FEM×DEM multi‐scale model for cemented granular materials: Inter‐ and intra‐granular cracking induced strain localisation

Abstract

In this paper, we present a multi‐scale model that combines the finite element method (FEM) and the discrete element method (DEM) to study the behaviour of cemented granular materials (CGM) at both the sample (macroscopic) and particle (microscopic) scales, taking into account inter‐ and intra‐granular cracking. At the microscopic scale, the material is made up of Volume Elements (VE) composed of particles. Their mechanical behaviour is modelled by the DEM. In the VEs, we find circular grains (single particles) and meso‐grains; the meso‐grains being made up of clusters of particles linked by strong cohesive bonds. All particles interact via normal/tangential contact and rolling resistance laws with cohesive bonds. At the macroscopic scale, the sample is modelled using the FEM. A VE is assigned to each Gauss point of the mesh; the mechanical response of the VE is used to numerically derive the constitutive response of the material to the strain increment exerted at each point. In the models reported in this paper, localised failures in shear band mode are observed at the macroscopic scale. By performing a microscopic analysis, the results show that the occurrence and development of shear bands give rise locally to a strong evolution of microscopic characteristics such as void ratio, number of contacts (total and debonding contacts), and remarkably by inter‐ and intra‐granular cracking in the case of meso‐grains. Furthermore, the stress and strain tensors non‐coaxiality is clearly demonstrated inside the shear band, but almost negligible outside.

Variability and loss of uniqueness of numerical solutions in FEM×DEM modeling with second gradient enhancement

“…The motion of particles is obtained by solving the equation of second Newton's law with a time discretization and a third order predictor‐corrector scheme 41 . It should be noted that considering a more complicated grain–grain interaction law is straightforward and already implemented in our multi‐scale FEM×DEM model, as recently demonstrated in Nguyen et al 26 …”

“…This coupling method provides a cross-scale tool and has given rise to several publications. [26][27][28][29][30][31] In this way, the DEM-based constitutive model can be considered as a traditional constitutive law with thousands of state variables, represented by grain coordinates and geometry, contact list, contact orientation, interaction forces, etc. However, it was observed that the mesh dependency was also encountered in FEM×DEM calculation when strain localization occurred.…”

“…Consequently, the development and implementation of DEM‐based constitutive model in conventional FEM method bridges the gap between the range of applications between two popular methods, FEM and DEM. This coupling method provides a cross‐scale tool and has given rise to several publications 26–31 . In this way, the DEM‐based constitutive model can be considered as a traditional constitutive law with thousands of state variables, represented by grain coordinates and geometry, contact list, contact orientation, interaction forces, etc.…”

Modelling the time‐dependent mechanical behaviour of clay rocks based on meso‐ and micro‐structural viscous properties

“…Considering the complex mesostructure of clay rocks, one possibility to evaluate their impacts on mesoscale and macroscale overall mechanical responses is to use a multi‐scale approach. Such approach allows to study a statistically equivalent Representative Elementary Volume (REV for 3D case) or Representative Elementary Area (REA for 2D case) to determine the behaviour of the equivalent homogeneous medium 24–33 . In particular, computational homogenisation methods, 34–36 also known as multi‐scale analyses, have emerged.…”

“…Such approach allows to study a statistically equivalent Representative Elementary Volume (REV for 3D case) or Representative Elementary Area (REA for 2D case) to determine the behaviour of the equivalent homogeneous medium. [24][25][26][27][28][29][30][31][32][33] In particular, computational homogenisation methods, [34][35][36] also known as multi-scale analyses, have emerged. They include, among others, finite element squared (FE 2 ) 24,[37][38][39][40] and FEM-DEM 41 methods for continuous media at the large scale and heterogeneous/discrete material at the small scale.…”