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Micromechanical modelling of granular materials and FEM simulations

  • Progress in Mechanics of Soils and General Granular Flows
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Abstract

We present a micro-mechanical continuum model used for the description of dilatant granular materials with incompressible rotating grains for which the kinetic energy, in addition to the usual translational one, consists of other two terms owing to microstructural motions: in particular, it includes the dilatational expansions and contractions of the granules relative to one another, as well as the rotation movements of each grain compared to the others. Next, we propose a linear theory in which the representations of constitutive functionals are linear with respect to both the volume fraction and the micro-rotation gradients, and to the dissipative variables. At the end we test the linear model on a two-dimensional domain, in which the arising system of partial differential equations is solved using the finite element method; thus we obtain a numerical solution in the case of a simplified granular micromechanics. The obtained computations of the early granular dynamics are consistent with theoretical insights as deduced from the proposed model. Viscous and rotational contributions to the granular dynamics have been identified and compared each others.

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References

  1. Ahmadi G (1982) A generalized continuum theory for granular materials. Int J Non-Linear Mech 17:21–33

    Article  MATH  Google Scholar 

  2. Amoddeo A (2015) Adaptive grid modeling for cancer cells in the early stage of invasion. Comput Math Appl 69:610–619

    Article  MathSciNet  Google Scholar 

  3. Amoddeo A (2015) Oxygen induced effects on avascular tumour growth: a preliminary simulation using an adaptive grid algorithm. J Phys Conf Ser 633:012088. https://doi.org/10.1088/1742-6596/633/1/012088

    Article  Google Scholar 

  4. Amoddeo A (2015) Moving mesh partial differential equations modelling to describe oxygen induced effects on avascular tumour growth. Cogent Phys 2:1050080. https://doi.org/10.1080/23311940.2015.1050080

    Article  Google Scholar 

  5. Amoddeo A, Barberi R, Lombardo G (2012) Surface and bulk contributions to nematic order reconstruction. Phys Rev E 85:061705. https://doi.org/10.1140/epje/i2012-12032-y

    Article  ADS  Google Scholar 

  6. Amoddeo A, Barberi R, Lombardo G (2013) Nematic order and phase transitions dynamics under intense electric fields. Liquid Cryst 40:799–809

    Article  Google Scholar 

  7. Bedford A, Drumheller DS (1983) On volume fraction theories for discretized materials. Acta Mech 48:173–184

    Article  MATH  Google Scholar 

  8. Billet G, Giovangigli V, de Gassowski G (2008) Impact of volume viscosity on a shock-hydrogen-bubble interaction. Combust Theor Model 12:221–248

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Buyevich YA, Shchelchkova IN (1978) Flow of dense suspensions. Prog Aerospace Sci 18:121–150

    Article  ADS  Google Scholar 

  10. Capriz G (1989) Continua with microstructure. Springer tracts in natural philosophy. Springer, Berlin, p 35

    Google Scholar 

  11. Capriz G, Mullenger G (1995) Extended continuum mechanics for the study of Granular flows. Rendiconti Accademia Lincei, Matematica 6:275–284

    MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Chen KC, Lan JY, Tai YC (2009) Description of local dilatancy and local rotation of granular assemblies by microstretch modeling. Int J Sol Struct 46:3882–3893

    Article  MATH  Google Scholar 

  14. Colombo G (1986) Manuale dell’Ingegnere, vol 1, 81st edn. Ulrico Hoepli, Milan

    Google Scholar 

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

    MATH  Google Scholar 

  16. Cowin SC (1974) A theory for the flow of granular materials. Powder Technol 9:61–69

    Article  Google Scholar 

  17. Cramer MS (2012) Numerical estimates for the bulk viscosity of ideal gases. Phys Fluids 24:066102. https://doi.org/10.1063/1.4729611

    Article  ADS  Google Scholar 

  18. Ehlers W, Ramm E, Diebels S, D’Addetta GA (2003) From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses. Int J Solids Struct 40:6681–6702

