Fully Discrete Approximations and an a Priori Error Analysis of a Two–Temperature Thermo–Elastic Model with Microtemperatures

1Abo-Dahab, S.M. (2020). A two-temperature generalized magneto-thermoelastic formulation for a rotating medium with thermal shock under hydrostatic initial stress, Continuum Mechanics and Thermodynamics 32(3): 883–900.Search in Google Scholar

2Ali, G. and Romano, V. (2017). Existence and uniqueness for a two-temperature energy-transport model for semiconductors, Journal of Mathematical Analysis and Applications 449(2): 1248–1264.Search in Google Scholar

3Barabasz, B., Gajda-Zagórska, E., Migórski, S., Paszyński, M., Schaefer, R. and Smołka, M. (2014). A hybrid algorithm for solving inverse problems in elasticity, International Journal of Applied Mathematics and Computer Science 24(4): 865–886, DOI: 10.2478/amcs-2014-0064.Search in Google Scholar

4Bazarra, N., Campo, M. and Fernández, J.R. (2019). A thermoelastic problem with diffusion, microtemperatures, and microconcentrations, Acta Mechanica 230: 31–48.Search in Google Scholar

5Bazarra, N., Fernández, J.R., Magaña, A. and Quintanilla, R. (2020). Numerical analysis of a dual-phase-lag model involving two temperatures, Mathematical Methods in the Applied Sciences 43(5): 2759–2771.Search in Google Scholar

6Campo, M., Copetti, M.I., Fernández, J.R. and Quintanilla, R. (2022). On existence and numerical approximation in phase-lag thermoelasticity with two temperatures, Discrete and Continuous Dynamical Systems B 27(4): 2221–2247.Search in Google Scholar

7Campo, M., Fernández, J., Kuttler, K., Shillor, M. and Viano, J. (2006). Numerical analysis and simulations of a dynamic frictionless contact problem with damage, Computer Methods in Applied Mechanics and Engineering 196(1): 476–488.Search in Google Scholar

8Chen, P. and Gurtin, M.E. (1968). On a theory of heat involving two temperatures, Zeitschrift fur angewandte Mathematik und Physik 19: 614–627.Search in Google Scholar

9Chen, P., Gurtin, M.E. and Williams, W. (1969). On the thermodynamics of non-simple materials with two temperatures, Zeitschrift fur angewandte Mathematik und Physik 20: 107–112.Search in Google Scholar

10Chen, P., Gurtin, M.E. and Williams, W.O. (1968). A note on non-simple heat conduction, Zeitschrift fur angewandte Mathematik und Physik 19: 969–970.Search in Google Scholar

11Ciarlet, P. (1993). Basic error estimates for elliptic problems, in P.G. Ciarlet and J. Lions (Eds), Handbook of Numerical Analysis, Springer-Verlag, Berlin, pp. 17–351.Search in Google Scholar

12Clement, P. (1975). Approximation by finite element functions using local regularization, RAIRO Mathematical Modeling and Numerical Analysis 9(R2): 77–84.Search in Google Scholar

13Cowin, S. (1985). The viscoelastic behavior of linear elastic materials with voids, Journal of Elasticity 15: 185–191.Search in Google Scholar

14Cowin, S. and Nunziato, J. (1983). Linear elastic materials with voids, Journal of Elasticity 13: 125–147.Search in Google Scholar

15D’Apice, C., Zampoli, V. and Chiriţă, S. (2020). On the wave propagation in the thermoelasticity theory with two temperatures, Journal of Elasticity 140(2): 257–272.Search in Google Scholar

16Feng, B. and Apalara, T.A. (2019). Optimal decay for a porous elasticity system with memory, Journal of Mathematical Analysis and Applications 470(2): 1108–1128.Search in Google Scholar

17Feng, B. and Yin, M. (2019). Decay of solutions for a one-dimensional porous elasticity system with memory: The case of non-equal wave speeds, Mathematics and Mechanics of Solids 24(8): 2361–2373.Search in Google Scholar

18Fernández, J.R. and Quintanilla, R. (2021a). Two-temperatures thermo-porous-elasticity with microtemperatures, Applied Mathematical Letters 111: 106628.Search in Google Scholar

19Fernández, J.R. and Quintanilla, R. (2021b). Uniqueness and exponential instability in a new two-temperature thermoelastic theory, AIMS Mathematics 6(6): 5440–5451.Search in Google Scholar

20Grot, R. (1969). Thermodynamics of a continuum with microstructure, International Journal of Engineering Science 7(8): 801-–814.Search in Google Scholar

21Gruais, I. and Poliševski, D. (2017). Model of two-temperature convective transfer in porous media, Zeitschrift fur angewandte Mathematik und Physik 68(6): 143.Search in Google Scholar

22Ieşan, D. (2007). Thermoelasticity of bodies with microstructure and microtemperatures, International Journal of Solids and Structures 44(25–26): 8648–8653.Search in Google Scholar

