Continuum media

Khludneva E.Yu. (1990b) Contact of a nonelastic plate with a rigid punch. In Dynamics of Continuum Media. Novosibirsk (98), 67-79 (in Russian). 

The continuum electrostatic approximation is based on the assumption that the solvent polarization density of the solvent at a position r in space is linearly related to the total local electric field at that position. The Poisson equation for macroscopic continuum media... 

In such a case, the mass balances may be described by equations of continuum media. However, if one looks the bed closer e.g. at the particle scale as represented by the close-up of Figure 6, the representation of the flow process is completely different. The liquid is distributed between different channels according to the local geometrical features of the... 

The introduction of the concepts of stress and deformation at a point has been a fundamental concept in the development of the mechanics of continuum media. From a physical point of view, only the displacement is a real quantity, while stress imphes an idealized situation that is not directly measurable the value of a stress can only be inferred from its effects. The effects of the force at a point P depend on the orientation of the element surface SA comprising the point, which in turn is characterized by a vector rij (j =1,2,3) normal to the surface at P, as shown in Figure 4.1. The stress vector at the point P can be written as... 

When comparing the kinematic limit of Eq. [21] with Eq. [24], we note similitude both derive from a conservation law and from a structural relation and both apply when gravity driven flow is dominant. But the first one is limited to macroscopic continuum media where the hydraulic conductivity is well defined everywhere. The other one is more general as the structural relation between the flux and the volumetric content is not dependent on the existence of a REV. Equation [24] does not account for waterfront dispersion, nevertheless dispersive effects have been experimentally observed for low input intensities (Di Pietro Lafolie,1991). 

In order to perform the large scale hydraulic flow and transport simulations using a continuum media based model, the first step is to examine the possibility of approximating hydraulic fracture network flow as continuum media at some small scale [Long et al. (1982) and Cacas et al. (1992)]. 

The main hypothesis of dam concrete and its foundation rock are as follows a) the dam and its rock foundation are isotropic continuum media in... 

Jointed rock mass can be regarded as equivalent continuum media if the distribution of fractures are in uniform style, a RVE (representative volume element) can be established, and the fracture flow controls the seepage of rock masses. A hydro-mechanical model of jointed rock mass coupled with damage is developed in this paper, based on the equivalent continuum assumption. From eqations (l) (i4), a FEM formulation can be established as below... 

Landau LD, Lifshitz EM, Pitaevskii LP (1948) Electrodynamics of continuum media. Pergamon, Oxford... 

In the FEM simulations, pinned support (Ux = 0, Uy = 0, and Uz = 0) was assumed at the base of the walls. The Plastic Damage model available in ABAQUS was also adopted in the simulations. The materials used in the walls were concrete (concrete brick tmits) and mortar (head/bed joints between bricks). Both materials have been considered as a continuum media and simulated using the continuum damage model. The only material properties available in (Li et al. 2005) were the concrete and mortar ultimate compression strengths. Other needed data was extracted using ACI formulas. Some of the parameters needed in the Plastic... 

This chapter studies the local and global structures of polymer networks. For the local structure, we focus on the internal structure of cross-Unk junctions, and study how they affect the sol-gel transition. For the global structure, we focus on the topological connectivity of the network, such as cycle ranks, elastically effective chains, etc., and study how they affect the elastic properties of the networks. We then move to the self-similarity of the structures near the gel point, and derive some important scaling laws on the basis of percolation theory. Finally, we refer to the percolation in continuum media, focusing on the coexistence of gelation and phase separation in spherical coUoid particles interacting with the adhesive square well potential. 

Physically, in the catalyst layer both carbon and electrolyte phases are porous structures. However, in modelling, porous carbon and electrolyte clusters are usually replaced by the two interpenetrating continuum media (phases). The idea of representing a porous distribution of potentials by continuous functions of space is usually referred to as a rnacrohomogeneous approach. 