    Article  MathSciNet  MATH  Google Scholar 

  19. Ehlers W, Scholz B (2007) An inverse algorithm for the identification and the sensitivity analysis of the parameters governing micropolar elasto-plastic granular material. Arch Appl Mech 77:911–931

    Article  MATH  Google Scholar 

  20. Emanuel G (1992) Effect of bulk viscosity on a hypersonic boundary layer. Phys Fluids A 4:491–495

    Article  ADS  MATH  Google Scholar 

  21. Emanuel G (1998) Bulk viscosity in the Navier–Stokes equations. Int J Eng Sci 36:1313–1323

    Article  Google Scholar 

  22. Einstein A (1906) Eine neue Bestimmung der Moleküldimensionen. Ann der Phys 324(2):289–306

    Article  ADS  MATH  Google Scholar 

  23. Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications! Table of Absolute viscosities of gases. Web site: www.engineeringtoolbox.com/gases-absolute-dynamic-viscosity-d_1888.html

  24. Eringen AC (1968) Mechanics of micromorphic continua. In: Kröner E (ed) Proc. IUTAM symposium on mechanics of generalized continua (Freudenstadt and Stuttgart 1967), Springer, Berlin, pp 18–35

  25. Fang C, Wang Y, Hutter K (2006) Shearing flows of a dry granular material—hypoplastic constitutive theory and numerical simulations. Int J Numer Anal Meth Geomech 30:1409–1437

    Article  MATH  Google Scholar 

  26. Fisher HL (1957) Chemistry of natural and synthetic rubbers. Reinhold, New York

    Google Scholar 

  27. Giovine P (1999) Nonclassical thermomechanics of granular materials. Math Phys Anal Geom 2:179–196

    Article  MathSciNet  MATH  Google Scholar 

  28. Giovine P (2008) An extended continuum theory for granular media. In: Capriz G, Giovine P, Mariano PM (eds) Mathematical models of granular matter, series: lecture notes in mathematics, vol 1937. Springer, Berlin, pp 167–192

    Chapter  Google Scholar 

  29. Giovine P (2010) Remarks on constitutive laws for dry granular materials. In: Giovine P, Goddard JD, Jenkins JT (eds) IUTAM-ISIMM Symposium on mathematical modeling and physical instances of granular flows, AIP Conf Proc Series, vol 1227. New York, pp 314–322

  30. Giovine P (2017) Extended granular micromechanics. In: Radjai F, Nezamabadi S, Luding S, Delenne JY (eds) Powders and grains 2017—8th international conference on micromechanics on granular media, EPJ Web of Conferences, France, 140, 11009. https://doi.org/10.1051/epjconf/201714011009

  31. Giovine P, Oliveri F (1995) Dynamics and wave propagation in dilatant granular materials. Meccanica 30:341–357

    Article  MathSciNet  MATH  Google Scholar 

  32. Giovine P, Speciale MP (2001) On interstitial working in granular continuous media. In: Ciancio V, Donato A, Oliveri F, Rionero S (eds) Proceedings on 10\(^{\rm th}\) International conference on waves and stability in continuous media (WASCOM’99), Vulcano (Messina), World Scientific, Singapore,pp 196–208

  33. Godano C, Oliveri F (1999) Nonlinear seismic waves: a model for site effects. Int J Non-linear Mech 34:457–468

    Article  MATH  Google Scholar 

  34. Goddard JD, Didwania AK (1998) Computations of dilatancy and yield surfaces for assemblies of rigid frictional spheres. Quat J Mech Appl Math 51:15–43

    Article  MathSciNet  MATH  Google Scholar 

  35. Goodman MA, Cowin SC (1971) Two problems in the gravity flow of granular materials. J Fluid Mech 45:321–339

    Article  ADS  MATH  Google Scholar 

  36. Goodman MA, Cowin SC (1972) A continuum theory for granular materials. Arch Rat Mech An 44:249–266

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Happel J, Brenner H (1965) Low Reynolds number hydrodynamics: with special applications to particulate media. Englewood Cliffs, N. J. Prentice-Hall, London

