Advertisement

Journal of Elasticity

, Volume 133, Issue 2, pp 155–199 | Cite as

On the Use of the Continuum Mechanics Method for Describing Interactions in Discrete Systems with Rotational Degrees of Freedom

  • Elena A. IvanovaEmail author
Article
  • 127 Downloads

Abstract

Elastic interactions in a system of two body-points possessing both translational and rotational degrees of freedom are studied for the most general case of motion in 3D space. The continuum mechanics method is used as a theoretical foundation for describing the interactions. A definition of strain measures for the discrete system is given by analogy with that in continuum mechanics. Constitutive equations for force and moment vectors are derived based on the energy balance equation. Several new interaction potentials are suggested.

Keywords

Discrete systems Constitutive equations Interaction potentials Rotational motion Moment interactions Continuum mechanics method Molecular dynamics Discrete element method Proppant packing 

Mathematics Subject Classification (2010)

70E17 70E55 70E99 70F99 70G99 74E15 74E40 74E99 74M25 

Notes

Acknowledgements

The author is deeply grateful to A.K. Belyaev and V.A. Kuzkin for useful discussions on the paper.

This work was supported by Ministry of Education and Science of the Russian Federation within the framework of the Federal Program “Research and development in priority areas for the development of the scientific and technological complex of Russia for 2014–2020” (activity 1.2), grant No. 14.575.21.0146 of September 26, 2017, unique identifier: RFMEFI57517X0146.

