Yield function

The total equation can now be decomposed into a set of equations for each atomic species, which are re-ananged to yield functions equal to zero. Thus... 

If the material is perfectly plastic, i.e., if the yield function is independent of k and a, then = 0 and the magnitude of the plastic strain rate cannot be determined from (5.81). Only its direction is determined by the normality condition (5.80), its magnitude being determined by kinematical constraints on the local motion. 

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... 

The yield function is usually taken in the simple form due to von Mises... 

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. 

BF3-Et20, NaCNBHs, THF, reflux 4-24 h, 65-98% yield. Functional groups such aryl ketones and nitro compounds are reduced and electron-rich phenols tend to be alkylated with the released benzyl carbenium ion. The use of BF3 Et20 and triethylsilane as a cation scavenger is also effective." ... 

Functional amines, e.g., m-aminophenol, 2-methyl-2-chloro-3-aminophenol, and 4-chloro-3-aminophenol, react with 4-methoxybut-3-en-2-one to yield functional lepidine derivatives 230 (62AG161). 

The epoxidation reaction usually takes place under mild conditions and with good to very good yield. Functional groups that are sensitive to oxidation should not be present in the starting material with carbonyl groups a Baeyer-Villiger reaction may take place. 

Analogous to the allylsilane cyclizations, alkynylsilanes can also be used to synthesize 1,2-fused bicyclic compounds. The intramolecular addition of 4-[5-(trimethylsilyl)-3-pentynyl]-2-cyclo-hexenone proceeded smoothly in the presence of various Lewis acids, yielding functionalized cw-fused octahydro-5/f-inden-5-ones containing the synthetically useful terminal allene unit45. 

More general processes rely on variations of the Guareschi-Thorpe reaction [14] where condensations between 1,3-dicarbonyls and cyanoacetamide yield functionalized monocyclic 2(lff)-pyridones (a and b. Scheme 2) [15, 16]. Unless the carbonyls are sufficiently different in reactivity, the reaction suffers from poor regioselectivity. The use of 3-alkoxy or 3-amino enones instead of 1,3-dicarbonyls has proven to be a versatile and reliable synthetic methodology where the 1,4-addition controls the regioselective outcome (c and d. Scheme 2) [17-19]. 

The oxophilicity of silicon is also the driving force that allows the reaction of the same silylated reagent 29 with oxiranes [134], aldehydes and ketones [135,136], and enones [136] for yielding functionalized AT-silylphosphazenes (Scheme 31). 

The diversity of the GABAa subunits (Figure 3.2d) is reflected in a very complex pharmacology. Expression of the subunits in heterologous systems shows that the combinations a, (3, and y can yield functional receptors, indicating that the limitation in subunit combination is defined by expression levels and most likely cell-dependent assembly mechanisms also. The pj to p3 subunits mainly co-assemble with each other to form the GABAC receptors. 

Katritzky offers a general one-pot alternative approach to polysubstituted pyrroles utilizing disubstituted olefins of which a wider variety is commercially available compared to acetylenes . Thus, thioamides 32 were subjected to Mannich condensation with aldehydes and BtH to yield functionalized thioamides 33 which were then treated with base... 

The intrinsic complexity and the multiplicity of cholinergic receptors became evident upon elucidation of their primary structures. In the CNS, at least nine different sequences of a subunits and three different sequences of (3 subunits of the nicotinic receptor have been identified [10, 11]. Expression of the cloned genes encoding certain subunit combinations yields functional receptors with different sensitivities toward various toxins and agonists. 

The palladium-catalyzed C-Se bond formation79-81 and the platinum-catalyzed carboselenation of alkynes with selenoesters have been reported in analogy with the thiolate chemistry. An electron-withdrawing or electron-donating group on the aromatic residue leads to the formation of the desired carboselenation product with acceptable yields. Functionalities like benzyl, hydroxyl, or nitrile group are tolerant with the reaction conditions. This method provides a new access to the functionalized vinylselenide 60,82 molecules of interest as key intermediate (Scheme 19) 85... 

