Principal consolidation stress

Construction of the Dynamic Internal Yield Locus. The dynamic yield locus represents the steady state deformation, as opposed to the static yield locus which represents the incipient failure. The dynamic yield locus is constructed by plotting on a (a, t) plane the principal Mohr circles obtained for various consolidation stresses. The dynamic yield locus will be the curve or straight line tangent to all circles, as shown in Figure 17. The dynamic angle of internal friction S and cohesion C are independent of the consolidation stress. S and Q are obtained as the slope and the intercept at er=0 of the dynamic yield locus of the powder. 

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... 

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)... 

In general, a material flows poorly if its strength is large relative to the consolidating stress o-j. Jenike called this ratio of o-i to, the flow fimetion (F = ai/f ). It is generally not constant because, as a rule, the compressive strength is not proportional to the principal stress 0-1. This ratio esm be used to make a classification of the flow according... 

Fig. 13 shows the value of the principal normal stress Sigma 1 under which the sample has been consolidated its value is obtained by drawing Mohr s circle through the end point of the yield locus (point at no volume change), tangential to the locus. 

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. 

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... 

Fig. 23. Critical stresses

The principle of these testers is that the specimen can be subjected to controlled stresses in two orthogonal directions (biaxial testers) or three orthogonal directions (triaxial testers). In the case of the triaxial testers, two of the orthogonal stresses are usually equal, normally generated by liquid pressure in a pressure chamber. The specimen is placed in a cylindrical rubber membrane and enclosed by rigid end cups. The specimen is consolidated isotropically, i.e. by the same pressure in all three directions which leads to volumetric strain but little or no shear strain. This is followed by anisotropic stress conditions, whereby a greater axial stress is imparted on the specimen by mechanical force through the end cups. In the evaluation of results it is assumed that the principal stresses act on horizontal and vertical planes, and Mohr circles can be easily drawn for the failure conditions. 

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). 

In a state of incipient failure, the yield locus is tangent to the Mohr circle. The Mohr circle graphically represents the equilibrium stress condition at a particular point at any orientation for a system in a condition of static equilibrium in a two-dimensional stress field. The equilibrium static conditions can also be applied to sufficiently slow steady flows. The maximum principal stress in Fig. 6.4(b) is called the unconfined yield strength. This is the maximum normal stress, under incipient failure conditions, at a point where the other principal stress becomes zero. Such a situation occurs on the exposed surface of an arch or dome in a feed hopper at the moment of failure see Fig. 7.5(b). In the analysis of bridging in feed hoppers, the unconfined yield strength becomes a very important parameter. The magnitude of the unconfined yield strength is determined by the YL and depends, therefore, on the consolidation pressure and time. 

The stress acting on an exposed surface is also the only non-zero principal stress, because the exposed surface is assumed self-supporting and traction-free (i.e., no shear stresses acting on the surface). The flow factor is determined by the geometry of the hopper and the properties of the bulk material. Another function used by Jenike is the flow function . This flow function is the ratio of the consolidating pressure CTi to the unconfined yield strength as defined in Section 6.1.2 ... 

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. 13, The mean principal stress during the consolidation for each experiment (X and Y stresses are averaged).

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

Fig. 11 Critical state line and consolidation behavior of Batch 7 silty sand from Lower San Fernando Dam (From Olson and Stark 2002 with permission from Canadian Science Publishing). Critical state data from Baziar and Dobry (1995) and Vasquez and Dobry (1989) range of in situ void ratios from Castro et al. (1989) cr 3 = minor effective principal stress...

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