Boltzmann Integral form

In order to simulate the experimental data, we calculated Apc(0, 6) using the Shockley-Chambers tube-integral form of the Boltzmann transport... 

The TPE-HNC/MS theory reduces to an integral form of the nonlinear Poisson-Boltzmann equation in the limit of point ions [8,44]. Hence, in that limit agreement between the two methods is exact. For a 0.1 M, 1 1 electrolyte separating plates with surface potentials of 70 mV, Lozada-Cassou and Diaz-Herrera [8] show excellent agreement between the TPE-HNC/MS theory and the Poisson-Boltzmann equation. The agreement becomes very poor, however, at a higher concentration of 1 M. In addition, like the Monte Carlo and AHNC results, the TPE-HNC/MS theory predicts attractive interactions at sufficiently high potentials and/or salt concentrations, and such effects are missed entirely by the Poisson-Boltzmann equation. 

In this case of uniaxial tension (compression), only nine independent orthotropic tensor components are involved, the three shear components being equal to zero. StiU, this time-dependent Young s modulus is a rather complex function. As in the Hnear elastic case, the inverse form of the Boltzmann integral can be used this would constitute the compHance formulation. 

GPa, respectively, with relaxation time r 5 s. The pofymer is subjected to a constant rate of tensOe strain e = 10" s". Derive the stress-strain relation Boltzmann superposition principle. 

Use the integral form of the Boltzmann superposition principle to show that the creep compliance and stress relaxation modulus of any linear viscoelastic material are related through... 

The advantage of the Boltzmann formulation (11) is that it deals directly with the fluxes, and so can be applied, in principle, directly to any multiplying medium, finite, infinite, uniform, or non-uniform. For computational purposes, it is just about as difficult, in the general case, to deal directly with O [as in (11)] as it is to bother about H as in (1) or (10). Of course (1) and (11) are equivalent —(1) is the resolved or integral form of (11). 

Since the Zener model is a linear viscoelastic model, it obeys the Boltzmann superposition principle. In this problem we are concerned with a strain history which is a smoothly varying function of time, with y undergoing sinusoidal oscillations. Therefore the integral form of the BSP is the most straightforward one to apply... 

It is next required to obtain a quantitative description of stress relaxation and creep that will help to form a link with the original mathematical description in terms of the Boltzmann integrals. It is simple and instructive to do this by development of the Maxwell and Kelvin models. 

Non-linear viseoelastie theories can also be created by generalising the Boltzmann superposition prineiple (see Chapter 5). Leaderman [18], working on polymer fibres, was the first to do this and Findley and Lai [19] have adopted a similar approach. Non-linearity is introdueed into the Boltzmann integral by ineluding strain or stress dependence into the integrand. Leaderman s integral takes the form... 

There is an identical form of this infinite series for the stress in terms of the strain history. In both oases, the first term is recognisable as the Boltzmann integral (see Chapter 5). 

Now we will show that this expression can be transformed into the usual form of the Boltzmann collision integral with the quantum cross section. For this purpose we can use the following relations from the scattering theory ... 

With these relations it is easy to get the usual form of the quantum Boltzmann collision integral ... 

Due to the simple product form of the Maxwell-Boltzmann distribution, the derivations given above are easily generalized to the expression for the relative velocity in three dimensions. Since the integrand in Eq. (2.18) (besides the Maxwell-Boltzmann distribution) depends only on the relative speed, we can simplify the expression in Eq. (2.18) further by integrating over the orientation of the relative velocity. This is done by introducing polar coordinates for the relative velocity. The full three-dimensional probability distribution for the relative speed is... 

Instead of velocity gradients, displacement gradients can be used in relation (8.38). In this form, relations of the kind (8.38) are established on the basis of the phenomenological theory of so-called simple materials (Coleman and Nolle 1961). To put the theory into practice, function (8.38) should be, for example, represented by an expansion into a series of repeated integrals, so that, in the simplest case, one has the first-order constitutive relation (8.37). Let us note that the first person who used functional relations of form (8.38) for the description of the behaviour of viscoelastic materials was Boltzmann (see Ferry 1980). 

Up to this point, the theory has been presented in a form suitable only for adsorption systems which can be treated by classical statistical mechanics. When the quantization of the motion of the atoms is important, one must replace the Boltzmann factors in the integrals for ZN by the appropriate Slater sums. After integrating over the coordinates, one obtains... 

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