Generalized Fractional Flow Rule and Its Modelling of the Monotonic and Cyclic Behavior of Granular Soils

Cyclic Fractional Plastic Model for Granular Soils

“…in which n v and n s are the flow directions induced by compression and shearing, respectively. Due to the distinct formulae of the fractional derivatives in describing loading and unloading, a generalized plastic flow rule suitable for both cases can be derived by substituting Equation ( 12) into Equations ( 1) and ( 2) (Sun et al, 2018a), such that:…”

“…(17) The stress-dilatancy relationship for loading and unloading can be therefore obtained by using Equations ( 16) and ( 17), where the classical modified Cam-clay stress-dilatancy relationship can be also achieved by using α = 1. Detailed derivations of Equations ( 15) and ( 16) can be found in Sun et al (2018a) and thus not repeated here for simplicity. In addition, the loading tensor m is assumed to be the same as the flowing tensor, such that:…”

“…To capture the state-dependent non-associated plastic behavior of sand, Been and Jefferies (1985) firstly suggested the concept of state dependence for sands, which was then promptly developed in various constitutive models (Li and Dafalias, 2000;Yang and Li, 2004;Sun et al, 2018aSun et al, ,b,c, 2019a. However, most of these models usually assumed a plastic flow rule that was only suitable for monotonic loads.…”