Effective yield locus

Figure 8.9. Effective yield locus and evaluation of Iq and a for steady hopper flows.

In general, this Ck)ulomb yield criterion can be used to determine what stress will be required to cause a ceramic powder to flow or deform. All that is needed are the two characteristics of the ceramic powder the angle of friction, 8, and the cohesion stress, c, for each particular void fraction. With these data, the effective yield locus can be determined, from which the force required to deform the powder to a particular void fraction (or density) can be determined. This Coulomb yield criterion, however, gives no information on how fast the deformation will take place. To determine the velocity that occurs durii flow or deformation of a dry ceramic powder, we need to solve the equation of motion. The equation of motion requires a constitutive equation for the powder. The constitutive equation gives the shear and normal states of stress in terms of the time derivative of the displacement of the material. This information is unavailable for ceramic powders, and the measurements are particularly difficult [76, p. 93]. 

There exists a critical state line, also referred to as the effective yield locus. The effective yield locus represents the relationship between shear stress and applied normal stress for powders always in a critically consolidated state. That is, the powder is not over- or undercompacted but rather has obtained a steady-state density. This density increases along the line with increases in normal stress, and bed porosity decreases. 

A given yield locus generally has an envelope shape the initial density for all points forming this locus prior to shear is constant. That is, the locus represents a set of points all beginning at the porosity this critical state porosity is determined by the intersection with the effective yield locus. 

Points to the left of the effective yield locus are in a state of overconsolidation, and they dilate upon shear. If sheared long enough, the density and shear stress will continue to drop until reaching the effective yield locus. Points to the right are underconsolidated and compact with shear. 

Under the hnear Mohr-Coloumb approximation, if parallel yield loci are assumed with constant angle of internal friction, and with zero intercept of the effective yield locus, the flow function is a straight line through the origin D, given by... 

For a cohesive particulate material, each YL curve ends at a point where the normal stress equals the consolidation pressure. Mohr circles can now he drawn that are tangent to the end point of the various yield loci. The envelope of these circles is called the effective yield locus (EYL). This is generally a straight line passing through the origin see Fig. 6.5. 

Walker [23] made a more rigorous analysis of the pressure distribution in vertical bins. He assumed a plastic equilibrium in the particulate solids with the Mohr circles representing the stress condition at a certain level touching the effective yield locus. Walker derived the following expression for the pressure profile in a vertical cylinder ... 

Figure 10.14 Definition of effective yield locus and effective angle of internal friction, <5...

In the chapter Design of Silos for Flow" it was shown that the knowledge of the Flow Function, the Time Flow Functions, the angle (pe of the effective yield locus and the wall fiction angle epy is necessary to design a silo properly. Having estimates of the Flow Function only (see Fig. 5) uncertainties remain and assumptions are necessary to get reliable flow. These assumptions are hard to check. 

Fig. 8. Horizontal stress ratio X versus angle of effective yield locus (pe (41 bulk solids).

The Flow Function, the Time Flow Fimctions, the angle (peof the effective yield locus and the angle of wall friction (p have to be known exactly. 

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