Brief Note A variational analysis of the equilibrium turbulent closure models of a dry granular densified motion with weak turbulent intensity

A variational principle is proposed to derive the equilibrium expressions of the turbulent closure models of an isothermal dry granular dense matter in weak turbulent motions. It is demonstrated that the equilibrium equations and the natural boundary conditions coincide with those derived by the thermodynamic analysis (Fang and Wu, in Acta Geotechnica, in press). The current work serves as a supplementary variational verification of the turbulent closure models proposed in Fang and Wu (in Acta Geotechnica, in press).