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Numerical analysis of a type III thermo-porous-elastic problem with microtemperatures

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Abstract

In this work, we consider, from the numerical point of view, a poro-thermoelastic problem. The thermal law is the so-called of type III and the microtemperatures are also included into the model. The variational formulation of the problem is written as a linear system of coupled first-order variational equations. Then, fully discrete approximations are introduced by using the classical finite-element method and the implicit Euler scheme. A discrete stability property and an a priori error estimates result are proved, from which the linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some one- and two-dimensional numerical simulations are presented to show the accuracy of the approximation and the behavior of the solution.

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References

  • Almeida Júnior DS, Ramos AJA (2017) On the nature of dissipative Timoshenko systems at light of the second spectrum of frequency. Zeitschrift für Angewandte Mathematik und Physik 68:145

    MathSciNet  MATH  Google Scholar 

  • Aouadi M, Ciarletta M, Passarella F (2018) Thermoelastic theory with microtemperatures and dissipative thermodynamics. J Thermal Stress 41:522–542

    Google Scholar 

  • Bazarra N, Fernández JR (2018) Numerical analysis of a contact problem in poro-thermoelasticity with microtemperatures. Zeitschrift für Angewandte Mathematik und Mechanik 98:1190–1209

    MathSciNet  Google Scholar 

  • Campo M, Fernández JR, Kuttler KL, Shillor M, Viaño JM (2006) Numerical analysis and simulations of a dynamic frictionless contact problem with damage. Comput Methods Appl Mech Eng 196(1–3):476–488

    MathSciNet  MATH  Google Scholar 

  • Casas P, Quintanilla R (2005) Exponential stability in thermoelasticity with microtemperatures. Int J Eng Sci 43:33–47

    MathSciNet  MATH  Google Scholar 

  • Casas P, Quintanilla R (2005) Exponential decay in one-dimensional porous-thermoelasticity. Mech Res Commun 32:652–658

    MATH  Google Scholar 

  • Cattaneo C (1958) On a form of heat equation which eliminates the paradox of instantaneous propagation. Comptes Rendus de l’Académie de Sciences de Paris 247:431–433

    MATH  Google Scholar 

  • Chirita S, Ciarletta M, D’Apice C (2013) On the theory of thermoelasticity with microtemperatures. J Math Anal Appl 397:349–361

    MathSciNet  MATH  Google Scholar 

  • Ciarlet PG (1993) Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds., vol II, 17-351

  • Ciarletta M, Straughan B, Tibullo V (2010) Structural stability for a rigid body with thermal microstructure. Int J Eng Sci 48:592–598

    MathSciNet  MATH  Google Scholar 

  • Cowin SC (1985) The viscoelastic behavior of linear elastic materials with voids. J Elasticity 15:185–191

    MATH  Google Scholar 

  • Cowin SC, Nunziato JW (1983) Linear elastic materials with voids. J Elasticity 13:125–147

    MATH  Google Scholar 

  • Green AE, Naghdi PM (1992) On undamped heat waves in an elastic solid. J Thermal Stress 15:253–264

    MathSciNet  Google Scholar 

  • Eringen AC (1999) Microcontinuum field theories. I. Foundations and solids. Springer, New York

    MATH  Google Scholar 

  • Feng B, Apalara TA (2019) Optimal decay for a porous elasticity system with memory. J Math Anal Appl 470:1108–1128

    MathSciNet  MATH  Google Scholar 

  • Feng B, Yin M (2019) Decay of solutions for a one-dimensional porous elasticity system with memory: the case of non-equal wave speeds. Math Mech Solids 24:2361–2373

    MathSciNet  Google Scholar 

  • Fernández JR, Masid M (2016) Analysis of a problem arising in porous thermoelasticity of type II. J Thermal Stress 39:513–531

    Google Scholar 

  • Fernández JR, Masid M (2017) A porous thermoelastic problem: an a priori error analysis and computational experiments. Appl Math Comput 305:117–135

    MathSciNet  MATH  Google Scholar 

  • Fernández JR, Masid M (2017) Numerical analysis of a thermoelastic diffusion problem with voids. Int J Numer Anal Model 14:153–174

    MathSciNet  MATH  Google Scholar 

  • Green AE, Naghdi PM (1993) Thermoelasticity without energy dissipation. J Elast 31:189–208

    MathSciNet  MATH  Google Scholar 

  • Grot R (1969) Thermodynamics of a continuum with microstructure. Int J Eng Sci 7:801–814

    MATH  Google Scholar 

  • Ieşan D (2007) Thermoelasticity of bodies with microstructure and microtemperatures. Int J Solids Sruct 44:8648–8653

    MATH  Google Scholar 

  • Iesan D (2018) On a theory of thermoelasticity without energy dissipation for solids with microtemperatures. Zeitschrift für Angewandte Mathematik und Mechanik 98:870–885

    MathSciNet  Google Scholar 

  • Iesan D, Quintanilla R (2000) On a theory of thermoelasticity with microtemperatures. J Therm Stresses 23:195–215

    MathSciNet  Google Scholar 

  • Ieşan D, Quintanilla R (2009) On thermoelastic bodies with inner structure and microtemperatures. J Math Anal Appl 354:12–23

