We study a mathematical model of mass points longitudinally oscillating between thermoelastoplastic springs. It is derived as a discrete version of a continuous model of longitudinal oscillations of a one-dimensional object. The problem is formulated as a system of nonlinear ordinary differential equations with Prandtl-Ishlinskii type of nonlinearity, subsequently simplified using the first integral of the energy. We show that the system is asymptotically directed to one of the many possible steady states, where all movements cease and temperatures equalize.
Citation: Jana Kopfová, Petra Nábělková. Thermoelastoplastic oscillator with Prandtl-Ishlinskii operator[J]. Mathematics in Engineering, 2025, 7(3): 264-280. doi: 10.3934/mine.2025012
We study a mathematical model of mass points longitudinally oscillating between thermoelastoplastic springs. It is derived as a discrete version of a continuous model of longitudinal oscillations of a one-dimensional object. The problem is formulated as a system of nonlinear ordinary differential equations with Prandtl-Ishlinskii type of nonlinearity, subsequently simplified using the first integral of the energy. We show that the system is asymptotically directed to one of the many possible steady states, where all movements cease and temperatures equalize.
| [1] | J. Kopfová, P. Nábělková, T. Černíková, Thermoelastic oscillator as a simple model of the heat death of the universe, 2024. |
| [2] | P. Krejčí, Hysteresis, convexity and dissipation in hyperbolic equations, Tokyo: Gakkotosho, Vol. 8, 1996. |
| [3] |
F. Shakeriaski, M. Ghodrat, J. Escobedo-Diaz, M. Behnia, Recent advances in generalized thermoelasticity theory and the modified models: a review, J. Comput. Des. Eng., 8 (2021), 15–35. https://doi.org/10.1093/jcde/qwaa082 doi: 10.1093/jcde/qwaa082
|
| [4] |
C. A. Egan, C. H. Lineweaver, A larger estimate of the entropy of the universe, Astrophys. J., 710 (2010), 1825. https://doi.org/10.1088/0004-637X/710/2/1825 doi: 10.1088/0004-637X/710/2/1825
|
| [5] | L. Smolin, Time, laws, and the future of cosmology, Phys. Today, 67 (2014), 38–43. https://doi.org/10.1063/PT.3.2310 |
| [6] |
M. Zak, Modelling 'Life' against 'heat death', Int. J. Astrobiol., 17 (2018), 61–69. https://doi.org/10.1017/S147355041700009X doi: 10.1017/S147355041700009X
|
| [7] | A. Bermúdez de Castro, Continuum thermomechanics, Progress in Mathematical Physics, Vol. 43, Birkhäuser Basel, 2005. https://doi.org/10.1007/3-7643-7383-0 |
| [8] | A. Y. Ishlinskii, Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR, Techn. Ser., 9 (1944), 583–590. |
| [9] |
L. Prandtl, Ein gedankenmodell zur kinetischen theorie der festen Körper, Z. Ang. Math. Mech., 8 (1928), 85–106. https://doi.org/10.1002/zamm.19280080202 doi: 10.1002/zamm.19280080202
|
| [10] | M. A. Krasnosel'skii, A. V. Pokrovskii, Systems with hysteresis, Springer, 1989. https://doi.org/10.1007/978-3-642-61302-9 |
| [11] | M. Brokate, J. Sprekels, Hysteresis and phase transitions, Applied Mathematical Science, Vol. 121, Springer-Verlag, 1996. https://doi.org/10.1007/978-1-4612-4048-8 |
| [12] | P. Hartman, Ordinary differential equations, Baltimore, 1973. |
| [13] | C. H. Edwards, D. E. Penney, Elementary differential equations and boundary value problems, New Jersey: Pearson, 2008. |
Figures(2)
Jana Kopfová, Petra Nábělková. Thermoelastoplastic oscillator with Prandtl-Ishlinskii operator[J]. Mathematics in Engineering, 2025, 7(3): 264-280. doi: 10.3934/mine.2025012
DownLoad: