Isotropic continua

This is possible because the elastieitj standard relations must hold for all isotropic continua, whether heterogeneous on the mieroscale or not [Torquato 2002]. It will be shown below, that in the case of the dense alumina-zirconia composite ceramics (Poisson ratios 0.23 and 0.31 for alumina and zirconia, respectively) the deviation of the Voigt bound Ey... 

From the standpoint of simplicity, it is convenient to restrict this discussion to homogeneous and isotropic continua. The former concept implies that the material properties do not depend... 

In nature there are reasonable approximations in the undeformed state to homogeneous isotropic continua. Any elastomer, natural or synthetic, can he prepared reasonably homogeneous, barring a few tenths of a percent of curing agent redsonahly continuous, having... 

The basic assumptions of fracture mechanics are (1) that the material behaves as a linear elastic isotropic continuum and (2) the crack tip inelastic zone size is small with respect to all other dimensions. Here we will consider the limitations of using the term K = YOpos Ttato describe the mechanical driving force for crack extension of small cracks at values of stress that are high with respect to the elastic limit. 

In this model, the binding electrons are often assimilated to a negatively charged, isotropic continuum ( jellium ) surrounding the positively charged ions (the ionic... 

Because the solvent molecules are usually of a similar size to the reactants, the assumption that reactants diffuse in a structureless and isotropic continuum is not very satisfactory. Liquids possess short-range order. Solvent molecules are several times more likely to be separated by a distance equal to their diameter than separated by about one and a half diameters. More details are revealed by the radial distribution function [see Figs. 38 (p. 216) and 44 (p. 235)]. This implies that there is an... 

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... 

The drying process by vdiich a latex transforms from a colloidal dispersion to a polymer film which, in one ideal, is a homogeneous, isotropic continuum of the... 

Continuum shell models used to study the CNT properties and showed similarities between MD simulations of macroscopic shell model. Because of the neglecting the discrete nature of the CNT geometry in this method, it has shown that mechanical properties of CNTs were strongly dependent on atomic structure of the tubes and like the curvature and chirality effects, the mechanical behavior of CNTs cannot be calculated in an isotropic shell model. Different from common shell model, which is constmcted as an isotropic continuum shell with constant elastic properties for SWCNTs, the MBASM model can predict the chirality induced anisotropic effects on some mechanical behaviors of CNTs by incorporating molecular and continuum mechanics solutions. One of the other theory is shallow shell theories, this theory are not accurate for CNT analysis because of CNT is a... 

This equation contains the basic model used by M. Newton in his pioneering calculations of the solvated electron [7]. If the charge density is replaced by a classical unit charge at the origin of the sphere, the RF potential obtained after integration of Eq. (26) corresponds to Born s model for a metalized sphere immersed in an isotropic continuum. 

In this and the next three chapters, we model the medium as a polarizable isotropic continuum. In treating a medium as a continuum, we neglect its atomic structure and focus on its larger-scale properties. In treating a medium as isotropic, we assume that its polarizability is the same in all directions. By treating a medium as polarizable, we assume that the charge redistributes in response to an electric held, even if the medium is neutral overall. 

The field la then the same for all molecules and parallel to . as are the induced moments m taken to be given by m > o where o is a simple scalar polarizability The gist of the Lorentz argument (7) is that the resultant field at any one molecule i from all the other dipoles j in a sphere, surrounding the one vanishes as it is given by the sumr Tl - 3 cos (R j z)]a over lattice distances which is zero for cubic symmetry (or an isotropic continuum) leaving s the field of charges external to the sphere if it is in a vacuum. The macroscopic in the sphere from Eo and the macroscopic is by electrostatics - - (AtT/3) and the Lorentz field is... 

The two-phase N + I droplet texture formed on cooling an I phase, with droplets of N in an isotropic continuum. (DSCG/water, x300, crossed polars+ 1A red plate.)... 

The problem was solved under the following assumptions. Particle-reactants A and B are in the state of thermal equilibrium with the medium, which is considered as a continuous isotropic continuum. These particles diffuse according to the laws of macroscopic diffusions, i.e., in agreement with Pick s law, which is valid in the absence of high gradients. This, however, is violated for the convergence and interaction of particles A and B, which react rapidly. The boundary conditions are determined by the chemical reaction occurred in the system. Particle B is considered as fixed, and particles A migrate with the diffusion coefficient D = D/ + D. The concentration c of particles A in the vicinity of particle B, which is considered as a sphere of the radius / = + r, depends on the distance r and time t and is described... 

All the theory developed up to this point has been limited in the sense that translational motion (the continuum degree of freedom) has been restricted to one dimension. In this section we discuss the generalization of this to three dimensions for collision processes where space is isotropic (i.e., collisions in homogeneous phases, such as in a... 

If we now transfer our two interacting particles from the vacuum (whose dielectric constant is unity by definition) to a hypothetical continuous isotropic medium of dielectric constant e > 1, the electrostatic attractive forces will be attenuated because of the medium s capability of separating charge. Quantitative theories of this effect tend to be approximate, in part because the medium is not a structureless continuum and also because the bulk dielectric constant may be an inappropriate measure on the molecular scale. Eurther discussion of the influence of dielectric constant is given in Section 8.3. 

First to be considered is the isotropic Pq coefficient. The parameter is proportional to the integrated cross-section, a [Eq. (11)]. In fact, the preceding arguments show that when j = 0, I m = Im, and so there are no interference cross terms in this case. Consequently, as is already widely recognized, the integrated cross-section displays no dependence on the relative phase of the final continuum channels. 

Wood and Blundy (2001) developed an electrostatic model to describe this process. In essence this is a continuum approach, analogous to the lattice strain model, wherein the crystal lattice is viewed as an isotropic dielectric medium. For a series of ions with the optimum ionic radius at site M, (A(m))> partitioning is then controlled by the charge on the substituent (Z ) relative to the optimum charge at the site of interest, (Fig. 10) ... 

Fig. 2.2 Self-Consistent Reaction Field (SCRF) model for the inclusion of solvent effects in semi-empirical calculations. The solvent is represented as an isotropic, polarizable continuum of macroscopic dielectric e. The solute occupies a spherical cavity of radius ru, and has a dipole moment of p,o. The molecular dipole induces an opposing dipole in the solvent medium, the magnitude of which is dependent on e.

---