Yield surfaces

Meantime, the stresses lie inside the given yield surface, namely... 

The normality conditions (5.56) and (5.57) have essentially the same forms as those derived by Casey and Naghdi [1], [2], [3], but the interpretation is very different. In the present theory, it is clear that the inelastic strain rate e is always normal to the elastic limit surface in stress space. When applied to plasticity, e is the plastic strain rate, which may now be denoted e", and this is always normal to the elastic limit surface, which may now be called the yield surface. Naghdi et al. by contrast, took the internal state variables k to be comprised of the plastic strain e and a scalar hardening parameter k. In their theory, consequently, the plastic strain rate e , being contained in k in (5.57), is not itself normal to the yield surface. This confusion produces quite different results. 

Since the yield function is independent of p, the yield surface reduces to a cylinder in principal stress space with axis normal to the 11 plane. If the work assumption is made, then the normality condition (5.80) implies that the plastic strain rate is normal to the yield surface and parallel to the II plane, and must therefore be a deviator k = k , k = 0. It follows that the plastic strain is incompressible and the volume change is entirely elastic. Assuming that the plastic strain is initially zero, the spherical part of the stress relation (5.85) becomes... 

Figure 5.7, Isotropic yield surfaces in (a) stress space and (b) strain space.

From (A.81), /3T, = k, and this equation implies that the yield surface in stress space is a circular cylinder of radius k, shown in a FI plane projection in Fig. 5.7(a). The corresponding yield surface in strain space may be obtained by inserting the deviatoric stress relation (5.86) into the yield function (5.92)... 

The yield surface in strain space is a ciruclar cylinder normal to the 11 plane with radius k/2fi and axis offset from the origin by e ", as shown in Fig. 5.7(b). It may be seen that, if the yield function in stress space is independent of pressure, then the yield function in strain space is independent of volume change and vice versa. 

Naghdi, P.M. and Trapp, J.A., On the Nature of Normality of Plastic Strain Rate and Convexity of Yield Surfaces in Plasticity, J. Appl. Mech. 62, 61-66 (1975). 

In many cases, some boundary conditions are not well known or not known at all. Temperature boundary conditions can be obtained from thermal building-dynamics programs that allow the capture of spatial mean temperatures during a time period as long as a whole year. Some of these programs yield surface temperature values (e.g., TRNSYS), which can be used as temperature boundary conditions at the time of CFD study. 

The simplified failure envelopes differ little from the concept of yield surfaces in the theory of plasticity. Both the failure envelopes (or surfaces) and the yield surfaces (or envelopes) represent the end of linear elastic behavior under a multiaxial stress state. The limits of linear elastic... 

Alkali metal trimetaphosphates react with primary amines in water at a pH value between 7 and 10, and at a temperature between 40 and 90°C, to yield surface-active alkylamidotriphosphates [64-66] see Eq. (20) ... 

Dialkyl phosphites react with acyl halides such as lauroyl chloride to yield surface-active acid esters of acylphosphonic acid [84-87] see Eq. (40). 

Aliphatic amines yield surface areas typically around 100 m g (Table 19.1, 1-4). The physisorption isotherms of samples 1-3 are shown in Fig. 19.1. 

The results of experiments with crude oil fractions in this study also suggest that several species were present in reaction interface. There are mainly long chain carboxylic acids. The difference in size and structure is expected to give them different pka values. As a result, different surface activity (i.e., IFT value) is obtained with different levels of alkali concentration. Crude oil fractions with lower surface activity only yield surface inactive salts that may appear as precipitates at the interface. 

The essential features of the above picture are shown schematically in Fig. 6. Hydrogen or deuterium adsorbs and desorbs rapidly on these sites (half-time less than 1 min) hence, the slower H2-D2 equilibration (half-time about 8 min) appears to be determined by the site-to-site migration required for exchange. Poisoning experiments show that water also prefers these sites in agreement with the limited IR data, Fig. 6 shows the adsorbed water yields surface hydroxyls (10). 

Step (18) in the above is the analog of step (8), which is required for H2—D2 equilibration it is a necessary step if we view the jr-allyl as an immobile species on the surface. The products of step (19) can be viewed as propylene in the form of a loosely held w-complex which on desorption yields isomerized propylene. Readsorption of the isomerized propylene or further reaction of the x-complex would yield surface OD groups. When equilibrium is achieved, the concentration of surface OD groups should equal 40% of the initial concentration of OH groups. Figure 21 shows a plot versus time of the intensity (multiplied by a scale factor to yield concentration) of the surface OH and OD. The expected equilibrium points are indicated by arrows. Corresponding data for CD3—CH=CH2 are also shown. Except for the OH species from CD3—CH=CH2, which is a relatively weak band on the side of a surface hydroxyl, the curves approach the expected value. 

Auger spectroscopy can also be used in a scanning mode (scanning Auger microscopy, or SAM) to yield surface topographical and elemental distribution data (30). 

ISS and UPS—Preference of one of the these techniques over another is difficult to determine. ISS could yield surface structural comparisons with... 

The character of the radioactive debris from a land surface explosion is determined largely by the extent of mixing between the extraneous debris injected into the cloud and the fission product radioactivities. Within the early cloud there is a well developed toroidal circulation (5), which is clearly evident in the case of air bursts and large yield surface bursts. In low yield surface explosions it may be obscured quickly by the dirt cloud and by rapid damping of a systematic circulation. 

The specific activity for a refractory chain in cloud and fallout samples of lower yield surface bursts may not be too different. Data exist for Johnie Boy (10) and for Lacrosse (7) which bear this out. Apparently mixing of the fission products with the debris is much more efficient in low yield explosions where the toroidal circulation is weaker and less persistent and where cloud rise is heavily damped by the atmosphere. This is the basis for drawing the specific activity of the refractory chain at low particle diameters more flatly for lower yield cases, as illustrated schematically in Figure 1. 

The present study is on a system Co(II)-H20-Si02 for which it was expected that there would be minimal adsorption of polynuclear species of the metal ion but the possibility of surface catalysis to yield surface polymers of the hydroxide. 

It remains to be determined to what extent the dye adsorption technique is applicable to other substrates. No evidence was obtained for Pseudocyanine adsorption to Mn02, Fe2Os or to pure silver surfaces, although this dye can be bound to mica, lead halides, and mercury salts with formation of a /-band (61). Not only cyanines but other dye classes can yield surface spectra which may be similarly analyzed. This is specifically the case with the phthalein and azine dyes which were recommended by Fajans and by Kolthoff as adsorption indicators in potentio-metric titrations (15, 30). The techniques described are also convenient for determining rates and heats of adsorption and surface concentrations of dyes they have already found application in studies of luminescence (18) and electrophoresis (68) of silver halides as a function of dye coverage. 

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