Major consolidation stress

It is convenient to introduce the concepts of material flow function, FF, and flow factor, ff. The material flow function, FF, relates the unconfined yield stress, To, to the corresponding major consolidating stress, cri, and is determined experimentally from the yield locus of the material, as shown in Fig. 8.9. The material flow function is presented as a plot of To versus flow factor, ff, is defined by... 

Solution The kinematic angle of internal friction can be determined from the Mohr circle, which is tangential to the yield locus at the end point. This Mohr circle yields the major consolidating stress o and minor consolidating stress <73. Thus, % is found to be 30°, either from Eq. (8.27) or from a tangent of the Mohr circle which passes through the origin, as shown in Fig. E8.1. 

The shear stress that occurs in a deforming (i.e.. flowing) bulk solid is dependent upon the major consolidating stresses (pressures) acting on the bulk solid but independent of the rate of shear. Conversely, for a liquid, generally the shear stress is dependent upon the rate of shear and independent of the major consolidating pressure. 

Figure 1.17 Yield loci for the determination of Jenike effective angle of friction (S) (a) and internal angle of friction (5e) (b) from the critical state line (CSL) (fc unconfined yield stress and cti major consolidation stress).

Major consolidation stress a ) This is the principal normal stress (cti) under which the sample has been consolidated in the principal stress plane. The major consolidation stress should not be confused with the initial compaction stress, which is the stress that compacts the powder bed. Each different compaction stress, (7c, leads to a different yield locus and becomes one of a family of yield loci at different densifications. The major consolidation stress is obtained by drawing a Mohr semi-circle through the equilibrium or end point of the yield locus and tangential to the yield locus. 

In order to achieve unique numbers for various and different degrees of flowability Jenike calculated the failure function at a specific value of the unconfined stress. The specific value of the ratio of the major consolidation stress to the unconfined stress was taken as 6.5 Ibf/ft (3.11 kPa) in the initial, Jenike designed, 4 inch diameter shear cell, because at this value the relationship between fa and a showed only small deviations from linearity. Currently a value of 3 kPa is generally used with a standard shear cell tester. 

The Flow Function as the dependence of the unconfmed yield strength Oc on the major consolidation stress oi (at steady state flow) can ordy be determined using testers where both stress states can be realized. Steady state flow can be realized in Jenike s tester, in annular shear cells, in a torsional shear cell, in the true biaxial shear tester and in a very specialized triaxial cell [2]. The unconfined yield strength Oe can be determined by running tests in Jenike s tester, in an annular shear cell [10], in uniaxial testers and in the true biaxial shear tester. Therefore, only Jenike s tester, annular shear cells and the true biaxial shear tester can guarantee the measurement of Flow Functions Oc (cf ) without further assumptions. 

Fig. 11 shows a cr, t-diagram. The curve represents the maximum shear stress x the sample can support under a certain normal stress o it is called the yield locus. Parameter of a yield locus is the bulk density Ai,. With higher preconsolidation loads the bulk density Ai, increases and the yield loci move upwards. Each yield locus terminates at point E in direction of increasing normal stresses a. Point E characterizes the steady state flow which is the flow with no change in stresses and bulk density. Two Mohr stress circles are shown. The major principal stresses of the two Mohr stress circles are charcteristic of a yield locus, Oi is (he major principal stress at steady state flow, called major consolidation stress, and cTc is the... 

As discussed in Ref [6], the bulk density will increase an amount Apfrom the initial loaded condition to the running condition as illustrated in Fig. 2. At the load point. L, the bulk density corresponds to the major consolidation stress oil -... 

Using Eq. (4), for soft loading, the bulk density at the load point is 740 kg/m corresponding to the major consolidation stress of 1.82 kPa. Thus the percentage increase in bulk density due to load settlement is Ap = 13.5%. 

As outlined in section 7.4.4.5, the static angle of internal friction and the cohesion of a granular material are a function of the consolidation stress. Therefore, they can be expressed also as a function of the major principal stress ct Table 13 reports the values obtained for the effective angle of internal friction and the cohesion for all the powders... 

Fig. 23. Critical stresses

The major principal or consolidating stress (o-j) depends on the stress condition just before the arching. The flowing powder in the dipleg is considered to be in passive state of stress. The axial profile of the average vertical stress over horizontal cross-section, crh, has been given by Li (1990)... 

FIG. 2 The free surface of a powder under consolidation represents the conditions of the minor Mohr circle. Under minor principal stress a3 = 0 the major principal stress is defined as the unconfined yield strength fc and represents the strength of the powder at the free surface of the arch (adapted from Bell, 2001). 

The Jenike effective angle of friction is the angle of the straight line drawn through the origin of a normal stress-shear stress plot and tangential to the Mohr semi-circle, which inscribes the equilibrium, or end point of the yield locus when failure occurs at no sample volume change. The Mohr semi-circle represents the stresses in a powder consolidated under a major principal stress. 

Fig. 5. Unconfmed yield strength Cc versus major at steady state flow ai(Flow Function) and versus major principal stress at consolidation 0, c (limestone x50= 4,8...

Fig. 9 shows the unconfined yield strength for various vibration velocities versus the major principle stress during consolidation oi. The dashed lines show the range of effective wall stresses 0 = 0]/ff of a cohesive powder arch for common values of the flow factor ff. The intersection point of Oc and oT delivers the so-called critical unconfined yield strength ac.crii-Eventually, the minimum outlet diameter to avoid bridging in a mass flow hopper bmin is directly proportional to Cc,crit- In the example, shown in Fig. 9, the critical unconfined yield strength (and hence bmin) can be strongly reduced in presence of vibrations. 

Fig. 9. Unconfmed yield strength vs. major principle stress during consolidation for limestone powder at various vibration velocities...

Consolidating major principal stress Fig. 4 The effect of basis on the failure function... 

By the turn of the century, major American industries were being consolidated through mergers and acquisitions, partly because of the stress of international competition. Frank A. Vanderlip observed in Scribner s Magazine in 1905 that as combinations are made in the industrial field, the possibility of employing highly trained technical experts rapidly increases. 

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