In 2001, Ferrigno and Girault proposed a hydrodynamic approach to ion-transfer reactions [122]. In the same way that the friction factor for the diffusion coefficient in the Stokes-Einstein equation can be given by solving the Navier-Stokes equation for a sphere in a laminar flow, the Navier-Stokes equation was solved numerically to account for the passage of a sharp boundary between two continuum media. These data show that the friction coefficient varies from to in a continuous manner over a distance one order of magnitude larger... 

Basilevsky, M. V., Chudinov, G. E. and Napolov, D. V. Calculation of the rate constant for the reaction chloride + chloromethane —> C1CH3 + Cl - in the framework of the continuum medium model,... 

The electrostatic potential in the cavity (in) can be separated into the direct contribution from solute (< soiute) and continuum medium (< medium)- The electrostatic energy between the solute and the continuum is calculated as... 

In a recent upsurge of studies on electron transfer kinetics, importance was placed on the outer shell solvent continuum, and the solvent was replaced by an effective model potential or a continuum medium with an effective dielectric constant. Studies in which the electronic and molecular structure of the solvent molecules are explicitly considered are still very rare. No further modem quantum mechanical studies were made to advance the original molecular and quantum mechanical approach of Gurney on electron and proton (ion) transfer reactions at an electrode. 

Defect diffusion traditionally is treated as a process in continuum medium. However, discreteness of the crystalline lattice becomes important in particular situations, e.g., when defect recombination occurs in several hops (nearest neighbour recombination) [3, 4] or even for nearest-site hops of defects if their recombination is controlled by the tunnelling whose probability greatly changes on a scale of lattice constant [45, 46],... 

We restrict ourselves to the case of classical particles and we thus disregard all quantum effects. The particle motion (if any) occurs by thermally-activated hops in continuum medium. 

An analysis of the limiting expressions for the three different terms contributing to the friction/diffusion is presented, considering the solute to be large compared to the solvent. When the solute becomes very large, physically one expects the solvent to appear as a continuum medium to the solute. Thus the microscopic details of the solvent becomes unimportant. 

For Equations (3.6) and (3.7), Fourier s law can be used to relate the heat flux to the temperature gradient in a continuum medium ... 

The dilational rheology behavior of polymer monolayers is a very interesting aspect. If a polymer film is viewed as a macroscopy continuum medium, several types of motion are possible [96], As it has been explained by Monroy et al. [59], it is possible to distinguish two main types capillary (or out of plane) and dilational (or in plane) [59,60,97], The first one is a shear deformation, while for the second one there are both a compression - dilatation motion and a shear motion. Since dissipative effects do exist within the film, each of the motions consists of elastic and viscous components. The elastic constant for the capillary motion is the surface tension y, while for the second it is the dilatation elasticity e. The latter modulus depends upon the stress applied to the monolayer. For a uniaxial stress (as it is the case for capillary waves or for compression in a single barrier Langmuir trough) the dilatational modulus is the sum of the compression and shear moduli [98]... 

A more sophisticated description of the solvent is achieved using an Apparent Surface Charge (ASC) [1,3] placed on the surface of a cavity containing the solute. This cavity, usually of molecular shape, is dug into a polarizable continuum medium and the proper electrostatic problem is solved on the cavity boundary, taking into account the mutual polarization of the solute and solvent. The Polarizable Continuum Model (PCM) [1,3,7] belongs to this class of ASC implicit solvent models. 

When activated ET involves both discrete molecular (m) and continuum medium (s) modes, one may assume additivity of A contributions (where we suppress the i, f subscripts) ... 

Fig. 2.3 Solvent screening of electronic couplings between chromophores in the four photo-syntletic proteins PE545 (pink triangles), PC645 (blue inverted triangles), PSII/LHCII (green circles). The protein medium is modeled as a dielectric continuum medium with a relative static constant of estat = 15 and optical dielectric constant of n2 = 2. Calculated values for the solvent screening factor s for the various chromophores pairs. The Forster value I jnl and the Onsager value 3(2n1 + 1) are indicated by the horizontal line...

Fig. 4.2 Homogenization of discretized microstructural features into a continuum medium...

