Numerical integration of an elasto-plastic critical state model for soils under unsaturated conditions

This paper presents the complete set of incremental equations for the numerical integration of the Glasgow Coupled Model (GCM) and a comprehensive algorithm for its numerical integration. The incremental formulation proposed is expressed in terms of strain and suction increments (i.e. strain-driven) and defines an initial value problem (IVP) that can be solved once the initial state and the pair of increments of the driven variables are known. The numerical integration of this IVP is carried out by extending to unsaturated condition, the well-known explicit substepping formulation with automatic error control widely used for saturated soils. A notable feature of the substepping integration scheme presented is that it integrates simultaneously the model equations for both mechanical and water retention responses. Hence, the estimate of the local truncation error to automatically adjust the size of the integration step is not only affected by the local error in stresses and mechanical hardening parameter (as in a saturated soil model) but, additionally, by the local error incurred in the integration of the water retention relations (i.e. degree of saturation and water retention hardening parameter). The correctness of the integration scheme is then verified by comparison of computational outcomes against analytical/reference solutions. This research has benefitted from the Marie-Skłodowska Curie project “COUPLED” funded from the H2020 programme of the EC (MSCA-IF-2015-706712). Support from the project “TERRE” (ETN-GA-2015-675762) of the EU is also acknowledged.