Stress consolidation

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. 

It is required to design a mass flow conical hopper with the volume capacity of 100 m3 to store a cohesionless material of bulk density 1,700 kg/m3 and an angle of internal friction 40°. Four sets of shear tests have been conducted on the material and results for the unconfined yield strength and the corresponding consolidating stress are as follows ... 

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. 

Table 11 Worked example E5 Consolidation stresses in internal yield loci experiments...

For each powder, four static yield loci are obtained by applying the normal loads reported in Table 11, to which the corresponding consolidation stresses, ctc, are calculated from Equation (41). Following the procedure described in section 7.4.4.7, the values of the normal stresses <3i applied for each consolidation stress ctc, are reported in Table 12 and also shown in Figure 22. 

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 unconfined yield strength of the powder,, is a powder characteristic which decides whether it can resist flow under gravity from a certain dipleg configuration, and is a function of the consolidating stress (c ) present during its preparation as shown in Fig. 25. 

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 practice, however, the value of FF is not a constant but increases with consolidating stress. It is determined by the experimental A ai curve as shown in Fig. 25. The A — al curve can approximately be expressed by Eq. (76). In this case a critical radius Rc of the dipleg can be calculated. 

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... 

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. 

Critically consolidated. If a powder is sheared sufficiently, it will obtain a constant density or critical porosity e for this consolidation normal stress Gc- This is defined as the critical state of the powder, discussed below. If a powder in such a state is sheared, initially the material will deform elastically with shear forces increasing linearly with displacement or strain. Beyond a certain shear stress, the material will fail or flow, after which the shear stress will remain approximately constant as the bulk powder deforms plastically Depending on the type of material, a small peak may be displayed originating from differences between static and dynamic density. Little change in density is observed during shear, as the powder has already reached the desired density for the given applied normal consolidation stress a . 

In practice, following the filhng of a cell, the powder is in an underconsolidated state. A set of shear steps is performed under a chosen consolidation stress in the consolidaton stage to increase its density and bring it into a critical state. A set of shears is then performed at small normal stresses in the shear stage to determine the strength of... 

Other workers assume a linear form with a nonzero intercept. This implies a minimum powder strength in the absence of gravity or any other applied consolidation stresses. As described above, the flow function is often curved, likely due to the angles of friction being a function of applied stress, and various fitting relations are extrapolated to zero to determine. While this is a typi( practice, it has questionable basis as the flow function may have pronounced curvature at low stress. 

Uniaxial Compression - Williams Method This method was developed by Williams, Birks and Bhatta-charya24. A compact is first formed in a split mould by applying an axial compressive force, the mould is then removed to leave a cylindrical specimen with its axis vertical. The compressive vertical stress needed to cause failure of the specimen is then found and this is the unconfined yield stress for the consolidating stress used in the compaction of the specimen. The failure function is found by forming a number of compacts under different consolidating stresses and finding the unconfined yield stress for each specimen. 

The limitations of the Jenike shear cell are that it is not very useful for measuring bulk solids with large shear deformations, e.g., plastic powders. The level of consolidation stresses required are inappropriate for pharmaceutical materials, and the quantity of material required is often beyond that available in the early stages of development. Alternative shear cells that have been used include annular shear cells (Nyquist and Brodin 1982 Irono and Pilpel 1982) and ring shear testers (Schulze 1996). 

Different vertical loads can be applied to a bulk solid sample of known mass, and compression of the sample is recorded electronically (Thomson, 1997). With these data, powder contact volume versus compressive force or stress can also be represented. Bulk density of a solid is a function of consolidation stress and changes during flow as the stress changes. Because the mass consolidating load and volume are known, the relationship can be plotted as shown in Figure 6. 

Typical plot of bulk density versus consolidating stress. 

Generally, the compressibility of many powders has been correlated with internal cohesion C and to some extent particle deformability. High compressibility is related to low flowability (expressed in terms of cohesion C) under high consolidation stress conditions (Peleg, 1978 Schubert, 1987). Particularly, compressibility has been found to correlate with cohesion C... 

Conversely, Ehlermann and Schubert (1987) sustained that compressibility results from materials of different composition cannot be compared and that flowability characterization through compressibility must be made specifically for each food variety. Moreover, confined uniaxial compression is a simple compression test that provides an approximate measure of the flowability of powders. Therefore, it is not suitable for silo design but may prove to be a convenient method for process control in any food laboratory (e.g., to evaluate particle cohesion). Table II offers a range value definition for flowability classification by comparing flow function (ratio between the maximum consolidation stress and unconfined yield stress) with compressibility. 

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