23Ieşan, D. and Quintanilla, R. (2000). On a theory of thermoelasticity with microtemperatures, Journal of Thermal Stresses 23(3): 195–215.Search in Google Scholar

24Kumar, K., Prasad, R. and Kumar, R. (2020). Thermoelastic interactions on hyperbolic two-temperature generalized thermoelasticity in an infinite medium with a cylindrical cavity, European Journal of Mechanics A: Solids 82: 104007.Search in Google Scholar

25Leseduarte, M., Magaña, A.M. and Quintanilla, R. (2010). On the time decay of solutions in porous-thermo-elasticity of type II, Discrete and Continuous Dynamical Systems B 13(2): 375–391.Search in Google Scholar

26Liu, Z. and Zheng, S. (1999). Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton.Search in Google Scholar

27Magaña, A., Miranville, A. and Quintanilla, R. (2020). Exponential decay of solutions in type II thermo-porous-elasticity with quasi-static microvoids, Journal of Mathematical Analysis and Applications 492(2): 124504.Search in Google Scholar

28Magaña, A. and Quintanilla, R. (2018). Exponential stability in type III thermoelasticity with microtemperatures, Zeitschrift fur angewandte Mathematik und Physik 69(5): 129.Search in Google Scholar

29Magaña, A. and Quintanilla, R. (2021). Decay of quasi-static porous-thermo-elastic waves, Zeitschrift fur angewandte Mathematik und Physik 72: 125.Search in Google Scholar

30Makki, A., Miranville, A. and Sadaka, G. (2019). On the nonconserved Caginalp phase-field system based on the Maxwell–Cattaneo law with two temperatures and logarithmic potentials, Discrete and Continuous Dynamical Systems Series B 24(3): 1341–1365.Search in Google Scholar

31Makki, A., Miranville, A. and Sadaka, G. (2021). On the conserved Caginalp phase-field system with logarithmic potentials based on the Maxwell–Cattaneo law with two temperatures, Applied Mathematics and Optimations 84(2): 1285–1316.Search in Google Scholar

32Miranville, A. and Quintanilla, R. (2016). On the Caginalp phase-field systems with two temperatures and the Maxwell–Cattaneo law, Mathematical Methods in the Applied Sciences 39(15): 4385–4397.Search in Google Scholar

33Miranville, A. and Quintanilla, R. (2019). Exponential decay in one-dimensional type III thermoelasticity with voids, Applied Mathematical Letters 94: 30–37.Search in Google Scholar

34Miranville, A. and Quintanilla, R. (2020). Exponential decay in one-dimensional type II thermoviscoelasticity with voids, Journal of Computational and Applied Mathematics 368: 112573.Search in Google Scholar

35Mukhopadhyay, S., Picard, R., Trostorff, S. and Waurick, M. (2017). A note on a two-temperature model in linear thermoelasticity, Mathematics and Mechanics of Solids 22(5): 905–918.Search in Google Scholar

36Nunziato, J. and Cowin, S.C. (1979). A nonlinear theory of elastic materials with voids, Archive for Rational Mechanics and Analysis 72: 175–201.Search in Google Scholar

37Pamplona, P., Muñoz-Rivera, J.M. and Quintanilla, R. (2011). On the decay of solutions for porous-elastic systems with history, Journal of Mathematical Analysis and Applications 379(2): 682—705.Search in Google Scholar

38Passarella, F., Tibullo, V. and Viccione, G. (2017). Rayleigh waves in isotropic strongly elliptic thermoelastic materials with microtemperatures, Meccanica 52: 3033–3041.Search in Google Scholar

39Riha, P. (1975). On the theory of heat-conducting micropolar fluids with microtemperatures, Acta Mechanica 23: 1–8.Search in Google Scholar

40Riha, P. (1976). On the microcontinuum model of heat conduction in materials with inner structure, International Journal of Engineering Science 14(6): 529–535.Search in Google Scholar

41Sarkar, N. and Mondal, S. (2020). Thermoelastic plane waves under the modified Green–Lindsay model with two-temperature formulation, Zeitschrift fur Angewandte Mathematik und Mechanik 100(11): e201900267.Search in Google Scholar

42Sellitto, A., Carlomagno, I. and Domenico, M.D. (2021). Nonlocal and nonlinear effects in hyperbolic heat transfer in a two-temperature model, Zeitschrift fur Angewandte Mathematik und Mechanik 72(1): 7.Search in Google Scholar

43Warren, W. and Chen, P.J. (1973). Wave propagation in two temperatures theory of thermoelaticity, Acta Mechanica 16: 83–117.Search in Google Scholar

44Youssef, H. and Elsibai, K.A. (2015). On the theory of two-temperature thermoelasticity without energy dissipation of Green–Naghdi model, Applicable Analysis 94(10): 1997–2010.Search in Google Scholar