American Institute of Mathematical Sciences

Advanced
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

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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.   Google Scholar

[3]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Translated from the Romanian, (1976).   Google Scholar

[4]

N. Bourbaki, "Éléments de Mathématiques,", Première Partie, (1961).   Google Scholar

[5]

H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité,, Ann. Inst. Fourier (Grenoble), 18 (1968), 115.  doi: 10.5802/aif.280.  Google Scholar

[6]

H. Brézis, Problèmes unilatéraux,, J. Math. Pures et Appli, 51 (1972), 1.   Google Scholar

[7]

F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces,, in, (1968), 1.   Google Scholar

[8]

O. Chau, R. Oujja and M. Rochdi, A mathematical analysis of a dynamical frictional contact model in thermoviscoelasticity,, Discrete and Continuous Dynamical Systems, 1 (2008), 61.   Google Scholar

[9]

O. Chau, D. Motreanu and M. Sofonea, Quasistatic frictional problems for elastic and viscoelastic materials,, Applications of Mathematics, 47 (2002), 341.  doi: 10.1023/A:1021753722771.  Google Scholar

[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.   Google Scholar

[11]

P. G. Ciarlet, "Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, 20 (1988).   Google Scholar

[12]

M. Cocu, E. Pratt and M. Raous, Formulation and approximation of quasistatic frictional contact,, Int. J. Engn. Sci, 34 (1996), 783.  doi: 10.1016/0020-7225(95)00121-2.  Google Scholar

[13]

G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics,", Grundlehren der Mathematischen Wissenschaften, 219 (1976).   Google Scholar

[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.  doi: 10.1142/S0218202599000038.  Google Scholar

[15]

H. Fraysse and J. M. Arnaudiès, "Cours de Mathématiques,", Dunod, (1999).   Google Scholar

[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 (2003).   Google Scholar

[17]

I. Hlaváček, J. Haslinger, J. Necăs and J. Lovíšek, "Solution of Variational Inequalities in Mechanics,", Applied Mathematical Sciences, 66 (1988).   Google Scholar

[18]

W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis,", Interdisciplinary Applied Mathematics, 9 (1999).   Google Scholar

[19]

J. Jarušek, Dynamic contact problems with given friction for viscoelastic bodies,, Czechoslovak Mathematical Journal, 46 (1996), 475.   Google Scholar

[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 (1988).   Google Scholar

[21]

J. L. Lions and G. Stampacchia, Variational inequalities,, Commun. Pure Appl. Math., 20 (1967), 493.  doi: 10.1002/cpa.3160200302.  Google Scholar

[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.  doi: 10.1016/0362-546X(87)90055-1.  Google Scholar

[23]

J. Nečas and I. Hlavaček, "Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction,", Elsevier, (1981).   Google Scholar

[24]

P. D. Panagiotopoulos, "Inequality Problems in Meechanical and Applications. Convex and Nonconvex Energy Functions,", Birkhäuser Boston, (1985).   Google Scholar

[25]

P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering,", Springer-Verlag, (1993).   Google Scholar

[26]

M. Raous, M. Jean and J. J. Moreau, eds., "Contact Mechanics,", Plenum Press, (1995).   Google Scholar

[27]

M. Rochdi, M. Shillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction,, Journal of Elasticity, 51 (1998), 105.  doi: 10.1023/A:1007413119583.  Google Scholar

[28]

E. Zeidler, "Nonlinear Functional Analysis and its Applications,", Springer-Verlag, (1997).   Google Scholar

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.   Google Scholar

[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.   Google Scholar

[3]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Translated from the Romanian, (1976).   Google Scholar

[4]

N. Bourbaki, "Éléments de Mathématiques,", Première Partie, (1961).   Google Scholar

[5]

H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité,, Ann. Inst. Fourier (Grenoble), 18 (1968), 115.  doi: 10.5802/aif.280.  Google Scholar

[6]

H. Brézis, Problèmes unilatéraux,, J. Math. Pures et Appli, 51 (1972), 1.   Google Scholar

[7]

F. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces,, in, (1968), 1.   Google Scholar

[8]

O. Chau, R. Oujja and M. Rochdi, A mathematical analysis of a dynamical frictional contact model in thermoviscoelasticity,, Discrete and Continuous Dynamical Systems, 1 (2008), 61.   Google Scholar

[9]

O. Chau, D. Motreanu and M. Sofonea, Quasistatic frictional problems for elastic and viscoelastic materials,, Applications of Mathematics, 47 (2002), 341.  doi: 10.1023/A:1021753722771.  Google Scholar

[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.   Google Scholar

[11]

P. G. Ciarlet, "Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, 20 (1988).   Google Scholar

[12]

M. Cocu, E. Pratt and M. Raous, Formulation and approximation of quasistatic frictional contact,, Int. J. Engn. Sci, 34 (1996), 783.  doi: 10.1016/0020-7225(95)00121-2.  Google Scholar

[13]

G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics,", Grundlehren der Mathematischen Wissenschaften, 219 (1976).   Google Scholar

[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.  doi: 10.1142/S0218202599000038.  Google Scholar

