Dynamic contact problems with velocity conditions

Oanh Chau; Viorica Motreanu

International Journal of Applied Mathematics and Computer Science (2002)

  • Volume: 12, Issue: 1, page 17-26
  • ISSN: 1641-876X

Abstract

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We consider dynamic problems which describe frictional contact between a body and a foundation. The constitutive law is viscoelastic or elastic and the frictional contact is modelled by a general subdifferential condition on the velocity, including the normal damped responses. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of second-order evolution variational inequalities. We show that the solutions of the viscoelastic problems converge to the solution of the corresponding elastic problem as the viscosity tensor tends to zero and when the frictional potential function converges to the corresponding function in the elastic problem.

How to cite

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Chau, Oanh, and Motreanu, Viorica. "Dynamic contact problems with velocity conditions." International Journal of Applied Mathematics and Computer Science 12.1 (2002): 17-26. <http://eudml.org/doc/207565>.

@article{Chau2002,
abstract = {We consider dynamic problems which describe frictional contact between a body and a foundation. The constitutive law is viscoelastic or elastic and the frictional contact is modelled by a general subdifferential condition on the velocity, including the normal damped responses. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of second-order evolution variational inequalities. We show that the solutions of the viscoelastic problems converge to the solution of the corresponding elastic problem as the viscosity tensor tends to zero and when the frictional potential function converges to the corresponding function in the elastic problem.},
author = {Chau, Oanh, Motreanu, Viorica},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {maximal monotone operator; nonlinear hyperbolic variational inequality; weak solution; dynamic process; subdifferential boundary condition; elastic; viscoelastic},
language = {eng},
number = {1},
pages = {17-26},
title = {Dynamic contact problems with velocity conditions},
url = {http://eudml.org/doc/207565},
volume = {12},
year = {2002},
}

TY - JOUR
AU - Chau, Oanh
AU - Motreanu, Viorica
TI - Dynamic contact problems with velocity conditions
JO - International Journal of Applied Mathematics and Computer Science
PY - 2002
VL - 12
IS - 1
SP - 17
EP - 26
AB - We consider dynamic problems which describe frictional contact between a body and a foundation. The constitutive law is viscoelastic or elastic and the frictional contact is modelled by a general subdifferential condition on the velocity, including the normal damped responses. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of second-order evolution variational inequalities. We show that the solutions of the viscoelastic problems converge to the solution of the corresponding elastic problem as the viscosity tensor tends to zero and when the frictional potential function converges to the corresponding function in the elastic problem.
LA - eng
KW - maximal monotone operator; nonlinear hyperbolic variational inequality; weak solution; dynamic process; subdifferential boundary condition; elastic; viscoelastic
UR - http://eudml.org/doc/207565
ER -

References

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  1. Amassad A., Shillor M. and Sofonea M. (1999): Aquasistatic contact problem for an elastic perfectly plastic body with Tresca's friction. - Nonlin. Anal., Vol. 35, No. 1, pp. 95-109. Zbl0923.73054
  2. Andrews K.T., Shillor M. and Kuttler K.L. (1997a): On the dynamic behaviorof a thermoviscoelastic body in frictional contact. -Europ. J. Appl. Math., Vol. 8, No. 4, pp. 417-436. Zbl0894.73135
  3. Andrews K.T., Klarbring A., Shillor M. and Wright S. (1997b): A dynamic contact problem with friction and wear. -Int. J. Eng. Sci., Vol. 35, No. 14, pp. 1291-1309. Zbl0903.73065
  4. Awbi B., Essoufi El.H. and Sofonea M. (2000): A viscoelastic contact problem with normal damped response and friction. - Annales Polonici Mathematici, Vol. 75, No. 3, pp. 233-246. Zbl0994.74051
  5. Barbu V. (1976): Nonlinear Semigroups and Differential Equations in Banach Spaces. - Leyden: Editura Academiei, Bucharest-Noordhoff. Zbl0328.47035
  6. Chau O., Han W. and Sofonea M. (2001a): Analysis and approximation of a viscoelastic contact problem with slip dependent friction. - Dynam. Cont. Discr. Impuls. Syst., Series B: Vol. 8, No. 2, pp. 153-174. Zbl1013.74053
  7. Chau O., Motreanu D. and Sofonea M. (2001b): Quasistatic Frictional Problems for Elastic and Viscoelastic Materials. -Applications of Mathematics, (to appear). Zbl1090.74041
  8. Duvaut G. and Lions J. L. (1976): Inequalities in Mechanics and Physics - Berlin: Springer-Verlag. Zbl0331.35002
  9. Han W. and Sofonea M. (2000): Evolutionary variational inequalities arising in viscoelastic contact problems. - SIAMJ. Num. Anal., Vol. 38, No. 2, pp. 556-579. Zbl0988.74048
  10. Han W. and Sofonea M. (2001): Time-dependent variational inequalities for viscoelastic contact problems. - J. Comput. Appl. Math. (to appear). Zbl1001.74087
  11. Jaruv sek J. and Eck C. (1999): Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions. -Math. Models Meth. Appl. Sci., Vol. 9, No. 1, pp. 11-34. Zbl0938.74048
  12. Kavian O. (1993): Introduction à la theorie des points critique set applications aux equations elliptiques. - Berlin: Springer. Zbl0797.58005
  13. Kuttler K. L. and Shillor M. (1999): Set-valued pseudomonotone maps and degenerate evolution inclusions -Comm. Contemp. Math., Vol. 1, No. 1, pp. 87-123. Zbl0959.34049
  14. Martins J.A.C. and Oden T.J. (1987), Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. - Nonlin. Anal., Vol. 11, No. 3, pp. 407-428. Zbl0672.73079
  15. Nečas J. and Hlavaček I. (1981): Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction. - Amsterdam: Elsevier. Zbl0448.73009
  16. Panagiotopoulos P.D. (1985), Inequality Problems in Mechanical and Applications. - Basel: Birkhauser. Zbl0579.73014
  17. Rochdi M. and Shillor M. (2001c), A dynamic thermoviscoelastic frictional contact problem with damped response (submitted). 

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