Coefficient of pressure at rest

Using continuum mechanics, ceramic powder packing can be analyzed for its mold filling capability. Powder packings are different from liquids in one important way, which is described by the coefficient of pressure at rest. 

FIGURE 12.34 Schematic of snap-throu buckle of a particle arch during pressing of a ceramic powder. The force exerted by the particles on the wall in the vertical direction is a result of the coefficient of pressure at rest. 

The coefficient of pressure at rest is a descriptive index used in soil mechanics. It shows that the flow of powder packings cannot be regarded as analogous to the behavior of liquids. Let us consider a die with a large square cross-section. They axis points vertically downward. Let be the horizontal and Tyy the vertical compressive stress at any given depth acting edong the x and y axes, respectively. For reasons of symmetry, these stresses are also principal stresses. The coefficient of pressure at rest is = r hyy. 

TABLE 12 The Effect of Particle Size, Particle Morphology, and Binder Content on the Coefiicimt of Pressure at Rest, Xo, for Zirconia Powder 

Data from D. Bortzmeyer, Dry Pressing of Ceramic Powders, NKV Summer School, September 1991, Pfetten, The Netherlands. 

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The following variables can affect wall friction values of a bulk soHd. (/) Pressure as the pressure acting normal to the wall increases, the coefficient of sliding friction often decreases. (2) Moisture content as moisture increases, many bulk soHds become more frictional. (3) Particle size and shape typically, fine materials are somewhat more frictional than coarse materials. Angular particles tend to dig into a wall surface, thereby creating more friction. (4) Temperature for many materials, higher temperatures cause particles to become more frictional. (5) Time of storage at rest if allowed to remain in contact with a wall surface, many soHds experience an increase in friction between the particles and the wall surface. (6) Wall surface smoother wall surfaces are typically less frictional. Corrosion of the surface obviously can affect the abiUty of the material to sHde on it. 

Change in state of stress on a saturated clay element upon sampling, (a) Initial field condition (b) state of stress on an undisturbed sample. (From Noorany, I., and Seed, H.B., /. Soil. Meek. Found. Div., 91,49-80,1965. Reprinted with permission of ASCE.) total vertical stress total horizontal stress in situ pore water pressure All, change in pore water pressure K , coefficient of earth pressure at rest A pore pressure coefficient expressing the change in pore water pressure due to release of the ground deviator stress isotropic effective stress acting on a completely undisturbed sample P effective overburden pressure. 

For an NC clay or a sandy clay in the grormd, a relation between S, tive friction angle (( ) ) can be derived as follows Consider a soil element at A in Figure 8.37. The major and minor effective principal stresses at A can be given by and K, a vo/ respectively. K is the coefficient of earth pressure at rest. Let the soil element be subjected to an unconsolidated rmdrained (UU) triaxial test. The total and effective stress Mohr s circles for this test, at failure, are shown in Figure 8.38. A review of this figure shows that at failure the total major principal stress is Oj = -I- AOj the total minor principal stress... 

Value of k actually varies with depth. At the pile head, its value is about the same as Rankine coefficient of passive kp), while at the pile tip, its value is similar to ko, coefficient of earth pressure at rest. This coefficient also depends on method of pile construction. Table 5.3 shows the values of k for a number of pile types. 

Dynamic losses also result from acceleration of air at rest. Most often this occurs at the entrance into an exhaust system. Turbulence at the entry to the system adds to dynamic losses. The coefficient of entry Q is a measure of the efficiency at the entry of a hood or pipe. The efficiency indicates how well static pressure converts to velocity pressure. 

Dorgunoglu and Mitchell, 1975 presented a method to determine the effective friction angle for a silica sand based on the cone resistance, the effective vertical stress and the coefficient of active earth pressure at rest K. ... 

Coefficient of active earth pressure at rest Horizontal seismic coefficient Vertical seismic coefficient Yield coefficient Earthquake magnitude Constrained Modulus Local magnitude Surface wave magnitude Moment magnitude of earthquake Porosity Initial porosity... 

Figure 15. Variation in Dxh for garnet versns reciprocal temperature for experimental data sources listed in Table lb at a variety of pressures n = 33). A distinction is made between mantle solidus partition coefficients (Salters and Longhi 1999 Salters et al. 2002 McDade et al. 2003a,b) and the rest. Note the strong temperatnre dependence, which is qnalitatively similar to that incorporated in Equation (25b). The scatter is due to additional compositional controls.

The Borda mouthpiece is of interest because it is one device for which the contraction coefficient can be very simply calculated. For all other orifices and tubes, there is a reduction of the pressure on the wall adjacent to the opening, but the exact pressure values are unknown. However, for the reentrant tube, the fluid is at rest on the wall around the tube hence, the pressure must be exactly that due to the depth below the surface. The only unbalanced pressure is that on an equal area opposite to the tube, and its value is whAo. The time rate of change of momentum due to the flow out of the tube is (W/g)V = wA V2lg, where A is the area of the jet. Equating force to time rate of change of momentum, whA0 - wA V2/g. Ideally, V2 = 2gh, and thus ideally Cc = A Ao = 0.5. If it is assumed that Cv = 0.98, then the actual values will be Cc = 0.52 and Cd = 0.51. 

After the specimen has been encased in a rubber membrane and set up within the triaxial chamber, the lateral pressure is applied. This may be any value consistent with actual field loading conditions. If field loading conditions are not known, use an at-rest coefficient of 0.4. In other words, set the lateral pressure to 40% of the axial load to be applied. Axial load may be applied immediately after applying the lateral pressure. 

The first term on the right side of (8-163) represents a hydrostatic buoyant force due to fluid pressure that acts in the direction opposite gravity. This force remains operative when the fluid is at rest. The second term on the right side of (8-163) represents dynamic contributions from fluid pressure (i.e., 33%) and viscous shear stress (i.e., 67%) which act in the direction of the approach velocity. This dynamic force vanishes under hydrostatic conditions. The friction coefficient which is the inverse of fluid mobility, is given by... 

The magnitude of the reflection coefficient at normal resting heart rate is about 0.4, similar for many mammalian species (Table 17.3). The resolution of pressure and flow waveforms into their respective forward and reflected components [Li, 2000,2004] are shown in Figure 17.2. 

Peck et al. (1972) proposed closed-form solutions in terms of thrust, bending moments, and displacement under external loading. The lining response was a function of structure compressibility and flexibility ratios, in situ overburden pressure, and at-rest earth coefficient. To adapt to seismic loading, the free-field shear stress replaces the in situ overburden pressure and earth coefficient. The stiffness of the tuimel relative to the ground is measured by the compressibility (C) and flexibility (F) ratios. Those are the extensional stiffness and flexural stif iess of the medium relative to the lining. 

Since the system is at constant volume and temperature, the number of moles must be directly proportional to the pressure. Use the coefficients of the reaction to determine stoichiometric ratios. (Recall Chapter 3.) Fill in the rest of the row labeled AP with 0.10 atm multiplied by the stoichiometric ratios ... 

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