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Buckling of granular systems with discrete and gradient elasticity Cosserat continua

Abstract

The stability of a granular column composed of a finite number of grains is investigated through an exact and some approximated continuum models. Shear and rotational interactions are taken into account at the rigid grain interfaces. This system can be considered as a discrete Cosserat chain with two independent degrees-of-freedom, namely the deflection and the rotation of each grain. The buckling of this discrete granular system on elastic foundation with translational and rotational stiffness (to account for some possible transversal grain interactions) is calculated whatever the number of grains. The formulation of the discrete boundary value problem is based on the exact resolution of a fourth-order linear difference equation. This solution is compared to the one of a continuous Cosserat chain asymptotically obtained for an infinite number of grains. In this last case, the asymptotic solution converges towards the one of a Bresse–Timoshenko beam under Winkler–Pasternak foundation. A more refined Cosserat continuum is built by continualization of the difference equations valid for the discrete Cosserat medium. It is shown that this more refine continuous model can be classified as a gradient elasticity Cosserat continuum, which is able to reproduce the scale effects observed for the buckling of the discrete granular system. These scale effects are related to the grain size, as compared to the structural length of the granular system. The key role played by the shear interaction in the instabilities of granular structural system is revealed, especially when the bending interaction can be neglected.

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Challamel, N., Lerbet, J., Darve, F. et al. Buckling of granular systems with discrete and gradient elasticity Cosserat continua. Ann. Solid Struct. Mech. 12, 7–22 (2020). https://doi.org/10.1007/s12356-020-00065-5

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  • DOI: https://doi.org/10.1007/s12356-020-00065-5

Keywords

  • Granular system
  • Buckling
  • Lattice formulation
  • Cosserat continuum
  • Discrete Cosserat formulation
  • Gradient elasticity
  • Difference equations
  • Enriched continua