We study the steady plane shear flow of a dense assembly of frictional, inelastic disks using discrete simulation and prescribing the pressure and the shear rate. We show that, in the limit of rigid grains, the shear state is determined by a single dimensionless number, called inertial number I, which describes the ratio of inertial to pressure forces. Small values of I correspond to the quasistatic regime of soil mechanics, while large values of I correspond to the collisional regime of the kinetic theory. Those shear states are homogeneous, and become intermittent in the quasi-static regime. When I increases in the intermediate regime, we measure an approximately linear decrease of the solid fraction from the maximum packing value, and an approximately linear increase of the effective friction coefficient from the static internal friction value. From those dilatancy and friction laws, we deduce the constitutive law for dense granular flows, with a plastic Coulomb term and a viscous Bagnold term. We also show that the relative velocity fluctuations follow a scaling law as a function of I. The mechanical characteristics of the grains (restitution, friction and elasticity) have a very small influence in this intermediate regime. Then, we explain how the friction law is related to the angular distribution of contact forces, and why the local frictional forces have a small contribution to the macroscopic friction. At the end, as an example of heterogeneous stress distribution, we describe the shear localization when gravity is added.

Abstract. Mechanical and/or chemical removal of material from the subsurface may generate large subsurface cavities, the destabilisation of which can lead to ground collapse and the formation of sinkholes. Numerical simulation of the interaction of cavity growth, host material deformation and overburden collapse is desirable to better understand the sinkhole hazard but is a challenging task due to the involved high strains and material discontinuities. Here, we present 2-D distinct element method numerical simulations of cavity growth and sinkhole development. Firstly, we simulate cavity formation by quasi-static, stepwise removal of material in a single growing zone of an arbitrary geometry and depth. We benchmark this approach against analytical and boundary element method models of a deep void space in a linear elastic material. Secondly, we explore the effects of properties of different uniform materials on cavity stability and sinkhole development. We perform simulated biaxial tests to calibrate macroscopic geotechnical parameters of three model materials representative of those in which sinkholes develop at the Dead Sea shoreline: mud, alluvium and salt. We show that weak materials do not support large cavities, leading to gradual sagging or suffusion-style subsidence. Strong materials support quasi-stable to stable cavities, the overburdens of which may fail suddenly in a caprock or bedrock collapse style. Thirdly, we examine the consequences of layered arrangements of weak and strong materials. We find that these are more susceptible to sinkhole collapse than uniform materials not only due to a lower integrated strength of the overburden but also due to an inhibition of stabilising stress arching. Finally, we compare our model sinkhole geometries to observations at the Ghor Al-Haditha sinkhole site in Jordan. Sinkhole depth ∕ diameter ratios of 0.15 in mud, 0.37 in alluvium and 0.33 in salt are reproduced successfully in the calibrated model materials. The model results suggest that the observed distribution of sinkhole depth ∕ diameter values in each material type may partly reflect sinkhole growth trends.

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Abstract. The 2-D distinct element method (DEM) code (PFC2D_V5) is used here to simulate the evolution of subsidence-related karst landforms, such as single and clustered sinkholes, and associated larger-scale depressions. Subsurface material in the DEM model is removed progressively to produce an array of cavities; this simulates a network of subsurface groundwater conduits growing by chemical/mechanical erosion. The growth of the cavity array is coupled mechanically to the gravitationally loaded surroundings, such that cavities can grow also in part by material failure at their margins, which in the limit can produce individual collapse sinkholes. Two end-member growth scenarios of the cavity array and their impact on surface subsidence were examined in the models: (1) cavity growth at the same depth level and growth rate; (2) cavity growth at progressively deepening levels with varying growth rates. These growth scenarios are characterised by differing stress patterns across the cavity array and its overburden, which are in turn an important factor for the formation of sinkholes and uvala-like depressions. For growth scenario (1), a stable compression arch is established around the entire cavity array, hindering sinkhole collapse into individual cavities and favouring block-wise, relatively even subsidence across the whole cavity array. In contrast, for growth scenario (2), the stress system is more heterogeneous, such that local stress concentrations exist around individual cavities, leading to stress interactions and local wall/overburden fractures. Consequently, sinkhole collapses occur in individual cavities, which results in uneven, differential subsidence within a larger-scale depression. Depending on material properties of the cavity-hosting material and the overburden, the larger-scale depression forms either by sinkhole coalescence or by widespread subsidence linked geometrically to the entire cavity array. The results from models with growth scenario (2) are in close agreement with surface morphological and subsurface geophysical observations from an evaporite karst area on the eastern shore of the Dead Sea.

In recent years, civil engineers have started to use discrete-element modelling to simulate large-scale soil volumes thanks to technological improvements in both hardware and software. However, existing procedures to prepare ‘representative elementary volumes’ (REV) are unsatisfactory in terms of computational cost and sample homogeneity. In this work, a simple but efficient procedure to initialise large-scale discrete-element models is presented. Periodic cells are first generated with a sufficient number of particles (enough to consider the cell an REV) matching the desired particle-size distribution and equilibrated at the desired stress state, porosity and coordination number. When the cell is in equilibrium, it is replicated in space to fill the problem domain, and when the model is filled, only a small number of mechanical cycles is needed to equilibrate a large domain. The result is an equilibrated homogeneous sample at the desired initial state in a large volume.