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Notes on Constitutive Relations for Porous Solids

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Views on Microstructures in Granular Materials

Part of the book series: Advances in Mechanics and Mathematics ((ACM,volume 44))

Abstract

The thermo-mechanical balance and constitutive equations for a porous material with large irregular pores are investigated by imposing internal kinematic and thermal constraints on ellipsoidal microstructures, in order to involve in the constitutive relations higher-order derivatives of macro- and micro-displacements and/or temperature avoiding classical incompatibilities.

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References

  1. Amoddeo, A., Giovine, P.: Micromechanical modelling of granular materials and FEM simulations. Meccanica 54, 609–630 (2019)

    Article  MathSciNet  Google Scholar 

  2. Barbu, V., Da Prato, G., Röckner, M.: Stochastic Porous Media Equations. Springer, Berlin (2016)

    Book  MATH  Google Scholar 

  3. Bear, J.: Dynamics of fluids in porous media. Dover Publications Inc., New York (1988)

    MATH  Google Scholar 

  4. Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)

    Article  MATH  Google Scholar 

  5. Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J. Acoust. Soc. Am. 28, 168–178 (1956)

    Article  Google Scholar 

  6. Bridgman, P.W.: Reflections on thermodynamics. Proc. Am. Acad. Arts Sci. 82, 301–309 (1953)

    Article  Google Scholar 

  7. Capriz, G.: Continua with latent microstructure. Arch. Ration. Mech. Analysis 90(1), 43–56 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Capriz, G.: Continua with Microstructure. Springer Tracts in Natural Philosophy, vol. 35. Springer-Verlag, Berlin (1989)

    Google Scholar 

  9. Capriz, G., Giovine. P.: On microstructural inertia. Math. Mod. Meth. Appl. Sciences 7, 211–216 (1997)

    MathSciNet  Google Scholar 

  10. Capriz, G., Giovine. P.: Remedy to omissions in a tract on continua with microstructure. Atti XIII° Congresso AIMETA ‘97, Siena, Meccanica Generale, Vol. I, pp.1–6 (1997)

    Google Scholar 

  11. Capriz, G., Giovine, P.: Weakly nonlocal effects in mechanics. In: Contributions to Continuum Theories, Krzysztof Wilmanski’s Anniversary Volume. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Report n.18, pp. 37–44 (2000). ISSN 0946-8838

    Google Scholar 

  12. Capriz, G., Podio Guidugli, P.: Discrete and continuous bodies with affine structure. Annali Mat. Pura Appl. (IV) 111, 195–211 (1976)

    Google Scholar 

  13. Capriz, G., Podio Guidugli, P.: Formal structure and classification of theories of oriented materials. Annali Mat. Pura Appl. (IV) 115, 17–39 (1977)

    Google Scholar 

  14. Capriz, G., Podio Guidugli, P.: Materials with spherical structure. Arch. Ration. Mech. Anal. 75, 269–279 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Capriz, G., Podio Guidugli, P.: Internal constraints. In: Truesdell, C.: Rational Thermodynamics, Appendix 3A, 2nd edition. Springer-Verlag, New York, pp. 159–170 (1984)

    Google Scholar 

  16. Capriz, G., Podio Guidugli P., Williams, W.: On balance equations for materials with affine structure. Meccanica 17, 80–84 (1982)

    Article  MATH  Google Scholar 

  17. Capriz, G., Virga, E.G.: On singular surfaces in the dynamics of continua with microstructure. Quart. Appl. Math. 52, 509–517 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chadwick P., Currie P.K.: The propagation of acceleration waves in heat conducting elastic materials. Arch. Rational Mech. Anal. 49, 137–158 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cheng, A.H.-D.: Poroelasticity. Springer, Berlin (2016)

    Book  MATH  Google Scholar 

  20. Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal. 13, 245–261 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cosserat, E., Cosserat, F.: Théorie des corps déformables. Hermann, Paris (1909)

    MATH  Google Scholar 

  22. Cowin, S.C.: Stress functions for Cosserat elasticity. Int. J. Solids Struct. 6, 389–398 (1970)

    Article  MATH  Google Scholar 

  23. Cowin, S.C., Weinbaum, S., Zeng, Y.: A case for bone canaliculi as the anatomical site of strain generated potentials. J. Biomechanics 28, 1281–1297 (1995)

    Article  Google Scholar 

  24. de Boer, R.: Theory of Porous Media. Highlights in the Historical Development and Current State. Springer, Berlin (2000)

