American Institute of Mathematical Sciences

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2012, 2(1): 91-104. doi: 10.3934/naco.2012.2.91

Solvability of a class of thermal dynamical contact problems with subdifferential conditions

Samir Adly 1, , Oanh Chau 2, and  Mohamed Rochdi 2,

1. 

Institut d'Ingénierie Informatique de Limoges and LACO, URA-1586, 123 Avenue A. Thomas, 87060 Limoges Cedex
2. 

Université de La Réunion, Département Maths-Info, BP 7151, 15 Avenue René Cassin, 97715 Saint Denis Messag, cedex 09, La Réunion

Received  January 2011 Revised  August 2011 Published  March 2012

We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic materials, which can be put into a general model of system defined by a second order evolution inequality, coupled with a first order evolution equation. We present and establish an existence and uniqueness result, by using general results on first order evolution inequality, with monotone operators and fixed point methods.
Keywords: sub-differential contact condition, Long memory thermo-visco-elasticity, dynamical process, evolution inequality., fixed point.
Mathematics Subject Classification: Primary: 74M10, 74M15, 74F25, 74H20, 74H25, 34G2.
Citation: Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91
References:
[1]

B. Awbi and O. Chau, Quasistatic Thermovisoelastic Frictional Contact Problem with Damped Response, Applicable Analysis, 83 (2004), 635-648. doi: 10.1080/00036810410001657233.  Google Scholar

[2]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Editura Academiei, Bucharest-Noordhoff, Leyden, 1976. Google Scholar

[3]

H. Brézis, Problèmes unilatéraux, J. Math. Pures et Appli., 51 (1972), 1-168. Google Scholar

[4]

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. Google Scholar

[5]

P. G. Ciarlet, "Mathematical Elasticity, Vol. I, Three-Dimensional Elasticity," North-Holland, 1988.  Google Scholar

[6]

M. Cocou and G. Scarella, Existence of a solution to a dynamic unilateral contact problem for a cracked viscoelastic body, C.R. Acad. Sci. Paris, Ser. I, 338 (2004), 341-346.  Google Scholar

[7]

G. Duvaut and J. L. Lions, "Les Inéquations en Mécanique et en Physique," Dunod, Paris, 1972.  Google Scholar

[8]

Ch. Eck, J. Jarusek and M. Krbec, "Unilateral Contact Problems, Variational Methods and Existence Theorems," Chapman and Hall, 2005. doi: 10.1201/9781420027365.  Google Scholar

[9]

D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, "Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics," Kluwer Academic Publishers, 2003.  Google Scholar

[10]

J. Jarušek and Ch. Eck, 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.  Google Scholar

[11]

N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity," SIAM, Philadelphia, 1988. doi: 10.1137/1.9781611970845.  Google Scholar

[12]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod et Gauthier-Villars, 1969. Google Scholar

[13]

J. Necas, J. Jarusek and J. Haslinger, On the solution of variational inequality to the Signorini problem with small friction, Bolletino U.M.I., 17 (1980), 796-811.  Google Scholar

[14]

J. Nečas and I. Hlaváček, "Mathematical Theory of Elastic and Elastoplastic Bodies: An introduction," Elsevier, Amsterdam, 1981. Google Scholar

[15]

P. D. Panagiotopoulos, "Inequality Problems in Mechanics and Applications," Birkhäuser, Basel, 1985. doi: 10.1007/978-1-4612-5152-1.  Google Scholar

[16]

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

[17]

Eberhard Zeidler, "Nonlinear Functional Analysis and its Applications, II/A, Linear Monotone Operators," Springer Verlag, 1997. Google Scholar

show all references

References:
[1]

B. Awbi and O. Chau, Quasistatic Thermovisoelastic Frictional Contact Problem with Damped Response, Applicable Analysis, 83 (2004), 635-648. doi: 10.1080/00036810410001657233.  Google Scholar

[2]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Editura Academiei, Bucharest-Noordhoff, Leyden, 1976. Google Scholar

[3]

H. Brézis, Problèmes unilatéraux, J. Math. Pures et Appli., 51 (1972), 1-168. Google Scholar

[4]

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. Google Scholar

[5]

P. G. Ciarlet, "Mathematical Elasticity, Vol. I, Three-Dimensional Elasticity," North-Holland, 1988.  Google Scholar

[6]

M. Cocou and G. Scarella, Existence of a solution to a dynamic unilateral contact problem for a cracked viscoelastic body, C.R. Acad. Sci. Paris, Ser. I, 338 (2004), 341-346.  Google Scholar

[7]

G. Duvaut and J. L. Lions, "Les Inéquations en Mécanique et en Physique," Dunod, Paris, 1972.  Google Scholar

[8]

Ch. Eck, J. Jarusek and M. Krbec, "Unilateral Contact Problems, Variational Methods and Existence Theorems," Chapman and Hall, 2005. doi: 10.1201/9781420027365.  Google Scholar

[9]

D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, "Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics," Kluwer Academic Publishers, 2003.  Google Scholar

[10]

J. Jarušek and Ch. Eck, 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.  Google Scholar

[11]

N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity," SIAM, Philadelphia, 1988. doi: 10.1137/1.9781611970845.  Google Scholar

[12]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod et Gauthier-Villars, 1969. Google Scholar

[13]

J. Necas, J. Jarusek and J. Haslinger, On the solution of variational inequality to the Signorini problem with small friction, Bolletino U.M.I., 17 (1980), 796-811.  Google Scholar

[14]

J. Nečas and I. Hlaváček, "Mathematical Theory of Elastic and Elastoplastic Bodies: An introduction," Elsevier, Amsterdam, 1981. Google Scholar

[15]

P. D. Panagiotopoulos, "Inequality Problems in Mechanics and Applications," Birkhäuser, Basel, 1985. doi: 10.1007/978-1-4612-5152-1.  Google Scholar

[16]

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

[17]

Eberhard Zeidler, "Nonlinear Functional Analysis and its Applications, II/A, Linear Monotone Operators," Springer Verlag, 1997. Google Scholar

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