Proof of Incompleteness of Critical State Theory in Granular Mechanics and Its Remedy

Abstract

According to classical critical state theory (CST) of granular mechanics, two conditions on the stress ratio and void ratio are satisfied when reaching and maintaining a critical state (CS). Therefore, CST dictates the necessity of these two conditions although their sufficiency has not been demonstrated, but only assumed. The present work challenges this assumption based on the results of a virtual two-dimensional (2D) discrete element method (DEM) experiment. The virtual sample is first brought to CS and then rotation of the principal axes (PA) of stress is imposed while keeping stress principal values fixed. The rotation induces a void ratio reduction and thus, abandonment of CS, despite the fact the two CST conditions are satisfied at the initiation of the rotation process, since the stress principal values remain fixed and the void ratio is at its critical state value for the given fixed pressure. The recently proposed anisotropic critical state theory (ACST) remedies this incompleteness of CST by enhancing its two conditions by a third, related to the critical state value of a fabric anisotropy variable, defined as the trace of the product of the fabric anisotropy tensor and the loading direction tensor. This third condition is violated by the stress PA rotation and can explain the aforementioned void ratio reduction. ACST can also explain various other response characteristics that cannot be addressed by classical CST with no fabric anisotropy consideration. In conclusion, the three conditions of ACST are shown to be both necessary and sufficient for reaching and maintaining CS.