Mathematical programming-based discrete element method in geomechanics

Geomaterials, such as rocks and soils, are complex and discontinuous in nature. To capture their non-linear mechanical behaviour, complex constitutive models containing a number of parameters or internal variables have often been developed. In contrast, the complex behaviour of geomaterials can be simulated via simple micro-level particle interactions using the discrete element method (DEM). Pre-existing discontinuities and growth of cracks can be represented explicitly and conveniently in DEM. Therefore, DEM is being used increasingly in geomechanics research. A mathematical programming-based DEM is proposed for modelling geomaterials in this thesis. The formulations developed naturally lead to convex optimisation problems that can be solved using efficient off-the-shelf solvers. An implicit time discretisation scheme is adopted, which allows a larger time step to be employed in the solution process. Furthermore, a purely static formulation is derived which requires no artificial damping parameters. Additionally, the new formulation is more general than the classical DEM methods because perfectly rigid particles can be employed to model stiff materials. The question of whether static formulations are capable of reproducing strain localisation and the shear behaviour of granular materials has been examined. A series of numerical biaxial compression tests were conducted and, to represent flexible boundary conditions, the tensile strength and cohesion were applied to the membrane particles. It was found that the numerical results are consistent with common experimental observations and that the method is computationally very efficient. To model cemented granular materials, variational formulations for a bond model were developed. Bonds were created between particles that can transmit tensile forces, shear forces and rolling moments up to certain thresholds. After the bond fails, the interaction between the particles is considered to be governed by frictional contact. The model has been applied to model cemented granular materials under static conditions and to the failure of slopes, thus showing its potential uses. Although circular (or spherical) particles are commonly used in DEM, polygonal particles may provide a better representation of the mineral structure in rocks. Therefore, the approach was developed further with polygonal particles based on the rigid-body-spring network method. It was employed to model the process of failure in rock, and its results were consistent with experimental observations. Finally, the approach was generalised to model jointed rock masses and applied to investigate jointed rock slope failure. Furthermore, the numerical results from the soft-particle and hard-particle models were compared. It was shown that the hard-particle model is more efficient than the soft-particle model, but the latter model is more versatile. When applying the hard-particle model to closely-packed blocks, unrealistic results may be obtained due to indeterminacy in redundant contacts.