Stress Distribution in a Steady Hopper Flow

For a slow and steady motion of powders, the local stresses inside the powder are approximated to be pseudostatic and, consequently, are governed by the equation of 

The axisymmetric nature of conical hoppers results in es = 0 and, according to Eq. (2.20), cre = ( rr + jz)v. To close the problem, one more relation is required. In a continuum solid, this relation is known as the compatibility requirement, i.e., the relationship of strains. This relation, with the aid of constitutive relations between stress and strain (e.g., Hooke s law), provides an additional equation for stress so that the problem can be closed. However, the compatibility relation for a continuum solid may not be extendable to the cases of powders. Thus, additional assumptions or models are needed to yield another equation for stresses in powders. 

and the modified Mohr-Coulomb yield (or failure) criterion, Eq. (8.27). It should be noted that other yield criteria, such as the von Mises criterion, are used to model the flow of bulk solids in hoppers, and more conditions may need to be imposed, such as the Levy flow rule, in order to close the system of equations [Cleaver and Nedderman, 1993], 

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