Quasi-static theory

Another condition, related to the possibility of simply interpreting the temperature dependence of the wing profile, is that one point in the spectrum should be related to one single point of the potential curve. In the present case this is always verified except in the neighborhood of extrema of the difference AV between the potentials of the upper and lower states of the transition. However this region, which corresponds to the so-called satellite, is not amenable to interpretation by the quasi static theory and will be considered later on. 

This result can be easily modified to describe an assembly of N perturbing particles provided that the actual frequency shift is attributed only to the effect of the nearest neighbour. However, this is only satisfactory for those interactions which have a relatively short range, i.e. n s 6. For Stark broadening, n=2 and 4, it is necessary to include the simultaneous effect of all the perturbers, which greatly complicates the calculation. Experimental results illustrating the application of the quasi-static theory will be discussed in section 8.7.5 below. 

Firstly, we are going to demonstrate how branch interference may be taken into account within the quasi-classical impact theory. Then we shall analyse a quasi-static case, when the exchange frequency between branches is relatively small. An alternative case, when exchange is intensive and the spectrum collapses, has been already considered in Chapter 2. Now it will be shown how the quasi-static spectrum narrows with intensification of exchange. The models of weak and strong collisions will be compared with each other and with experimental data. Finally, the mutual agreement of various theoretical approaches to the problem will be considered. 

Pressure induced broadening and narrowing of a whole spectrum are described by the quasi-static approximation and the perturbation theory, correspondingly. Comparing inequalities (6.13) and (2.53) one can see that the border between the stages is determined by the criterion coij 1,... 

Before treating specific faradaic electroanalytical techniques in detail, we shall consider the theory of electrolysis more generally and along two different lines, viz., (a) a pragmatic, quasi-static treatment, based on the establishment of reversible electrode processes, which thermodynamically find expression in the Nernst equation, and (b) a kinetic, more dynamic treatment, starting from passage of a current, so that both reversible and non-reversible processes are taken into account. 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

Two theories exist for the lack of isotope fractionation in chondmles. Both rely on the concept of quasi-statical (i.e., almost equilibrium) exchange of isotopes between liquid and gas followed by permanent loss of rock-forming elements to the escaping gas phase. One model (Alexander 2003) implies that the chondmles formed in such close proximity to one... 

Generally, a.c. elcetric field could not unambiguously be characterized by any potential function. However, in our theory of collision models, which is applied for description of a low-frequency spectrum, we may employ the quantity Ub, which has a sense of a quasi-static potential energy. 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

The basic theories of elastic deformations associated with various contact forces under static contact conditions have been introduced in the last section. Assuming that an impact process of two solids can be regarded as quasi-static, the theories given in 2.3 are used directly to link the dynamic deformations of the colliding solids with the impact forces. In this section, the collisions of elastic spheres are described. 

Collisions between particles with smooth surfaces may be reasonably approximated as elastic impact of frictionless spheres. Assume that the deformation process during a collision is quasi-static so that the Hertzian contact theory can be applied to establish the relations among impact velocities, material properties, impact duration, elastic deformation, and impact force. 

The fractures on a plane surface, created by the collisions of hard spherical particles at low-impact velocities, may form a conical crack according to the Hertzian quasi-static stress theory. In a multiple-impact situation, the conical cracks meet those extending from neighboring impact sites, and then the brittle material becomes detached. Once appreciable damage is done, the cracking mechanism may be altered because the particles no longer strike on a plane surface nevertheless the brittle removal continues by the successive formation and intersection of cracks. 

The approaches are divided into those which do not take frequency into account (except for relaxation phenomena) and therefore are "static" theories and those which take frequency into account, generally by a scattering approach. The latter, "dynamic" theories, when reduced to the low frequency or quasi-static limit, usually compare favorably with the static theories. Some approaches take multiple scattering into account and cannot be solved in closed form. These require elaborate computer number crunching techniques (5). 

The Kerner equation, a three phase model, is applicable to more than one type of inclusion, Honig (14,15) has extended the Hashin composite spheres model to include more than one inclusion type. Starting with a dynamic theory and going to the quasi-static limit, Chaban ( 6) obtains for elastic inclusions in an elastic material... 

Similarly, the effects of bubbles in viscoelastic materials were studied by preparing rubber samples containing microvoids. Microvoids were used in order to avoid the effects of bubble resonance and to compare theories in their quasi-static limit. 

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