This paper presents a study of thermoelasticity theory grounded in the recently developed two-temperature Moore–Gibson–Thompson (2TMGT) generalized thermoelasticity theory. We derive the fundamental governing equations for a homogeneous and isotropic medium within the context of 2TMGT. A primary objective is to establish key theoretical results, beginning with the uniqueness theorem for a mixed initial-boundary value problem in linear thermoelasticity under the current framework. Furthermore, we formulate a variational principle based on an appropriate functional corresponding to the governing equations of motion. This variational formulation offers deeper insights into the interactions between the mechanical and thermal aspects of the system. In addition, a reciprocity theorem is derived through the application of the Laplace transform method. The generalized results presented herein not only extend classical thermoelastic theorems but also enhance their applicability to a broader range of physical scenarios.
GreenAENaghdiP. A re-examination of the basic postulates of thermomechanics. Proc Royal Soc Lond A1991; 432: 171–194.
12.
GreenAENaghdiP. On undamped heat waves in an elastic solid. J Therm Stress1992; 15(2): 253–264.
13.
GreenAENaghdiP. Thermoelasticity without energy dissipation. J Elast1993; 31(3): 189–208.
14.
ChenPJGurtinME. On a theory of heat conduction involving two temperatures. J Appl Math Phys1968; 19: 614–627.
15.
ChenPJWilliamsWO. A note on non-simple heat conduction. ZAMP1968; 19: 969–970.
16.
ChenPJGurtinMEWilliamsWO. On the thermodynamics of non-simple elastic materials with two temperatures. J Appl Math Phys1969; 20: 107–112.
17.
QuintanillaR. Moore–Gibson–Thompson thermoelasticity. Math Mech Solid2019; 24(12): 4020–4031.
18.
ContiMPataVQuintanillaR. Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature. Asymptot Anal2020; 120(1–2): 1–21.
19.
JangidKMukhopadhyayS. A domain of influence theorem under mgt thermoelasticity theory. Math Mech Solid2021; 26(2): 285–295.
20.
JangidKMukhopadhyayS. A domain of influence theorem for a natural stress–heat-flux problem in the Moore–Gibson–Thompson thermoelasticity theory. Acta Mech2021; 232(1): 177–187.
21.
QuintanillaR. Moore–Gibson–Thompson thermoelasticity with two temperatures. Appl Eng Sci2020; 1: 100006.
22.
WeinerJH. A uniqueness theorem for the coupled thermoelastic problem. Q Appl Math1957; 15: 102–105.
23.
SheriefHHDhaliwalRS. A uniqueness theorem and a variational principle for generalized thermoelasticity. J Therm Stress1980; 3(2): 223–230.
24.
IgnaczakJ. Uniqueness in generalized thermoelasticity. J Therm Stress1979; 2(2): 171–175.
25.
IgnaczakJ. A note on uniqueness in thermoelasticity with one relaxation time. J Therm Stress1982; 5(3–4): 257–263.
26.
IgnaczakJ. A completeness problem for stress equations of motion in the linear elasticity theory. Arch Mech Stos1963; 15: 225–234.
27.
GurtinME. Variational principles for linear elastodynamics. Arch Rat Mech Anal1964; 16: 34–50.
28.
NickellRSackmanJ. Variational principles for linear coupled thermoelasticity. Q Appl Math1968; 26: 11–26.
29.
QuintanillaR. On uniqueness and stability for a thermoelastic theory. Math Mech Solids2017; 22: 1387–1396.
30.
MayselVM. The temperature problem of the theory of elasticity. Kiev: Ukrainian Academy of Sciences, 1951.
31.
PredeleanuPM. On thermal stresses in viscoelastic bodies. Bull Math Soc Sci Math Phys1959; 3(51): 223–228.
32.
Ionescu-CazimirV. Problem of linear thermoelasticity: Theorems on reciprocity. Bull Pol Acad Sci Ser Sci Tech1964; 12: 473–480.
33.
NowackiW. Dynamic problems of thermoelasticity. Warszawa: PWN–Polish Scientific Publishers, 1975.
34.
KhomyakevichEPRudenkoMK. Generalized reciprocity problem of thermoelasticity. Sov Appl Mech1974; 10(7): 767–771.
35.
ChandrasekharaiahDSSrikantaiahKR. Some theorems in temperature-rate-dependent theory of thermoelasticity. Proc Math Sci1983; 92(3): 135–141.
36.
WangJDhaliwalRSMajumdarSR. Some theorems in the generalized theory of thermoelasticity for prestressed bodies. Ind J Pure Appl Math1997; 28: 267–276.
37.
ChandrasekharaiahDS. Variational and reciprocal principles in micropolar thermoelasticity. Int J Eng Sci1987; 25(1): 55–63.
38.
KumarRPrasadRMukhopadhyayS. Variational and reciprocal principles in two-temperature generalized thermoelasticity. J Therm Stress2010; 33(3): 161–171.
39.
