Towards a constitutive law for the unsteady contact stress in granular media

In order to build a unified modelling for granular media by means of Eulerian averaged equations, 8 it is necessary to study two contributions in the effective Cauchy stress tensor: the first one concerns solid and 9 fluid matter, including contact and collisions between grains; the second one focuses on the random movements 10 of grains and fluid, similar to Reynolds stress for turbulent flows. It is shown that the point of view of piecewise 11 continuous media already used for two phase flows allows one to derive a constitutive equation for the first 12 contribution, under the form of a partial differential equation. Similarly as for the Reynolds stress in turbulent 13 flows, this equation cannot be written only in terms of averaged quantities without adequate approximations. 14 The structure of the closed equation is discussedwith respect to irreversible thermodynamics, and in connection 15 with some already existingmodels. It is emphasised that numerical simulations by the discrete elementsmethod 16 can be used in order to validate these approximations, through numerical experiments statistically considered. 17 Finally an extension of this approach to the second contribution of the effective Cauchy stress tensor is briefly 18 discussed, showing how the modelling of both contributions have to be coupled.