    Google Scholar 

  39. Hutter K, Rajagopal KR (1994) On flows of granular materials. Contin Mech Thermodyn 6:81–139

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Jenkins JT, La Ragione L (2010) Microstructure and particle-phase stress in a dense suspension. In: Giovine P, Goddard JD, Jenkins JT (eds) IUTAM-ISIMM symposium on mathematical modeling and physical instances of granular flows. vol 1227, AIP Conf Proc Series, New York, pp 41–49

  41. Johnson PC, Jackson R (1987) Frictional-collisional constitutive relations for granular materials, with applications to plane shearing. J Fluid Mech 176:67–93

    Article  ADS  Google Scholar 

  42. Kanatani K-I (1979) A micropolar continuum theory for the flow of granular materials. Int J Eng Sci 17:419–432

    Article  MATH  Google Scholar 

  43. Love AEH (1926) A treatise on the mathematical theory of elasticity, 4th edn. Dover Publications, New York

    Google Scholar 

  44. Mariano PM (2002) Multifield theories in mechanics of solids. Adv Appl Mech 38:1–93

    Article  Google Scholar 

  45. Pan S, Johnsen E (2017) The role of bulk viscosity on the decay of compressible, homogeneous, isotropic turbulence. J Fluid Mech 833:717–744

    Article  ADS  MathSciNet  Google Scholar 

  46. Quarteroni A (2009) Numerical models for differential problems. Springer, Milan

    Book  MATH  Google Scholar 

  47. Reynolds O (1885) On the dilatancy of media composed of rigid particles in contact. Phil Mag 20:469–481

    Article  Google Scholar 

  48. Rueda MM, Auschera MC, Fulchiron R, Périé T, Martin G, Sonntag P, Cassagnau P (2017) Rheology and applications of highly filled polymers: a review of current understanding. Prog Polym Sci 66:22–53

    Article  Google Scholar 

  49. Savage SB (1979) Gravity flow of cohesionless granular materials in chutes and channels. J Fluid Mech 92:53–96

    Article  ADS  MATH  Google Scholar 

  50. Simha R, Somcynsky T (1965) The viscosity of concentrated spherical suspensions. J Colloid Sci 20:278–281

    Article  Google Scholar 

  51. Simpson CJSM, Bridgman KB, Chandler TRD (1968) Shock-tube study of vibrational relaxation in carbon dioxide. J Chem Phys 49:513–522

    Article  ADS  Google Scholar 

  52. Wang Y, Hutter K (1999) Shearing flows in a Goodman–Cowin type granular material—theory and numerical results. Part Sci Technol 17:97–124

    Article  ADS  Google Scholar 

  53. Zienkiewicz OC, Taylor RL (2002) The finite element method, 5th edn. Butterworth-Heinemann, Oxford

    Google Scholar 

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Acknowledgements

This research was supported by the “Gruppo Nazionale di Fisica Matematica (GNFM)” of the Italian “Istituto Nazionale di Alta Matematica (INDAM)”. We also thank the support of the Department of Civil, Energy, Environment and Materials Engineering (DICEAM) of the University ‘Mediterranea’ of Reggio Calabria.

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  1. Department of Civil Engineering, Energy, Environment and Materials (DICEAM), University “Mediterranea” of Reggio Calabria, Via Graziella, 1 - Locality Feo di Vito, 89122, Reggio Calabria, Italy

    Antonino Amoddeo & Pasquale Giovine

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  1. Antonino Amoddeo
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  2. Pasquale Giovine
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Correspondence to Pasquale Giovine.

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Author AA has no conflict of interest to report. Author PG is one of the Guest Editors of this special issue “Progress in mechanics of soils and general granular flows”.

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Amoddeo, A., Giovine, P. Micromechanical modelling of granular materials and FEM simulations. Meccanica 54, 609–630 (2019). https://doi.org/10.1007/s11012-018-00927-8

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