Superposition approximation theories, exact solutions

As we have mentioned in Chapter 2, the accuracy of the kinetic equations derived using the superposition approximation cannot be checked up in the framework of the same theory. It is the analysis of the limiting case of the infinitely diluted system, no —> 0, which nevertheless permits us to compare approximate results obtained in the linearized approximation with the exact solution of the two-particle problem (Chapter 3). 

A different situation arises if one calculates the same constant in the superposition approximation (SA) competing with encounter theory. This approximation was first employed in Refs. 139 and 140 and then applied to a number of particular problems [141-145], However, the present problem is an exceptional one because it has an exact solution that allows the examination of all others. To make this possible, the original SA was employed for this very problem in Ref. 136, and the following Stem-Volmer constant was obtained ... 

Helfand and Stillinger have recently employed this theory to obtain a suitable "local density correction of the superposition theory for the hard sphere system. While the numerical solution of the modified integral equation is not available at this time, the equation is unique among other approximate theories in )delding the exact value of the fourth virial coefficient. 

Whitney and Drzal [87] presented an analytical model to predict the stresses in a system consisting of a broken fiber surrounded by an unbounded matrix. The model (Fig. 8) is based on the superposition of the solutions to two axisymmetric problems, an exact far-field solution and an approximate transient solution. The approximate solution is based on the knowledge of the basic stress distribution near the end of the broken fiber, represented by a decaying exponential function multiplied by a polynomial. Equilibrium equations and the boundary conditions of classical theory of elasticity are exactly satisfied throughout the fiber and matrix, while compatibility of displacements is only partially satisfied. The far-field solution away from the broken fiber end satisfies all the equations of elasticity. The model also includes the effects of expansional, hygrothermal strains and considers orthotropic fibers of the transversely isotropic class. 

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