Implicit Algorithm of the SBSP-R Model for Predicting the Non-Unique Critical State of Soils

The non-unique critical state represents the distance between the critical state line (CSL) and the isotropic consolidation line (ICL) that significantly varies with stress paths and particle size distribution of soils. A structural bounding surface plasticity model with spacing ratio r (SBSP-R model) was implemented using an explicit algorithm. However, the explicit algorithm did not well capture the non-unique critical state of soils with a large spacing ratio r, which prevented the soil mechanics research on non-unique critical state via finite element analysis. To overcome the limitation, the implicit algorithm of the SBSP-R model is formulated, and it mainly includes elastic prediction and plastic correction. The plastic correction is realized using the Newton–Simpson scheme with a controlling equation set related to consistency condition, plastic flow, hardening parameter, structural bounding surface, plastic modulus, and mapping rule. Case studies indicate that the implicit algorithm of the SBSP-R model is right and stable in predicting non-unique critical states. Comparisons between predicted and tested results indicate that the implicit algorithm of the SBSP-R model not only captures the critical state, stress-strain, and stress paths of various soils but also shows higher computational accuracy and efficiency compared with the previous explicit algorithm. These results indicate that the formulated implicit algorithm of the SBSP-R model is an alternative approach to the previous explicit algorithm.