A critical state line is presented for a crushable numerical soil, which is parallel to the isotropic normal compression line. A previous theory for the normal compression line, which correctly predicts the slope as a function of the size-effect on particle strength is extended to justify the slope of the critical state line. The micro mechanics behind critical states are examined, leading to a theory for a relationship between the volume of smallest particles and mean effective stress. A unique relationship exists for crushed states, leading to a two-dimensional interpretation of the state boundary surface for soils looser than critical.
de Bono, J. P., & McDowell, G. R. (2018). Micro mechanics of the critical state line at high stresses. Computers and Geotechnics, 98, 181–188. https://doi.org/10.1016/j.compgeo.2018.02.016