References

  1. 1.
    Tersoff, J.: New empirical model for the structural properties of sellicon. Phys. Rev. Lett. 56(6), 632–635 (1986) ADSCrossRefGoogle Scholar
  2. 2.
    Tersoff, J.: New empirical approach for the structure and energy of covalent systems. Phys. Rev. B 37(12), 6991–7000 (1988) ADSCrossRefGoogle Scholar
  3. 3.
    Brenner, D.W.: Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys. Rev. B 42(15), 9458–9471 (1990) ADSCrossRefGoogle Scholar
  4. 4.
    Zhao, H., Alurua, N.R.: Temperature and strain-rate dependent fracture strength of graphene. J. Appl. Phys. 108, 064321 (2010) ADSCrossRefGoogle Scholar
  5. 5.
    Savin, A.V., Kivshar, Y.S., Hu, B.: Suppression of thermal conductivity in graphene nanoribbons with rough edges. Phys. Rev. B 82, 195422 (2010) ADSCrossRefGoogle Scholar
  6. 6.
    Gupta, S.S., Barta, R.C.: Elastic properties and frequencies of free vibrations of single-layer graphene sheets. J. Comput. Theor. Nanosci. 7, 1–14 (2010) CrossRefGoogle Scholar
  7. 7.
    Mindlin, R.D.: Elasticity, piezoelectricity and crystal lattice dynamics. J. Elast. 2(4), 217–282 (1972) CrossRefGoogle Scholar
  8. 8.
    Askar, A.: Molecular crystals and the polar theories of the continua. Experimental values of material coefficients for KNO3. Int. J. Eng. Sci. 10, 293–300 (1972) CrossRefGoogle Scholar
  9. 9.
    Pouget, J., Maugin, G.A.: Nonlinear dynamics of oriented elastic solids. I. Basic equations. J. Elast. 22(2–3), 135–155 (1989) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Pouget, J., Maugin, G.A.: Nonlinear dynamics of oriented elastic solids. II. Propagation of solitons. J. Elast. 22(2–3), 157–183 (1989) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Clarendon Press, Oxford (1987) zbMATHGoogle Scholar
  12. 12.
    Moreno-Razo, J.A., Sambriski, E.J., Koenig, G.M., Díaz-Herrera, E., Abbotta, N.L., de Pablo, J.J.: Effects of anchoring strength on the diffusivity of nanoparticles in model liquid-crystalline fluids. Soft Matter 7, 6828–6835 (2011) ADSCrossRefGoogle Scholar
  13. 13.
    Price, S.L., Stone, A.J., Alderton, M.: Explicit formulae for the electrostatic energy, forces and torques between a pair of molecules of arbitrary symmetry. Mol. Phys. 52(4), 987–1001 (1984) ADSCrossRefGoogle Scholar
  14. 14.
    Allen, M.P., Germano, G.: Expressions for forces and torques in molecular simulations using rigid bodies. Mol. Phys. 104(20), 3225–3235 (2006) ADSCrossRefGoogle Scholar
  15. 15.
    Coleman, B.D., Olson, W.K., Swigon, D.: Theory of sequence-dependent DNA elasticity. J. Chem. Phys. 118(15), 7127–7140 (2003) ADSCrossRefGoogle Scholar
  16. 16.
    Moakher, M., Maddocks, J.H.: A double-strand elastic rod theory. Arch. Ration. Mech. Anal. 177(1), 53–91 (2005) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ivanova, E.A., Krivtsov, A.M., Morozov, N.F., Firsova, A.D.: Description of crystal particle packing considering moment interactions. Mech. Solids 38(4), 101–117 (2003) Google Scholar
  18. 18.
    Ivanova, E.A., Kirvtsov, A.M., Morozov, N.F.: Macroscopic relations of elasticity for complex crystal latices using moment interaction at microscale. Appl. Math. Mech. 71(4), 543–561 (2007) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kuzkin, V.A., Krivtsov, A.M.: Description for mechanical properties of graphene using particles with rotational degrees of freedom. Dokl. Phys. 56(10), 527–530 (2011) ADSCrossRefGoogle Scholar
  20. 20.
    Kuzkin, V.A., Asonov, I.E.: Vector-based model of elastic bonds for simulation of granular solids. Phys. Rev. E 86, 051301 (2012) ADSCrossRefGoogle Scholar
  21. 21.
    Kuzkin, V.A., Krivtsov, A.M.: Enhanced vector-based model for elastic bonds in solids. Lett. Mater. 7(4), 455–458 (2017) CrossRefGoogle Scholar
  22. 22.
    Bagi, K.: Microstructural stress tensor of granular assemblies with volume forces. J. Appl. Mech. 66(4), 934–936 (1999) ADSCrossRefGoogle Scholar
  23. 23.
    Kruyt, N.P.: Statics and kinematics of discrete Cosserat-type granular materials. Int. J. Solids Struct. 40(3), 511–534 (2003) CrossRefGoogle Scholar
  24. 24.
    Murdoch, A.I.: On the microscopic interpretation of stress and couple stress. J. Elast. 71(1–3), 105–131 (2003) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Bagi, K.: Analysis of microstructural strain tensors for granular assemblies. Int. J. Solids Struct. 43(10), 3166–3184 (2006) CrossRefGoogle Scholar
  26. 26.
    Murdoch, A.I.: On molecular modelling and continuum concepts. J. Elast. 100(1–2), 33–61 (2010) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Cundall, P.A.: A computer model for simulating progressive large scale movements in blocky rock systems. In: Proceedings Symposium Int. Soc. Rock Mech., Nancy Metz, vol. 1. (1971). S. Paper II-8 Google Scholar
  28. 28.
    Cundall, P.A., Strack, O.D.L.: A distinct element model for granular assemblies. Geotechnique 29, 47–65 (1979) CrossRefGoogle Scholar
  29. 29.
