Skip to main content

Moving interfaces that separate loose and compact phases of elastic aggregates: a mechanism for drastic reduction or increase in macroscopic deformation

Abstract

Granular materials such as sand may be viewed as continuous bodies composed of much smaller elastic bodies. The multiscale geometry of structured deformations captures the contribution at the macrolevel of the smooth deformation of each small body in the aggregate (deformation without disarrangements) as well as the contribution at the macrolevel of the non-smooth deformations such as slips and separations between the small bodies in the aggregate (deformation due to disarrangements). When the free energy response of the aggregate depends only upon the deformation without disarrangements, is isotropic, and possesses standard growth and semi-convexity properties, we establish (i) the existence of a compact phase in which every small elastic body deforms in the same way as the aggregate and, when the volume change of macroscopic deformation is sufficiently large, (ii) the existence of a loose phase in which every small elastic body expands and rotates to achieve a stress-free state with accompanying disarrangements in the aggregate. We show that a broad class of elastic aggregates can admit moving surfaces that transform material in the compact phase into the loose phase and vice versa and that such transformations entail drastic changes in the level of deformation of transforming material points.

This is a preview of subscription content, access via your institution.

References

  1. Del Piero G., Owen D.R.: Structured deformations of continua. Arch. Ration. Mech. Anal. 124, 99–155 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  2. Capriz, G., Giovine, P., Mariano, P.M. (eds.): Mathematical Models of Granular Materials, Lecture Notes in Mathematics, vol. 1937, Springer, Berlin, Heidelberg (2008)

  3. Deseri L., Owen D.R.: Toward a field theory for elastic bodies undergoing disarrangements. J. Elast. 70, 197–236 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  4. Deseri L., Owen D.R.: Submacroscopically stable equilibria of elastic bodies undergoing dissipation and disarrangements. Math. Mech. Solids 15, 611–638 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  5. Khakhar D.: Rheology and mixing of granular materials. Macromol. Mater. Eng. 296, 278–289 (2011)

    Article  Google Scholar 

  6. Bićanić B.: Discrete element methods. In: Stein, I., de Borst, R., Hughes, T. (eds.) Encyclopedia of Computational Mechanics, vol. I, chap. 11, Wiley, London (2007)

    Google Scholar 

  7. Soroush A., Ferdowsi B.: Three dimensional discrete element modeling of granular media under cyclic constant volume loading: a micromechanical perspective. Powder Technol. 212, 1–16 (2011)

    Article  Google Scholar 

  8. Alam M., Luding S.: First normal stress difference and crystallization in a dense sheared granular fluid. Phys. Fluids 15, 2298–2312 (2003)

    ADS  Article  Google Scholar 

  9. Mueggenburg, N.: Behavior of granular materials under cyclic shear. Phys. Rev. E 71, 031301-0313010 (2005)

    Google Scholar 

  10. Ciarlet, P.G.: Mathematical Elasticity, vol. I: Three Dimensional Elasticity. Studies in Mathematics and its Applications. vol. 20, North Holland, Amsterdam (1988)

  11. Abeyaratne R., Knowles J.K.: Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114, 119–154 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  12. Owen D.R.: Field equations for elastic constituents undergoing disarrangements and mixing. In: Šilhavý, M. (ed.) Mathematical Modelling of Bodies with Complicated Bulk and Boundary Behavior. Quaderni di Matematica 20, pp. 101–133. Aracne, Rome (2007)

    Google Scholar 

  13. Mizel V.J.: On the ubiquity of fracture in nonlinear elasticity. J. Elast. 52, 257–266 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  14. Šilhavý, M.: Energy minimization for isotropic nonlinear elastic bodies. In: Del Piero, G., Owen, D.R. (eds.) Multiscale Modeling in Continuum Mechanics and Structured Deformations, pp. 1–51, CISM Courses and Lectures No. 447, Springer, Heidelberg (2004)

  15. Gajo A., Bigoni D., Muir-Wood D.: Multiple shear band development and related instabilities in granular materials. J. Mech. Phys. Solids 52, 2683–2724 (2004)

    MathSciNet  ADS  MATH  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to David R. Owen.

Additional information

Communicated by Francesco dell'Isola and Samuel Forest.

In honor of Gianpietro Del Piero, our teacher, colleague, and mentor, with admiration and enduring friendship.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Deseri, L., Owen, D.R. Moving interfaces that separate loose and compact phases of elastic aggregates: a mechanism for drastic reduction or increase in macroscopic deformation. Continuum Mech. Thermodyn. 25, 311–341 (2013). https://doi.org/10.1007/s00161-012-0260-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-012-0260-y

Keywords

  • Granular materials
  • Phase transitions
  • Elastic aggregates
  • Structured deformations