    Google Scholar 

  25. de Fabritiis, C., Mariano, P.M.: Geometry of interactions in complex bodies. J. Geom. Physics 54, 301–323 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dunn, J.E., Serrin, J.: On the thermomechanics of interstitial working. Arch. Ration. Mech. Anal. 88, 95–133 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dunn, J.E.: Interstitial working and a nonclassical continuum thermodynamics. In: Serrin, J. (ed) New Perspectives in Thermodynamics. Springer-Verlag, Berlin, pp. 187–222 (1986)

    Chapter  Google Scholar 

  28. Ehlers, W.: Continuum and numerical simulation of porous materials in science and technology. In: Capriz, G., Ghionna, V.N., Giovine, P. (eds), Modeling and Mechanics of Granular and Porous Materials. Series on Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, pp. 245–292 (2002)

    Google Scholar 

  29. Eringen, A.C.: Mechanics of micromorphic continua. In: Kröner, E. (ed) Proc. IUTAM Symposium on Mechanics of Generalized Continua, Freudenstadt and Stuttgart, 1967 Springer, Berlin Heidelberg New York, pp. 18–35 (1968)

    Google Scholar 

  30. Esposito, R., Badescu, K., Steele-MacInnis, M., Cannatelli, C., De Vivo, B., Lima, A., Bodnar, R.J., Manning, C.E.: Magmatic evolution of the Campi Flegrei and Procida volcanic fields, Italy, based on interpretation of data from well-constrained melt inclusions. Earth-Science Reviews 185, 325–356 (2018)

    Article  Google Scholar 

  31. Forest, S., Sievert, R.: Nonlinear microstrain theories. Int. J. Solids Struct. 43, 7224–7245 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Giacobbe, V., Giovine, P.: Plane waves in linear thermoelastic porous solids. In: Ganghoffer, J.-F., Pastrone, F. (eds) Mechanics of Microstructured Solids: Cellular Materials, Fibre Reinforced Solids and Soft Tissues. Series: Lecture Notes in Applied and Computational Mechanics. Springer Verlag, Berlin, 46, pp. 71–80 (2009)

    Google Scholar 

  33. Giovine, P.: Porous solids as materials with ellipsoidal structure. In: Batra, R.C., Beatty, M.F. (eds) Contemporary Research in the Mechanics and Mathematics of Materials. CIMNE, Barcelona, pp. 335–342 (1996)

    Google Scholar 

  34. Giovine, P.: A linear theory of porous elastic solids. Transp. Porous Media 34, 305–318 (1999)

    Article  MathSciNet  Google Scholar 

  35. Giovine, P.: Nonclassical thermomechanics of granular materials. Math. Phys. Analys. Geom. 2(2), 179–196 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Giovine, P.: Remarks on constitutive laws for dry granular materials. In: Giovine, P., Goddard, J.D., Jenkins, J.T. (eds), IUTAM-ISIMM Symposium on Mathematical Modeling and Physical Instances of Granular Flows, Amer. Inst. Physics Conference Proceed. Vol. 1227, AIP, New York, USA, pp. 314–322 (2010)

    Google Scholar 

  37. Giovine, P.: Linear wave motions in continua with nano-pores. In: Giovine, P. (ed) Wave Processes in Classical and New Solids, Intech, Rijeka, pp. 61–86 (2012)

    Chapter  Google Scholar 

  38. Giovine, P.: Nano-pores in a thermo-elastic continuum with strain gradient effects. In: Ascione, L., Berardi, V.P., Feo, L., Fraternali F., Tralli, A.M. (eds) Proceedings of the XXIII Conference of the Italian Association of Theoretical and Applied Mechanics (AIMETA), September 2017, Salerno, Italy, Vol. 1, pp. 50–59 (2017)

    Google Scholar 

  39. Giovine, P.: Strain gradient effects in a thermo-elastic continuum with nano-pores. J. Mech. Engng. Automation 7(7), 348–355 (2017)

    Google Scholar 

  40. Giovine, P.: Internal constraints in the theories of immiscible mixtures for soils. Int. J. Solids Struct., 187, 3–22 (2020)

    Article  Google Scholar 

  41. Giovine, P., Margheriti, L., Speciale, M.P.: On wave propagation in porous media with strain gradient effects. Comp. Math. Applications 55, 307–318 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Goodman, M.A., Cowin, S.C.: A continuum theory for granular materials. Arch. Ration. Mech. Anal. 44, 249–266 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  43. Green, A.E., Naghdi, P.M., Trapp, J.A.: Thermodynamics of a continuum with internal constraints. Int. J. Eng. Sci. 8, 891–908 (1970)

