Generalized yield conditions

Under more general conditions of multi-axial principal stresses a, 02,o the same condition is reached when the equivalent stress a (or defined as 

This generalized yield condition, which is usually referred to as the von Mises yield condition, has a simple geometrical visualization in principal stress space of o i,o 2,o 3 shown in Fig. 3.2. There the line making equal angles with the three principal stress axes represents the locus of pure mean normal stress a, along which all deviatoric stresses vanish and no plastic flow can occur. Thus, plastic flow requires a critical deviation from this line in the radial direction away from it 

The condition of the associated-flow rule is depicted as a vector def parallel to the outward normal to the yield surface where a — Y. 

The von Mises (or simply Mises) yield condition of yield for the most general case when all stress elements are present becomes 

Most mechanisms of local plastic strain production are best understood in simple shear, i.e., under a local shear stress such as 022,. When this is the case, the yield condition becomes 

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Figure 1 shows the effect of this so-called backstress-tensor a as the translation of the surface center. Variants (a) and (b) can be seen as theoretical extremes because only case (c) is observed in the experiment. The general yield condition for such kind of material behavior looks like... 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

Extension of this reaction to other substrates, however, revealed that it is more complex, and that side products are formed depending on (1) the nature of the substrate, (2) the reaction conditions, e.g. temperature and solvent,and (3) the method of work-up." Thus, in addition to the desired substitution products, primary and secondary hydroxy steroids generally yield esters and ethers and undergo simple dehydration as well as dehydration accompanied by rearrangement. 

With the exception of the parent compounds, where the Michael adducts are isolated, acrylic esters [see, e.g. 6,7,31,105,111 ] and nitriles [6,7], and vinyl ketones [26, 113, 115] generally yield the cyclopropanes (Table 7.6) under the standard Makosza conditions with chloroform. Mesityl oxide produces a trichlorocyclopropy-lpropyne in low yield (10%) [7]. When there is no substituent, other than the electron-withdrawing group at the a-position of the alkene, further reaction occurs with the trichloromethyl anion to produce spiro systems (35-48%) (Scheme 7.12) [7, 31]. Under analogous conditions, similar spiro systems are formed with a,p-unsaturated steroidal ketones [39]. Generally, bromoform produces cyclo adducts with all alkenes. Vinyl sulphones are converted into the dichlorocyclopropane derivatives either directly or via the base-catalysed cyclization of intermediate trichloromethyl deriva-... 

Whereas it was reported in CHEC-II(1996) <1996CHEC-II(7)229> that examples of this system were rare, the increase in synthetic activity since then has been significant. Such compounds can be obtained using either a thiophene or a pyrazine precursor. Virtually all of the molecules prepared from thiophene precursors follow the pathway shown in Equation (185). The appropriate diaminothiophenes 491, usually obtained by reduction of the corresponding nitro groups, are condensed with the desired 1,2-dicarbonyl compound under generally mild conditions to yield 492. 

Both fumaric and maleic acid give rise to coumarins on reaction with phenols. Initially, this route was considered different from the Pechmann reaction, but experimental evidence has been accumulated which suggests that it is simply a variant (73AJC899). Under the acidic conditions, the unsaturated acid could well produce some malic acid, which would then lose carbon monoxide and water as usual. Generally yields in the two approaches to coumarins are much the same. 

In turn, equations (4.1.9) and (4.1.10) yield a general initial condition necessary for an analysis of the transition from the monomolecular to bimolecular kinetics. When the initial particle distribution is unknown (which is usually the case, e.g., after prolonged irradiation), it is assumed, as a rule, to be the simplest, i.e., the Poisson one, equation (4.1.12). 

Moreover, under a particular set of conditions, the extraction efficiencies for the SC-C02 system were typically found to be very similar to the corresponding values in dodecane. The strong correlation between the SFE and conventional SX results for the two ions suggests that the solvation behavior of SC-C02 is similar to that of dodecane for the TBP system. Substitution of a stronger Lewis base, such as triph-enyl- (TPPO), tributyl- (TBPO), or trioctylphosphine oxide (TOPO) for TBP, generally yielded higher extraction efficiencies for both uranium and thorium. For both TBPO and TOPO, in fact, extraction was nearly quantitative over the entire range of... 

Denmark has developed a practical dioxirane-mediated protocol for the catalytic epoxidation of alkenes, which uses Oxone as a terminal oxidant. The olefins studied were epoxidized in 83-96% yield. Of the many reaction parameters examined in this biphasic system, the most influential were found to be the reaction pH, the lipophilicity of the phase-transfer catalyst, and the counterion present. In general, optimal conditions feature 10 mol% of the catalyst l-dodecyl-l-methyl-4-oxopiperidinium triflate (30) and a pH 7.5-8.0 aqueous-methylene chloride biphasic solvent system [95JOC1391]. 

We have seen that the appropriate choice of the catalyst and/or the olefin substitution pattern can dramatically shift the equilibrium in favor of the macrocyclic product. Other factors are important as well, such as the general experimental conditions. Danishefsky and coworkers have highlighted the effects of solvent, temperature, and concentration on the yield of macrocyde RCM for the synthesis of analogs of epothilone 490 [14]. As shown in Scheme 2.4, RCM of compound 7 in... 

Integral equation theories of g(r) do in general yield only an approximate estimate of this quantity, and hence they are, to more or less extent, thermodynamically inconsistent. In practice, instead of Eq. (16), one prefers to apply the pressure-compressibility (P — %T) condition expressed by... 

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