Poisson Term

The last term has been denoted the Poisson term [72] and corresponds to V E evaluated at the expansion point and is therefore zero when the sources of the field are at an appreciable distance from the expansion point (154). In this particular case, the energy of interaction with the electric field gradient, static... 

The linearised Poisson-Boltzmann equation is obtained by taking only the first term in the expansion, giving ... 

The Poisson-Boltzmann equation is a modification of the Poisson equation. It has an additional term describing the solvent charge separation and can also be viewed mathematically as a generalization of Debye-Huckel theory. 

We propose to describe the distribution of the number of fronts crossing x by the Poisson distribution function, discussed in Sec. 1.9. This probability distribution function describes the probability P(F) of a specific number of fronts F in terms of that number and the average number F as follows [Eq. (1.38)] ... 

Poisson s ratio, is the negative of the ratio of the strain transverse to the fiber direction, 8, and the strain ia the fiber direction, S, when the lamina is loaded ia the fiber direction and can also be expressed ia terms of the properties of the constituents through the rule of mixtures. 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

The modulus term in this equation can be obtained in the same way as in the previous example. However, the difference in this case is the term V. For elastic materials this is called Poissons Ratio and is the ratio of the transverse strain to the axial strain (See Appendix C). For any particular metal this is a constant, generally in the range 0.28 to 0.35. For plastics V is not a constant. It is dependent on time, temperature, stress, etc and so it is often given the alternative names of Creep Contraction Ratio or Lateral Strain Ratio. There is very little published information on the creep contraction ratio for plastics but generally it varies from about 0.33 for hard plastics (such as acrylic) to almost 0.5 for elastomers. Some typical values are given in Table 2.1 but do remember that these may change in specific loading situations. 

Example 2.7 A nylon ring with a nominal inside diameter of 30 mm, an outer diameter of SO mm and a width of S mm is to be made an interference fit on a metal shaft of 30 mm diameter as shown in Fig. 2.17. The design condition is that the initial separation force is to be 1 kN. Calculate (a) the interference on radius needed between the ring and the shaft and (b) the temperature to which the nylon must be heated to facilitate easy assembly. What will be the maximum stress in the nylon when it is in position on the shaft The coefficient of friction between nylon and steel is 0.2S. The short-term modulus of the nylon is 1 GN/m, its Poisson s ratio is 0.4 and its coefficient of thermal expansion is 100 X 10- °C- . 

In practical terms the above analysis is tcx) simplistic, particularly in regard to the assumption that the stresses in the fibre and matrix are equal. Generally the fibres are dispersed at random on any cross-section of the composite (see Fig. 3.8) and so the applied force will be shared by the fibres and matrix but not necessarily equally. Other inaccuracies also arise due to the mis-match of the Poisson s ratios for the fibres and matrix. Several other empirical equations have been suggested to take these factors into account. One of these is the Halpin-Tsai equation which has the following form... 

Thus, three reciprocal relations must be satisfied for an orthotropic material. Moreover, only 2, V13, and V23 need be further considered because V21, V31, and V32 can be expressed in terms of the first-mentioned group of Poisson s ratios and the Young s moduli. The latter group of Poisson s ratios should not be forgotten, however, because for some tests they are what is actually measured. 

For isotropic materials, certain relations between the engineering constants must be satisfied. For example, the shear modulus is defined in terms of the elastic modulus, E, and Poisson s ratio, v, as... 

The only difference between the maximum strain failure criterion. Equation (2.125), and the maximum stress failure criterion, Equation (2.118), is the inclusion of Poisson s ratio terms in the maximum strain failure criterion. 

