Towards parameter limits of displacement boundary value problems for Mohr-Coulomb models

To solve problems in geotechnical engineering often numerical methods such as the Finite Element Method (FEM) are used. This method can be applied for example for the calculation of the strength of dikes, the determination of the stability of (rail)road embankments, the prediction of deformations due to landfills, or the analysis of subsurface constructions such as foundations, excavation pits and tunnels. Executing these numerical calculations frequently unreliable results are observed, which are the consequence of non-converging or unstable solutions. Indeed, often the source of the unexpected behaviour remains unknown. The present research aims to explain one of the possible causes, i.e. the influence of the applied material model on the behaviour of the numerical solution. In soil mechanics the elasto-plastic Mohr-Coulomb material model (including hardening and softening) is very commonly used. In this research the equations of static equilibrium, on which the FEM formulation is based, are analysed and solved completely analytically. For this purpose the method of separation of variables is used, in an adapted and extended version, which allows for the solution of a larger class of problems than generally assumed. Using this method the complete analytical solution is derived for linear elasticity as well as for Mohr-Coulomb elasto-plasticity. The necessary and sufficient conditions for uniqueness and stability of the solution are determined. These conditions allow for the determination and clarification of the parameter limits for the applicability of those material models. Using the results of this research the limits of applicability of the two considered material models can be determined for numerical applications as e.g. the Finite Element Method