A parabolic system modeling the thermoelastic contact of two rods
Authors:
Kevin T. Andrews, Peter Shi, Meir Shillor and Steve Wright
Journal:
Quart. Appl. Math. 53 (1995), 53-68
MSC:
Primary 73B30; Secondary 35K60, 35Q72, 73T05
DOI:
https://doi.org/10.1090/qam/1315447
MathSciNet review:
MR1315447
Full-text PDF Free Access
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Abstract: We consider an initial-boundary value problem for a nonlinear parabolic system that arises naturally in modeling behavior of two (possibly dissimilar) thin homogeneous rods. Each rod is held fixed at one end and is free to expand or contract at the other, as a result of the evolution of its temperature and stress fields. The two rods may also come into contact at their free ends. We establish the existence and uniqueness of a strong solution to the system, assuming that the thermal expansion coefficients are small. The proofs rely on a priori estimates, the functional method of Ladyzhenskaya, and Schauder’s fixed-point theorem.
K. T. Andrews, A. Mikelić, P. Shi, M. Shillor, and S. Wright, One-dimensional thermoelastic contact with a stress-dependent radiation condition, SIAM J. Math. Anal. 23, 1393–1416 (1992)
K. T. Andrews, P. Shi, M. Shillor, and S. Wright, Thermoelastic contact with Barber’s heat exchange condition, Appl. Math. Optim. 28, 11–48 (1993)
K. T. Andrews and M. Shillor, A parabolic initial-boundary value problem modeling axially symmetric thermoelastic contact, Nonlinear Analysis (to appear)
J. R. Barber, Stability of thermoelastic contact, IMech. E. Internat. Conf. Tribology, Mechanical Engineering Publications Ltd., London, 1987, pp. 981–986
J. R. Barber and R. Zhang, Transient behaviour and stability for the thermoelastic contact of two rods of dissimilar materials, Internat. J. Mech. Sci. 30, 691–704 (1988)
D. E. Carlson, Linear thermoelasticity, Handbuch der Physik, Vol. VIa/2 (S. Flugge, ed.), Springer-Verlag, Berlin, 1972, pp. 297–345
C. Cheng and M. Shillor, Numerical solutions to the problem of thermoelastic contact of two rods, Math. and Comput. Modelling 17, no. 10, 53–71 (1993)
M. I. M. Copetti and C. M. Elliott, A one-dimensional quasi-static contact problem in linear thermoelasticity, European J. Appl. Math. 4, 151–174 (1993)
W. A. Day, Heat Conduction with Linear Thermoelasticity, Springer-Verlag, New York, 1985
R. P. Gilbert, P. Shi, and M. Shillor, A quasistatic contact problem in linear thermoelasticity, Rend. Mat. 10, 785–808 (1990)
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York, 1985
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc., Providence, RI, 1968
J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, New York, 1972
T. Roubiček, A coupled contact problem in nonlinear thermo-visco-elasticity, preprint
P. Shi and M. Shillor, Uniqueness and stability of the solution to a thermoelastic contact problem, European J. Appl. Math. 1, 371–387 (1990)
P. Shi and M. Shillor, A quasistatic contact problem in thermoelasticity with a radiation condition for the temperature, J. Math. Anal. Appl. 172, 147–165 (1993)
P. Shi and M. Shillor, Existence of a solution to the N-dimensional problem of thermoelastic contact, Comm. Partial Differential Equations 17, 1597–1618 (1992)
P. Shi, M. Shillor, and X. L. Zou, Numerical solutions to one-dimensional problems of thermoelastic contact, Comput. Math. Appl. 22, no. 10, 65–78 (1991)
R. Zhang and J. R. Barber, Effect of material properties on the stability of static thermoelastic contact, J. Appl. Mech. 57, 365–369 (1990)
X. Zou, Existence and uniqueness of a solution to the axially symmetric thermoelastic contact problem, M. S. Thesis, Oakland Univ., Rochester, MI, 1991
X. Zou, Existence and uniqueness of a solution to a singular thermoelastic contact problem, Applicable Anal. 51, no. 2, 139 (1993)
K. T. Andrews, A. Mikelić, P. Shi, M. Shillor, and S. Wright, One-dimensional thermoelastic contact with a stress-dependent radiation condition, SIAM J. Math. Anal. 23, 1393–1416 (1992)
K. T. Andrews, P. Shi, M. Shillor, and S. Wright, Thermoelastic contact with Barber’s heat exchange condition, Appl. Math. Optim. 28, 11–48 (1993)
K. T. Andrews and M. Shillor, A parabolic initial-boundary value problem modeling axially symmetric thermoelastic contact, Nonlinear Analysis (to appear)
J. R. Barber, Stability of thermoelastic contact, IMech. E. Internat. Conf. Tribology, Mechanical Engineering Publications Ltd., London, 1987, pp. 981–986
J. R. Barber and R. Zhang, Transient behaviour and stability for the thermoelastic contact of two rods of dissimilar materials, Internat. J. Mech. Sci. 30, 691–704 (1988)
D. E. Carlson, Linear thermoelasticity, Handbuch der Physik, Vol. VIa/2 (S. Flugge, ed.), Springer-Verlag, Berlin, 1972, pp. 297–345
C. Cheng and M. Shillor, Numerical solutions to the problem of thermoelastic contact of two rods, Math. and Comput. Modelling 17, no. 10, 53–71 (1993)
M. I. M. Copetti and C. M. Elliott, A one-dimensional quasi-static contact problem in linear thermoelasticity, European J. Appl. Math. 4, 151–174 (1993)
W. A. Day, Heat Conduction with Linear Thermoelasticity, Springer-Verlag, New York, 1985
R. P. Gilbert, P. Shi, and M. Shillor, A quasistatic contact problem in linear thermoelasticity, Rend. Mat. 10, 785–808 (1990)
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York, 1985
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and quasilinear equations of parabolic type, Amer. Math. Soc., Providence, RI, 1968
J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, New York, 1972
T. Roubiček, A coupled contact problem in nonlinear thermo-visco-elasticity, preprint
P. Shi and M. Shillor, Uniqueness and stability of the solution to a thermoelastic contact problem, European J. Appl. Math. 1, 371–387 (1990)
P. Shi and M. Shillor, A quasistatic contact problem in thermoelasticity with a radiation condition for the temperature, J. Math. Anal. Appl. 172, 147–165 (1993)
P. Shi and M. Shillor, Existence of a solution to the N-dimensional problem of thermoelastic contact, Comm. Partial Differential Equations 17, 1597–1618 (1992)
P. Shi, M. Shillor, and X. L. Zou, Numerical solutions to one-dimensional problems of thermoelastic contact, Comput. Math. Appl. 22, no. 10, 65–78 (1991)
R. Zhang and J. R. Barber, Effect of material properties on the stability of static thermoelastic contact, J. Appl. Mech. 57, 365–369 (1990)
X. Zou, Existence and uniqueness of a solution to the axially symmetric thermoelastic contact problem, M. S. Thesis, Oakland Univ., Rochester, MI, 1991
X. Zou, Existence and uniqueness of a solution to a singular thermoelastic contact problem, Applicable Anal. 51, no. 2, 139 (1993)
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Article copyright:
© Copyright 1995
American Mathematical Society
