Necessary and sufficient conditions for reaching and maintaining critical state

Abstract

Summary According to classical critical state theory (CST) of granular mechanics, two analytical conditions on the ratio of stress invariants and the void ratio are postulated to be necessary and sufficient for reaching and maintaining critical state (CS). The present work investigates the sufficiency of these two conditions based on the results of a virtual three‐dimensional discrete element method experiment, which imposes continuous rotation of the principal axes of stress with fixed stress principal values at CS. Even though the fixity of the stress principal values satisfies the two analytical CST conditions at the initiation of rotation, contraction and abandonment of CS occur, which proves that these conditions may be necessary but are not sufficient to maintain CS. But if fixity of stress and strain rate directions in regard to the sample is considered at CS, the two analytical conditions of CST remain both necessary and sufficient. The recently proposed anisotropic critical state theory (ACST) turned this qualitative requirement of fixity into an analytical condition related to the CS value of a fabric anisotropy variable A defined in terms of an evolving fabric tensor and the plastic strain rate direction, thus, enhancing the two CST conditions by a third. In this way, the three analytical conditions of ACST become both necessary and sufficient for reaching and maintaining CS. In addition, the use of A explains the observed results by relating the stress‐strain response, in particular the dilatancy, to the evolution of fabric by means of the relevant equations of ACST.

On the attraction power of critical state in granular materials

“…k n /D s is given as 300 MPa, where D s = 2R 1 R 2 /(R 1 + R 2 ) and R 1 , R 2 are the radii of particles in a given contact and k t /k n is 0.5. If the pressure of 100 kPa is considered, the average ratio of contact overlap and particle size < u n > /d 50 is around 0.16%, which is close to the value (10 −3 ) in literature [18,41]. For sample preparation, two schemes can be used to obtain an isotropic compression of the specimen: the boundary moving scheme and the internal compacting scheme [45].…”

“…As recalled in the introduction, the critical state (CS) refers to a state where stresses, void ratios and fabrics tend to be steady at relatively large deformation when shear strain further increases [41]. Biaxial tests used as references have been simulated to define the critical state line which refers to a collection of critical states obtained at different confining pressures.…”

“…The concept of critical state was first proposed by Casagrande [3] and developed by Roscoe et al and Schofield et al [34,37] in the form of a stationary state where stress and volume tend to be constant under continuous shear strain. A steady stress ratio and a steady void ratio are two conditions for this classical definition of critical state [41].…”

“…The coaxiality between the plastic strain rate direction and the fabric anisotropy proved to be a relevant state variable in defining critical state [22,41]. A third condition that quantifies the role of fabric anisotropy in terms of its intensity and its relative orientation with respect to loading direction should also be taken into account, in addition to the aforementioned two conditions of constant stress ratio and void ratio [22,41].…”

Use of a bounding surface model in predicting element tests and capacity in boundary value problems

“…With respect to the relevance of the present work, it is worth noting that recent discrete element model studies have found that: i) the fabric tensor evolves towards a critical state value with plastic deformation (Li and Li 2009) and ii) large volume contractions occur on imposition of stress rotation at the critical state (Theocharis et al 2017;Theocharis et al 2019). Capturing the above effects compatibly under evolving fabric anisotropy can be achieved through the recently developed framework of ACST (Anisotropic Critical State Theory) (Li and Dafalias 2012), wherein an anisotropy dependent additive term was used to modify ψ to control contractant versus or dilatant behaviour rather than the multiplicative factor adopted here (i.e.…”