The examples treated in this chapter have shown that only the Coulom-bic polarization of the solvent, represented as a continuum medium, is responsible for significant changes of the behaviour of reacting systems in passing from the gas phase to the solution. Dispersion, Pauli repulsion and cavitation do not lead to significant effects. 

While the effective dielectric constant e is expected to be small near the surface dipoles, it is expected to acquire the bulk value after a few molecular distances. When calculating the net field generated by the surface dipoles in water, it will be assumed that effective dielectric constants can be used only for the interactions with the water molecules close to the surface (the first water layer). The interactions with more distant water molecules will be considered as in a continuum medium hence, the net field will vanish on average. 

Table 6.14 with computational results. Harmonic and PT2 anharmonic frequencies have been computed by B3LYP/N07D and CAM-B3LYP/N07D computational models. Since low-temperature experimental frequencies in Argon and Neon are very close, matrix effects have been taken into account by continuum medium... 

Typical values calculated from the previous equations are 0.1 to 1 s. In reality, coagulation times are longer. This can be attributed to the fact that for large distances the solvent acts as a continuum medium, so the opposition of the medium to the particle movement is incorporated in the equations with the viscosity coefficient. However, for smaller separation distances the solvent cannot... 

The use of a Cl description of the solute wavefunction, instead of MO treatment, has allowed a further development of the continuum medium model. In fact, in its traditional versions, solvent electrons are considered in terms of the fast polarization field defined by the solute charge density. The limitations of this classical description become clear if we consider... 

Malvern LE (1969) Introduction to the Mechanics of a Continuum Medium. Prentice-Hall Inc, Englewood Cliffs... 

A is a constant independent of the propagation direction (when the continuum medium is ideal) is the initial propagation phase... 

Alkali halides are compounds with a strong ionic character and without a solvent they are stabilized by the strong electrostatic interaction between the cation and the anion. In the present study we limit attention to the dissociation potentials comparing the curves obtained for the free molecule and for the molecule in water solution. The solvent is treated only as a continuum medium and then in this way we cannot consider the formation of complexes between the ions and the water molecules which instead are extensively studied by means of Monte Carlo (MC) and molecular dynamics (MD) simulations (see for example [13]). Although it could be possible to include some water molecules with the sodium chloride as a more complicated solute, we have preferred to focus attention on the solvent effect on the electronic structure of the simplest solute, this effect being the most important in the next two examples. 

The original formal theory is expressed in terms of quanttun electrodynamics with the continuum mediwn characterized by its spectnun of complex dielectric frequencies. A more recent formulation, derived from this theory, is based on the extension of the reaction field concept to a dipole subject to fluctuations exclusively electric in origin. Another procedme has been formulated starting, as for the repulsion contribution, from the theory of intermolecular forces. Following the scheme commonly exploited to derive the electrostatic contribution to the interaction energy, the molecule B is substituted by a continuum medium, the solvent S, described by a surface charge density as induced by the solute transition densities of M (the equivalent of A) and spreading on the cavity surface. 

Among the several available models, those that represent the solvent molecules through a continuum medium are by far the most extensively used in and solvation energy calculations. Thus, we will concentrate the discussion on the continuum approaches but since this is not the main objective of this chapter only the basic ideas will be presented. For more comprehensive discussions, the reader is referred to the excellent reviews of Tomasi and Persico [69], Cramer and Thrular [76], and Orozco and Luque [77]. 

Rocky material has the unique physical mechanical nature that obviously shows characteristics of particle agglomeration under microscopic perspective of microscope, usually deemed to the kind of porous medium material between continuum medium and discontinuous medium. Failure mechanism of the material is urgently needed to be recognized along with extension of human engineering activities. In this situations, the methods, such as energy, discrete element simulation, acoustic emission, are applied to further study of failure mechanism. 

The dependence of minimal value of average diameter of disperse phase particles that subject to deformation at the expense of hydrodynamic effect of continuum medium was received by numerical experiment data analysis ... 

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