10.1016/0020-7225(95)00121-2.
[13]	G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.
[14]	C. Eck and J. Jarušek, Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions, Mathematical models and Methods in Applied Sciences, 9 (1999), 11-34. doi: 10.1142/S0218202599000038.
[15]	H. Fraysse and J. M. Arnaudiès, "Cours de Mathématiques," Dunod, 1999.
[16]	D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, "Variational and Hemivariational Inequalities: Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics," Nonconvex Optimization and its Applications, 69, Kluwer Academic Publishers, Boston, MA, 2003.
[17]	I. Hlaváček, J. Haslinger, J. Necăs and J. Lovíšek, "Solution of Variational Inequalities in Mechanics," Applied Mathematical Sciences, 66, Springer-Verlag, New York, 1988.
[18]	W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis," Interdisciplinary Applied Mathematics, 9, Springer-Verlag, New York, 1999.
[19]	J. Jarušek, Dynamic contact problems with given friction for viscoelastic bodies, Czechoslovak Mathematical Journal, 46 (1996), 475-487.
[20]	N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," SIAM Studies in Applied Mathematics, 8, SIAM, Philadelphia, PA, 1988.
[21]	J. L. Lions and G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519. doi: 10.1002/cpa.3160200302.
[22]	J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlin. Anal., 11 (1987), 407-428. doi: 10.1016/0362-546X(87)90055-1.
[23]	J. Nečas and I. Hlavaček, "Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction," Elsevier, Amsterdam, 1981.
[24]	P. D. Panagiotopoulos, "Inequality Problems in Meechanical and Applications. Convex and Nonconvex Energy Functions," Birkhäuser Boston, Inc., Boston, MA, 1985.
[25]	P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering," Springer-Verlag, Berlin, 1993.
[26]	M. Raous, M. Jean and J. J. Moreau, eds., "Contact Mechanics," Plenum Press, New York, 1995.
[27]	M. Rochdi, M. Shillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction, Journal of Elasticity, 51 (1998), 105-126. doi: 10.1023/A:1007413119583.
[28]	E. Zeidler, "Nonlinear Functional Analysis and its Applications," Springer-Verlag, 1997.

[1]	Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10
[2]	Xiaoqiang Dai, Wenke Li. Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem. Electronic Research Archive, 2021, 29 (6) : 4087-4098. doi: 10.3934/era.2021073
[3]	Linglong Du. Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect. Networks and Heterogeneous Media, 2018, 13 (4) : 549-565. doi: 10.3934/nhm.2018025
[4]	Mustapha El Jarroudi, Youness Filali, Aadil Lahrouz, Mustapha Er-Riani, Adel Settati. Asymptotic analysis of an elastic material reinforced with thin fractal strips. Networks and Heterogeneous Media, 2022, 17 (1) : 47-72. doi: 10.3934/nhm.2021023
[5]	Maria-Magdalena Boureanu, Andaluzia Matei, Mircea Sofonea. Analysis of a contact problem for electro-elastic-visco-plastic materials. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1185-1203. doi: 10.3934/cpaa.2012.11.1185
[6]	Moncef Aouadi, Imed Mahfoudhi, Taoufik Moulahi. Approximate controllability of nonsimple elastic plate with memory. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1015-1043. doi: 10.3934/dcdss.2021147
[7]	Tuan Anh Dao, Hironori Michihisa. Study of semi-linear $ \sigma $-evolution equations with frictional and visco-elastic damping. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1581-1608. doi: 10.3934/cpaa.2020079
[8]	Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall. Networks and Heterogeneous Media, 2008, 3 (3) : 651-673. doi: 10.3934/nhm.2008.3.651
[9]	Marita Thomas, Sven Tornquist. Discrete approximation of dynamic phase-field fracture in visco-elastic materials. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 3865-3924. doi: 10.3934/dcdss.2021067
[10]	Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 255-278. doi: 10.3934/naco.2021004
[11]	Habib ur Rehman, Poom Kumam, Yusuf I. Suleiman, Widaya Kumam. An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in Hilbert spaces. Numerical Algebra, Control and Optimization, 2022 doi: 10.3934/naco.2022007
[12]	Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649
[13]	Alessia Berti, Claudio Giorgi, Elena Vuk. Free energies and pseudo-elastic transitions for shape memory alloys. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 293-316. doi: 10.3934/dcdss.2013.6.293
[14]	Sergei A. Avdonin, Boris P. Belinskiy. On controllability of a linear elastic beam with memory under longitudinal load. Evolution Equations and Control Theory, 2014, 3 (2) : 231-245. doi: 10.3934/eect.2014.3.231
[15]	Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of a frictional contact problem for viscoelastic materials with long memory. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 687-705. doi: 10.3934/dcdsb.2011.15.687
[16]	Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205
[17]	Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1521-1543. doi: 10.3934/cpaa.2021031
[18]	Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793
[19]	Zijia Peng, Cuiming Ma, Zhonghui Liu. Existence for a quasistatic variational-hemivariational inequality. Evolution Equations and Control Theory, 2020, 9 (4) : 1153-1165. doi: 10.3934/eect.2020058
[20]	Jiaohui Xu, Tomás Caraballo, José Valero. Asymptotic behavior of nonlocal partial differential equations with long time memory. Discrete and Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021140

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Khalid Addi Oanh Chau Daniel Goeleven
on Google Scholar
Khalid Addi Oanh Chau Daniel Goeleven

[Back to Top]