On stress ratio equations in three-dimensional stress space for modelling soil behaviour

Constitutive models and failure criteria of soils, rocks, and other materials often need to be extended beyond the triaxial state where they are usually defined, for plane strain, axisymmetric, or three-dimensional analyses. This extension is commonly done by making the stress ratio a function of the Lode angle. This process turns two-dimensional yield, plastic potential, bounding, dilatancy and similar surfaces into three-dimensional shapes such as cones and bullets. The equations used have a range of shapes on the octahedral or pi-plane between Mohr-Coulomb's irregular hexagon and Drucker-Prager's circle. Nine stress ratio generalization equations popular in soil mechanics are evaluated based on numerical stability, agreement with available data, and ease of implementation. The computed limits on their convexity, and the flexibility they offer in the calibration process are discussed. At the end, a new equation that satisfies all these criteria while remaining simple and easy to calibrate is proposed, implemented in a finite element model, and demonstrated to improve numerical stability and efficiency.