    Deng, Sh., Li, H., Ma, G., Huang, H., Li, X.: Simulation of shale-proppant interaction in hydraulic fracturing by the discrete element method. Int. J. Rock Mech. Min. Sci. 70, 219–228 (2014) CrossRefGoogle Scholar
  30. 30.
    Kuzkin, V.A., Krivtsov, A.M., Linkov, A.M.: Computer simulation of effective viscosity of fluid-proppant mixture used in hydraulic fracturing. J. Min. Sci. 50(1), 1–9 (2014) CrossRefGoogle Scholar
  31. 31.
    Kuzkin, V.A., Krivtsov, A.M., Linkov, A.M.: Comparative study of rheological properties of suspension by computer simulation of Poiseuille and Couette flows. J. Min. Sci. 50(6), 1017–1025 (2014) CrossRefGoogle Scholar
  32. 32.
    Basu, D., Das, K., Smart, K., Ofoegbu, G.: Comparison of Eulerian-granular and discrete element models for simulation of proppant flows in fractured reservoirs. In: Fluids Engineering Systems and Technologies. ASME International Mechanical Engineering Congress and Exposition, vol. 7B, p. V07BT09A012 (2015).  https://doi.org/10.1115/IMECE2015-50050. ASME CrossRefGoogle Scholar
  33. 33.
    Bancewicz, M., Poła, J., Koza, Z.: Simulations of proppant transport in microfractures. In: 19th EGU General Assembly, EGU2017, Proceedings from the Conference, 23–28 April 2017, Vienna, Austria, p. 16538 (2017) Google Scholar
  34. 34.
    Zhang, G., Gutierrez, M., Li, M.: A coupled CFD-DEM approach to model particle- fluid mixture transport between two parallel plates to improve understanding of proppant micromechanics in hydraulic fractures. Powder Technol. 308, 235–248 (2017) CrossRefGoogle Scholar
  35. 35.
    Zhilin, P.A.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12, 635–648 (1976) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Altenbach, H., Naumenko, K., Zhilin, P.A.: A micro-polar theory for binary media with application to phase-transitional ow of fiber suspensions. Contin. Mech. Thermodyn. 15(6), 539–570 (2003) ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Zhilin, P.A.: Advanced Problems in Mechanics, vol. 2. Institute for Problems in Mechanical Engineering, St. Petersburg (2006) Google Scholar
  38. 38.
    Zhilin, P.A.: Rational Continuum Mechanics. Polytechnic University Publishing House, St. Petersburg (2012). (In Russian) Google Scholar
  39. 39.
    Van Zon, R., Schofield, J.: Event-driven dynamics of rigid bodies interacting via discretized potentials. J. Chem. Phys. 128, 154119 (2008) ADSCrossRefGoogle Scholar
  40. 40.
    Ivanova, E.A., Krivtsov, A.M., Morozov, N.F., Firsova, A.D.: Inclusion of the moment interaction in the calculation of the flexural rigidity of nanostructures. Dokl. Phys. 48(8), 455–458 (2003) ADSCrossRefGoogle Scholar
  41. 41.
    Byzov, A.P., Ivanova, E.A.: Mathematical modelling of the moment interactions of particles with rotary degrees of freedom. In: Scientific and Technical Sheets of St. Petersburg State Technical University. No. 2, pp. 260–268 (2007). (In Russian) Google Scholar
  42. 42.
    Zhilin, P.A.: Rigid Body Dynamics. Polytechnic University Publishing House, St. Petersburg (2015). (In Russian) Google Scholar
  43. 43.
    Altenbach, H., Maugin, G.A., Erofeev, V. (eds.): Mechanics of Generalized Continua. Springer, Berlin (2011) zbMATHGoogle Scholar
  44. 44.
    Altenbach, H., Forest, S., Krivtsov, A. (eds.): Generalized Continua as Models for Materials with Multi-scale Effects or Under Multi-field Actions. Springer, Berlin (2013) Google Scholar
  45. 45.
    Altenbach, H., Forest, S. (eds.): Generalized Continua as Models for Classical and Advanced Materials. Springer, Berlin (2016) Google Scholar
  46. 46.
    Zhilin, P.A.: Rigid body oscillator: a general model and some results. Acta Mech. 142, 169–193 (2000) CrossRefGoogle Scholar
  47. 47.
    Zhilin, P.A.: A new approach to the analysis of free rotations of rigid bodies. Z. Angew. Math. Mech. 76(4), 187–204 (1996) MathSciNetCrossRefGoogle Scholar
  48. 48.
    Zhilin, P.A.: Rotations of rigid body with small angles of nutation. Z. Angew. Math. Mech. 76(2), 711–712 (1996) MathSciNetzbMATHGoogle Scholar
  49. 49.
    Zhilin, P.A., Sorokin, S.A.: The motion of gyrostat on nonlinear elastic foundation. Z. Angew. Math. Mech. 78(2), 837–838 (1998) zbMATHGoogle Scholar
  50. 50.
    Zhilin, P.A.: Dynamics of the two rotors gyrostat on a nonlinear elastic foundation. Z. Angew. Math. Mech. 79(2), 399–400 (1999) Google Scholar
  51. 51.
    Noll, W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal. 2(1), 197–226 (1958) MathSciNetCrossRefGoogle Scholar
  52. 52.
    Grekova, E.F., Maugin, G.A.: Modelling of complex elastic crystals by means of multi-spin micromorphic media. Int. J. Eng. Sci. 43(5), 494–519 (2005) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dept. of Theoretical MechanicsPeter the Great St. Petersburg Polytechnic UniversitySt. PetersburgRussia
  2. 2.Institute for Problems in Mechanical Engineering of Russian Academy of SciencesSt. PetersburgRussia

Personalised recommendations