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.

Red-bed soft rock is widely distributed in the upper Yellow River. To investigate the strain-rate effect of saturated red-bed soft rock, a series of triaxial tests was conducted on argillaceous siltstone. Specifically, fifteen samples were tested by controlling confining pressures (5 MPa, 10 MPa, 15 MPa, 20 MPa, and 25 MPa) and strain rates (6.0%/min, 0.6%/min, and 0.06%/min) via triaxial tests, scanning electron microscopy (SEM), X-ray diffraction (XRD), Computed Tomography (CT), and 3D laser scanner. The results show that the peak strength of argillaceous siltstone increases with the rise in strain rate, being highly sensitive to the variation in strain rate under low confining pressure while relatively insensitive under high confining pressure. The tangent modulus of argillaceous siltstone is significantly affected by strain rate, and the influence of strain rate on tangent modulus weakens with the increase in confining pressure. Strain rate exerts an influence on the fracture mode and the shear zone roughness of soft rock. That is, with the increase in strain rate, the roughness increases, and the fracture mode transforms from pure shear fracture to tension–shear coupled fracture. With the increase in confining pressure, the number of shear cracks increases, and the shear surfaces tend to distribute approximately in parallel. The research results provide a scientific reference for the reservoir site selection of pumped storage power station in the upper Yellow River.