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The aim of this paper is to understand how the shear modulus of a particulate composite, manufactured with hollow spheres surrounded by a graded interphase in a matrix, depends on the properties of the interphase zone and the inclusion wall. Considering composites with small volume fractions of the hollow inclusions and assuming isotropic phases with a radial power law behavior for the shear modulus in the inhomogeneous interphase, the boundary value problems used to determine shear modulus bounds with the Composite Sphere Assemblage method, are studied. Adopting Hashin’s approach, the shear elastic bound expressions are analytically obtained, numerically investigated and compared with results present in literature. Parametric analysis allows us to highlight the effects of a soft or stiff interphase and shows that the shear bounds are close to each other for inclusions with thin wall thickness thereby leading in some cases to a unique prediction value for the shear modulus. To this end, an analytical expression supplying the thickness of the inclusion wall that gives rise to a match of the two shear modulus bounds for assigned phase property ratio and volumetric fraction is presented.

In this paper some analytical solutions are given for a two-dimensional advection equation with anisotropic dispersion. Chemical decay or adsorption-like reaction inside the liquid phase is considered. Bessel function expansion is used to solve the second order PDE model with different initial conditions, corresponding to usual experimental practices. Cylindrical geometry is considered since large columns could be adopted to investigate both anisotropic dispersion and adsorption/desorption kinetic mechanisms.