The theory of generalized thermoelasticity with fractional order strain for dipolar materials with double porosity
Source abstract
Abstract
This article continues the work on dipolar thermoelastic materials, which are a special case of multipolar continuum mechanics. This theory allows a double-porous structure: a macro-porosity related to pores in the material and a microporosity, which shows fissures in the porous skeleton. This paper constructs a mathematical model for dipolar materials, which have a double-porosity structure by considering a fractional order Duhamel–Neumann stress–strain relation. The heat conduction is described by Cattaneo’s equations. The results are the constitutive equations of the linear theory of thermoelasticity with fractional order strain. The equations are valid for anisotropic materials and are called the Duhamel–Neumann equations with fractional order. Finally, the isotropic case is considered under the conditions of plane strain in order to perform some numerical simulations for samples of porous copper.
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Research context
OpenAlex topics: Thermoelastic and Magnetoelastic Phenomena · Nonlocal and gradient elasticity in micro/nano structures · Nanofluid Flow and Heat Transfer
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References (37)
Show source-provided references
1. Atangana A, Baleanu D (2017) Application of fixed point theorem for stability analysis of a nonlinear Schrödinger with Caputo–Liouville derivative. Filomat 31(8):2243–2248. https://www.doi.org/10.1117/12.32329510.1117/12.323295 ↗
2. Berryman JG, Wang HF (1998) Double-porosity modeling in elastic wave propagation for reservoir characterization. In: Proceedings of mathematical methods in geophysical imaging V, vol. 3453. https://doi.org/10.1117/12.32329510.1117/12.323295 ↗
3. Bhatti MM, Rashidi MM (2016) Effects of thermo-diffusion and thermal radiation on Williamson nanofluid over a porous shrinking/stretching sheet. J Mol Liq 221:567–57310.1016/j.molliq.2016.05.049 ↗
4. Chirilă A (2017) Generalized micropolar thermoelasticity with fractional order strain. Bull Transilvania Univ Braşov Ser III: Math Inf Phys 10(1):83–90
5. Chirilă A, Agarwal RP, Marin M (2017) Proving uniqueness for the solution of the problem of homogeneous and anisotropic micropolar thermoelasticity. Bound Value Probl 2017:3. https://doi.org/10.1186/s13661-016-0734-010.1186/s13661-016-0734-0 ↗
6. Ciarletta M, Ieşan D (1993) Non-classical elastic solids, Pitman research notes in mathematics series. Longman Scientific and Technical, London
7. Cosserat E, Cosserat F (1909) Sur la théorie des corps deformables. Dunod, Paris
8. Eringen AC (1999) Microcontinuum field theories, I. Foundations and solids. Springer, New York10.1007/978-1-4612-0555-5 ↗
9. Green AE (1965) Micro-materials and multipolar continuum mechanics. Int J Eng Sci 3:533–53710.1016/0020-7225(65)90033-9 ↗
10. Green AE, Rivlin RS (1965) Multipolar continuum mechanics: functional theory I. Proc R Soc Lond A 284:303–324. https://doi.org/10.1098/rspa.1965.006510.1098/rspa.1965.0065 ↗
11. Grote MJ, Mitkova T (2012) Explicit local time-stepping methods for time-dependent wave propagation. arXiv:1205.0654v2:1-32
12. Hetnarski RB (1996) Thermal stresses IV. Elsevier, Amsterdam
13. Hilfer R (2000) Applications of fractional calculus in physics. World Scientific, Singapore10.1142/3779 ↗
14. Ieşan D (1980) Mecanica generalizată a solidelor. Universitatea “Al. I. Cuza”, Centrul de multiplicare, Iaşi
15. Jiang C, Davey K, Prosser R (2017) A tessellated continuum approach to thermal analysis: discontinuity networks. Contin Mech Thermodyn 29:145–18610.1007/s00161-016-0523-0 ↗
