Continuum analysis

Application of the exact continuum analysis of dispersion forces requires significant calculations and the knowledge of the frequency spectmm of the material dielectric response over wavelengths X = 2irc/j/ around 10-10 nm. Because of these complications, it is common to assume that a primary absorption peak at one frequency in the ultraviolet, j/uv. dominates the dielectric spectrum of most materials. This leads to an expression for the dielectric response... 

In this chapter, the motion of solute and solvent molecules is considered in rather more detail. Previously, it has been emphasised that this motion approximates to diffusion only over times which are long compared with the velocity relaxation time (see Chap. 8, Sect. 2.1). At times comparable with or a little longer than the velocity relaxation time, the diffusion equation does not provide a satisfactory description of molecular motion. An alternative approach must be sought. This introduces considerable complications to a theoretical analysis of very fast reactions in solution. To develop an understanding of chemical reactions occurring over very short time intervals, several points need to be discussed. Which reactions might be of interest and over what time scale What is known of the molecular motion of solute and solvent molecules Why does the Markovian (hydrodynamic) continuum analysis fail and what needs to be done to develop a better theory These points will be considered in further detail in this chapter. 

Lifshitz more recent continuum analysis of the van der Waals interaction (Israelaehvili and Tabor, 1973) leads to approximately the same distance dependence hut to a more accurate physical interpretation of Hamaker s constant A. However, this distinction becomes unimportant to the present analysis if A is treated as a measurable property. 

While Lifshitz continuum analysis leads to a more accurate physical interpretation of the Hamaker constant A, both Lifshitz result and Eq. K] predict that [Pg.106]

The exact laws, based on continuum analysis of the fibers and the matrix, would be very complicated. The analysis would involve equilibrium of stresses around, and in, the fibers and compatibility of matrix deformation with the fiber strains. Furthermore, end and edge effects near the free surfaces of the composite material would introduce complications. However, a simplified model can be developed for the interior of the composite material based on the notion that the fibers and the matrix interact only by having to experience the same longitudinal strain. Otherwise, the phases behave as two uniaxially stressed materials. McLean5 introduced such a model for materials with elastic fibers and he notes that McDanels et al.6 developed the model for the case where both the fibrous phase and the matrix phase are creeping. In both cases, the longitudinal parameters are the same, namely... 

For most hydrocarbons burning in air, coo,oe/v is only about 0.07, and WF.eff is about 2, so that the pseudo-homogeneous source term in Equation 6 depends only weakly on wq. For this case, i/r plays the role of a zero-order Thiele modulus (13). Thus, the present continuum analysis clearly reveals that the dimensionless parameter which dictates the onset of incipient group combustion is simply a Damkohler niunber (Thiele modulus) which takes on a critical value (Equation 15) dependent only on the ambient oxidizer mass fraction, on the fuel vapor mass fraction at the droplet surface, and on the stoichiometric oxidizer/fuel mass ratio. 

Gethin DT, Ransing RS, Lewis RW, et al. Numerical comparison of a deformable discrete element model and an equivalent continuum analysis for the compaction of ductile porous material. Comput Struct 2001 79 1287-1294. 

Despite the existence of powerful analytical tools that allow for explicit solution of certain problems of interest, in general, the modeler cannot count on the existence of analytic solutions to most questions. To remedy this problem, one must resort to numerical approaches, or further simplify the problem so as to refine it to the point that analytic progress is possible. In this section, we discuss one of the key numerical engines used in the continuum analysis of boundary value problems, namely, the finite element method. The finite element method replaces the search for unknown fields (i.e. the solutions to the governing equations) with the search for a discrete representation of those fields at a set of points known as nodes, with the values of the field quantities between the nodes determined via interpolation. From the standpoint of the principle of minimum potential energy introduced earlier, the finite element method effects the replacement... 

An alternative perspective on the subject of point defects to the continuum analysis advanced above is offered by atomic-level analysis. Perhaps the simplest microscopic model of point defect formation is that of the formation energy for vacancies within a pair potential description of the total energy. This calculation is revealing in two respects first, it illustrates the conceptual basis for evaluating the vacancy formation energy, even within schemes that are energetically more accurate. Secondly, it reveals additional conceptual shortcomings associated with... 

Thus, while the continuum analysis helps visualize transport on the scale of the chamber dimension, to predict the OVPD growth of patterns whose size and resolution are on the order of X, the molecular nature of transport near the substrate must be considered. 

A pile in a group deflects much more than the same pile isolated when subjected to the same lateral load (Cooke et al., 1979 Matlock et al., 1980). The analysis of pile group behavior is by means of an incremental superposition of the nonlinear single-pile behavior based on a p-y analysis and the interaction from the other piles in the group based on an elastic continuum analysis (Poulos, 1981a). 

Tadmor EB, Ortiz M, Phillips R. Quasi-continuum analysis of defects in solids. Philos. Mag. A... 

In a continuum analysis, Li et al. (2002) adopted an approach whereby the nonlinear elastic response of the material was based on the known homogeneous deformation response of the single crystal of interest. The stress field throughout the performing elastic crystal is then determined incremen-... 

By combining continuum analysis of inelastic deformation, such as that described in Section 7.4, with the mechanistic models of the foregoing subsections, Shen and Suresh (1996) have identified the variation of equi-biaxial film stress through the thickness of the film during steady-state creep deformation. An appealing feature of such an approach is that the evolution of substrate curvature, spatial variation of residual stress and relative... 

So far the discussion is general, hut to proceed further the nature of the constraints must be made explicit. The most powerful mathematics available is that of continuum analysis i.e. one should use differential equations rather than matrices in a first attack. It is therefore convenient (though not at all essential) to represent the polymer molecules by curves in space, being... 

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