A number of problems in random media are studied. The effective trapping rate of diffusion-controlled reaction processes is evaluated for polydispersed spherical traps and oriented spheroidal traps. Rigorous bounds are derived for polydispersed overlapping spherical traps, and an efficient random-walk technique is employed to determine "exact" results for each microgeometry; results are compared to bounds as well as to theory in the polydispersed case.;Monte-Carlo methods are used to evaluate a key integral ;Improved rigorous bounds on the effective elastic and transport properties of a transversely isotropic fiber-reinforced material composed of oriented, infinitely long, multisized circular cylinders distributed throughout a matrix are computed. Specifically, bounds are evaluated for the effective axial and transverse shear moduli and the effective transverse bulk modulus, and provide significant improvement over bounds which incorporate only volume-fraction information. Generally, increasing the degree of polydispersivity in cylinder radius increases the effective axial shear modulus and transverse bulk modulus, and decreases (slightly) the effective transverse shear modulus for cases in which the fibers are stiffer than the matrix.;A Monte-Carlo simulation is presented for the fragmentation behavior of a two-phase solid composed of non-reactive inclusions in a reactive matrix and undergoing a reaction at the surface of the solid. The morphological hindering of the fragment release following their separation of the parent solid is considered. Percolation scaling theories are used to show scaling behavior does occur when selected fragmentation objects are measured in terms of various subobjects; the interesting phenomenon of "virtual" criticality arising as a scaling parameter is introduced.
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