This study investigates several granular interaction laws used in the modelling of discrete granular media and the static response of a small assembly of 4 identical grains (diamond pattern). In the considered model, each grain interacts with its neighbour with a coupled shear-normal interaction law. The analysis is performed in a geometrically exact framework allowing large rotation and displacement evolutions, without any geometrical approximations (see also [1] for the granular elastica problem). It is shown that most of the granular interaction laws available in the literature are classified as hypoelastic interaction laws [2], [3] (such as the initial interaction models of Serrano and Rodriguez-Ortiz, 1973 or the popular model of Cundall and Strack, 1979 which gives birth to Particle Flow Codes). The interaction is weakly hypoelastic if an integral form exists, whereas it remains strongly hypoelastic when only an incremental formulation is available. Hyperelastic interaction laws may be also considered, that avoid possibly artificial dissipation (model of McNamara et al [4] or model of Turco et al [5]). We also show that along specific loading paths for which the normal and tangential laws are uncoupled, is the behaviour hyperelastic for all the studied models. For the three types of interactions, the modes of instability are then characterized for large displacement of the diamond pattern. We discuss the discrepancies between each granular model during the deformation process of some displacement-based loading tests.

References

[1] Challamel, N. and Kocsis, A. (2021) Geometrically exact bifurcation and post-buckling analysis of the granular elastica, Int. J. Non-linear Mech., 136, 103772, 1-15.

[2] Truesdell C. (1955) Hypoelasticity, J. Rational Mech. Anal., 4, pp. 83-133, 1955.

[3] Lerbet J., Challamel N., Nicot F. and Darve F. (2018) Coordinate free nonlinear incremental discrete mechanics, Zeischrift für Angewandte Mathematik und Mechanik, 98, 10, 1813-1833, 2018.

[4] McNamara S., Garcia-Rojo R. and Hermann H.J. (2008) Microscopic origin of granular ratcheting, Physical Review E, 77, 031304, 1-12, 2008.

[5] Turco E., dell'Isola F. and Misra A. (2019) A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations, Int. J. Anal. Num. Meth. Geomech., 43, 1051-1079.