    Article  MATH  Google Scholar 

  44. Grioli, G.: Microstructures as a refinement of Cauchy theory. Problems of physical concreteness. Continuum Mech. Thermodyn. 15, 441–450 (2003)

    Article  MATH  Google Scholar 

  45. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces Arch. Rational Mech. Anal. 57, 291–323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  46. Gurtin, M.E., Podio-Guidugli, P.: The thermodynamics of constrained materials. Arch. Rational Mech. Anal. 51, 192–208 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ichikawa, Y., Selvadurai, A.P.S.: Transport phenomena in porous media: aspects of micro/macro behaviour. Springer, Berlin-Heidelberg (2012)

    Book  Google Scholar 

  48. Korteweg, D.J.: Sur la forme que prennent les equations du mouvement des fluides si l’on tient compte des forces capillaires causees par des variations de densite considerables mais connues et sur la theorie de la capillarite dans l’hypothese d’une variation continue de la densite. Arch. Neerl. Sci. Exactes Nat. 6(2), 1–24 (1901)

    MATH  Google Scholar 

  49. Kunin, I.A.: Elastic media with microstructure II. Springer Series in Solid-State Sciences, Vol. 44. Springer-Verlag, Berlin (1983)

    Google Scholar 

  50. Lakes, R.S.: Size effects and micromechanics of a porous solid. J. Mat. Science 18, 2572–2580 (1983)

    Article  Google Scholar 

  51. Lakes, R.S.: Experimental microelasticity of two porous solids. Int. J. Solids Struct. 22, 55–63 (1986)

    Article  Google Scholar 

  52. Mariano, P.M.: Multifield theories in mechanics of solids. Adv. Appl. Mech. 38, 1–93 (2002)

    Article  Google Scholar 

  53. Mariano, P.M.: Second-neighbor interactions in classical field theories: invariance of the relative power and covariance. Math. Meth. Appl. Sci. 40, 1316–1332 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  54. Mariano, P.M.: Finite speed heat propagation as a consequence of microstructural changes. Continuum Mech. Thermodyn. 29, 1241–1248 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  55. Mariano, P.M., Augusti, G.: Multifield description of microcracked continua: a local model. Math. Mech. Solids 3(2), 183–200 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  56. Mariano, P.M., Stazi, F.L.: Computational aspects of the mechanics of complex materials. Arch. Comput. Meth. Engng. 12, 391–478 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  57. Müller, I.: On the entropy inequality. Arch. Ration. Mech. Anal. 26, 118–141 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  59. Ponte-Castañeda, P., Zaidman, M.: Constitutive models for porous materials with evolving microstructure. J. Mech. Phys. Solids 42, 1459–1497 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  60. Toupin, R. A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17, 85–112 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  61. Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. 2nd edition, Springer-Verlag, Berlin (1992)

    Book  MATH  Google Scholar 

  62. Truesdell, C., Toupin, R.A.: The Classical Field Theories. In: Flügge, S. (ed) Handbuch der Physik, Vol. III/1. Springer-Verlag, Berlin, pp. 226–793 (1960)

    Google Scholar 

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Acknowledgements

This research is part of the activities of the PRIN Project 2017J4EAYB: “Multiscale innovative materials and structures (MIMS)”. The support of the “Gruppo Nazionale di Fisica Matematica” of the “Istituto Nazionale di Alta Matematica ‘F. Severi’ (GNFM-INDAM)” is also acknowledged.

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Authors and Affiliations

  1. Department of Civil, Energy, Environment, and Materials Engineering (DICEAM), University ‘Mediterranea’ of Reggio Calabria, Reggio Calabria, Italy

    Pasquale Giovine

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  1. Pasquale Giovine
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Correspondence to Pasquale Giovine .

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Editors and Affiliations

  1. Department of Civil, Energy, Environment, and Materials Engineering (DICEAM), University ‘Mediterranea’ of Reggio Calabria, Reggio Calabria, Italy

    Pasquale Giovine

  2. Department of Civil and Environmental Engineering, University of Florence, Firenze, Italy

    Paolo Maria Mariano

  3. Department of Civil, Energy, Environment, and Materials Engineering (DICEAM), University of Reggio Calabria, Reggio Calabria, Italy

    Giuseppe Mortara

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Giovine, P. (2020). Notes on Constitutive Relations for Porous Solids. In: Giovine, P., Mariano, P.M., Mortara, G. (eds) Views on Microstructures in Granular Materials. Advances in Mechanics and Mathematics(), vol 44. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-49267-0_4

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