Fhe value of Poisson s ratio, v, for the composite material has been derived explicitly as Fqtiatinir(3.6i) lTiiis, the upper bound on E can be obtained by substituting the expression for v. Equation (3.61), in the expression for the upper bound on E in terms of v. Equation (3.57). However, the algeb gj jji ittjxiessy, so an explicit expression for the upper bound on E is not presented. In practical applications, the value of V can be calculated from Equation (3.61) and then substituted in Equation (3.57) to obtain E. For the special casg in ch v j Vdj the expression for v, Equation (3.61), reduces to... 

To describe the simple phenomena mentioned above, we would hke to have only transparent approximations as in the Poisson-Boltzmann theory for ionic systems or in the van der Waals theory for non-coulombic systems [14]. Certainly there are many ways to reach this goal. Here we show that a field-theoretic approach is well suited for that. Its advantage is to focus on some aspects of charged interfaces traditionally paid little attention for instance, the role of symmetry in the effective interaction between ions and the analysis of the profiles in terms of a transformation group, as is done in quantum field theory. 

Schadler, G. H., 1992, Solution of Poisson s equation for arbitrary shaped overlapping or nonoverlapping charge densities in terms of multipole moments, PAy. Rev. 545 11314. 

Poison s ratio It is the proportion of lateral strain to longitudinal strain under conditions of uniform longitudinal stress within the proportional or elastic limit. When the material s deformation is within the elastic range it results in a lateral to longitudinal strain that will always be constant. In mathematical terms, Poisson s ratio is the diameter of the test specimen before and after elongation divided by the length of the specimen before and after elongation. Poisson s ratio will have more than one value if the material is not isotropic... 

The shot noise process is defined in terms of the Poisson process by means of the formula... 

Our first objective is going to be the determination of the finite order probability density function of Y(t) in terms of the known finite order probability densities for the increments of N(t). In preparation for this, we first note that, since N(s) — N(t) is Poisson distributed with parameter.n(s — t) for s > t, it follows that... 

Equation (20) yields the elastic modulus, Ec, of the composite in terms of the moduli and Poisson s ratios of the phases. If the Ec-modulus and vc-value are accurately measured and the Ef- and Em-moduli and the Poisson rations vf and vm are known, the average modulus of the mesophase, Ef, may be determined. Poisson s ratio of the mesophase may be found from the simple relation ... 

Henry [ 157] solved the steady-flow continuity and Navier-Stokes equations in spherical geometry, neglecting inertial terms but including pressure and electrical force terms, coupled with Poisson s equation. The electrical force term in Henry s analysis consisted of the sum of the externally applied electric field and the field due to the double layers. His major assumptions are low surface potential (i.e., potentials less than approximately 25 mV) and undistorted double layers. The additional parameter ku appearing in the Henry... 

The relation between the spatial potential distribntion and the spatial distribution of space-charge density can be stated, generally, in terms of Poisson s dilferential equation. 

The Monte Carlo method as described so far is useful to evaluate equilibrium properties but says nothing about the time evolution of the system. However, it is in some cases possible to construct a Monte Carlo algorithm that allows the simulated system to evolve like a physical system. This is the case when the dynamics can be described as thermally activated processes, such as adsorption, desorption, and diffusion. Since these processes are particularly well defined in the case of lattice models, these are particularly well suited for this approach. The foundations of dynamical Monte Carlo (DMC) or kinetic Monte Carlo (KMC) simulations have been discussed by Eichthom and Weinberg (1991) in terms of the theory of Poisson processes. The main idea is that the rate of each process that may eventually occur on the surface can be described by an equation of the Arrhenius type ... 

The fluid model is a description of the RF discharge in terms of averaged quantities [268, 269]. Balance equations for particle, momentum, and/or energy density are solved consistently with the Poisson equation for the electric field. Fluxes described by drift and diffusion terms may replace the momentum balance. In most cases, for the electrons both the particle density and the energy are incorporated, whereas for the ions only the densities are calculated. If the balance equation for the averaged electron energy is incorporated, the electron transport coefficients and the ionization, attachment, and excitation rates can be handled as functions of the electron temperature instead of the local electric field. 

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