KumarRPrasadRMukhopadhyayS. Some theorems on two-temperature generalized thermoelasticity. Arch Appl Mech2011; 81: 1031–1040.
40.
IesanD. Principes variationnels dans la th’eorie de la thermo’elasticit’e coupl’ee. Ann Sci Univ “Al I Cuza” Iasi Math1966; 12: 439–456.
41.
IesanD. On some reciprocity theorems and variational theorems in linear dynamic theories of continuum mechanics. Turin: Memorie dell’Accademia delle Scienze di Torino, Classe di Scienze Fisiche, Matematiche e Naturali, 1974.
42.
LebonG. Variational principles in thermomechanics. Wien and New York: Springer, 1980.
43.
CarlsonDE. Linear thermoelasticity. In: TruesdellC (ed.) Fl”ugge’s handbuch der physik. Berlin: Springer-Verlag, 1972, pp. 297–345.
44.
HetnarskiRBIgnaczakJ. Mathematical theory of elasticity. New York, NY: Taylor & Francis, 2004.
45.
HetnarskiRBEslamiMR. Thermal stresses: Advanced theory and applications, Solid Mechanics and Its Applications, vol. 158. Dordrecht: Springer, 2010.
46.
KumariBMukhopadhyayS. Some theorems on linear theory of thermoelasticity for an anisotropic medium under an exact heat conduction model with a delay. Math Mech Solid2017; 22(5): 1177–1189.
47.
ShivayONMukhopadhyayS. Some basic theorems on a recent model of linear thermoelasticity for a homogeneous and isotropic medium. Mathematics and Mechanics of Solids2019; 24(8): 2444–2457.
48.
JangidKMukhopadhyayS. Variational and reciprocal principles on the temperature-rate dependent two-temperature thermoelasticity theory. J Therm Stress2020; 43(7): 816–828.
49.
MollaMAKMallikSH. Variational principle uniqueness and reciprocity theorems for higher order time-fractional four-phase-lag generalized thermoelastic diffusion model. Mech Bas Des Struc Mach2021; 51(4): 1–16.
50.
SachanSKumarRPrasadR. Reciprocal and variational theorems on the two-temperature Green–Lindsay thermoelasticity theory. Arch Appl Mech2023; 93: 1555–1563.
51.
MahdyAMS. Stability, existence, and uniqueness for solving fractional glioblastoma multiforme using a caputo–fabrizio derivative. Math Method Appl Sci2023;.
52.
RanjanAGKumarPPrasadR. On uniqueness and somigliana theorem for a thermoelastic theory. J Therm Stress2024; 47: 1010–1027.
53.
KumarPPrasadRKumarV, et al. Some basic theorems on the modified Green–Lindsay model of linear generalized thermoelasticity. Wave Rand Comp Med2024. Epub ahead of print 19 August. DOI: 10.1080/17455030.2024.2391499.
54.
AbouelregalAEAhmadHNofalTA, et al. Moore–Gibson–Thompson thermoelasticity model with temperature-dependent properties for thermo-viscoelastic orthotropic solid cylinder of infinite length under a temperature pulse. Phys Scr2021; 96(10): 105201.
55.
AttaDAbouelregalAESedighiHM, et al. Thermodiffusion interactions in a homogeneous spherical shell based on the modified Moore–Gibson–Thompson theory with two time delays. Mech Time-Depend Mater2024; 28: 617–638.
56.
AbouelregalAEAttaD. A rigid cylinder of a thermoelastic magnetic semiconductor material based on the generalized Moore–Gibson–Thompson heat equation model. Appl Phys A2022; 128: 118.
57.
AbouelregalAEMarinMOchsnerA. The influence of a non-local Moore–Gibson–Thompson heat transfer model on an underlying thermoelastic material under the model of memory-dependent derivatives. Continuum Mech Thermodyn2023; 35: 545–562.
58.
El-SapaSAlmoneefAALotfyK, et al. Moore–Gibson–Thompson theory of a non-local excited semiconductor medium with stability studies. Alex Eng J2022; 61(12): 11753–11764.
59.
El-SapaSBecheikhNChtiouiH, et al. Moore–Gibson–Thompson model with the influence of moisture diffusivity of semiconductor materials during photothermal excitation. Front Phys2023; 11: 1224326.
60.
IbrahimEEl-SapaSChteouiR, et al. Four-phase lags in a generalized thermoelastic rotational diffusive plate with laser pulse emission. Mech Solids2023; 58: 2412–2423.
61.
El-SapaSEl-BaryAAAlbalawiW, et al. Modelling Pennes’ bioheat transfer equation in thermoelasticity with one relaxation time. J Electromagn Waves Appl2023; 38(1): 105–121.
62.
KumarPPrasadR. Theorems in the generalized thermoelasticity based on the thermomass motion in the modified Green–Lindsay theory. Acta Mech2025. Epub ahead of print 8 November. DOI: 10.1007/s00707-025-04574-5.
