Self-consistent model

A variation on the exact soiutions is the so-caiied seif-consistent modei that is explained in simpiest engineering terms by Whitney and Riiey [3-12]. Their modei has a singie hollow fiber embedded in a concentric cylinder of matrix material as in Figure 3-26. That is, only one inclusion is considered. The volume fraction of the inclusion in the composite cylinder is the same as that of the entire body of fibers in the composite material. Such an assumption is not entirely valid because the matrix material might tend to coat the fibers imperfectiy and hence ieave voids. Note that there is no association of this model with any particular array of fibers. Also recognize the similarity between this model and the concentric-cylinder model of Hashin and Rosen [3-8]. Other more complex self-consistent models include those by Hill [3-13] and Hermans [3-14] which are discussed by Chamis and Sendeckyj [3-5]. Whitney extended his model to transversely isotropic fibers [3-15] and to twisted fibers [3-16]. 

Theessence of the procedure is that Halpin and Tsai [3-17] showed that Hermans solution [3-14] generalizing Hill s self-consistent model [3-13] can be reduced to the approximate form... 

R. Hill, Theory of Mechanical Properties of Fibre-Strengthened Materials - III. Self-Consistent Model, Journal of the Mechanics and Physics of Solids, August 1965, pp. 189-198. 

In order to solve the system of the above-described equations, and which are derived by applying the self-consistent model, applied for composites by Budiansky 7), it is necessary to evaluate experimentally the moduli of elasticity (tension, shear, bulk) and Poisson s ratios of the constituent phases and the composite. Thus, the only unknown are the radius r of the mesophase layer and its mechanical properties and thermal expansion coefficient, which are then derived. 

A three-layer model for fiber composites may be developed, based on the theory of self-consistent models and adapting this theory to a three-layered cylinder, delineating the representative volume element for the fiber composite. 

Indeed, the self-consistent model averages the stresses and strains in either phase of a two-phase material, and it determines them, by solving separate problems, whose superposition yields the final configuration of the model 7). 

In the self-consistent model the matrix material outside the inclusion is assumed as possessing the effective macroscopic properties of the composite. Moreover, two consecutive problems were solved by assuming either phase of the composite as occupying its position and surrounded by this average material. In both cases the average values of the composite are determined from the values of the characteristic... 

An important variation of the self-consistent model is the three-phase model, introduced by Kerner 20), according to which the inclusion is enveloped by a matrix annulus, which in turn is embedded in an infinite medium with the unknown macroscopic properties of the composite. 

Newns DM. 1969. Self consistent model of hydrogen chemisorption. Phys Rev 178 ... 

Although the extended ab initio atomistic thermodynamics approach provides an exact expression for the interfacial stability, the formalism requires self-consistent modeling of the entire electrochemical system, or electrode/electrolyte interface, exceeding presently available computational capabilities. Therefore, certain assumptions had to be made that reduce the effort to the calculation of the electrode surface only. Even with this simplified approach, which has been applied to the two examples discussed in this chapter, the qualitative behavior can be reproduced. 

The main reason for using a 1D model is the reduction of the computational effort compared to the higher-dimensional models. This reduction to one dimension is acceptable because it is possible to study the sustaining mechanisms and the chemistry in the discharge with a ID self-consistent model. 

Recent research now concentrates on the more physical models involving theories of rotation. The long-term aim of these attempts are to provide fully self-consistent models which include stellar evolution, rotation, transport of angular momentum and of chemical species. The key players in this field are P. Denis-senkov [11,12] and C. Charbonnel and coworkers (for their approach, see the contributions by Charbonnel and Palacios in this volume). Both groups employ the theoretical description of rotation by Zahn and Maeder [27,19]. 

Rotation remains the best candidate for extra-mixing in RGB stars. The present modelling of the rotational mixing is however still incomplete, and agreement between self-consistent models and observations might be achieved by improving the description of the hydrodynamics related to rotation. 

E. Bakker, R.K. Meruva, E. Pretsch, and M.E. Meyerhoff, Selectivity of polymer membrane-based ion-selective electrodes — self-consistent model describing the potentiometric response in mixed ion solutions of different charge. Anal. Chem. 66, 3021—3030 (1994). 

