Analysis of a viscoelastic antiplane contact problem with slip-dependent friction

Thierry-Vincent Hoarau-Mantel; Andaluzia Matei

International Journal of Applied Mathematics and Computer Science (2002)

  • Volume: 12, Issue: 1, page 51-58
  • ISSN: 1641-876X

Abstract

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We study a mathematical problem modelling the antiplane shear deformation of a viscoelastic body in frictional contact with a rigid foundation. The contact is bilateral and is modelled with a slip-dependent friction law. We present the classical formulation for the antiplane problem and write the corresponding variational formulation. Then we establish the existence of a unique weak solution to the model, by using the Banach fixed-point theorem and classical results for elliptic variational inequalities. Finally, we prove that the solution converges to the solution of the corresponding elastic problem as the viscosity converges to zero.

How to cite

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Hoarau-Mantel, Thierry-Vincent, and Matei, Andaluzia. "Analysis of a viscoelastic antiplane contact problem with slip-dependent friction." International Journal of Applied Mathematics and Computer Science 12.1 (2002): 51-58. <http://eudml.org/doc/207568>.

@article{Hoarau2002,
abstract = {We study a mathematical problem modelling the antiplane shear deformation of a viscoelastic body in frictional contact with a rigid foundation. The contact is bilateral and is modelled with a slip-dependent friction law. We present the classical formulation for the antiplane problem and write the corresponding variational formulation. Then we establish the existence of a unique weak solution to the model, by using the Banach fixed-point theorem and classical results for elliptic variational inequalities. Finally, we prove that the solution converges to the solution of the corresponding elastic problem as the viscosity converges to zero.},
author = {Hoarau-Mantel, Thierry-Vincent, Matei, Andaluzia},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {viscoelastic law; slip-dependent friciotn law; antiplane problem; convergence; vanishing viscosity; antiplane shear deformation; classical formulation; variational formulation; existence; unique weak solution; Banach fixed-point theorem; elliptic variational inequalities},
language = {eng},
number = {1},
pages = {51-58},
title = {Analysis of a viscoelastic antiplane contact problem with slip-dependent friction},
url = {http://eudml.org/doc/207568},
volume = {12},
year = {2002},
}

TY - JOUR
AU - Hoarau-Mantel, Thierry-Vincent
AU - Matei, Andaluzia
TI - Analysis of a viscoelastic antiplane contact problem with slip-dependent friction
JO - International Journal of Applied Mathematics and Computer Science
PY - 2002
VL - 12
IS - 1
SP - 51
EP - 58
AB - We study a mathematical problem modelling the antiplane shear deformation of a viscoelastic body in frictional contact with a rigid foundation. The contact is bilateral and is modelled with a slip-dependent friction law. We present the classical formulation for the antiplane problem and write the corresponding variational formulation. Then we establish the existence of a unique weak solution to the model, by using the Banach fixed-point theorem and classical results for elliptic variational inequalities. Finally, we prove that the solution converges to the solution of the corresponding elastic problem as the viscosity converges to zero.
LA - eng
KW - viscoelastic law; slip-dependent friciotn law; antiplane problem; convergence; vanishing viscosity; antiplane shear deformation; classical formulation; variational formulation; existence; unique weak solution; Banach fixed-point theorem; elliptic variational inequalities
UR - http://eudml.org/doc/207568
ER -

References

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  2. Campillo M. and Ionescu I.R. (1997): Initiation of antiplane shear instability under slip dependent friction. - J.Geophys. Res., Vol. B9, pp. 363-371. 
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  5. Horgan C.O. (1995): Decay estimates for boundary-value problems in linear and nonlinear continuum mechanics, In: Mathematical Problems in Elasticity (R. Russo, Ed.). - Singapore: World Scientific, pp. 47-89. Zbl0843.73014
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  7. Ionescu I.R. and Paumier J.-C. (1996): On the contact problem with slip displacement dependent friction in elastostatics. - Int. J.Eng. Sci., Vol. 34, No. 4, pp. 471-491. Zbl0900.73682
  8. Kuttler K.L. and Shillor M. (1999): Set-valued pseudomonotone maps and degenerate evolution equations. - Comm. Contemp. Math., Vol. 1, No. 1, pp. 87-123. Zbl0959.34049
  9. Matei A., Motreanu V.V. and Sofonea M. (2001): A Quasistatic Antiplane Contact Problem With Slip Dependent Friction. - Adv. Nonlin. Variat. Ineq., Vol. 4, No. 2, pp. 1-21. Zbl1205.74132
  10. Motreanu D. and Sofonea M. (1999): Evolutionary variational inequalities arising in quasistatic frictional contact problems for elastic materials. - Abstr. Appl. Anal., Vol. 4, No. 3, pp. 255-279. Zbl0974.58019
  11. Ohnaka M. (1996): Nonuniformity of the constitutive law parameters for shear rupture and quasistatic nucleation to dynamic rupture: A physical model of earthquake generation model. - Earthquake Prediction: The Scientific Challange, Irvine, CA: Acad. of Sci. 
  12. Rabinowicz E. (1965): Friction and Wear of Materials. -New York: Wiley. 

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