    MathSciNet  MATH  Google Scholar 

  • Ieşan D, Quintanilla R (2014) On a theory of thermoelastic materials with double porosity structure. J Therm Stresses 37:1017–1036

    Google Scholar 

  • Iesan D, Quintanilla R (2018) Qualitative properties in strain gradient thermoelasticity with microtemperatures. Math Mech Solids 23:240–258

    MathSciNet  MATH  Google Scholar 

  • Kothari S, Mukhopadhyay S (2012) Study of harmonic plane waves in rotating thermoelastic media of type III. Math Mech Solids 17:824–839

    MathSciNet  Google Scholar 

  • Kumar R, Vohra R (2017) Effect of hall current in thermoelastic materials with double porosity structure. Int J Appl Mech Eng 22:303–319

    Google Scholar 

  • Kumar R, Vohra R (2019) Forced vibrations of a thermoelastic double porous microbeam subjected to a moving load. J Theor Appl Mech 57:155–166

    Google Scholar 

  • Kumar R, Vohra R, Gorla M (2016) Reflection of plane waves in thermoelastic medium with double porosity. Multidiscip Model Mater Struct 12:748–778

    Google Scholar 

  • Leseduarte MC, Magaña A, Quintanilla R (2010) On the time decay of solutions in porous-thermo-elasticity of type II. Discr Contin Dyn Syst B 13:375–391

    MathSciNet  MATH  Google Scholar 

  • Magaña A, Quintanilla R (2006) On the exponential decay of solutions in one-dimensional generalized porous-thermo-elasticity. Asympt Anal 49:173–187

    MathSciNet  MATH  Google Scholar 

  • Magaña A, Quintanilla R (2017) On the time decay of solutions in porous-elasticity with quasi-static microvoids. J Math Anal Appl 331:617–630

    MathSciNet  MATH  Google Scholar 

  • Magaña A, Quintanilla R (2018) Exponential stability in type III thermoelasticity with microtemperatures. Z Angew Math Phys 69(5):129(1)–129(8)

    MathSciNet  MATH  Google Scholar 

  • Magaña A, Quintanilla R (2020) Exponential stability in three dimensionaltype III thermo-porous-elasticity with microtemperatures. J Elast 139:153–161

    MATH  Google Scholar 

  • Miranville A, Quintanilla R (2019) Exponential decay in one-dimensional type III thermoelasticity with voids. Appl Math Lett 94:30–37

    MathSciNet  MATH  Google Scholar 

  • Miranville A, Quintanilla R (2020) Exponential decay in one-dimensional type II thermoviscoelasticity with voids. J Comput Appl Math 368:112573

    MathSciNet  MATH  Google Scholar 

  • Nunziato JW, Cowin S (1979) A nonlinear theory of elastic materials with voids. Arch Ration Mech Anal 72:175–201

    MathSciNet  MATH  Google Scholar 

  • Quintanilla R (2011) On the growth and continuous dependence in thermoelasticity with microtemperatures. J Therm Stresses 34:911–922

    Google Scholar 

  • Quintanilla R (2013) On the logarithmic convexity in thermoelasticity with microtemperatures. J Therm Stresses 36:378–386

    Google Scholar 

  • Ramos AJA, Almeida-Júnior DS, Freitas MM, DosSantos MJ (2020) A new exponential decay result for one-dimensional porous dissipation elasticity from second spectrum viewpoint. Appl Math Lett 101:106061

    MathSciNet  MATH  Google Scholar 

  • Riha P (1975) On the theory of heat-conducting micropolar fluids with microtemperatures. Acta Mech 23:1–8

    MATH  Google Scholar 

  • Riha P (1976) On the microcontinuum model of heat conduction in materials with inner structure. Int J Eng Sci 14:529–535

    Google Scholar 

  • Verma PDS, Singh DV, Singh K (1979) Hagen-Poiseuille flow of microthermopolar fluids in a circular pipe. Acta Technical CSAV 24:402–412

    MATH  Google Scholar 

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Acknowledgements

The work of J.R. Fernández was partially supported by Ministerio de Ciencia, Innovación y Universidades under the research project PGC2018-096696-B-I00 (FEDER, UE). The work of R. Quintanilla was supported by projects “Análisis Matemático de Problemas de la Termomecánica” (MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness, and “Análisis matemático aplicado a la termomecánica” (Ref. PID2019-105118GB-I00), (AEI/FEDER, UE) of the Spanish Ministry of Science, Innovation and Universities.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación Campus as Lagoas Marcosende s/n, 36310, Vigo, Spain

    Noelia Bazarra & José R. Fernández

  2. Departament de Matemátiques, Universitat Politècnica de Catalunya, C. Colom 11, 08222, Terrassa, Barcelona, Spain

    Ramón Quintanilla

Authors
  1. Noelia Bazarra
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  2. José R. Fernández
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  3. Ramón Quintanilla
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Correspondence to José R. Fernández.

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Communicated by Cassio Oishi.

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Bazarra, N., Fernández, J.R. & Quintanilla, R. Numerical analysis of a type III thermo-porous-elastic problem with microtemperatures. Comp. Appl. Math. 39, 242 (2020). https://doi.org/10.1007/s40314-020-01248-x

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  • DOI: https://doi.org/10.1007/s40314-020-01248-x

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