Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Some properties of solutions in linear theory for semi-strongly elliptic porous elastic materials

Abstract

This paper is concerned with the linear theory of elastodynamics for homogeneous, isotropic, porous elastic materials with memory effects for the intrinsic equilibrated body forces. We are able to relax the conditions on constitutive coefficients and to determine the wider class of materials for which the internal energy is positive semi-definite, when boundary conditions are homogeneous. We found the class of semi-strongly elliptic porous elastic materials. For this class of materials, the above conditions may be relaxed without loss of some well-posedness properties of the solutions. In particular, we obtain uniqueness of the solutions and we study the spatial behavior problem.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2

References

  1. 1.

    Gurtin ME (1972) The linear theory of elasticity. In: Truesdell C (ed) Handbuch der physik, vol VIa/2. Springer, Berlin, pp 1–292

  2. 2.

    Ericksen JL, Toupin RA (1956) Implications of Hadamard’s conditions for elastic stability with respect to uniqueness theorems. Can J Math 8:432–436. https://doi.org/10.4153/CJM-1956-051-2

  3. 3.

    Lakes RS (1987) Foam structures with a negative Poisson’s ratio. Science 235:1038–1040. https://doi.org/10.1126/science.235.4792.1038

  4. 4.

    Caddock BD, Evans KE (1989) Microporous materials with negative Poisson’s ratios. I. Microstructure and mechanical properties. J Phys D Appl Phys 22(12):1877–1882. https://doi.org/10.1088/0022-3727/22/12/012

  5. 5.

    Park J, Lakes RS (2007) Biomaterials: an introduction. Springer, New York. https://doi.org/10.1007/978-0-387-37880-0. ISBN 978-0-387-37879-4

  6. 6.

    Lee T, Lakes RS (1997) Anisotropic polyurethane foam with Poisson’s ratio greater than 1. J Mater Sci 32(9):2397–2401. https://doi.org/10.1023/A:1018557107786

  7. 7.

    Wang Y, Lakes RS (2002) Analytical parametric analysis of the contact problem of human buttocks and negative Poisson’s ratio foam cushions. Int J Solids Struct 39(18):4825–4838. https://doi.org/10.1016/S0020-7683(02)00379-7

  8. 8.

    Ciarletta M, Passarella F, Tibullo V (2015) Plane harmonic waves in strongly elliptic thermoelastic materials with microtemperatures. J Math Anal Appl 424(2):1186–1197. https://doi.org/10.1016/j.jmaa.2014.11.065

  9. 9.

    Passarella F, Tibullo V (2016) On the propagation of Rayleigh waves in a strongly elliptic thermoelastic material with microtemperatures. J Therm Stress 39(9):1111–1118. https://doi.org/10.1080/01495739.2016.1192861

  10. 10.

    Merodio J, Ogden RW (2003) A note on strong ellipticity for transversely isotropic linearly elastic solids. Q J Mech Appl Math 56(4):589–591. https://doi.org/10.1093/qjmam/56.4.589

  11. 11.

    Merodio J, Ogden RW (2005) On tensile instabilities and ellipticity loss in fiber-reinforced incompressible non-linearly elastic solids. Mech Res Commun 32(3):290–299. https://doi.org/10.1016/j.mechrescom.2004.06.008

  12. 12.

    Tibullo V, Vaccaro M (2008) Spatial behaviour for constrained motion of a cylinder made of a strongly elliptic anisotropic material. J Mech Mater Struct 3(5):983–993. https://doi.org/10.2140/jomms.2008.3.983

  13. 13.

    Passarella F, Tibullo V, Zampoli V (2011a) On the strong ellipticity for orthotropic micropolar elastic bodies in a plane strain state. Mech Res Commun 38(7):512–517. https://doi.org/10.1016/j.mechrescom.2011.06.006

  14. 14.

    Gurtin ME, Sternberg E (1961) A note on uniqueness in classical elastodynamics. Q Appl Math 19(2):169–171. https://doi.org/10.1090/qam/129226

  15. 15.

    Gurtin ME, Toupin RA (1965) A uniqueness theorem for the displacement boundary-value problem of linear elastodynamics. Q Appl Math 23(1):79–81. https://doi.org/10.1090/qam/177557

  16. 16.

    Borchardt CW (1873) Ueber die transformation der elasticitätsgleichungen in allgemeine orthogonale coordinaten. J Reine Angew Math 1873(76):45–58. https://doi.org/10.1515/crll.1873.76.45

  17. 17.

    Thomson W (1888) XLVI. On the reflexion and refraction of light. Philos Mag 26(162):414–425. https://doi.org/10.1080/14786448808628288

  18. 18.

    Nunziato JW, Cowin SC (1979) A nonlinear theory of elastic materials with voids. Arch Ration Mech Anal 72(2):175–201. https://doi.org/10.1007/BF00249363

  19. 19.

    Cowin SC, Nunziato JW (1983) Linear elastic materials with voids. J Elast 13(2):125–147. https://doi.org/10.1007/BF00041230

  20. 20.