16. Kilbas AA, Marichev OI, Samko SG (1993) Fractional integrals and derivatives (theory and applications). Gordon & Breach, Switzerland
17. Kong Q, Feng W, Zhu X, Sun C, Ma J, Wang X (2017) Fabrication and characterization of bulk nanoporous Cu with hierachical pore structure. J Mater Sci 52:12445–12454. https://doi.org/10.1007/s10853-017-1356-310.1007/s10853-017-1356-3 ↗
18. Kumar R, Vohra R (2015) State space approach to plane deformation in elastic material with double porosity. Mater Phys Mech 24:9–17
19. Marin M (1997) On weak solutions in elasticity of dipolar bodies with voids. J Comput Appl Math 82(1–2):291–29710.1016/S0377-0427(97)00047-2 ↗
20. Marin M, Ellahi R, Chirilă A (2017) On solutions of Saint-Venant’s problem for elastic dipolar bodies with voids. Carpathian J Math 33(2):199–21210.37193/CJM.2017.02.09 ↗
21. Marin M (2010) Harmonic vibrations in thermoelasticity of microstretch materials. J Vib Acoust ASME 132(4):044501-1–044501-610.1115/1.4000971 ↗
22. Marin M, Nicaise S (2016) Existence and stability results for thermoelastic dipolar bodies with double porosity. Contin Mech Thermodyn 28:1645–165710.1007/s00161-016-0503-4 ↗
23. Masin D, Herbstova V, Bohac J (2005) Properties of double porosity clayfills and suitable constitutive models. In: 16th International conference on soil mechanics and geotechnical engineering rotterdam, Millpress Rotterdam, Netherlands
24. Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York
25. Mindlin RD (1963) Microstructure in linear elasticity. Columbia University, New York10.21236/AD0424156 ↗
26. Plavšić M, Naerlović-Veljković N (1979) Thermodiffusion in dipolar elastic materials. In: Bulletin T. LXIV de l’Academie serbe des Sciences et des Arts, pp 29–39
27. Podlubny I (1999) Fractional differential equations. An introduction to fractional derivatives, fractional differential equations some methods of their solution and some of their applications. Academic Press, New York
28. Rice RW (2005) Use of normalized porosity in models for the porosity dependence of mechanical properties. J Mater Sci 40:983–989. https://doi.org/10.1007/s10853-005-6517-010.1007/s10853-005-6517-0 ↗
29. Sheikholeslami M, Bhatti MM (2017) Active method for nanofluid heat transfer enhancement by means of EHD. Int J Heat Mass Transf 109:115–12210.1016/j.ijheatmasstransfer.2017.01.115 ↗
30. Tao XF, Zhang LP, Zhao YY (2007) Mechanical response of porous copper manufactured by lost carbonate sintering process. Mater Sci Forum 539–543:1863–1867
31. Wagh AS (1993) Porosity dependence of thermal conductivity of ceramics and sedimentary rocks. J Mater Sci 28:3715–3721. https://doi.org/10.1007/BF0035316910.1007/BF00353169 ↗
32. Xiao Z (2013) Heat transfer, fluid transport and mechanical properties of porous copper manufactured by lost carbonate sintering. Ph.D. thesis, University of Liverpool
33. Xiao Z, Zhao Y (2013) Heat transfer coefficient of porous copper with homogeneous and hybrid structures in active cooling. J Mater Res 28(17):2545–255310.1557/jmr.2013.190 ↗
34. Youssef H (2010) Theory of fractional order generalized thermoelasticity. J Heat Transf 132(6):1–710.1115/1.4000705 ↗
35. Youssef H (2016) Theory of generalized thermoelasticity with fractional order strain. J Vib Control 22(18):3840–385710.1177/1077546314566837 ↗
36. Yu YJ, Tian XG, Lu TJ (2013) On fractional order generalized thermoelasticity with micromodeling. Acta Mech 224(12):2911–292710.1007/s00707-013-0913-3 ↗
37. Zervos A (2008) Finite elements for elasticity with microstructure and gradient elasticity. Int J Numer Meth Eng 73:564–59510.1002/nme.2093 ↗