Mathematical and numerical approaches to a one-dimensional dynamic thermoviscoelastic contact problem. (English) Zbl 1326.74096
Summary: We propose numerical schemes for solving a nonlinear system which consists of a coupled partial differential equations and two conditions, called normal compliance contact condition and Barber’s heat exchange condition. The convergence of numerical trajectories is shown by using a time discretization and passing the limit of the time step size. The uniqueness of the weak solution is proved as well. We derive the extensive form of an energy balance which will be a criterion to examine numerical stability. An example is provided to present and discuss numerical results.
MSC:
| 74M15 | Contact in solid mechanics |
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 74D05 | Linear constitutive equations for materials with memory |
| 74F05 | Thermal effects in solid mechanics |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 74H20 | Existence of solutions of dynamical problems in solid mechanics |
Keywords:
dynamic contact; thermoviscoelasticity; normal compliance; Barber’s heat exchange conditionReferences:
| [1] | Carlson, D. E., Linear Thermoelasiticity, Handbuch der Physik. Bd., vol. Vla/2 (1972), Springer-Verlag: Springer-Verlag Berlin |
| [2] | Sofonea, M.; Shillor, M.; Telega, J. J., Models and Analysis of Quasistatic Contact, Lecture Notes in Physics (2004), Springer: Springer Berlin · Zbl 1180.74046 |
| [3] | Frémond, M., Non-Smooth Thermomechanics (2002), Springer: Springer Berlin · Zbl 0990.80001 |
| [4] | Alan Day, William, Heat Conduction within Linear Thermoelasiticity, vol. 30 (1885), Springer-Verlag: Springer-Verlag New York |
| [5] | Klarbring, A.; Mikelić, A.; Shillor, M., Frictional contact problems with normal compliance, Int. J. Engng. Sci., 26, 8, 811-832 (1988) · Zbl 0662.73079 |
| [6] | Kuttler, Kenneth L.; Shillor, Meir, Dynamic contact with normal compliance wear and discontinuous friction coefficient, SIAM J. Math. Anal., 34, 1, 1-27 (2002) · Zbl 1029.74033 |
| [7] | Martins, J. A.C; Oden, J. T., Models and computational methods for dynamic frcition phenomena, Comput. Meth. Appl. Engng., 52, 527-634 (1985) · Zbl 0567.73122 |
| [8] | Martins, J. A.C; Oden, J. T., Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Anal. TMA, 11, 3, 407-428 (1987) · Zbl 0672.73079 |
| [9] | Barber, J. R., Contact problems invloving a cooled punch, J. Elasticity, 8, 4, 409-423 (1978) · Zbl 0392.73100 |
| [10] | Barber, J. R.; Dundurs, J.; Comninou, M., Stability consideration in thermoelastic contact, J. Appl. Mech., 47, 871-874 (1980) · Zbl 0447.73006 |
| [11] | Andrews, K. T.; Mikelić, Andero; Shi, P.; Shillor, M.; Wright, S., One-dimensional thermoelastic contact with stress-dependent radiation condition, SIAM J. Math. Anal., 23, 6, 1396-1416 (1992) · Zbl 0770.35035 |
| [12] | Andrews, K. T.; Shi, P.; Shillor, M.; Wright, S., Themroelastic contact with Barber’s heat exchange conditions, Appl. Math. Opt, 28, 1, 11-48 (1993) · Zbl 0807.35064 |
| [13] | Andrews, Kevin T.; Mikelic, Andro; Shi, Peter; Shillor, Meir; Wright, Steve, One-dimensional themroelastic contact with a stress-dependent radiation condition, SIAM J. Math. Anal., 23, 6, 1393-1416 (1992) · Zbl 0770.35035 |
| [14] | Shi, P.; Shillor, M.; Wright, S., A quasistatic contact in thermoelasticity with radiation conditions for the temperature, J. Math. Anal. Appl., 172, 147-165 (1993) · Zbl 0771.73078 |
| [15] | Shi, Peter; Shillor, Meir; Zou, Xiu-Lin, Numerical solutions to one-dimensional problems of themroelastic contact, Comput. Math. Appl., 22, 10, 65-78 (1991) · Zbl 0747.73055 |
| [17] | Gilbert, R. P.; Shi, P.; Shillor, M., A quasistatic contact problem in linear thermoelasticity, Rend. Mat., 7, 10, 785-808 (1990) · Zbl 0748.73011 |
| [18] | Shi, P.; Shillor, M., Uniqueness and stability of the solution to a thermoelastic contact problem, European J. Appl. Math., 1, 371-387 (1990) · Zbl 0722.73058 |
| [19] | Duvaut, G., (Free Boundary Problems Connected with Thermoeelasticity and Unilateral Contact. Free Boundary Problems Connected with Thermoeelasticity and Unilateral Contact, Free Boundary Problems: Proceedings of a Seminar held in Pavia, vol. II (1979), Instituto nazionale: Instituto nazionale Roma) |
| [20] | Wloka, J., Partial Differential Equations (1987), Cambridge University Press · Zbl 0623.35006 |
| [21] | Kuttler, Kenneth L., Modern Analysis (1998), CRC Press: CRC Press Boca Raton, FL · Zbl 0893.46001 |
| [22] | Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1978), North Holland: North Holland Amsterdam · Zbl 0387.46033 |
| [23] | Ahn, Jeongho; Stewart, David E., A viscoelastic Timoshenko beam with dynamic frictionless impact, Discrete Contin. Dyn. Syst. Ser. B, 12, 1, 1-22 (2009) · Zbl 1167.74034 |
| [24] | Evans, Lawrence C., (Partial Differential Equations. Partial Differential Equations, Graduate Studies in Mathematics, vol. 19 (1998), AMS: AMS Providence, Rhode Island) · Zbl 0902.35002 |
| [25] | Ahn, Jeongho; Kuttler, Kenneth L.; Shillor, Meir, Dynamic contact of two gao beams, Electron. J. Differential Equations, 2012, 194, 1-42 (2002) · Zbl 1302.74116 |
| [26] | Kuttler, Kenneth L.; Shillor, Meir, Vibrations of a beam between two stops, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 8, 1, 93-110 (2001) · Zbl 1013.74033 |
| [27] | Knops, R. J.; Villaggio, Piero, On oblique impact of a rigid rod against a winkler foundation, Continum Mech. Thermodyn., 24, 559-582 (2012) · Zbl 1258.74127 |
| [28] | Shi, Peter, The restitution coefficient for a linear elastic rod, Comput. Model., 28, 4-8, 427-435 (1998) · Zbl 1122.74429 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
