Yield locus

Powder Mechanics Measurements As opposed to fluids, powders may withstand applied shear stress similar to a bulk solid due to interparticle friction. As the applied shear stress is increased, the powder will reach a maximum sustainable shear stress T, at which point it yields or flows. This limit of shear stress T increases with increasing applied normal load O, with the functional relationship being referred to as a yield locus. A well-known example is the Mohr-Coulomb yield locus, or... 

Here, [L is the coefficient of internal friction, ( ) is the internal angle of friction, andc is the shear strength of the powder in the absence of any applied normal load. The yield locus of a powder may be determined from a shear cell, which typically consists of a cell composed of an upper and lower ring. The normal load is applied to the powder vertically while shear stresses are measured while the lower half of the cell is either translated or rotated [Carson Marinelli, loc. cit.]. Over-... 

This line represents the critical shear stress that a powder can withstand which has not been over or underconsolidated, i.e., the stress typically experienced by a powder which is in a constant state of shear, when sheared powders also experience fiiciion along a wall, this relationship is described by the wall yield locus, or... 

Figure 16.5 shows the loci of general yielding and fast fracture plotted against crack size. The yield locus is obviously independent of crack size, and is simply given by ct = (jy. The locus of fast fracture can be written as... 

In general, a yield locus is obtained that relates the shear strength of the powder bed to the consolidation load and reduced load. The yield locus has been found to take the following form [71] ... 

In a slightly different form, Eq. (6) is commonly referred to as the Warren spring equation. Representative yield loci determined utilizing the simplified shear cell are shown in Fig. 7 for spray-dried lactose, bolted lactose, and sucrose. The yield locus for each material relates the shear strength to the applied load. 

Figure 11 Sample wall yield locus generated from wall friction test data.

The Mohr-Coulomb failure criterion can be recognized as an upper bound for the stress combination on any plane in the material. Consider points A, B, and C in Fig. 8.4. Point A represents a state of stresses on a plane along which failure will not occur. On the other hand, failure will occur along a plane if the state of stresses on that plane plots a point on the failure envelope, like point B. The state of stresses represented by point C cannot exist since it lies above the failure envelope. Since the Mohr-Coulomb failure envelope characterizes the state of stresses under which the material starts to slide, it is usually referred to as the yield locus, YL. 

A rigid-plastic powder which has a linear yield locus is called a Coulomb powder. Most powders have linear yield loci, although, in some cases, nonlinearity appears at low compressive stresses. A relation between the principal stresses in a Coulomb powder at failure can be found from the Mohr circle in Fig. 8.4 as... 

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

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. 

Jenike carried out many experimental measurements on free-flowing and cohesive materials. He found that the yield locus of a dry material would be a straight line passing through the origin, as shown in... 

Shear stress chart Static yield locus... 

Figure 16(a). The term Cohesionless was therefore used to describe materials which have a negligible shear strength under zero normal load (an = 0). On the other hand, Jenike found that the yield loci of cohesive materials differ significantly from a straight line and have a nonzero intercept, indicated by C. Moreover, the position of the locus for a cohesive powder is a strong function of the interstitial voidage of the material. Fig 16(b) shows the typical yield locus for cohesive materials. 

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. 

Figure 22 shows a typical experimental shear stress chart, obtained for A2, using a compaction with normal stress equal to 1106.2 Pa. Figure 23 shows the corresponding yield locus from which the static angle of internal friction, cp, was worked out from the slope of the yield locus. The cohesion, C, is obtained from the intercept with the shear stress axis. 

Figure 23 Worked example E5 experimental static yield locus - Alumina A2, uc — 1106.2 Pa...

Double-grooved specimens were used to study the failure of PC, PC/PE, PET, ABS, and HIPS during transitions from plane stress to plane strain. The yield behavior of PC is consistent with a von Mises-type yield criterion plane strain reduces its elongation. The yield behavior of PC/PE is consistent with a Tresca-type yield criterion plane strain appears to be relieved by voiding around the PE particles. PET undergoes a ductile-to-brittle transition its behavior is consistent with a von Mises-type yield locus intersected by a craze locus. The yield behavior of ABS and HIPS is not significantly affected by the plane-stress-to-plane-strain transition. Plane strain alone does not necessarily cause brittleness. 

The yield criteria of polymers have been reviewed by Ward (7) and more recently by Raghava et al. (8). Except for the craze yield criteria of Sternstein and Ongchin (9) and Bowden and Oxborough (10), most of the yield data can be described by a pressure-modified, von Mises-yield criterion. The corresponding yield surface is everywhere convex. A typical yield locus on the [Pg.103]

Figure 2. Von Mises-type polymer yield locus with tensile plane strain (e1 = 0) indicated. Convention for coordinates is also shown.

The tensile yield stress variation as a function of W for a material which has a von Mises-type yield locus is illustrated schematically in Figure 5. This variation is caused by the fact that as the width of the specimen increases, the biaxiality also increases toward the asymptotic value at plane strain. If the material obeys the von Mises yield criterion exactly, the plane strain yield stress should be 15% higher than it would be for simple tension. On the other hand, if the material obeys the Tresca yield criterion, the plane strain yield stress should be identical... 

PC/PE. In the case of PC/PE, plane strain alone does not produce significant changes in the yield stress and the deformation behavior. Its yield locus in the tension-tension quadrant is therefore either very nearly a quarter circle or similar to a Tresca locus. The exact shape of the locus can be determined only by much more elaborate biaxial tests. This material is not very notch sensitive compared with PC. The energy to break in a notched Izod impact test is 15 ft-lb/inch for Vs-inch thick bars and 11 ft-lb/inch for 4-inch bars whereas for PC the latter figure is about 2 ft-lb/inch. This reduction in notch sensitivity over pure PC appears to be related to the material s ability to void internally, probably relieving the plane strain. 

PET. The behavior of crystalline PET at plane strain can be explained if its yield locus is similar to that of PS and PMMA (9, 10) where a craze locus intercepts the shear yield locus. The transition at plane strain to a craze locus would account for the brittleness. This transition, which takes place quite sharply at W/t = 23 (W/b = 8), is probably the cause for the low impact strength (< 1 ft-lb/inch) of the Vs-inch thick notched bars. The plane strain brittleness can be avoided if the geometric constraints can be removed, such as making the notch less sharp or making the test bar thinner. In fact, unnotched bars of PET, equivalent to having an infinite notch radius, are quite tough. The notch sensitivity of PET is typical of crystalline polymers. 

Jenike developed the idea that no single line represents the yield but rather a curve called the yield locus. The yield behavior depends on the packing density of the powder when it is caused to flow under the action of normal and shear stress. Figure 12.36 shows a yield locus for a given porosity, e. A Mohr circle for the stage when yielding starts is characterized by the principal stresses i and 2-The points at the end of the yield locus lies on the Mohr circle pertains to... 

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