Scaling laws for quasi-static granular sand at critical state

The critical state theory of soils describes the correspondence between effective stress, shear strength and soil density. Numerous soil mechanics experiments have also revealed a correlation between soil strength and loading rate. Considering that the granular matter is the actual medium of natural soils, a quasi-static inertia number is proposed, i.e., Q=ϕ₀ln(I)+α, for the granular soils considering the particle volume fraction. Based on the classical triaxial test data of soils, the scaling laws of quasi-static deforming sand at the critical state from the perspective of granular physics are explored, and a simple linear relationship i.e., μ=ξQ, is found between the friction coefficient and the quasi-static particle inertia number. The newly established scaling laws can quantitatively describe the influences of the volume fraction, shear rate, confining pressure and particle size on the frictional properties of sand when reaching the critical state. In addition, to quantify the volumetric deformation laws of sand under quasi-static shear, a correlation is obtained between the particle volume fraction ϕ at the critical state and the quasi-static inertia number Q. In attempt to characterize the scaling laws of the three-dimensional stress state, a new dimensionless number (i.e., the intermediate principal stress number) is defined to reveal the influences of the intermediate principal stress on the frictional properties. Thus, the scaling laws are extended.