In this paper, we investigate a model of poro-thermoelasticity with microtemperatures, where the behavior of the body is influenced by the history of both temperature and microtemperatures. Mathematically, this translates into a system of partial integro-differential equations. Under suitable condition on the tensors appearing in the model, we prove that the resulting system is well posed. In the one-dimensional case, the exponential decay of the energy is proved.
CosseratECosseratF. Thorie des Corps Deformables. Paris: Hermann, 1909.
2.
EringenAC. Microcontinuum field theories. I. Foundations and solids. New York: Springer, 1999.
3.
IeşanD. Thermoelastic models of continua. Dordrecht: Springer, 2004.
4.
CowinSC. The viscoelastic behavior of linear elastic materials with voids. J Elast1985; 15: 185–191.
5.
CowinSCNunziatoJW. Linear elastic materials with voids. J Elast1983; 13: 125–147.
6.
NunziatoJWCowinSC. A nonlinear theory of elastic materials with voids. Arch Ration Mech Anal1979; 72: 175–201.
7.
StraughanB. Stability and wave motion in porous media. New York: Springer, 2008.
8.
BazarraNFernándezJR. Numerical analysis of a contact problem in poro-thermoelasticity with microtemperatures. Z Angew Math Mech2018; 98: 1190–1209.
9.
FernándezJRMasidM. Mathematical analysis of a problem arising in porous thermoelasticity of type II. J Therm Stresses2016; 39: 513–531.
10.
FernándezJRMasidM. A porous thermoelastic problem: an a priori error analysis and computational experiments. Appl Math Comput2017; 305: 117–135.
11.
StraughanB. Mathematical aspects of multi-porosity continua. Cham: Springer, 2017.
12.
CasasPQuintanillaR. Exponential stability in thermoelasticity with microtemperatures. Int J Eng Sci2005; 43: 33–47.
13.
FengBApalaraTA. Optimal decay for a porous elasticity system with memory. J Math Anal Appl2019; 470: 1108–1128.
14.
FengBYinM. Decay of solutions for a one-dimensional porous elasticity system with memory: the case of non-equal wave speeds. Math Mech Solids2019; 24: 2361–2373.
15.
MagañaAQuintanillaR. Exponential stability in type III thermoelasticity with microtemperatures. Z Angew Math Phys2018; 69: 1291–1298.
16.
MiranvilleAQuintanillaR. Exponential decay in one-dimensional type II thermoviscoelasticity with voids. J Comput Appl Math2020; 368: 112573.
17.
PamplonaPXMuñoz-RiveraJEQuintanillaR. Analyticity in porous-thermoelasticity with microtemperatures. J Math Anal Appl2012; 394: 645–655.
18.
SantosMLCampeloADSAlmeida JúniorDS. On the decay rates of porous elastic systems. J Elast2017; 127: 79–101.
19.
GrotR. Thermodynamics of a continuum with microstructure. Int J Eng Sci1969; 7: 801–814.
20.
RihaP. On the theory of heat-conducting micropolar fluids with microtemperatures. Acta Mech1975; 23: 1–8.
21.
RihaP. On the microcontinuum model of heat conduction in materials with inner structure. Int J Eng Sci1976; 14: 529–535.
22.
VermaPDSSinghDVSinghK.Poiseuille flow of microthermopolar fluids in a circular pipe. Acta Tech CSAV1979; 24: 402–412.
23.
IeşanDQuintanillaR. On a theory of thermoelasticity with microtemperatures. J Therm Stresses2000; 23: 195–215.
24.
IeşanD. Thermoelasticity of bodies with microstructure and microtemperatures. Int J Solids Struct2007; 44: 8648–8653.
25.
IeşanD. On a theory of thermoelasticity without energy dissipation for solids with microtemperatures. Z Angew Math Mech2018; 98: 870–885.
26.
IeşanDQuintanillaR. On thermoelastic bodies with inner structure and microtemperatures. J Math Anal Appl2009; 354: 12–23.
27.
IeşanDQuintanillaR. Qualitative properties in strain gradient thermoelasticity with microtemperatures. Math Mech Solids2018; 23: 240–258.
28.
JaianiGBitsadzeL. On basic problems for elastic prismatic shells with microtemperatures. Z Angew Math Mech2016; 96: 1082–1088.
29.
MiranvilleAQuintanillaR. Exponential decay in one-dimensional type III thermoelasticity with voids. Appl Math Lett2019; 94: 30–37.
30.
BazarraNFernandezJRQuintanillaR. Lord Shulman thermoelasticity with microtemperatures. Appl Math Optim2021; 84: 1667–1685.
31.
LiuZQuintanillaRWangY. Dual-phase-lag heat conduction with microtemperatures. Z Angew Math Mech2021; 101: e202000167.
32.
GurtinMEPipkinAC. A general theory of heat conduction with finite wave speeds. Arch Ration Mech Anal1968; 31: 113126.
33.
AmendolaGFabrizioMGoldenJM. Thermodynamics of materials with memory: theory and applications. New York: Springer, 2012.
34.
ContiMPataVQuintanillaR. Thermoelasticity of Moore-Gibson-Thompson type with history dependence in the temperature. Asymptotic Anal2020; 120: 1–21.
35.
GiorgiCNasoMGPataV. Exponential stability in linear heat conduction with memory: a semigroup approach. Commun Appl Anal2001; 5: 121–134.
36.
NasoMGVegniFM. Asymptotic behavior of the energy to a thermo-viscoelastic Mindlin–Timoshenko plate with memory. Int J Pure Appl Math2005; 21: 175–198.