Perhydrooxazolo[3,2- ]pyridines 338 are excellent precursors of iminium ions 339 obtained after treatment of the oxazolidine with either a Bronsted or Lewis acid. Trapping of these intermediate iminium ions with nucleophiles then allows for substitution at the C-8a position together with ring opening, yielding functionalized piperidines 340 (Scheme 93). 

Another type of acyllithium synthons was generated in situ from chloroimines. The naphthalene-catalyzed (4%) lithiation of chloroimines 93 in THF at —78 °C was followed by filtration of the excess of lithium, being then treated with an electrophile and finally hydrolyzed, to yield functionalized imines 94 (Scheme 40) . ... 

Dihydrobenzofuran (327) reacted with lithium and a catalytic amount (5%) of DTBB in TFIF at 0°C to give mainly or exclusively the corresponding intermediate 328, which by treatment with an electrophile at the same temperature and final hydrolysis yielded functionalized phenols 329 (Scheme 96) . ... 

The a-phenyl substituted six-membered sulfur-containing material 399 was reduc-tively cleaved using DTBB (5%) as the arene catalyst in THF at —78°C. The resulting intermediate 400 was then reacted with an electrophile followed by hydrolysis, yielding functionalized thiols 401 (Scheme 112) . 

Finally, the ring opening of substituted thioisocromans (402) with lithium and a catalytic amount of DTBB (5%) in THF at —78 °C gave the expected benzylic intermediates 403 which, after reaction with electrophiles at the same temperature and final hydrolysis with water, yielded functionalized thiols 404 (Scheme 113) . 

The remaining results in Fig. 9 demonstrate that when a small amount of molecular oxygen is mixed in the Ar layer condensed on -hexane [Fig. 9(e)] or deposited onto an isolated Ar layer [Fig. 9(d)], the P hi resonance reappears in the Ar desorption yield function. Since the -hexane spacer inhibits Ar decay by electron transfer to Pt(l 11), the presence of Ar resonance in Fig. 9(c) and (d) was therefore interpreted [164] as due to electron transfer to O2 leading to the formation of O2 in its ground-state Og. With the electron affinity of O2 being of the order of the binding energy of the first electronically excited state of Ar, the decay of Ar P into lowest bulk excitons is possible by electron transfer to O2. 

The production of stable anions by low-energy electron impact has been studied by ESD [22,41,43,44,122,147,154,156,157,166-219] and charge-trapping [154,156,217,220-236] experiments. Below 20 eV, the desorption of stable anions from condensed systems is attributed to DEA, which produces oscillatory structures in the anion yield functions, and... 

Figure 10 0 yield functions for electron impact on (a) gaseous O2, (b) 4 ML of O2, and (c-g) 0.15 ML of O2 condensed on 4 ML of the indicated substrate. The dashed lines and bars indicate the known positions of the Oj resonances. The gain factor over each curve is relative to curve (c), and the shifted baselines correspond to the zero anion intensity levels. Figure based on the data of Ref. 41. 

Also shown in Fig. 10(c)-(g) are the anion yield functions for submonolayer quantities of O2 deposited onto various multilayer atomic and molecular solids. The data represent part of a study [41] on the environmental factors involved in the DEA process. As can be seen, the yield of desorbed ions can vary greatly with substrate composition. Such variations can be attributed to the so-called extrinsic factors that modify the ESD process at times before attachment and after dissociation, for example, electron energy-loss processes in the substrate and postdissociation interactions (PDI) of ions with the surrounding medium [41]. These processes can be contrasted with intrinsic factors, which... 

In early studies of anion desorption from solid 02/hydrocarbon mixtures [246], several observations were made concerning the desorbed signal of OH . (1) The OH yield functions resembled more closely the yield functions for 0 with kinetic energies 1.5 eV from pure O2 films than the signal associated with DEA to the hydrocarbon molecule. (2)... 

Figure 15 (a) The yield function from 4-ML-thick films of an O2/C4H10 mixture of increasing percentage volume concentration of Oj. (b) The H signal from a 4-ML-thick mixture of 10% O2 in C4H10 can be resolved into two components one centered at 10 eV, associated with production via DEA to the alkane (dotted line), and another at 7.8 eV, associated with reactive scattering of 0 ions (bold line). Also shown is the yield of OH ions from the same film. (From Ref. 257.)... 

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