Stress Ratio–Controlled Hardening Rule in the NorSand Framework

The NorSand model is a Cambridge-type soil model based on critical state theory, designed to simulate the monotonic behavior of sand. Cambridge-type models use an associated flow rule, with their yield surface obtained by integrating the stress-dilatancy relationship. The size of the yield surface is controlled by the amount of mean effective stress. In the case of NorSand, its hardening rule links the variation in this mean stress to incremental plastic deviatoric strain. However, the current model’s hardening rule, referred to as NorSand-OHR, provides accurate simulation results only for drained conditions, failing to capture undrained behavior, particularly for loose samples. In contrast, bounding surface models with stress ratio–based hardening rules effectively simulate sand behavior under both drained and undrained loading conditions. To address this limitation, this paper incorporates a stress ratio–based concept into NorSand’s framework, improving control over evolution of the yield surface and resolving the issues with NorSand-OHR. The modified hardening rule, called NorSand-ηHR, calibrates its hardening parameter to capture flow liquefaction in loose samples under undrained loading. NorSand-ηHR does not introduce additional properties; instead, it adopts hardening properties compatible with it. This new hardening rule was introduced in a recently modified version of the NorSand model with a nonlinear state-dilatancy law. The model’s validity is confirmed by comparing its simulation results with both drained and undrained test data from the Toyoura sand database.