Existence and uniqueness for a quasistatic frictional bilateral contact problem in thermoviscoelasticity
Authors:
M. Rochdi and M. Shillor
Journal:
Quart. Appl. Math. 58 (2000), 543-560
MSC:
Primary 74M15; Secondary 35Q72, 74D10, 74F05, 74G25, 74G30, 74M10
DOI:
https://doi.org/10.1090/qam/1770654
MathSciNet review:
MR1770654
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Abstract: We prove the existence and uniqueness of the weak solution for a quasistatic thermoviscoelastic problem which describes bilateral frictional contact between a deformable body and a moving rigid foundation. The model consists of the heat equation for the temperature, the elliptic viscoelasticity system for the displacements, the SJK-Coulomb law of friction and frictional heat generation condition. The proof is accomplished in two steps. First, the existence of solutions for a regularized problem is established and a priori estimates obtained. Then the limit function, which is the weak solution of the original problem, is shown to be the unique fixed point of the solution operator when the friction coefficient is small.
- Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
- Lars-Erik Andersson, A quasistatic frictional problem with normal compliance, Nonlinear Anal. 16 (1991), no. 4, 347–369. MR 1093846, DOI https://doi.org/10.1016/0362-546X%2891%2990035-Y
- Kevin T. Andrews, Peter Shi, Meir Shillor, and Steve Wright, Thermoelastic contact with Barber’s heat exchange condition, Appl. Math. Optim. 28 (1993), no. 1, 11–48. MR 1208796, DOI https://doi.org/10.1007/BF01188756
- K. T. Andrews, K. L. Kuttler, and M. Shillor, On the dynamic behaviour of a thermoviscoelastic body in frictional contact with a rigid obstacle, European J. Appl. Math. 8 (1997), no. 4, 417–436. MR 1471601, DOI https://doi.org/10.1017/S0956792597003173
- Kevin T. Andrews, M. Shillor, S. Wright, and A. Klarbring, A dynamic thermoviscoelastic contact problem with friction and wear, Internat. J. Engrg. Sci. 35 (1997), no. 14, 1291–1309. MR 1488812, DOI https://doi.org/10.1016/S0020-7225%2897%2987426-5
- Jean-Pierre Aubin, Approximation of elliptic boundary-value problems, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1972. Pure and Applied Mathematics, Vol. XXVI. MR 0478662
- Marius Cocu, Elaine Pratt, and Michel Raous, Formulation and approximation of quasistatic frictional contact, Internat. J. Engrg. Sci. 34 (1996), no. 7, 783–798. MR 1397607, DOI https://doi.org/10.1016/0020-7225%2895%2900121-2
- G. Duvaut, Loi de frottement non locale, Indian J. Pure Appl. Math. 13 (1982), no. 8, 73–78 (French, with English summary). MR 670346
- G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972 (French). Travaux et Recherches Mathématiques, No. 21. MR 0464857
- I. Figueiredo and L. Trabucho, A class of contact and friction dynamic problems in thermoelasticity and in thermoviscoelasticity, Internat. J. Engrg. Sci. 33 (1995), no. 1, 45–66. MR 1309740, DOI https://doi.org/10.1016/0020-7225%2894%29E0042-H
- R. J. Gu, K. L. Kuttler, and M. Shillor, Frictional wear of a thermoelastic beam, J. Math. Anal. Appl. 242 (2000), no. 2, 212–236. MR 1737847, DOI https://doi.org/10.1006/jmaa.1999.6652
R. J. Gu and M. Shillor, Thermal and wear finite elements analysis of an elastic beam in sliding contact, Internat. J. Solids Structures, to appear
- Lars Johansson and Anders Klarbring, Thermoelastic frictional contact problems: modelling, finite element approximation and numerical realization, Comput. Methods Appl. Mech. Engrg. 105 (1993), no. 2, 181–210. MR 1220080, DOI https://doi.org/10.1016/0045-7825%2893%2990122-E
- Kenneth L. Kuttler Jr., Time-dependent implicit evolution equations, Nonlinear Anal. 10 (1986), no. 5, 447–463. MR 839357, DOI https://doi.org/10.1016/0362-546X%2886%2990050-7
- K. L. Kuttler, Y. Renard, and M. Shillor, Models and simulations of dynamic frictional contact of a beam, Comput. Methods Appl. Mech. Engrg. 177 (1999), no. 3-4, 259–272. Computational modeling of contact and friction. MR 1710454, DOI https://doi.org/10.1016/S0045-7825%2898%2900384-3
- Kenneth L. Kuttler and Meir Shillor, Set-valued pseudomonotone maps and degenerate evolution inclusions, Commun. Contemp. Math. 1 (1999), no. 1, 87–123. MR 1681814, DOI https://doi.org/10.1142/S0219199799000067
- N. Kikuchi and J. T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, vol. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. MR 961258
A. Klarbring, A. Mikelic, and M. Shillor, A global existence result for the quasistatic frictional contact problem with normal compliance, pp. 85–111 in Unilateral Problems in Structural Mechanics IV, eds. G. DelPiero and F. Maceri, Birkhäuser, Boston, 1991