Such ambiguity and also the low structural resolution of the method require that the spectroscopic properties of protein fluorophores and their reactions in electronic excited states be thoroughly studied and characterized in simple model systems. Furthermore, the reliability of the results should increase with the inclusion of this additional information into the analysis and with the comparison of the complementary data. Recently, there has been a tendency not only to study certain fluorescence parameters and to establish their correlation with protein dynamics but also to analyze them jointly, to treat the spectroscopic data multiparametrically, and to construct self-consistent models of the dynamic process which take into account these data as a whole. Fluorescence spectroscopy gives a researcher ample opportunities to combine different parameters determined experimentally and to study their interrelationships (Figure 2.1). This opportunity should be exploited to the fullest. 

These studies were with polycrystalline copper, but more detailed results (Fig. 9) were obtained subsequently (44) with Cu(100), Cu(l 11), and Cu(l 10) surfaces XPS, UPS, LEED, and mass spectrometric data being combined to provide a self-consistent model. The surface species formed with their associated N(ls) and 0(ls) binding energies (eV) on exposing copper surfaces to NO(g) are listed in Table I. 

The use of an ideal-solution model meant that there were a number of instances where calculated and experimental results were quantitatively at variance. However, the approach very successfully predicted the general form of most of the phase diagrams, for example whether they were peritectic or eutectic, and accounted for the appearance of intermediate phases in systems such as Cr-Rh. That the approach could do this using such simple and internally self-consistent models is a demonstration of the inherent power of CALPHAD methods. The importance of this first step therefore cannot be overestimated, although its significance was not... 

To explain dendritic encapsulation, one needs to know something about dendrimer conformations and dynamics. These can be complex and difficult to map out. This problem leads to a chicken and egg conundrum (to crack a bad encapsulating joke). Ideally, one has information about structure that can be used to rationalize encapsulation behavior. However, as it is difficult to get structural (particularly conformational) information about dendrimers, structures are often proposed based on observed encapsulation behaviors. Thus, a central theme of this chapter will be that the redox unit, as it is perturbed by the structure of the dendrimer, acts as both embedded reporter and performer. This approach runs the risk of generating a circuitous argument. However, in combination with other methods of assessing structure, it hopefully can iterate to a self-consistent model. 

The most drastic effect on the losses of the thermal energy is due to dissociation of molecular hydrogen. According to Fox and Wood (1985) as much as a half of the thermal energy behind the shock front is absorbed due to dissociation of Hg molecules. At the same time photodissociation of Hg molecules in the precursor causes retardation of the collisional ionization in the relaxation zone, whereas the precursor structure is very sensitive to the radiation flowing from the wake (Gillet and Lafon 1983 1984). So, the self-consistent model of the radiative shock is urgently needed. 

At the close of this conference, we do not seem to have a completely self-consistent model in which the basic features of the light curve and spectra are satisfied by calculations of the theoretical spectra and light curves at all epochs. There is much left to be learned about the structure of the progenitor and the manner in which it exploded. 

The former postulate is the more attractive in that it allows more freedom of assignment, and is the currently favoured model (Hotta and Anderson, 1959 Piette, 1960). The latter has been discussed by two groups simultaneously in terms of restricted rotations within the radical (Abraham et al., 1958 Ingram et al., 1958). However, in neither instance nor in subsequent discussion (Symons, 1959) was it possible to obtain a really self-consistent model for the whole spectrum. 

Now we turn to calculation of the susceptibility component Xst( ) in Eq. (17). To extract it from Eq. (14c), one should replace there p for and account for an inhomogeneity of the induced distribution F(y). The latter is determined by a chosen collision model. Such models are described in detail in GT, Section IV.B, and in VIG, Section VI, where they are separated into the self-consistent and non-self-consistent models. For one simple example they are considered also in Section VII.C... 

The simplest, self-consistent model of the diffuse-ion swarm near a planar, charged surface like that of a smectite is modified Gouy-Chapman (MGQ theory [23,24]. The basic tenets of this and other electrical double layer models have been reviewed exhaustively by Carnie and Torrie [25] and Attard [26], who also have made detailed comparisons of model results with those of direct Monte Carlo simulations based in statistical mechanics. The postulates of MGC theory will only be summarized in the present chapter [23] ... 

The determination of plasma parameters using He-like spectra is based on a self-consistent modeling of the theoretical spectra. The following variables take part in the variation procedure based on least-squares fitting electron and ion temperatures, toroidal plasma velocity, concentrations of H-, He-and Li-like ions. In addition, a background function was used to subtract the plasma background from the experimental spectra. The background consists of continuum radiation from the plasma and detector noise. 

Scott D. R. and Stevenson D. J. (1989) A self-consistent model of melting, magma migration, and buoyancy-driven circulation beneath mid-ocean ridges. J. Geophys. Res. 94,... 

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