[15]

H. Fraysse and J. M. Arnaudiès, "Cours de Mathématiques,", Dunod, (1999).   Google Scholar

[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 (2003).   Google Scholar

[17]

I. Hlaváček, J. Haslinger, J. Necăs and J. Lovíšek, "Solution of Variational Inequalities in Mechanics,", Applied Mathematical Sciences, 66 (1988).   Google Scholar

[18]

W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis,", Interdisciplinary Applied Mathematics, 9 (1999).   Google Scholar

[19]

J. Jarušek, Dynamic contact problems with given friction for viscoelastic bodies,, Czechoslovak Mathematical Journal, 46 (1996), 475.   Google Scholar

[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 (1988).   Google Scholar

[21]

J. L. Lions and G. Stampacchia, Variational inequalities,, Commun. Pure Appl. Math., 20 (1967), 493.  doi: 10.1002/cpa.3160200302.  Google Scholar

[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.  doi: 10.1016/0362-546X(87)90055-1.  Google Scholar

[23]

J. Nečas and I. Hlavaček, "Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction,", Elsevier, (1981).   Google Scholar

[24]

P. D. Panagiotopoulos, "Inequality Problems in Meechanical and Applications. Convex and Nonconvex Energy Functions,", Birkhäuser Boston, (1985).   Google Scholar

[25]

P. D. Panagiotopoulos, "Hemivariational Inequalities, Applications in Mechanics and Engineering,", Springer-Verlag, (1993).   Google Scholar

[26]

M. Raous, M. Jean and J. J. Moreau, eds., "Contact Mechanics,", Plenum Press, (1995).   Google Scholar

[27]

M. Rochdi, M. Shillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction,, Journal of Elasticity, 51 (1998), 105.  doi: 10.1023/A:1007413119583.  Google Scholar

[28]

E. Zeidler, "Nonlinear Functional Analysis and its Applications,", Springer-Verlag, (1997).   Google Scholar

[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]

Linglong Du. Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect. Networks & Heterogeneous Media, 2018, 13 (4) : 549-565. doi: 10.3934/nhm.2018025
[3]

Maria-Magdalena Boureanu, Andaluzia Matei, Mircea Sofonea. Analysis of a contact problem for electro-elastic-visco-plastic materials. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1185-1203. doi: 10.3934/cpaa.2012.11.1185
[4]

Tuan Anh Dao, Hironori Michihisa. Study of semi-linear $ \sigma $-evolution equations with frictional and visco-elastic damping. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1581-1608. doi: 10.3934/cpaa.2020079
[5]

Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall. Networks & Heterogeneous Media, 2008, 3 (3) : 651-673. doi: 10.3934/nhm.2008.3.651
[6]

Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649
[7]

Alessia Berti, Claudio Giorgi, Elena Vuk. Free energies and pseudo-elastic transitions for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 293-316. doi: 10.3934/dcdss.2013.6.293
[8]

Sergei A. Avdonin, Boris P. Belinskiy. On controllability of a linear elastic beam with memory under longitudinal load. Evolution Equations & Control Theory, 2014, 3 (2) : 231-245. doi: 10.3934/eect.2014.3.231
[9]

Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of a frictional contact problem for viscoelastic materials with long memory. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 687-705. doi: 10.3934/dcdsb.2011.15.687
[10]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020205
[11]

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 & Continuous Dynamical Systems - A, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793
[12]

Valeria Danese, Pelin G. Geredeli, Vittorino Pata. Exponential attractors for abstract equations with memory and applications to viscoelasticity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2881-2904. doi: 10.3934/dcds.2015.35.2881
[13]

Zijia Peng, Cuiming Ma, Zhonghui Liu. Existence for a quasistatic variational-hemivariational inequality. Evolution Equations & Control Theory, 2020, 9 (4) : 1153-1165. doi: 10.3934/eect.2020058
[14]

Shujuan Lü, Zeting Liu, Zhaosheng Feng. Hermite spectral method for Long-Short wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 941-964. doi: 10.3934/dcdsb.2018255
[15]

Francesco Maddalena, Danilo Percivale, Franco Tomarelli. Adhesive flexible material structures. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 553-574. doi: 10.3934/dcdsb.2012.17.553
[16]

Vittorino Pata. Exponential stability in linear viscoelasticity with almost flat memory kernels. Communications on Pure & Applied Analysis, 2010, 9 (3) : 721-730. doi: 10.3934/cpaa.2010.9.721
[17]

Toyohiko Aiki. On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 1-13. doi: 10.3934/dcdss.2012.5.1
[18]

G. Idone, A. Maugeri. Variational inequalities and a transport planning for an elastic and continuum model. Journal of Industrial & Management Optimization, 2005, 1 (1) : 81-86. doi: 10.3934/jimo.2005.1.81
[19]

H. Thomas Banks, Kidist Bekele-Maxwell, Lorena Bociu, Marcella Noorman, Giovanna Guidoboni. Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models. Mathematical Control & Related Fields, 2019, 9 (4) : 623-642. doi: 10.3934/mcrf.2019044
[20]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences