Strengthening of materials caused by dispersion of doughnut-like second phase

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We have made the following studies to understand the change of mechanical properties of materials caused by dispersion of doughnut-like second phase.Cu-Si alloys are internally oxidized at various of Si atoms with oxygen, film-like SiO_2 precipitates formed on grain boundaries (GBs) is observed. After the reaction of Si atoms with oxygen, film-like SiO_2 precipitates with holes which expose GB of Cu are initially formed. At temperatures used for the internal oxidation, the holes rapidly grow and the morphology of the film-like SiO_2 evolves to a particulate equilibrium shape through formation of two-dimensional SiO_2 network.Theoretical calculations to understand effects of doughnut-like inclusions on the mechanical properties of materials are made. The averaged Eshelby tensors that relate uniform eigenstrains and average total strains in doughnut-like inclusions are obtained. These are compared with the Eshelby tensors of oblate spheroidal and rod-like inclusions. We show that the averaged Eshelby tensors of the doughnut-like inclusions on a plane are almost the same as the average of the Eshelby tensors of randomly oriented rod-like inclusions on the same plane. This is satisfied for both "thin" doughnut-like inclusions. The strain field caused by uniform and purely dilatational eigenstrains is also discussed for a doughnut-like inclusion. Near the doughnut-like inclusion, there are two points where all components of the strains are null. Using the results results obtained, we have discussed the effects of the doughnut-like inclusions on the mechanical properties of materials.