    Ieşan D (1985) Some theorems in the theory of elastic materials with voids. J Elast 15(2):215–224. https://doi.org/10.1007/BF00041994

  21. 21.

    Ieşan D (1986) A theory of thermoelastic materials with voids. Acta Mech 60(1–2):67–89. https://doi.org/10.1007/BF01302942

  22. 22.

    Ciarletta M, Ieşan D (1993) Non-classical elastic solids, volume 293 of Research Notes in Mathematics. CRC Press

  23. 23.

    Ieşan D (2004) Thermoelastic models of continua, volume 118 of solid mechanics and its applications. Springer, Dordrecht. ISBN 978-90-481-6634-3. https://doi.org/10.1007/978-1-4020-2310-1

  24. 24.

    Passarella F, Tibullo V, Zampoli V (2011b) On the heat-flux dependent thermoelasticity for micropolar porous media. J Therm Stress 34(8):778–794. https://doi.org/10.1080/01495739.2011.564041

  25. 25.

    Hassan M, Marin M, Ellahi R, Alamri SZ (2018) Exploration of convective heat transfer and flow characteristics synthesis by Cu–Ag/Water hybrid-nanofluids. Heat Transf Res 49(18):1837–1848. https://doi.org/10.1615/HeatTransRes.2018025569

  26. 26.

    Othman MIA, Marin M (2017) Effect of thermal loading due to laser pulse on thermoelastic porous medium under G-N theory. Results Phys 7:3863–3872. https://doi.org/10.1016/j.rinp.2017.10.012

  27. 27.

    Chiriţǎ S, Ciarletta M (1999) Time-weighted surface power function method for the study of spatial behaviour in dynamics of continua. Eur J Mech A Solids 18(5):915–933. https://doi.org/10.1016/S0997-7538(99)00121-7

  28. 28.

    Scalia A (2001) Spatial and temporal behavior in elastic materials with voids. Acta Mech 151(1–2):47–60. https://doi.org/10.1007/BF01272524

  29. 29.

    Chiriţǎ S (2004) Further results on the spatial behavior in linear elastodynamics. An Stiint Univ Al I Cuza Iasi Mat 50:289–304

  30. 30.

    Ciarletta M, Chiriţă S, Passarella F (2005) Some results on the spatial behaviour in linear porous elasticity. Arch Mech 57(1):43–65

  31. 31.

    D’Apice C, Tibullo V, Chiriţǎ S (2005) On the spatial behavior in the dynamic theory of mixtures of thermoelastic solids. J Therm Stress 28(1):63–82. https://doi.org/10.1080/014957390523732

  32. 32.

    Passarella F, Zampoli V (2006) Some results on the spatial behaviour for elastic mixtures. Eur J Mech A Solids 25(6):1031–1040. https://doi.org/10.1016/j.euromechsol.2006.01.004 ISSN 0997-7538

  33. 33.

    Passarella F, Zampoli V (2009a) Spatial estimates for transient and steady-state solutions in transversely isotropic plates of Mindlin-type. Eur J Mech A Solids 28(4):868–876. https://doi.org/10.1016/j.euromechsol.2009.01.004 ISSN 0997-7538

  34. 34.

    Passarella F, Zampoli V (2009b) Some results concerning the state of bending for transversely isotropic plates. Math Methods Appl Sci 32(14):1828–1843. https://doi.org/10.1002/mma.1113

  35. 35.

    Passarella F, Tibullo V, Zampoli V (2010) Decay properties of solutions of a Mindlin-type plate model for rhombic systems. J Mech Mater Struct 5(2):323–339. https://doi.org/10.2140/jomms.2010.5.323

  36. 36.

    Ghiba ID (2013) On the spatial behaviour in the bending theory of porous thermoelastic plates. J Math Anal Appl 403(1):129–142. https://doi.org/10.1016/j.jmaa.2013.02.023

  37. 37.

    Chiriţă S, Ghiba ID (2010) Strong ellipticity and progressive waves in elastic materials with voids. Proc R Soc Lond Ser A 466(2114):439–458. https://doi.org/10.1098/rspa.2009.0360

  38. 38.

    Passarella F (2018) New results for semi-strongly elliptic materials in linear elastodynamics. Mech Res Commun 94:53–57. https://doi.org/10.1016/j.mechrescom.2018.09.002

  39. 39.

    Passarella F, Tibullo V (2019) New decay results in linear thermoelastodynamics. Math Methods Appl Sci 42(4):1114–1121. https://doi.org/10.1002/mma.5417

Download references

Author information

Correspondence to Vincenzo Tibullo.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Amendola, A., Passarella, F. & Tibullo, V. Some properties of solutions in linear theory for semi-strongly elliptic porous elastic materials. Meccanica 55, 103–112 (2020). https://doi.org/10.1007/s11012-019-01102-3

Download citation

Keywords

  • Porous materials
  • Memory effect
  • Semi-strong ellipticity
  • Uniqueness result
  • Spatial behavior