Path-dependent effects

We also want to point out the difference between simple rate-dependent phenomena and path-dependent effects. Simple rate dependence means that the internal micromechanical state (as possibly represented by some meso-scale variables) depends only on the current deformation and current rate of deformation the material has no memory of the past. In terms of dislocation dynamics and (7.1), a simple rate-dependent constitutive description would be one in which... 

Two examples of path-dependent micromechanical effects are models of Swegle and Grady [13] for thermal trapping in shear bands and Follansbee and Kocks [14] for path-dependent evolution of the mechanical threshold stress in copper. 

Although the difference in final strength f, integrated through both the actual shock wave and the computational shock wave, will be mitigated by dynamic recovery (saturation) processes, this is still a substantial effect, and one that should not be left to chance. These are very important practical considerations in dealing with path-dependent, micromechanical constitutive models of all kinds. 

Compton profile. Furthermore the contribution of the multiple scattered photons to the measured spectra has to be taken into account (for example by a Monte Carlo simulation [6]). Additionally one has to take heed of the fact that the efficiency of the spectrometer is energy dependent, so the data must be corrected for energy dependent effects which are the absorption in the sample and in the air along the beam path, the vertical acceptance of the spectrometer and the reflectivity of the analyzing crystal. 

Table 3.5 compares estimates made by the different methods the procedure of Helgeson et al. (1978) effectively leads to the best estimates. According to Robinson and Haas (1983), however, the result of this procedure is path-dependent, because it is conditioned by the selected exchange components different exchange reactions (with different reference components) can lead to appreciably different results for the same component. 

The electron-ion recombination in high-mobility systems has also been analyzed in terms of the fractal theory [24,25]. It was postulated that even when the fractal dimension of particle trajectories is not equal to 2, the motion of particles is still described by diffusion but with a distance-dependent effective diffusion coefficient D r) = D(H-// where the parameter / is proportional to the mean free path X [24]. However, when the fractal dimension of trajectories is not equal to 2, the motion of particles is not described by orthodox diffusion. 

If regulatory control is used, the cure cycles should be developed as efficiently and effectively as possible. The cost of cure cycle development is open-ended because there is no limit on the number of possible variations to cure cycles. There is no guarantee either that the results will be transferable to other processing equipment or materials because the relationship between primary and secondary variables is unpredictable in such complex, path-dependent processes. 

As there exists a phase equilibrium both phases must have reached in the internal thermodynamic equilibrium with respect to the arrangement and distribution of the molecules the measuring time. Therefore, no time effects or path dependencies of the thermodynamic properties in the liquid crystalline phase should be expected. To check this point for the l.c. polymer, a cut through the measured V(P) curves at 2000 bar has been made (Fig. 6) and the volume values are inserted at different temperatures in Fig. 7, which represents the measured isobaric volume-temperature curve at 2000 bar 38). It can be seen from Fig. 7 that all specific volumes obtained by the cut through the isotherms in Fig. 6 he on the directly measured isobar. No path dependence can be detected in the l.c. phase. From these observations we can conclude that the volume as well as other properties of the polymers depend only on temperature and pressure. The liquid crystalline phase of the polymer is a homogeneous phase, which is in its internal thermodynamic equilibrium within the normal measuring time. 

Related chemistry of a cationic rf-a- yrm Mo(ll) complex is shown in Scheme 28. Hydride addition to the cationic reactant complex gives a neutral allyl-Mo that is oxidized by pyridinum dichromate (PDC) to a cation. Nucleophilic addition of water and oxidative decomplexation of the Mo fragment gives an enone. Substituted a-pyran hgands follow one of two paths depending on the electronic and steric effects of the substituent, as shown. 

The results presented in the previous sections show that the anodic reactions on a silicon electrode may proceed via different paths depending on the conditions and that those in HF solutions and those in KOH solutions are rather different. They also show that the mechanistic models proposed for the reactions in HF and KOH solutions from the many studies in the literature are largely separated. However, in both HF and KOH solutions, the silicon/electrolyte interface is fundamentally similar differing only in the concentrations of hydroxyl and fluoride ions. Thus, a reaction scheme must be coherent with respect to the experimental observations in both FIF and KOH solutions. For comparison. Table 5.8 summarizes the characteristic features of the reactions occurring on silicon in FIF and KOH, in terms of nature of the reaction, rate, effective dissolution valence, photoeffect, and uniformity of the surface. 

An illustrative example of the effect of counterions on selectivity is the work by Brinchi et al, in which they demonstrated that reaction of sulphonate esters in the presence of equimolar amounts of bromide and hydroxyl ions took completely different paths depending on whether the reaction was performed in a micellar system based on cationic surfactant or in a homogeneous solution, see Fig. 5.5 [29]. When there is no surfactant present, attack by the hydroxyl ion dominates. In the micellar solutions, on the other hand,... 

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