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, 1969
M. Rochdi and M. Shillor, A dynamic thermoviscoelastic frictional contact problem with damped response, preprint, 1998
- M. Rochdi, M. Shillor, and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction, J. Elasticity 51 (1998), no. 2, 105–126. MR 1664496, DOI https://doi.org/10.1023/A%3A1007413119583
- M. Rochdi, M. Shillor, and M. Sofonea, A quasistatic contact problem with directional friction and damped response, Appl. Anal. 68 (1998), no. 3-4, 409–422. MR 1701715, DOI https://doi.org/10.1080/00036819808840639
M. Shillor, editor, Contact Mechanics, a special issue of Computer and Math. Modelling 28, 4–8 (1998)
- Meir Shillor and Mircea Sofonea, A quasistatic viscoelastic contact problem with friction, Internat. J. Engrg. Sci. 38 (2000), no. 14, 1517–1533. MR 1763038, DOI https://doi.org/10.1016/S0020-7225%2899%2900126-3
- Niclas Strömberg, Lars Johansson, and Anders Klarbring, Derivation and analysis of a generalized standard model for contact, friction and wear, Internat. J. Solids Structures 33 (1996), no. 13, 1817–1836. MR 1392130, DOI https://doi.org/10.1016/0020-7683%2895%2900140-9
R. S. Adams, Sobolev Spaces, Pure and Applied Mathematics, vol. 65, Academic Press, New York and London, 1975
L.-E. Andersson, A quasistatic frictional problem with normal compliance, Nonlinear Anal. 16(4), 347–369 (1991)
K. T. Andrews, P. Shi, M. Shillor, and S. Wright, Thermoelastic contact with Barber’s heat exchange condition, Appl. Math. Opt. 28(1), 11 48 (1993)
K. T. Andrews, K. L. Kuttler, and M. Shillor, On the dynamic behaviour of a thermoviscoelastic body in frictional contact with a rigid obstacle, European J. Appl. Math. 8, 417–436 (1997)
K. T. Andrews, A. Klarbring, M. Shillor, and S. Wright, On the the dynamic behavior of a thermoviscoelastic contact problem with friction and wear, Internat. J. Engrg. Sci. 35(14), 1291–1309 (1997)
J.-P. Aubin, Approximation of Elliptic Boundary Value Problems, John Wiley and Sons, New York, 1972
M. Cocu, E. Pratt, and M. Raous, Formulation and approximation of quasistatic frictional contact, Internat. J. Engrg. Sci. 34(7), 783–798 (1996)
G. Duvaut, Loi de frottement non locale, J. Méc. Thé. Appl., Special issue, 73-78 (1982)
G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, 1972
I. Figueiro and L. Trabucho, A class of contact and friction dynamic problems in thermoelasticity and in thermoviscoelasticity, Internat. J. Engrg. Sci. 33(1), 45–66 (1995)
R. J. Gu, K. L. Kuttler, and M. Shillor, Frictional wear of a thermoelastic beam, J. Math. Anal. Appl. 242, 212–236 (2000)
R. J. Gu and M. Shillor, Thermal and wear finite elements analysis of an elastic beam in sliding contact, Internat. J. Solids Structures, to appear
L. Johansson and A. Klarbring, Thermoelastic frictional contact problems: Modelling, finite element approximation and numerical realization, Comput. Methods Appl. Mech. Engrg. 105, 181–210 (1993)
K. L. Kuttler, Time-dependent implicit evolution equations, Nonlinear Anal. 10(5), 447–463 (1986)
K. L. Kuttler, Y. Renard, and M. Shillor, Models and simulations of dynamic frictional contact of a beam. Computational modeling of contact and friction, Comput. Methods Appl. Mech. Engrg. 177, 259–272 (1999)
K. L. Kuttler and M. Shillor, Set-valued pseudomonotone maps and degenerate evolution inequalities, Commun. Contemp. Math. 1, 87–123 (1999)
N. Kikuchi and T. J. Oden, Contact Problems in Elasticity, SIAM, 1988
A. Klarbring, A. Mikelic, and M. Shillor, A global existence result for the quasistatic frictional contact problem with normal compliance, pp. 85–111 in Unilateral Problems in Structural Mechanics IV, eds. G. DelPiero and F. Maceri, Birkhäuser, Boston, 1991
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, 1969
M. Rochdi and M. Shillor, A dynamic thermoviscoelastic frictional contact problem with damped response, preprint, 1998
M. Rochdi, M. Shillor, and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction, J. Elasticity 51, 105–126 (1998)
M. Rochdi, M. Shillor, and M. Sofonea, A quasistatic contact problem with directional friction and damped response, Applicable Analysis 68(3–4), 409–422 (1998)
M. Shillor, editor, Contact Mechanics, a special issue of Computer and Math. Modelling 28, 4–8 (1998)
M. Shillor and M. Sofonea, A quasistatic viscoelastic contact problem with friction, Internat. J. Engrg. Sci., to appear
N. Strömberg, L. Johansson, and A. Klarbring, Derivation and analysis of a generalized standard model for contact friction and wear, Internat. J. Solids Structures 33(13), 1817–1836 (1996)
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© Copyright 2000
American Mathematical Society