Mohr circle

Figure 8.4. Mohr circle and Mohr-Coulomb failure envelope.

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 8.5. Mohr circles for two cases (a) Active failure (b) Passive failure.

Figure 8.8. Mohr circle for the state of stresses near the hopper wall.

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. 

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. 

The basic character of non-fluidized gas-particle flow is the existence of contact pressure (or stress) among particles and between particles and the pipe wall. Both theoretical analyses and experimental results (Terzaghi, 1954 Johanson and Jenike, 1972 Li and Kwauk, 1989) showed that the pressure of the interstitial fluid in particulate material neither compresses nor increases the shear resistance of the particulate material. After Walker (1966) and Walters (1973), Li and Kwauk (1989) analyzed the stresses in a vertical pneumatic moving-bed transport tube by using the stress theory of particulate media mechanics and Mohr circles, shown in Figs. 19 and 20, and gave the following stress ratios at any point in the flow field ... 

These equations show the relationship between the normal and shear t3q>es of stress on the principal types of stress and the angle a that represents a force balance for the volume element. The equations can be represented the Mohr stress circle (Figure 12.35) with a radius of (o-j — o-2)/2 and its center at (ci + 0 2)/ on the abscissa. The shear stress T y is plotted on the ordinate and the normal forms of stress r and Tyy are plotted on the abscissa. Tbe points of the circle intersection with the abscissa give the principal t3q>es of stress ai and a2- The normal stress and the associated shear stress, t, are fixed by the radius arm which is rotated through an angle 2a from the abscissa. The normal stress r, now appears as the projection of the radius arm onto the abscissa, and the shear stress appears as the projection onto the ordinate. It can be seen from the Mohr circle that the shear stress has its maximum values for a = 45° and 135°. 

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... 

For each porosity, there is a particular yield locus, a family of three 3deld loci is shown in Figure 12.38. Many experiments [72] have established that the envelope of the Mohr circles through the points Ei that lead to steady state flow for different porosities is, to a veiy close... 

Fig. 5 Mohr circle construction to obtain the major normal stress, (T v the unconfined field stress, fc. (From Ref. l)...

Fig. 5. Mohr-circle construction of precursory R- and R -shears that form early during upward-propagation of normal fault through ductile shale bed. The maximum shear stress is assumed to occur on fault-parallel planes the shear orientations are obtained through the use of...

Flow Functions and Flowabilily Indices Consider a powder compacted in a mold at a compaction pressure Oi. When it is removed from the mold, we may measure the powder s strength, or unconflned miiaxial compressive yield stress L (Fig. 21-38). The unconfined yield and compaction stresses are dietermined directly from Mohr circle constructions to yield loci measurements (Fig. 21-36). This strength increases with increasing previous compaction, with this relationship referred to as the powder s flow function FF. 

The actual value of Nh is not known, but if the soil has not failed it can be plotted as shown as long as the Mohr circle of stress does not touch the failure line. If some event such as nearby excavation occurs, the soil mass will tend to expand horizontally, and the magnitude of Nh will decrease. The circle will thus grow in size until it touches the failure line. At this point failure is incipient. The value of Nh cannot decrease any further since this... 

Figure 2.6 Mohr circles for active and passive pressures in cohesionless soils.

Summary. The variation of compressive stress with direction in a principal plane can be represented in three equivalent ways—by an ellipse, a sine-wave or a Mohr circle. The variation is fixed by just the two extreme values, tjy and (T3 or 02 and (T3 or and ffj- The different diagrams are simply visualizations of the original statement... 

Figure 6.5 The normal stresses on a set of planes as shown in Figure 6.4a. (a) Variation of normal stress with orientation a. (b) The same range of normal stresses shown by a Mohr circle.

Sine-wave, Mohr-circle, and ellipsoid representations... 

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). 

Figure 5. The Mohr circle technique to determine stress regime. (Reproduced with permission from reference 30. Copyright 1981 Society of Professional Well Log Analysts.)...

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